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OptiSystemComponent Library

Optical Communication System Design Software

Version 3.0for Windows® 2000/XP

OptiSystemComponent LibraryOptical Communication System Design Software

Copyright © 2003 Optiwave CorporationAll rights reserved.

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DisclaimerOptiwave Corporation makes no representation or warranty with respect to the adequacy of this documentation or the programs which it describes for any particular purpose or with respect to its adequacy to produce any particular result. In no event shall Optiwave Corporation, its employees, its contractors or the authors of this documentation, be liable for special, direct, indirect, or consequential damages, losses, costs, charges, claims, demands, or claim for lost profits, fees, or expenses of any nature or kind.

9/30/03

Technical support

If you purchased Optiwave software from a distributor that is not listed here, please send technical questions to your distributor.

Optiwave Corporation Canada/USTel (613) 224-4700 E-mail [email protected]

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Table of contents

Transmitters Library

Pulse GeneratorsElectrical

Duobinary Pulse Generator........................................................................................3Electrical Jitter............................................................................................................5Noise Source..............................................................................................................7RZ Pulse Generator ...................................................................................................9NRZ Pulse Generator...............................................................................................13Gaussian Pulse Generator.......................................................................................15Hyperbolic-Secant Pulse Generator.........................................................................17Sine Generator.........................................................................................................19Triangle Pulse Generator .........................................................................................21Saw-Up Pulse Generator .........................................................................................23Saw-Down Pulse Generator.....................................................................................25Impulse Generator ...................................................................................................27Raised Cosine Pulse Generator...............................................................................29Sine Pulse Generator...............................................................................................31Measured Pulse .......................................................................................................33Measured Pulse Sequence ......................................................................................35Bias Generator .........................................................................................................37

Optical

Optical Gaussian Pulse Generator...........................................................................39Optical Sech Pulse Generator..................................................................................43Optical Impulse Generator .......................................................................................47Measured Optical Pulse ...........................................................................................51Measured Optical Pulse Sequence..........................................................................55Time Resolve Chirp (TRC) Measurement Data .......................................................59

Optical SourcesCW Laser .................................................................................................................63Laser Rate Equations...............................................................................................67Laser Measured .......................................................................................................71LED ..........................................................................................................................77White Light Source...................................................................................................79Pump Laser..............................................................................................................81Pump Laser Array ....................................................................................................83CW Laser Array........................................................................................................87CW Laser Array ES..................................................................................................91CW Laser Measured ................................................................................................95Directly Modulated Laser Measured ......................................................................101WDM Transmitter ...................................................................................................107

Bit Sequence GeneratorsPseudo-Random Bit Sequence Generator.............................................................115User-Defined Bit Sequence Generator ..................................................................117Mach-Zehnder Modulator.......................................................................................119

ModulatorsOptical

Electroabsorption Modulator ..................................................................................121Amplitude Modulator ..............................................................................................123Phase Modulator ....................................................................................................125Frequency Modulator .............................................................................................127Dual Drive Mach-Zehnder Modulator Measured ....................................................129Electroabsorption Modulator Measured .................................................................133Single Drive Mach-Zehnder Modulator Measured .................................................137Dual Port Dual Drive Mach-Zehnder Modulator Measured ....................................141LiNbO3 Mach-Zehnder Modulator..........................................................................145

Optical Fibers Library

Optical fiber data ....................................................................................................151Optical fiber ............................................................................................................159Linear Multimode fiber............................................................................................187Nonlinear Dispersive fiber ......................................................................................193

Receivers Library

RegeneratorsElectrical

Clock Recovery ......................................................................................................211Data Recovery .......................................................................................................2123R Regenerator......................................................................................................214

OpticalDemodulators

Ideal Frequency Demodulator................................................................................217Ideal Phase Demodulator.......................................................................................219

PhotodetectorsPhotodetector PIN..................................................................................................221Photodetector APD ................................................................................................228

Amplifiers Library

OpticalRaman

Raman Amplifier.....................................................................................................235Raman Amplifier—Average power model ..............................................................251Raman Amplifier—Dynamic model ........................................................................261

EDFA

EDFA Black Box.....................................................................................................271EDF Dynamic — Full model...................................................................................283EDF Dynamic — Analytical model .........................................................................289EDFA......................................................................................................................295EDFA Ideal.............................................................................................................303EDFA Measured.....................................................................................................309Erbium doped fiber.................................................................................................315Er-Yb codoped fiber ...............................................................................................353Er-Yb codoped waveguide amplifier ......................................................................363

SOA

Semiconductor Optical Amplifier ............................................................................381

Electrical

Limiting Amplifier....................................................................................................387Electrical Amplifier..................................................................................................389Transimpedance Amplifier .....................................................................................391AGC Amplifier ........................................................................................................393

Filters Library

OpticalOptical IIR filter.......................................................................................................397Measured Optical filter ...........................................................................................400Measured Group Delay Optical filter ......................................................................404Rectangle Optical filter ...........................................................................................409Trapezoidal Optical filter ........................................................................................411Gaussian Optical filter ............................................................................................413Butterworth Optical filter.........................................................................................415Bessel Optical filter ................................................................................................417Fabry Perot Optical filter ........................................................................................420Acousto Optical filter ..............................................................................................422Mach-Zehnder Interferometer ................................................................................425Inverted Optical IIR filter.........................................................................................427Inverted Rectangle Optical filter .............................................................................430Inverted Trapezoidal Optical filter ..........................................................................432Inverted Gaussian Optical filter ..............................................................................434Inverted Butterworth Optical filter...........................................................................436Inverted Bessel Optical filter ..................................................................................438

FBGFiber Bragg Grating (FBG).....................................................................................440Uniform Fiber Bragg Grating ..................................................................................446Ideal Dispersion Compensation FBG.....................................................................448

ElectricalLow Pass IIR filter (Electrical) ................................................................................453Low Pass Rectangle filter (Electrical).....................................................................456Low Pass Gaussian filter (Electrical)......................................................................458Low Pass Butterworth filter (Electrical) ..................................................................460Low Pass Bessel filter (Electrical) ..........................................................................462Low Pass Chebyshev filter (Electrical) ...................................................................464Low Pass RC filter (Electrical)................................................................................466Low Pass Raised Cosine filter (Electrical) .............................................................468Low Pass Cosine Roll Off filter (Electrical).............................................................470Low Pass Squared Cosine Roll Off filter (Electrical) ..............................................472

Band Pass IIR filter (Electrical) ..............................................................................474Measured filter (Electrical) .....................................................................................477Band Pass Rectangle filter (Electrical) ...................................................................481Band Pass Gaussian filter (Electrical)....................................................................483Band Pass Butterworth filter (Electrical).................................................................485Band Pass Bessel filter (Electrical) ........................................................................487Band Pass Chebyshev filter (Electrical) .................................................................489Band Pass RC filter (Electrical)..............................................................................491Band Pass Raised Cosine filter (Electrical)............................................................493Band Pass Cosine Roll Off filter (Electrical) ...........................................................495Band Pass Square Cosine Roll Off filter (Electrical) ..............................................497S Parameters Measured filter (Electrical) ..............................................................499

Filter AnalyzersOptical Filter analyzer ............................................................................................503Electrical Filter analyzer .........................................................................................505

WDM Multiplexers Library

Add and DropWDM Add...............................................................................................................509WDM Drop .............................................................................................................511WDM Add and Drop ...............................................................................................513

DemultiplexersWDM Demux 1x2 ...................................................................................................517WDM Demux 1x4 ...................................................................................................521WDM Demux 1x8 ...................................................................................................525WDM Demux..........................................................................................................529WDM Demux ES ....................................................................................................533Ideal Demux ...........................................................................................................535

MultiplexersWDM Mux 2x1........................................................................................................536WDM Mux 4x1........................................................................................................539WDM Mux 8x1........................................................................................................543WDM Mux ..............................................................................................................547WDM Mux ES.........................................................................................................551Ideal Mux................................................................................................................553

Network Library

Optical SwitchesDynamic Y Select Nx1 Measured ..........................................................................557Dynamic Y Switch 1xN Measured..........................................................................560Dynamic Y Switch 1xN...........................................................................................563Dynamic Y Select Nx1 ...........................................................................................567Dynamic Space Switch Matrix NxM Measured ......................................................571Dynamic Space Switch Matrix NxM .......................................................................575Optical Switch ........................................................................................................579Digital Optical Switch .............................................................................................581Optical Y Switch .....................................................................................................583Optical Y Select......................................................................................................585Ideal Switch 2x2 .....................................................................................................587Ideal Y Switch ........................................................................................................589Ideal Y Select .........................................................................................................591Ideal Y Switch 1x4..................................................................................................593Ideal Y Select 4x1 ..................................................................................................595Ideal Y Switch 1x8..................................................................................................597Ideal Y Select 8x1 ..................................................................................................599Ideal Y Select Nx1..................................................................................................601Ideal Y Switch 1xN .................................................................................................603

Frequency ConvertersIdeal Frequency Converter.....................................................................................605

Passives Library

ElectricalElectrical Signal Time Delay ..................................................................................609

OpticalOptical Attenuator ..................................................................................................611Phase Shift .............................................................................................................613PMD Emulator........................................................................................................615Time Delay .............................................................................................................619

Couplers

X Coupler ...............................................................................................................621Pump Coupler Co-Propagating ..............................................................................623Pump Coupler Counter-Propogating......................................................................625

Power Splitters

Power Splitter 1x2 ..................................................................................................627Power Splitter 1x4 ..................................................................................................629Power Splitter 1x8 ..................................................................................................631Power Splitter.........................................................................................................633

Power Combiners

Power Combiner 2x1..............................................................................................635Power Combiner 4x1..............................................................................................637Power Combiner 8x1..............................................................................................639Power Combiner ....................................................................................................641

Polarization

Linear Polarizer ......................................................................................................643Circular Polarizer....................................................................................................645Polarization Attenuator...........................................................................................647Polarization Combiner............................................................................................649Polarization Controller............................................................................................651Polarization Rotator................................................................................................653Polarization Splitter ................................................................................................655

Isolators

Isolator ...................................................................................................................657Ideal Isolator...........................................................................................................659

Circulators

Circulator................................................................................................................661Ideal Circulator .......................................................................................................663

Signal Processing Library

ArithmeticElectrical

Electrical Gain ........................................................................................................667Electrical Adder ......................................................................................................668Electrical Subtractor ...............................................................................................669Electrical Multiplier .................................................................................................670Electrical Bias.........................................................................................................671Electrical Norm.......................................................................................................672Electrical Differentiator ...........................................................................................673Electrical Integrator ................................................................................................674Electrical Limiter.....................................................................................................675

Optical

Optical Gain ...........................................................................................................677Optical Adder .........................................................................................................678Optical Subtractor ..................................................................................................679Optical Bias ............................................................................................................680Optical Multiplier.....................................................................................................681

ToolsOptical

Merge Optical Signal Bands...................................................................................682Convert to Parameterized ......................................................................................683Convert to Noise Bins ............................................................................................684

LogicBinary

Binary NOT ............................................................................................................685Binary AND ............................................................................................................686Binary OR...............................................................................................................687Binary XOR ............................................................................................................688Binary NAND..........................................................................................................689Binary NOR ............................................................................................................690Binary XNOR..........................................................................................................691Binary Delay...........................................................................................................692Duobinary precoder................................................................................................693

Tools Library

Switch.....................................................................................................................697Select .....................................................................................................................699Fork 1x2 .................................................................................................................701Loop Control...........................................................................................................702Ground ...................................................................................................................703Buffer Selector .......................................................................................................704Fork 1xN.................................................................................................................705Binary Null..............................................................................................................706Optical Null .............................................................................................................707Electrical Null .........................................................................................................708Binary Delay...........................................................................................................709Optical Delay..........................................................................................................710Electrical Delay ......................................................................................................711Optical Ring Controller ...........................................................................................713Electrical Ring Controller........................................................................................715Limiter ....................................................................................................................719

Initializer .................................................................................................................721Load from file .........................................................................................................724Command Line Application ....................................................................................725

Optiwave Software Tools

OptiAmplifier...........................................................................................................731IFO_Gratings..........................................................................................................739WDM_Phasar Demux 1xN .....................................................................................743WDM_Phasar Mux Nx1..........................................................................................745OptiBPM Component NxM.....................................................................................747

MATLAB Library

MATLAB Filter Component ....................................................................................753MATLAB Optical Filter Component ........................................................................757MATLAB Component .............................................................................................761

EDA Cosimulation Library

Save ADS File........................................................................................................777Load ADS File ........................................................................................................779Save Spice Stimulus File .......................................................................................783Load Spice CSDF File............................................................................................789Triggered Save Spice Stimulus File .......................................................................793Triggered Load Spice CSDF File ...........................................................................797

Cable Access Library

Carrier GeneratorsCarrier Generator ...................................................................................................803Carrier Generator Measured ..................................................................................805

TransmittersModulators

Electrical Amplitude Modulator (AM)......................................................................807Electrical Frequency Modulator (FM) .....................................................................809Electrical Phase Modulator ....................................................................................811Quadrature Modulator ............................................................................................813PAM Modulator ......................................................................................................815QAM Modulator ......................................................................................................817PSK Modulator .......................................................................................................819

DPSK Modulator ....................................................................................................821OQPSK Modulator .................................................................................................823MSK Modulator ......................................................................................................825FSK Modulator .......................................................................................................827CPFSK Modulator ..................................................................................................829

Pulse Generators

M-ary Pulse Generator...........................................................................................831PAM Pulse Generator ............................................................................................833QAM Pulse Generator............................................................................................835PSK Pulse Generator.............................................................................................839DPSK Pulse Generator ..........................................................................................841OQPSK Pulse Generator .......................................................................................843MSK Pulse Generator ............................................................................................845

Sequence Generators

PAM Sequence Generator .....................................................................................849QAM Sequence Generator.....................................................................................853PSK Sequence Generator......................................................................................857DPSK Sequence Generator ...................................................................................861

ReceiversDemodulators

Electrical Amplitude Demodulator ..........................................................................865Electrical Phase Demodulator................................................................................867Electrical Frequency Demodulator .........................................................................869Quadrature Demodulator .......................................................................................871

Decoders

PAM Sequence Decoder........................................................................................873QAM Sequence Decoder .......................................................................................877PSK Sequence Decoder ........................................................................................881DPSK Sequence Decoder......................................................................................885

Detectors

M-ary Threshold Detector ......................................................................................889

Visualizer Library

OpticalOptical Spectrum Analyzer (OSA)..........................................................................893Optical Time Domain Visualizer (OTDV)................................................................897Optical Power Meter Visualizer ..............................................................................903WDM Analyzer (WDMA) ........................................................................................905Dual Port WDM Analyzer (DPWDMA) ...................................................................911

ElectricalOscilloscope Visualizer ..........................................................................................919RF Spectrum Analyzer (RFSA) .............................................................................923Eye Diagram Analyzer ...........................................................................................927BER Analyzer.........................................................................................................937Electrical Power Meter ...........................................................................................951Electrical Carrier Analyzer (ECAN) ........................................................................953Electrical Constellation Visualizer ..........................................................................959

Transmitters LibraryThis section contains information on the following transmitters.

Pulse Generators

Electrical

• Duobinary Pulse Generator• Electrical Jitter• Noise Source• RZ Pulse Generator• NRZ Pulse Generator• Gaussian Pulse Generator• Hyperbolic-Secant Pulse Generator• Sine Generator• Triangle Pulse Generator• Saw-Up Pulse Generator• Saw-Down Pulse Generator• Impulse Generator• Raised Cosine Pulse Generator• Sine Pulse Generator• Measured Pulse• Measured Pulse Sequence• Bias Generator

Optical

• Optical Gaussian Pulse Generator• Optical Sech Pulse Generator• Optical Impulse Generator• Measured Optical Pulse

1

• Measured Optical Pulse Sequence• Time Resolve Chirp (TRC) Measurement Data

Optical Sources

• CW Laser• Laser Rate Equations• Laser Measured• LED• White Light Source• Pump Laser• Pump Laser Array• CW Laser Array• CW Laser Array ES• CW Laser Measured• Directly Modulated Laser Measured• WDM Transmitter

Bit Sequence Generators

• Pseudo-Random Bit Sequence Generator• User-Defined Bit Sequence Generator

Modulators

Optical

• Mach-Zehnder Modulator• Electroabsorption Modulator• Amplitude Modulator• Phase Modulator• Frequency Modulator• Dual Drive Mach-Zehnder Modulator Measured• Electroabsorption Modulator Measured• Single Drive Mach-Zehnder Modulator Measured• Dual Port Dual Drive Mach-Zehnder Modulator Measured• LiNbO3 Mach-Zehnder Modulator

Duobinary Pulse Generator

Used for duobinary modulation schemes. It is equivalent to a subsystem based on an electrical delay and adder. It can be used together with any electrical pulse generator.

Ports

Parameters

Simulation

Name and description Port type Signal type

Input Input Electrical

Clock Input Binary

Output Output Electrical

Name and description Default value Units Value range

Enabled

Determines whether or not the component is enabled

True — True, False

DUOBINARY PULSE GENERATOR

Technical backgroundThe equivalent subsystem is:

Figure 1 Duobinary Pulse Generator subsystem

4

Electrical Jitter

Inserts jitter in the input signal.

Ports

Parameters

Main

Simulation



Clock Input Binary


Name and description Default value

Default unit Units Value range

FrequencyJitter frequency

100 MHz Hz, MHz, GHz, THz

[0,+INF[

Jitter amplitudeJitter amplitude range

0.1 UI — [0,+INF[


Units Value range

Enabled



ELECTRICAL JITTER

Technical backgroundThe jitter is a short-term, non-cumulative variation of the significant instants of a digital signal from their positions in time. Jitter amplitude is measured in unit intervals (UI), where 1 UI is the phase deviation of one clock period. The peak-to-peak UI deviation of the phase function with respect to time is referred as jitter amplitude. The output signal is:

where A is the jitter amplitude, B is the bit rate, and f is the jitter frequency.

Eout t( ) Ein t A2B------- 2πft( )sin+

=

6

NOISE SOURCE

Noise Source

Source of thermal noise.

Ports

Parameters

Main

Simulation





PSDDetermines whether the power is defined as PSD or as the average power in time

True — — True, False

Noise PowerValue of the PSD or the average power

–60 dBm W, mW, dBm ]-INF,+INF[



EnabledDetermines whether or not the component is enabled


IterationsNumber of times to repeat the calculation

Iterations — — [1,+INF[

Sample rateFrequency simulation window

Sample rate Hz Hz, GHz, THz ]0,+INF[

7

NOISE SOURCE

Noise

Random numbers

Technical backgroundThe average output Power or Power spectral density are parameters that you specify. This model generates electrical sampled signals or electrical sampled noise according to:

A Gaussian distribution describes the probability density function for the real and imaginary part of E. P is the average power when PSD parameter is false, if PSD is true then P is calculated from the power spectral density multiplied by the Sample rate.


Units Value range

Add noise to signalDetermines whether the noise will propagate separately from the signal or will be added to the signal

False — True, False


Units Value range

Generate random seedDetermines if the seed is automatically defined and unique


Random seed indexUser-defined seed index for noise generation

0 — [0, 4999]

Eout x t( ) jy t( )+[ ] P 2⁄=

8

RZ PULSE GENERATOR

RZ Pulse Generator

Generates a Return to Zero (RZ) coded signal.

Ports

Parameters

Main


Bit sequence Input Binary



Default unit Value range

Rectangle shapeDetermines the shape for the edges of the pulse

Exponential — Exponential, Gaussian, Linear, Sine

AmplitudePeak-to-peak amplitude of the pulse

1 a.u. ]-INF,+INF[

BiasDC Offset of the pulse

0 a.u. ]-INF,+INF[

Duty cycleDuration of the high level bit

0.5 bit [0,1]

Position 0 bit

Rise timeDefined as the time from when the rising edge reaches 10% of the amplitude to the time it reaches 90% of the amplitude

0.05 bit [0,1]

Fall timeDefined as the time from when the falling edge reaches 90% of the amplitude to the time it reaches 10% of the amplitude

0.05 bit [0,1]

9

RZ PULSE GENERATOR

Simulation


Default units

Unit Value range





10

RZ PULSE GENERATOR

Technical backgroundAccording to the parameter Rectangle shape, this model can produce pulses with different edge shapes:

Exponential

Gaussian

Linear

E t( )

1 et cr⁄( )–

0 t t1<≤,–

1 t1 t t2<≤,

et cf⁄( )–

t2 t tc<≤,

0 tc t T<≤,

=

E t( )

1 et cr⁄( )2–

0 t t1<≤,–

1 t1 t t2<≤,

et cf⁄( )2–

t2 t tc<≤,

0 t, c t T<≤

=

E t( )

t cr 0 t t1<≤,⁄

1 t1 t t2<≤,

t cf t2 t tc<≤,⁄

0 tc t T<≤,

=

11

RZ PULSE GENERATOR

Sine

where cr is the rise time coefficient and cf is the fall time coefficient. t1 and t2, together with cr and cf , are numerically determinate to generate pulses with the exact values of the parameters Rise time and Fall time. tc is the duty cycle duration, and T is the bit period.

E t( )

π.t cr⁄( ) 0 t t1<≤,sin

1 t1 t t2<≤,

π.t c⁄ f( ) t2 t tc<≤,sin

0 tc t T<≤,

=

12

NRZ PULSE GENERATOR

NRZ Pulse Generator

Generates a Non Return to Zero (NRZ) coded signal.

Ports

Parameters

Main






Rectangle shapeDetermines the shape for the edges of the pulse

Exponential — Exponential, Gaussian, Linear, Sine


1 a.u. ]-INF,+INF[


0 a.u. ]-INF,+INF[

Position 0 bit


0.05 bit [0,1]


0.05 bit [0,1]

13

NRZ PULSE GENERATOR

Simulation

Technical backgroundAccording to the parameter Rectangle shape, this model can produce pulses with different edge shapes:

Exponential

Gaussian

Linear

Sine

where cr is the rise time coefficient and cf is the fall time coefficient. t1 and t2, together with cr and cf, are numerically determined to generate pulses with the exact values of the parameters Rise time and Fall time, and T is the bit period.


Default units

Unit Value range





E t( )

1 et cr⁄( )–

0 t t1<≤,–

1 t1 t t2<≤,

et cf⁄( )–

t2 t T<≤,

=

E t( )

et cr⁄( )2–

0 t t1<≤,

1 t1 t t2<≤,

et cf⁄( )2–

t2 t T<≤( ),

=

E t( )

t cr 0 t t1<≤,⁄

1 t1 t t2<≤,

t cf t2 t T<≤,⁄

=

E t( )

π.t cr⁄( ) 0 t t1<≤,sin

1 t1 t t2<≤,

π.t c⁄ f( ) t2 t T<≤,sin

=

14

GAUSSIAN PULSE GENERATOR

Gaussian Pulse Generator

Generates an electrical Gaussian-pulsed signal.

Ports

Parameters

Main







1 a.u. ]-INF,+INF[


0 a.u. ]-INF,+INF[

WidthFWHM of the pulse amplitude

0.5 bit [0,1]

Position 0 bit

OrderOrder of the function

1 — [1,100]

TruncatedDetermines whether or not the pulses overlap with each other


15

GAUSSIAN PULSE GENERATOR

Simulation

Technical backgroundThis model generates Gaussian or super-Gaussian electrical pulses according to the bit sequence at the input. For each bit

where Ap is the parameter peak-to-peak Amplitude, and Abias is the parameter Bias. B is the bit value (1 or 0) and depends on the input bit sequence. k is the fitting coefficient determined numerically to generate pulses with the exact values of the parameter Width TFWHM, and N is the Order of the Gaussian (N=1) or super-Gaussian pulses (N>1).


Default units

Unit Value range





E t( ) B. Ap.e

12--- t.k

TFWHM----------------

2N–

Abias+

=

16

HYPERBOLIC-SECANT PULSE GENERATOR

Hyperbolic-Secant Pulse Generator

Generates a hyperbolic-secant pulsed signal.

Ports

Parameters

Main







1 a.u. ]-INF,+INF[


0 a.u. ]-INF,+INF[


0.5 bit [0,1]

Position 0 bit

TruncatedDefinea whether or not the pulses overlap with each other


17

HYPERBOLIC-SECANT PULSE GENERATOR

Simulation

Technical backgroundThis model generates electrical pulses according to the bit sequence at the input. For each bit:

where Ap is the parameter peak-to-peak Amplitude, and Abias is the parameter Bias. B is the bit value (1 or 0) and depends on the input bit sequence. k is the fitting coefficient determined numerically to generate pulses with the exact values of the parameter Width, TFWHM.


Default units

Unit Value range





E t( ) B. Apt.k

TFWHM-----------------

2cosh⁄ Abias+

=

18

SINE GENERATOR

Sine Generator

Generates an electrical sine waveform signal.

Ports

Parameters

Main





FrequencyFrequency simulation window

32 GHz Hz, MHz, GHz, THz

]0,+INF[


1 a.u. — ]-INF,+INF[



PhaseInitial phase of the signal

0 deg — ]-INF,+INF[

19

SINE GENERATOR

Simulation









20

TRIANGLE PULSE GENERATOR

Triangle Pulse Generator

Generates an electrical triangle-pulsed signal.

Ports

Parameters

Main







1 a.u. ]-INF,+INF[


0 a.u. ]-INF,+INF[


0.5 bit [0,1]

Position 0 bit



21

TRIANGLE PULSE GENERATOR

Simulation


Default units

Unit Value range





22

SAW-UP PULSE GENERATOR

Saw-Up Pulse Generator

Generates a saw-up signal.

Ports

Parameters

Main







1 a.u. ]-INF,+INF[


0 a.u. ]-INF,+INF[


0.5 bit [0,1]

Position 0 bit



23

SAW-UP PULSE GENERATOR

Simulation


Default units

Unit Value range





24

SAW-DOWN PULSE GENERATOR

Saw-Down Pulse Generator

Generates a saw-down pulsed signal.

Ports

Parameters

Main







1 a.u. ]-INF,+INF[


0 a.u. ]-INF,+INF[


0.5 bit [0,1]

Position 0 bit [-1, 1]



25

SAW-DOWN PULSE GENERATOR

Simulation


Default units

Unit Value range





26

IMPULSE GENERATOR

Impulse Generator

Generates an electrical signal composed by a sequence of Impulses.

Ports

Parameters

Main

Simulation







1 a.u. ]-INF,+INF[


0 a.u. ]-INF,+INF[

PositionRelative position of the impulse

0.5 bit [0,1]


Default units

Unit Value range





27

IMPULSE GENERATOR

Notes:

28

RAISED COSINE PULSE GENERATOR

Raised Cosine Pulse Generator

Generates a raised-cosine pulsed signal.

Ports

Parameters

Main







1 a.u. ]-INF,+INF[


0 a.u. ]-INF,+INF[


0.5 bit [0,1]

Position 0 bit



29

RAISED COSINE PULSE GENERATOR

Simulation


where Ap is the parameter peak-to-peak Amplitude, and Abias is the parameter Bias. B is the bit value (1 or 0) and depends on the input bit sequence. k is the fitting coefficient determined numerically to generate pulses with the exact values of the parameter Width, TFWHM.


Default unit Unit Value range





E t( ) B. Ap. t.kTFWHM-----------------

2cos Abias+

=

30

SINE PULSE GENERATOR

Sine Pulse Generator

Generates a sine-pulsed signal.

Ports

Parameters

Main







1 a.u. ]-INF,+INF[


0 a.u. ]-INF,+INF[


0.5 bit [0,1]

Position 0 bit



31

SINE PULSE GENERATOR

Simulation


where Ap is the parameter peak-to-peak Amplitude, and Abias is the parameter Bias. B is the bit value (1 or 0) and depends on the input bit sequence. k is the fitting coefficient numerically determinate to generate pulses with the exact values of the parameter Width TFWHM.


Default units

Unit Value range





E t( ) B. Ap. t.kTFWHM-----------------

cos Abias+ =

32

MEASURED PULSE

Measured Pulse

Generates an electrical pulse based on measurements according to the bit sequence at the input port.

Ports

Parameters

Main

Numerical




Name and description Default value Default unit Value range


1 a.u. ]-INF,+INF[


0 a.u. ]-INF,+INF[

Position 0 bit

FilenameFilename with the measured data

Pulse.dat — —


InterpolationDetermines the interpolation algorithm for the measured data

Linear — Linear, Cubic

33

MEASURED PULSE

Simulation

Graphs


Default units

Unit Value range





Name and description X Title Y Title

Measured data Time period (a.u.) Amplitude (a.u.)

34

MEASURED PULSE SEQUENCE

Measured Pulse Sequence

Generates an electrical signal based on measurements.

Ports

Parameters

Main

Numerical




ScaleFactor to scale the signal amplitude

1 a.u. ]-INF,+INF[

Start timeInitial part of the signal to be skipped

0 s [0,+INF[


Sequence.dat — —




35

MEASURED PULSE SEQUENCE

Simulation

Graphs

Technical backgroundThis model generates electrical signal loading measurements from a file. The input file is formatted containing two values per line, the time in seconds and signal amplitude in arbitrary units. For example, the file representing one measurement has the following form:


Default unit

Units Value range








Measured data Time (s) Amplitude (a.u.)

0 0

1e-6 0.5

2e-6 0.5

3e-6 0

...

36

BIAS GENERATOR

Bias Generator

A d.c. source.

Ports

Parameters

Main

Simulation




Units Value range

AmplitudeAmplitude of the signal output

1 a.u. ]-INF,+INF[









37

BIAS GENERATOR

Notes:

38

OPTICAL GAUSSIAN PULSE GENERATOR

Optical Gaussian Pulse Generator

Generates a Gaussian-pulsed optical signal.

Ports

Parameters

Main



Output Output Optical



Frequency Emission frequency

193.1 THz Hz, THz, nm [0,+INF[

PowerPeak-to-peak power of the pulse

0 dBm W, mW, dBm ]-INF,+INF[




0.5 bit — [0,1]

Position 0 bit —

OrderOrder of the function

1 — — [1,100]


False — — True, False

39


Chirp

Polarization

Simulation



Chirp definition Linear — Linear, Measured

Chirp factor 0 rad/s

Alpha parameter 0 rad/W

Adiabatic chirpResults from changes in the steady state carrier densities

0 1/s [0,1]



AzimuthAzimuth angle of output polarization

0 deg ]-90,90]

EllipticityEllipticity angle of output polarization

0 deg [-45,45]





Parameterized Parameterized — —



40


Technical backgroundThis model generates Gaussian or super-Gaussian optical pulses according to the bit sequence at the input. For each bit, the output optical power is:

where Ap is the parameter peak-to-peak Power, and Abias is the parameter Bias. B is the bit value (1 or 0) and depends on the input bit sequence. k is the fitting coefficient determined numerically to generate pulses with the exact values of the parameter Width, TFWHM, and N is Order of the Gaussian (N=1) or super-Gaussian pulses (N>1).

The chirp is modeled using:

where ϕ is the signal phase, αe is the parameter Linewidth enhancement factor, and is the parameter Adiabatic chirp.

The output is multiplied with a complex vector considering the state of polarization:

where the power splitting k and the phase difference θ are related to the parameters Azimuth and Ellipticity as:

P t( ) B. Ap.e

12--- t.k

TFWHM----------------

2N–

Abias+

=

dϕdt------

αe2------ d

dt----- P t( ) κP t( )+ln=

κ

EX t( )

EY t( ) 1 k–

kejθ

P t( )⋅=

α ε

2α( )tan 2 k 1 k–( ) θ( )cos1 2.k–

-----------------------------------------=

2ε( )sin 2 k 1 k–( ) θ( )sin=

41


Notes:

42

OPTICAL SECH PULSE GENERATOR

Optical Sech Pulse Generator

Generates a hyperbolic-secant pulsed optical signal.

Ports

Parameters

Main


Bit sequences Input Binary











0.5 bit — [0,1]

Position 0 bit —



43


Chirp

Polarization

Simulation



Chirp definition Linear — Linear, Measured

Chirp factor 0 rad/s



0 1/s [0,1]


Units Value range


0 deg ]-90,90]


0 deg [-45,45]








44


Technical backgroundThis model generates optical pulses according to the bit sequence at the input. For each bit, the output optical power is:

where Ap is the parameter peak-to-peak Power, and Abias is the parameter Bias. B is the bit value (1 or 0) and depends on the input bit sequence. k is the fitting coefficient determined numerically to generate pulses with the exact values of the parameter Width, TFWHM.




The power splitting k and the phase difference θ are related to the parameters Azimuth and Ellipticity as:

P t( ) B. Apt.k

TFWHM

------------- Abias+cosh⁄

=

dϕdt------

αe2------ d

dt----- P t( ) κP t( )+ln=

κ

EX t( )

EY t( ) 1 k–

kejθ

P t( )⋅=

αε

2α( )tan 2 k 1 k–( ) θ( )cos1 2.k–

-----------------------------------------=

2ε( )sin 2 k 1 k–( ) θ( )sin=

45


Notes:

46

OPTICAL IMPULSE GENERATOR

Optical Impulse Generator

Generates an optical signal composed by a sequence of Impulses.

Ports

Parameters

Main

Chirp












PositionRelative position of the impulse

0 bit [0,1]


Units Value range



0 1/s [0,1]

47


Polarization

Simulation


Units Value range


0 deg ]-90,90]


0 deg [-45,45]








48


Technical backgroundThis model generates optical pulses according to the bit sequence at the input. For each bit, the output optical power is:

where Ap is the parameter peak-to-peak Power, and Abias is the parameter Bias. B is the bit value (1 or 0) and depends on the input bit sequence. δ is the impulse function and tP is the parameter Pulse position.





P t( ) B. Apδ t tp–( ) Abias+( )=

dϕdt------

αe2------ d

dt----- P t( ) κP t( )+ln=

κ

EX t( )

EY t( ) 1 k–

kejθ

P t( )⋅=

αε

2α( )tan 2 k 1 k–( ) θ( )cos1 2.k–

-----------------------------------------=

2ε( )sin 2 k 1 k–( ) θ( )sin=

49


Notes:

50

MEASURED OPTICAL PULSE

Measured Optical Pulse

Generates a pulse based on measurements.

Ports

Parameters

Main












Position 0 bit —


Optical pulse.dat

— — —

File formatDetermines the format of the file with the measurements

Power — — Power, Power Phase, Real Imag, Phase

51


Polarization

Numerical

Simulation

Graphs


Units Value range


0 deg ]-90,90]


0 deg [-45,45]


Units Value range











Measured magnitude data Time period (a.u.) Amplitude (V)

Measured phase data Time period (a.u.) Phase (rad)

52


Technical backgroundThe input file is formatted containing two items per line — the time in seconds and the signal measurement (Power in watts, Phase in radians, Real and Imag in Volts). According to the parameter File format, the second item can be one value (Power or Phase), or two values (Power and Phase or Real and Imag).

Power (Phase will be set to zero)

Power Phase

Real Imag

0 0

1e-6 0.5

2e-6 0.5

3e-6 0

...

0 0 0

1e-6 0.5 3.14

2e-6 0.5 3.14

3e-6 0 0

...

0 0 0

1e-6 –0.5 7.9e-4

2e-6 –0.5 7.9e-4

3e-6 0 0

...

53


Phase (Power will be set to one)

This model generates optical pulses according to the bit sequence at the input. For each bit, the output optical power is:

where Ap is the parameter peak-to-peak Power, and Abias is the parameter Bias. B is the bit value (1 or 0) and depends on the input bit sequence. M is the measured data.



0 0

1e-6 3.14

2e-6 3.14

3e-6 0

...

P t( ) B. ApM t( ) Abias+( )=

EX t( )

EY t( ) 1 k–

kejθ

P t( )⋅=

αε

2α( )tan 2 k 1 k–( ) θ( )cos1 2.k–

-----------------------------------------=

2ε( )sin 2 k 1 k–( ) θ( )sin=

54

MEASURED OPTICAL PULSE SEQUENCE

Measured Optical Pulse Sequence

Generates an optical signal based on measurements.

Ports

Parameters

Main

Polarization










0 s — [0,+INF[


Sequence.dat — — —

File formatDetermines the format of the file with the measurements

Power — — Power, Power Phase, Real Imag, Phase


Units Value range


0 deg ]-90,90]

55


Numerical

Simulation

Graphs


0 deg [-45,45]


Units Value range









Parameterized Parameterized — — Sampled, Parameterized




Measured magnitude data Time (s) Amplitude (V)

Measured phase data Time (s) Phase (rad)


Units Value range

56


Technical backgroundThis model generates optical signal loading measurements from a file.

The input file is formatted containing two items per line — the time in seconds and signal measurement (Power in watts, Phase in radians, Real and Imag in Volts). According to the parameter File format, the second item can be one value (Power or Phase) or two values (Power and Phase or Real and Imag).

Power (Phase will be set to zero)

Power Phase

Real Imag

0 0

1e-6 0.5

2e-6 0.5

3e-6 0

...

0 0 0

1e-6 0.5 3.14

2e-6 0.5 3.14

3e-6 0 0

...

0 0 0

1e-6 –0.5 7.9e-4

2e-6 –0.5 7.9e-4

3e-6 0 0

...

57


Phase (Power will be set to one)



0 0

1e-6 3.14

2e-6 3.14

3e-6 0

...

EX t( )

EY t( ) 1 k–

kejθ

P t( )⋅=

αε

2α( )tan 2 k 1 k–( ) θ( )cos1 2.k–

-----------------------------------------=

2ε( )sin 2 k 1 k–( ) θ( )sin=

58

TIME RESOLVE CHIRP (TRC) MEASUREMENT DATA

Time Resolve Chirp (TRC) Measurement Data

This component is an interface between OptiSystem and time resolve chirp (TRC) [1] measurement instruments, such as the OSA Agilent 86146B with TRC option.

Ports

Parameters

Main

Polarization










0 s — [0,+INF[


Sequence.dat — — —


Units Value range


0 deg ]-90,90]


0 deg [-45,45]

59


Numerical

Simulation

Graphs

Technical backgroundThis component generates optical signal loading measurements from a file. These measurements are TRC data that describe the power and chirp evolution of the optical signal in time [1].

TRC provides frequency vs time information about a modulated lightwave signal. Also called dynamic chirp, the TRC graph provides useful information on the ability of a modulated signal to propagate over long distances in optical fiber.

Using measurement equipment such as the Agilent 86146B, with the filter mode capability, Agilent 86100 Infinium Digital Communications Analyzer (DCA) dedicated software (86146B Option TRL), and a personal computer, the time resolved chirp (TRC) of a modulated laser can be calculated.

From the measurement, a file with the TRC data is generated. OptiSystem can load this file and the effect of laser chirp on a wide variety of system performance metrics


Units Value range













Measured power data Time (s) Power (W)

BER measured chirp data Time (s) Chirp (Hz)

60


- such as the effect on the performance of a long-haul dense wavelength division multiplexed (DWDM) system with EDFA and Raman optical amplification and dispersion compensation - can be studied across an unlimited range of system designs.

The input file is formatted containing three items per line - the time in seconds, the signal power is Watt (Linear scale) or dBm, and the signal chirp (Hz).



Time Signal power (W or dBm) Signal chirp

0 1.27617e-006 -7.80425e+009

6.25e-012 1.139e-006 -4.94806e+009

1.25e-011 1.46161e-006 -6.57706e+009

1.875e-011 1.33136e-006 -6.10874e+009

2.5e-011 1.54705e-006 -2.89844e+009

3.125e-011 1.03595e-006 -7.38826e+009

. . . . . . . . .

EX t( )

EY t( ) 1 k–

kejθ

P t( )⋅=

αε

2α( )tan 2 k 1 k–( ) θ( )cos1 2.k–

-----------------------------------------=

2ε( )sin 2 k 1 k–( ) θ( )sin=

61


Reference:[1] Agilent Technologies, “Making Time-Resolved Chirp Measurements Using the Optical

Spectrum Analyzer and Digital Communications Analyzer”, Agilent Application Note 1550-7, 2002.

62

CW LASER

CW Laser

Generates a continuous wave (CW) optical signal.

Ports

Parameters

Main

Polarization






193.1 THz Hz,THz, nm [0,+INF[

Power 0 dBm W, mW, dBm ]-INF,+INF[

Linewidth 10 MHz — [0,+INF[

Initial phase 0 deg — ]-INF,+INF[

MxN next generation — — — —

StringParameter — — — —


Units Value range

Azimuth Azimuth angle of output polarization

0 deg ]-90,90]


0 deg [-45,45]

63

CW LASER

Simulation

Noise

Random numbers












Noise bandwidthBandwidth to create noise bins

0 THz Hz, THz, nm [0,+INF[

Noise thresholdMinimum value for adaptation of noise bins

–100 dB — ]-INF,+INF[

Noise dynamicThreshold ratio for adaptation of noise bins

3 dB — ]-INF,+INF[


Units Value range




0 — [0,4999]

64

CW LASER

Technical backgroundIn the CW case, the average output Power is a parameter that you specify. Laser phase noise is modeled using the probability density function:

where is the phase difference between two successive time instants and dt is the time discretization. A Gaussian random variable for the phase difference between two successive time instants with zero mean and a variance equal to has been assumed, with as the laser Linewidth.


where the power splitting k and the phase difference θ are related to the parameters Azimuth and Ellipticity as follows:

f ∆ϕ( ) 12π ∆fdt---------------------- e

∆ϕ2

4π∆fdt------------------–

⋅=

∆ϕ

2π ∆f ∆f

EX t( )

EY t( ) 1 k–

kejθ

P t( )⋅=

α ε

2α( )tan 2 k 1 k–( ) θ( )cos1 2.k–

--------------------------------------------=

2ε( )sin 2 k 1 k–( ) θ( )sin=

65

CW LASER

Notes:

66

LASER RATE EQUATIONS

Laser Rate Equations

Utilizes the rate equations to simulate a DFB laser.

Ports

Parameters

Main

Physical


Modulation Input Electrical




Frequency 193.1 THz Hz, THz, nm [30,3e5]

Calculate current True — — True, False

Power 0 dBm W, mW, dBm [-1e100, 1e100]

Power at bias current –30 dBm W, mW, dBm

Bias current 38 mA — [0, 1000]

Modulation peak current 28 mA — [0, 1000]

Threshold current 33.4572 mA — [0, 1000]

Threshold power 0.0155558 mW — [0, 1000]



Active layer volume 1.5e-010 cm3 0, 1e-3

Quantum efficiency 0.4 — 0, 1

Spontaneous emission factor 3e-005 — 2e-5, 20e-5

67


Simulation

Noise

Random numbers

Gain compression coefficient 1e-017 cm3 0.5e-17, 10e-17

Carrier density at transparency 1e+018 cm-3 0, 100e18

Differential gain coefficient 2.5e-016 cm2 0, 50e-16

Group velocity 8.5e+009 cm/s 0, 100e9

Linewidth enhancement factor 5 — –20, 20

Mode confinement factor 0.4 — 0, 1

Carrier lifetime 1e-009 s 0, 50e-9

Photon lifetime 3e-012 s 0, 50e-9


Units Value range

Enabled True — True, False



Units Value range

Include noise True — True, False

Include phase noise True — True, False


Units Value range




0 — [0,4999]



68


Technical backgroundThe modulation dynamics of the laser are modeled by coupled rate equations which describe the relation between the carrier density , photon density , and optical phase

:

where go is the gain slope constant,

The optical power and chirp response of the semiconductor laser to a current waveform is determined by the above equations. A Runge-Kutta algorithm is used to numerically integrate the coupled first order differential equations (2-4). These

(2)

(3)

(4)

ao is the active layer coefficient

vg is the group velocity

is the gain compression factor

Nt is the carrier density at transparency

is the fraction of spontaneous emission coupled into the lasing mode

is the mode confinement factor

V is the active layer volume

is the photon lifetime

is the electron lifetime

is the linewidth enhancement factor

N t( ) S t( )φ t( )

dN t( )dt

-------------- I t( )q V⋅----------- N t( )

τn----------– go N t( ) Nt–( ) 1

1 ε S t( )⋅+( )------------------------------- S t( )⋅ ⋅ ⋅–=

dS t( )dt

------------- Γ go N t( ) Nt–( ) 11 ε S t( )⋅+( )

------------------------------- S t( ) S t( )τp

---------– Γ β N t( )⋅ ⋅τn

--------------------------+⋅ ⋅ ⋅ ⋅=

dφ t( )dt

------------- 12--- α Γ go N t( ) Nt–( ) 1

τp-----–⋅ ⋅⋅ ⋅=

go vg ao⋅=

ε

β

Γ

τp

τn

α

( )tI

69


equations apply to a noiseless laser oscillating in a single longitudinal mode above threshold. The photon and electron densities within the active region of the laser are assumed to be uniform. The linewidth enhancement factor and the nonlinear gain compression parameter are taken to be constant for a given structure.

The time variations for the optical and laser chirp are:

where is the differential quantum efficiency

References

[1] J. C. Cartledge and G. S. Burley, “The Effect of the Laser Chirping on Lightwave System Performance”, J. Lightwave Technology, vol. 7, pp. 568-573, March 1989.

(5)

(6)

v is the optical frequency

h is the Planck’s constant

PS V ηo h v⋅ ⋅ ⋅ ⋅

2 Γτp⋅------------------------------------=

∆v 12 π⋅---------- dφ

dt------⋅=

ηo

70

LASER MEASURED

Laser Measured

Extracts values of the rate equation parameters using measurements and simulates a DFB laser.

Ports

Parameters

Main

Measurements







Calculate current True — — True, False

Power 0 dBm W, mW, dBm [-1e100, 1e100]

Power at bias current –30 dBm W, mW, dBm [-1e100, 1e100]

Bias current 23 mA — [0, 1000]

Modulation peak current 28 mA — [0, 1000]



Damping factor 10.28 1e9 s-1 — ]0, 1000]

Resonance frequency factor 6.43 1e20 Hz2 — ]0, 1000]

Threshold current 18 mA — [0, 1000]

Power bias 0.4 mW W, mW, dBm [1e-100, 1e100]

71

LASER MEASURED

Physical

Initial estimate

Simulation


Calculate parameters True —

Active layer volume 2e-011 cm3 0, 1e-3

Quantum efficiency 0.2 — 0, 1

Spontaneous emission factor 0.0001 — 2e-5, 20e-5

Gain compression coefficient 1.5e-017 cm3 0.5e-17, 10e-17

Carrier density at transparency 1e+018 cm-3 0, 100e18

Differential gain coefficient 1.765e-016 cm2 0, 50e-16

Mode confinement factor 0.2 — 0, 1

Carrier lifetime 1e-009 s 0, 50e-9

Photon lifetime 1e-012 s 0, 50e-9

Group velocity 8.5e+009 cm/s 0, 100e9

Linewidth enhancement factor 5 — –20, 20


Active layer volume estimation 2e-011 cm3 0, 1e-3

Quantum efficiency estimation 0.2 — 0, 1

Spontaneous emission factor estimation 0.0001 — 2e-5, 20e-5

Gain compression coefficient estimation 1.5e-017 cm3 0.5e-17, 10e-17

Carrier density at transparency estimation 1e+018 cm-3 0, 100e18

Differential gain coefficient estimation 1.765e-016 cm2 0, 50e-16

Mode confinement factor estimation 0.2 — 0, 1

Carrier lifetime estimation 1e-009 s 0, 50e-9

Photon lifetime estimation 1e-012 s 0, 50e-9





72

LASER MEASURED

Noise

Random numbers


Include noise True — True, False

Include phase noise True — True, False





0 — [0,4999]

73

LASER MEASURED

Technical backgroundThe laser measured model extracts values of the rate equation parameters using measurements of the threshold current, optical power, resonance frequency, and damping factor to simulate a DFB laser.

Based on the results featured in [1] , the values of the rate equation parameters are calculated in a way that parameters simultaneously yield the measured values of Y (damping factor), Z (resonance frequency factor), Ith (threshold current), and P (Power bias). The parameter extraction procedure is based on minimization of the sum of squared errors between the measured values of (Y, Z, Ith, P) and values calculated from rate equation parameters. The minimization is over the values of the rate equation parameters which are:

Damping factor

Resonance frequency factor

Threshold current

Power bias

where go is the gain slope constant,

ao is the active layer coefficient

is the gain compression factor

Nt is the carrier density at transparency

is the fraction of spontaneous emission coupled into the lasing mode

is the mode confinement factor

is the differential quantum efficiency

V is the active layer volume

Y g0S

1 ε S⋅+( )------------------------ 1

τn----- Γ g0 N Nt–( ) 1

1 ε S⋅+( )2

-------------------------- 1τp-----+⋅–+=

Z g0S

1 ε S⋅+( )------------------------ 1

τp----- β 1–( ) Γ

g0τn-----g0 N Nt–( ) 1

1 ε S⋅+( )2

-------------------------- 1τp τn⋅--------------+⋅ ⋅+⋅=

Ithq V⋅

τn-----------

1 Nt Γ go τp⋅ ⋅ ⋅+Γ go τp⋅ ⋅

------------------------------------------⋅=

PS V η0 h v⋅ ⋅ ⋅ ⋅

2 Γτp⋅------------------------------------=

go vg ao⋅=

ε

β

Γ

η0

74

LASER MEASURED

The minimization routine finds a local minimum for the equation

where are the measured values and are the calculated values using the initial estimates of the rate equation parameters.

The parameters available in the main tab allow the user to enter the values for current, or for power in steady state. Using these numbers, the model will estimate the values of the current.

Note: It is recommended to enter the values for current, rather than power, when using the measured laser (as this is the realistic case).

The parameters in the measured tab are used to extract the physical/geometrical properties of the laser. This extraction is completely independent of the parameters in the main tab (current/power).

After finding the rate equation parameters, the laser measured works similarly to the DFB laser model.

is the photon lifetime

is the electron lifetime

are the steady-state values of the carrier and photon densities corresponding to the bias current of the laser

v is the unmodulated optical frequency

vg is the group velocity


τp

τn

NandS

Func Ymea Ycal–( )2 zmea zcal–( )2 Pmea Pcal–( )2 Imea Ical–( )2+ + +=

Ymea Zmea Pmea Imea,,,( ) Ycal Zcal Pcal Ical,,,( )

75

LASER MEASURED

Reference:[1] Cartledge, J. C. and Srinivasan, R. C. “Extraction of DFB laser rate equation parameters for

system simulation purposes”, J. Light. Techn., 15, 852-860, (1997).

76

LED

LED

Simulates a modulated LED.

Ports

Parameters

Main

Simulation







Electron lifetime 1e-009 s — ]0, 1]

RC constant 1e-009 s — ]0, 1]

Quantum efficiency 0.05 — — ]0, 1]

Bandwidth 6 THz Hz, THz, nm ]0, INF ]


Units Value range




Iterations Iterations — [1, 1e+009]

77

LED

Random numbers

Technical backgroundIn this model, the mean of the optical power is a function of the modulation current (input signal). The conversion of the current into optical power is described by the responsivity of the LED:

where is the quantum efficiency

The modulated characteristics depend of the electron lifetime and the device of the diode, and are modeled by the transfer function applied to the current:

where is the Electron life time and is the RC constant.

If the parameter Parameterized is selected, the output consist of a single value representing the average LED output at the frequency output.

Note: The noise bins signals are not produced by this modulator.


Units Value range


Yes — True, False


0 — [0,4999]


f is the emission frequency

q is the electron charge

i(t) is the modulation current signal

P η h f i t( )q

--------⋅ ⋅ ⋅=

η

H f( ) 11 j 2 π f τn τrc+( )⋅ ⋅ ⋅ ⋅+------------------------------------------------------------=

τn τrc

78

WHITE LIGHT SOURCE

White Light Source

Generates a gaussian distributed optical white noise.

Ports

Parameters

Main

Simulation







PSDDetermines whether the Power is the PSD (/Hz) or the average power


PowerAverage output powers










79

WHITE LIGHT SOURCE

Noise

Random numbers

Technical backgroundThe average output Power or Power spectral density and Frequency are parameters that you specify. This model generates noise bins or sampled signals at the output according to:

A Gaussian distribution has been assumed to describe the probability density function for the real and imaginary part of Ex and Ey. P is the average power when PSD parameter is false. If PSD is true, then P is calculated from the power spectral density multiplied by the Sample rate.



Noise bins spacing 10 GHz Hz, GHz, THz, nm

[1, 100000]

Convert noise binsDetermines if the generated noise bins are incorporated into the signal

Convert noise bins

— — —


Units Value range




0 — [0,4999]

80

PUMP LASER

Pump Laser

Generates an optical parameterized signal to be used for optical amplifier pumping.

Ports

Parameters

Main

Polarization






980 nm Hz, THz, nm [0,+INF[

PowerAverage output powers

100 mW W, mW, dBm [0,+INF[


Units Value range


0 deg ]-90,90]


0 deg [-45,45]

81

PUMP LASER

Simulation

Technical backgroundIn the CW Laser case, average output Power is a parameter that you specify. This model generates only parameterized signal at the output.


where the power splitting k and the phase difference θ are related to the parameters Azimuth and Ellipticity as follows:


Units Value range




Iterations — [1, 1e+009]

EX t( )

EY t( ) 1 k–

kejθ

P⋅=

α ε

2α( )tan 2 k 1 k–( ) θ( )cos1 2.k–

--------------------------------------------=

2ε( )sin 2 k 1 k–( ) θ( )sin=

82

PUMP LASER ARRAY

Pump Laser Array

An array of pump lasers.

Ports

Parameters

Main

Frequency


Output 1 Output Optical









Number of output ports 8 — [1, 1000]



Frequency[0]Center frequency for pump 0

1405 nm Hz, THz, nm [100, 2000]


1412.5 nm Hz, THz, nm [100, 2000]

83

PUMP LASER ARRAY

Power


1420 nm Hz, THz, nm [100, 2000]


1427.5 nm Hz, THz, nm [100, 2000]


1435 nm Hz, THz, nm [100, 2000]


1442.5 nm Hz, THz, nm [100, 2000]


1450 nm Hz, THz, nm [100, 2000]


1457.5 nm Hz, THz, nm [100, 2000]



Power[0]Ouptut power for pump 0


















84

PUMP LASER ARRAY

Polarization

Simulation



0 deg ]-90,90]


0 deg [-45,45]






85

PUMP LASER ARRAY

Notes:

86

CW LASER ARRAY

CW Laser Array

This component is an array of CW lasers.

Ports

Parameters

Main










Name and description Default value Default Unit Value range


Linewidth 10 MHz [0, 1e+009[

Initial phase 0 deg [-1e+100,1e+100]

87

CW LASER ARRAY

Frequency

Power



Frequency[0]Center frequency for laser 0

193.1 THz Hz, THz, nm [30, 300000]


193.2 THz Hz, THz, nm [30, 300000]


193.3 THz Hz, THz, nm [30, 300000]


193.4 THz Hz, THz, nm [30, 300000]


193.5 THz Hz, THz, nm [30, 300000]


193.6 THz Hz, THz, nm [30, 300000]


193.7 THz Hz, THz, nm [30, 300000]


193.8 THz Hz, THz, nm [30, 300000]



Power[0]Ouptut power for laser 0












88

CW LASER ARRAY

Polarization

Simulation

Noise






Units Value range


0 deg ]-90,90]


0 deg [-45,45]







Parameterized Parameterized — — —








–100 dB — ]-INF,+INF[





89

CW LASER ARRAY

Random numbers


Units Value range




0 — [0,4999]

90

CW LASER ARRAY ES

CW Laser Array ES

This component is an array of CW lasers. The emission frequencies are equally spaced (ES).

Ports

Parameters

Main












FrequencyEmission frequency of the first laser

193.1 THz, Hz, nm [30,+INF[

Frequency spacingFrequency spacing between adjacent lasers

100 GHz, THZ, Hz, nm

]-INF,+INF[



91

CW LASER ARRAY ES

Power

Polarization




















Units Value range


0 deg ]-90,90]


0 deg [-45,45]

92

CW LASER ARRAY ES

Simulation

Noise

Random numbers















–100 dB — ]-INF,+INF[




Units Value range




0 — [0,4999]

93

CW LASER ARRAY ES

Technical BackgroundThe CW Laser Array ES is equivalent to the conventional CW Laser Array component. However, The CW Laser Array ES model is easier to set up for WDM systems, because it only requires the initial laser emission frequency and the spacing. The signal output power is the same for all the output signals.

94

CW LASER MEASURED

CW Laser Measured

Generates a continuous wave (CW) optical signal based on measurements. You can enter parameters such as linewidth, side mode suppression, and relative intensity noise (RIN).

Ports

Parameters

Main

Side Mode












Calculate side modeDetermines if the signal output will have one side mode

False — — —

95

CW LASER MEASURED

RIN

Polarization

SeparationMode frequency separation from the laser center frequency

75 GHz Hz, GHz, THz, nm

[0,+INF[

Suppression ratioAttenuation of the side mode relative to the output power

30 dB — [0,+INF[

Independent side modeWhen enabled, the side mode has an independent power value that can change the total average power

False — — —



RINRelative intensity noise value

–130 dB/Hz — ]-INF,+INF[

Include RINDetermines if the RIN will be added to the output signal


Measured powerValue of the power during the measurement of RIN

10 dBm W. mW, dBm ]-INF,+INF[


Units Value range


0 deg ]-90,90]


0 deg [-45,45]

Polarization filterDetermines the polarization of the filter

None — None, Polarization X, Polarization Y



96

CW LASER MEASURED

Simulation

Noise

Random numbers












Noise bandwidthBandwidth to increase noise bins

1 THz Hz, THz, nm [1e-100, 1e-100]

Noise bins spacingDetermines noise bins spacing


[1, 1000]


Convert noise bins

— — [0, 0]


Units Value range




0 — [0,4999]

97

CW LASER MEASURED

Technical backgroundThis model is similar to the CW Laser — however, it includes additional effects, such as a side mode and RIN.

If the you enable the parameter Calculate side mode, the side mode will be generated according to:

where P is laser output power, s is the parameter Suppression ratio in linear scale, and is defined by the parameter Separation.

If the parameter Independent side mode is enabled, the average signal power will be greater than P, since it includes the contribution from the side mode. If this parameter is disabled, the output power will be P. This means that the signal will be scaled in order to give the same average power. The signal phase and polarization is calculated in the same way as the CW laser.

If the parameter Include RIN is enabled, the model generates noise bins with bandwidth and spacing that you define. The parameter RIN is the ratio of the mean-square optical intensity noise to the square of the average power [1][2]:

where is the mean-square optical intensity fluctuation at a specific frequency and is the parameter Measured power. This models estimates based on the parameters RIN and Measured power.

The signal phase and polarization is calculated in the same way as the CW laser, where the laser phase noise is modeled using a Gaussian random variable for the phase difference between two successive time instants with zero mean and a variance equal to , where is the laser Linewidth.

The probability density function is:

where is the phase difference between two successive time instants and dt is the time discretization.


Eout t( ) P 1 s 2π∆ft( ) s 2π∆ft–( )ejϕcos+cos+[ ]=

∆f

RIN ∆P2⟨ ⟩

Pm2

---------------dB Hz⁄=

∆P2⟨ ⟩ Pm2

∆P2⟨ ⟩

2π ∆f ∆f

f ∆ϕ( ) 12π ∆fdt---------------------- e

∆ϕ2

4π∆fdt------------------–

⋅=

∆ϕ

EX t( )

EY t( ) 1 k–

kejθ P t( )⋅=

98

CW LASER MEASURED

The power splitting k and the phase difference are calculated from the parameters Azimuth and Ellipticity :

θα ε

2α( )tan 2 k 1 k–( ) θ( )cos1 2.k–

-----------------------------------------=

2ε( )sin 2 k 1 k–( ) θ( )sin=

99

CW LASER MEASURED

References:[1] Lau, K. Y. and Yariv, A., "Ultra-High Speed Semiconductor Laser", J. Quant. Elect., 21, 121-136,

(1985).

[2] Agrawal, G.P., Fiber-Optic Communication Systems, Second edition. John Wiley & Sons, Inc., N.Y., (1997).

100

DIRECTLY MODULATED LASER MEASURED

Directly Modulated Laser Measured

Directly modulated laser that allows you to specify the dynamic of the laser based on measured parameters. You can also enter parameters such as linewidth, chirp, side mode, suppression and relative intensity noise (RIN).

Ports

Parameters

Main

Measurements






FrequencyEmission frequency



Extinction ratioSteady state power ratio between marks and spaces

10 dB — [0,+INF[




Default unit

Units Value range

OvershootPercentage of overshoot during the transition from 0 to 1 relative to the steady state power

30 % — [0,+INF[

101


Side Mode

UndershootPercentage of undershoot during the transition from 0 to 1 relative to the steady state power

30 % — [0,+INF[

Rise timeDefined as the time from when the rising edges reaches 0% of the amplitude to the time it reaches 100% of the amplitude

1/(Bit rate) * 0.05 s s, ms, ns, ps [0,+INF[

Fall timeDefined as the time from when the falling edges reaches 100% of the amplitude to the time it reaches 0% of the amplitude


Damping time leading edgeRelaxation time when the signal overshoot reaches 1/e of the max value during the transition from 0 to 1


Damping time trailing edgeRelaxation time when the signal undershoot reaches 1/e of the min value during the transition from 1 to 0


Resonant frequency leading edgeFrequency of the oscillations in the transition from 0 to 1

(Bit rate) * 5 Hz Hz, MHz, GHz, THz

[0,+INF[

Resonant frequency trailing edgeFrequency of the oscillations in the transition from 1 to 0


[0,+INF[







[0,+INF[


30 dB — [0,+INF[


Default unit

Units Value range

102


RIN

Chirp

Polarization

Simulation






False — — —





Alpha parameter 0 — [-100, 100]


0 1/(W.s) ]-INF,+INF[


Units Value range


0 deg ]-90,90]


0 deg [-45,45]




Units Value range



Parameterized Parameterized — [1,+INF[

103


Noise

Random numbers

Technical backgroundThis model is a different from the Laser Measured, where you can enter measured parameters and the model calculates the rate equation parameter by using sophisticated optimization routines. Here you can enter measured parameters that describe the laser dynamics by building the laser output signal.

The range of the amplitude of the signal input is normalized between 0 and 1. This means that this model converts the input signal to a sequence of squared pulses.

The parameter Power is the steady state value of the output power at the 1 level. The steady-state value for the power at the 0 level is calculated from the parameter Extinction ratio:

where P1 is the parameter Power, Er is the parameter Extinction ratio, and P0 is the steady-state power at the 0 level.




1 THz Hz, THz, nm [1e-100, 1e-100]



[1, 1000]


Convert noise bins

— — [0, 0]


Units Value range




0 — [0,4999]

Er 10 P1 P0⁄( )log=

104


The measured parameters will be used to build P(t) (see Figure 1).

Figure 1 Measured parameters used to build P(t)

If you enable the parameter Calculate side mode, the side mode is generated according to:

where P is laser output power, s is the parameter Suppression ratio in linear scale, and is defined by the parameter Separation.

If the parameter Include RIN is enabled, the model will generate noise bins with bandwidth and spacing that you define. The parameter RIN is the ratio of the mean-square optical intensity noise to the square of the average power [1][2]:

where is the mean-square optical intensity fluctuation at a specific frequency and is the parameter Measured power.

This model estimates based on the parameters RIN and Measured power.

Eout t( ) P t( ) 1 s 2π∆ft( ) s 2π∆ft–( )ejϕcos+cos+[ ]=

∆f

RIN ∆P2⟨ ⟩

Pm2

---------------dB Hz⁄=

∆P2⟨ ⟩ Pm2

∆P2⟨ ⟩

105



where is the signal phase, is the parameter Alpha parameter or linewidth enhancement factor, and is the parameter Adiabatic chirp.

The signal phase and polarization is calculated in the same way as the CW laser, where the laser phase noise is modeled using a Gaussian random variable for the phase difference between two successive time instants with zero mean and a variance equal to , where is the laser Linewidth. The probability density function is:

where is the phase difference between two successive time instants and dt is the time discretization.


The power splitting k and the phase difference is calculated from the parameters Azimuth and Ellipticity :

References:[1] Lau, K. Y. and Yariv, A., "Ultra-High Speed Semiconductor Laser", J. Quant. Elect., 21, 121-136,

(1985).

[2] Agrawal, G.P., Fiber-Optic Communication Systems, Second edition. John Wiley & Sons, Inc., N.Y., (1997).

dϕdt------

αe2

------ ddt-----InP t( ) κP t( )+=

ϕ αeκ

2π ∆f ∆f

f ∆ϕ( ) 12π ∆fdt---------------------- e

∆ϕ2

4π∆fdt------------------–

⋅=

∆ϕ

EX t( )

EY t( ) 1 k–

kejθ P t( )⋅=

θα ε

2α( )tan 2 k 1 k–( ) θ( )cos1 2.k–

-----------------------------------------=

2ε( )sin 2 k 1 k–( ) θ( )sin=

106

WDM TRANSMITTER

WDM Transmitter

This component is a WDM transmitter.

Ports

Parameters

Main












FrequencyEmission frequency of the first laser

193.1 THz, Hz, nm [30,+INF[

Frequency spacingFrequency spacing between adjacent lasers

100 GHz, THZ, Hz, nm

]-INF,+INF[

Power 0 dBm W, mW, dBm

Extinction ratio 10 dB [0,1000]



107

WDM TRANSMITTER

PRBS

Coding

Enhanced


Bit rate Bit rate Bits/s

MBits/s

GBits/s

[0, 1e+012]

OrderlogOrder of the PRBS generator

(Sequence length)/log(2) — [2,30]

Number of leading zeros 1 — [0,+INF[

Number of trailing zeros 1 — [0,+INF[


Modulation typeDefines the signal modulation type

NRZ Off, NRZ, RZ

Duty cycleOrder of the PRBS generator

—


0.05 bit [0,1]


0.05 bit [0,1]


Transmitter type EML — EML, DML

OvershootPercentage of overshoot during the transition from 0 to 1 relative to the steady state power

30 % —

UndershootPercentage of undershoot during the transition from 0 to 1 relative to the steady state power

30 % —

108

WDM TRANSMITTER

Side Mode

RIN

Damping time leading edgeRelaxation time when the signal overshoot reaches 1/e of the max value during the transition from 0 to 1

1/(Bit rate) * 0.5 s s, ms, ns, ps

Damping time trailing edgeRelaxation time when the signal undershoot reaches 1/e of the min value during the transition from 1 to 0

1/(Bit rate) * 0.5 s s, ms, ns, ps

Resonant frequency leading edgeFrequency of the oscillations in the transition from 0 to 1


Resonant frequency trailing edgeFrequency of the oscillations in the transition from 1 to 0








[0,+INF[


30 dB — [0,+INF[






False — — —




109

WDM TRANSMITTER

Chirp

Polarization

Simulation



Alpha parameter 0 rad/W [-1000, 1000]


0 1/s [-1000, 1000]


Units Value range


0 deg ]-90,90]


0 deg [-45,45]












110

WDM TRANSMITTER

Noise

Random numbers


Units Value range


Sample rate THz Hz, THz, nm


Sample rate GHz Hz, GHz, THz, nm


Convert noise bins

— —


Units Value range




0 — [0,4999]

111

WDM TRANSMITTER

Technical BackgroundWDM systems require multiple transmitters and different parameters for each one of them. In addition, they also require different modulation schemes and formats. By using multiple components, users can customize designs, but it is time consuming. The WDM Transmitter encapsulates different components, allowing users to select different modulation formats and schemes for multiple channels in one single component. It is a transmitter array that allows for different modulation types and schemes.

The block diagram for each WDM channel transmitter is shown below.

The first stage is the PRBS; the same engine used in the Pseudo-Random Bit Sequence Generator component is used in this stage. Parameters Bit rate, Order, Number of leading and trailing zeros are used in the internal Pseudo-Random Bit Sequence Generator. A different seed will be used for each bit sequence for each WDM channel. The operation and parameters of the PRBS component is described in the technical background of the Pseudo-Random Bit Sequence Generator.

The second stage is the Coding/Modulation; the parameter Modulation type has three options: RZ, NRZ and Off. RZ and NRZ coding is generated by the engines of the RZ Pulse Generator and NRZ Pulse Generator respectively. A CW operation of the

112

WDM TRANSMITTER

transmitter is possible by selecting Off as modulation type. The Duty cycle parameter is used when modulation type RZ is selected. The operations and parameters of the electrical pulse generators are described in the technical background of the RZ and NRZ Pulse Generators.

The last stage is the optical source and modulation scheme; by using the parameter Transmitter type the user can select between a external modulated laser scheme (EML) or a directly modulated laser scheme (DML). The laser engine used in this stage is the same used in the Directly Modulated Laser Measured component. The operation and parameters of this component are described in the technical background of the Directly Modulated Laser Measured.

By using 3R regenerators, it is possible to recover the original bit sequence and electrical signals for all the WDM channels:

113

WDM TRANSMITTER

Notes:

114

PSEUDO-RANDOM BIT SEQUENCE GENERATOR

Pseudo-Random Bit Sequence Generator

Generates a Pseudo Random Binary Sequence (PRBS) according to different operation modes. The bit sequence is designed to approximate the characteristics of random data.

Ports

Parameters

Main


Bit sequence Output Binary



MBits/s

GBits/s

[0, 1e+012]

Operation mode Order — Probability, Order, Alternate, Ones, Zeros

OrderlogOrder of the PRBS generator

(Sequence length)/log(2) — [2,30]

Mark probabilityProbability of ones in the sequence

0.5 — [0,1]

Number of leading zeros (Time window * 3 / 100 ) * Bit rate — [0,+INF[

Number of trailing zeros (Time window * 3 / 100 ) * Bit rate — [0,+INF[

115

PSEUDO-RANDOM BIT SEQUENCE GENERATOR

Simulation

Random numbers

Technical backgroundThis model generates a sequence of N bits:

Tw is the global parameter Time window and Br is the parameter Bit rate.

The number of bits generated is . and are the Number of leading zeros and the Number of trailing zeros.

Operation mode controls the algorithm used to generate the bit sequence:• Probability: Random number generator is used, with parameter Mark probability

specifying the probability of ones in the sequence• Order: PRBS generator[1] with Order k is used to generate a sequence with

period of 2k-1• Alternate: Alternate sequence of ones and zeros is generated• Ones: A sequence of ones is generated• Zeros: A sequence of zeros is generated

References[1] Press, W. H., Flannery, B. P., Teukolsky, S. A., and Vetterling, W. T., Numerical Recipes in C.

Cambridge University Press, (1991).


Units Value range






Units Value range



Random seed indexUser-defined seed index for bit generation

0 — [0,4999]

where N TwBr=

NG N nl– nt–=

NG nl nt

116

USER-DEFINED BIT SEQUENCE GENERATOR

User-Defined Bit Sequence Generator

Generates a bit sequence that is user-defined.

Ports

Parameters

Main





MBits/s

GBits/s

[0,+INF[

Load from fileDetermines whether or not the component will load the bit sequence from the file


FilenameFile with the bit sequence

Sequence.dat — Filename

Bit sequenceUser-defined bit sequence

0101101110 — String

Number of leading zeros (Time window * 3 / 100 ) * Bit rate — [0, 1000]

Number of trailing zeros (Time window * 3 / 100 ) * Bit rate — [0, 1000]

117

USER-DEFINED BIT SEQUENCE GENERATOR

Simulation

Technical backgroundYou can enter the string Bit sequence or choose Load from file. In this, case the parameter Filename is enabled.

All bit files are formatted containing one bit per line, e.g. the bit file representing the sequence "01011..." has the following form:

The sequence length is defined by:

N = TwBr

Tw is the global parameter Time window and Br is the parameter Bit rate. If the user-defined sequence is shorter than the N, the sequence will be repeated until the length is equal to N.


Units Value range




1 — [1, 1e+009]

0

1

0

1

1

118

MACH-ZEHNDER MODULATOR

Mach-Zehnder Modulator

Simulates a Mach-Zehnder modulator using an analytical model.

Ports

Parameters

Main

Simulation

Technical backgroundThe Mach-Zehnder modulator is an intensity modulator based on an interferometic principle. It consists of two 3 dB couplers which are connected by two waveguides of equal length (see Figure 1). By means of an electro-optic effect, an externally applied voltage can be used to vary the refractive indices in the waveguide branches.



Carrier Input Optical



Extinction ratio 30 dB [0,+INF[

Negative signal chirp False — True, False

Symmetry factor –1 — [-1,1[




119

MACH-ZEHNDER MODULATOR

The different paths can lead to constructive and destructive interference at the output, depending on the applied voltage. Then the output intensity can be modulated according to the voltage.

Figure 1 Mach-Zehnder modulator

The equations that describe the behavior of the MZ modulator are:

where is the phase difference between the two branches and is defined as:

with

and

is the signal phase change defined as:

where the parameter SC is –1 if negative signal chirp is true, or 1 if negative signal chirp is false. extract is the extinction ratio, SF is the symmetry factor, and modulation(t) is the electrical input signal. The electrical input signal is normalized between 0 and 1.

For parameterized and noise bins signals, the average power is calculated according to the above.

Eout t( ) Ein t( ) ∆θ t( )( )cos⋅ j ∆φ t( )⋅( )exp⋅=

∆θ

∆θ t( ) π2--- 0.5 ER Modulation t( ) 0.5–( )⋅–( )⋅=

ER 1 4π--- arc 1

extrat-------------------

tan⋅–=

∆φ

∆φ t( ) SC ∆θ t( ) 1 SF+( ) 1 SF–( )⁄⋅⋅=

120

ELECTROABSORPTION MODULATOR

Electroabsorption Modulator

Simulates an Electro-absorption modulator using an analytical model.

Ports

Parameters

Main

Simulation






Modulation index 0.95 — [0 ,1[

Chirp factor 0 — ]-INF, +INF[




121

ELECTROABSORPTION MODULATOR

Technical backgroundIn this model, the optical carrier is modulated externally by the electrical modulation signal, (see Figure 1).

Figure 1 EA modulator

Assuming that the optical input signal is Ein, the following equation describes the behavior of the model:

where Eout(t) is the output optical signal, is the chirp factor, and Mod(t) is defined as

where MI is the modulation index and modulation(t) is the electrical input signal. The electrical input signal is normalized between 0 and 1.


Eout t( ) Ein t( ) Mod t( ) jα2--- Mod t( )( )ln⋅

exp⋅ ⋅=

α

Mod t( ) 1 MI–( ) MI+ modulation t( )⋅=

122

AMPLITUDE MODULATOR

Amplitude Modulator

Simulates an ideal amplitude modulator.

Ports

Parameters

Main

Simulation






Modulation index 1 — [0,1]



Yes — True, False

123

AMPLITUDE MODULATOR

Technical backgroundIn this model, the optical carrier is modulated externally by the electrical modulation signal. Assuming that the optical input signal is Ein, the following equations describe the behavior of the model:

where Eout(t) is the output optical signal and Mod(t) is defined as

where MI is the modulation index and modulation(t) is the electrical input signal. The electrical input signal is normalized between 0 and 1.


Eout t( ) Ein t( ) Mod t( )⋅=

Mod t( ) 1 MI–( ) MI modulation t( )⋅+=

124

PHASE MODULATOR

Phase Modulator

Simulates an ideal phase modulator.

Ports

Parameters

Main

Simulation






Phase deviation 90 deg ]-INF,+INF[




125

PHASE MODULATOR

Technical backgroundIn this model, the electrical modulation signal imposes a phase modulation on an optical carrier. Assuming that the optical input signal is Ein, the following equation describes the behavior of the model.

where Eout(t) is the output optical signal, is the phase deviation, and modulation(t) is the electrical input signal. The electrical input signal is normalized between 0 and 1.

The parameterized and noise bins signals are not affected by this modulator.

Eout t( ) Ein t( ) j ∆φ modulation t( )⋅ ⋅( )exp⋅=

∆φ

126

FREQUENCY MODULATOR

Frequency Modulator

Simulates an ideal frequency modulator.

Ports

Parameters

Main

Simulation







Frequency deviation 10 GHz Hz, GHz, THz [0,+INF[


Units Value range



127

FREQUENCY MODULATOR

Technical backgroundIn this model, the electrical modulation signal imposes a frequency modulation on an optical carrier. Assuming that the optical input signal is Ein, the following equation describes the behavior of the model:

where Eout(t) is the output optical signal, is the frequency deviation, and modulation is the electrical input signal. The electrical input signal is normalized between 0 and 1.

The parameterized and noise bins signals are not affected by this modulator.

Eout t( ) Ein t( ) j 2π ∆f modulation τ( ) 0.5–( )⋅ τd⋅0

t

∫⋅

exp⋅=

∆f τ( )

128

DUAL DRIVE MACH-ZEHNDER MODULATOR MEASURED

Dual Drive Mach-Zehnder Modulator Measured

Simulates a Mach-Zehnder modulator with dual-drive modulation using measured parameters.

Ports

Parameters

Main

Simulation






Splitting Ratio 1.3 — [0,10000]

Modulator Type Phase-Shift — Conventional, Phase-Shift

Bias Voltage 1 –2.8 V ]-INF, +INF[


Normalize electrical signal True — True, False

Modulation Voltage12 1.2 V [0 , +INF[

Absorption / Phase FilenameFile with the measured absorption and phase

AbsorptionPhase.dat

— —




129


Graphs

Technical backgroundIn this model, you can specify the dependence of the measured absorption and phase on applied voltage for a Mach-Zehnder modulator. You can use the default characteristics curves or choose to load from Filename.

For a modulator with the same input and output Y-branch splitting ratios, the output signal is:

where SR = P1/P2 is the Y-branch power splitting ratio

is defined as:

for the normalized case

where is the bias voltage, is the peak-to-peak voltage, and is the normalized modulation waveform with a peak-to-peak amplitude of 1 and an average value of 0. The electrical input signal can be normalized between 0.5 and -0.5.

for the non-normalized case.

The model utilizes a Dual drive (push and pull) modulation ( .


Measured absorption Voltage (V) Absorption (dB)

Measured phase Voltage (V) Phase (radians)

is the attenuation constant

is the phase constant

L is the interaction length of the modulator arm

is 0 radians for a conventional modulator and π radians for phase-shift modulator

V1 and V2 are voltages applied to arms 1 and 2, respectively

I is the intensity of the optical signal

is the phase

E V1 V2,( )E0

1 SR+---------------- SR

∆αa V1( )2

--------------------- j ∆β V1( )⋅+ L–

∆αa V2( )2

--------------------- j ∆β V2( )⋅+ L– j φ0⋅–

exp+exp⋅=

E V1 V2,( ) I V1 V2,( ) j Φ V1 V2,( )⋅( )exp⋅≡

∆αa 2⁄

∆β

φ0

Φ

Vi i 1 2,=( )

Vi t( ) Vbi Vmod12 v t( )⋅+=

Vbi Vmod12 v t( )

Vi t( ) Vbi Vmod± t( )=

∆V1 ∆V2–=

130


The model has stored default curves characteristics of a Mach-Zehnder modulator. The dependence of the measured absorption and phase of the optical signal on applied voltage for each arm of a modulator is illustrated in Figure 1.

Figure 1 Default characteristics of absorption and phase in the Dual Mach-Zehnder model

131


Reference:[1] Cartledge, J. C., “Combining self-phase modulation and optimum modulation conditions to

improve performance of 10 Gb/s transmission systems using MQW Mach-Zehnder modulators”, J. Light. Techn., 18, 647-654, (2000).

132

ELECTROABSORPTION MODULATOR MEASURED

Electroabsorption Modulator Measured

Simulates an Electro-absorption modulator using measured parameters.

Ports

Parameters

Main

Simulation






Bias voltage –1 V ]-INF, +INF[


Modulation voltage (peak-to-peak) 2 V [0 , +INF[

Absorption / Alpha FilenameFile with the measured absorption and α-parameter αm

AbsorptionAlpha.dat — —




133


Graphs

Technical backgroundIn this model, you can specify the dependence of the measured absorption and -parameter- on the applied voltage for an EA modulator. You can use the default characteristic curves or choose to load from file. In this case, the parameter Filename is enabled.

In the case of the EA modulator, the output signal response to an applied voltage is:

where IV is the voltage-dependent intensity of the signal.

While Equation 1 is an accurate result, it is not in the most convenient form for simulation purposes when empirical equations for and are obtained from a fitting to measured results. The determination of the argument of the exponential function in Equation 1 requires function evaluation and integration.

The modulator output signal given by Equation 1 can also be written in the convenient form using a voltage-dependent parameter as:

A comparison of the phase terms in Equation and Equation 2 yields

Equation 3 shows how the attenuation constant and α-parameter- jointly combine to determine . Using Equation 2, with determined from measurements of and , the evaluation of the argument of the exponent only requires function evaluation.



Measured alpha-parameter Voltage (V) Alpha-parameter

Calculated alpha-parameter Voltage (V) Alpha-parameter

(1)

(2)

(3)

ααm

E V( ) I V( ) j12--- αm V( ) I V( )( )lnd∫

exp=

αm V( ) I V( )

I 1 jα+( ) 2⁄ αr V( )

E V( ) I V( )1 jαr V( )+( ) 2⁄

=

αr V( ) 1γ V( )----------- αm V( ) γ V( )d⋅∫=

γ V( ) αm V( )αr V( ) αr V( )

αm V( ) I V( )

134


The default characteristics curves stored in the component, the dependence of the measured absorption, and α-parameter- on applied voltage, is illustrated in Figure 1.

Figure 1 Dependence of the absorption and on the applied voltage for an MQW-EAM

For this component, the electrical input signal can be normalized between 0.5 and -0.5. Then, the voltage applied to the modulator is given by:

where Vb is the bias voltage, Vmod is the peak-to-peak voltage, and v(t) is the normlized modulation waveform (electrical input signal ) with a peak-to-peak amplitude of 1 and an average value of 0.

(4)

αm V( )

αm

V t( ) Vb Vmod v t( )⋅+=

135


Notes:

136

SINGLE DRIVE MACH-ZEHNDER MODULATOR MEASURED

Single Drive Mach-Zehnder Modulator Measured

Simulates a Mach-Zehnder modulator with single drive modulation using measured parameters.

Ports

Parameters

Main

Simulation







Modulator Type Phase-Shift — Conventional, Phase-Shift




Modulation Voltage 1.5 V [0 , +INF[

Operation mode Change in V2 = 0 — Change in V1 = 0, Change in V2 = 0


AbsorptionPhase.dat — —




137


Graphs



where SR = P1/P2 is the Y-branch power splitting ratio

Vi(i=1,2) is defined as:


where Vbi is the bias voltage, Vmodi is the peak-to-peak voltage, v(t) is the normalized modulation waveform with a peak-to-peak amplitude of 1 and an average value of 0. The electrical input signal is normalized between 0.5 and -0.5.

for the non-normalized case

The model utilizes a single drive modulation, i.e., is 0 in one of the arms.











is the phase

E V1 V2,( )E0

1 SR+---------------- SR

∆αa V1( )2

--------------------- j ∆β V1( )⋅+ L–

∆αa V2( )2

--------------------- j ∆β V2( )⋅+ L– j φ0⋅–

exp+exp⋅=

E V1 V2,( ) I V1 V2,( ) j Φ V1 V2,( )⋅( )exp⋅≡

∆αa 2⁄

∆β

φ0

Φ

Vi t( ) Vbi Vmodi v t( )⋅+=

Vi t( ) Vbi Vmod± t( )=

Vmod

138


Figure 1 Default characteristics of absorption and phase in the Single Mach-Zehnder mode

139



improve performance of 10 Gb/s transmission systens using MQW Mach-Zehnder modulators”, J. Light. Techn., 18, 647-654, (2000).

140

DUAL PORT DUAL DRIVE MACH-ZEHNDER MODULATOR MEASURED

Dual Port Dual Drive Mach-Zehnder Modulator Measured

Simulates a Mach-Zehnder modulator with dual-drive modulation using two ports with measured parameters.

Ports

ParametersMain

Simulation


Modulation 1 Input Electrical

Modulation 1 Input Electrical





Modulator Type Phase-Shift — Conventional, Phase-Shift,




Modulation Voltage12 1.2 V [0 , +INF[


AbsorptionPhase.dat — —




141


Graphs



where is the Y-branch power splitting ratio

is defined as:


where is the bias voltage, is the peak-to-peak voltage, and is the normalized modulation waveform with a peak-to-peak amplitude of 1 and an average value of 0. The electrical input signal is normalized between 0.5 and -0.5.

for the non-normalized case.

The model utilizes a Dual drive (push and pull) modulation ( .










is the phase

E V1 V2,( )E0

1 SR+---------------- SR

∆αa V1( )2

--------------------- j ∆β V1( )⋅+ L–

∆αa V2( )2

--------------------- j ∆β V2( )⋅+ L– j φ0⋅–

exp+exp⋅=

E V1 V2,( ) I V1 V2,( ) j Φ V1 V2,( )⋅( )exp⋅≡

SR P1 P2⁄=

∆αa 2⁄

∆β

φ0

Φ

Vi i 1 2,=( )

Vi t( ) Vbi Vmodi± v t( )⋅=

Vbi Vmodi v t( )

Vi t( ) Vbi Vmodi± t( )=

∆V1 ∆V2–=

142



Figure 1 Default characteristics of absorption and phase in the Dual Mach-Zehnder model

143



improve performance of 10 Gb/s transmission systems using MQW Mach-Zehnder modulators”, J. Light. Techn., 18, 647-654, (2000).

144

LINBO3 MACH-ZEHNDER MODULATOR

LiNbO3 Mach-Zehnder Modulator

This component simulates a Lithium Niobate Mach-Zehnder modulator based on basic parameters.

Ports

Parameters

Main


Input 1 Input Optical

Input 2 Input Electrical




Extinction ratio 20 dB [0,+INF[

Switching bias voltage

DC voltage required to turn the modulator from the OFF state to the ON state, or vice versa

4 V [0,+INF[

Switching RF voltage

RF voltage required to turn the modulator from the OFF state to the ON state, or vice versa

4 V ]-INF,+INF[

Bias voltage1 0 V ]-INF,+INF[

Bias voltage2 4 V ]-INF,+INF[

Insertion loss 5 dB [0,+INF[


Modulation voltage1 0 V ]-INF,+INF[

Modulation voltage2 4 V ]-INF,+INF[

145


Bandwidth Response

Simulation

Technical BackgroundThe Mach-Zehnder structure consists of an input optical branch, which splits the incoming light into two arms, followed by two independent optical arms, which are subsequently recombined by the output optical branch. Application of an electrical signal to one of the optical arms controls the degree of interference at the output optical branch and therefore controls the output intensity.

The optical field at the output of the modulator is given by:

where is the input signal

and are the RF modulating electrical voltage

and are the DC bias voltage applied to arm one and two, respectively


Load transfer functionDetermines whether you want to load a modulator transfer function or use an ideal one.


File frequency unitDetermines the frequency unit of the file.

Hz — Hz, THz

File formatDetermines the format of the file.

Power — Power; Phase; Power Phase; Real, Imag.

Linear scaleDetermines whether or not the data is in linear scale.


HF filenameFile with the transfer function (S21)

Filter.dat — —




EO t( )Ein t( )

10 insertionloss 20⁄( )--------------------------------------------- γ e

j π v2 t( ) VπRF⁄ j π vbias2⋅ ⋅+ VπDC⁄⋅ ⋅( ) ⋅ 1 γ–( ) e

j π v1 t( ) VπRF⁄ j π vbias1⋅ ⋅+ VπDC⁄⋅ ⋅( )⋅+( )⋅=

Ein t( )

v1 t( ) v2 t( )

vbias1 vbias2

146


denotes the power splinting (combining) ration of arm two for the input (output, respectively) Y-branch waveguide, and is given by:

where .

and , the DC bias voltages, are included separately as parameters due to the possibility of the (Switching Bias Voltage) to be different from the Switching RF Voltage.

If the Switching Bias Voltage is equal to the Switching RF Voltage, and the Normalize Electrical Signal parameter is False, the bias voltage can be included in the electrical signal.

The optical power and phase of the modulator output are determined in response to the modulating voltage waveforms. The modulator transfer function relates the effective drive voltage to the applied drive voltage. This component can also load the modulator transfer function data from file or consider an ideal transfer function.

The file is formatted containing two items per line, the frequency and filter measurement. The parameter File frequency unit determines the frequency unit of the first item; it can be Hz or THz.

According to the parameter File format, the second item can be one value (Power or Phase) or two values (Power and Phase or Real and Imag):

Power (Phase is set to zero, assuming frequency units THz)

Power Phase

193.10 0

193.11 0.5

193.12 0.5

193.13 0

193.14 0 0

193.15 0.5 3.14

193.16 0.5 3.14

193.17 0 0

γ

γ 1 1εr

--------–

2⁄=

εr 10ExtRatio 10⁄=

vbias1 vbias2VπDC

147


Real Imag

Phase (Power is set to one)

When the Normalize electrical signal parameter is True, the electrical signals of port1 and port2 are normalized between -0.5 and 0.5. In this case, the amplitude of each RF electrical signal considered in and will be the values in the modulation voltage parameters divided by 2.

References[1] Cartledge, J. C., Rolland, C., Lemerle, S., and Solheim, A., “Theoretical performance of 10 Gb/s

lightwave systems using a III-V semiconductor Mach-Zehnder modulator.”, IEEE Phot. Techn. Letters., 6, 282-284, (1994).

[2] Cartledge, J.C., "Performance of 10 Gb/s lightwave systems based on lithium niobate Mach-Zehnder modulators with asymmetric Y-branch waveguides". IEEE Phot. Techn. Letters., 7, 1090 -1092, (1995).

193.18 0 0

193.19 -0.5 7.9-e-4

193.20 -0.5 7.9-e-4

193.21 0 0

193.22 0

193.23 3.14

193.24 3.14

193.253 0

v1 t( ) v2 t( )

148

Optical Fibers LibraryThis section contains information on the following optical fibers.

• Optical fiber data• Optical fiber• Linear Multimode fiber• Nonlinear Dispersive fiber (obsolete)

149

Notes:

150

OPTICAL FIBER DATA

Optical fiber data

SMF-28The SMF-28 model used in OptiSystem has the following characteristics:

Figure 1 Attenuation

Figure 2 Group Velocity Dispersion

151

OPTICAL FIBER DATA

Figure 3 Effective Area

Figure 4 Group Delay

Attenuation curve shows a minimum of for a wavelength of .

GVD curve reveals a dispersion of at with a dispersion slope of .

Effective area at is .

Group delay is .

This model can be varied in any way because you have the ability to change any particular parameter. Create a new file and then load it into the appropriate section, or just set the parameter to 'Constant' and enter a value. The Nonlinear Fiber model is very flexible, because it has the ability to model practically every manufactured fiber that exists on the market today.

0.185 dBm 1550 nm

16.5 ps/nm/km 1550 nm0.05 ps/nm2 km⁄

1550 nm 76.5 µm2

4897650 ps/km

152

OPTICAL FIBER DATA

+D NZDSF modelThe +D NZDSF model used in OptiSystem has the following characteristics:



153

OPTICAL FIBER DATA





The effective area at is .

Group delay is .

0.185 dBm 1550 nm

4.5 ps/nm/km 1550 nm0.01 ps/nm2 km⁄

1550 nm 71.5 µm2

4895870 ps/km

154

OPTICAL FIBER DATA

-D NZDSF modelThe -D NZDSF model used in OptiSystem has the following characteristics:

Figure 9 Attentuation


155

OPTICAL FIBER DATA






Group delay is .

0.185 dBm 1550 nm

7.5 ps/nm/km– 1550 nm0.18 ps/nm2 km⁄

1550 nm 92 µm2

4890750 ps/km

156

OPTICAL FIBER DATA

CDF (Standard)The DCF model used in OptiSystem has the following characteristics:



157

OPTICAL FIBER DATA






Group delay is .

0.3 dBm 1600 nm

82 ps/nm/km– 1550 nm4.5 ps/nm2 km⁄

1550 nm 32 µm2

4914000 ps/km

158

OPTICAL FIBER

Optical fiber

The optical fiber component simulates the propagation of an optical field in a single-mode fiber with the dispersive and nonlinear effects taken into account by a direct numerical integration of the modified nonlinear Scrödinger (NLS) equation (when the scalar case is considered) and a system of two, coupled NLS equations when the polarization state of the signal is arbitrary. The optical sampled signals reside in a single frequency band, hence the name total field [1]. The parameterized signals and noise bins are only attenuated.

Ports

Parameters

Main


Input Input Optical


Name and description Symbol Default value Default unit Value range

User defined reference wavelength

If TRUE, frequency value of “Reference wavelength” is used internally as ‘zero’ (or reference) frequency in spectrum of signal envelope. Values of parameters (attenuation, dispersion) are assumed to correspond to this frequency. If parameters are wavelength-dependent (from files), they are evaluated at this frequency. If FALSE, central frequency of simulated band is used.

TRUE — TRUE/FALSE

Reference wavelength

Value of user defined/specified reference wavelength.

1550 nm [100, 2000]

Length

Fiber length

50 km [0, 100,000]

λ0

L

159

OPTICAL FIBER

Dispersion

Attenuation effect

If TRUE, attenuation effect is enabled.

— TRUE — TRUE/FALSE

Attenuation data type

Defines the attenuation as a fixed constant value or as a wavelength dependent curve taken from a file. If ‘constant’, value from “Attenuation” tab in component is used.

— Constant — Constant/ From File

Attenuation

Specified value is used if “Attenuation data type” is set to ‘constant’. If ‘from file’, the value is ignored.

0.2 dB/km [0, 1010]

Attenuation vs. wavelength

Defines the attenuation as a wavelength dependent curve in a file.

— — — —


Group velocity dispersion

If TRUE, the GVD effect is enabled.


Third order dispersion

If TRUE, the TOD effect is enabled.


Frequency domain parameters

Defines domain in which dispersion parameters are specified. If TRUE, frequency domain is used and dispersion effect is specified in terms of and

. Otherwise, wavelength domain is used ( and ).

Dispersion data type

Defines if dispersion parameter valuesare read from component tabs, or taken from a file

Constant — Constant/ From File

Beta 2

Value of the GVD parameter in the frequency domain

-20 ps2/km [-10100, 10100]

Beta 3

Value of the GVD parameter in the frequency domain

-20 ps3/km [-10100, 10100]


α

β2β3 D

S

β2

β3

160

OPTICAL FIBER

The parameter “Frequency domain parameters” refers to the alternative definitions:

and

of the dispersion parameters, but not to the argument of these functions, which is always assumed to be the wavelength. All the parameters in the component (including and ) are given as functions of wavelength (not frequency). This is also the case when or are specified from a file - the first column of the file contains wavelength values ( ) and the second column - the corresponding values of .

Dispersion

Value of the GVD parameter in the wavelength domain

16.75 — [-10100, 10100]

Dispersion slope

Value of dispersion slope parameter.

— 0.075 [-10100, 10100]

Dispersion file format

Determines contents of dispersion file: group delay or dispersion vs. wavelength. If “Dispersion vs. wavelength” and “Frequency domain parameters” are selected, it is assumed that file contains

. If “Frequency domain parameters” is disabled, component assumes that file contains

. If “Group delay vs wavelength”, the file contains .

— Dispersion vs wavelengtht

— Dispersion vs wavelength/ Group delay vs wavelength

Dispersion file name

Specifies file containing dispersion data

— — — —


Dps

nm( ) km( )-------------------------

ps

nm( )2 km( )---------------------------

β2 λ( )

D λ( )β1 λ( )

D ∂β1

∂λ-------- S, ∂D

∂λ------- (wavelength domain definition)= =

β2∂β1

∂ω-------- β3

∂β2

∂ω--------= (frequency domain definition),=

β2 β3

β1 β2

λβ1 λ( ) or β2 λ( )

161

OPTICAL FIBER

PMD

Nonlinearities


Birefringence type

Defines the birefringence. If “Deterministic”, both the strength of birefringence and principal axes are assumed constant, hence random mode coupling is disabled. If “Stochastic”, random mode coupling is enabled.

— Deterministic — Deterministic/ Stochastic

Differential group delay

If Birefringence type is “Deterministic”, this is the value of the differential group delay. If “Stochastic”, parameter is disabled.

0.2 [-10100, 10100]

PMD coefficient

Polarization mode dispersion coefficient. If Birefringence type is “Stochastic”, this is the value of the PMD parameter. If “Deterministic”, parameter is disabled.

0.5 [0,10100]

Mean scattering section length

Averaged value of fiber length at which the polarization state of the signal is randomized by applying the scattering matrix.

500 [0,10100]

Scattering section dispersion

Dispersion of the scattering section length.

100 [0,10100]


Self-phase modulation

Determines if the self-phase modulation (SPM) effect will be taken into account. If FALSE all the nonlinear effects - self-steepening, SRS - are disabled. In the vector case enabling this effect enables also the cross-phase modulation between the orthogonal polarization components.


Effective area data type

Defines is effective area parameter value is read from the component tab or from a file. If “Constant”, the value from the component is used.

Constant — Constant/ From File

Effective area

Defines the value of the effective area parameter. This value is used if “Effective area data type” is set to “Constant”. Otherwise, the value is ignored.

80 [0,1010]

ddω------- ∆β( )

pskm-------

Dppskm

-----------

Lscattm

σscatt m

Aeff µm2

162

OPTICAL FIBER

Effective area vs. wavelength

If “Effective area data type” is “From file”, this tab specifies the file containing the effective area data.

— — — —

n2 data typeDetermines if parameter (nonlinear index of refraction) value is read from the component tab or from a file. If “Constant”, value is taken from component.

— Constant — Constant/ From File

n2

The value of the parameter (nonlinear index of refraction). If data type is set to “Constant”, this value is used, otherwise the value is ignored.

2.6 X 10-20 [0,10100]

Self-steepening

Specifies whether self-steepening effect is taken into account. Can be enabled only after enabling the SPM, and is taken into account only in the scalar case (if Model type is set to Scalar), and if Full Raman response parameter is FALSE.

— FALSE — FALSE/TRUE

Full Raman response

Defines the stimulated Raman scattering (SRS) effect representation in the model. If TRUE, SRS is represented through the convolution integrals of the fields with the Raman susceptibilities [6, 18-21]. Intrapulse Raman scatterins is disabled.


Intrapulse Raman scattering

Defines the stimulated Raman scattering (SRS) for [19-21]. Can be enabled if Full Raman response is FALSE. If both Full Raman response and Intrapulse Raman scattering are FALSE, SRS effect is not taken into account in the simulation.


Raman self-shift time 1

Value of the Raman self-shift time parameter associated with the parallel SRS effect

t

Units are such that [19-21].

14.2 [0,10100]

Raman self-shift time 2

Units are such that [18, 20, 21].

3 [0,10100]


n2

n2

n2 m2

W------

τR1 dImχ1111 ω( ) dω⁄( )ω 0==

Re χ1111 ω 0=( )( ) 1=

τR1 fs

τR2 dImχ1122 ω( ) dω⁄( )ω 0==

Re χ1111 ω 0=( )( ) 1=

τR2 fs

163

OPTICAL FIBER

Numerical

Fractional Raman contribution

Fraction of the nonlinear polarization, related to the stimulated Raman scattering effect [2].

0.18 — [0, 1]

Orthogonal Raman factor

Units are such that .

0.75 — [0, 1]


Model type

Defines model type used for simulation. Depends on polarization state of signal. If “Vector” selected, signal can have arbitrary polarization state and a system of two coupled equations (17) is solved. If “Scalar” selected, the signal preserves its polarization state and a single equation is solved (1). In the following cases, vector simulation is performed regardless of value of model type parameter:• Two polarization components are detected at

fiber input• PMD effect is “Stochastic”.

— Scalar — Scalar/Vector

Propagator type

Method used to apply nonlinear propagator in the split-step Fourier method. “Exponential” corresponds to standard implementation [2], “Runge-Kutta 4th (2nd) order” uses Runge-Kutta 4th (2nd) order (see [3]) to apply nonlinearity operator. Exponential cannot be used when Model type is set to Vector, and SRS effect is enabled. The default selection is Runge-Kutta 2nd order.

Exponential — Exponential Runge-Kutta 4th order

Exponential Runge-Kutta 2nd order

Calculation type

Specifies implementation of split-step Fourier method [2, 4] when Propagator type is “Exponential”.

— Iterative — Iterative/ Noniterative

Number of iterations

Switch On/Off the dispersion slope (the third-order dispersion)

2 — [2, 1010]


ρ

αf Re χ1122 ω 0=( )( )=

Re χ1111 ω 0=( )( ) 1=

αf

164

OPTICAL FIBER

Step Size

Specifies whether variable or fixed step-size simulation is used. If “Variable”, step size is adaptively changed depending on value of “Max. nonlinear phase shift” parameter, and solution itself. If “Constant”, step size is evaluated once at the beginning of simulation. In some cases, the fixed step size calculation executes faster, due to the smaller number of calculations per step, but the variable step size calculation is more flexible and can be faster if the peak power of the waveform varies considerably in (for example, in the presence of strong attenuation).

— Variable — Variable/ Constant

Max. Nonlinear phase shift

Maximum (over the time window) phase shift induced by the self-phase modulation effect per step.

3.14 [0,10100]

Boundary conditions

Specifies type of boundary conditions used in simulation.

— Periodic — Periodic/ Absorbing

Filter steepness

If “Boundary conditions” option is set to “Absorbing”, the “Filter steepness” parameter determines the absorption/reflection properties of the time window boundaries.

— 0.5 — [0,10100]

Lower/Upper calculation limit

Set the spectral range in which the simulation is performed. Any spectral components outside the range is ignored.

— [1400, 1700] [100, 2000]


z

ϕmaxNL mrad

nm

165

OPTICAL FIBER

Graphs

Note: The rest of the parameters in the Graphs tab of the component determine which graphs are plotted after the simulation is complete.

Simulation


Calculate graph

Enables/disables 3D graphs. If disabled, no graphs are plotted and no data are stored.


Number of distance steps

Number of snapshots used to construct a 3D plot. If this value is increased, the fidelity of the plot is improved only if the value is below the number of actual steps in . The number of snapshots stored cannot be bigger than the number of steps in taken by the simulation to obtain the solution. The latter is determined by the maximum nonlinear phase-shift parameter (numerical tab).

— 200 — [1, 100000000]

Number of wavelength/time steps

Number of stored points per snapshot. If this value is increased, the fidelity of the plot is improved only if the value is below the actual number of points in the time (frequency) domain used by the simulation to obtain the solution. The latter is related to the number of samples, which is a global parameter.

— 200 — [1, 100000000]

Linear scale

Determines axis type (linear or logarithmetic) for the dependent variable. If TRUE, the axis type is linear.



Enabled

Determines whether or not the component is enabled. If FALSE, all input signals reach the output port of the component without any changes.


zz

166

OPTICAL FIBER

Noise

Random numbers


Convert noise bins

If TRUE, each noise bin within the bandwidth of the signal is converted to a Gaussian white noise, with the correct power spectral density, and the noise is added to the signal.



Generate random seed

Determines how random number generator is initialized (seeded). If TRUE, the seed index used for the initialization is the random number itself. Otherwise, a user specified number is used.


Random seed index

If “Generate random seed” is FALSE, this value specifies the seed index. The generated pseudo-random sequence is the same if the seed index is not changed. The value of the “Random seed index” is ignored if “Generate random seed” is TRUE.

— 0 — [0, 4999]

167

OPTICAL FIBER

Technical Background

Scalar approach

Basic equation

When the optical field is assumed to maintain its polarization along the fiber length, the evolution of a slowly varying electric field envelope can be described by a single nonlinear Schrödinger (NLS) [2] equation (the scalar approach, Model type parameter from the "Numerical" tab is set to "Scalar") of the form:

In Equation 2, is the electric field envelope. A frame moving at the group velocity ( ) is assumed.

The derivatives of the propagation constant of the fiber mode , ( is the mode effective index), with respect to frequency

.

and are the first and the second group velocity dispersion (GVD) parameters, respectively, and is the reference frequency of the signal, related to the parameter "Reference wavelength" ("Main" category of the components tool-box) through with being the light speed in vacuum.

The physical meaning of the terms in Equation 2 is the following. The first term takes into account the slow changes of the electric field along the fiber length. The second term is the (first-order) group velocity dispersion. This is the effect responsible for the pulse broadening. (See "Group velocity dispersion" from the Tutorials). The third term is the second-order GVD, known also as third-order dispersion (TOD). This effect becomes important for a signal with a broad spectrum (e.g. femtosecond pulses or WDM systems with many channels). The pulse shape becomes asymmetric due to the effect of TOD. (See "Third order dispersion" from the Tutorials). The parameters

and are denoted as "frequency domain parameters" in the interface of the component (see the "Dispersion" category in the Parameters table). The following

(1)∂E∂z------ αE iβ2 ω0( )∂2E

∂T 2-------- β3 ω0( )

6----------------∂3E

∂T 3--------–+ + iγ E 2E i

ω0------ ∂

∂T------ E2 E( ) ρτR1E∂ E 2

∂T-----------–+

=

E E z T,( )=T t z vg⁄ t β1 z–≡–=

β ω( ) β ω( )c( ) ω⁄

βn∂nβ ω 0( )

∂ωn------------------- n, 1 2 3, ,= =

β2( ) β3( )ω0

ω02πcλ0

---------= c

β2( ) β3( )

168

OPTICAL FIBER

relations are used internally to convert between them and the commonly used wavelength domain parameters (dispersion) and (dispersion slope).

The parameter is given by:

In Equation 3, is the nonlinear refractive index coefficient and is the fiber effective area. The first term in the right-hand side in Equation 1 accounts for the self-phase modulation effect. It is responsible for the broadening of the pulse spectra and, in the presence of anomalous GVD, for the formation of optical solitons (See "Self-phase modulation" and "Self-phase modulation and group velocity dispersion" from the Tutorials). The second term in the right-hand side of Equation 1 takes into account the self-steepening effect. It leads to an asymmetry in the SPM-broadened spectra of ultrashort (femtosecond) pulses [2] and is responsible for the formation of optical shocks (see "Self-steepening" in the Tutorials). This effect will be taken into account only if the "Full Raman response" parameter is set to False. The last term in Equation 1 accounts for the intra-pulse Raman scattering effect with the parameter

being the parallel Raman self-shift time. The intra-pulse Raman scattering is an approximation to the actual Raman response of the material which is valid provided that signal spectrum is narrow compared to the Raman-gain spectrum. The parameter is related to the slope of the imaginary part of the Raman susceptibility

at zero frequency offset [2]. The parameter is the fractional contribution of the delayed response of the material to the total nonlinearity [2]. The intra-pulse Raman scattering effect is responsible for the self-frequency shift i.e. energy transfer from higher to lower spectral components. It leads to a decay of higher order solitons into its constituents (see "Intrapulse Raman scattering" in the Tutorials). The intrapulse Raman scattering plays the most important role among the higher order nonlinear effects [2].

In a WDM system, the stimulated Raman scattering is responsible for an energy transfer from higher to lower frequency channels (crosstalk). The Raman induced crosstalk can be neglected when the following relation is satisfied [5]:

where is the total effective length, is the fiber loss, is the amplifier spacing, the link length, is the total optical power, and is the total optical bandwidth.

(2)

(3)

(4)

D S

D dβ1

dλ-------- 2πc

λ2---------– β2= =

β3λ

2πc---------

2

λ2S 2λD+( ) S, dDdλ-------= =

γ

γ ω0n2

cAeff-----------=

n2 Aeff

τR1

τR

Im χ1111 ω( )( ) ρ

PTOTBTOTLE 9mWTHzMm ,<

LE z Lampα( )⁄≈ α Lamp

z PTOT BTOT

169

OPTICAL FIBER

Full Raman response

By selecting the option "Full Raman response" from the Numerical tab, the component can simulate the SRS effect even if the signal spectrum is much narrower than the Raman gain spectrum. In this case Equation 1 is replaced by:

Contained within Equation (4a) is which is the (time-domain) Raman response function [2], [20]. It is the Fourier-transform of the of the Raman susceptibility . In this case the self-steeping effect is neglected.

Numerical solution

In dimensionless form, Equation 1 reduces to:

where the coefficients are given by:

The new quantities are introduced according to::

In Equation 7, is the time window size and is the maximum (over the time window) of the electric field intensity .

The symmetrized split-step Fourier method [2, 4] is used to solve Equation 5. The solution is advanced from to ( is the step-size, related to the value of the Max. nonlinear phase shift parameter ) according to:

(4a)

(5)

(6)

(7)

(8)

∂E∂z------ αE iβ2 ω0( )∂2E

2--------------------------- ∂2E

∂T 2-------- β3 ω0( )

6----------------∂3E

∂T 3--------–+ + iγ 1 ρ–( ) E 2E ρE h1111 s( ) E T s–( ) 2 sd

0

∞

∫+

=

h1111 t( )

χ1111 ω( )

i∂U∂ξ------- D2

∂2U∂t2--------- N1 U 2U+ + iD3

∂3U∂t3--------- N2U ∂ U 2

∂t------------ iN3

∂∂t---- U 2U( )– iAU ,–+=

D2sign β2( )

2--------------------- D3,

LDsign β3( )LD'

---------------------------- N1,= LD

L------

NLN2,

LD

L------

NLτR' N3

LD

LNL-------- s .=,= = =

LDT 0

2

β2-------- LNL, 1

γP0-------- LD',= T 0

3

β3-------- s, 1

ω0T0----------- τR',

τR

To----- E, P0U T, T0t z, ξLD .= = = = = = =

T0 P0

E z 0 T,=( ) 2

ξ ξ h+ hϕmax

NL max U 2h( )=

U ξ h t,+( ) h2---D

N ξ'( ) ξ'dξ

ξ h+( )

∫ h

2---D

U ξ t,( ) ,expexpexp=

170

OPTICAL FIBER

where the dispersion and nonlinearity operators are given by:

and

The different options available from the "Numerical" tab specify the details of the implementation of Equation 8 and Equation 10 (see Figure 1). The simplest (and the fastest) implementation corresponds to "Propagator type" set to "Exponential" and "Calculation type" set to "Noniterative". In this case, the following approximation is used:

Figure 1 Component “Numerical" tab

(9)

(10)

(11)

D N

D iD2∂2

∂t2------ D3

∂3

∂t3------ A–+=

N iN1 U 2 iN– 2∂ U 2

∂t------------ N3

∂ U 2

∂t------------ U∗ ∂U

∂t------- +

–=

N ξ'( ) z'dξ

ξ h+

∫ hN h 2⁄( )D[ ]exp U ξ t,( )( ) .≈

171

OPTICAL FIBER

According to Equation 11, the half-step propagated field, with the nonlinear effects ignored, is used in turn to evaluate the nonlinearity operator. The dispersion operator is evaluated in the frequency domain according to:

where means fast Fourier transform. If, in addition the "Step size" option is set to "Constant" ("Propagator type", "Exponential", and "Calculation type" are set to "Noniterative"), the number of operations per step decreases because the first and the last Fourier transform for each step cancels each other out (dispersion operators combine) (see Equation 13).

When the "Propagator Type" is set to "Runge-Kutta 4th order" (or "Runge-Kutta 2nd order") (RK4 or RK2), the exponent with the nonlinearity operator in Equation 8 is replaced by the direct integration of the following system of coupled ordinary differential equations:

by means of the standard RK4 (or RK2) routine (see example in [3]). The application of the dispersion operator is the same.

Note: The Runge-Kutta (2nd or 4th order) implementations in the fiber component enable modeling the stimulated Raman scattering effect with the optical signal having an arbitrary polarization ("Model type" parameter set to "Vector"). However, due to the larger number of operations per step, they are executed slower and are not recommended otherwise (in "scalar" simulations or when the Raman effect is not included in a vector simulation) because the "Exponential" implementation of the nonlinearity provides faster execution.

(12)

(13)

(14)

UD ξ h2---+

FFT 1– h2---D iω( )

FFT U ξ t,( )[ ]exp ,=

FFT

U ξ 2h t,+( ) h2---D

N ξ'( ) ξ'dξ

ξ h+( )

∫ h

2---D

h2---D

exp N ξ'( ) ξ'dξ

ξ h+( )

∫ h

2---D

U ξ t,( ) =expexpexpexpexp=

h2---D

N ξ'( ) ξ'dξ

ξ h+( )

∫

hD( ) N ξ'( ) ξ'dξ

ξ h+( )

∫ h

2---D

U ξ t,( )expexpexpexpexp

∂U∂z-------

NL

NU=

172

OPTICAL FIBER

If the "Propagator type" is set to "Exponential" and "Calculation type" to "Iterative", Equation 11 is replaced by [2], [4]:

The symbol means . Since is unknown at , it is necessary to follow an iterative procedure that is initiated by replacing by

(see [2], [4] for the details). Working with two iterations gives a reasonable combination between accuracy and speed, as recommended in [2].

Figure 2 Evolution of for N=3 soliton over 15 soliton periods with different calculation modes

Note: In the three cases presented, , constant step size.

A comparison between the "Iterative" and "Noniterative" approaches is presented in Figure 2. Evolution of N=3 soliton over 15 soliton periods is presented. The "Step size" is kept "Constant" with the "Max. nonlinear phase shift" parameter is equal to 27.6. mrad. The noniterative approach is the fastest but not accurate enough at this step size. The development of spurious, numerical instability, which breaks the periodicity of the soliton evolution [2], is evident at the end of the run. For the same step size the iterative implementation of the split-step Fourier method suppresses the

(15)N ξ'( ) ξ' h2--- N ξ( ) N ξ h+( )+( )≈d

ξ

ξ h+

∫

N ξ( ) N E ξ( )( ) N ξ h+( ) ξ h 2⁄+N ξ h+( )

N ξ( )

E ξ t, 0=( ) 2

ϕmaxNL 27.6mrad=

173

OPTICAL FIBER

instability, thus improving the quality of the results, however this improvement is at the expense of increased computation time.

The step size in the component is determined through the value of the parameter . In the case of the constant step size calculation, it is

calculated once, using the input signal to obtain the maximum value of the intensity. In the case of variable step size calculation such an evaluation is performed at each step.

Figure 3 Variable step size, value of is

In Figure 3, the calculation presented in Figure 2 is repeated using variable step size. This calculation takes longer in comparison to the "Noniterative" case presented in Figure 2, but less than in the case where two iterations are used. Depending on the behavior of the solution, variable step size calculation can take less time compared to the constant step size, although the fixed step size calculation performs a smaller number of operations per step (see Equation 13). In the presence of considerable attenuation, the importance of nonlinear effects decreases along the fiber length, which would permit the use of a larger step size. In this case, the use of variable step size will reduce the computation time. The variable step size calculation is more flexible, because different tasks can be handled keeping the value of constant. For the case presented in Figure 3, this value is double the size of the one used in Figure 2, but the results are even better (refer to compare with Figure 2, "Noniterative").

The split-step scheme used in the model is locally second order accurate which means that the local error is proportional to the . However, the global error (after N steps) is proportional to [22]. Thus, increasing the fiber length might require decrease of the step size to maintain the same accuracy.

The use of FFT implies periodic boundary conditions. In some cases a part of the pulse energy may spread eventually hitting the time window boundaries. When the energy reaches one of the edges of the time window it automatically reenters from the other edge perturbing the solution. This can be avoided using the absorbing type of

hϕmax

NL γmax E 2( )h=

ϕmaxNL ϕmax

NL 50mrad=

ϕmaxNL

h3

Nh3 Lh2=

174

OPTICAL FIBER

boundary conditions. To achieve this at each step the optical field is multiplied in the time domain [10] by:

where indicates the nearest edge. The effect of periodic and absorbing boundary conditions is shown in Figure 4 where the results presented in Figure 3 from "Birefringence and solitons" (propagation distance is equal to 1262.34km) are redisplayed. However here the time window is reduced to show the effect of the periodic boundary conditions. The oscillatory tail developed by the solution in the case when periodic boundary conditions are used is an unphysical effect, resulting from the interference of the radiation that has reentered the time window and the solution. In the case when absorbing boundary conditions are used the radiation that has separated from the solution is removed. The smaller the value of the filter steepness parameter the better the time window boundaries absorb (and do not reflect), however the larger part of the time window becomes absorbing (see Equation 16.

Figure 4 Periodic (left plot) and absorbing with filter steepness 0.05 (right plot) boundary conditions

(16)Γ t( ) 1 FilterSteepnes t tedge–( )( ) ,sech–=

tedge

175

OPTICAL FIBER

Vector approach

When the polarization state of the incident light is not preserved during its propagation inside an optical fiber the scalar approach is no longer applicable and Equation 1 is replaced by [2], [6] - [10]:

Equation 17, contains the Raman response functions [6], [18]. Their Fourier transformations and Raman susceptibilities , are shown in Figure 4.1. The convolution integrals in Equation 17 are evaluated in the frequency domain, by multiplying the spectra of the electric fields with the Raman susceptibilities and then performing the inverse FFT.

Figure 4.1 Raman susceptibilities for fused quartz [6, 18]

(17)

∂EX

∂z--------- β1X

∂EX

∂t--------- iβ2

2-------∂2EX

∂t2----------- β3

6-----∂3EX

∂t3----------- = iγ 1 ρ–( ) EX

2 23--- EY

2+ EX–+ +

+ iγρEX h1111 s( ) EX t s–( )2 s h1122 s( ) EY t s–( )2 sd0

∞

∫+d0

∞

∫

+ iγρEY h1212 s( )EX t s–( )EY∗ t s–( ) sd0

∞

∫

∂EY

∂z--------- β1X

∂EY

∂t--------- iβ2

2------- ∂2EY

∂t2----------- β3

6-----∂3EY

∂t3----------- = iγ 1 ρ–( ) EY

2 23--- EX

2+ EY–+ +

+ iγρEY h1111 s( ) Eγ t s–( )2 s h1122 s( ) EX t s–( ) 2 sd0

∞

∫+d0

∞

∫

+ iγρEX h1212 s( )Eγt s– E∗X t s–( ) sd0

∞

∫

hijkl t( )χijkl v( )

176

OPTICAL FIBER

The SRS effect is represented by "Intrapulse Raman scattering" (Equation 17) is replaced by [20]:

Note: In the case of Equation 17 or Equation 17a, due to the orthogonal Raman gain terms (the last sections in Equation 17 or Equation 17a), the "Exponential" option for the "Propagator type" is not applicable. The component automatically selects "Runge Kutta 2nd order" when the model type is set to "Vector", and the Raman effect ("Intrapulse Raman scattering" or "Full Raman response" options are selected. Due to the increased number of convolutions performed at each step the fiber component can be slow when solving Equation 17.

In normalized units and when the SRS effect is neglected ( ) Equation 17 reads as:

The quantities and are the inverse group velocities for the and polarization components respectively.

(17a)

(18)

∂EX

∂z--------- β1X

∂EX

∂t--------- iβ2

2-------∂2EX

∂t2----------- β3

6-----∂3EX

∂t3----------- = –+ +

iγ EX2 2

3--- 1 ρ–( ) ρ

1 α f+2

--------------+ + EY

2 ρτR1∂ EX

2

∂t--------------– ρτR2

∂ EY2

∂t--------------– EX

iγρτR1 τR2–

2--------------------∂ EXEY∗( )

∂t-----------------------EY–

∂EY

∂z--------- β1Y

∂EY

∂t--------- iβ2

2------- ∂2EY

∂t2----------- β3

6-----∂3EY

∂t3----------- = –+ +

iγ EY2 2

3--- 1 ρ–( ) σ

1 α f+2

--------------+ EX

2

+ στR1∂ EY

2

∂t-------------- ρ– τR2

∂ EX2

∂t--------------– EY

iγρτR1 τR2–

2--------------------∂ EYEX∗( )

∂t-----------------------EX–

ρ 0=

i ∂u∂ξ------ δ∂u

∂t------+

D2∂2u∂τ2-------- iD3

∂3u∂τ3--------– N1 u 2 2

3--- v 2+

u+ + 0=

i ∂v∂ξ------ δ∂v

∂t-----+

D2∂2v∂τ2------- iD3

∂3v∂τ3-------– N1 v 2 2

3--- u 2+

v+ + 0=

β1X β1Y X Y

177

OPTICAL FIBER

Figure 5 Optical fiber as a concatenation of trunks

Note: The arrows represent the principal axes.

The parameter is given by , where is the value of the differential group delay parameter entered from the "PMD" tab, in the case where "Deterministic" mode is selected for the birefringence effect (see "Birefringence and solitons" from the Tutorials). The effects of four-wave mixing between the orthogonal polarization components are not taken into account due to their negligible contribution for typical values of the birefringence [9], [10]. The normalized time is introduced according to where

. All the other parameters have the same meaning as in the scalar case.

The "coarse-step method" [11] is used to simulate the PMD effects in the "Stochastic" mode. The fiber is represented by a concatenation of trunks and the propagation of light in each trunk is simulated by the split-step Fourier method described in the previous section. The lengths of the trunks are random numbers with a Gaussian distribution [12]. The average and the dispersion of this distribution are the "Scattering section length" and "Scattering section dispersion" parameters:

It is recommended [12] that the dispersion is 20% of the average value. The birefringence of each trunk is given by [11] (see the related PMD examples in the tutorials):

where is the PMD coefficient. The principal axes of the trunks are randomly oriented with respect to each other (see Figure 4). To simulate the random mode coupling at the end of each trunk the following transformation is applied [11], [13]:

(19)

(20)

(21)

δ δ β1X β1Y–( )T0 2 β2( )⁄= β1X β1Y–

τ τ t β1z–( ) T0⁄=β1 β1X β1Y+( ) 2⁄=

Lscatt σscatt

f Lscatti( ) 1

2πσscatt

---------------------- Lscatti Lscatt–( )2–2σscatt

2--------------------------------------exp=

ddω------- ∆β( ) DP

Lscatti

---------------=

DPMD

EX'EY'

α α iϕ( )expsincosα i– ϕ( ) cosαexpsin–

EX

EY

=

178

OPTICAL FIBER

In Equation 20, and are random numbers uniformly distributed in the interval .

Wavelength dependent parameters

The file that specifies the wavelength dependence of the parameters consists of two columns with the left column being the wavelength in nanometers and the right column containing the corresponding values of the parameters (see Table 1 ). The sampling interval is not necessarily be constant. The parameter values must be given in the units specified in the "Units" tab of the table.

Table 1 Wavelength dependence of the attenuation parameter

The values of the parameters in Equation 1 and Equation 17 are evaluated at the reference wavelength.

Note: The reference wavelength must be within the wavelength interval covered by the files for all the wavelength dependent parameters specified.

The reference wavelength can be either user-specified or "automatic". In the last case the wavelength corresponding to the central frequency of the spectrum of the signal is assumed by the component to be the reference wavelength. Linear interpolation is used to calculate the values of the attenuation, effective area and parameters at this wavelength. For the dispersion parameters the following procedure is used. The wavelength dependence specified by the file is fitted internally using the five-term Sellmeier formula [14]. The higher-order dispersion parameters are then obtained by analytically differentiating this expression. If the option frequency domain parameter is unchecked, the file may give either the group delay or dispersion (depending on the choice made in the "Dispersion file format" tab), and if the frequency domain parameters option is selected, either or can be supplied, again determined by the value of the "Dispersion file format" parameter. If the wavelength dependence of the group delay is given by the user, two successive differentiations are applied to its Sellmeier fit. Differentiating the analytical fit instead of using a direct numerical differentiation of the data provides the advantage of being able to produce reasonable results even in the case where the supplied data is noisy (see Appendix 1).

1400 0.31405

1402.5 0.30246

1405 0.29276

1407.5 0.28457

1410 0.27757

1412.5 0.27153

α ϕ0 2π,[ ]

λ nm[ ] α dB km⁄[ ]

n2

β1 λ( ) D λ( )

β1 λ( ) β2 λ( )

179

OPTICAL FIBER

Note: The accuracy of the Sellmeier fit depends on the type of the fiber. This is shown in Figure 6, where the results obtained for dispersion flattened and dispersion shifted fibers are shown.

Figure 6 Comparison between the original dispersion data and their fits for two fiber types

Guidelines for using the component for WDM simulations

Periodic boundary conditions are required for simulating the propagation of long bit sequences at different carrier wavelengths, which is the case when WDM systems are designed.

To avoid the aliasing phenomena (see e.g. [3]), the sample rate is chosen to be at least three times bigger (Figure 7) than the bandwidth occupied by the simulated channels (see e.g. [15]).

Figure 7 WDM channels and their four-wave mixing products

Any frequency component outside the frequency range (Fc-SR/2, Fc+SR/2), where SR is the sample rate and Fc is the reference frequency is falsely translated (aliased) into that range by the very act of discrete sampling [3]. If the sample rate is bigger than

180

OPTICAL FIBER

the bandwidth occupied by the WDM channels (so it can accommodate all the channels) but less than three times that value in the presence of nonlinear effect the four-wave mixing products resulting from the nonlinear interaction between the channels (spurious waves [16]) will be aliased. In [16], to minimize the amount of aliased power the requirement that the value of the power spectrum at the boundary of the available spectral range be -40 dB of its peak value is used.

The longitudinal step size depends on the importance of the nonlinear effects for the particular simulation. If all the nonlinear effects are disabled step size equal to the fiber length will be used. The increase of the impact of nonlinearity will require decrease of the step size (decrease of the value of the max. nonlinear phase shift parameter) to maintain the same accuracy.

Figure 8 Output spectra corresponding to

Note: The propagation distance is 100km. Input configuration is given in "Cross-phase modulation" in the Tutorials.

Values in the order of a few miliradians (one [15] and three [17])) are used with this parameter in a WDM system simulation. The effect of an improperly chosen step size is shown in Figure 8, where the output spectra corresponding to an interaction of two Gaussian pulses with carrier wavelengths one nm spaced are shown (see "Cross-phase modulation" from the Tutorials). While the correct result that the four-wave mixing products (or spurious waves) should disappear when the pulses are no longer over-lapped (in the absence of any loss and gain [16]) is reached when the step-size is small enough, in the opposite case, the spurious frequencies present in the output spectra are still evident. The improperly chosen step size (too big) tends to exaggerate the four wave mixing products (see [22] and references therein).

To increase the accuracy, you can switch from a "Noniterative" to an "Iterative" calculation type, keeping the step size the same (with the same step size, the "Iterative" implementation is more accurate, (see Figure 2), or alternatively, to keep working in the "Noniterative" mode and decrease the step size, or the value of the "Max. nonlinear phase shift" parameter. With respect to saving computational time, the latter strategy is better. It should be noted that computational time will not be saved by simultaneously increasing the number of iterations and the step size.

ϕmaxNL 50mrad and ϕmax

NL 3mrad= =

181

OPTICAL FIBER

Appendix 1

Dispersion fitting according to the Sellmeier formulaWhen the option "Dispersion from file” is selected, the dispersion data are internally fitted according to the five-term Sellmeier formula [14], namely:

where is the group delay (per unit fiber length) or, respectively:

where is the dispersion [ps/nm/km]. The user supplies data either for the dispersion or the group delay that are then fitted according to Equation 2A or Equation 1A, and the slope and/or dispersion are calculated by differentiating Equation 1A and Equation 2A analytically.

The least-square fitting associated with Equation 2A amounts to minimizing:

where is the number of points. Using:

the following linear system is obtained:

(1)A

(2)A

(3)A

(4)A

(5)A

τ c1λ 4– c2λ 2– c3 c4λ2 c5λ4+ + + +=

τ

D dτdλ------ c1'λ

5– c2'λ3– c4'λ c5'λ

3+ + += =

D

Q c1λ i5– c2λ i

3– c4λ i c5λ i3 Di–+ + +( )2

i 1=

N

∑ min= =

N

∂Q∂ci------- 0 i, 1…4 ,= =

λi10–∑ λi

8–∑ λi4–∑ λi

2–∑λi

8–∑ λi6–∑ λi

2–∑ N

λi4–∑ λi

2–∑ λi2∑ λi

4∑λi

2–∑ N λi4∑ λi

6∑

C1

C2

C4

C5

Diλ i5–∑

Diλ i3–∑

Diλ i∑Diλ i

3∑

=

182

OPTICAL FIBER

which is solved by LU-decomposition [3].

In the case when the user supplies a group delay data file, Equation 1A is used and Equation 5A transforms into Equation 6A.

The fitting procedure is useful when/if noisy data is supplied by the user, as the following example shows. Figure 1A shows dispersion-versus-wavelength dependence of SMF-28 and the corresponding "exact" results for dispersion parameters are displayed below the graph.

Figure 1A Lambda = 1550.75nm beta2=-2.08625e-026 s2/m beta3=1.27246e-040 s3/m

D= 1.63411e-005 s/m2 S= 56.9931 s/m3

To assess the influence of noise on the results from the calculation some noise is added to the data presented in Figure 1A with the resulting graph presented in Figure 2A. Supplying the data from Figure 2A to the Nonlinear Dispersive Fiber Total

(6)A

λi8–∑ λi

6–∑ λi4–∑ λi

2–∑ N

λi6–∑ λi

4–∑ λi2–∑ N λi

2∑λi

4–∑ λi2–∑ N λi

2∑ λi4∑

λi2–∑ N λi

2∑ λi4∑ λi

6∑N λi

2∑ λi4∑ λi

6∑ λi8∑

C1

C2

C3

C4

C5

τiλ i4–∑

τiλ i2–∑

τi∑τiλ i

2∑τiλ i

4∑

=

183

OPTICAL FIBER

Field component gives the results for the dispersion parameters presented under Figure 2A.

Figure 2A Lambda = 1550.75nm beta2=-2.10115e-026 s2/m beta3=1.32966e-040 s3/m

D= 1.64578e-005 s/m2 S= 60.3521 s/m3

184

OPTICAL FIBER

References:[1] G. P. Agrawal, "Applications of nonlinear fiber optics", Academic press, 3rd edition, 2001.

[2] G. P. Agrawal, "Nonlinear fiber optics", Academic press, 3rd edition, 2001.

[3] W. H. Press, et al., "Numerical Recipes: The Art of Scientific Computing", 2nd Edition, Cambridge University Press, 1992.

[4] M. Lax, J. H. Batteh and G. P. Agrawal, Journ. Appl. Phys. 52 , 109, (1981).

[5] F. Matera and M. Settembre, Journ. Lightwave Technol. 14, 1 (1996).

[6] R. W. Hellwarth, Prog. Quant. Electr. 5, 1 (1977).

[7] E. A. Golovchenko and A. N. Pilipetskii, JOSA B, 11, 92 (1994).

[8] P. T. Dinda, G. Millot, and S. Wabnitz JOSA B, 15, 1433 (1998).

[9] C. R. Menyuk, Opt. Lett., 12, p. 614 (1987).

[10] C. R. Menyuk, JOSA B, 5, p. 392(1988).

[11] D. Marcuse, C. R. Menyuk and P. K. A. Wai JLT, vol. 15, No. 9, pp. 1735 (1997).

[12] C. H. Prola Jr., J. A. Pereira da Silva, A. O. Dal Forno, R. Passy, J. P. Von der Weid, and N. Gisin IEEE Phot. Technol. Letters, 9, No. 6, 842 (1997).

[13] P. K. A. Wai, C. R. Menyuk, and H. H. Chen , Opt. Lett. 16 1231 (1991).

[14] L. G. Cohen, Journ. Lightwave Technol. 3, 958, (1985).

[15] M. I. Hayee and A. E. Willner, IEEE Phot. Technol. Lett. 11, No. 8, (1999).

[16] D. Marcuse, A. R. Chraplyvy, and R. W. Tkach, Journ. Lightwave Technol, 9, 121 (1991).

[17] R. W. Tkach, A. R. Chraplyvy, F. Forghieri, A. H. Gnauck, and R. M. Derosier, Journ. Lightwave Technol, 13, 841 (1995).

[18] P. Tchofo Dinda, G. Millot, and S. Wabnitz, JOSA B, 15, 1433, (1998).

[19] R.H.Stolen, J.P.Gordon, W.J. Tomlinson and H.A. Haus, JOSA B, 6, 1159 (1989).

[20] C.R.Menyuk, M.N.Islam and J.P.Gordon, Optics Letters, 16 566, (1991).

[21] K.J. Blow and D. Wood, IEEE J. Quant. Electr., 25, 2665, (1989).

[22] O. Sinkin, R. Holzlohner, J. Zweck and C. R. Menyuk, Journ Lightwave Technol. 21, 61 (2003).

185

OPTICAL FIBER

Notes:

186

LINEAR MULTIMODE FIBER

Linear Multimode fiber

This component is a multimode fiber that assumes the fiber has sufficient mode mixing due to imperfections or splices; in this case the modal transfer function approaches a Gaussian function. It also includes first- and second-order chromatic dispersion.

Ports

Parameters

Main


Input Input Optical



Frequency

Reference center frequency

850 nm, Hz, THz [100, 2000]

Length

Fiber length

1 km [0,+INF[

Attenuation

Fiber attenuation

2.61 dB/km [0,+INF[

Modal bandwidth

Fiber modal bandwidth

1324 MHz.km [0,+INF[

Cutback factor

Cutback factor

1 - [0,+INF[

187


Chromatic dispersion

Simulation


Include chromatic dispersion

Defines whether the model includes chromatic dispersion effects

False nmHz, THz True, False

Use Sellmeier approximations

Defines whether the user enters data sheet parameters for zero dispersion wavelength or at the reference wavelength


Zero dispersion wavelength

Wavelength at zero dispersion

1354 nm [100, 2000]

Zero dispersion slope

Dispersion slope at zero dispersion

0.097 ps / (nm2.km) ]-INF,+INF[

Dispersion

Dispersion at reference frequency

–100 ps / (nm.km) ]-INF,+INF[

Dispersion slope

Dispersion slope at reference frequency

0.5 ps / (nm2.km) ]-INF,+INF[

Spectral width

Source spectral width

0.4 nm [0, 2000]



Enabled



Resample

Determines if the filter will down sample the signal bandwidth to the filter sample rate


Sample rate

New output signal sample rate

500 GHz Hz, GHz, THz ]0,+INF[

188


Noise

Technical backgroundThe optical fiber has three dominant dispersion mechanisms, modal, and first-order chromatic and second-order chromatic. We can assume that modal dispersion and chromatic dispersion mechanisms act independently and can be treated separately[1][2].

Modal dispersionPersonick has shown that if a multimode fiber has sufficient mode mixing due to imperfections or splices, in this case the modal transfer function approaches a Gaussian function [3][4][5]

(1)

where is the angular baseband frequency and is the RMS impulse response width.

In this model, the modal dispersion is characterized by the 6 dB half of the optical power frequency:

(2)

where is defined by the parameter Modal bandwidth and is the fiber parameter Length. is the cutback factor, that takes into account the mode coupling, mixing and concatenation effects.

Rewriting Equation 1 and Equation 2 in terms of frequency and bandwidth:

(3)


Units Value range

Noise threshold

Minimum value for adaptation of noise bins

–100 dB ]-INF,+INF[

Noise dynamic

Threshold ratio for adaptation of noise bins

3 dB ]-INF,+INF[

HM ω( ) eω2σ2

2-------------–

=

ω σ

B6dB2 1n 2( )⋅2 π σ⋅ ⋅

-------------------------- BM1L---

γ

= =

BM Lγ

HM f( ) e

1n 2( ) f 2⋅

BM1L---

γ ⋅

2---------------------------------- –

=

189


Chromatic dispersionSince most of the injection-lasers used in optical fiber communications have a Gaussian line shape [1][2][6][7], we can use the solution for the chromatic transfer function for a perfect Gaussian linewidth case:

(4)

where and are abbreviations for

(5)

where the parameter is defined by Spectral width, S is the parameter Dispersion slope, D is the Dispersion, is the reference center wavelength calculated from the parameter Frequency, and L is the fiber length.

The parameter Use Sellmeier approximations defines whether you will enter D and S directly, or if they will be calculated from the Sellmeier approximations [2]:

(6)

HD ω( ) 11 iω ω2⁄+( )1 2⁄

------------------------------------eω ω1⁄( )2

2 1 iω ω2⁄+( )----------------------------------–

=

ω1 ω2

ω1 σλ D L( ) 1––=

ω1 σλ2 S 2 D λr⁄+( )L[ ] 1–=

σλλr

D S0

4---- λr

λ04

λr3

-----– =

S S0

4---- 1 3λ0

4

λr4

-----+ =

190


References

[1] C. Yabre, "Comprehensive Theory of Dispersion in Graded-Index Optical Fibers", Journal of Lightwave Technology, Vol. 18, No. 2, pp. 166-176, February 2000.

[2] G.D. Brown, "Bandwidth and Rise Time Calculations for Digital multimode Fiber-Optical Data Links", Journal of Lightwave Technology, Vol. 10, No. 5, pp. 672-678, May 1992.

[3] S.D.Personick "Baseband Linearity and Equalization in Fiber Optic Digital Communication Systems", The Bell System Technical Journal, pp. 1174-1194, September 1973.

[4] D.G.Duff, "Computer-Aided Design Of Digital Lightwave Systems", IEEE Journal on Selected Areas in Communications, Vol. SAC-2, No. 1, pp. 171-185, January 1984.

[5] D.O.Harris, J.R. Jones "Baud Rate Response: Characterizing Modal Dispersion for Digital Fiber Optic Systems", Journal of Lightwave Technology, Vol. 6, No. 5, pp. 668-677, May 1988.

[6] J.L.Gimlett, N,K,Cheung "Dispersion Penalty Analysis for LED/Single-Mode Fiber Transmission Systems", Journal of Lightwave Technology, Vol. LT-4, No. 9, pp. 1381-1391, September 1986.

[7] T. Pfeiffer, M. Witte, B. Deppisch "High-Speed Transmission of Broad-Band Thermal Light Pulses Over Dispersion Fibers", IEEE Photonic Technology Letters, Vol. 11, No. 3, pp. 385-387, March 1999.

191


Notes:

192

NONLINEAR DISPERSIVE FIBER

Nonlinear Dispersive fiber

Fundamental model of a generic nonlinear and dispersive optical fiber. Its power and versatility allow it to represent any type of real fiber by a suitable combination of parameters.

Ports

Parameters

Main


Input Input Optical



Length

Fiber length

50 km [0, INF]

Attenuation data type

Defines the attenuation as a fixed constant value or as a wavelength dependent curve taken from a file

Constant — Constant, Wavelength Dependent/ From File

Attenuation – constant

Defines the attenuation as a fixed constant value, the same for all channels

0.25 dB/km [0, INF]

Attenuation vs. wavelength

Defines the attenuation as a wavelength dependent curve in a file

AtnVsLambda.dat — [0, INF]

Input coupling loss

Overall input coupling loss resulting from mode mismatch, Fresnel reflections, etc.

-1 dB [-INF,0]

Output coupling loss

Overall output coupling loss resulting from mode mismatch, Fresnel reflections, etc.

-0.022 dB [-INF,0]

193


Dispersion


Group delay data type

Defines the group delay as a fixed constant value, or as a wavelength dependent curve taken from a file


Group delay – constant

Defines the group delay as a fixed constant value, the same for all channels

4.9e+006 ps/km [-INF,INF]

Group delay vs. wavelength

Defines the group delay as a wavelength dependent curve in a file

GroupVs Lambda.dat — [-INF,INF]

GVD data type

Defines the group-velocity dispersion as a fixed constant value, or as a wavelength dependent curve taken from a file


GVD – constant

Defines the group-velocity dispersion as a fixed constant value, the same for all channels

4.5 ps/nm/km [-INF,INF]

GVD vs. wavelength

Defines the group-velocity dispersion as a wavelength dependent curve in a file

GVDvsLambda.dat — [-INF,INF]

Dispersion slope data type

Defines the dispersion slope as a fixed constant value, or as a wavelength dependent curve taken from a file


Dispersion slope – constant

Defines the dispersion slope as a fixed constant value, the same for all channels

0.11 ps/nm2/km [-INF,INF]

Dispersion slope vs. wavelength

Defines the dispersion slope as a wavelength dependent curve in a file

DispSlope vs. Lambda.dat

— [-INF,INF]

Effective refractive index vs. wavelength

Defines the effective refractive index as a dispersive curve vs. the wavelength in a file

EffRIVsLambda.dat — [0,INF]

194


Birefringence


Birefringence data type

Defines the birefringence (the mismatch between the propagation constants of the two orthogonal polarization modes) as a fixed constant value, or as a wavelength dependent curve taken from a file


Birefringence – constant

Defines the birefringence as a fixed constant value, the same for all channels

6.2832e-005 rad/m [-1,1]

Birefringence vs. wavelength

Defines the birefringence as a wavelength dependent curve in a file

BirefringenceVs Lambda.dat

— [-1,1]

Coupling length of polarization mixing

Coupling length of polarization scrambling

0.1 km [0,INF]

PMD coefficient

Polarization mode dispersion coefficient

0.07 ps/km1/2 [0,INF]

DGD data type

Defines the differential group delay between the two orthogonal polarization modes as a fixed constant value, or as a wavelength dependent curve taken from a file


DGD – constant

Defines the differential group delay as a fixed constant value, the same for all channels

3 ps/km [-INF,INF]

DGD vs. wavelength

Defines the differential group delay as a wavelength dependent curve in a file

DGDVsLambda.dat — [-INF,INF]

195


Nonlinearities


Effective area data type

Defines the effective area of the fiber as a fixed constant value, or as a wavelength dependent curve taken from a file.


Effective area – constant

Defines the effective area as a fixed constant value, the same for all channels.

72 µ2 [0,INF]

Effective area vs. wavelength

Defines the effective area as a wavelength dependent curve in a file.

EffAreaVsLambda.dat — [0,INF]

n2 data type

Defines the nonlinear refractive index as a fixed constant value, or as a wavelength dependent curve taken from a file.


n2 – constant

Define the nonlinear refractive index as a fixed constant value, the same for all channels.

2.6e-020 m2/W [-INF,INF]

n2 vs. wavelength

Defines the nonlinear refractive index as a fixed constant value, or as a wavelength dependent curve taken from a file.

N2VsLambda.dat — [-INF,INF]

Raman-resonant n2 dispersion

Defines the Raman-resonant dispersion of the third-order nonlinear susceptibility as a frequency dependent curve in a file

RamanResN2Vs Freq.dat

— [-INF,INF]

Peak Raman gain coef

The peak Raman gain coefficient at certain pump wavelength

9.9e-014 m/W [0,INF]

Pump Wavelength of Peak Raman gain coef

The pump wavelength corresponding to the above peak Raman gain coefficient

1000 nm [0,INF]

Raman Gain Spectrum

Defines the Raman gain spectrum vs. frequency in a file

RamanGainVsFreq.dat — [0,INF]

Raman self-shift Time

The characteristic Raman self-frequency shifting time

5 fsec [0,INF]

196


Effects On/Off; Model Details


Attenuation

Switch On/Off the attenuation

ON — [ON, OFF]

Group velocities mismatch

Switch On/Off the group velocities mismatch

ON — [ON, OFF]

GVD (Group velocity dispersion)

Switch On/Off the group velocity dispersion

ON — [ON, OFF]

GVD Slope (third-order dispersion)

Switch On/Off the dispersion slope (the third-order dispersion)

ON — [ON, OFF]

Polarization evolution

Specify the polarization maintaining capabilities of the fiber and the polarization evolution models to use

Hi-Bi PM fiber, no PMD, fixed DGD

— Hi-Bi PM fiber, no PMD, fixed DGD, Non-PM fiber, PMD, stochastic DGD, Averaged polarizations

Independent pol. mode mixing of WDM channels

In the case of non-PM fiber, determines whether the polarization scrambling follows the same pattern for all the channels or is completely independent

OFF [ON, OFF]

n2 polarization factor 1 dimensionless [0.5, 1]

Raman Gain polarization factor 1 dimensionless [0.5, 1]

Birefringence

Switch On/Off the birefringence

ON — [ON, OFF]

SPM (Self-phase modulation)

Switch On/Off the SPM (Self-phase modulation)

ON — [ON, OFF]

XPM (Cross-phase modulation)

Switch On/Off the XPM (Cross-phase modulation)

ON — [ON, OFF]

XPM of orthogonally polarized modes

Switch On/Off the XPM of orthogonally polarized modes

ON — [ON, OFF]

FWM (four-wave mixing)

Switch On/Off the FWM (four-wave mixing)

OFF — [ON, OFF]

FWM of orthogonally polarized modes

Switch On/Off the XPM of orthogonally polarized modes

OFF — [ON, OFF]

197


Simulation details

Maximal phase-mismatch

FWM generated waves with phase-mismatches larger than this value are neglected

100 radian [-1e+100, 1e+100]

SRS (stimulated Raman scattering) ON — [ON, OFF]

SRS with pump wave depletion

Switch On/Off the effect of pump wave depletion in SRS

ON — [ON, OFF]

RSFS (Raman self-frequency shifting)

Switch On/Off the RSFS (Raman self-frequency shifting)

OFF — [ON, OFF]


Enabled

Enable the calculations

ON — [ON, OFF]

Number of steps

Number of longitudinal steps

25 — [0,INF]

Step defined as:

Choose one of the three alternative ways of defining the step size

Fixed = Main Channel Initial Nonlinear length/Number of Steps

— Fixed = Full length/Number of Steps

Fixed = Main Channel Initial Nonlinear length/Number of Steps

Variable = Main Channel Current Nonlinear length/Number of Steps

Time-window boundaries

Choose the type of the time-window boundary conditions

Absorbing — Periodic, Absorbing

Random Phases

Randomize the phase offsets of the channels at input

OFF — [ON,OFF]

Random Phases Seed

The seed of the random phases

1 — [0, 65535]


198


3D Graphics selection

Graphs


Power spectrum of channels

Displays the average power spectrum of the channels

OFF — [ON,OFF]

Unit of power spectra

Displays the average power spectrum of channels or the PSD of a selected channel in [mW] or [dBm]

dBm — mW, dBm

Bandwidth spectrum of channels

Displays the rms bandwidths of the channels

OFF — [ON,OFF]

Monitor central sampled channel

Monitors the center most channel if described as a sampled waveform

ON — [ON,OFF]

Wavelength of sampled channel to monitor

Monitors an arbitrary sampled channel, defined by its central wavelength

0 nm [0,INF]

Waveform

Displays the waveform of the selected sampled channel

ON — [ON,OFF]

Chirp

Displays the chirp of the selected sampled channel

OFF — [ON,OFF]

PSD

Displays the PSD of the selected sampled channel

OFF — [ON,OFF]

Spectral Delay

Displays the spectral delay of the selected sampled channel

OFF — [ON,OFF]

Number of 2D snapshots in the 3D graphics

Defines the number of 2D snapshots forming the selected 3D graphics

50 — [2, 1000]


Fiber 3D Graph EmptyX EmptyY

199


Parameters—Detailed descriptionsIn the following section, the parameters descriptions are further elaborated. There are descriptions of features pertaining to multiple parameters, and also extended descriptions of individual parameters.

Note: Many parameters pertaining to the NDF can be defined as either constant or wavelength dependent/from file values. The first option is used usually for rapid development of simple designs. If a parameter is wavelength dependent (arb. curve ) you have to prepare a text file with (Wavelength ParameterValue) data pairs, and create the parameter in the appropriate Component properties dialog box. This option is recommended for detailed, quantitatively precise designs. Many parameters of the NDF, such as losses, dispersion, and effective fiber area, can be defined in both ways - as constants or curves loaded from a file. When a parameter is defined as a curve, the format of the text file is as follows:

The units of wavelength are nanometers ( ). The units and the value ranges of the parameter values are the same as those of the respective 'constant' parameters.

For example, when a loss spectrum is loaded from file it might look like:

or:

Wavelength_1 ParameterValue_1



......

Wavelength_N ParameterValue_N

1500 1.99E-01

1525 1.92E-01

1550 1.89E-01

1575 1.93E-01

1600 2.05E-01

1500 0.199

1525 0.192

1550 0.189

1575 0.193

1600 0.205

nm

200


An arbitrary number of points (file lines) are permitted, except 0 (empty file). The column separator can be an arbitrary number (except 0) of either spaces or tabs. The files are opened using the standard Windows "File Open" dialog box.

Technical background

Origin of the nonlinearityAt high optical intensities for intense electromagnetic fields, the dielectric medium behaves as a nonlinear medium. This is also the case for the fiber material. Under the influence of intense electromagnetic fields, the motion of bound electrons becomes anharmonic and, as a result, the induced polarization P from the electric dipoles becomes nonlinear function of the electric field E:

where χ(j) (j =1,2,3, …) denotes the jth order of susceptibility. The lowest order nonlinearity in optical fibers originates from the third order susceptibility χ(3).

Nonlinear effects in optical fibersThe following nonlinear effects in optical fibers are caused by the third-order nonlinear susceptibility and are included in the numerical engine of the component:• Self-phase modulation (SPM)• Cross-phase modulation (XPM)• Cross-phase modulation between the orthogonal modes of a birefringent fiber

(PXPM)• Four-wave mixing (FWM)• Four-wave mixing between the orthogonal modes of a birefringent fiber (PFWM)• Interchannel Stimulated Raman scattering (SRS) and intrachannel Raman self-

shifting (RSS)

OptiSystem currently supports several different models specialized for different signal representations and/or combinations of parameters.

Model IaThis model has been derived for the separated channels signal representation. It also accounts explicitly for the nonlinear interactions and mixing of the orthogonal polarization modes in an SM fiber. It is a system of 2N coupled modified nonlinear Schrödinger equations (NLSE).

This model accounts for:• background loss and linear dispersion up to third order• birefringence and PMD• nonlinearities — SPM, XPM, FWM, SRS, RSS, PXPM, and PFWM

P ε0 x 1( ).E+x 2( ):EE+x 3( ):EEE+...[ ]=

201


For Sampled signals, the following effects are accounted for: XPM, XPM of orthogonally polarized modes, Raman, FWM, and SSFS.

Whereas, for Parameterized signals and ASE noise bins, we account for Raman and FWM.

There are 3 types of polarization evolution that could be taken into account:

Hi-Bi PM fiber, no PMD, fixed DGD

In the case of polarization maintaining fiber, we have to specify the birefringence and DGD of the fiber.

Non PM fiber, PMD, stochastic DGD

In this case the correlation length Lcorr and PMD coefficient have to be specified. The component allows the calculation for PMD of any order. To see the effect of PMD, the following effects must also be selected under the Effects tab: Birefringence and Group velocity mismatch

Averaged polarizations

In this case, the effect of the Kerr nonlinearity is averaged over the Poincare sphere, and is taken into account with a coefficient value of 8/9. The effect of nonlinear PMD [2] is not taken into account.

The intrapulse Raman scattering (or Raman Self Shifting) effect, which leads to soliton self frequency shift, has to be considered for very short optical pulses with duration ~ picosecond or smaller.

The model has the following form:

∂Aix

∂z---------- β1ix

∂Aix

∂t---------- i

2---β2i

∂2Aix

∂t2----------- 1

6---β3i

∂3Aix

∂t3-----------– 1

2---αiAix+ +± =

i Mγx µ v ρ, , , δ ωk ωl ωj– ωi–+( )fijkl

fii-------Ajµ∗AkvAlρ i∆βz( ) +exp

j k l, , 1=

j k l i≠, ,

µ x=

v ρ, x y,=

N

∑

i gRj gR

n ωj ωi–( )fij

fii---- Ajx

2Aix igRi gR

n ωi ωj–( )fij

fii---- Ajx

2Aix

j 1=

j 1≠

ωj ωi<

N

∑–j 1=

j 1≠

ωj ωi>

N

∑

13---iγAiy

2Aix∗ 2i∆βxyz–( ) –exp

202


(1)

where Aix, Aiy are the slowly varying complex electric field amplitudes of the radiation in the respective x/y polarization mode of the i’th WDM channel,

and are the inverses of the group velocities of the pol. modes,

evaluated at the respective carrier frequency of the channels. is the GVD coefficient, related to the dispersion parameter as:

is the third-order dispersion coefficient, related to the dispersion slope as:

(2)

αi is the loss coefficient for the respective carrier frequency of the channel

is the normalized Raman gain function taken from reference [1], Figure 8.1 on page 300.

γ = ωi n2 / c Aeff is the nonlinear coefficient ( ≈ 1-10 W-1km-1 )

n2 is the nonlinear refractive index equal to 3 χxxxx / 8 neff ( ≈ 3.10-16 cm2/ W )

γ xµνρ = ωi (3 χxµνρ /8 neff )/ (c Aeff ) is the nonlinear coefficient of the four-wave interactions and is proportional to the relevant component of the χ tensor.

iγ Aix2Aix 21γ

fij

fii---- Ajx

2

j 1=

j 1≠

N

∑ Aix+ +

13---iγ Aiy

2Aix23---iγ fij

fii---- Ajy

2

j 1=

j 1≠

N

∑ Aix+ +

iγTR∂ Aix

2

∂t--------------Aix

β1ix1 vg⁄( )ix= β1iy

1 vg⁄( )iy=

β2 i

D 2πcβ2i

λ2----------------–=

β3i

S 2πcλ2

--------- 2

β3i4πcλ3

--------- β2 i+=

gRn

203


Aeff is the effective area:

(3)

where F(x,y) is the modal field distribution of the fiber mode.

The overlap integrals fij are defined by:

(4)

M is the multiplicity factor. Its value is 2 if all three waves are different — otherwise, its value is 1.

The overlap integral fijkl is:

(5)

where the angle brackets denote integration over the transverse coordinates x and y.

Aeff =

fii =

fijkl=

F x y,( )( ) 2 xd yd∞–

∞

∫∞–

∞

∫

F x y,( ) 4 xd yd∞–

∞

∫∞–

∞

∫-----------------------------------------------------

Fi x y,( ) 2 Fj x y,( ) 2 xd yd∞–

∞

∫∞–

∞

∫

Fi x y,( ) 2 xd yd∞–

∞

∫∞–

∞

∫ Fj x y,( ) 2 xd yd∞–

∞

∫∞–

∞

∫----------------------------------------------------------------------------------------------

Fi∗Fj∗FkFl⟨ ⟩

Fi2⟨ ⟩ Fj

2⟨ ⟩ Fk2⟨ ⟩ Fl

2⟨ ⟩[ ]1 2⁄--------------------------------------------------------------------------

204


Also

(6)

where

(7)

are the propagation constant mismatches of the processes of FWM and (PFWM) and TR ~ 5 fsec is the slope of the Raman gain curve.

Model IbSimilar to Model Ia, but disregards the polarization evolution of the signal and uses the average power of the two polarization modes. It consists of a system of only N coupled modified nonlinear Schrödinger equations (NLSE) with correspondingly adjusted nonlinear coefficients.

Model IDerived for the case of the total field signal representation. All sampled signals are in a single frequency band. This is the basic method used for modeling WDM systems.

It also accounts explicitly for the mixing of the orthogonal polarization modes in an SM fiber. It is a system of two coupled modified nonlinear Schrödinger equations (NLSE).

This model accounts for:• background loss and linear dispersion up to third order• birefringence and PMD• nonlinearities - SPM, XPM, FWM, SRS, RSS, PXPM

It works with all types of signals: Sampled, Parameterized and ASE noise bins. For parameterized and ASE noise bins, only linear losses are taken into account.

'Total field approach' automatically accounts the XPM and FWM effects. There is no possibility to switch off these effects.

'Total field approach for both polarizations' will additionally account for PXPM of orthogonally polarized signals' and PFWM of orthogonally polarized signals'.

The model for the case of one polarization has the following form:

(8)

All the parameters in the above equation have been explained, along with the Model Ia.

∆β ωknk ωlnl ωjnj– ωini–+( ) c⁄=

∆βxy βy βx–=

∂A∂z------ β1

∂A∂t------ i

2---β2

∂2A∂t2-------- 1

6---β3

∂3A∂t3--------– 1

2---αAx+ +± iγ A 2A iγTR

∂ A 2

∂t-----------A–=

205


Numerical Methods

The three models (Model la, Model lb, and Model l) are solved by a scalar or vectorial version of the split-step Fourier transform method:

(9)

with symmetrized step size [1].

In addition, the step size can be controlled along the propagation.

Step size selection rulesThe user can choose one of the following three ways to calculate the step size:• Fixed• Initial Nonlinear Length / Number of Steps• Current Nonlinear Length / Number of Steps

FixedIn this case the step size is simply , where is the length of the fiber and is the user defined number of steps.

Initial Nonlinear Length / Number of StepsOne of the well known strategies for guaranteeing accurate split-step calculations is to limit the value of the accumulated nonlinear phase-shift per step.

This is equivalent to set

where is the nonlinear length at the input of the fiber (a measure of the distance needed for considerable nonlinear distortions to occur), and

is the user specified number of steps per .

Another limitation imposed is that the maximum temporal displacement of the channels due to group-velocity mismatch per step is less than 1% of the bit period.

Current Nonlinear Length / Number of StepsIn this case, the nonlinear length is periodically recalculated along the fiber:

.

In this way, the possible changes in due to loss or gain are taken into account.

The term indicates the channel used in the calculations above. When the separate channels signal representation is used, it is either the channel with the highest power or the central channel. If we use only one continuous spectral band, as in the total field signal representation, there can be only one main channel.

∂A∂z------ D N+[ ]A=

∆z L N⁄= LN

∆z LNL NLNL⁄=

LNL 1 γP 0( )⁄=

NLNL LNL

LNL z( ) 1 γP⁄ z( )=

LNL z( )

206


References[1] Agrawal, G.P., “Nonlinear Fiber Optics, 3rd Edition”, Academic Press, 2001.

[2] Marcuse, D., Menyuk, C.R., and Wai, P.K.H., "Application of the Manakov - PMD Equation to Studies of Signal Propagation in Optical Fibers with Randomly Varying Birefringence", Journ. Light. Technol.,15, 1735-1746 (1997).

[3] Tchofo Dinda, P., Milot, G., and Wabnitz, S. "Polarization Switching and Suppression of Stimulated Raman Scattering in Birefringent Optical Fibers", JOSA B, 15, 1433-1441 (1998).

207


Notes:

208

Receivers Library

This section contains information on the following receivers.

Regenerators

Electrical

• Clock Recovery• Data Recovery• 3R Regenerator

Optical

Demodulators

• Ideal Frequency Demodulator• Ideal Phase Demodulator

Photodetectors

• Photodetector PIN• Photodetector APD

209

Notes:

210

CLOCK RECOVERY

Clock RecoveryCompensates the time delay between the original signal at the reference port and the signal that is received at the input port.

Ports

Parameters

Simulation

Results


The time delay is calculated from cross-correlation of the reference signal and the received signal. The signal is then shifted in time.


Reference Input Electrical






Name and description Units

Signal delay s

Signal delay samples

211

DATA RECOVERY

Data RecoveryThis component recovers the binary data from the electrical signal. It can be used in 3R generators for the data recovery stage.

Ports

Parameters

Main




Name and description Default value Default unit Units Value range

Reference bit rateReference bit rate to use for the decision instant calculation

Bit rate Bits/s Bits/sMBits/sGBits/s

[0,+INF[

User defined delayDefines whether the user can define the delay compensation or not

False — — True, false

Delay compensationDelay to apply to the signal input

0 s s, ms, ns ]-INF,+INF[

User defined decisionDefines whether the component will automatically calculate the decision instant or it will be defined by the user


Decision instantValue for the decision instant to use when recovering the bit sequence

0.5 Bit — [0,1]

User defined thresholdDefines whether the component will be automatically calculated or willbe user-defined


Absolute thresholdValue for the threshold to use when recovering the bit sequence

0.5 a.u. — ]-INF,+INF[

212

DATA RECOVERY

Random numbers


This component allows the user to recover a bit sequence from an electrical signal. In order to recover the bit sequence, the user should provide the signal bit rate, given by the parameter Reference bit rate. The decision instant and the threshold level can be defined by the user or automatically calculated by this component. If the parameter User defined decision is disabled, the model automatically estimates the decision instant by generating internally an eye diagram and searching for the maximum opening for the eye amplitude. The time instant with the maximum opening is the decision instant, this method is valid for RZ and NRZ modulation types. The user can disable the searching and enter directly the value of the decision instant by disabling User defined decision and entering the instant using the parameter Decision instant.

If the parameter User defined threshold is disabled, the threshold is calculated at the decision instant, by searching for the maximum eye opening. The threshold value will be at the center of the maximum eye opening. The user can disable the searching and enter directly the value of the threshold by disabling User defined threshold and entering the threshold using the parameter Absolute threshold.

The parameter Delay compensation allows the user to compensate the propagation delays of the input signal by enabling the parameter User defined delay. If the parameter User defined delay is disable, the delay will be estimated by comparing the input signal with a signal generated by the internal clock.





0 — [0,4999]

213

3R REGENERATOR

3R RegeneratorThis component regenerates an electrical signal.

Ports

Parameters

Main




Reference signal Output Electrical




Bit rate Bits/s Bits/sMBits/sGBits/s

[0,+INF[

User defined delayDefines whetherthe u;ser can define the delay compensation or not




User defined decisionDefines whether thecomponent will automatically calculate the decision instant or it will be defined by the user


Decision instantValue for the decision instant to use when recovering thebit sequence

0.5 Bit — [0,1]

User defined thresholdDefines whether the component will be automatically calculated or willbe user-defined


214

3R REGENERATOR


This component regenerates an electrical signal. It generates the original bit sequence, and a modulated electrical signal to be used for BER analysis. It is a subsystem based on the Data Recovery component and a NRZ Pulse Generator.

This first output port is the bit sequence, the second one is a modulated NRZ signal and the last output is a copy of the input signal. These three signals can be connected directly to the BER Analyzer, avoiding additional connections between transmitter and the receiver stage.

The following system shows a conventional connection between the BER Analyzer in the receiver stage with the transmitter stage, 2 additional connections are required between the transmitter and the BER Analyzer.

Absolute thresholdValue for the threshold to use when recovering the bit sequence

0.5 a.u. — ]-INF,+INF[


215

3R REGENERATOR

By using the 3R Regenerator, there is no need for connections between the transmitter and the BER Analyzer. This is especially important for WDM systems, where you have with multiple transmitters, receivers and BER Analyzers. For more information, see “WDM Transmitter” on page 107.

216

IDEAL FREQUENCY DEMODULATOR

Ideal Frequency DemodulatorConverts the received optical signal phase into electrical signal amplitude.

Ports

Parameters

Main

Downsampling


Input Input Optical



Min. amplitudeMinimum electrical signal amplitude at the output port

0 a.u. ]-INF,+INF[

Max. amplitudeMaximum electrical signal amplitude at the output port

1 a.u. ]-INF,+INF[


Centered at max powerDetermines whether the internal filter will be centered at the maximum amplitude of the signal or if it will be user-defined


Center frequencyUser-defined center frequency for the internal filter

193.1 THz Hz, THz, nm [30, 3e5]

Sample rateInternal filter bandwidth

5*(Sample rate) Hz Hz, GHz, THz, nm

[0,+INF[

217

IDEAL FREQUENCY DEMODULATOR

Polarization

Random numbers


The incoming optical signal and noise bins are filtered by an ideal rectangle filter to reduce the number of samples in the electrical signal. The new sample rate is defined by the parameter Sample rate. You can define the center frequency, or it can be calculated automatically by centering the filter at the optical channels with maximum power.

Optical noise bins are converted to gaussian noise inside the signal bandwidth. You must supply the polarization for the frequency extraction. The signal frequency is then normalized in the range between the parameters Min. and Max. amplitude.

Figure 1 Filtered signal

The converter resamples the signal and converts the noise bins. They are added in time domain.


PolarizationDetermines if the frequency from the polarization X or Y of the optical signal will be converted to amplitude

X — X, Y





0 — [0,4999]

218

IDEAL PHASE DEMODULATOR

Ideal Phase DemodulatorConverts the received optical signal phase into electrical signal amplitude.

Ports

Parameters

Main

Downsampling


Input Input Optical



Min. amplitudeMinimum electrical signal amplitude at the output port

0 a.u. [-1e+100, -1e+100]

Max. amplitudeMaximum electrical signal amplitude at the output port

1 a.u. [-1e+100, -1e+100]





193.1 THz Hz, THz, nm [30, 3e5]

Sample rateInternal filter bandwith

5*(Sample rate) Hz Hz, GHz, THz, nm

[0,+INF[

219

IDEAL PHASE DEMODULATOR

Polarization

Random numbers


The incoming optical signal and noise bins are filtered by an ideal rectangle filter to reduce the number of samples in the electrical signal. The new sample rate is defined by the parameter Sample rate. You can define the center frequency, or it can be calculated automatically by centering the filter at the optical channels with maximum power.

Optical noise bins are converted to gaussian noise inside the signal bandwidth. You must supply the polarization for the phase extraction. The signal phase is then normalized in the range between the parameters Min. and Max. amplitude.

Figure 1 Converted noise bins enabled

The converter resamples the signal and converts the noise bins. They are added in time domain.


PolarizationDetermines if the frequency from the polarization X or Y of the optical signal will be converted to amplitude

X — X, Y





0 — [0,4999]

220

PHOTODETECTOR PIN

Photodetector PINPIN photodiode.

Ports

Parameters

Main

Downsampling


Input Input Optical



Responsivity 1 A/W [0,100]

Dark current 10 nA [0,+INF[





193.1 THz Hz, THz, nm [30,3e5]

Sample rate 5*(Sample rate) Hz Hz, GHz, THz, nm

[1e-3,+INF[

221

PHOTODETECTOR PIN

Noise

Random numbers


Noise calculation type Numerical — — Analytical, Numerical, Numerical - Convert noise bins

Add signal-ASE noise True — — True, False

Add ASE-ASE noise True — — True, False

Add thermal noise True — — True, False

Thermal noise 0 W/Hz — [0,+INF[

Add shot noiseDetermines if shot noise is added to the signal


Shot noise distributionDetermines the distribution used to generate the shot noise

Gaussian — — Poisson, Gaussian





0 — [0,4999]

222

PHOTODETECTOR PIN

Technical backgroundThe incoming optical signal and noise bins are filtered by an ideal rectangle filter to reduce the number of samples in the electrical signal. The new sample rate is defined by the parameter Sample rate. You can define the center frequency, or it can be calculated automatically by centering the filter at the optical channel with maximum power.

If the noise calculation type in Numerical:

Optical noise bins are converted to Gaussian noise inside of the signal bandwidth. The combined optical field is then converted to optical power. If the option Numerical — Convert Noise Bins is selected, the output noise and signal are combined. This means that you cannot see the the separate contributions of the noise. However, if you select Numerical only, the signal and noise are separated and you can select the different contributions of the noise.

Figure 1 Convert noise bins enabled

The PIN resamples the signal and converts the noise bins when Convert Noise Bins is enabled.

Gaussian shot noise distribution

If the parameter Add shot noise is enabled and the Shot noise distribution parameter is Gaussian, the optical power is converted to electrical current by:

where is(t) is the optical signal calculated from the responsivity r:

where ith(t) is the thermal noise current calculated from the power spectral density defined by the parameter Thermal noise, and id is the dark current.

(1)

(2)

i t( ) is t( ) ith t( ) id ish t( )+ + +=

is t( ) rPs t( )=

223

PHOTODETECTOR PIN

The shot noise current ish(t) is calculated according to the power spectral density [1]:

Poisson shot noise distribution

If the parameter Add shot noise is enabled and Shot noise distribution parameter is Poisson, the electrical current is calculated according to [2]:

where ne denotes the number of electrons generated in the time instant ∆t. The average number of generated electrons (equal to the average number of detected photons) within the time interval is given by:

The number of generated electrons is the Poisson random variable with mean and variance equal .

If the noise calculation type is Analytical:

In this case, the signal and the noise components are calculated independently. The noise components are the variance and the noise PSD.

(3)

(4)

(5)

Nsh q is id+( )=

i t( ) qne

∆t-------- ith t( )+=

ne⟨ ⟩ ∆t

ne⟨ ⟩is t( )

q----------∆t id

q---∆t .+=

nene⟨ ⟩

224

PHOTODETECTOR PIN

Figure 2 Convert noise bins disabled

In Figure 2, the PIN resamples the signal and does not convert the noise bins if Convert Noise Bins is disabled.

225

PHOTODETECTOR PIN

)

)

)

The output electrical signal is:

Note: This signal does not include the noise components. The noise components are calculated by the noise variance and by the power spectral density.

For the noise variances:

where is the signal shot noise:

where is the electrical bandwidth.

and is the signal ASE beating:

For the noise PSD components:

where PTH(f) is the thermal noise and PASE-ASE(f) is the beating of ASE-ASE:

and the ASE shot noise is:

Defining sensitivity

The sensitivity of a receiver can be defined by optimizing the receiver parameters.

A typical way of doing this is to optimize the thermal noise in your receiver, to obtain a specific BER .

(6)

(7)

(8)

(9)

(10

(11

(12

i t( ) rP t( ) id+=

σ2 t( ) σsh2 t( ) σs ASE–

2 t( )+=

σsh2 t( )

σsh2 t( ) qis t( )Be=

Be

σs ASE–2 t( )

σs ASE–2 t( ) 4r2PASE t( )Ps t( )=

P f( ) PTH f( ) PASE ASE– f( ) PASEsh f( )+ +=

PASE ASE– f( ) r2 PASE f( )∗PASE f( )( )=

PASEsh f( ) qrPASE f( )Be=

1 10 9–×( )

226

PHOTODETECTOR PIN

References:[1] Agrawal, G.P., Fiber-Optic Communication Systems. John Wiley & Sons, New York, (1997).

[2] Jeruchim, M.C., Balaban, P., Shanmugan, K., Simulation of Communication Systems: Modeling, Methodology, and Techniques. Plenum Press, New York, (1997).

227

PHOTODETECTOR APD

Photodetector APDFilter with a square cosine roll off frequency transfer function.

Ports

Parameters

Main

Downsampling


Input Input Optical



GainAvalanche multiplication factor

3 — — [0,+INF[

Responsivity 1 A/W — [0,100]

Ionization ratioIonization factor

0.9 — — ]0,1]

Dark currentDark current amplified by the avalanche effect

10 nA — [0,+INF[





193.1 THz Hz, THz, nm [30,3e5]

Sample rate 5*(Sample rate) Hz Hz, GHz, THz, nm

[1e-3,+INF[

228

PHOTODETECTOR APD

Noise

Random numbers


The incoming optical signal and noise bins are filtered by an ideal rectangle filter to reduce the number of samples in the electrical signal. The new sample rate is defined by the parameter Sample rate. You can define the center frequency, or it can be calculated automatically by centering the filter at the optical channel with maximum power.

If the noise calculation type in Numerical:

Optical noise bins are converted to Gaussian noise inside of the signal bandwidth. The combined optical field is then converted to optical power. If the option Numerical — Convert Noise Bins is selected, the output noise and signal are combined. This means that you cannot see the the separate contributions of the noise. However, if you select Numerical only, the signal and noise are separated and you can select the different contributions of the noise.


Noise calculation type Numerical — — Analytical, Numerical, Numerical - Convert noise bins

Add signal-ASE noise True — — True, False

Add ASE-ASE noise True — — True, False

Add thermal noise True — — True, False

Thermal noise 0 W/Hz — [0,+INF[

Add shot noiseDetermines if shot noise is added to the signal


Shot noise distributionDetermines the distribution used to generate the shot noise

Gaussian — — [WMC, Gaussian]





0 — [0,4999]

229

PHOTODETECTOR APD

Figure 1 Convert noise bins enabled

The APD resamples the signal and converts the noise bins when Convert Noise Bins is enabled.

If the parameter Add shot noise is enabled and Shot noise distribution parameter is Gaussian, the optical power is converted to electrical current:

where is(t) is the optical signal calculated from the responsivity r and the gain M as:

and ith(t) is the thermal noise current calculated from the power spectral density defined by the parameter Thermal noise and id is the additive dark current.

The shot noise current ish(t) is calculated according to the power spectral density:

where idm is the dark current and F depends on M:

where k is the Ionization ratio.

(1)

(2)

(3)

(4)

i t( ) is t( ) ith t( ) id ish t( )+ + +=

is t( ) MrPs t( )=

Nsh t( ) qM2F rPs t( ) idm+( )=

F M( ) kM 2 1 M⁄–( ) 1 k–( )+=

230

PHOTODETECTOR APD

If the noise calculation type is Analytical:

In this case, the signal and the noise components are calculated independently. The noise components are the variance in time and the noise PSD.

Figure 2 Convert noise bins disabled

The PIN resamples the signal and does not convert the noise bins if Convert Noise Bins is disabled.

The output electrical signal is:

The noise variances are:

where is the signal shot noise:

where is the electrical bandwidth.

and is the signal ASE beating:

(5)

(6)

(7)

(8)

i t( ) rP t( ) id+=

σ2 t( ) σsh2 t( ) σs ASE–

2 t( )+=

σsh2 t( )

σsh2 t( ) qM2Fis t( )Be=

Be

σs ASE–2 t( )

σs ASE–2 t( ) 4r2M2PASE t( )Ps t( )=

231

PHOTODETECTOR APD

)

)

The noise PSD components are:

where PTH(f) is the thermal noise and PASE-ASE(f) is the beating of ASE-ASE:

and the ASE shot noise is:

Reference:

[1] Agrawal, G.P., Fiber-Optic Communication Systems. John Wiley & Sons, New York, (1997).

(9)

(10

(11

P f( ) PTH f( ) PASE ASE– f( ) PASEsh f( )+ +=

PASE ASE– f( ) r2M2 PASE f( )∗PASE f( )( )=

PASEsh f( ) qM2FrPASE f( )Be=

232

Amplifiers Library

This section contains information on the following amplifiers.

Optical

Raman

• Raman Amplifier• Raman Amplifier—Average power model• Raman Amplifier—Dynamic model

EDFA

• EDFA Black Box• EDF Dynamic — Full model• EDF Dynamic — Analytical model• EDFA• EDFA Ideal• EDFA Measured• Erbium doped fiber• Er-Yb codoped fiber• Er-Yb codoped waveguide amplifier

SOA

• Semiconductor Optical Amplifier

233

AMPLIFIERS LIBRARY

Electrical

• Limiting Amplifier• Electrical Amplifier• Transimpedance Amplifier• AGC Amplifier

234

RAMAN AMPLIFIER

Raman Amplifier

Fundamental model of a generic fiber Raman Amplifier. Its power and versatility allow it to represent virtually any type of Raman amplifier — employing any type of fiber, with arbitrary pump configuration and multiple sections — with a suitable combination of parameters.

Ports

Parameters

Fiber







Fiber length 10 km [0, INF]

Attenuation data type Constant — Constant, Wavelength Dependent /From File

Attenuation – constant 0.25 dB/km [0, INF]

Attenuation vs. wavelength AtnVsLambda.dat — [0, INF]

Forward Input Coupling Loss 1 dB [0, 106]

Forward Output Coupling Loss 0.022 dB [0, 106]

Backward Input Coupling Loss 1 dB [0, 106]

Backward Output Coupling Loss

0.022 dB [0, 106]

Effective area data type Constant — Constant, Wavelength Dependent/From File

Effective area – constant 72 µm2 [0, INF]

Effective area vs. wavelength EffAreaVsLambda.dat — [0, INF]

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RAMAN AMPLIFIER

Raman effect

Rayleigh effect

Reflections


Peak Raman gain coef 9.9e-14 m/W [0, INF]

Pump wavelength of peak Raman gain coef

1000 nm [0, INF]

Raman gain spectrum vs. freq. RamanGainVsFreq.dat — —

Raman gain polarization factor 0.5 — [0, INF]

Temperature 300 K [0, INF]

Name and description Default value Unit Value range

Rayleigh coef. data type Constant — Constant, Wavelength Dependent/From File

Rayleigh coef. — constant 5e-005 1/km [0, INF]

Rayleigh coef. vs. wavelength RayleighGainvsLambda.dat — [0, INF]


Left end reflection data type Constant — Constant, Wavelength Dependent/From File

Left end reflection — constant –30 dB [-INF, 0]

Left end reflection vs. wavelength

NearEndReflVsLambda.dat — [-INF, 0]

Right end reflection data type Constant — Constant, Wavelength Dependent/From File

Right end reflection — constant

–30 dB [-INF, 0]

Right end reflection vs. wavelength

FarEndReflVsLambda. dat — [-INF, 0]

236

RAMAN AMPLIFIER

Other nonlinearities

Effects on/off


Brillouin gain coef 5e-011 m/W [0, INF]

Brillouin bandwidth data type Constant — Constant, Wavelength Dependent/From File

Brillouin bandwidth — constant

40 MHz [0, INF]

Brillouin bandwidth vs. wavelength

FarEndReflVsLambda.dat — [0, INF]

Brillouin Stokes shift 11 GHz [0, INF]

Nonlinear refr. index data type Constant — Constant, Wavelength Dependent/From File

Nonlinear refr. index — constant

3e-020 m2/W [0, INF]

Nonlinear refr. index vs. wavelength

N2VsLambda.dat — [0, INF]

Raman-resonant n2 dispersion RealHiRezVsLambda.dat — [–INF, INF]

Eff. refr. index vs. wavelength EffRIVsLambda.dat — [0, INF]

Group velocity dispersion 5 ps/nm/km [0, INF]

Dispersion slope 0.1 ps/nm2/km [0, INF]

Name and description Value range Dependence

Attenuation ON [ON, OFF]

Rayleigh backscattering gain ON [ON, OFF]

SRS gain(Stimulated Raman scattering gain)

ON [ON, OFF]

SpRS gainSpontaneous Raman scattering gain)

OFF [ON, OFF]

Pump depletion in SRS ON [ON, OFF]

Double Rayleigh scattering OFF [ON, OFF]

Left end reflection OFF [ON, OFF]

Right end reflection OFF [ON, OFF]

Polarisation maintaining fiber OFF [ON, OFF]

237

RAMAN AMPLIFIER

Simulation details

Noises

SBS gain(Stimulated Brillouin scattering gain)

OFF [OFF]

Pump depletion in SBS OFF [OFF]

FWM(four-wave mixing)

OFF [OFF]

Nonlinear contribution to the phase-mismatch OFF [OFF]

Note: The last four effects will be implemented soon — they are currently disabled.


Enable ON — [ON, OFF]

Parameter set Default — Default, Auto, User

Upper Pump wavelength 1450 nm [0, INF]

Power accuracy 0.001 — [0, INF]

Max. number of iterations 100 — [1, 10000]

Number of power iterations 4 — [1, 10000]

ODE integration method 5th-order Runge-Kutta with step size control

— 5th-order Runge-Kutta with step size control, Gear's stiff eq. solver with step size control

ODE integrator accuracy 1e-006 — [0, 1]

Max. number of steps per iteration

100000 — [1, 10000]

Number of longitudinal points 256 — [10, 100000]

Background noise PSD level 1e-100 W/Hz [0, 10000]

Inphase noise ratio 0 — [0, 1]

Calculate 3D graphics ON — [ON,OFF]

3D graphics resolution 10 — [1, 100]

Name and description Default value Default unit Unit Value range

Noise center frequency 193.1 THz Hz, THz, nm [30, 3e+006]

Noise bandwidth 30 THz Hz, THz, nm [0, INF]

Name and description Value range Dependence

238

RAMAN AMPLIFIER

Random numbers

Results

Graphs

Noise bins spacing 1000 GHz Hz, GHz, THz, nm [0, INF]

Noise threshold –100 dB — [-INF,+INF]

Noise dynamic 3 dB — [0, INF]

Convert noise bins Convert noise bins — — [ON, OFF]


Unit Value range


ON — [ON,OFF]


0 — [0, 4999]


Unit Value range

Lower limit of Region of Interest 1550 nm [0, INF]

Upper limit of Region of Interest 1600 nm [0, INF]


Forward Output Power Spectrum [dBm] Wavelength [nm] Power [dBm]

Forward Output Gain [dB] Wavelength [nm] Gain [dB]

Forward Output OSNR [dB] Wavelength [nm] OSNR [dB]

Forward Output Multiple Rayleigh Scattering Spectrum [dBm]

Wavelength [nm] Power [dBm]

Backward Output Power Spectrum [dBm] Wavelength [nm] Power [dBm]

Backward Output Gain [dB] Wavelength [nm] Gain [dB]

Backward Output OSNR [dB] Wavelength [nm] OSNR [dB]

Backward Output Multiple Rayleigh Scattering Spectrum [dBm]

Wavelength [nm] Power [dBm]

Forward Power Spectrum [dBm] Wavelength [nm] Fiber Length [km]

Forward Gain [dB] Wavelength [nm] Fiber Length [km]


239

RAMAN AMPLIFIER

When a parameter is defined as a curve loaded from a file, the format of the file is:

The unit of the wavelengths is always [nm]. The units of the parameter values are given in the table above, and are the same as the units of the respective Constant parameter. Arbitrary number of points (file lines) are allowed, except 0 (empty file).

ResultsThe component produces the following results:

Maximum Forward Gain [dB]

Maximum Forward On/Off Gain [dB]

Forward Gain Flatness [dB]

Maximum Forward Power [dB]

Wavelength of Maximum Forward Power [dB]

Minimum Forward Effective Noise Figure [dB]

Forward Effective Noise Figure Flatness [dB]

Maximum Backward Gain [dB]

Maximum Backward On/Off Gain [dB]

Backward Gain Flatness [dB]

Forward Gain Coefficient [dB/km] Wavelength [nm] Fiber Length [km]

Forward OSNR [dB] Wavelength [nm] Fiber Length [km]

Forward Double Rayleigh Scatt. Spectrum [dBm] Wavelength [nm] Fiber Length [km]

Backward Power Spectrum [dBm] Wavelength [nm] Fiber Length [km]

Backward Gain [dB] Wavelength [nm] Fiber Length [km]

Backward Gain Coefficient [dB/km] Wavelength [nm] Fiber Length [km]

Backward OSNR [dB] Wavelength [nm] Fiber Length [km]

Backward Double Rayleigh Scatt. Spectrum [dBm] Wavelength [nm] Fiber Length [km]




......

Wavelength_N ParameterValue_N


240

RAMAN AMPLIFIER

Maximum Backward Power [dB]

Wavelength of Maximum Backward Power [dB]

Minimum Backward Effective Noise Figure [dB]

Backward Effective Noise Figure Flatness [dB]

These results are calculated for the wavelength range defined in the Results tab of the Component Properties dialog box.

Forward and Backward are names used to distinguish the characteristics pertaining to the left and right ends of the fiber respectively. They have nothing to do with the frequently used terms forward / (backward) Raman amplification, meaning amplifier configuration having co-propagating / (counter-propagating) pump and signals.

GraphicsThe Raman Amplifier presents the results of the calculations in a variety of both 2D and 3D graphics.

2D graphicsThe following 2D graphs are available:• Forward Output Power Spectrum [dBm]• Forward Output Gain [dB]• Forward Output On/Off Gain [dB]• Forward Output OSNR [dB]• Forward Double Rayleigh Scattering Spectrum [dBm]• Forward Eff. Noise Figure Spectrum [dB]• Backward Output Power Spectrum [dBm]• Backward Output Gain [dB]• Backward Output On/Off Gain [dB]• Backward Output OSNR [dB]• Backward Double Rayleigh Scattering Spectrum [dBm]• Backward Eff. Noise Figure Spectrum [dB]

3D graphicsThe following 3D graphs are available:• Forward Power Spectrum [dBm]• Forward Gain [dB]• Forward Gain Coefficient [dB/km]• Forward OSNR [dB]• Forward Double Rayleigh Scattering Spectrum [dBm]• Backward Power Spectrum [dBm]

241

RAMAN AMPLIFIER

• Backward Gain [dB]• Backward Gain Coefficient [dB/km]• Backward OSNR [dB]• Backward Double Rayleigh Scattering Spectrum [dBm]

2D/3D graphicsThe following 2D/3D graphics are available:• Forward Power Spectrum [dBm]• Forward On/Off Gain [dB]• Forward Gain [dB]• Forward Gain Coefficient [dB/km]• Forward OSNR [dB]• Forward Double Rayleigh Scattering Spectrum Power [dBm]• Forward Eff. Noise Figure [dB]• Backward Power Spectrum [dBm]• Backward On/Off Gain [dB]• Backward Gain [dB]• Backward Gain Coefficient [dB/km]• Backward OSNR [dB]• Backward Double Rayleigh Scattering Spectrum Power [dBm]• Backward Eff. Noise Figure [dB]

Forward and Backward are names used to distinguish the characteristics pertaining to the overall optical spectra propagating from the left end to the right end of the fiber respectively, and vice-versa. They have nothing to do with the frequently used terms forward / (backward) Raman amplification, meaning amplifier configuration having co-propagating / (counter-propagating) pump and signals.

242

RAMAN AMPLIFIER


IntroductionThe most promising technology to support almost unlimited bandwidth employs the nonlinear effect of stimulated Raman scattering (SRS) in hybrid EDFA + Raman fiber amplifiers (HRA) or purely Raman fiber amplifiers (RFA) [1,2]. The most important advantage of this effect is that the pump wavelength λp does not need to be tied to a particular energy level/absorption band, as it is in EDFAs. Raman amplification is readily obtainable in any spectral region and in any type of fiber, provided a practical pump source with wavelength 80-100 nm shorter than that of the signal and with sufficiently high power is available. Given the progress in the manufacturing of high-power pump lasers in the infrared [3] along with the seemingly limitless demand for amplification bandwidth, Raman amplification will play an increasingly important role in WDM networks.

SRS is among the best-understood third-order nonlinear processes, observed experimentally for the first time in 1962 in bulk media [4] and in 1972 in optical fibers [5]. It manifests itself as an exponential growth of a signal (Stokes) wave in the field of a shorter wavelength-intensive pump. As mentioned above, SRS is a non-resonant effect with respect to pump wavelength, which may lie anywhere in the transparency windows of the medium. On the other hand, the frequency difference ωp-ωs between the pump and signal waves should be resonant with one of the vibrational modes ωR of the host. SRS does not require phase-matching, and for CW pumps, it allows both forward (pump and signal co-propagating) and backward (counter-propagating) pumping configurations. The most important characteristics of SRS in telecom-grade fibers are [6 (and references)]:• The SRS gain spectrum peaks at 13.2 THz (≈ 100 nm at λp =1.55 µm), but

extends up to 30 THz.• The 3dB bandwidth of the gain spectrum is 6-7 THz (≈ 50 nm at λp =1.55 µm).• The peak gain gR

peak (λ) coefficient is 6.4x10-13 m/W for λp = 1.55 [µm], and is inversely proportional to λp.

• Both the shape of the spectrum and the value of gRpeak (λ) depend on the

concentration of the dopants; the peak gain coefficient of pure GeO2 is 8 times larger than that of fused silica. Figure 1 shows the zero temperature Raman gain coefficient spectra of pure fused silica, pure fused GeO2, and silica doped with 25 mol.% GeO2. The spectra are scaled to the peak gain coefficient of silica.

• The SRS effect is in principle highly polarization-dependent. Raman gain is negligible for orthogonal polarizations of the pump and signal. However, in non-polarization maintaining fibers, the gain becomes polarization independent due to mode-scrambling. In this case gR

peak(λ) is reduced by a factor of 2.

243

RAMAN AMPLIFIER

Figure 1 Zero temperature Raman coefficient spectra

The arbitrary choice of pump(s) wavelength(s) allows for a key new feature in all types of hybrid and Raman fiber amplifiers: the possibility to arrange several pumps in a finite pump band and to amplify the WDM signals in their extended aggregate gain spectrum. Gain-equalization is achieved by a proper choice of the wavelengths and powers of the individual pumps.

On the device level, the HRA and FRA come in a variety of configurations: backward-, forward- and bidirectionally pumped, discrete or distributed, single- or multi-stage. The ubiquitous nature of the Raman effect allows numerous types of fibers to be used as the SRS–active media — from standard transmission fibers in distributed FRA to short (5-8 km) DCFs or highly nonlinear heavily-doped fibers with small effective areas [9]. Typically, several hundred milliwatts of pump power are required.

The challenges in modeling and optimizing FRAs are related mainly to the nonlinear, inefficient nature of SRS, requiring high pump powers and long fibers, and to the different pump mechanism.• All participating optical waves interact with each other. The shorter wavelengths

transfer power to the longer wavelengths (all long wavelengths deplete all short wavelengths), resulting in a complex longitudinal distribution of gain coefficients and noise powers.

• Other third-order nonlinear processes among the pumps take place — SPM and XPM, FWM, and stimulated Brillouin scattering (SBS).

• Considerable noise powers and crosstalk are generated by multi-path Rayleigh scattering.

An additional challenge is the requirement to build a model that is both quantitatively and qualitatively precise. While the general features of any of the effects above are well known [6], it is the complex interplay of the details that matters if such a model is

244

RAMAN AMPLIFIER

to be used as a versatile design tool by the photonics industry. As a result, some of the simplifications (usually found in the literature) should be rejected:• The Raman spectrum of pure fused silica must be used with care. For discrete

FRAs, the magnitude and the spectrum of the Raman gain coefficient must always be defined in dependence on the concentration of the dopants [14]. The dispersion of the real part of the Raman-resonant nonlinear susceptibility must also be accounted for [15].

• The assumption that the fiber parameters, such as effective areas/overlap integrals, losses, and Brillouin gain bandwidth, are constants. In the wavelength region of 1.4-1.65 [µm], the effective areas of SMF-28TM and a typical DSF vary by 25% and 50% respectively.

The comprehensive model described here uses the unified spectral signal representation illustrated in Figure 2. It features arbitrary number and location of pumps, signals and ASE bands, and complete forward / backward symmetry. Each forward propagating wave has a backward counterpart at the same wavelength and vice-versa.

Figure 2 Unified spectral signal representation

245

RAMAN AMPLIFIER

Formulation of the modelAs an example, the power and the phase change of any type of wave (pump, signal, or ASE) with central carrier frequency ωk as PF,B(z, ωk) and θF,B (z, ωk) respectively, where the subscripts F and B discriminate against the forward and backward propagating waves at the same wavelength. The system of coupled differential equations describing the operation of a FRA or the Raman sub-unit of a HRA has the form:

(1)

dPF z ωk,( )dz

------------------------- α ωk( )PF z ωk,( )–=

ρ ωk( )PB z ωk,( )+

+ gR ωk ω1,( ) PF z ω1,( ) PB z ω1,( )+[ ] PF z ωk,( ) Psp ω1 ωk T Bk,,,( )+[ ]l k 1+=

N

∑

gR ωl ωk,( ) PF z ωl,( ) PB z ωl,( )+[ ]PF z ωk,( )l 1=

k 1–

∑–

2Ftotalsp ωk T,( )PF z ωk,( )–

+ BBr

BBr Bk+( )-------------------------gBrPB z ωk ∆ωBr+,( ) PF z ωk,( ) Psp ωk ∆ωBr ωk T Bk,,,+( )+[ ]

BBr

BBr Bk+( )-------------------------gBr PB z ωk ∆ωBr–,( ) Psp ωk ωk ∆ωBr T Bk,,–,( )+[ ]PF z ωk,( )–

gR ωk ωl ωm ωn,,,( ) Ψ z( )[ ] 4γ ωk ωl ωm ωn,,,( ) Ψ z( )[ ]sin–cos{ }n 1=

N

∑m 1=

N

∑l 1=

N

∑+

ωk ωl ωm ωn–+=

x PF z ωk,( )PF z ωl,( )( )PF z ωm,( )PF z ωn,( )

246

RAMAN AMPLIFIER

z( )]

(2)

The equations describing the evolution of and are obtained by alternative interchanging of subscripts F and B.

The notations are explained in Table 1.

Table 1 Description of notations

Notation Description

N Number or pumps+signals+ASE bands in each direction

2N Total number of interacting waves

Total losses

Rayleigh scattering coefficient

Raman gain coefficient

Peak Raman gain coefficient, depending on the frequency of the current pump wave. In fused silica, it is downshifted by

= 13.2 THz from the respective pump.

Normalized Raman gain spectrum of the fiber, as dependent on the type and concentration of the dopant.

Mode overlap integrals; for definitions see, for example [6] (chap. 7 and 10)

dθF z ωk,( )dz

------------------------- γ ωk ω, l ω, k ωl,( ) 2 δkl–( )PF z ωl,( ) 2PB z ωl,( )+[ ]l 1=

N

∑=

+ 2γ ωk ωl ωm ωn,,,( ) Ψ z( )[ ]gR ωk ωl ωm ωn,,,( )

2------------------------------------------ Ψ[sin+cos

n 1=

N

∑m 1=

N

∑l 1=

N

∑

ωk ωl ωm ωn–+=

XPF z ωl,( )PF z ωm,( )PF z ωn,( )

PF z ωk,( )--------------------------------------------------------------------

PB z ωk,( ) θB z ωk,( )

α ωk( )

ρ ωk( )

gR ωk ω1,( ) f ωk ω1,( )gpeakR ω1( )gnorm

R ω1 ωk–( )=

gpeakR ω1( )

∆ω ωR=

gnormR ∆ω( )

f ωl ωk,( ) f ωk ωl ωm ωn,,,( );

247

RAMAN AMPLIFIER

Power generated by spontaneous Raman and Brillouin scattering of the wave with carrier frequency into the bandwidth of the wave with carrier . Although the forms of these terms are identical, their values are different:

for SpRS, while for SpBS.

A factor (with dimension of length) determining the integrated total power lost by the current wave via spontaneous Raman scattering into all possible lower frequencies, as depending on the Raman spectrum and the temperature.

B Bandwidth of the respective wave.

h, K, T Planck bar constant, Boltzmann constant, Temperature.

gBr, BBr, Br Brillouin gain coefficient, line width, and Stokes shift.

.z Total phase difference between the nonlinearly mixed waves

Input phase mismatch

Kroneker delta

Notation Description

Psp ωl ωk T B,,,( ) =

2hωkB 1 1 e

h ω1 ωk–( )

KT---------------------------

1–⁄+

=

ω1ωk

Psp ω1 ωk T B,,,( ) 2hωkB≈Psp ω1 ωk T B,,,( ) 2hωkB»

Ftotalsp ωk T,( ) =

2πhωk gR ω ωk,( ) 1 1 e

h ωk ω–( )KT

-------------------------⁄ 1–+

ωd0

ωk

∫

∆ω

Ψ z( ) θ1 z( ) θm z( ) θn z( )– θk z( )– ∆k–+=

∆k

δlm

248

RAMAN AMPLIFIER

References[1] H. Masuda, S. Kawai, IEEE Photonics Technology Letters, Vol. 11, p. 647, 1999.

[2] T. Nielsen, P. Hansen, A. Stentz, M. Aquaro, J. Pedrazzani, A. Abramov, and R. Espindola, IEEE Photonics Technology Letters, Vol. 10, p. 1492, 1998.

[3] Laser Focus World, January 2000; SDL Press Release, http://www.sdli.com/investor/releases/19990630_BROADENS.html

[4] E. Woodbury and W. Ng, Proc. IRE, Vol. 50, p. 2347, 1962.

[5] R. Stolen, E. Ippen, and A. Tynes, Applied Physics Letters, Vol. 20, p. 62, 1972.

[6] G. Agrawal, “Nonlinear Fiber Optics,” 2nd Edition, Academic Press Inc., San Diego, California, 1995.

[7] F.L. Galeener, J.C. Mikkelsen Jr., R.H. Geils, and W.J. Mosby, Applied Physics Letters, Vol. 32, p. 34, 1978.

[8] Y. Emori, K. Tanaka, and S. Namiki, Electronics Letters, Vol. 35, p. 1355, 1999.

[9] T. Hosaka, S. Sudo, H. Itoh, and K. Okamoto, Electronics Letters, Vol. 24, p. 770, 1988.

[10] H. Kidorf, K. Rottwitt, M. Nissov, M. Ma, and E. Rabarijaona, IEEE Photonics Technology Letters, Vol. 11, p. 530, 1999.

[11] M. Nissov, K. Rottwitt, H. Kidorf, and M. Ma, Electronics Letters, Vol. 35, p. 997, 1999.

[12] Y. Chen, Journal of the Optical Society of America, Vol. B7, p. 43, 1990.

[13] B. Foley, M. Dakss, R. Davies, and P. Melman, Journal of Lightwave Technology, Vol. 7, p. 2024, 1989.

[14] S. Davey, D. Williams, B. Ainslie, W. Rothwell, and B. Wakefield, IEE Proceedings, Vol. 136, p. 301, 1989.

[15] R. Hellwarth, Progress of Quantum Electronics, Vol.5 , p. 1, 1977.

[16] Y. Shen, “The Principles of Nonlinear optics,” J. Wiley & Sons Inc., 1984.

[17] A. Uchida, M. Takeoka, T. Nakata, and F. Kannari, Journal of Lightwave Technology, Vol. 16, p. 92, 1998.

[18] S. Evangelides, L. Mollenauer, J. Gordon, and N. Bergano, Journal of Lightwave Technology, Vol. 10, p. 28, 1992.

249

RAMAN AMPLIFIER

Notes:

250

RAMAN AMPLIFIER—AVERAGE POWER MODEL

Raman Amplifier—Average power model

This component simulates a Raman amplifier based on the average power approach [1], [2].

Ports

Parameters

Main






Name and description Symbol

Default value Default unit Value range

LengthAmplifier length.

L 10 km ]0; 1,000,000[

Attenuation data typeDefines if attenuation is entered as scalar, used for all wavelengths, or if it is wavelength dependent / downloaded from a file.

Constant — Constant, From File

AttenuationConstant attenuation value

0.2 dB/km [0,+INF[

Attenuation fileAttenuation value dependent on wavelength.

- FiberLoss.dat nm - dB/km -

Effective area data typeDefines if effective area is entered as scalar, used for all wavelengths, or if it is wavelength dependent/ downloaded from a file.

- Constant — Constant, From File

α

251


Enhanced

Effective interaction areaConstant effective area.

72 µm2 [0, INF[

Effective interaction area fileEffective area dependent on wavelength.

- EffectiveArea.dat nm - µm2 -

Raman gain typeDefines type of Raman gain. If Raman gain efficiency is selected, effective area is disabled, and value is

. Otherwise, it is normalized multiplied by Raman gain peak

(see below).

- Raman gain - Raman gain, Raman gain efficiency

Raman gain peakNormalized Raman gain is multiplied by Raman gain peak. Formula is detailed later in this section.

- 1e-013 - [0,+INF[

Raman gain reference pumpValue used for Raman gain calculation. Formula is detailed later in this section.

- 1000 nm [0,+INF[

Gain X frequencyFile that defines Raman gain or the Raman gain efficiency.

RG.dat THZ - normalized Raman gain

-



TemperatureAbsolute temperature at which fiber is operating. Used for noise consideration.

T 300 K [0,500]

Polarization factorActual value depends on relative polarization of fields of channels i and j. Equals 1 if fields of both channels are polarization-aligned, and 2 for totally scrambled polarization [4].

2 - [1,2]



Aeff

gr / Aeffgr

gr

Keff

252


Numerical

Rayleigh back scattering data typeDefines whether Rayleigh back scattering coefficient is entered as scalar, used for all wavelengths, or wavelength dependent/downloaded from a file.

- Constant - Constant, From File

Rayleigh back scattering

Constant Rayleigh back scattering.

- 5.0e-005 1/km [0, INF[

Rayleigh back scattering file

Rayleigh back scattering dependent on wavelength.

- Rayleigh.dat nm - 1/km -

Upper pump reference

Used for convergence test. All wavelengths below this value are considered pump, and are not taken into account for the convergence test.

- 1450 nm [0,3000]

Enable dispersion

Enables the linear chromatic dispersion application for the signals.

- False - True, False

Dispersion

Value of the GVD (Group Velocity Dispersion) parameter in wavelength domain.

- 16.75 ps/nm/km ]-INF,+INF[

Dispersion slope

Value of the dispersion slope parameter.

- 0.075 ps/nm2/km ]-INF,+INF[


Used internally as “zero” (or reference) frequency in spectrum of signal envelope. Attenuation value is assumed to correspond to this frequency.

- 1550 nm [100,2000]


ToleranceUsed to check convergence of the model. Based on gain of the signals.

0.01 - ]0,+INF[



γ

253


Graphs

Simulation

Number of divisionsNumber of divisions (in space) of the fiber.

50 - [1;50,000]

Number of iterationsMaximum number of iterations executed. If convergence is not reached in this number of iterations, model returns the calculated values anyway.

50 - [1;50,000]

Check convergence using:Defines if convergence is checked using “All signals” or “First signal”.

All signals - All signals, First signal


Unit Value range

Calculate graphsDefines if graphs are calculated or not. If False, component graphs are not represented.

False - True, False


Number of distance steps considered for graph generation.

20 - [1,1e8]

Number of wavelength steps

Number of wavelength steps considered for graph generation.

20 - [1,1e8]

Linear scale

Defines if a linear scale (Watt) or logarithmic one (dBm) is used.

True - True, False

Minimum value

If a logarithmic scale is used, this parameter defines the minimum value for the power that is displayed on the graph.

-50 dBm ]-INF,+INF[


EnabledDefines whether the component is enabled or not.

True - True, False


254


Noise

Random numbers

Technical BackgroundIn recent years, Raman amplifiers have become one of the most promising technologies for the next generation of fiber amplifiers, mostly due to their flexibility in bandwidth design.

Nevertheless, the simulation techniques that are commonly used for RA's have demanded exhaustive computational time, mainly due to the use of direct integration of the coupled differential equations that describe the RA behavior [3].

The coupled differential equations have the shape observed in Equation 1. A similar set of equations, describing the backward propagation, is solved at the same time we solve the forward equations written below.


Noise center frequencyDetermines noise center frequency.

193.4 THz Hz, THz, nm [30, 30e5]

Noise bandwidthBandwidth to create noise bins.

13 THz Hz, THz, nm ]0,+INF[

Noise bins spacingSpecifies the noise bins spacing.

125 GHz Hz, GHz, THz, nm [1,1000[

Noise thresholdMinimum value for adaption of noise bins.

-100 dB — ]-INF,0[

Noise dynamicThreshold ratio for adaption of noise bins.

3 dB — [0,+INF[

Convert noise binsDetermines if generated noise bins are incorporated into the signal.

Convert noise bins — — True, False


Unit Value range




0 — [0, 4999]

255


where

In the equations, the following physical effects were taken into account:• pump-to-pump, signal-to-signal, and pump-to-signal Raman interactions• spontaneous Raman emission and its temperature dependency

(1)

Symbol Definition

frequencies (Hz)

fiber attenuation [N/m]

Rayleigh back scattering coefficient [N/m]

Raman gain coefficient for frequency difference ( ) [m/W]

backward propagating power [W]

effective core area [m2]

polarization factor

frequency interval

Plank’s constant

Boltzman’s constant

temperature [K]

dPf z υ,( )dz

---------------------- α υ( )Pf z υ,( ) γ υ( )Pb z υ,( ) ++=

Pf z υ,( )gr υ ζ–( )KeffAeff

---------------------- Pf z ζ,( ) Pb z ζ,( )+[ ] +v ζ<∑

h∆υgr υ ζ–( )

Aeff---------------------- Pf Pb+[ ] 1 h ζ υ–( ) kT⁄[ ] 1–( ) 1–exp+[ ] –

v ζ<∑


----------------------υζ--- Pf z ζ,( ) Pb z ζ,( )+[ ] –

v ζ>∑

2hυ∆υPf z υ,( )gr υ ζ–( )

Aeff---------------------- 1 h υ ζ–( ) kT⁄[ ] 1–( ) 1–exp(+[ ]

v ζ>∑

υ ζ,α υ( )γ υ( )

gr υ ζ–( )υ ζ–( )

Pb z υ,( )

AeffKeffδυhkT

256


• stimulated Raman scattering• pump depletions due to Raman energy transfer• high-order Stokes generation• multiple Rayleigh back scattering• fiber loss• spontaneous emission noise

A very interesting approach that considerably reduces the computational time for simulating RA is the one used for this component. The idea behind this technique is first to split the amplifier into a concatenation of small segments, and then to use the small-signal-traveling wave solution in each section (see Equation 3). In order to eliminate the dependence in a small segment length, average powers in each section are introduced (see Equation 4). So, basically, we rearrange some terms of the original Equation 1 and reduce the propagation equations to a simpler form.

This new form, suitable for the purpose of average power analyses, can be written as [2]:

if we substitute , , in (2a), (2b) in each lump by average powers in the lump,coefficients , are independent of (within the lump, , and the solution of Equation 2 can be written as:

where is the length of the lumps. Within each lump, powers , must be replaced by average powers

(2)

(2a)

(2b)

(3)

(4)

z

dPf z v,( )dz

--------------------- A z v,( )Pf z v,( ) B z v,( )+=

where

A z v,( ) α– υ( )gr υ ζ–( )KeffAeff

---------------------- Pf z ζ,( ) Pb z ζ,( )+[ ] υζ---

gr υ ζ–( )KeffAeff

---------------------- P[ f z ζ,( ) Pb z ζ,( ) ] +v ζ>∑–

v ζ<∑+=

2– hυ∆υgr υ ζ–( )

Aeff---------------------- 1 1

h υ ζ–( ) kT⁄[ ] 1– 1–exp-----------------------------------------------------------+

v ζ>∑

B z υ,( ) γ υ( )Pb= z υ,( ) hυ∆υgr υ ζ–( )

Aeff---------------------- Pf z ζ,( )[ Pb z ζ,( ) ] 1 1

h υ ζ–( ) kT⁄[ ] 1– 1–exp-----------------------------------------------------------++

v ζ<∑+

Pf z ζ,( ) Pb z ζ,( )

A z v,( ) B z v,( ) z A υ( ) B υ( )

Pf z0 H υ,+( ) Pf z0 υ,( ) A υ( )H( ) B υ( )A υ( )------------ A υ( )H( ) 1–( )exp[ ]+exp=

H Pf z ζ,( ) Pb z ζ,( )

Pf b, v( )⟨ ⟩ Pf b,in G 1–

1nG------------- B v( )

A v( )----------- G 1–

1nG------------- 1–+=

257


where are forward and backward propagating input powers to the lump, .

The user is responsible to guarantee that the term does not become zero. For example, it is impossible to simulate the chromatic dispersion of just one signal if the attenuation is not considered, once the term will become zero.

Numerical approachThe relaxation method is used in order to satisfy the boundary conditions of the two-point boundary problem with given accuracy.

There are two different iteration procedures, for both forward and backward directions. Forward direction is from Input port 1 to Output port 1, and backward is from Input port 2 to Output port 2.

The first procedure, the innermost one, is intended to evaluate the self-consistent convergence for the average powers used in Equation 4 for every amplified segment. When a certain tolerance is reached (10-12), the average powers are considered good enough to be used as an approximation of the desired functions.

In the outermost one, or second procedure, the convergence is checked after the integration in forward direction is performed. If the variance in the gain is less than the tolerance desired (see “Numerical” on page 253) , the simulation is considered finished. Otherwise, the component runs for the maximum number of iterations set by the user.

The reason for the reduction in computational time is that direct numerical integration of Equation 1 is replaced by algebraic operations.

The user can choose the signals that will be used in the convergence checking. There are two available choices: All signals and First signal. When the First signal option is chosen, just the signal with the smallest wavelength is used in checking the convergence by the given tolerance. Otherwise, if the All signals option is chosen, all signals are used in the checking. In the case where there a signal has not been transmitted, the convergence test is performed based on the pumps.

FilesSome data necessary for this model may be downloaded from a file. In general, these files are in the ASCII format and follow Optiwave's standard format. For clarity, the units of each column in the files are listed in the following table.

Field First column Second column

Attenuation Wavelength (nm) Attenuation (dB/km)

Effective area Wavelength (nm) Effective area (µm2)

Pf b,in

G A υ( )H( )exp=A v( )

A v( )

258


When a file with the normalized Raman gain is used, it must be provided values for the Raman gain peak and Raman gain reference pump to use in the calculation of the Raman gain used in the simulation. The following formula is used:

where is the Raman gain, is the Raman gain peak, is the gain reference pump and is the normalized Raman Gain.

The unit of Raman gain is given in .

ComparisonAs stressed in the beginning of the technical description, the average power model is intended to decrease the computational time required to solve the Raman Amplifier differential equations by simplifying the way the equations are written.

In fact, the model shows a reduction in computation time of over two orders of magnitude [2] compared to the model using direct integration approach (fourth-order Runge-Kutta). However, in some cases, it is known that the model fails in converging (for example, when the total pump becomes very high).

Therefore, based on the characteristics presented, this model is very useful in getting a first approximation for a network under certain limits. Once the rough estimation is reached, the system could be generalized using the full steady state model.

A validation example for this model is presented in Lesson: "Raman amplifier - Average power model" in the tutorials section.

Raman gain X frequency Frequency shift (THz) Normalized Raman gain

Raman gain efficiency X frequency

Frequency shift (THz) Raman gain efficiency

Rayleigh back scattering Wavelength (nm) Back scattering (1/km)


m2

W------

1W m⋅-------------

gRPRλp------ gN=

gR PR λpgN

mW-----

259


References:[1] M. Karasek, M. Menif, "Protection of surviving channels in pump-controlled gain-locked Raman

fibre amplifier", Optics Communications 210 (2002) 57-65.

[2] B. Min, W. J. Lee, N. Park, "Efficient Formulation of Raman Amplifier Propagation Equations with Average Power Analysis", IEEE Photonics Technology Letters, Vol. 12, No. 11, November 2000.

[3] E. Desurvire, "Erbium-doped fiber amplifiers: principles and applications", Wiley-Interscience, 1994.

[4] S. Tariq, J.C. Palais, "A Computer Model of Non-Dispersion-Limited Stimulated Raman Scattering in Optical Fiber Multiple-Channel Communications", IEEE Journal of Lightwave Technology, Vol. 11, No. 12, December 1993.

260

RAMAN AMPLIFIER—DYNAMIC MODEL

Raman Amplifier—Dynamic model

This component simulates a Raman amplifier using a dynamic model based on direct integration of the differential equations that describe it.

Ports

Parameters

Main








LengthAmplifier length.

L 10 km ]0; 1,000,000[

Attenuation data typeDefines if attenuation is entered as scalar, used for all wavelengths, or if it is wavelength dependent / downloaded from a file.

— Constant — Constant, From file

AttenuationConstant attenuation value

0.2 dB/km [0,+INF[

Attenuation fileAttenuation value dependent on wavelength.

— FiberLoss.dat nm - dB/km —

α

261


Enhanced

Effective area data typeDefines if effective area is entered as scalar, used for all wavelengths, or if it wavelength dependent/downloaded from a file.


Effective interaction areaConstant effective area.

72 µm2 ]0, +INF[

Effective interaction area fileEffective area dependent on wavelength.

— EffectiveArea.dat nm - µm2 —

Raman gain typeDefines type of Raman gain. If Raman gain efficiency is selected, effective area is disabled, and value is

. Otherwise, is is normalized multiplied by Raman gain peak

(see below).

— Raman gain — Raman gain, Raman gain efficiency

Raman gain peakNormalized Raman gain is multiplied by Raman gain peak. Formula is detailed later in this section.

— 1e-013 — [0,+INF[

Raman gain reference pumpValue used for Raman gain calculation. Formula is detailed later in this section.

— 1000 nm [0,+INF[

Gain X frequencyFile that defines Raman gain or the Raman gain efficiency.

RG.dat THZ - normalized Raman gain

—



TemperatureAbsolute temperature at which fiber is operating. Used for noise consideration.

T 300 K [0,500]



Aeff

gr / Aeffgr

gr

262


Polarization factorActual value depends on relative polarization of fields of channels i and j. Equals 1 if fields of both channels are polarization-aligned, and 2 for totally scrambled polarization [4].

2 — [1,2]

Rayleigh back scattering data typeDefines whether Rayleigh back scattering coefficient is entered as scalar, used for all wavelengths, or wavelength dependent/downloaded from a file.


Rayleigh back scattering

Constant Rayleigh back scattering.

- 5.0e-005 1/km [0, +INF[

Rayleigh back scattering file

Rayleigh back scattering dependent on wavelength.

— Rayleigh.dat nm - 1/km —

Upper pump reference

Used for convergence test. All wavelengths below this value are considered pump, and are not taken into account for the convergence test.

— 1450 nm [0,3000]

Enable dispersion

Enables the linear chromatic dispersion application for the signals.

— False — True, False

Dispersion

The value of the GVD (Group Velocity Disperion) parameter in the wavelength domain.

— 16.75 ps/nm/km ]-INF, +INF[

Dispersion slope

The value of the dispersion slope parameter.

— 0.075 ps/nm2/km -INF, +INF[


This value is used internally as a “zero” or reference frequency in the spectrum of the signal envelope. The attenuation value is assumed to correspond to this frequency.

— 1550 nm [100, 2000]



Keff

γ

263


Numerical

Graphs

Group delay data type

Defines if the group delay is entered as a scalar used for all wavelengths, or if it wavelength dependent/entered from a file.


Group delay

Constant group delay

1/Vg(v) 4900000 ps/km [0, 1010]

Group delay file — GroupDelay.dat ns—ps/km —


ToleranceUsed to check convergence of the model. Based on gain of the signals.

0.01 — ]0,+INF[

Number of divisionsNumber of divisions (in space) of the fiber.

50 — [1;50,000]

Number of iterationsMaximum number of iterations to be executed. If convergence is not reached in this number of iterations, model returns the calculated values regardless.

50 — [1;50,000]

Check convergence using:Defines if convergence is checked using “All signals” or “First signal”.

All signals - All signals, First signal


Unit Value range

Calculate graphsDefines if graphs are calculated or not. If False, component graphs are not represented.

False - True, False


Number of distance steps considered for graph generation.

20 - [1,1e8]

Number of wavelength steps

Number of wavelength steps considered for graph generation.

20 - [1,1e8]



264


Simulation

Noise

Random numbers

Linear scale

Defines if a linear scale (Watt) or logarithmic one (dBm) is used.

True - True, False

Minimum value

If a logarithmic scale is used, this parameter defines the minimum value for the power that is displayed on the graph.

-50 dBm ]-INF,+INF[


EnabledDefines whether the component is enabled or not.

True - True, False


Noise center frequencyDetermines noise center frequency.

193.4 THz Hz, THz, nm [30, 30e5]

Noise bandwidthBandwidth to create noise bins.


Noise bins spacingSpecifies the noise bins spacing.

125 GHz Hz, GHz, THz, nm [1,1000[

Noise thresholdMinimum value for adaption of noise bins.

-100 dB — ]-INF,0[

Noise dynamicThreshold ratio for adaption of noise bins.

3 dB — [0,+INF[

Convert noise binsDetermines if generated noise bins are incorporated into the signal.

Convert noise bins — — True, False


Unit Value range




Unit Value range

265


Technical backgroundIt is widely believed that Raman Amplifier (RA) will play an increasing role in future optical fiber communication (OFC) systems [1]. They have already been widely utilized due to their fundamental advantages [1]:• there is amplification at any wavelength, provided the appropriate pump sources

are available.• a fiber itself can be used as an active medium.• a pump spectrum determines a gain spectrum.

The equations that describe a Raman amplifier are [2]:

where


0 — [0, 4999]

(1)

Symbol Definition

frequencies (Hz)

fiber attenuation [N/m]

Rayleigh back scattering coefficient [N/m]


Unit Value range

dPf z υ,( )dz

---------------------- α υ( )Pf z υ,( ) γ υ( )Pb z υ,( ) ++=


---------------------- Pf z ζ,( )[ ] Pb z ζ,( ) ++v ζ<∑

h∆υgr υ ζ–( )

Aeff---------------------- Pf z ζ,( )[ ] 1 h ζ υ–( ) kT⁄[ ] 1–( ) 1–exp+[ ] –

v ζ<∑


----------------------υζ--- Pf z ζ,( )[ ] Pb z ζ,( ) –+

v ζ>∑

2hυ∆υPf z υ,( )gr υ ζ–( )

Aeff---------------------- 1 h υ ζ–( ) kT⁄[ ] 1–( ) 1–exp(+[ ]

v ζ>∑

υ ζ,α υ( )γ υ( )

266


In these equations, the following physical effects were taken into account:• pump-to-pump, signal-to-signal and pump-to-signal Raman interactions;• spontaneous Raman emission and its temperature dependency;• stimulated Raman scattering;• pump depletions due to Raman energy transfer;• high-order stokes generation;• multiple Rayleigh backscattering;• fiber loss;• spontaneous emission noise.

In this component, the equations in Equation 1 (forward and backward) are solved spatially through direct integration using a standard classical fourth-order Runge-Kutta formula without adaptive step size [3].

Numerical approachThe convergence of the model is checked in two directions: forward and backward. An iterative forward and backward integration of propagation equations must be applied because backward propagating ASE powers and a counter-directional pumping scheme may be defined, and the possibility of counter directional signal propagation [2]. The forward direction is from Input Port 1 to Output Port 1 and backward is from Input Port 2 to Output Port 2.

The iterative scheme is started with a forward integration of forward signals, propagating ASE spectral components, and pumps. The backward pumps and backward ASE powers are set to zero. At each backward integration, the final results

of the previous forward integration, together with the boundary conditions for the backward pump, backward ASE powers, and backward signals, are used as starting conditions.

Raman gain coefficient for frequency difference ( ) [m/W]

backward propagating power [W]

effective core area [m2]

polarization factor

frequency interval

Plank’s constant

Boltzman’s constant

temperature [K]

gr υ ζ–( )υ ζ–( )

Pb z υ,( )

AeffKeffδυhkT

P+ z( L υ ),=

267


Similarly, the results of the previous backward integration together with the boundary conditions for forward signal channels, pumps, and forward ASE, are used as starting conditions for each forward integration [2].

The convergence checking is done after integration in the forward direction is complete. If the variance in the gain is less than the tolerance desired (see “Numerical” on page 264) , the simulation is considered complete. Otherwise, the component runs for the maximum number of iterations set by the user.

The user can choose the signals that will be used in the convergence checking. There are two available choices: All signals and First signal. When the First signal option is chosen, just the signal with the smallest wavelength is used in checking the convergence by the given tolerance. Otherwise, if the All signals option is chosen, all signals are used in the checking. In the case where there a signal has not been transmitted, the convergence test is performed based on the pumps.

After the spatial integration is complete, the time evolution of pumps, signals, and amplified spontaneous emission waves is performed by direct integration with Equation 1, starting with the steady-state solution for longitudinal distribution of individual powers along the Raman fiber. To avoid possible oscillations of the solution in time domain, care must be taken in the selection of bin widths used in space ( ), and time ( ) discretization schemes. Stable solutions has been obtained when the time bin ( ) is equal to or less than the propagation time through a space bin .

In order to determine the rise/fall times of the surviving channel power transients with sufficient resolution, the ratio of time and space bins should be independently kept for the Raman fiber length, as in the examples.

Some data necessary for this model may be downloaded from a file. In general, these files are in the ASCII format and follow Optiwave's standard format.

For clarity, the units of each column in the files are listed in the following table.


Attenuation Wavelength (nm) Attenuation (dB/km)

Effective area Wavelength (nm) Effective area (µm2)

Raman gain X frequency Frequency shift (THz) Normalized Raman gain

Raman gain efficiency X frequency

Frequency shift (THz) Raman gain efficiency

P_ z( 0 υ ),=

∆z∆t

∆t∆t ∆z Vg⁄≤

∆t ∆z⁄ 4 10 9–× s m⁄[ ]=

m2

W------

1W m⋅-------------

268


When a file with the normalized Raman gain is used, it must be provided values for the Raman gain peak and Raman gain reference pump to use in the calculation of the Raman gain used in the simulation. The following formula is used.

where is the Raman gain, is the Raman gain peak, is the gain reference pump and is the normalized Raman Gain.

The unit of Raman gain is given in .

Rayleigh back scattering Wavelength (nm) Back scattering (1/km)


gRPRλp------ gN=

gR PR λpgN

mW-----

269


References:[1] E. M. Dianov, "Advances in Raman Fibers", Journal of Lightwave Technology, Vol. 20, No. 8,

August 2002.

[2] M. Karasek, M. Menif, "Protection of surviving channels in pump-controlled gain-locked Raman fibre amplifier", Optics Communications 210 (2002) 57-65.

[3] W. H. Press, et al., "Numerical Recipes: The Art of Scientific Computing", 2nd Edition, Cambridge University Press, 1992.

[4] S. Tariq, J.C. Palais, "A Computer Model of Non-Dispersion-Limited Stimulated Raman Scattering in Optical Fiber Multiple-Channel Communications", IEEE Journal of Lightwave Technology, Vol. 11, No. 12, December 1993.

270

EDFA BLACK BOX

EDFA Black Box

Designs erbium doped fiber amplifiers (EDFAs) pumped by 980 nm or 1480 nm. Requires just the experimental characterization of a practical device such as the gain spectrum and noise figure under non-saturated and saturated conditions. Details about erbium-doped fiber specifications and elements in the layout are not required to perform the simulations.

The amplifier is specified to operate under conditions required by wavelength division multiplex (WDM) systems.

Ports

Parameters

Main


Input Input Optical




Operation modeAmplifier operation mode

Saturation — — Gain control, Power control, Saturation

GainDetermines the signal gain

20 dB — [0,100]

PowerDetermines the signal output power

5 dBm W, mW, dBm [-100,100]

271

EDFA BLACK BOX

Measurements

Numerical

Polarization


Units Value range

File wavelength unitDetermines the wavelength unit

m — nm, m, Hz, THz

First gain spectrum file nameFilename with the gain spectra measurements

Gain1.dat — —

Second gain spectrum file nameFilename with the gain spectra measurements

Gain2.dat — —

OSA bandwidthSet the bandwidth of the Optical Spectrum Analyzer

0.1 nm [0.0001,10]

Noise typeSelect the noise type

Power dBm Power, Spectral density, Noise figure

Noise spectrum file nameFilename concerning the noise spectra

Noise.dat — —

Saturation wavelengthDetermines the saturation wavelength

1540 nm [800,1700]

Saturation file nameFilename concerning the saturation spectra

Saturation.dat — —


Units Value range

Relative errorDetermines the relative error acceptable in each calculation

0.1 dB ]0,100]

Interpolation algorithmDetermines the interpolation algorithm for the measured data

Cubic — Linear, Cubic


Units Value range



272

EDFA BLACK BOX

Simulation

Noise

Random numbers


Units Value range





Noise bins spacingSpecifies the noise bins spacing


[1,1000]


–100 dB — ]-INF,0[


3 dB — [0,+INF]


Convert noise bins

— — True, False


Units Value range


Determines if the seed is automatically defined and unique


Random seed index

User-defined seed index for noise generation

0 — [0,4999]

273

EDFA BLACK BOX

Graphs

Technical backgroundUsing input parameters as the characterization of the gain spectrum and noise figure under non-saturated and saturated conditions measured to a practical device is sufficient for designing the amplifier performance using this black box model. Additional information concerning fiber specifications, such as absorption and emission cross-sections, erbium doping, core radius, or details related to the amplifier layout, is not required in this case. Curves containing the amplifier characterization are available internally as a default in the software, which makes it possible for you to perform simulations.

Three different control modes are available that allow you to perform the amplifier analysis under distinct points of view. Each mode control – gain, power control, and saturation – defines a different amplifier operating condition.

The black box model enables passive components, such as optical isolators, equalizer filters, and several types of couplers, to be included in the amplifier design while considering an additional loss variation along the amplifier due to the active and passive components. The gain and the noise characterization measured to distinct states of operation are provided as input files that enable the complete modelling of the amplifier performance.

Operation Mode descriptionFiber amplifiers used in WDM systems usually make use of control systems as power or gain. It is useful to include the option in the EDFA Black Box to select one of three operation modes:• Gain mode: Based on a control of the amplifier gain (Gain Control) relating the

input and output signals (with or without the generated amplified spontaneous emission (ASE).

• Power Control mode: Considers the value of the output power to control the amplifier performance (Power Control).

• Saturation mode: Considers the amplifier operating in a saturated condition (Saturation).


First gain spectrum Wavelength (m) Gain (dB)

Second gain spectrum Wavelength (m) Gain (dB)

Noise spectrum Wavelength (m) Power (dBm)

Saturation Input power (dBm) Gain (dB)

274

EDFA BLACK BOX

Gain Control modeIn this mode, you set the gain amplifier (Gain in dB units). The gain calculation is performed using Equation 1, where the total input (Pin) and the output (Pout) power spectra are considered. The specified amplifier gain (Gspeci) is given by the ratio of the total output power and input total power, with or without the generated ASE.

SASE(f) represents the spectral density of the amplified spontaneous emission integrated on the optical frequency f.

Note: You can include the noise by selecting the noise type as power, spectral density, or noise figure in the EDFA Black Box Properties dialog box.

Power Control modeThe value that you define in the power control mode is the desired amplifier output power (Power in dBm units). The specified amplifier output power (Pspeci) that includes the spectral ASE is:

where is the amplifier gain.

The option to select the noise type that will be included in the simulations is also available in this control mode.

Saturation modeIn the saturation mode, the gain is the specified parameter. The noise type can be selected in this mode, and two experimental gain curves are inserted as input files considering two different saturation conditions. The gain curve in a saturated condition is provided in a file format containing two columns. The first column refers to the signal output power given in [dBm] units. The second column gives the gain in [dB] units.

(1)

(2)

Gspeci

Pout λ( ) SASE f( ) fd∞–

+ ∞

∫+λ∑

Pin λ( )λ∑

---------------------------------------------------------------=

Pspeci G Pin λ( ) SASE f( )

∞–

+ ∞

∫–λ∑× df=

G

275

EDFA BLACK BOX

An example of the saturated gain input file is:

where the signal output power is given in [dBm] units and the gain is in [dB] units. There is no limit of rows or power spacing previously defined.

Basic equationsThe black box model considers a two-level Er3+ system assumption that is usually adopted to model erbium-doped fiber amplifiers [1]. The propagating equation written as a function of the absorption and emission coefficients, α(λ) and γ(λ) respectively, is [2]:

I(z) represents the fraction of active ions in the excited state, P(λ,z) describes the propagating power at a specific wavelength and fiber position, and PASE

eq is the term that includes the amplified spontaneous emission (ASE) as an equivalent ASE power.

The solution to Equation is:

where L is the total Er-doped fiber length and P(λ,0) represents the power at the wavelength λ and at the fiber input. Considering the scope of this approximation, PASE

eq(λ) works as an independent source of amplified spontaneous emission.

The total gain along the erbium-doped fiber is:

where is the updated term that represents the detailed evolution of the population inversion along the erbium-doped fiber.

The black box model takes into account a multiple-stage amplifier, where all amplifier stages use the same type of erbium-doped fiber (the same absorption and emission coefficients are considered). Figure 1 shows a sketch of an amplifier set up in two sections, containing passive elements such as optical isolators, couplers, taps and

Signal output power (dBm) Gain (dB)

–40 28.82

–30 28.83

–20 28.82

–10 28.81

0 28.72

...

(3)

(4)

(5)

dP λ z,( )dz

------------------- α λ( ) γ λ( )+[ ]I z( ) α λ( )–{ }P λ z,( ) γI z( )PASEeq λ( )+=

P λ L,( ) G λ( ) P λ 0,( ) PASEeq λ( )+[ ]=

Gz λ( ) α λ( ) γ λ( )+{ }Iz α λ( )– z×{ }exp=

Iz

276

EDFA BLACK BOX

one filter. The total Er-doped fiber length and the total gain are L and G respectively.

Figure 1 Erbium doped amplifier set up in multiple stages, where the black box parameters G , IL , ILin are indicated

If the insertion loss is included in the analysis, the gain G is written as:

G = [GA + GB ]/ IL .

Amplifier GainIn order to model the gain of the amplifier, two different states of operation are considered where each state has a characteristic population inversion. The amplifier gain expression is given as a function of a reference gain value, (for example, [2]):

where and specifies the wavelength and the gain at a reference amplifier operating point.

The term is named tilt function and is obtained by the ratio of the gain curves measured in the two states of operation. One acts as a reference curve (for example,

).

The tilt function is given by the analytical expression:

where and are the gain measured to the state1 and state2 respectively of the amplifier operation. The experimental gain, measured at these two states of operation, is provided as input file in the black box model.

It is convenient to introduce this concept of tilt function in the model, since it considers the interdependence between the ratio of the characteristic gain and the absorption

(6)

(7)

λ( )

λ( ) λ( )λ( )

λ( )

λ( ) λ( ) λ( ) λ( )

G λ( )log Tλrefλ( ) G λref( ) Gref λref( )log–log[ ] Gref λ( )log+=

λref Gref λref( )

Tλrefλ( )

Gref λ( )

T λ( )G1 λ( ) G2 λ( )log–log

G1 λref( ) G2 λref( )log–log----------------------------------------------------------------=

G1 λ( ) G2 λ( )

277

EDFA BLACK BOX

and emission coefficients. On the other side, as the internal losses IL(λ) caused by passive elements modify both G1(λ) and G2(λ) in the same manner, the tilt function isn’t affected by optical circuitry variations.

By choosing G2(λ) equal to G(λ), the expression for the amplifier at the operation point is:

where specifies the gain difference between or . The term is a free parameter and may be altered to adjust

the gain.

Gain measurementThe gain curves are critical to the black box model operation. The best way to obtain these values used as input files in the model is by measuring them in a practical amplifier. It is important to note that the precision of these measurements defines the accuracy of the simulated results. However, the model alternatively accepts curves generated by a simulated amplifier that supplies gain and ASE curves as the output files.

Obtaining Gain Curve G1The first gain profile is acquired with the amplifier operating in a constant saturated regime that assures a specific population inversion. This condition can be obtained by coupling a large signal input power to the amplifier, typically −10 dBm, at the wavelength λref (e.g., 1540 nm), which is maintained constantly.

A small signal with power equal to −30 dBm (for instance) is added to the amplifier input as a probe signal. Its frequency (probe signal) is scanned through the range defined by the two-limit frequencies, which is written in terms of signal wavelength and usually varies from 1530 nm to 1570 nm. This scan over the probe signal allows you to obtain the spectral gain for one specific saturated condition.

This method was checked by analyzing a series of gain curves measured at the same saturated conditions, and a nominally identical population inversion was recorded [2].

Obtaining Gain Curve G2Analogous measurement procedure is repeated to obtain the second gain profile. However, in this case, the probe signal input is enlarged to –20 dBm, and the reference signal at a selected wavelength (1540 nm) can be varied. This new signal input combination results in a different population inversion condition, which characterizes the gain .

(8)

(9)

G λ( )log G1 λ( ) Tλrefλ( ) G1 λref( ) Gref λref( )log–log[ ]×–log=

G λ( )log G2 λ( ) Tλrefλ( ) ∆Glog×+log=

∆Glog Gref λref( ) G2 λref( )log–logG1 λref( ) G2 λref( )log–log ∆Glog

G2 λ( )

278

EDFA BLACK BOX

The difference is that the added signal test presents larger potency, typically −20 dBm, which causes a change in the gain curve profile by saturating the amplifier. With the value obtained for the gain in each wavelength, the gain curve profile is obtained. The high signal power, with the same λref, can also be altered, since the total sum of the power is larger than the sum of the power to generate the curve G1.

The experimental gain curves must be provided in files containing two columns. The first column refers to the wavelength specified in [nm], [m], [Hz] or [THz] units. The second column gives the gain in [dB] units.

As an example of the gain input file is:

where the wavelength is given in [nm] units and the gain is in [dB] units. There is no limit of rows or wavelength spacing previously defined.

Amplifier Noise FigureThe noise figure is the figure of merit that usually describes the amplifier noise performance. In order to evaluate the noise figure, three different options are available. You can select the input format of noise that will be considered to perform the calculations.

The first option is to select the noise input in terms of ASE power. In this case, the ASE noise spectral density is written as:

where P(λ) is the ASE power measured at each wavelength range and ∆f is the bandwidth considered in the ASE spectrum acquisition.

Another option to evaluate the amplifier noise performance is to select the ASE spectral density. In this case, the spectral density S(λ) is required as input file and is written as:

where h is the Planck constant, f is the optical frequency, and the exponent NF(λ) is the noise figure as a function of the signal wavelength.

Wavelength [nm] G [dB]

1535.58 38.17

1538.95 34.09

1542.11 33.35

1545.26 33.17

...

(10)

(11)

Spower λ( ) P λ( )∆f

------------=

S λ( ) hf 10NF λ ) 10⁄( G λ( ) 1–×[ ]=

279

EDFA BLACK BOX

The model will internally calculate the noise figure considering the noise curve provided as input file. Rewriting Equation 11 in terms of noise figure produces [3]:

The third option is to select the noise figure value given as a function of the signal wavelength. In this case, the ASE spectrum is modeled considering the provided noise figure value.

It is also possible to evaluate the noise figure considering different amplifier state operation that means to consider distinct gain values. In this case, the spectral density given by Equation 11 is rewritten including the gain variation (∆G in linear units or log∆G in dB units).

The new spectral noise density is dependent on the amplifier gain and is:

where can be calculated from Equation 8 and Equation 9.

Equivalent ASE Noise Measurement The experimental ASE noise curves complement the measured parameters required by the black box model.

Obtaining Equivalent ASE NoiseThe third input to obtain (experimental) is the amplified spontaneous emission. In the ASE acquisition curve, only the saturating signal must be maintained turned-on and operating with a constant power at a specified signal wavelength (1540 nm as suggested in the previous measurement descriptions). This is sufficient to produce population inversion along the Er-doped fiber.

The spectrum obtained at the fiber output registers the amplified spontaneous emission observed along the whole wavelength range considered (1530 nm to 1570 nm, typically).

The experimental gain curves must be provided in files containing two columns. The first column refers to the wavelength specified in [nm], [m], [Hz] or [THz] units. The second column gives the ASE noise curve in [dBm] units.

An example of input file:

(12)

(13)

Wavelength [nm] ASE [dBm]

1543 –25.13

1544 –25.20

1546 –25.42

NF λ( ) 10 S λ( ) hf+hf G λ( )×------------------------log=

S λ ∆Glog,( ) hf 10NF λ ) 10⁄( G λ ∆Glog,( ) 1–×=

∆Glog

280

EDFA BLACK BOX

where the wavelength is in [nm] units and the gain is in [dB] units. There is no limit of rows or wavelength spacing previously defined.

1551 –26.43

Wavelength [nm] ASE [dBm]

281

EDFA BLACK BOX

References[1] E. Desurvire, “Erbium-Doped Fiber Amplifiers – Principles and Applications”, John Wiley &

Sons, Inc., USA, 1994.

[2] J. Burgmeier, A. Cords, R. März, C. Schäffer, B. Stummer “A black box model of EDFA’s operating in WDM systems”, J. Lightwave Technol., Vol. 16, N. 7, pp. 1271-1275, 1998.

[3] S. P. Bastien, H. R. D. Sunak, B. Sridhar, V. E. Kalomiris “Temporal, spatial and spectral modeling of erbium doped fiber amplifiers”, SPIE – Physic and Simulations of Optoelectronic Devices, pp. 2-11, 1992.

282

EDF DYNAMIC — FULL MODEL

EDF Dynamic — Full model

Incorporates time-varying input signal and pump powers that enable simulating dynamic effects presented by erbium-doped amplifiers inserted in a fiber link. This powerful tool solves the full rate and propagation equations in the time and spatial domain. The powers and population densities are calculated as a function of the time variation at each point of the z fiber. This model is specifically designed to simulate cascaded amplifiers in a long fiber link, considering multiple signal input.

Ports

Parameters

Main








Core radiusDetermines the Er-doped fiber core radius

2.2 µm — [0.1,10]

Er doping radiusSpecifies the Er-doped radius

2.2 µm — [0.1,10]

Er metastable lifetimeDetermines the Er metastable lifetime

10 ms — [0.01,100]

Numerical apertureSpecifies the numerical aperture of the Er-doped fiber

0.24 — — [0.1,1]

Er ion densitySpecifies the Er doping in the Er-doped fiber

1e+025 m–3 m–3~ppm-wt ~wt%

[1,+INF[

283


Cross-sections

Numerical

Loss at 1550 nmDetermines the fiber loss at 1550 nm

0.1 dB/cm — [0,100]


0.15 dB/cm — [0,100]

LengthDetermines the Er-doped fiber length

5 m — [0,10000]

Reference timeDetermines the instant of time used to take the powers to use as input powers in the fiber to solve the steady-state regime that will determine the initial values for the population levels.


Units Value range

OptiAmplifier formatDetermines the format of the OptiAmplifier file


File frequency unitDetermines the frequency unit of the file with the measurements

nm — nm, m, Hz, THz

cross-section file nameDetermines the cross-section file

Erbium.dat — —


Units Value range

Relative errorDetermines the relative error acceptable in each calculation for the steady-state solution used as initial condition for the dynamic behavior

0.0001 — ]0,1]

Max. number of iterationsSpecifies the maximum number of times to repeat the longitudinal integrations for the powers when solving the steady-state equations used as initial condition for the dynamic behavior

50 — [10,10000]

Longitudinal stepsDetermines the number of longitudinal steps in the calculation

100 — [10,10000]



284


Simulation

Noise

Random numbers

Graphs


Units Value range





Noise center frequencyDetermines the noise center frequency

193.4 THz Hz, THz, nm [30,30e5]





[1,1000]


–100 dB — ]-INF,0[


3 dB — [0,+INF[


Convert noise bins

— — True, False


Units Value range




Random seed index


0 — [0,4999]


Absorption parameters Wavelength (m) Cross-section (m2)

Emission parameters Wavelength (m) Cross-section (m2)

285


Technical backgroundDifferent solutions to the problem of transient fluctuations due to gain cross-saturation observed in EDFAs inserted in multi-wavelength networks have been suggested. Gain cross-saturation in fiber amplifiers induces transients in the surviving channels remaining as a consequence of the process of adding or removing channels in the network reconfiguration. Although this perturbation will be small in a single amplifier, it becomes considerable along cascaded amplifiers. As a result, a tool that enables analyzing the effects of addition and/or dropping wavelength channels in a multi-wavelength optical network containing EDFAs is important.

In opposition to the steady-state model (EDF module), the EDF Dynamic enables you to calculate the variation of signals and pumps power with the time when sampled channels are present in the layout. The dynamic behavior of cascaded EDFAs can be simulated as well. The results will help you design cascaded amplifier systems with suppression of both transient and steady state signal power fluctuations due to channel addition/removal.

The numerical EDF Dynamic uses a two-level system approximation and is based on the solution of the propagation and rate equations for transitions between the upper and lower levels. These equations are given by Equation 1, Equation 2, and Equation 3, which are also in the technical background for the Erbium doped fiber [1]:

(1)

(2)

(3)

where the optical powers are expressed in units of number of photons per unit time, is the metastable spontaneous emission lifetime, N is the number of channels taken into account in the simulation (including signals, pumps, and ASE bins), is the number density of the active erbium ions, is the attenuation coefficient (which takes into account the background loss of the fiber), is the frequency step used in the simulation to resolve the ASE spectrum, and Aeff is the effective doped area given by , where is the Er doping radius (it is considered a uniform distribution of erbium ions in the area given by the Er doping radius region).

∂N2 z t,( )∂t

--------------------N2 z t,( )

τ-----------------– 1

Aeff------- Γn σn

e σna+( )N2 z t,( ) σn

a–[ ]{ } Pn+ z t,( ) Pn

– z t,( )+[ ]n 1=

N

∑–=

N2 N1+ 1=

∂Pn± z t,( )∂z

----------------------- un ρΓn σne σn

a+( )N2 z t,( ) σna– α–[ ]{ }Pn

± z t,( ) 2ρ∆νN2Γnσne+=

τ

ρα

∆νπ b2× b

286


The nth channel of wavelength has optical power Pn(z,t) at location z and time t, with emission and absorption cross-section and respectively, and confinement factor . The superscript symbols + and – are used to indicate channels traveling in forward (from 0 to L) and backward (from L to 0) directions, respectively. For beams traveling in the forward direction and for beams in the opposite direction . The overlap integrals between the LP01 mode intensity (which is used in this program) distribution doped region area are given by:

(4)

where E(r, ) gives the electric field intensity.

This model assumes that the signal and pump powers change slowly compared to the optical transit time in the fiber. This assumption is valid since the typical time that the light takes to pass by one 100 m fiber (one EDFA does not use fibers larger than that) is 500 ns. The time scales we deal with are always on the order of microseconds or longer.

Numerical solutionThe solution of the time-dependent rate equations and the propagation equations is based on the assumption that the atomic populations remain constant during a time step , typically microseconds. This assumption is acceptable since the metastable lifetime is relatively long (around 10 ms) and the transit time of photons through the Er3+-doped fiber is short.

Initial values for the population of the upper level in each point of the fiber of the program first solves the steady-state case. The parameter reference time determines the instant of time used to take the powers that will be used as input powers in the fiber to solve the steady-state regime that will determine the initial values for the population levels. When the calculation of the dynamic behavior for the sampled signal and pump channels starts at t=0, the program assumes that the population inversion is already different from zero, and the value of the population of the upper level at each point of the fiber (N2(z)) is given as t=0 by the powers at the reference time.

Generally speaking, you will be interested in the behavior of the amplifier in scales of times that go from a few microseconds to some tens of milliseconds. It is important to set the bit rate and the sequence length of the simulations in such a way that the time windows obey this requirement. If the time windows in your simulation are too short (for example, by a few nanoseconds), the gain of the EDF Dynamic amplifier will be given at almost all instants by the gain that one amplifier operating in the steady-state regime with the input powers given by the reference time would have, because the time response scales in EDFA are always on the order of microseconds or longer.

λn

σne σn

a

Γn

un 1=un 1–= Γn

Γn ν( )

E r ν,( ) 2r rd0

b

∫

E r ν,( ) 2r rd0

∞

∫

---------------------------------=

ν

δt

287


The parameterized channels and noise bins input powers are considered constant in time. The output powers for these channels are average in time. This means that during the calculation, the program saves the output powers that each one of these channels would have at each sample point, and then gives as output power the sum of the power at each sample divided by the total number of samples.

References

[1] C.R. Giles and E. Desurvire, “Modeling erbium-doped fiber amplifiers,” Journal of Lightwave Technology, Vol. 9, N. 2, pp. 271-283, 1991.

288

EDF DYNAMIC — ANALYTICAL MODEL

EDF Dynamic — Analytical model

Enables you to simulate the dynamic response of an EDF for input powers that vary in time. In opposition to the EDF Dynamic — Full model component, it doesn't solve the full rate and propagation equation. Neglecting ASE these equations can be solved analytically, which is described in this module. An additional approximation which considers the population of the upper level constant for the propagation equations is used to include the ASE effects on the behavior of the amplifier. The results using analytical solutions are achieved faster than using the EDF Dynamic — Full model, but the results are less accurate. The model which you use depends on the trade off between time and accuracy.

Ports

Parameters

Main









2.2 µm — [0.1,10]


2.2 µm — [0.1,10]


10 ms — [0.01,100]


0.24 — — [0.1,1]

289


Cross-sections

Numerical


1e+025 m–3 m–3~ppm-wt ~wt%

[1,+INF[


0.1 dB/cm — [0,100]


0.15 dB/cm — [0,100]


5 m — [0,10000]

Reference timeDetermines the instant of time used to take the powers to use as input powers in the fiber to solve the steady-state regime that will determine the initial values for the population levels.


Units Value range





Cross-section file nameDetermines the cross-section file

Erbium.dat — —


Units Value range

Relative errorDetermines the relative error acceptable in each calculation for the steady-state solution used as initial condition for the dynamic behavior

0.0001 — ]0,1]

Max. number of iterationsSpecifies the maximum number of times to repeat the longitudinal integrations for the powers when solving the steady-state equations used as initial condition for the dynamic behavior

50 — [10,10000]



290


Simulation

Noise

Random numbers


100 — [10,10000]


Units Value range






193.4 THz Hz, THz, nm [30,30e5]





[1,1000]


–100 dB — ]-INF,0[


3 dB — [0,+INF[


Convert noise bins

— — True, False


Units Value range




Random seed index


0 — [0,4999]


Units Value range

291


Graphs

Technical backgroundThis model uses analytical solutions for the same rate and propagation as [1], Equation 2, and Equation 3 described in Erbium doped fiber. These equations consider a two-level system interacting with light. As with the EDF Dynamic — Full model, it's assumed in this analytical model that the signal and pump powers change slowly compared to the optical transit time along the fiber.

Neglecting the ASE and the background loss in Equation 1, Equation 2, and Equation 3 for the Erbium doped fiber [1]:

(1)

and

(2)

where all the parameters were defined in Erbium doped fiber. Integrating Equation 1 and Equation 2 over z from 0 to L and defining as the total number of erbium ions in the upper state:

(3)




∂N2 z t,( )∂t

--------------------N2 z t,( )

τ-----------------– 1

ρAeff----------- uj

∂Pn± z t,( )∂z

-----------------------n 1=

N

∑–=

∂Pn z t,( )∂z

-------------------- un ρΓn σne σn

a+( )N2 z t,( ) σna–[ ]{ }Pn z t,( )=

N2

N2 t( ) ρAeff N2 z t,( ) zd0

L

∫=

292


we have

(4)

and

(5)

where

(6)

A further approximation enables us to estimate the ASE effects on this model. Considering N2(z) constant at each instant of time (which is a good approximation for strongly inverted EDFA), the propagation equations have an analytical solution which gives [2]:

(7)

where

(8)

is called the spontaneous emission factor. Substituting Equation 8 for Equation 4, we finally obtain:

(9)

This module uses Equation 5 and Equation 9 to simulate the dynamic behavior of the amplifier. Once given an initial value for the total number of excited ions, that is,

dN2 t( )dt

---------------- N2 t( )–τ

---------------- Pnout± t( ) Pn

in± t( )–n 1=

N

∑–=

Pnout± t( ) Pn

in± Gn 1–( )–

Gn Γn σne σn

a+( )N2 z t,( ) ρσna–[ ]L{ }exp=

Pnout± t( ) Pn

in± t( )– Pnin± t( ) Gn t( ) 1–[ ] 2nn

sp Gn t( ) 1–[ ]∆νASE+=

nnsp N2 t( )σn

e

σne σn

a+( )N2 σnaρ–

----------------------------------------------=

dN2 t( )dt

---------------- N2 t( )–τ

---------------- Pnin± t( ) Gn t( ) 1–[ ] 4nn

sp Gn t( ) 1–[ ]∆νASE

n

N

∑+n 1=

N

∑–=

N2

293


(t=0), and the input powers at each time, these coupled equations can be solved with an interactive loop between them.

Numerical solutionAs initial values for the total population of the upper level, the program solves the steady-state case. The parameter reference time determines the instant of time used to take the powers that will be used as input powers in the fiber in order to solve the equations in the steady-state regime. The obtained results will determine the initial value for the total number of excited erbium ions at t=0 ( (t=0)). In this way, when the calculation of the dynamic behavior to the sampled signal and pump channels starts at t=0, the program assumes that the population inversion is already different from zero, and the value of the upper level population is given at t=0 by the powers at the reference time.

Generally speaking, it is interesting to determine the behavior of the amplifier in scales of time that go from a few microseconds to tens of milliseconds. It is important to set the bit rate and the sequence length of the simulations in such a way that the time windows obey this requirement. If the time windows in your simulations are too short (for example, by a few nanoseconds), the gain of the EDF Dynamic amplifier will be given at almost all instants by the gain that one amplifier operating in the steady-state regime with the inputs powers given by the reference time would have, because the time response scales in EDFA are always in the order of microseconds or longer.

The parameterized channels and noise bins input powers are considered constant in time. The output powers for these channels are calculated averaging in time . This means that during the calculation, the program saves the values of at each instant of time and then calculates the medium value . Equation 4 and are then used to calculate the output powers of the parameterized and noise channels.

References[1] Y. Sun, J.L. Zyskind, and A.K. Srivastava, "Average Inversion Level, Modeling, and Physics of

Erbium-Doped Fiber Amplifiers," Journal of Selected Topics in Quantum Electronics, Vol. 3, N. 4, pp. 991-1006, 1997.

[2] T. Georges and E. Delevaque, "Analytical Modeling of High-Gain Erbium-Doped Fiber Amplifiers," Optics Letters, Vol. 17, N. 16, pp. 1113-1115, 1992.

N2

N2N2

N2⟨ ⟩ N2⟨ ⟩

294

EDFA

EDFA

Designs Er-doped fiber amplifiers by considering numerical solutions of the rate and the propagation equations under stationary conditions. The model includes amplified spontaneous emission (ASE) as observed in the amplifier Erbium Doped Fiber. However, this module allows you to select forward and/or backward pump, as well as the pump power values.

Ports

Parameters

Main


Input Input Optical



Default unit

Units Value range


2.2 µm — [0.1,10]


2.2 µm — [0.1,10]


10 ms — [0.01,100]


0.24 — — [0.1,1]


1e+025 m–3 m–3~ppm-wt~wt%

[1,+INF[


0.1 dB/cm — [0,100]


0.15 dB/cm — [0,100]

295

EDFA

Pumping

Cross-sections

Numerical


5 m — [0,10000]



Forward pump powerDetermines the co-propagating pump power


Backward pump powerDetermines the counter-propagating pump power


Forward pump wavelength Determines the co-propagating pump wavelength

980 nm — [700,1600]

Backward pump wavelengthDetermines the counter-propagating pump wavelength

980 nm — [700,1600]


Units Value range





cross-section file nameDetermines the cross-section file

Erbium.dat — —


Units Value range

Relative errorDetermines the relative error acceptable in each calculation

0.0001 — ]0,1]

Max. number of iterationsSpecifies the maximum number of times to repeat the calculation

50 — [10,10000]


Default unit

Units Value range

296

EDFA

Polarization

Simulation

Noise


100 — [10,10000]


Units Value range




Units Value range


Yes — [0, 0]




193.4 THz Hz, THz, nm [30, 30]


13 THz Hz, Thz, nm [1e-100, 1e-100]



[1,1]


–100 dB — [-1e+100, -1e+100]


3 dB — [0, 0]


Convert noise bins

— — [0, 0]


Units Value range

297

EDFA

Random numbers

Graphs

Technical backgroundThe routines in EDFA numerically solve the rate equations coupled with propagating equations under stationary conditions. It is assumed a two-level Er system approximation and the rate equations are based on the energy level diagram. The same expressions described in the module Erbium doped fiber are adopted by this model.

The main difference is related to the amplifier pump scheme selection. You can choose co-propagating, counter-propagating, or bi-directional pump schemes with the option to set wavelength and pump power. Geometrical Er-doped fiber parameters and cross-section curves are required as input files. As output files, you can access gain, output power values, and noise figure determined in the ASE bandwidth set as noise input data.

Er Doped Fiber Rate and Propagation EquationsThe lifetime transition from level 4I11/2 is of the order of microseconds for silicate hosts. Therefore, it is reasonable to neglect the population density N3 in the rate equations description. A two-level system approximation is used in this case. Under the assumption of the normalized population densities N1 and N2 at the ground and metastable energy level, 4I15/2 and 4I13/2 populations are calculated by numerically solving the rate and propagation equations[1]:

(1)


Units Value range


Yes — [0, 0]


0 — [0, 0]




∂N2 z t,( )∂t

--------------------N2 z t,( )

τ-----------------– 1

Aeff------- Γn σn

e σna+( )N2 z t,( ) σn

a–[ ]{ } Pn+ z t,( ) Pn

– z t,( )+[ ]n 1=

N

∑–=

298

EDFA

(2)

(3)

where the optical powers are expressed in units of number of photons per unit time, is the metastable spontaneous emission lifetime, N is the number of channels taken into account in the simulation (including signals, pumps, and ASE bins), is the number density of the active erbium ions, is the attenuation coefficient (which takes into account the background loss of the fiber), is the frequency step used in the simulation to resolve the ASE spectrum, and Aeff is the effective doped area given by , where is the Er doping radius (it is considered a uniform distribution of erbium ions in the area given by the Er doping radius region).

The nth channel of wavelength has optical power Pn(z,t) at location z and time t, with emission and absorption cross-section and respectively, and confinement factor . The superscript symbols + and – are used to indicate channels traveling in forward (from 0 to L) and backward (from L to 0) directions, respectively. For beams traveling in the forward direction and for beams in the opposite direction . The overlap integrals between the LP01 mode intensity (which is used in this program) distribution doped region area are given by:

(4)

where E(r, ) gives the electric density field.

Solving Equation 1, Equation 2, and Equation 3 under stationary conditions allows you to determine the amplifier performance features. The fiber parameters such as core and Er doping radius, Er metastable lifetime, numerical aperture, Er ion density, loss at 980 nm and 1550 nm, and the fiber length are required as input values. The absorption and emission cross-section are also required as input files.

Absorption and Emission cross-sectionsThere are two options available to you to prepare the cross-section file, which is specified in an ASCII file. The first option is to provide the cross-section input file in three columns. The first column refers to the wavelength in [m], [nm], [Hz] or [THz] units. The second column gives the absorption cross- section in [m2] units. The third

N2 N1+ 1=

∂Pn± z t,( )∂z

----------------------- un ρΓn σne σn

a+( )N2 z t,( ) σna– α–[ ]{ }Pn

± z t,( ) 2ρ∆νN2Γnσne+=

τ

ρα

∆νπ b2× b

λn

σne σn

a

Γn

un 1=un 1–= Γn

Γn ν( )

E r ν,( ) 2r rd0

∫

E r ν,( ) 2r rd0

∞

∫

---------------------------------=

ν

299

EDFA

column gives the emission cross-section in [m2] units. In this case, the cross-section file format is:

The second option is to consider the absorption and emission coefficients (or Giles parameters) as input parameters that are converted to cross-section by internal routines in the software. This is especially interesting when only Giles parameters are measured to the Er-doped fiber. The file format in this case contains three columns. The first column refers to the wavelength in [m], [nm], [Hz] or [THz] units. The second column gives the absorption coefficient in [dB/m] units. The third column gives the emission coefficient in [dB/m] units. An example of this input file is:

(nm)

929.982 9.28e-27 0

930.172 7.05e-27 0

.

.

.

1029.972 2.85e-27 0

1030.072 3.59e-27 0

1450.6 2.086e-26 1.726e-27

1450.8 2.186e-26 1.823e-27

.

.

.

1649.8 1.540e-26 8.228e-26

1650.0 1.540e-26 8.280e-26

(nm) (dB/m) g* (dB/m)

929.982 0.39168 0

930.172 0.2856 0

.

.

.

1029.972 –0.05508 0

1030.072 –0.14484 0

1450.6 1.8075 0.35599973

1450.8 1.815 0.360619883

λ σa m2[ ] σe m2[ ]

λ α

300

EDFA

where the wavelength is given in [nm] units, absorption and emission coefficients are in [dB/m].

.

.

.

1649.8 0.005 0.484116259

1650.0 –0.0175 0.477803876

(nm) (dB/m) g* (dB/m)λ α

301

EDFA

Reference:

[1] C.R. Giles and E. Desurvire, “Modeling erbium-doped fiber amplifiers,” Journal of Lightwave Technology, Vol. 9, N. 2, pp. 271-283, 1991.

302

EDFA IDEAL

EDFA Ideal

Enables designing EDFAs considering pre-defined operation conditions that means to specify previously the expected gain, the noise figure and the amplifier output power. The EDFA Ideal presents the same facilities as a black box model, which enables you to select the operation mode with gain control, power control, or performing simulations under saturated condition, as well as to define the expected amplifier performance. It is specially indicated for the prompt performance analysis of one or cascaded amplifiers present in a long-haul system.

Ports

Parameters

Main


Input Input Optical




Operation modeAmplifier operation mode

Gain control — — Gain control, Power control, Saturation

GainDetermines the signal gain

20 dB — [0,100]

PowerDetermines the signal output power

10 dBm W, mW, dBm [-100,100]

Saturation powerSpecifies the optical power at the gain compressed by 3 dB

10 dBm — [-100,100]

Saturation portDetermines the amplifier saturation port

Output — — Input, Output

Include noise YES — — —

303

EDFA IDEAL

Polarization

Simulation

Noise

Random numbers

Noise figureDetermines the amplifier noise figure

4 dB — [3,100]


Units Value range




Units Value range





Noise center frequency 193.4 THz Hz, THz, nm [30, 3e+006]

Noise bandwidth 13 THz Hz, THz, nm [1e-100, 1e+100]



[1,1000]


Convert noise bins

— — True, False


Units Value range




Random seed index


0 — [0,4999]



304

EDFA IDEAL

Technical backgroundThe simulation of the ideal amplifier is performed in the opposite way than that used by the previous described models. In this case, the desired amplifier performance given by the gain, the output power, the saturated output power, and the noise figure values are used as input parameters to design the amplifier.

The input data are related by the propagation equation written in terms of the parameter required in each mode selected. There are three different mode controls — Gain control, Power Control, and Saturation. Large and small input signal can be considered in this amplifier model. The concept of the ideal amplifier enabling you to define the device performance makes this model flexible to design amplifiers considering different applications in a system such as booster, in-line, and pre-amplifier.

The amplified spontaneous emission is included in the model of the EDFA Ideal and it is built from the noise figure input value.

Mode Controls Description The EDFA Ideal subsystem enables three operation modes, which you can select in the EDFA Ideal Properties dialog box by clicking on Main/Operation Mode/Value. The first option is the Gain Control that maintains the gain constant and allows you to include (or not include) the amplified spontaneous emission in the calculations. In the second operation mode option, Power Control, the value of the output power is maintained constantly. The third operation mode, Saturation, considers the amplifier operating in a saturated condition — operating in an output signal power correspondent to a gain 3 dB lower than the saturated gain.

Gain Control ModeIn this mode, you set the desired amplifier gain (in dB units), which is given by the ratio of the total output power (Pout) and total input power (Psin), including (or not including) the generated ASE (PASE), as given by Equation 1. There are no additional iterations or complicated equation solutions in this mode. The set amplifier input parameters as gain and noise figure give the performance of this sub-system to be inserted in the global system.

(1)

Power Control ModeThe value that you define in the power control mode is the desired amplifier output power (in dBm units), which is maintained constantly. Analogous with the gain-controlled mode, there is no additional calculation involved in the designed amplifier. The output power set as input parameter defines the amplifier performance to be considered in the system where this amplifier is inserted. The ASE, which basically computes the noise introduced by the amplifier into the system, can be included (or

GPout PASE–( )

Psin---------------------------------=

305

EDFA IDEAL

not included) in the amplifier performance. Note that the specified output power is not degraded by the ASE noise included in the amplifier subsystems — however, this noise source is computed in the global system analysis.

Saturation ModeIn the saturation mode, it is assumed that the pump power is constant, causing the amplifier to operate in a saturated regime. The saturation power, gain, and noise figure are the parameters required by this mode. The saturation power is the input parameter maintained constant in this mode selection, and in an ASE-free model can be related with the gain (G), output power (Pout), and intrinsic saturation power (Psat

int) by the expression:

(2)

where G0 is the small-signal gain or unsaturated gain.

The intrinsic saturation power is written as:

(3)

where A is the mode-field area, h is the Planck’s constant, ν is the frequency at the propagating signal, σa is the absorption cross-section, and τ is the Er metastable lifetime in silica.

These fiber specifications are not required in this Ideal Amplifier module, since the intrinsic saturation power will be related to the amplifier saturation power under the gain compression condition.

Under the 3 dB gain compression, the output power is proportional to the intrinsic saturation power. This relation is:

(4)

ASE CalculationThe ASE noise spectrum is built in this model from the noise figure provided as input parameter, considering the expression that relates spectral ASE noise with noise figure. The noise figure (NF) evaluated at a specific signal wavelength is:

(5)

The term 1/G corresponds to the shot noise, Sout is the output ASE spectral density at the signal wavelength, and hν is the photon energy. In practical cases, there is ASE

G G0G 1–

G-------------

Pout

Psatint

----------–exp=

Psatint Ahv

σaτ----------=

Pout Compressed3dB In 2( )Psat

int=

NF 1G----

SoutGhv----------+=

306

EDFA IDEAL

present at the input of the doped fiber so that the amplified input ASE must be added to the output ASE spectral density. The output ASE can be written as:

(6)

where Samp is the spectral density ASE generated by the doped fiber.

Correcting for the input ASE gives the signal-spontaneous beat noise limited noise figure as a function of the signal gain, and input and output ASE spectral densities:

(7)

In the signal-spontaneous beat noise limited regime, with high gain and negligible input coupling, the noise figure of the optical preamplifier approaches a theoretical limit of [1]:

(8)

where the spontaneous emission factor, nsp, is defined as:

(9)

where

(10)

Since nsp ≥ 1, an EDFA at high gain has a minimum noise figure of 3 dB. This is derived by assuming that the input signal is shot noise limited and the output noise is signal-spontaneous beat noise limited. In practical situations, the noise figure is degraded by the amplifier input coupling loss.

Noise figureThis lists the signal-spontaneous beat noise limited noise figure. For each signal wavelength, the noise figure is:

(11)

where is the output ASE spectral density (W/Hz) at the signal wavelength, and is the input ASE spectral density at the signal wavelength.

Sout Samp Sin G×+=

NF 1G----

SoutGhv----------

Sinhv------–+=

NFoptσsig sp–

2

σsig sh–2 in( )G2

----------------------------------- 2nsp= =

nsp v z,( )N2 z( )

N2 z( ) N1 z( )ε v( )–--------------------------------------------=

ε v( )σa v( )σe v( )-------------=

NoiseFigure dB( ) 10 101G----

Sout λs( )Ghv

-------------------Sin λs( )

hv-----------------–+log×=

Sout λs( ) Sin λs( )

307

EDFA IDEAL

Rewriting the ASE spectral density as a function of noise figure value, the noise spectrum can be generated considering the noise figure input parameter. Therefore, the ASE spectrum is obtained from the expression:

(12)

References

[1] T. Okoshi, "Exact Noise-Figure Formulas for Optical Amplifiers and Amplifier-Fiber Cascaded Chains," IEEE/OSA Topical Meeting on Optical Amplifiers and their Applications, Monterrey, PDP11, 1990.

Sout λs( ) G hv 10NoiseFigure dB( )

10------------------------------------- 1

G----–

Sin λs( )hv

-----------------–×=

308

EDFA MEASURED

EDFA Measured

Enables you to design EDFAs considering pre-defined operation conditions that mean to specify previously the measured gain, noise figure, and amplifier output power. It is specially indicated for the prompt performance analysis of one or cascaded amplifiers present in a long-haul system. It can be also used for flat gain amplifiers.

Ports

Parameters

Main


Input Input Optical




Measured gain and noise figureWavelength, gain, and NF table with the measured data

(nm dB dB) — — —

Gain and noise figure file nameFilename with the measured data

GainAndNF.dat — — —

Max. output powerDetermines the total signal output power

25 dBm W, mW, dBm [-100,+100]

Include noiseDetermines if the component add noise to the output signal


309

EDFA MEASURED

Polarization

Simulation

Noise

Random numbers


Units Value range




Units Value range


Yes — [0, 0]




193.4 THz Hz, THz, nm [30, 30]


13 THz Hz, Thz, nm [1e-100, 1e-100]



[1, 1]


–100 dB — [-1e+100, -1e+100]


3 dB — [0, 0]


Convert noise bins

— — [0, 0]


Units Value range


Yes — [0, 0]

310

EDFA MEASURED

Technical backgroundThe simulation of the ideal amplifier is performed in similar way to the EDFA Ideal. In this case, the desired amplifier performance given by the measured gain, noise figure, and maximum output power. Large and small input signals can be considered in this amplifier model. The concept of the measured amplifier enabling you to define the device performance makes this model flexible to design amplifiers for different applications in a system such as booster, in-line, pre-amplifier, gain flat, and noise flat. It can also load measurements from other software tools such as Optiwave's OptiAmplifier.

This maximum output power can be limited when the total output power is greater then the parameter Max. output power. The calculation engine reduces the amplifier gain in order to have the total output power equal to the parameter Max. output power.

ASE CalculationThe ASE noise spectrum is built in this model from the noise figure provided as input parameter, considering the expression that relates spectral ASE noise with noise figure. The noise figure (NF) evaluated at a specific signal wavelength is:

(1)

The term 1/G corresponds to the shot noise, Sout is the output ASE spectral density at the signal wavelength, and hν is the photon energy.

In practical cases, there is ASE present at the input of the doped fiber so that the amplified input ASE must be added to the output ASE spectral density. Therefore, the output ASE can be written as:

(2)

where Samp is the spectral density ASE generated by the doped fiber.

Correcting for the input ASE gives the signal-spontaneous beat noise limited noise figure as a function of the signal gain, and input and output ASE spectral densities:


0 — [0, 4999]


Units Value range

NF 1G----

SoutGhv----------+=

Sout Samp Sin G×+=

NF 1G----

SoutGhv----------

Sinhv------–+=

311

EDFA MEASURED

(3)

In the signal-spontaneous beat noise limited regime, with high gain and negligible input coupling, the noise figure of the optical preamplifier approaches a theoretical limit of [1]:

(4)

where the spontaneous emission factor, nsp, is defined as:

(5)

where

(6)

Since nsp ≥ 1, an EDFA at high gain has a minimum noise figure of 3 dB. This is derived by assuming that the input signal is shot noise limited and the output noise is signal-spontaneous beat noise limited.

In practical situations, the noise figure is degraded by the amplifier input coupling loss.

Noise figureThis lists the signal-spontaneous beat noise limited noise figure. For each signal wavelength, the noise figure is:

(7)

where is the output ASE spectral density (W/Hz) at the signal wavelength, is the input ASE spectral density at the signal wavelength.

Rewriting the ASE spectral density as a function of noise figure value, the noise spectrum can be generated considering the noise figure input parameter. Therefore, the ASE spectrum is obtained from the expression:

(8)

NFoptσsig sp–

2

σsig sh–2 in( )G2

----------------------------------- 2nsp= =

nsp v z,( )N2 z( )

N2 z( ) N1 z( )ε v( )–--------------------------------------------=

ε v( )σa v( )σe v( )-------------=

NoiseFigure dB( ) 10 101G----

Sout λs( )Ghv

-------------------Sin λs( )

hv-----------------–+log×=

Sout λs( ) Sin λs( )

Sout λs( ) G hv 10NoiseFigure dB( )

10------------------------------------- 1

G----–

Sin λs( )hv

-----------------–×=

312

EDFA MEASURED

MeasurementsYou can provide the measurements in the parameter Measured gain and noise figure. Alternatively, the measurements can be loaded from a file using the parameter Gain and noise figure file name. The gain and noise figure curves must be provided in the file containing three columns. The first column refers to the wavelength specified in [nm] units. The second column gives the gain noise curve in [dB] units. The third column gives the noise figure in [dB] units.

Example of input file:

Wavelength ([nm] Gain [dB] NF [dB]

1500.00 20.00 4.00

1510.00 20.00 4.00

1520.00 20.00 4.00

1530.00 20.00 4.00

1540.00 20.00 4.00

1550.00 20.00 4.00

313

EDFA MEASURED

Reference:

[1] T. Okoshi, "Exact Noise-Figure Formulas for Optical Amplifiers and Amplifier-Fiber Cascaded Chains," IEEE/OSA Topical Meeting on Optical Amplifiers and their Applications, Monterrey, PDP11, 1990.

314

ERBIUM DOPED FIBER

Erbium doped fiber

This component simulates a bidirectional Erbium doped fiber considering ESA, Raleigh scattering, ion-ion interactions, and temperature dependence effects. The component solves numerically the rate and propagation equations in the steady-state case, assuming a two-level Erbium system for an inhomogeneous and homogeneous approach.

Ports

Parameters

Main


Input1 Input Optical

Output1 Output Optical



Name and description Symbol Default value Default unit

Units Value range

LengthSpecifies the doped fiber length

L 5 m — [0, 1e4]

Er metastable lifetimeSpecifies the Erbium metastable lifetime

10 ms — ]0, +INF[

Input dataDetermines if saturation parameter is used or not

— Fiber specification — — Fiber specification, Saturation parameter

Saturation parameterSpecifies value of saturation parameter

4.4e+015 1/(s.m) — [1e-10, +INF[

Core radiusSpecifies the fiber core radius

a 2.2 µm — [0,1, 10]

τ

ζ

315

ERBIUM DOPED FIBER

Cross-sections

Enhanced

Er doping radiusSpecies the Erbium doped radius

b 2.2 µm — [0.1, 10]

Er ion densitySpecifies the Erbium doping in the fiber

1e+25 m-3 m-3 , ~ppm-wt, ~wt%

[1e23, +INF[

Numerical apertureSpecifies the numerical aperture of the fiber

NA 0.24 — — [0.1,1]



OptiAmplifier formatDetermines if format of cross-section file is an OptiAmplifier file


File frequency unitDetermines frequency unit of the file with the cross sections

nm — — nm, m, Hz, THz

Cross-section file nameSpecifies Erbium cross-section file name

Erbium.dat — — —


Units Value range

Background loss data typeDetermines if the loss will be calculated from the loss at 1310nm (constant) or it will be loaded from a file

Constant — — Constant, From file

Loss at 1310 nm

Specifies the fiber loss at 1310nm

3 dB/Km [0, +INF[

Background loss file name

Specifies loss file name

— Loss.dat — — —

Include Rayleigh backscattering

Determines if Rayleigh scattering effect is included or not



Units Value range

nt

l λ( )

l1310

316

ERBIUM DOPED FIBER

Rayleigh ConstantSpecifies the value of the Rayleigh constant

150 — dB/Km [0, 1000]

Backscattering capture fractionDetermines if capture fraction values are calculated by the component or loaded from a file

Calculate — — Calculate, From file

Rayleigh capture file nameSpecifies the capture file name

— Capture.dat — — —

Includes ion-ion interaction effectsDetermines whether Er-Er ion interaction effects are included or not

— False — — True, False

Ion-Ion interaction effectDetermines which kind of Er-Er ion interaction is considered

— Homogeneous — — Homogeneous, Inhomogeneous, Combined

Upconversion coefficientSpecifies the two-particle upconversion coefficient

1e-022 [0, 1000]

Ions per clusterSpecifies number of ions in a cluster

2 — — [0, 500]

Relative number of clustersSpecifies the relative number of clusters

K 12 — % [0, 100]

Include Temperature EffectsDetermines if temperature dependence is taken into account


TemperatureSpecifies the current temperature

T 20 — C [-273, 500]

Cross-section TemperatureSpecifies the temperature when the cross-section was measured

Tm 20 — C [-273, 500]

Include ESA EffectDetermines if excited stated absorption is taken into account


ESA Cross-section file nameSpecifies the ESA cross-section file name

— ESAErbium.dat — — —


Units Value range

KR

C λ( )

Uc m3 s⁄

mk

317

ERBIUM DOPED FIBER

Numerical

Extract ESA from emissionDetermines if the component has to extract the ESA cross-section from the loaded file



Units Value range

Calculation algorithmDetermines algorithm to be used in simulation

— Giles — Saleh, Jopson, Giles, Inhomogeneous

Relative errorSpecifies maximum acceptable difference between two consecutive iterations to complete the iteration process

0.0001 — — [1e-100, 1]

Max. number of iterationsSpecifies the maximum number of iterations executed

100 — — [1, 1e8]

Number of longitudinal stepsSpecifies the minimum number of longitudinal steps in the fiber

50 [1, 1e8]

Inhomogeneous accuracyIf the inhomogeneous model is selected, this parameter specifies the accuracy in the convolution integrals

— 0.001 — — [1E-10, 0.1]

Overlap factor dataDetermines whether overlap factor values are calculated by the component or loaded from a file

Calculate — — Calculate, From file

Geometrical modelDetermines whether the component calculates the overlap factor using one of the Gaussian approximations, or the LP01 mode

— LP01 — — Marcuse Gaussian, Whitley Gaussian, Desurvire Gaussian, Myslinski Gaussian, LP01

Overlap factorDetermines if overlap factor calculations takes into account the signal and pump power

— Power independent

— — Power independent, Power dependent


Units Value range

ζ

Nmax

Γ

318

ERBIUM DOPED FIBER

Simulation

Noise

Nr. of transverse integrationsIf PowerDependent is selected for Overlap factor, specifies the number of times that the overlap factor is calculated over the fiber length

— 2 — — [1, 50]

Overlap factor file nameSpecifies the overlap factor file name

— Confinement.dat — — —

Generate homogeneous cr.Generate the homogeneous cross- sections


Inhomogeneous linewidthSpecifies the Erbium-doped fiber inhomogeneous linewidth

11.5 nm — ]0, 100]

Number of gaussiansDetermines number of gaussians used in generation of the homogeneous cross-sections

17 — — [8, 28]


Units Value range



Enable reflectionsDetermines whether or not the component launches reflections due to backscatterig in the output





193.4 THz Hz, THz, nm [30,30e5]




125 THz Hz, GHz, THz, nm

[1,1000]


Units Value range

∆λinh

nG

319

ERBIUM DOPED FIBER

Random numbers

Graphs


–100 dB — ]-INF,0[


3 dB — [0,+INF[


Convert noise bins

— — True, False


Units Value range




Random seed index


0 — [0,4999]



Calculate graphs False — — True, False

Number of distance steps 20 — — [1,1e8]

Number of wavelength steps 20 — — [1,1e8]

Linear scale True — — True, False

Minimum value -50 — dBm ]1e-100, 1e100[

Pump reference wavelength 1400 nm [100, 1900]



320

ERBIUM DOPED FIBER

Technical backgroundThis module presents a rapid numerical solver for the EDF rate and propagation equations for signals, pumps and amplified spontaneous emission (ASE) considering the steady-state case. The propagation and rate equations of a two level system are used to model the Erbium-doped fiber. Several effects are considered, including

interactions, excited state absorption, temperature dependence, and background loss. Furthermore, the component assumes the possibility of considering the inhomogeneous broadening in the EDF.

Propagation and Rate EquationsThe Erbium Doped Fiber component is based on the solution of the rate and propagation equations assuming a two-level model. The use of a two-level model for the amplifier is justified, as for pumping into the 980nm absorption band, the lifetime transition from level is of the order of microseconds for silicate hosts and is reasonable to neglect the population density in the rate equations description. At 1480nm, the pumping is direct to the upper sub-levels of the metastable manifold.

Rate equations are based on energy levels and describe the effects of absorption, stimulated emission, and spontaneous emission on the populations of the ground ( ) and metastable ( ) states.

For a two-level system with optical beams, the rate equations are given by:

where is the Planck constant, is the metastable lifetime parameter, is the frequency, and is the power of the th beam. The absorption and emission cross-section of the th beam are and , respectively, and is the local erbium ion density. The normalized optical intensity is defined as

, where is light intensity distribution of the th beam.

(1)a

(1)b

Er+3 Er-3–

4 11 2⁄N3

n1 n2

k

dn1dt

--------–dn2dt

--------σa vk( )

hvk---------------- ik r φ,( ) Pk z( ) n1 r φ z, ,( )

σe vk( )hvk

---------------- ik r φ,( ) Pk z( ) n2 r φ z, ,( ) 1τ--- n2 r φ z, ,( )⋅–⋅⋅⋅

k∑–⋅⋅⋅

k∑= =

n1 r φ z, ,( ) n2 r φ z, ,( ) nt r φ z, ,( )=+

h τ vkPk k

k σa vk( ) σe vk( ) ntik r φ,( )

ik r φ,( ) Ik r φ z, ,( ) Pk z( )⁄= Ik r φ z, ,( )k

321

ERBIUM DOPED FIBER

The propagation equations describe the propagation of the beams through the doped fiber, and are given by:

where each beam propagates in the forward ( ) or backward ( ) direction, and means the spontaneous emission contribution from the local metastable population . , where the normalized number of modes m is normally 2, and is the noise bandwidth.

Setting the time derivative in Equation 1a to zero and using Equation 1b, the problem is reduced to the steady-state case and the metastable population is defined as:

(2)

(3)

dPkdt

--------- uk σe vk( ) Pk z( ) P0k+( ) n2 r φ z, ,( ) ik r φ,( ) r r φ uk σa vk( ) Pk z( ) .⋅⋅–d⋅d⋅ ⋅⋅

0

∞

∫0

2π

∫⋅ ⋅⋅=

. n1 r φ z, ,( ) ik r φ,( ) r r φd⋅d⋅ ⋅⋅

0

∞

∫0

2π

∫

uk 1= uk 1–=P0k

n2 P0k m h vk ∆vk⋅ ⋅ ⋅=∆vk

n2 r φ z, ,( ) nt

σa vk( ) τ⋅hvk

----------------------- ik r φ,( ) Pk z( )⋅⋅

k 1=

n

∑

σa vk( ) σe vk( )+( ) τ⋅hvk

-------------------------------------------------- ik r φ,( ) Pk z( ) 1+⋅⋅

k 1=

n

∑

--------------------------------------------------------------------------------------------------------------⋅=

322

ERBIUM DOPED FIBER

With the specified boundary conditions at and , Equation 2 and Equation 3 can be integrated over space and frequency.

Figure 1 Example of absorption and emission cross-sections

It is important realize to that the transverse shape of the optical mode and its overlap with the erbium ion distribution profile are very important. It can be parameterized by a factor known as overlap integral factor.

Considering a steady-state case, and substituting Equation 1b in Equation 1a, the rate equation becomes:

Integrating Equation 4 over space:

(4)

z 0= z L=

σa vk( )hvk

---------------- ik r φ,( ) Pk z( ) nt r φ z, ,( )σa vk( )

hvk---------------- ik r φ,( ) Pk z( ) n2 r φ z, ,( ) 1

τ--- n2 r φ z, ,( ) –⋅–⋅⋅⋅

k∑–⋅⋅⋅

k∑

σe vk( )

hvk---------------- ik r φ,( ) Pk z( ) n2 r φ z, ,( ) 1

τ--- n2 r φ z, ,( ) ⋅–⋅⋅⋅

k∑–

1τ--- n2 r φ,( ) π beff

2 σa vk( )hvk

---------------- Pk z( ) nt

ik r φ,( ) nt r φ,( ) r r φd⋅d⋅ ⋅⋅

0

∞

∫0

2π

∫

nt

----------------------------------------------------------------------------- σa vk( )

hvk---------------- Pk z( ) n2 .⋅⋅

k 1=

n

∑–⋅ ⋅⋅k∑=⋅ ⋅⋅

.

ik r φ,( ) n2 r φ,( ) r r φd⋅d⋅ ⋅⋅

0

∞

∫0

2π

∫

n2

------------------------------------------------------------------------------σe vk( )

hvk---------------- Pk z( ) n2 .

ik r φ,( ) n2 r φ,( ) r r φd⋅d⋅ ⋅⋅

0

∞

∫0

2π

∫

n2

------------------------------------------------------------------------------⋅⋅

k 1=

n

∑–

323

ERBIUM DOPED FIBER

where is considered the average density, and is given by:

and is the equivalent radius of the doped region:

when the ion density population is uniform, the effective radius is equal to the doped radius, .

For an effective doped radius , the effective cross-sectional area is .

Then, the overlap integral or confinement factor for the level can be defined as:

If the erbium ions are well confined to the center of the optical modes, then and are nearly equal, and can be replaced with the single constant .

Therefore, using the definition of overlap integral, the average population density for the level 2 is given by:

(5)

(6)

(7)

ni

ni z( )

ni r φ,( ) r r φd⋅d⋅⋅

0

∞

∫0

2π

∫

π beff2⋅

------------------------------------------------------=

beff

beff 2nt r( )nt 0( )------------ r rd⋅ ⋅

0

π

∫

12---

==

b

beffAeff π beff

2⋅=

ith

Γkj z( )

ik r φ,( ) ni r φ,( ) r r φd⋅d⋅ ⋅⋅

0

∞

∫0

2π

∫

ni

----------------------------------------------------------------------------=

Γk 1,Γk 2, Γk

n2 z( )

σa vk( )hvk

---------------- Pk z( ) nt Γk⋅ ⋅⋅

k 1=

n

∑

1τ--- Aeff

σa vk( ) σe vk( )+hvk

-------------------------------------- Pk z( ) Γk⋅⋅

k 1=

n

∑–⋅

----------------------------------------------------------------------------------------------------=

324

ERBIUM DOPED FIBER

and the propagation equation becomes:

Basically, Equation 7 and Equation 8 are the equations solved in the homogeneous case. Slight modifications are made to these equations in order to include other effects.

(8)dPkdz

--------- σe vk( ) σa vk( )+( ) Pk z( ) n2 Γk σa vk( ) Pk z( ) nt Γk P0k σe vk( ) n2 Γk⋅ ⋅⋅+⋅ ⋅⋅–⋅ ⋅⋅=

325

ERBIUM DOPED FIBER

Giles-Desurvire Propagation and Rate EquationsA simpler method of fiber characterization can be done by writing the amplifier equations in terms of absorption coefficient ( ), gain coefficient ( ), and a fiber saturation parameter ( ). These parameters can be obtained by conventional fiber measurement techniques [1].

The saturation parameter can be defined theoretically as:

and the absorption and gain coefficients are expressed in terms of distributions of the ions and optical modes:

For a uniform ion distribution the absorption and gain coefficients can be simplified as:

Giles and Desurvire in [1] rewrote the propagation Equation 8 in terms of saturation parameter, and absorption and emission coefficients:

where is the background loss.

(9)

Er+3 αk gkζ

ζ

ζ π beff2 nt τ⁄⋅ ⋅=

αk λk( ) σa λk( ) ik r φ,( ) nt r φ z, ,( ) r r φd⋅d⋅ ⋅⋅

0

∞

∫0

2π

∫⋅=

gk λk( ) σe λk( ) ik r φ,( ) nt r φ z, ,( ) r r φd⋅d⋅ ⋅⋅

0

∞

∫0

2π

∫⋅=

αk λk( ) Γ λk( ) nt σa λk( )⋅ ⋅=

gk λk( ) Γ λk( ) nt σe λk( )⋅ ⋅=

dPk z( )dz

---------------- uk Pk z( ) gk vk( ) αk vk( )+( )n2

nt

----- αk vk( )– lk–⋅

uk P⋅ 0k gk vk( )n2

nt

-----⋅⋅+⋅⋅=

lk

326

ERBIUM DOPED FIBER

In the same way, the steady-state solution of rate Equation 7 was rewritten as:

Note: The equation for sums over all forward and backward beams, including ASE.

Equation 9 and Equation 10 are referenced further as a Giles model. These equations are solved in the homogeneous line broadening case.

The Giles model provides a full spectral solution. The propagation Equation 9 is integrated back and forth along the fiber, in an iterative numerical process, until the solution converges, or the maximum number of iterations ( ) is reached.

The propagation equation solved by the Giles model can be slightly different from Equation 9, depending on which effects the user has considered in the simulation, such as ESA and Rayleigh scattering. Equation 10 can be different depending on whether the user takes into account the interactions.

Overlap IntegralsThe value of the overlap integral can be calculated using Equation 6. The transverse optical modes distributions are described by their normalized optical intensity.

For a single-mode fiber, the optical mode can be approximated by the mode distribution:

(10)

(11)

n2

nt

----- z( )

Pk z( ) αkvk⋅h vk ζ⋅ ⋅

-----------------------------k 1=

n

∑

1Pk z( ) αk vk( ) gk vk( )+( )⋅

hvk------------------------------------------------------------

k 1=

n

∑+

--------------------------------------------------------------------------------=

n2 z( )

Nmax

Er+3 Er+3–

LP01

i r φ,( )

1π---

vJ0 ur a⁄( )aVJ1 u( )

--------------------------2r a<

1π---

uK0 vr a⁄( )aVK1 v( )

---------------------------2r a≥

=

327

ERBIUM DOPED FIBER

where is the fiber core radius, the fiber number is , and are the eigenvalues found by matching

the solutions at , is the Bessel function of the first kind of order 0, is the Bessel function of the first kind of order 1, is the modified Bessel function of the second kind of order 0, and is the modified Bessel function of the second kind of order 1.

The mode distribution can also be approximated with a Gaussian function:

where the Gaussian mode radius, , has been given by various authors as:

The overlap integrals depend on:• the energy level occupied by the ions, because the distribution is different for

each level• the power, because the ion dopant distribution is power dependent• the wavelength, because the optical mode profile is wavelength dependent

In principle, the overlap integrals are also functions of , due to variations in doping level along the fiber, and mode coupling (if more than one mode is supported).

(12)

(13)

(14)

(15)

(16)

a VV 2 π a ncore

2 nclad2–( ) λ⁄⋅ ⋅ ⋅= u v

r a= J0 J1K0

K1

LP01

i r φ,( ) 2πw2---------- 2 r2⋅

w2------------–

exp=

wGauss

wGauss a 0.65 1.619V 1.5------------- 2.879

V 6-------------+ +

Marcuse=

wGauss a 0.616 1.66V 1.5---------- 0.987

V 6-------------+ +

Whitley=

wGauss a 0.759 1.289V 1.5------------- 1.041

V 6-------------+ +

Desurvire=

wGauss a 0.761 1.237V 1.5------------- 1.429

V 6-------------+ +

Myslinski=

z

328

ERBIUM DOPED FIBER

For a fundamental mode approximated by a Gaussian profile and a uniformly doped fiber with doped radius , the overlap of the mode with the total ion profile is given by:

In the low-power limit, all excited-state overlap integrals with the Gaussian approximation reduce to:

where Equation 18 is an approximated form of the upper levels (1, 2, 3 and 4).

For the mode approximation with a uniformly doped fiber and fiber doped radius , the overlap with the total ion distribution is given by:

Typically, the fiber doped radius is less than or equal to the core radius ( ), and for , the integrals also have weak power dependence [1]. For most cases, therefore, it is reasonable to assume that overlap integrals are power independent and are equal to for ions in all the energy levels.

(17)

(18)

(19)

b nt r φ z, ,( )

Γt 1 e

2b2–w2

-----------

–=

Γ1 2 3 4, , , P 0→( ) bw----

2

1 e

4b2

w2--------–

–

1 e

2b2

w– 2---------

–

--------------------≈

LP01b

Γtub

VaJ1 u( )-------------------

2J0

2 ub a⁄( ) J12 ub a⁄( )+[ ]=

b w 0.8≤⁄b w 0.8≤⁄

Γt

329

ERBIUM DOPED FIBER

Additional Effects

Background LossBackground loss in a fiber amplifier or laser is usually negligible compared to absorption coefficients and discrete losses. However, the background loss may be significant for lightly-doped fibers, for losses at the signal wavelength of a four level ion, for wavelengths far from absorption maxima, and for wavelengths beyond the low-loss region of the host glass. The actual fiber loss is composed of the Rayleigh backscattering loss, and losses from impurities.

Here, the excess loss, , is assumed to be wavelength-independent, and is given by:

where is the total loss at 1310nm and is the loss due the Rayleigh scattering effect at 1310nm.

The user specifies the total loss at 1310 nm ( ), from which the component calculates the excess loss. The loss at any other wavelength then adds an additional term to the propagation equations as:

The user has the possibility of considering the excess loss as wavelength dependent. In this case, a file has to be provided that contains the total loss characteristics for the band of interest. Then, the wavelength dependent excess loss will be defined as:

Note: The effects of background loss are only considered during the Giles algorithm calculation.

Rayleigh ScatteringRayleigh Backscattering is incorporated in the model by coupling each forward and backward traveling signal at a wavelength to a backward-traveling and forward-traveling signal at the same wavelength:

(20)

(21)

(22)

αEL

αEL l1310nm αRS 1310nm( )–=

l1310nm αRS 1310nm( )

a1310nm

dPk+

dz--------- σe vk( ) σa vk( )+( ) Pk

+ z( ) n2 Γk σa vk( ) Pk+ z( ) nt Γk σe vk( ) P0k n2 Γk αRS vk( ) αEL+( ) Pk

+⋅–⋅ ⋅ ⋅+⋅ ⋅⋅–⋅ ⋅⋅=

αEL vk( ) l vk( ) αRS vk( )–=

Pk+

Pk_

Prefk_

Prefk+

dPk+

dz--------- σe vk( ) σa vk( )+( ) Pk

+ z( ) n2 Γk σa vk( ) Pk+ z( ) nt Γk σe vk( ) P0k n2 Γk αRS vk( ) Pk

+⋅–⋅ ⋅ ⋅+⋅ ⋅⋅–⋅ ⋅⋅=

dPrefk_

dz---------------– σe vk( ) σa vk( )+( ) Prefk

_z( ) n2 Γk σa vk( ) Prefk

_z( ) nt Γk C αRS vk( ) Pk

+⋅⋅+⋅ ⋅⋅–⋅ ⋅⋅=

330

ERBIUM DOPED FIBER

where is the background loss caused by Rayleigh scattering, and is the backscattering capture fraction. The component has the option of loading the capture fraction from a file (wavelength dependent) or generating a theoretical capture fraction using the definition given by [2]:

Where is the fiber numerical aperture, is the refractive index of the fiber and depends on the refractive index profile. For single mode fibers a typical value for is 4.55.

The Rayleigh background loss in a fiber is given by [3]:

The first term (0.63 dB/km) is the scattering loss for pure silica fiber at 1000 nm, and the second term accounts for the material and geometrical dependence. The Raleigh constant parameter, , generally is equal to about 70 dB/km for Ge co-doped fiber, and about 150 dB/km for Aluminum co-doped fiber. The index difference can be derived from the numerical aperture, , as:

where it is assumed that the fiber refractive index is approximately 1.45.

In accordance with Equation 20 - Equation 23, the equation that gives the density population in the metastable level, Equation 10, was modified to take into account the reflected powers in the calculation for the steady state case.

(23)

(24)

(25)

(26)

dPk_

dz---------– σe vk( ) σa vk( )+( ) Pk

_z( ) n2 Γk σa vk( ) Pk

_z( ) nt Γk σe vk( ) P0k n2 Γk αRS vk( ) Pk

_⋅–⋅ ⋅ ⋅+⋅ ⋅⋅–⋅ ⋅⋅=

dPrefk+

dz--------------- σe vk( ) σa vk( )+( ) Prefk

+ z( ) n2 Γk σa vk( ) Prefk+ z( ) nt Γk C αRS vk( ) Pk

_⋅⋅+⋅ ⋅⋅–⋅ ⋅⋅=

αRS vk( ) C

C NAno--------

2 1mn------⋅=

NA nomnmn

αRS vk( )

αRS vk( ) 0.63 KR∆n+( ) 1000nmλ nm( )

-------------------- 4

=

KR∆n

NA

∆n NA2

2∗1.45----------------=

n2

331

ERBIUM DOPED FIBER

Double Rayleigh scatteringDouble Rayleigh scattering occurs when a portion of the backscattered signal is reflected again and it is recoupled to the forward direction. It is a problem because it creates paths of different lengths for signals to travel. It is considered in the model changing Equation 22 and Equation 24 by:

The buildup of backscattered light is always included in the Giles calculation, but it can be neglected by setting the capture fraction to zero or not including the Rayleigh scattering in the simulation. The degradation of EDFA performance from internal backscattering has been reported in [3] and [4].

Er3+ - Er3+ Interaction EffectsThe Erbium Doped Fiber Amplifier component allows the user to consider interactions between neighboring ions. The exchange of energy between neighboring ions is also known as "Concentration Quenching". The most important ion-ion interaction for EDFA is the stepwise up-conversion shown in Figure 2. Initially, there are two ions at the metastable level. Energy is transferred from the donor ion, which falls back to the ground level, and the acceptor ion, which returns to the metastable level by phonon transitions, after being excited to one of the upper levels. The net result is that two excited ions become one excited ion so that the quantum efficiency is reduced. Therefore, it has a negative impact on amplifiers.

Figure 2 Stepwise up-conversion

Stepwise up-conversion becomes stronger as the distance between the doped ions decreases, i.e. as the concentration increases. Depending on the fiber material, it

(27)

(28)

dPrefk_

dz---------------– σe vk( ) σa vk( )+( ) Prefk

_z( ) n2 Γk σa vk( ) Prefk

_z( ) nt Γk C αRS vk( ) Pk

+ Prefk++( )⋅⋅+⋅ ⋅⋅–⋅ ⋅⋅=

dPrefk+

dz--------------- σe vk( ) σa vk( )+( ) Prefk

+ z( ) n2 Γk σa vk( ) Prefk+ z( ) nt Γk C αRS vk( ) Pk

_Prefk

_+( )⋅⋅+⋅ ⋅⋅–⋅ ⋅⋅=

332

ERBIUM DOPED FIBER

becomes significant when the concentration is greater than about 1000 ppm. There are three models to account for stepwise up-conversion.

Homogeneous Upconversion

Considering that the ions are independent, i. e., if one ion is excited to the state this would not prevent a neighboring ion from also being excited to the state.

The upconversion fluorescence intensity can be calculated redefining Equation 1 as [5]:

Where is the branching ration between the - transition (980nm) and the - nonradiative transition; is the two-particle upconversion coefficient ( is concentration independent). In [7], the value found for the and

parameters were 1e4 and 1e-22 ( ), respectively. Considering the steady-state case, the rate equation (29) becomes:

Inhomogeneous Pair Induced Quenching

In this model [6] [7], erbium ions exist as two distinct species: single ions (no interaction with others) and clustered ions. The ions residing in each cluster can occupy only two energy levels: State 1 - all the ions in the ground state or State 2 - only one ion per cluster in the excited state. When more than one ion is excited in the cluster, the excitation energy is rapidly transferred from one ion to another, and the upconversion continues until all but one ion in the cluster occupies the metastable excited-state.

Note: It is assumed that all the clusters are of the same size and contain the same number of ions, .

For the total concentration of erbium ions, , the concentration of clustered ions is introduced as , where is the relative number of clusters and

is the percentage of ions in clusters. The concentration of single ions is .

(29)

(30)

I13 2⁄I13 2⁄

dn2 r φ z t, , ,( )dt

--------------------------------σa vk( )

hvk---------------- ik Pk z( ) n1 r φ z, ,( )

σe vk( )hvk

---------------- ik Pk z( ) n2 r φ z, ,( ) _⋅⋅ ⋅–⋅⋅ ⋅k∑=

n2 r φ z t, , ,( )τ

-----------------------------– 1 1 m⁄+( ) Ue n2 r φ z t, , ,( )⋅ ⋅–

m I11 2⁄ I15 2⁄I11 2⁄ I13 2⁄ Uc

Uc mUc m3 s⁄

n2 r φ z t, , ,( )

σa vk( )hvk

---------------- ik Pk z( ) nt r φ z, ,( )⋅⋅ ⋅k∑


-------------------------------------- ik Pk z( ) 1 1 m⁄+( )+⋅ ⋅k∑ Uc n2 r φ z t, , ,( )⋅ 1

τ---+⋅

---------------------------------------------------------------------------------------------------------------------------------------------------------------=

mk

ntnc mk k nt⋅ ⋅= k

mk k⋅ns l mk k⋅–( ) nt⋅=

333

ERBIUM DOPED FIBER

For single ions the rate equations is:

For the steady-state case:

For clustered ions, the rate equation is:

Then, the ion population in the metastable level is:

(31)

(32)

(33)

(34)

(35)

dn2Sdt

-----------σa vk( )

hvk---------------- ik r φ,( ) Pk z( ) n1S r φ z, ,( )

σe vk( )hvk

----------------k∑ ik r φ,( ) Pk z( ) n2S r φ z, ,( ) 1

τ--- n2S r φ z, ,( )⋅–⋅⋅ ⋅–⋅⋅ ⋅

k∑=

n1S n2S+ 1 mk k⋅( ) nt⋅–=

n2S r φ z, ,( )

σa vk( )hvk

---------------- ik r φ,( ) Pk z( ) 1 mk k⋅–( ) nt⋅ ⋅⋅ ⋅k∑


-------------------------------------- ik r φ,( ) Pk z( )⋅ ⋅k∑ 1

τ---+

---------------------------------------------------------------------------------------------------------=

dn2Cdt

------------σa vk( )

hvk---------------- ik r φ,( ) Pk z( ) mk k nt mk n2C⋅–⋅ ⋅( )

σe vk( )hvk

----------------k∑ ik r φ,( ) Pk z( ) n2C r φ z, ,( ) 1

τ--- n2C r φ z, ,( )⋅–⋅⋅ ⋅–⋅⋅ ⋅

k∑=

n2C r φ z, ,( )

σa vk( )hvk

---------------- ik r φ,( ) Pk z( ) mk k⋅ nt⋅ ⋅⋅ ⋅k∑

σe vk( )hvk

---------------- ik r φ,( ) Pk z( )⋅ ⋅k∑

σa vk( )hvk

---------------- ik r φ,( ) Pk z( ) mk k⋅ nt⋅ ⋅⋅ ⋅k∑+

-----------------------------------------------------------------------------------------------------------------------------------------------------------------=

+

σa vk( )hvk

---------------- ik r φ,( ) Pk z( ) 1 mk– k⋅( ) nt⋅ ⋅ ⋅ ⋅k∑

σe vk( )hvk

---------------- ik r φ,( ) Pk z( )⋅ ⋅k∑

σa vk( )hvk

---------------- ik r φ,( ) Pk z( ) mk⋅ 1τ---+⋅ ⋅

k∑+

----------------------------------------------------------------------------------------------------------------------------------------------------------

n2 r φ z t, , ,( ) n2S r φ z t, , ,( ) n2C r φ z t, , ,( )+

σa vk( )hvk

---------------- ik r φ,( ) Pk z( ) 1 mk– k⋅( ) nt⋅ ⋅ ⋅ ⋅k∑


-------------------------------------- ik r φ,( ) Pk z( ) 1τ---+⋅ ⋅

k∑

------------------------------------------------------------------------------------------------------ += =

334

ERBIUM DOPED FIBER

Homogeneous Upconversion and Inhomogeneous Pair Induced Quenching

This case is a combination of the cooperative upconversion and the pair induced upconversion. The combined model is similar to the inhomogeneous model, except that the single ions experience concentration quenching at the same rate as for the homogeneous model. Therefore, the population inversion in the steady-state becomes:

The first term on the right-hand side is for single ions and the second term is for clustered ions.

Temperature dependenceThe temperature dependence exhibited by an erbium doped fiber is mainly attributed to the variation in the occupation probability density of each manifold with temperature. In an EDFA, the gain is temperature dependent through the temperature dependence of the gain and absorption coefficients. Therefore, to represent the temperature dependence of an EDFA, the model needs properly represent the temperature dependence of and (or and ).

The temperature model in the erbium doped fiber amplifier component is based on physical intuition and use fitting parameters to generate modeling parameters at any temperature. It is assumed that the temperature dependence of an EDF is due to the variation in the occupation probability density. Using the Boltzmann's law for the level occupation and the definition that the sum of all occupation probabilities for all states of the manifold must equal unit, integral expressions for and were derived [8]. After a series of approximations, the following equations [8], outline an effective procedure for calculation of the temperature dependence of absorption and emission coefficients:

(36)

(37)

N2 r φ z t, , ,( ) N2S r φ z t, , ,( ) Nn2C r φ z t, , ,( )+=

N2 r φ z t, , ,( )

σa vk( )hvk

----------------k∑ ik Pk z( ) 1 mk– k⋅( ) n⋅ t r φ z, ,( )⋅⋅ ⋅


-------------------------------------- ik r φ,( ) Pk z( ) 1τ---+⋅ ⋅

k∑

------------------------------------------------------------------------------------------------------------ +=

+

σa vk( )hvk

----------------k∑ ik r φ,( ) Pk z( ) mk k⋅ n⋅ t⋅⋅ ⋅

σa vk( )hvk

---------------- ik r φ,( ) Pk z( )σa vk( )

hvk----------------

k∑ ik r φ,( ) Pk z( ) mk

1τ---+⋅ ⋅ ⋅+⋅ ⋅

k∑-----------------------------------------------------------------------------------------------------------------------------------------------------------

g λ( ) α λ( ) σe λ( ) σa λ( )

g λ( ) α λ( )

α λ T,( ) α λ ∞,( ) e

βa λ( )KT

--------------

⋅=

335

ERBIUM DOPED FIBER

where is the Boltzmann's constant, and is the temperature in degrees Kelvin. The fitting parameters and are both temperature independent and can be interpreted as the absorption and gain at "infinite" temperature when all energy levels of each manifolds are equally occupied, according with to Boltzmann statistics. However, a more appropriate interpretation of and is that they represent the absorption and gain coefficient when all levels of the relevant manifolds are uniformly occupied. The parameters and are expected to capture the thermal occupation probability of the initial energy level for the transition at a given wavelength.

In order to calculate functions , , , and , the component requires two sets of measurement data for and at different temperatures. One set of measured and for "infinite" temperature is provided by the component. Another set of measured data for and (or

and ) at a different temperature has to be provided by the user. With these two sets of data for and at different temperatures, the component is able to calculate the functions and . The values of and , at an arbitrary temperature defined by the user, will then be generated by the component in accordance to Equation 37 and Equation 38.

Note that the set of measured data for the gain and absorption coefficients at "infinite" temperature, and , provided by the component, are expected to represent accurately the dependence of EDF spectra for fibers with similar compositions only. However, in [9] is reported that only minor differences for a variety of silica-based, aluminum-codoped EDFs with a wide range of germanium and aluminum levels were observed [9][8]. More information about how temperature dependence can be simulated can be found in the tutorials.

(38)g λ T,( ) g λ ∞,( ) e

βe λ( )KT

-------------

⋅=

K Tα λ ∞,( ) g λ ∞,( )

α λ ∞,( ) g λ ∞,( )

βa λ( ) βe λ( )

α λ ∞,( ) g λ ∞,( ) βa λ( ) βe λ( )g λ( ) α λ( )

g λ( ) α λ( )g λ( ) α λ( )

σe λ( ) σa λ( )g λ( ) α λ( )

βa λ( ) βe λ( ) g λ( ) α λ( )

α λ ∞,( ) g λ ∞,( )

336

ERBIUM DOPED FIBER

Figure 3 Absorption and gain coefficients at infinite "temperature"

Excited-State Absorption Effect (ESA)The excited-state absorption can affect amplifiers in two ways; through parasitic absorption of pump photons, or signal photons. With pump ESA, the pump light at frequency is not absorbed from the ground level (1) of the rare earth ion, but from an excited level (2), due to the existence of a third level (3) whose energy gap

with level (2) happens to closely match the pump photon energy . This happens only if the ESA cross section overlaps with the ground state

absorption (pump absorption cross-section). In the case of signal ESA, the signal light of energy is absorbed from the metastable level (2) to a level (3), due to the same energy gap matching relation . This indicates that both pump and signal ESA result in an excess loss for the pump or the signal.

The ESA effect has been observed to occur in Er-doped fibers in several wavelength bands, but our main interest is in the 980 nm pumping band and in 1500-1620 nm signal band. In the first band, the pump ESA initiated from the metastable level

, is nonexistent near 980 nm [10]. However, pump ESA can be initiated from the energy short-lived level; where the terminal level is . Nevertheless, since the level population is rapidly damped by nonradiative decay, ESA from this level can occur only at high pump power levels [10]. Therefore, the ESA effect in the second band can be more serious in the degradation of amplifier performance, mainly in L-band amplifiers (see lesson about ESA in the tutorials) and it is taken into consideration in the Erbium doped fiber modeling.

α λ ∞,( ) g λ ∞,( )

vp

∆E E3 E2–=h vp⋅

h vs⋅∆E E3 E2 hvs≈( )–=

I413 2⁄

I411 2⁄ F4

7 2⁄

337

ERBIUM DOPED FIBER

To include the ESA effect in our two-level model, Equation 8 was modified to introduce the ESA cross-section :

Additional information about the modeling of the ESA effect can be found in [10].

Figure 4 ESA Cross-sections

Inhomogeneous BroadeningThe previous model considered only homogeneous broadening, which is satisfactory to predict the gain and noise performance of a majority of erbium doped fiber amplifiers. However, to accurately describe the saturation behavior of the amplifier and the effect of spectral-hole burning, inhomogeneous broadening has to be considered. The main assumption in the modeling of this effect is that the variation of the stark splitting from site to site due to the change of the ligand fields leads to randomization of central frequencies of the transition lines; the linewidths, the absorption and emission cross-sections, and the fluorescence lifetime do not change.

The density distribution for inhomogeneous broadening of central frequencies of the transition lines is given for a Gaussian function:

where is the inhomogeneous broadening spectral bandwidth and is the inhomogeneous line width.

(39)

(40)

σESA

dPkdz

--------- σe vk( ) σESA vk( ) σa vk( )+ +( ) Pk z( ) n2 Γk σa vk( ) Pk z( ) nt Γk P0k σa vk( ) n2 Γk⋅ ⋅⋅+⋅ ⋅⋅–⋅ ⋅⋅=

f ω( ) 4 1n 2( )⋅

π ∆ωi2⋅

---------------------- 4 1n 2( ) ω∆ωi---------

2⋅⋅–exp⋅=

∆ωi 2 π c ∆λinh λ2⁄⋅ ⋅ ⋅=∆λinh

338

ERBIUM DOPED FIBER

The observed (measured) inhomogeneous absorption and emission cross-sections, and , are the convolutions of the homogeneous absorption and

emission cross-sections, and , with the normalized inhomogeneous broadening distribution , and can be expressed by:

The description of the inhomogeneous broadening is based on the following form of the propagation equation suggested in [10]:

To include spontaneous emission, a noise source term is introduced in Equation 42.

In order to determine the homogeneous absorption and emission cross-section used in the propagation equation, a deconvolution procedure to resolve Equation 41 is applied.

In Homogeneous cross-sections, there is a description of the procedures used in the component to generate the homogeneous cross-sections.

Homogeneous cross-sectionsHomogeneous cross-sections can be derived from the experimental (inhomogeneous) cross-sections through an inversion Fourier transformation in Equation 41, (a) and (b). However, a direct deconvolution of Equation 41 has a unique

(41)a

(41)b

(42)

σaI v( ) σe

I v( )σa

H v( ) σeH v( )

f v( )

σeI v( ) f v v'–( ) σe

H v( ) v'd⋅⋅

∞–

∞

∫=

σaI v( ) f v v'–( ) σa

H v( ) v'd⋅⋅

∞–

∞

∫=

dP ωk( )dz

------------------ ρ Γk P ωk( ) ωd f ω( ) σaH ωk ω–( )

σeH ωk ω–( )

σaH ωk ω–( )

----------------------------

Pmh v Am⋅ ⋅---------------------σa

H ωm ω–( ) τ⋅m∑

1Pm

h v Am⋅ ⋅--------------------- σa

H ωm ω–( ) σeH ωm ω–( )+( ) τ⋅ ⋅

m∑+

-------------------------------------------------------------------------------------------------------------------------- _⋅ ⋅ ⋅ ⋅

∞–

∞

∫⋅ ⋅ ⋅=

ρ– Γk P ωk( ) ωd f ω( ) σaH ωk ω–( )

1Pm

h v Am⋅ ⋅---------------------σe

H ωm ω–( ) τ⋅m∑+

1Pm

h v Am⋅ ⋅--------------------- σa

H ωm ω–( ) σeH ωm ω–( )+( ) τ⋅ ⋅

m∑+

--------------------------------------------------------------------------------------------------------------------------⋅ ⋅ ⋅

∞–

∞

∫⋅ ⋅ ⋅

339

ERBIUM DOPED FIBER

solution only when the functions , and their evanescent tails are well defined analytically. This is not the case with experimental line shapes.

Nevertheless, there is a possibility of fitting the line shapes with a superposition of Gaussians functions such as

where , , and are the Gaussian line shapes parameters for the fitting. The parameter is the number of Gaussians.

Using this superposition of Gaussian functions; the deconvolution of Equation 41 can be calculated analytically. With the Gaussians functions line shapes parameters found in the numerical fitting, the homogeneous emission and absorption cross-sections can calculated in accordance with the inhomogeneous line width ( ) provided by the user using the definition [10]:

The Erbium-doped fiber component is able to do the fitting of the cross-sections provided by the user using the number of Gaussian functions ( ) determined by the Number of Gaussians parameter.

(43)

σaI λ( ) σe

I λ( )

I λ( ) ai 4 1n 2( )λ λi–( )2

∆λi2

---------------------⋅

⋅–

exp⋅

i

nG

∑=

ai λi ∆λinG

∆λinh

σa e,H λ( ) ai

a e, ∆λi

∆λi2 ∆λinh

2–--------------------------------- 4 1n 2( )

λ λi–( )2

∆λi2

---------------------⋅⋅–

exp⋅ ⋅

i

nG

∑=

nG

340

ERBIUM DOPED FIBER

Figure 5 Homogeneous (a) absorption and (b) emission cross-sections

341

ERBIUM DOPED FIBER

Approximations of Giles-Desurvire rate and propagation equations

Saleh modelThe Saleh model is an approximation of the propagation and rate equations for a two-level system in the steady-state case. This allows for an analytical solution of the equations by means of a transcendental equation, instead of N coupled differential equations [11]. This model could be successfully applied to the study of the small signal gain and saturated gain, optimum fiber length, and saturated power. The theory uses some simplifying assumptions. First, although spontaneous decay is accounted for, amplified spontaneous emission (ASE) is neglected. This is valid for fiber lasers above threshold and for fiber amplifiers when the input signal power is significantly above the equivalent ASE noise input power, as discussed in [11]. Second, it is assumed that there is no excited state absorption (ESA) at any of the pump or signal wavelengths. Third, it is assumed that field and ion distributions are independent of fiber position and power levels. These assumptions are satisfactory in the case of typical doped fibers that have a doped fiber radius less than the core fiber radius. Background loss is also neglected, as with three level ions such as erbium, the absorption by the rare earth ions is typically much greater than other losses.

Using the assumptions, Equation 8 could be integrated analytically from 0 to L [11]. The result is given by the following expression for the output photon flux :

where is the output photon flux for kth signal,

is the input photon flux for kth signal

is the total output photon flux

is the total input photon flux.

Summing Equation 44 over all k signals yields:

which is a implicit equation for the total output photon flux . Note that is completely determined, given the input flux, by the following four fiber parameters;

(fiber length). Solving Equation 45 for allows for the determination of the output fluxes of each individual signal through Equation 44.

Since the Saleh model neglects ASE, it becomes less accurate for cases in which ASE becomes significant, e.g. for low input powers (less than about -20 dBm,

(44)

(45)

Qk

Qkout Qk

in αkL–αk gk+( )

ζ---------------------- Qtot

in Qtotout–( )⋅+

exp=

Qkout Pk

out hvk( )⁄=

Qkin Pk

in hvk( )⁄=

Qtotout Qk

out

k∑=

Qtotin Qk

in

k∑=

Qtotout Qk

in αkL–αk gk+( )

ζ---------------------- Qtot

in Qtotout–( )⋅+

expk∑=

Qtotout Qtot

out

αk gk ζ and L, , , Qtotout

342

ERBIUM DOPED FIBER

depending on the gain and signal wavelengths). In these cases, the accuracy is improved by using an equivalent ASE input, which inputs effective input beams at both ends of the fiber with equivalent input powers:

where for the forward ASE, and for the backward ASE. is the spectral width of the noise beams. The spontaneous emission factor is given by:

where = is the ratio of cross-sections.

The Saleh model has the advantage that longitudinal integrations are not required, so it is much faster to solve. Note that unlike literature that typically uses one or two equivalent ASE beams centered at the spectral peaks near 1532 nm and 1555 nm, this component has an equivalent ASE beam for each of the bins defined in the Noise tab.

Jopson modelThe Saleh model only estimates the pump and signal powers, and equivalent ASE at the doped fiber output. These values are used to estimate the population inversion at the doped fiber ends. However, no information is obtained about the values along the fiber. Jopson and Saleh extended the Saleh model to obtain estimates of the powers and inversion levels along the fiber [12]. The photon flux in distance can be determined by:

where is defined by:

and it is computed from the transcendental equation:

In order to obtain the pump, signals, and equivalent ASE powers and population inversion along the fiber, starting from either end of the fiber, this equation can be solved for in every user-defined step.

(46)

(47)

(48)

(49)

(50)

hvkPkin 2nsp v zin,( )∆v hvk=

zin 0= zin L= ∆v

nsp v zin,( )n2 zin( )

n2 zin( ) n1 zin( )–---------------------------------------- ε v( )⋅=

ε v( ) σa σe⁄

Qk z

Qk z( ) Qk 0( ) ukαkz– ukαk gk+( )

ζ---------------------- Q 0( ) Q z( )–( )⋅+

exp=

Q z( )

Q z( ) ukQk z( )k∑=

Q z( ) ukQk z( )∑ eukαka–

euk Q 0( ) Q z( )–( ) αk gk+( ) ζ⁄⋅[ ]=

343

ERBIUM DOPED FIBER

NoiseThe spontaneous-emission noise at wavelength , of a single polarization, emitted in a single direction by a section of amplifier of length is given by:

where can be determined using Equation 50 and Equation 7.

The amplified spontaneous emission noise (ASE) emitted from the output or input end of the amplifier at wavelength can be obtained by multiplying the spontaneous emission from each section of the amplifier by the amplifier gain at from that section to the desired end of the amplifier. The gain is given by:

, where is the gain from the input ( ) to the length

and

, where is the gain from the length to the output .

λkdz

dP gk n2 z( ) ∆v dz⋅⋅⋅=

n2 z( )

λkλk

Gk 0 z,( ) eukαkz–

euk Q 0( ) Q z( )–( ) αk gk+( ) ζ⁄⋅[ ]= Gk 0 z,( )

z 0= z

Gk z L,( ) eukαk L z–( )–

euk Q z( ) Q L( )–( ) αk gk+( ) ζ⁄⋅[ ]= Gk z L,( )

z L

344

ERBIUM DOPED FIBER

Input Parameters DescriptionMost of the input parameters for the component were described in the sections before and they can be easily linked to a particular effect or equation. However, there are some parameters that were not described yet or they are load from files. In this section those parameter are explained,

Main tabThis tab contains the basic parameters of the erbium-doped fiber. All of them are well described in the technical description. However, there is a new parameter (Input data parameter) that gives the user the choice to enter the saturation parameter or to enter the fiber parameters (core radius, doped radius, numerical aperture, and erbium density population).

Cross-sections tabIn this tab the user defines which cross-section file has to be loaded and what characteristics it has. There are two options available to prepare the cross-section file, which is specified in an ASCII file. The first option is to provide directly the cross-section in an input file with three columns. The first column refers to the wavelength (or frequency) in [m], [nm], [Hz] or [THz] units; the File frequency unit parameter defines the unit of this column. The second column gives the absorption cross-section in [m2] units. The third column gives the emission cross-section file in [m2] units. The unit of the second and third column must be in [m2]. As an example, one possible cross-section file format is:

(nm)

975 1.95386E-25 0

976 2.07791E-25 0

977 2.20195E-25 0

978 2.26852E-25 0

979 2.13394E-25 0

980 1.99935E-25 0

981 1.86477E-25 0

982 1.73019E-25 0

983 1.5956E-25 0

:

:

1450 5.88956E-26 1.78862E-26

1451 6.19338E-26 1.87881E-26

1452 6.50958E-26 1.97301E-26

1453 6.83832E-26 2.06921E-26

λλ nm[ ] σa m2[ ] σe m2[ ]

345

ERBIUM DOPED FIBER

The second option is to provide the absorption and gain coefficients (or Giles parameters) as input parameters that are converted to cross-section by internal routines in the software. The file format in this case contains three columns. The first column refers to the wavelength (or frequency) in [m], [nm], [Hz] or [THz] units; the File frequency unit parameter defines the unit of this column. The second column gives the absorption coefficient in [dB/m] units. The third column gives the emission coefficient in [dB/m] units. The unit of the second and third column must be in [dB/m]. An example of this input file is:

When the EDF component load the cross-section file, it detects whether the file contain the Giles parameters ( and ) or cross-section parameters ( and ).

1454 7.17971E-26 2.16742E-26

1455 7.53386E-26 2.26767E-26

1456 7.90081E-26 2.37003E-26

1457 8.2806E-26 2.4746E-26

1458 8.67324E-26 2.58149E-26

1459 9.07873E-26 2.69085E-26

:

(nm)

977 5 0

978 5 0

979 5 0

980 5 0

981 5 0

:

:

1460 1.357 0.29

1461 1.417 0.309

1462 1.464 0.328

1463 1.525 0.35

1464 1.562 0.365

1465 1.562 0.387

1466 1.562 0.411

:

(nm)λλ nm[ ] σa m2[ ] σe m2[ ]

λλ nm[ ] α dB m⁄[ ] g∗ dB m⁄[ ]

g λ( ) α λ( )σa vk( ) σe vk( )

346

ERBIUM DOPED FIBER

The parameter OptiAmplifier format is used to allow the component load cross-sections files originated from the software OptiAmplier. Therefore, if the user wants to load a cross-section under the crs format (format used in the OptAmplifier software), the OptiAmplifier format parameter has to be set TRUE.

Enhanced tabThe enhanced tab defines the parameters related to the background loss, Rayleigh scattering, interaction effects, ESA, and temperature dependence. First, the user can choose the Background loss data type parameter that determines the background loss through the loss at 1310nm (Loss at 1310 nm parameter) or using a wavelength dependent background loss loaded from a file. In the second case, the user has to specify the name of the file contained the losses in the Background loss file name parameter. The format of this file must be similar to the following example:

The user can include the Rayleigh scattering effect or not in the simulations through the parameter Include Rayleigh scattering. If the Include Rayleigh scattering parameter is TRUE, then the user has to specify the value of the Rayleigh constant. The Backscattering capture parameter determines if the component will generate the capture fraction using Equation 25, or the user will provide a file with the capture fraction - in this case the user should specify the file name in the Rayleigh capture file name parameter and the file has to be in the format similar to the below:

In the case of interaction effects, the user has to decide to include or not this effect through the parameter Include ion-ion interaction effects. If the user chooses to include this effect, the parameter Ion-Ion interaction effect has to specify

(nm)

1460 10

1461 10.5

1462 10.2

1463 10.1

1464 10.3

(nm)

1460 -20

1461 -21.5

1462 -21

1463 -20.5

1464 -20.48

Er+3 Er+3–

λλ nm[ ] α dB km⁄[ ]

λλ nm[ ] C dB[ ]

Er+3 Er+3–

347

ERBIUM DOPED FIBER

which interaction effect will be considered in the simulations; Homogeneous upconversion, pair-induced quenching, or a combination of both. When the ion-ion effect is defined, then the parameters necessaries for that effect will be enabled. Upconversion coefficient, ions per cluster, and relative number of clusters are the parameters that have to be specified depending on the effect considered.

The user can include the temperature dependence in EDF model setting the parameter Include temperature dependence to TRUE. After this, the user has to define in which temperature, the cross-section defined in the cross-sections tab, was measured (Cross-section temperature parameter). With these parameters and the cross-section at infinite temperature stored in the component, it is possible to calculate the parameters and from Equation 37 and Equation 38. The other parameter to be defined is the temperature that will be considered in the simulation (Temperature parameter). For more information, refer to the tutorial about temperature dependence.

The ESA effect can be included in the EDF simulation. In this case the user has to set the parameter Include ESA effect to TRUE. After this, the user has to provide the ESA cross-section. Similar to the cross-sections in the cross-section tab, the ESA cross-section can be in the Giles format [ ] or cross-section format [ ]. The difference is the ESA cross-section file must have only two columns: (1) wavelength (or frequency) in [m], [nm], [Hz] or [THz] units and (2) the ESA cross-section. The unit of the wavelength column has to be the same as defined in the File frequency unit parameter (Cross-sections tab).

The last parameter is Extract ESA from emission. If this parameter is TRUE, it means that the second column of the ESA file contains the ESA cross-section and the emission cross-section together, so the component has to extract the ESA cross-section from this file. If the Extract ESA from emission parameter is FALSE, the component assumes that the second column contains only the ESA cross-section. An example of ESA file is:

(nm)

1449.91984 0.32257

1451.30261 0.35195

1452.68537 0.38317

1454.06814 0.4175

1455.4509 0.4571

:

Er+3 Er+3–

βa λ( ) βe λ( )

dB m⁄ m2

λλ nm[ ] g∗ dB m⁄[ ]

348

ERBIUM DOPED FIBER

For more information, refer to the tutorial about ESA.

Numerical tabThe numerical tab contains most of the options related to the different models or approximations used in the EDF model. In the Calculation algorithm parameter, the user can choose between the four possible models: (1) Saleh, (2) Jopson, (3) Giles, and (4) Inhomogeneous. These four possible models are described in the technical background. If a model is selected, for example the model number 3 (Giles model), the EDF component will start the simulation process from the first model (Saleh) until the model chose by the user (Giles model). Figure 6 details how the component works.

1571.60321 4.08152

1572.98597 3.81553

1574.36874 3.60032

1575.7515 3.37804

1577.13427 3.20419

1578.51703 3.05017

:

1648.98 1.43477

1649.23 1.4325

1649.48 1.49899

1649.73 1.42809

1649.98 1.42593

1650.23 1.49333

(nm)λλ nm[ ] g∗ dB m⁄[ ]

349

ERBIUM DOPED FIBER

Figure 6 Diagram describing the process order of the algorithm models

The EDF component's preprocessing is done to improve the speed of convergence in the model selected by the user. This preprocessing is done in accordance with the complexity of each model.

The user defines the parameter Relative error that indicates the threshold value which the component uses to decide if the results from the iterative process have converged. Another parameter is the Max. number of iterations. This parameter defines the maximum number of iterations allowed for the numerical method to reach the value determined by the Relative error parameter. The parameter Number of longitudinal steps defines the minimum number of steps in the fiber to be considered in the Jopson, Giles, and inhomogeneous method.

If the Inhomogeneous algorithm is chose, then the user has to specify the parameter Inhomogeneous accuracy. This parameter determines the tolerance of the numerical integration of Equation 42, and directly influences the simulation time. Some simulations have shown us that this parameter shoud be between 0.01 and 0.001 to obtain accurate results in a reasonable time.

The user can make their selection via the Overlap factor data parameter, by determining if the component will calculate the overlap integral or the component or load the overlap factor from a file. For the calculation case, the Geometrical model parameter has to be defined. The Geometrical model parameter indicates if the

350

ERBIUM DOPED FIBER

component will use one of the Gaussian approximations (Equation 13 - Equation 16) or the LP01 mode to calculate the overlap integral.

Another possible method to calculate the overlap integral is to consider the power dependence on it. The Overlap factor parameter determines if the power dependence has to be taken into consideration. In this case, Equation 6 is solved numerically for the LP01 mode and the number of integrations to be done in the fiber is defined by the Nr. of transverse integration parameter. In the other way, the confinement factor is calculated in accordance with the Geometrical model parameter.

If the overlap factor is loaded from a file, the user has to specify the file name in the Overlap factor file name parameter, and the file has to be the same as the format below:

If the Inhomogeneous algorithm is chose, then the homogeneous absorption and emission cross-sections are necessary for the inhomogeneous broadening model. In this case, the component generates the homogeneous cross-sections from the measured cross-sections, as explained in the technical background. For this purpose, the user has to specify the number of Gaussians to be used in the fitting and the value of the inhomogeneous linewidth.

(nm)

1449.91984 0.45

1451.30261 0.44

1452.68537 0.43

1454.06814 0.42

1455.4509 0.41

λλ nm[ ] Γ

351

ERBIUM DOPED FIBER

References[1] C. Randy Giles, and Emmanuel Desurvire, "Modeling Erbium-Doped Fiber Amplifiers". IEEE

Journal of Lightwave Technology, Volume: 9 Issue: 2, Feb. 1991, Page(s): 271 - 283.

[2] Fiber Optic Test and Measurement, Edited by Dennis Derickson, 1997.

[3] S. L. Hansen, K. Dybdal, and C. C. Larsen. "Gain Limited in Erbium-Doped Fiber Amplifiers Due to Internal Rayleigh Backscattering". IEEE Photonics Technology Letters, Volume 4, Issue 6, Jun. 1992.

[4] P. F. Wysocki, G. Jacobovitz-Veselka, D. S. Gasper, S. Kosinski, J. Costelloe, and S. W. Granlund. "Modeling, Measurement, and a Simple Analytic Approximation for the Return Loss of Erbium-Doped Fiber Amplifiers". IEEE Photonics Technology Letters, Volume: 7, Issue: 12, Dec. 1995.

[5] P. Blixt, J. Nilsson, T. Carlnas, and B. Jaskorzynska. "Concentration-Dependent Upconversion in Er3+-Doped Fiber Amplifiers: Experiments and Modeling". IEEE Photonics Technology Letters, Volume: 3 Issue: 11, Nov. 1991.

[6] P. Myslink, D. Nguyen, and J. Chrostowski. "Effects of Concentration on the Performance of Erbium-Doped Fiber Amplifiers". Journal of Lightwave Technology, volume 15, Issue 1, Jan. 1997.

[7] Blixt, P.; Jaskorzynska, B.; Nilsson, J. "Performance reduction and design modification of erbium-doped fiber amplifiers resulting from pair-induced quenching". IEEE Photonics Technology Letters , Volume: 5 Issue: 12 , Dec 1993.

[8] M. Bolshtyansky, P. F. Wysocki, N. Conti. "Model of Temperature Dependence for Gain Shape of Erbium-Doped Fiber". Journal of Lightwave Technology, volume 18, Issue 11, Dec 2000.

[9] P. F. Wysocki, N. Conti, and D. Holcomb. "Simple Modeling Approach for the Temperature Dependence of the Gain of Erbium-Doped Fiber Amplifiers". SPIE Conference on Optical Devices for Fiber Communication, Volume 3847, 1999.

[10] Emmanuel Desurvire. "Erbium-Doped Fiber Amplifier: Principles and Applications", John Wiley & Sons.

[11] A. A. M. Saleh, R. M. Jopson, J. D. Evankow, and J. Aspell. "Modeling of Gain in Erbium-Doped Fiber Amplifiers". IEEE Photonics Technology Letters, Volume: 2 Issue: 10, Oct. 1990, Page(s): 714 - 717.

[12] R. M. Jopson, A. A. M Saleh. "Modeling of Gain and Noise in Erbium-Doped Fiber Amplifiers". Fiber Laser Sources and Amplifiers, SPIE Volume: 1581, 1991, Page(s): 114 - 119.

[13] C. R. Giles, C. A. Burrus, D. J. DiGiovanni, N. K. Dutta, and G. Raybon. "Characterization of Erbium-Doped Fibers and Application to Modeling 980 nm and 1480 nm Pumped Amplifiers". IEEE Photonics Technology Letters, Volume: 3 Issue: 4, Apr. 1991, Page(s): 363 -365.

[14] "Rare-Earth-Doped Fiber Laser and Amplifiers", Edited by M. J. F. Digonnet, 2001.

[15] P. C. Becker, N. A. Olsson, and J. R. Simpson. "Erbium-Doped Fiber Amplifiers: Fundamentals and Technology". Optics and Photonics, 1999.

352

ER-YB CODOPED FIBER

Er-Yb codoped fiber

This component simulates a bidirectional Erbium-Ytterbium codoped fiber. The component solves numerically the rate and propagation equations for the steady-state case.

Ports

Parameters

Main








Length

Doped fiber length

1 m — [0,1000000]

Core radius

Doped fiber core radius

2 µm — [1,1e100]

Doping radius

Doped radius

2 µm — [1,1e100]

Numerical aperture

Specifies numerical aperture of fiber

0.15 — — [0.1,1]

Loss data type Constant Constant, FromFile

Signal loss

Fiber loss at signal range

0.10 dB/m — [0,1e100]

353

ER-YB CODOPED FIBER

Doping

Cross-sections

Pump loss

Fiber loss at pump range

0.15 dB/m — [0,1e100]

Loss vs. wavelength Loss.dat — — —



Er ion density

Specifies Erbium doping in the fiber

5.14e+025 m-3 — [1,1e100]

Yb ion density

Specifies Ytterbium doping in the fiber

6.2e+026 m-3 — [1,1e100]

Er metastable lifetime

Specifies the Erbium metastable lifetime

10 ms — [1e-100, 1e100]

Yb metastable lifetime

Specifies the Ytterbium metastable lifetime

1.5 ms — [1e-100, 1e100]



OptiAmplifier format

Determines if format of cross-section file is an OptiAmplifier file


File frequency unit



Er cross-section file name

Specifies Erbium cross-section file name


Yb cross-section file name

Specifies Ytterbium cross-section file name

Ytterbium.dat — — —



354

ER-YB CODOPED FIBER

Enhanced

Numerical


Default Unit Units Value range

Calculate upconversion

Component calculates C16 and C14 based on ion density


C14

Cross relaxation coefficient between level 1 and 4

5.2834e-024 — m-3/s [1e-100, 1e100]

C16

Cross relaxation coefficient between level 1 and 6

3.44e-022 — m-3/s [1e-100, 1e100]

Cup

Homogeneous upconversion coefficient from level 2

5.2834e-024 — m-3/s [1e-100, 1e100]

A32

Nonradiative emission rate from level 3 to level 2

1000000000 — 1/s [1,1e100]

A43

Nonradiative emission rate from level 4 to level 3

1000000000 — 1/s [1,1e100]



Relative error

Specifies maximum acceptable difference between two consecutive iterations to complete the iteration process

0.0001 — — [1e-100,1]

Maximum number of iterations

Specifies the maximum number of times for iteration process

150 — — [1,1e8]

Longitudinal steps

Specifies the number of longitudinal steps in the fiber

100 — — [1,1e8]

Radial steps

Specifies the number of radial steps for integration

50 — —

355

ER-YB CODOPED FIBER

Graphs

Simulation

Noise

Numerical solverDefines whether the numerical solver is used instead of analytical solutions for the rate equations.




Calculate graphs False — — True, False

Number of distance steps 20 — — [1,1e8]

Number of wavelength steps 20 — — [1,1e8]

Linear scale True — — True, False

Minimum value -50 — dBm ]1e-100, 1e100[

Pump reference wavelength 1400 nm [100, 1900]



Enabled

Defines whether the component is enabled or not


Longitudinal monitor True — — True, False

Number of monitors 10 — — [1,1000]



Noise center frequency

Determines the noise center frequency

193.4 THz Hz, THz, nm [30, 30e5]

Noise bandwidth

Bandwidth to create noise bins


Noise bins space

Specifies the noise bins spacing


[1,1000]



356

ER-YB CODOPED FIBER

Random numbers

Noise threshold


-100 dB — ]-INF, 0[

Noise dynamic

Threshold ratio for adaption of noise bins

3 dB — [0, +INF[

Convert noise bins

Determines if generated noise bins are incorporated into signal

Convert noise bins

— — True, False






Random seed index


0 — — [0, 4999]



357

ER-YB CODOPED FIBER


Er-Yb Codoped Fiber Propagation and Rate EquationsIn order to give flexibility to change the waveguiding parameters of the Er3+ - Yb3+ codoped fiber for large signal and high pump power applications the extended model for Er3+ - Yb3+ codoped fiber presented in [1] is used.

Figure 1 Energy levels for Er3+ - Yb3+ system

Population densities of the levels of have been included together with the upconversion from the pump level . The model takes into account propagation of the forward and backward amplified spontaneous powers for both the pump and the signal wavelength range.

Depending on the pump wavelength, pump energy can be absorbed by both the Er ions in the and by the ions in the ground levels. Ytterbium ions excited to the level transfer their energy to neighboring Erbium ions in the

ground level, exciting them to the pump level from where they rapidly relax to the metastable level. The backtransfer from the Er pump level to the Yb ground level is neglected.

4 11 2⁄ and 4 9 2⁄ Er3+

4 11 2⁄

4 15 2⁄ Yb3+ F27 2⁄

F25 2⁄

4 15 2⁄ 4 11 2⁄4 13 2⁄

358

ER-YB CODOPED FIBER

Let us denote the , , and the levels of as levels 1, 2, 3, and 4, and the and the levels of as levels 5 and 6, and their population densities as N1, N2, N3, N4, N5, and N6, respectively. The uniform upconversion mechanisms from the erbium metastable and pump levels are modeled by quadratic terms in N2 and N3, with a concentration dependent upconversion coefficient. The pair induced energy transfer process from to is described by a cross relaxation coefficient [2]. The rate equations for the above atomic populations are:

In these equations, the terms represent the stimulated transition rates between the i and j levels, , are the spontaneous emission lifetimes for and

levels, , are the nonradiative relaxation rates, and are the upconversion and cross-relaxation coefficients. The signal absorption, signal emission, pump absorption, and pump emission rates, are given by:

(1)

(2)

(3)

(4)

(5)

(6)

4 15 2⁄ 4 13 2⁄ 4 11 2⁄ 4 9 2⁄ Er3+

F27 2⁄ F2

5 2⁄ Yb3+

Yb3+ Er3+

∂N1∂t

--------- W12N1 W13N1–N2τEr------- W21N2 CupN2

2 C14N1N4– CupN32 CcrN1N6–+ + + +–=

∂N2∂t

--------- W12N1 W21N2–N2τEr-------– A32N3 2CupN2

2– 2C14N1N4+ +=

∂N3∂t

--------- W13N1 A32N3– A43N4 2CupN32– CerN1N6+ +=

∂N4∂t

--------- 2CupN22 C14N1N4– A43N4– CupN3

2+=

∂N6∂t

--------- W56N5 N6τYb--------– W65N6– CerN1N6–=

WijτEr τYb 4 13 2⁄

F25 2⁄ A32 A43 Cup C14 C16, ,

W12 W21 W13 W56 W65, , , ,

W12 r z,( )σ12 vs( )

hvs------------------Ps z( ) E r vs,( ) 2 σ12 v( )

hv---------------- PASE

+ z v,( ) PASE_

z v,( )+[ ] E r v,( ) 2⋅ vd0

∞

∫+=

359

ER-YB CODOPED FIBER

where are the frequency dependent and emission and absorption cross sections, respectively, is the

Planck’s constant, , are the forward and backward propagating optical powers at frequency in a frequency interval , and at a longitudinal fiber coordinate . They represent the forward and backward ASE powers due to the - transition at , and also the ASE powers due to the - transition at .

is the signal power, the pump power, are the signal and pump frequencies, and is the field distribution of the mode normalized according to

The total and ion density distributions , are assumed to be constant within the whole or a part of the fiber core, and along the fiber length (top hat shaped with the diameter of 2b).

(7)

(8)

(9)

(10)

(11)

W21 r z,( )σ21 vs( )

hvs------------------Ps z( ) E r vs,( ) 2 σ21 v( )

hv---------------- PASE

+ z v,( ) PASE_

z v,( )+[ ] E r v,( ) 2⋅ vd0

∞

∫+=

W13 r z,( )σ13 vp( )

hvp------------------Pp z( ) E r vp,( ) 2 σ13 v( )

hv---------------- PASE

+ z v,( ) PASE_

z v,( )+[ ] E r v,( ) 2⋅ vd0

∞

∫+=

W56 r z,( )σ56 vp( )

hvp------------------Pp z( ) E r vp,( ) 2 σ56 v( )

hv---------------- PASE

+ z v,( ) PASE_

z v,( )+[ ] E r v,( ) 2⋅ vd0

∞

∫+=

W65 r z,( )σ65 vp( )

hvp------------------Pp z( ) E r vp,( ) 2 σ65 v( )

hv---------------- PASE

+ z v,( ) PASE_

z v,( )+[ ] E r v,( ) 2⋅ vd0

∞

∫+=

σ21 v( ) σ65 v( ) σ12 v( ) σ13 v( ) and σ56 v( ),,,,Er3+ Yb3+ h

PASE+ z v,( ) PASE

_z v,( )

v ∆vz

4 13 2⁄ 4 15 2⁄ 1400nm λ 1650nm< <F2

5 2⁄ F27 2⁄ 850nm λ 1100nm< <

Ps z( ) Pp z( ) vs vp,E r v,( ) LP01

2π E r v,( ) 2r rd0

∞

∫ 1=

Er3+ Yb3+ NEr t NYb

t

360

ER-YB CODOPED FIBER

They satisfy the conservation equations

Propagation of the pump power along the active fiber is described by the following differential equation:

where is the radius of - codoped part of the fiber core. The signal power and the ASE powers in both the pump and the signal wavelength range are amplified according to:

where is the frequency dependent background loss of the active fiber and the emission and absorption factors , are determined from the corresponding emission and absorption cross sections as overlap integrals between the intensity distribution and the population densities of the , and

, levels defined in:

(12)

(13)

(14)

(15)

(16)

(17)

NEr t N1 r z,( ) N2 r z,( ) N3 r z,( ) N4 r z,( )+ + +=

NYb t N5 r z,( ) N6 r z,( )+=

∂Pp z vp,( )∂z

------------------------- 2π σ56 vp( )N5 r z,( ) σ13 vp( )N1 r z,( ) σ65 vp( )N6 r z,( )–+[ ] E r vp,( ) 2rdr α vp( )+0

b

∫ Pp z vp,( )=

b Er3+ Yb3+

∂Ps z vs,( )∂z

------------------------ ge z vs,( ) ga z vs,( )– α vs( )–[ ]Ps z vs,( )=

∂PASE ± z v,( )

∂z---------------------------- 2hv∆vge z vs,( ) ge z v,( ) ga z v,( )– α v( )–[ ]±± z vs,( )PASE

± z v,( )=

α v( )ge z v,( ) z v,( )

LP01 F25 2⁄ 4 13 2⁄F2

7 2⁄ 4 15 2⁄

ge z v,( )2πσ65 v( ) N6 r z,( ) E r v,( ) 2r r…850nm λ 1100nm< <d

0

b

∫

2πσ21 v( ) N2 r z,( ) E r v,( ) 2r r…1400nm λ 1650nm< <d0

b

∫

=

361

ER-YB CODOPED FIBER

These equations form a system of coupled differential equationsthat are solved by numerical integration along the active fiber, using the Runge-Kutta method. Population densities , , , , , and

are derived from the steady-state solutions to the rate equations [1] - [5] together with conservation laws, equations [12] and [13] are substituted. Due to quadratic terms appearing in the rate equations, it is not possible to eliminate densities , , , and analytically, and so the numerical approach must be used. It was assumed that and that the upconversion coefficient and the cross-relaxation coefficient are linearly increasing functions of and respectively.

References:[1] M. Karasek, "Optimum Design of Er3+ - Yb3+ Codoped Fibers for Large-Signal High-Pump-

Power Applications", IEEE Journal of Quantum Electronics, vol. 33, pp 1699-1705, 1997.

[2] M. Federighi, F. Di Pasquale, "The Effect of Pair-induced Energy Transfer on the Performance of Silica Waveguide Amplifiers with High Er3+-Yb3+ Concentrations", IEEE Photon. Technol. Lett., vol 7, pp. 303-305, 1995.

(18)

(19)

(20)

ga z v,( )2πσ56 v( ) N5 r z,( ) E r v,( ) 2r r…850nm λ 1100nm< <d

0

b

∫

2πσ12 v( ) N1 r z,( ) E r v,( ) 2r r…1400nm λ 1650nm< <d0

b

∫

=

N1 r z,( ) N2 r z,( ) N3 r z,( ) N4 r z,( ) N5 r z,( )N6 r z,( )

N1 r z,( ) N2 r z,( ) N5 r z,( ) N6 r z,( )C14 Cup=

Cup CcrNEr

1 NYb1

Cup 3.5 10 24–× 2.41 10 49–× NEr1 4.4 1025×–( )+=

Ccr 1.0 10 22–× 4.0 10 49–× NYb1 1.0 1025×–( )+=

362

ER-YB CODOPED WAVEGUIDE AMPLIFIER

Er-Yb codoped waveguide amplifier

This component simulates an Er-Yb codoped waveguide amplifier based on basic parameters.

Ports

Parameters

Main








Waveguide length 0.03 — m ]0,+INF[

Signal background loss

Represents the intrinsic material losses, given by the losses at 1300nm.

0 — db/m [0,+INF[

Pump background loss

Represents the intrinsic material losses, given by the losses at 1300nm.

0 db/m [0,+INF[

Refractive index data fileSame as OptiBPM’s refractive index file. Contains a uniform refractive index distribution and follows the format defined in OptiBPM. Also contains the number of points used to discretize the domain.

Index.rid — — —

Er ion density distribution fileSame as Refractive index data file, contains the Erbium ion density distribution. File must have the same discretization as the Refractive index data file, and must be filled with ones and zeros.

Erdensity.dat — — —

363


Doping

Yb ion density distribution fileSame as Er ion density distribution file, contains the Ytterbium ion density distribution. File must have the same discretization as the Refractive index data file, and must be filled with ones and zeros.

Ybdensity.dat — — —

Calculate mode in all wavelengthsIdentifies if a mode must be calculated for all the signal wavelengths. If selected, the mode solver is activated, using the refractive index distribution defined in a file for all the signal wavelengths.


Wavelength to calculate the modeIf Calculate mode in all wavelength is not selected, a signal mode, calculated at the defined wavelength, is shared for all signals. This selections makes the calculation faster after the part of the execution time is spent calculating the modes.

1550 nm [1490, 3000]

Recalculate modes every runningIdentifies if all the modes, in the pump and signal wavelengths, must be recalculated. It is suggested that this option not be selected, due to the excessive time spent recalculating the modes.

False — — False, True

Polarization for signal mode calculationPolarization used to calculate the signal modes.

TE — — TE, TM

Number of modes at pump wavelengthNumber of modes that are calculated at the pump wavelength. Read-only value. To change it, edit the “Power ratio for each pump mode” option.

2 — — [1, 10]

Polarization for pump mode calculationPolarization used to calculate the pump’s modes.

TE — — TE, TM

Power ratio for each pump modePower ratio for each pump mode. Number of elements in the list must be equal to the number of modes at the pump wavelength, and the sum of the ratios must be 1.

0.5 0.5 — — any string with numbers



Er ion densitySpecifies Erbium doping in the fiber

1e+025 m-3 [0, +INF[



364


Cross-sections

Er metastable lifetimeSpecifies the Erbium metastable lifetime

11 ms ]0, +INF[

Er signal excess lossRepresents the losses, due to the introduction of Erbium in the material by diffusion or by another implantation method, at the signal wavelength. Backscattering is a typical effect observed in this case. Note: This isn’t a commonly observed absorption loss in the 1550nm wavelength range.

0 dB/m [0, +INF[

Er pump excess lossRepresents the losses, due to the introduction of Erbium in the material by diffusion or by another implantation method, at the pump wavelength. Backscattering is a typical effect observed in this case. Note: This isn’t a commonly observed absorption loss at 980nm.

0 dB/m [0, +INF[

Yb ion densitySpecifies Ytterbium doping in the fiber

1e+025 m-3 [0, +INF[

Yb metastable lifetimeSpecifies the Ytterbium metastable lifetime

11 ms ]0, +INF[

Yb signal excess lossRepresents the losses, due to the introduction of Ytterbium in the material by diffusion or by another implantation method, at the signal wavelength. Backscattering is a typical effect observed in this case.

0 dB/m [0, +INF[

Yb pump excess lossRepresents the losses, due to the introduction of Ytterbium in the material by diffusion or by another implantation method, at the pump wavelength. Backscattering is a typical effect observed in this case.

0 dB/m [0, +INF[



EDFA design formatDetermines if format of cross-section file uses EDFA file format


File frequency unitDetermines if the filter will down sample the signal bandwidth to the filter sample rate




365


Enhanced

ESA cross section value at 1480 5.53-026 m2 ]0, +INF[

Er cross-section file nameSpecifies Erbium cross-section file name. File contains erbium absorption and emission cross sections.


Yb cross-section file nameSpecifies Ytterbium cross-section file name. File contains the ytterbium absorption and emission cross sections.

Ytterbium.dat — — —



Number of ASE modelsQuantity of excited modes by the ASE. Normally this number is the total number of modes (TE and TM) present in the waveguide at the signal wavelength.

2 — — [1, 1e+008]

A32Nonradiative emission rate from level 3 to level 2

1000000000 — 1/s ]0, +INF[

A43Nonradiative emission rate from level 4 to level 3

1000000000 — 1/s ]0, +INF[

Fraction of ion in pairFraction of ion in pair due to the pair-induced quenching PIQ phenomenon.

0 — — [0, 1]

Fast nonradiative upconversion lifetime 5e-006 s ]0, +INF[

Calculate upconversion effectsDefine whether the upconversion effects are calculated or just approximated.


CupHomogeneous upconversion coefficient from level 2

1e-022 — m3/s ]0, +INF[

C3Homogeneous upconversion coefficient from level 3

1e-022 — m3/s ]0, +INF[

C14Cross relaxation coefficient between level 1 and 4

7e-023 — m3/s ]0, +INF[



366


Numerical

Graphs

Simulation

Noise

C16Cross relaxation coefficient between level 1 and 6

7e-023 — m3/s ]0, +INF[



Relative errorSpecifies maximum acceptable error in solving the propagation equations

5e-007 — — ]0, 1]

Longitudinal stepsSpecifies the number of divisions necessary to discretize the waveguide length

130 — — [1,1e+008[



Calculate graphs True — — True, False

Longitudinal power graphs True — — True, False

Nornalized population density graphs True — — True, False



Enabled

Defines whether the component is enabled or not




Noise center frequency

Determines the noise center frequency

193.4 THz Hz, THz, nm [30, 30e5]

Noise bandwidth

Bandwidth to create noise bins




367


Technical BackgroundThe Er-Yr codoped waveguide amplifier solves the propagation of electromagnetic fields on Erbium doped, or on Erbium doped and Ytterbium co-doped waveguides. The pump wavelength must be in the region of 980 nm or 1480 nm, and can be co-and counter-propagating. Multiple co- and counter-propagating input signals may be considered in different wavelengths (DWDM).

In order to run this component, the following data must be provided: the Erbium and Ytterbium doping profiles, with their respective cross sections (parameters located in the cross-sections tab); the pump wavelength ( ) with the co- and counter-propagant pump powers ( ); and the WDM signal wavelengths ( ) with its respective powers. Notice that a signal is characterized by its wavelength, and may have different co and counter-propagant powers ( ).

The main characteristics of this component are:• co- and counter-propagant pump at 980nm region or 1480nm region;• multiple signals (co- and counter-propagant) at different wavelengths (DWDM);• multimode operation for the pump and signals;• co- and counter-propagant ASE noise due to Erbium concentration;• homogeneous upconversion (HUC) 1 from levels;• pair-induced quenching - PIQ;• nine energy levels considered.

Model implementationThis model is based on the solution of the propagation equations, using, directly, the solutions of the involved electromagnetic fields and the exact Erbium and Ytterbium transversal distributions.

Noise bins space

Specifies the noise bins spacing


[1,1000[


-100 dB — ]-INF, 0[

Noise dynamicThreshold ratio for adaption of noise bins

3 dB — [0, +INF[

Convert noise binsDetermines if generated noise bins are incorporated into signal

Convert noise bins

— — True, False



λpPp+ Pp_,

λs1…λs

WDM

λsi P s+

i and P s-i→

4 13 2⁄ e 4 11 2⁄

368


Propagation equations

The propagation equations describe the power evolution of the propagating electromagnetic fields in the optical amplifier and are described as:

This set of equations forms a system of 2+2WDM+2M coupled ones, and must be solved with the following boundary conditions:

where

is the device length

, , , and are the signals, pumps and ASE (Amplified Spontaneous Emission) longitudinal power distributions in the direction of propagation , with the signs (+) and (-) meaning, respectively, the co- and counter propagant direction;

and are the attenuation coefficients in the wavelengths for signal and pumping, respectively.

The index in refers to the -th signal, centered in the frequency , of a total number of WDM signals that can propagate simultaneously within the amplifier, as in systems with Dense Wavelength Division Multiplex - DWDM. The ASE± spectrum is discretized in M intervals (slots) with spectral width , centered in the frequencies

, in such a way, that (see Equation 3) refers to the -th spectral

(1)

(2)

(3)

(4)

(5)

(6)

dPp+_ z( )dz

---------------------- γp z( )Pp+_ z( ) αpPp+_ z( )+−+−=

dP s+_i z vs

i,( )

dz----------------------------- γ 21 z vs

i,( ) γ12 z vsi,( )–[ ]P s

i z vsi,( ) αpP s

i z vsi,( ) i,+−+− 1 … WDM, ,= =

dP jASE +_ z vj,( ) γ21 z vj,( ) γ12 z vj,( )–[ ]P j

ASE +_ z vs,( ) +±=

mhvj∆vjγ21 z vj,( ) αsPjASE +_ z vs,( ) j,±± 1 … M, ,=

Pp+ 0( ) Pp0 Pp_ L( ), PpL= =

P s+i 0 vs

i,( ) Ps0 vsi( ) and P s_

i L vsi,( ) PsL vs

i( ) i, 1 …WDM,= = =

PASE + 0 vs,( ) PASE_ L vj,( ) 0 j, 1 … M, ,= = =

L

P s+_i Pp+_ e P j

ASE +_

z

αs αp

i P s+_i i vs

i

∆vjvj P j

ASE +_ j

369


component of ASE±. Also in Equation 3, we have as the total number of modes present in the waveguide, and is the Planck constant. In Equation 1 - Equation 3, the gain coefficients , , and are given by:

where , , and , are the populations of Erbium ions of the ground ( ), meta-stable( ) and pump levels ( if pumped in 1480nm, or if pumped in 980nm). and are the populations of Ytterbium ions of levels

and .

The populations of three possible states of an excited pair exist:

- no ion excited

- one ion excited

- two ions excited

due to the phenomenon of pair-induced quenching - PIQ. , , , and are the absorption and emission cross sections of the Erbium doped material,

at the signal (12 and 21) and pump (13 and 31) wavelengths. The parameters and are the absorption and emission cross sections of the Ytterbium doped/co-doped material at the pump wavelength in the region of 980nm. When the amplifier is pumped at 980nm, the level 3 corresponds to the main level of the Stark Split. However, when the amplifier is pumped in the 1480nm region, the pump level is confounded with the main level . Thus, according to Equation 7a or

(7)a

(7)b

(8)

(9)

mh

γp γ12 γ21

γp z( ) Ψp x y,( ) σa13 N1 x y z, ,( ) 2N0p x y z, ,( ) N1p x y z, ,( )+ +( )[∫A∫=

σe31N3 x y z, ,( )– σa56N5 x y z, ,( ) σe65N6 x y z, ,( ) ]dxdy–+

γp z( ) Ψp x y,( ) σa13 N1 x y z, ,( ) 2N0p x y z, ,( ) N1p x y z, ,( )+ +( ) + [∫A∫=

σ– e31 N2 x y z, ,( ) N3 x y z, ,( ) 2N0p x y z, ,( ) N1p x y z, ,( )+ + +( ) ]dxdy

γ12 z vi,( ) Ψs x y,( )σa12 N1 x y z, ,( ) 2N0p x y z, ,( ) N1p x y z, ,( )+ +( ) xd yd∫A∫=

γ21 z vi,( ) Ψs x y,( )σe21 N2 x y z, ,( ) 2N2p x y z, ,( ) N1p x y z, ,( )+ +( ) xd yd∫A∫=

N1 N2 N3 4 15 2⁄4 13 2⁄ 4 13 2⁄ 4 11 2⁄

N5 N6F2

7 2⁄ F25 2⁄

N0p

N1p

N2p

σa13 σe31 σa12σe21

σa56σe65

4 11 2⁄

4 13 2⁄

370


Equation 7b, the coefficient of gain is taken when the amplifier is pumped at 980nm or 1480nm, respectively. In Equation 7 through to Equation 9, and

are the normalized intensity profiles obtained from the modal analysis of the waveguide (see the section on "Multimode operation"), in such a way that the intensity distributions of the signal, pump and ASE± can be written as:

where it should be noted that the same normalized intensity profile has been used for ASE and signal, because the difference between the central wavelengths of the intervals used to discretize the ASE and the signal wavelength is relatively small and may be considered . The correlation between the field distribution of the fundamental mode at 1530nm and 1650nm is higher than 95% for a typical optical fiber/waveguide.

Rate equationsThe populations , and also the population of the three possible states of an excited pair ( , , and ) in Equation 7 through to Equation 9, are the solutions of the rate equations for the energetic systems of Figure 1 or Figure 2, when it is considered the pumping in 980nm or 1480nm, respectively.

(10)

(11)

(12)

γpΨs x y,( )

Ψp x y,( )

Is x y z, ,( ) Ψs x y,( )Ps z( )=

Ip x y z, ,( ) Ψp x y,( )Pp z( )=

I jASE +_ x y z, ,( ) Ψs x y,( )P j

ASE +_ z( )=

M

ΨASEj Ψs

i≈

N1 N6→N0p N1p N2p

371


Figure 1 Coupled Er+3/Yb+3 system, pumped in 980nm region

Note: There are nine relevant energy levels considered (four levels due to the presence of the Erbium ions, two due to the Ytterbium dopant, and three due to the formation of Erbium paired-induced ions). 's and 's are the pump and signal stimulated rates,and the 's are the nonradioactive rate from level i to j. A21 is the fluorescent rate. Cup and C3 are the homogeneous upconversion coefficients. C14 and C16 are the cross-relaxation coefficients. Figure 1 also shows the population densities of the three possible states of an excited pair ( (no ions excited), (one ion excited), and (two ions excited), due to the PIQ effect.

In Figure 1, for the 980nm-pumping region, we have representations of the , , , and energy levels (due to the Erbium dopant), with

corresponding population densities of , respectively. The and energy levels are also shown, with population densities , due to

the Ytterbium dopant. The populations of the three possible states of an excited ion

Rij WijAij

N0p N1p N2p

4 15 2⁄4 13 2⁄ 4 11 2⁄ 4 9 2⁄

Ni i 1 4,=( ) F27 2⁄

F25 2⁄ Ni i 5 6,=( )

372


( , , and ) are also shown. is the fast nonradiative upconversion lifetime, and can be calculated as:

with being the distance between two ions in a pair. Experimental measurements report typical value of the order of few microseconds. We know that

‘s are pump rates (stimulated absorption and emission) between levels , and 's are the absorption and emission stimulated rates at the signals wavelength. The non-radiative rates between levels are represented by . A21 is the fluorescence rate. Cup and C3 are the homogeneous upconversion coefficients from levels 2 and 3. The homogeneous upconversion is modeled through the quadratic terms in N2 and N3 in the rate equations. These terms are dependent on the Erbium concentration, and can be calculated using reference [1]. C14 and C16 are the cross-relaxation coefficients between levels and . The cross-relaxation between levels is the main energy transfer mechanism between the Ytterbium and Erbium ions, and the approached value of the coefficient can be obtained from [1]. Due to the short lifetime of level 3 ( ), the back energy transfer process (from Erbium to Ytterbium ions) is not being considered in this model. However, the fact that all Erbium ions are surrounded by Ytterbium ions is taken into account. Special attention should be paid to the Ytterbium concentration in relation to the Erbium concentration. Geometrically, it is observed that the Ytterbium concentration must be in the interval . If the Ytterbium concentration ( ) is less than , the formation of clusters may occur and the energy transference form Ytterbium ions ( ) to the Erbium ones ( ) may not be so efficient. On the other hand, if the Ytterbium concentration is too high, Ytterbium clusters may form, which means there won't be any energy transference to the Erbium ions, the pump energy will be wasted, and consequently, the efficiency of the amplifier device will be reduced. It is believed that the homogeneous upconversion that occurs from level 3 doesn't reach level and relaxes very quickly to level 4 ( ).

(13)

N0p N1p N2p τ21p

τ21pdpair

3

Cup-----------=

dpair

Rij i j→Wij

i j→ Aij

4 1→ 6 1→6 1→

1 A32⁄

4NEr NYb 20NEr< <NYb 4NEr

Yb+3 Er+3

F47 2⁄ 4 9 2⁄

373


The rate equations for the energy system of Figure 2 is given by:

The presence of the crossed terms for the solutions of the population , suggests the use of a special numeric treatment due to its non-linear

nature. However, the system for the paired-induced population is a linear one and can be solved by a straightforward solution.

(14)

∂N1∂t

--------- W12N1– R13N1– R31N3 A21N2 W21N2 ++ + +=

+ CupN22 C14N1N4– C3N3

2 C16N1N6–+

∂N2∂t

--------- W12N1 A21N2– W21– N2 A22N3 + +=

2CupN22– 2C14N1N4+

∂N3∂t

--------- R13N1 R31N3– A32– N3 A43N4 2C3N32– C16N1N6++=

N1 N2 N3 N4+ + + 1 2p–( )NEr=

∂N5∂t

--------- R56– N5 A65N6 R65N6 C16N1N6+ + +=

N5 N6+ NYb=

∂N0p∂t

------------ 2R13– N0p A21N1p 2W12N0p– W21N1p+ +=

∂N1p∂t

------------ + 2R13N0p A– 21N1p 2+ W12N0p W– 21N1p 2A21N2p ++=

R13– N1p W12– N1p 2W21N2pN2pτ21p---------+ +

N0p N1p N2p+ + pNEr=

NiNjNi i 1 6→=( )

374


The solution for the paired-induced population when the amplifier is pumped at 980nm is given by:

Figure 2 shows the system of energy levels that are being taken into account for the 1480nm pumping wavelength, as well as the numbering of these levels. In this case the pump energy level belongs to the main level 2 ( ). However, due to the presence of the nonradioactive transitions inside the level , we have named the pump level as "level 3". Note that it should not be confused with the level , when the system is pumped at 980nm region.

Figure 2 Coupled Er+3/Yb+3 system, pumped at 1480nm region

(15)

N0p N( 1p pNEr– 3A21N1pτ21p N1pR13τ21p 2A21pNErτ21p– N1pτ21pW12+ + + +–=

3N1pτ21pW21 2pNErτ21pW21 ) 1 2A21τ21p 2R13τ21p– 2τ21pW12– 2τ21pW21+ +( )⁄–

N1p2N0p R13 W12+( )( )

A21 W21+( )----------------------------------------------=

N2p pNEr N0p– N1p–=

4 13 2⁄4 13 2⁄

4 11 2⁄

375


Note: There are nine relevant energy levels considered (four levels due to the presence of the Erbium ions, two due to the Ytterbium dopant, and three due to the formation of Erbium paired-induced ions). 's and 's are the pump and signal stimulated rates,and the 's are the nonradioactive rate from level i to j. A21 is the fluorescent rate. Cup and C3 are the homogeneous upconversion coefficients. C14 and C16 are the cross-relaxation coefficients. Figure 2 also shows the population densities of the three possible states of an excited pair ( (no ions excited), (one ion excited), and (two ions excited), due to the PIQ effect.

When the system is pumped in the 1480nm region, levels 5 and 6 (Ytterbium levels) are considered to be empty.

It is known that the effects of ESA cannot be disregarded when the pumping wavelength is at 1480nm region, because the ESA cross section is approximately 10% of the peak value of the absorption cross section. The non-radioactive rate embodies the non-radioactive rates between levels and

, in such a way that level is not considered. In this case, the system of rate equations is then described as:

(16)

Rij WijAij

N0p N1p N2p

A434 9 2⁄ 4 11 2⁄→

4 11 2⁄ 4 13 2⁄→ 4 11 2⁄

∂N1∂t

--------- W12N1– R13N1– R31N3 A21N2 W21N2 CupN22 C14N1N4–+ + + +=

∂N2∂t

--------- W12N1 A21N2– W21– N2 A32N3 2CupN22– 2C14N1N4 R24

ESAN2–++=

∂N3∂t

--------- R13N1 R31N3– A32– N3 A43N4+=

N1 N2 N3 N4+ + + 1 2p–( )NEr=

∂N0p∂t

------------ 2R13– N0p A21N1p 2W12N0p– W21N1p+ +=

∂N1p∂t

------------ + 2R13N0p R– 31N1p A– 21N1p 2W12N0p W– 21N1p 2A21N2p ++ +=

R13– N1p 2R31 N2p W– 12N1p 2W21N2pN2pτ21p---------+ + +

N0p N1p N2p+ + pNEr=

376


and are the simulated rates between levels , at the pump ( ) and signals ( ) wavelengths. are the non-radioactive rates between levels , and A21 is the fluorescence rate. Cup and C14 are the homogeneous upconversion and the cross-relaxation coefficients. is the ESA rate for level 2 and level 4. The presence of the crossed terms in Equation 16 suggests the use of a numeric solution for the populations . However, the system for the paired-induced population is a linear one and can be solved by a straightforward solution. The solution for the paired-induced population when the amplifier is pumped at 1480nm is given by:

In the stationary state, the solutions of the rate Equation 14 and Equation 16 are obtained by nullifying the left side of these equations. As we have previously stated, the systems of Equation 14 and Equation 16 are non-linear due to the presence of the crossed terms and , and must be solved numerically. The stimulated rates and are written as:

(17)

(18)a

Rij Wij i j→ RW Aij i j→

R24ESA

NiNjN1 N6→

N0p pNEr 1 2A21τ21p 2R31τ21p 2τ21pW21+ + +( )( ) 1–( 2A21 2R13τ21p 2R31τ21p– 2τ21pW21 _+ +–⁄–=

2 R13 W12+( ) 1 3A21τ21p R13τ21p 3R31τ21p τ21pW12 3τ21pW21+ + + + +( ) A21 R31 W21+ +( )⁄ )

N1p2N0p R13 W12+( )( )A21 R31 W21+ +( )

-----------------------------------------------=

N2p pNEr N0p– N1p–=

N1N4 N1N6Wij Rij

W12 x y z vs, , ,( )σa12

i vsi( )

hvsi

-------------------- Is+i x y z vs

i, , ,( ) Is_i x y z vsi, , ,( )+( ) +

i 1=

WDM

∑=

+ σa12

j v j( )

hv j-------------------- IASE+

j x y z v j, , ,( ) IASE_j x y z v j, , ,( )+( )

j 1=

M

∑

377


where when the pumping wavelength is in the 980nm region. When the pumping wavelength is in the 1480nm region, we have and . In Equation 18, it is presumed that the propagation of WDM signals with frequencies and intensities ; pumping intensities ; and ASE(Amplified Spontaneous Emission) with its spectrum discretized in slots of

(18)b

(18)c

(18)d

(18)e

(18)f

(18)g

W21 x y z vs vp, , , ,( )σe21

i vsi( )

hvsi

-------------------- Is+i x y z vs

i, , ,( ) Is_i x y z vsi, , ,( )+( ) +

i 1=

WDM

∑=

+ σep21 vp( )

hvp----------------------- Ip+ x y z vp, , ,( ) Ip_ x y z vp, , ,( )+( ) +

+ σe21

j v j( )

hv j------------------- IASE+


j 1=

M

∑

R13 x y z vp, , ,( )σa13 vp( )

hvp--------------------- Ip+ x y z vp, , ,( ) Ip_ x y z vp, , ,( )+( )=

R31 x y z vp, , ,( )σe31 vp( )

hvp-------------------- Ip+ x y z vp, , ,( ) Ip_ x y z vp, , ,( )+( )=

R56 x y z vp, , ,( )σa56 vp( )

hvp--------------------- Ip+ x y z vp, , ,( ) Ip_ x y z vp, , ,( )+( )=

R65 x y z vp, , ,( )σe65 vp( )

hvp-------------------- Ip+ x y z vp, , ,( ) Ip_ x y z vp, , ,( )+( )=

R24ESA x y z vs vp, , , ,( )

σa24 vsi( )

hvsi

-------------------- Is+i x y z vs

i, , ,( ) Is_i x y z vsi, , ,( )+( ) +

i 1=

WDM

∑=

+ σa24 vp( )

hvp--------------------- Ip+ x y z vp, , ,( ) Ip_ x y z vp, , ,( )+( ) +

+ σa24

j v j( )

hv j-------------------- IASE+


j 1=

M

∑

R24ESA σep21 0= =

σep21 σep31=R56 R65 0= =

vsi I

s +_ Ip +_M

378


width with intensity . The sign "+" refer to the co-propagant waves, and the sign "-" to the counter-propagant waves. The use of the homogeneous upconversion and the cross-relaxation coefficients and consideration of the PIQ phenomenon in Equation 14 and Equation 16 allows for the adequate modeling of Erbium doped and Ytterbium co-doped waveguides. In general, for Erbium concentrations in the order of 100ppm ("1024 ions/m3) these effects are not important. However, the present applications of optical amplifiers demand Erbium concentrations higher than 1000ppm, and, therefore, such effects cannot be ignored.

Multimode operationThe doped waveguide may present more than one mode at the pump or at the signal frequencies/wavelength. This is common in integrated optics, in which the discontinuity between the refraction index of the core and the cladding is raised on purpose to provoke a high confinement of the pump field and, thus, obtain higher gain [1].

We can presume that the device is externally excited by a beam with gaussian field distribution , with different spatial widths at the pump and signal wavelengths. This supposition is experimentally sustained when a beam that it is being coupled through a set of lenses excites an integrated optical device. Consider that at the wavelengths and (signal and pump wavelength, respectively), and modes with fields distributions can propagate. The input beam can then be described through a modal expansion of the modes present in the waveguide, that is:

where can assume and , and represents the coupling coefficient between the field of the gaussian input beam and the field of the corresponding -th mode. Then, the fraction of the total power allocated in each expansion mode for the pumping and for the signal will be:

(19)

(20)

∆v IASE +_

Φ x y ω, ,( )

λs λp NsNp φi x y λs p⁄, ,( )

Φ x y λs p⁄, ,( )gauss ciφ x y λs p⁄, ,( )ii 1=

Nq

∑=

Nq Ns Np cii

ηpicpicpi∗

cpjcpj∗

j 1=

Np

∑

-------------------------- ηsicsicsi∗

csjcsj∗

j 1=

Ns

∑

------------------------- ==

379


In this way, for multimode waveguides, the normalized intensity profile for the signals and the pump can be calculated as:

where and are the normalized intensity profiles at the signals and the pump wavelength, respectively.

References:[1] M. Federighi, F. Di Pasquale, "The Effect of Pair-Induced Energy Transfer on the Performance

of Silica Waveguide Amplifiers with High Er+3/Yb+3 Concentration", IEEE Photonics Technology Letters, Vol.7, No.3, pp.303-305, March 1995.

[2] S. Honkanen, S.I. Najafi e W.J. Wang, "Composite Rare-Earth Doped Glass Waveguides", IEEE Electronics Letters, Vol.28, No.8, pp.746-747, abril, (1992).

(21)Ψs p⁄ ηsi pi⁄ Ψs p⁄i

i 1=

Nq

∑=

Ψsi x y,( ) Ψp

i x y,( )

380

SEMICONDUCTOR OPTICAL AMPLIFIER

Semiconductor Optical Amplifier

Performs lumped amplification with traveling wave semiconductor optical amplifiers (SOA). The rate-equation approximation has been used in which the electrical field is described by the wave equation and the carrier density by means of the rate equation. Such model is applicable to describe the amplification of CW and optical pulsed signals.

Ports

Parameters

Main

Physical


Input Input Optical



Injection current 0.15 A [0,1]


Length 0.0005 m ]0,1e-3]

Width 3e-006 m ]0,500e-6]

Height 8e-008 m ]0,10e-6]

Optical confinement factor 0.15 — ]0,1]

Loss 4000 1/m [0,10e-4]

Differential gain 2.78e-020 m2 ]0,50e-20]

Carrier density at transparency 1.4e+024 m3 ]0,10e-25]

Linewidth enhancement factor 5 — [–30,30]

Recombination coefficient A 143e+008 1/s ]0,1e-15]

Recombination coefficient B 1e-016 m3/s ]0,1e-10]

381


Numerical

Simulation

Technical backgroundThis module performs lumped amplification with traveling wave semiconductor optical amplifiers (SOA) [AGR, 1993] and [SHI, 1994]. The rate-equation approximation has been used in which the electrical field is described by the wave equation and the carrier density by means of the rate equation [1-4]. Such model is applicable to describe the amplification of CW and optical pulse signals. The pulse widths have to be much larger than the intraband relaxation time that governs the dynamics of the induced polarization. Typically, the intraband relaxation time is 0.1 ps. Therefore, the model can be used for pulse widths larger than 1 ps [3-4].

The basic approximation done in the wave equation for the electrical field in the SOA is a linear dependence between the carrier induced susceptibility and the carrier density [6-8]. In the framework of this approximation the material gain coefficient gm is related to carrier density N(t) by,

(1)

where N0 is the carrier density at transparency point and Ag is the differential gain coefficient [2].

Recombination coefficient C 3e-041 m6/s [0,1e-30]

Initial carrier density 3e+024 m–3 ]0,10e-25]

Name and description Value Units Mode

Integration type Runge Kutta 4th order — Normal

Relative tolerance 1e-006 — Normal

Maximum number of steps 100000 — Normal

Interpolation type Polynomial — Normal

Order of polynomial 4 — Normal



True — True, False]


gm t( ) Ag N t( ) N0–[ ]=

382


The net gain coefficient g is related to the material gain gm by,

(2)where α is an effective loss coefficient which includes scattering and absorption losses and Γ is the optical confinement factor defined as a fraction of the mode power within the active layer.

It is also assumed that the amplifier supports a single wave-guide mode and it does not change the polarization state during the amplification. Linearly polarized input light is presumed. The group velocity dispersion in the SOA is neglected. The amplified spontaneous emission noise is not taken into account. In the framework of these assumptions, the gain G for a traveling wave SOA for a distance z is:

(3)

The carrier density rate equation expresses the conservation of carriers inside the active layer. It takes into account the current density and the net rate of carrier generation and recombination averaged over the active layer. The recombination rate consists of spontaneous and stimulated recombinations. The spontaneous recombination rate includes the radiative and nonradiative components. The nonradiative recombination takes into account the Auger recombination, which is generally the dominant nonradiative process in long wavelength lasers. The spontaneous recombination rate can be characterized by a quantity known as the carrier lifetime :

(4)

where RA is the non-radiative coefficient due to recombination at defects and traps, RB is the spontaneous radiative recombination coefficient, and RC is the Auger recombination coefficient.

Neglecting the carrier diffusion, the amplified spontaneous emission noise and the shot noise the equation for the carrier density N(t) is [3-4]:

(5)

where I is the light intensity, J is the injection current density, q is the electron charge, h is the Planck’s constant, f is the light frequency, t is the time, and d is the active layer thickness.

Equation 5 can be rewritten as:

(6)

where Ip is the pump current (or injection current), V = L w d is the volume of the active region, and L and w are the length and the width of the amplifier respectively.

g t( ) Γgm t( ) α–=

G t z,( ) e g t( )z[ ]=

τs

N t( )τs

---------- RAN t( ) RBN2 t( ) RcN3 t( )+ +=

dNdt------- J

qd------ N

τs----Ag N N0–( ) I

hf-----–=

dNdt-------

IpqV------- N

τs----– ΓAg N t( ) N0–( )P N t,( )L

Vhf--------------------–=

383


The amplifier power P(N,t), which is the average power over the length of the amplifier, is by:

(7)

The output optical field is:

(8)

where δ is the linewidth enhancement factor.

This parameter takes into account the coupling between the gain and refractive index of the amplifying medium. The output power to parameterized signals is:

(9)

To include multiple frequency bands, the term P(N,t) / f in Equation 6 should be substituted with:

(10)

where fk is the center frequency for each frequency band.

Basic physical effects described by the model for single wavelength channel are gain saturation, gain-saturation induced self-phase modulation, and gain recovery [3-5].

Gain-saturation induced self-phase modulation is responsible for important changes in the spectrum of amplified pulses:• appearance of multi-peak spectral structure• red shift of the spectrum• appearance of the positive chirp

In addition, the shape and the spectral pulse distortions depend on the shape and the initial frequency pulse modulation.

Gain saturation and gain recovery effects for Gaussian, super Gaussian, and chirped Gaussian pulses for an SOA are in OptiSystem Tutorials — Introduction to the

P N t,( ) P N z,( )L

----------------- zd0

L

∫PinG t z,( )

L-----------------------

0

L

∫ Pine g t( )L[ ] 1–

g t( )-----------------------= = =

Eout t( ) Ein t( )e1 jδ+( )g t( )L[ ]

2------------------------------

=

Pout Pine g t( )L[ ]=

Pk N t,( )fk

------------------k∑

384


basic gain saturation and gain recovery characteristics of the SOA. A strong agreement with [3-4] can be identified in this section.

Generally, gain saturation effect is a serious obstacle for an SOA as an inline amplifier. In the case of single-channel transmission, gain saturation effect leads to a pattern effect. Pattern effect is demonstrated for 10 Gb/s average soliton transmission over a 500 km SMF optical link in OptiSystem Tutorials — Basic application of the OSA as an inline amplifier.

In the case of multi-channel transmission, gain saturation effect leads to inter-channel crosstalk. Independent of the problems connected with applying an SOA as an inline amplifier, they are used near the 1.3 wavelength in SMF. The fundamental reason for this is the possibility of avoiding the large group velocity dispersion of SMF at 1.55 [6-11]. This idea following [11] is demonstrated in OptiSystem Tutorials — Basic application of the OSA as an inline amplifier.

Some undesirable properties of applying an SOA as an inline amplifier have found other applications. For example, the positive pulse chirp created during the process of amplification can be used for pulse compression if you can propagate the pulse in a dispersive media with a proper sign of the group velocity dispersion. Pulse compression with the help of SMF following [12] is described in OptiSystem Tutorials — Applying the gain saturation properties of the SOA to obtain pulse compression.

SOAs have found new applications as wavelength converters, fast switches for wavelength routing in WDM networks, and nonlinear elements for clock recovery and demultiplexing in TDM systems [5, 13-14]. In OptiSystem Tutorials — Application of the SOA as a wavelength converter, SOA wavelength conversion is demonstrated based on four-wave mixing and cross-saturation effects.

References

[1] M.J. Adams, H.J. Westlake, M.J. O’Mahony, I.D. Henning, “A Comparison of Active and Passive Optical Bistability in Semiconductors”, IEEE Journal of Quantum Electronics, Vol. QE-21, N 9, September 1985.

[2] M.J. O’Mahoney, “Semiconductor Laser Optical Amplifier for use in Fiber Systems,” Journal of Lightwave Technology, Vol. 6, N 4, April 1988.

[3] G.P. Agrawal and N.A. Olsson, “Self-Phase Modulation and Spectral Broadening of optical pulses in semiconductor Laser Amplifiers”, IEEE J. of Quantum Electronics, Vol. QE-25, N 11, pp. 2297-2306, November 1989.

[4] N.A. Olsson and G.P. Agrawal, “Spectral shift and distortion due to self-phase modulation of picosecond pulses in 1.5 mm optical amplifiers”, Appl. Phys. Lett. 55, N 1, pp. 13-15, July 1989.

[5] G.P. Agrawal, “Fiber-Optic Communication Systems”, Second edition, John Wiley & Sons, Inc. 1997.

µm

µm

385


[6] J.J. Reid, C.T.H.F. Liendenbaum, L.F. Tiemeijer, A.J. Boot, P.I. Kuindersma, I. Gabitov, and A. Mattheus, in Proceedings of the 20th European Conference on Optical Communication (Instituto Internationale delle Communicaziono, Genova, Italy, 1994).

[7] A. Mecozzi, “Optics Letters,” 20, 1616-1618, 1995.

[8] S. Wabnitz, “Optics Letters,” 20, 1979-1982, 1995.

[9] S.K. Turitsyn, Phys. Rev. E 54, R3125, 1996.

[10] I.M. Uzunov, M. Golles, and F. Lederer, “Optics Letters,” 22, 1406-1408, 1997.

[11] M. Settembre, F. Matera, V. Hagele, I. Gabitov, A.W. Mattheus, and S. Turitsyn, “Journal of Lightwave Technology,” Vol. 15, pp. 962-967, 1997.

[12] G.P. Agrawal and N.A. Olsson, “Optics Letters,” 14, 500-502, 1989.

[13] T. Durhuus, B. Mikkelsen, and K.E. Stubkjaer, “Journal of Lightwave Technology,” Vol. 10, pp. 1056-1065, 1992.

[14] T. Durhuus, B. Mikkelsen, C. Joergensen, S.L. Danielsen, and K.E. Stubkjaer, “Journal of Lighwave Technology,” Vol. 14, pp. 942-954, 1992.

Technical references

[AGR, 1993] G.P. Agrawal and N.K. Dutta, “Semiconductor lasers,” Second edition, International Thomson Publishing, Inc., 1993.

[BAS, 1992] S.P. Bastien, H. R. D. Sunak, B. Sridhar, V. E. Kalomiris “Temporal, spatial and spectral modeling of erbium doped fiber amplifiers”, SPIE – Physic and Simulations of Optoelectronic Devices, pp. 2-11, 1992

[BUR, 1998] J. Burgmeier, A. Cords, R. März, C. Schäffer, B. Stummer “A black box model of EDFA’s operating in WDM systems”, Journal of LIghtwave Technology, Vol. 16, N. 7, pp. 1271-1275, 1998

[DES, 1994] E. Desurvire, “Erbium-Doped Fiber Amplifiers – Principles and Applications”, John Wiley & Sons, Inc., USA, 1994

[GIL, 1991] C.R. Giles, E. Desurvire, "Modeling erbium-doped fiber amplifiers," Journal of LIghtwave Technology, Vol. 9, N. 2, pp. 271-283, 1991

[KAR, 1998] J. A. Vallés, “Analysis of channel addition/removal response in all-optical gain-controlled cascade of erbium-doped fiber amplifiers”, Journal of Lightwave Technology, Vol. 16, N. 10, pp. 1795-1803, 1998

[OKO, 1990] T. Okoshi, "Exact Noise-Figure Formulas for Optical Amplifiers and Amplifier-Fiber Cascaded Chains," IEEE/OSA Topical Meeting on Optical Amplifiers and their Applications, Monterrey, PDP11, 1990

[SHI, 1994] S. Shimada, H. Ishio, “Optical Amplifiers and their Applications”, John Wiley & Sons, Chichester, 1994.

386

LIMITING AMPLIFIER

Limiting Amplifier

This component is an electrical limiting amplifier. The minimum and maximum output signal values are user-defined parameters.

Ports

Parameters

Main

Simulation





Units Value range

Max. output voltageThe maximum value of the output signal.

0.5 Volt ]INF,+INF[

Min. output voltageThe minimum value of the output signal.

-0.5 Volt ]INF,+INF[


Units Value range


True —

387

LIMITING AMPLIFIER

Technical backgroundThis component measures the input signals and compares the amplitude with the parameters Max. output voltage and Min. output voltage. If the signal value is outside of the range between the min and max values, the signal will be clipped. This component does not affect the noise amplitude, only the signal amplitude.

388

ELECTRICAL AMPLIFIER

Electrical Amplifier

Electrical amplifier with additive thermal noise.

Ports

Parameters

Main

Simulation

Noise





Units Value range

Gain 10 dB [-1e+100, 1e+100]


Units Value range


True —



Include noise Yes — —

PSDDetermines whether the power is defined as PSD or the average power in time

Yes — —

389

ELECTRICAL AMPLIFIER

Random numbers

Noise powerValue of the PSD or the average power

–60 dBm W, mW, dBm [-1e+100, 1e+100]

Add noise to signal No — —


Units Value range




0 — [0,4999]



390

TRANSIMPEDANCE AMPLIFIER

Transimpedance Amplifier

DescriptionThis component is an electrical transimpedance amplifier with user defined noise figure. It has linear gain and additive thermal noise.

Ports

Parameters

Main

Simulation





Units Value range

Voltage gainThe linear gain of the amplifier.

600 Ohm, kOhm, dB

[0,+INF[

Include NoiseDefines whether the noise will included in the output

True — True, false

Noise figureAmplifer noise figure

6 dB [0,+INF]

Input noise densityMinumum input noise

4e-21 A/Hz-1, W/Hz, mW/Hz, dBm/Hz

[0,+INF]


Units Value range


True —

391

TRANSIMPEDANCE AMPLIFIER

Noise

Random numbers

Technical backgroundThis component amplifies the input electrical signal and adds thermal noise to the signal output. The value of the thermal noise is calculated from the input SNR and the user defined parameter Noise figure.

Since OptiSystem can have noiseless electrical signals, the parameter Input noise density assures a minimum value for the noise floor at the input signal.





Yes — —


–60 dBm W, mW, dBm [-1e+100, 1e+100]



Units Value range




0 — [0,4999]

392

AGC AMPLIFIER

AGC Amplifier

This component is an electrical limiting amplifier with user defined noise figure. It inputs signal dependent gain and additive thermal noise.

Ports

Parameters

Main

Simulation





Units Value range

Output voltageThe peak value of the output signal.

0.005 Volt [0,+INF[

Include NoiseDefines whether the noise will included in the output

False — True, false

Noise figureAmplifer noise figure

6 dB [0,+INF]

Input noise densityMinumum input noise

4e-21 A/Hz-1, W/Hz, mW/Hz, dBm/Hz

[0,+INF]


Units Value range


True —

393

AGC AMPLIFIER

Noise

Random numbers

Technical backgroundThis component amplifies/attenuates the input electrical signal and adds thermal noise to the signal output. The output signal will have a peak value defined by the parameter Output voltage. The value of the thermal noise is calculated from the input SNR and the user defined parameter Noise figure.

Since OptiSystem can have noiseless electrical signals, the parameter Input noise density assures a minimum value for the noise floor at the input signal.





Yes — —


–60 dBm W, mW, dBm [-1e+100, 1e+100]



Units Value range




0 — [0,4999]

394

Filters LibraryThis section contains information on the following filters.

Optical

• Optical IIR filter• Measured Optical filter• Measured Group Delay Optical filter• Rectangle Optical filter• Trapezoidal Optical filter• Gaussian Optical filter• Butterworth Optical filter• Bessel Optical filter• Fabry Perot Optical filter• Acousto Optical filter• Mach-Zehnder Interferometer• Inverted Optical IIR filter• Inverted Rectangle Optical filter• Inverted Trapezoidal Optical filter• Inverted Gaussian Optical filter• Inverted Butterworth Optical filter• Inverted Bessel Optical filter

FBG

• Fiber Bragg Grating (FBG)• Uniform Fiber Bragg Grating• Ideal Dispersion Compensation FBG

395

Electrical

• Low Pass IIR filter (Electrical)• Low Pass Rectangle filter (Electrical)• Low Pass Gaussian filter (Electrical)• Low Pass Butterworth filter (Electrical)• Low Pass Bessel filter (Electrical)• Low Pass Chebyshev filter (Electrical)• Low Pass RC filter (Electrical)• Low Pass Raised Cosine filter (Electrical)• Low Pass Cosine Roll Off filter (Electrical)• Low Pass Squared Cosine Roll Off filter (Electrical)• Band Pass IIR filter (Electrical)• Measured filter (Electrical)• Band Pass Rectangle filter (Electrical)• Band Pass Gaussian filter (Electrical)• Band Pass Butterworth filter (Electrical)• Band Pass Bessel filter (Electrical)• Band Pass Chebyshev filter (Electrical)• Band Pass RC filter (Electrical)• Band Pass Raised Cosine filter (Electrical)• Band Pass Cosine Roll Off filter (Electrical)• Band Pass Square Cosine Roll Off filter (Electrical)• S Parameters Measured filter (Electrical)

Filter Analyzers

• Optical Filter analyzer• Electrical Filter analyzer

396

OPTICAL IIR FILTER

Optical IIR filter

Infinite impulse response filter (IIR) for optical signals.

Ports

Parameters

Main

Numerator coefficients


Input Input Optical




Frequency

Filter center frequency


Filter sample rate

User-defined sample rate independent from the signal sample rate


[1e-9,+INF[

Additional loss

Loss applied to the signal after filtering

0 dB — [0,+INF[

Filter coefficients type

Type of numerator and denominator coefficients for the filter

Z domain — — Frequency domain, Poles and zeros, Z domain



Number of numerator coefficients

3 — [1,+INF[

Numerator[0].real 0.64 — ]-INF,+INF[

397

OPTICAL IIR FILTER

Denominator coefficients

Simulation

Numerator[0].imag 0 — ]-INF,+INF[







Number of denominator coefficients

3 — [1,+INF[

Denominator[0].real 5.05 — ]-INF,+INF[

Denominator[0].imag 0 — ]-INF,+INF[

Denominator[1].real –4.75 — ]-INF,+INF[






Enabled



Resample



Sample rate




398

OPTICAL IIR FILTER

Noise

Technical backgroundThe infinite impulse response filter is a recursive digital filter. The transfer function can be expressed in the z domain as:

where H(z) is the filter transfer function in the Z domain, α is the parameter for Additional loss, N is the parameter number of Numerator coefficients, an are the coefficients for the numerator, M is the parameter number of Denominator coefficients, and bm are the coefficients for the denominator.

Also,

where fc is the filter center frequency defined by the parameter Frequency, fs is the parameter Filter sample rate, and f is the frequency.

According to the parameter Filter coefficients type, the filter transfer function can be given in the z (Z domain) or in the frequency domain. In the second case, the filter is determined by the numerator and the denominator polynomial, which can be expressed by their roots (Poles and zeros) or by the polynomial coefficients (in Frequency domain).


Units Value range

Noise threshold



Noise dynamic


3 dB ]-INF,+INF[

H z( )

α anz n–

n 0=

N

∑

bmz m–

m 0=

M

∑------------------------=

z j2π f fc–( ) fs⁄( )exp=

399

MEASURED OPTICAL FILTER

Measured Optical filter

Filter based on measurements.

Ports

Parameters

Main

Measurements


Input Input Optical




User-defined frequency

Determines whether you can define the filter center frequency or use the value from the measurements


Frequency

User-defined filter center frequency



Units Value range

File frequency unit

Determines the frequency unit of the file with the measurements

Hz — Hz, THz, m, nm

File format

Determines the format of the file with the measurements

Power — Power, Power Phase, Real Imag, phase

Linear scale

Determines whether the measured data is in linear scale or not

True — ]-INF,+INF[

400


Numerical

Simulation

Noise

Filter filename

Filename with the measured data

Filter.dat — —


Units Value range

Interpolation

Determines the interpolation algorithm for the measured data




Enabled



Resample



Sample rate




Units Value range

Noise threshold



Noise dynamic


3 dB ]-INF,+INF[


Units Value range

401


Graphs

Technical backgroundThe input file is formatted containing two items per line — frequency and filter measurement. The parameter File frequency unit determines the frequency or wavelength unit of the first item. It can be in Hz, THz, m, or nm.

According to the parameter File format, the second item can be one value (Power or Phase) or two values (Power and Phase or Real and Imag):

Power (Phase is set to zero, assuming frequency unit is THz)

Power Phase


Filter transmission - real part Frequency (Hz) Amplitude (a.u.)

Filter transmission - imag part Frequency (Hz) Amplitude (a.u.)

193.10 0

193.11 0.5

193.12 0.5

193.13 0

...

193.10 0 0

193.11 0.5 3.14

193.12 0.5 3.14

193.13 0 0

...

402


Real Imag


The parameter User defined frequency determines if you can enter the center frequency. This means that the filter data is shifted from the measured center frequency to the user center frequency that you define in the parameter Frequency.

193.10 0

193.11 –0.5 7.9e-4

193.12 –0.5 7.9e-4

193.13 0 0

...

193.10 0

193.11 3.14

193.12 3.14

193.13 0

...

403

MEASURED GROUP DELAY OPTICAL FILTER

Measured Group Delay Optical filter

Loads files with the filter amplitude and group delay ripple measurements. This FBG was designed mainly for dispersion compensation.

Ports

Parameters

Main

Measurements


Input Input Optical

Transmission Output Optical






Frequency




Units Value range

File frequency unit


m — nm, m

Group delay unit

Determines the group delay unit of the file with the measurements

ps — s, pss

404


Numerical

Simulation

Noise

File format


Delay — Power, Power Delay, Delay

Linear scale

Determines whether or not the measured data is in linear scale


Filename


GroupDelay.dat — —


Units Value range

Interpolation Linear — Linear, Cubic



Enabled



Resample



Sample rate


500 GHz Hz, GHz, THz [1e-9,+INF[


Units Value range

Noise threshold



Noise dynamic


3 dB ]-INF,+INF[


Units Value range

405


Graphs

Technical backgroundThis model is a filter with measured group delay. The filter transfer function is

where f is the frequency dependence phase of the filter.

The group delay is defined by Equation 1:

Writing Equation 2 as a function of wavelength:

where c is the speed of light.

You define by entering the table with the measurements.


Filter transmission — Amplitude Wavelength (m) Amplitude (a.u.)

Filter transmission — Phase Wavelength (m) Phase (rad)

(1)

(2)

(3)

H f( ) ejφ f( )=

τ f( ) 12π------dφ

df------–=

τ λ( ) λ2

2πc---------dφ

dλ------–=

τ

406


Typically, this measurement looks like the graph in Figure 1, where X is the wavelength in nm and Y is the group delay in ps:

Figure 1 Group delay measurement

Calculate the phase from this curve in order to calculate the filter transfer function.

Phase calculationThe phase is calculated with Equation 3:

File formatThe input file is formatted with two items per line — the wavelength and the filter measurement. The parameter File frequency unit determines the wavelength unit of the first item, and can be in m or in nm. The parameter Group delay unit determines the group delay unit, and can be in s or in ps.

According to the parameter File format, the second item can be one value (Power or Delay) or two values (Power and Delay).

(4)φ 2πc τ λ( ) 1λ2----- λd∫–=

407


Example of input file:

Power (Delay is set to zero)

Power Delay

Delay (Power is set to one)

The parameter User defined frequency determines if you can enter the center frequency. This means that the filter data is shifted from the measured center frequency to the user center frequency that you define in the parameter Frequency.

References[1] Madsen, C. K. and Zhao, J H., Optical Filter Design and Analysis: A Signal Processing

Approach. John Wiley & Sons, USA, (1999).

1551 0

1551.1 0.5

1551.2 0.5

1551.3 0

...

1551 0 0

1551.1 0.5 –10

1551.2 0.5 –20

1551.3 0 –30

...

1551 0

1551.1 –10

1551.2 –20

1551.3 –30

...

408

RECTANGLE OPTICAL FILTER

Rectangle Optical filter

Optical filter with a rectangle frequency transfer function.

Ports

Parameters

Main


Input Input Optical




Frequency



Bandwidth

3 dB filter bandwidth


]0,+INF[

Insertion loss

Insertion loss of the filter

0 dB — [0,+INF[

Depth

Maximum attenuation value for the filter

100 dB — [0,+INF[

409

RECTANGLE OPTICAL FILTER

Simulation

Noise

Technical backgroundThe filter transfer function is:

where H(f) is the filter transfer function, α is the parameter Insertion loss, d is the parameter Depth, fc is the filter center frequency defined by the parameter Frequency, B is the parameter Bandwidth , and f is the frequency.



Enabled



Resample



Sample rate




Units Value range

Noise threshold



Noise dynamic


3 dB ]-INF,+INF[

H f( )α,d,

=fc B 2 f fc B 2⁄+< <⁄–

otherwise

410

TRAPEZOIDAL OPTICAL FILTER

Trapezoidal Optical filter

Optical filter with a trapezoidal frequency transfer function.

Ports

Parameters

Main


Input Input Optical




Frequency



Zero dB bandwidth

Filter bandwidth at 0 dB


]0,+INF[

Bandwidth

Filter bandwidth at cutoff magnitude


]0,+INF[

Cutoff magnitude

Attenuation at the filter bandwidth

3 dB — [0,+INF[

Insertion loss


0 dB — [0,+INF[

Depth


100 dB — [0,+INF[

411

TRAPEZOIDAL OPTICAL FILTER

Simulation

Noise


where

and H(f) is the filter transfer function, α is the parameter Insertion loss, fc is the filter center frequency defined by the parameter Frequency, B is the parameter Bandwidth at the cutoff magnitude, B0dB is the parameter Zero dB bandwidth, and f is the frequency.



Enabled



Resample



Sample rate




Units Value range

Noise threshold



Noise dynamic


3 dB ]-INF,+INF[

H f( )α.10

1 A–10B B0dB–------------------------------- f f2–( )

,α,

α.101 A–

10B B0dB–------------------------------- f f1–( )

,

=

f f2>

f1 fc f2< <

f f1<

f1 fc B0dB 2⁄–=f2 fc B0dB 2⁄+=

412

GAUSSIAN OPTICAL FILTER

Gaussian Optical filter

Optical filter with a Gaussian frequency transfer function.

Ports

Parameters

Main


Input Input Optical




Frequency



Bandwidth



]0,+INF[

Insertion loss


0 dB — [0,+INF[

Depth


100 dB — [0,+INF[

Order

Order of the function

1 — — [1,100]

413

GAUSSIAN OPTICAL FILTER

Simulation

Noise


where H(f) is the filter transfer function, α is the parameter Insertion loss, fc is the filter center frequency defined by the parameter Frequency, B is the parameter Bandwidth, N is the parameter Order, and f is the frequency.



Enabled



Resample



Sample rate




Units Value range

Noise threshold



Noise dynamic


3 dB ]-INF,+INF[

H f( ) αe 1n 2–

2 f fc–( )2N

B--------------------------

=

414

BUTTERWORTH OPTICAL FILTER

Butterworth Optical filter

Optical filter with a Butterworth frequency transfer function.

Ports

Parameters

Main


Input Input Optical




Frequency



Bandwidth



]0,+INF[

Insertion loss


0 dB — [0,+INF[

Depth


100 dB — [0,+INF[

Order


1 — — [1,100]

415

BUTTERWORTH OPTICAL FILTER

Simulation

Noise

Technical backgroundButterworth filters are a class of all-pole filters with maximally flat frequency response. The filter transfer function is:

where

and H(f) is the filter transfer function, α is the parameter Insertion loss, fc is the filter center frequency defined by the parameter Frequency, B is the parameter Bandwidth, N is the parameter Order, and f is the frequency.



Enabled



Resample



Sample rate




Units Value range

Noise threshold



Noise dynamic


3 dB ]-INF,+INF[

H f( ) α B 2⁄( )N

j f fc–( ) pk–( )k 0=

N 1–

∏------------------------------------------=

pkB2--- e

jπ2--- 1 2k 1+

N---------------+

⋅=

416

BESSEL OPTICAL FILTER

Bessel Optical filter

Optical filter with a Bessel frequency transfer function.

Ports

Parameters

Main


Input Input Optical




Frequency



Bandwidth



]0,+INF[

Insertion loss


0 dB — [0,+INF[

Depth


100 dB — [0,+INF[

Order


1 — — [1,100]

417


Simulation

Noise

Technical backgroundBessel filters have a transfer function of the form:

α is the parameter Insertion loss, N is the parameter Order, and

is a normalizing constant and BN(s) is an nth-order Bessel polynomial of the form

where



Enabled



Resample



Sample rate




Units Value range

Noise threshold



Noise dynamic


3 dB ]-INF,+INF[

H s( ) αd0

BN s( )-------------=

d02N( )!

2N N!⋅----------------=

BN s( ) dksk

k 0=

N

∑=

dk2N k–( )!

2N k– k! N k–( )!⋅---------------------------------------=

418


and

where fc is the filter center frequency defined by the parameter Frequency, B is the parameter Bandwidth, and Wb denotes the normalized 3 dB bandwidth and can be approximated by

for N≥ 3

s j 2 f fc–( ) wb⋅B

----------------------------- =

wb 2N 1–( ) 2ln⋅≈

419

FABRY PEROT OPTICAL FILTER

Fabry Perot Optical filter

Optical filter with a Fabry Perot frequency transfer function.

Ports

Parameters

Main


Input Input Optical




Frequency



Bandwidth



]0,+INF[

Free spectral range 500 GHz Hz, GHz, THz, nm

]0,+INF[

Insertion loss


0 dB — [0,+INF[

Depth


100 dB — [0,+INF[

420

FABRY PEROT OPTICAL FILTER

Simulation

Noise


where H(f) is the filter transfer function, α is the parameter Insertion loss, fc is the filter center frequency defined by the parameter Frequency, B is the parameter Bandwidth, and f is the frequency.

where

where FSR is the parameter Free spectral range.



Enabled



Resample



Sample rate




Units Value range

Noise threshold



Noise dynamic


3 dB ]-INF,+INF[

H f( ) α 1 R–

1 R∗e2πJ

f fc–( )

B-----------------

–

-----------------------------------=

R2 πB

FSR-----------

2

2 πBFSR-----------+

2

4––+

2---------------------------------------------------------------------------=

421

ACOUSTO OPTICAL FILTER

Acousto Optical filter

Optical filter with an Acousto optical frequency transfer function.

Ports

Parameters

Main

Channels


Input Input Optical

Transmission Ouput Optical

Reflection Output Optical



Bandwidth



]0,+INF[

Insertion loss


0 dB — [0,+INF[

Depth


100 dB — [0,+INF[



Number of channels

Number of filter channels

4 — — [1,+INF[

Frequency[0]

Filter center frequency 0


422


Simulation

Noise

Frequency[1]



Frequency[2]



Frequency[3]





Enabled



Resample



Sample rate




Units Value range

Noise threshold



Noise dynamic


3 dB ]-INF,+INF[



423


Technical backgroundThe filter transfer function is described using a sum of power transfer functions of the type

where k=2.78311475, Hn(f) is the filter transfer function for each channel, α is the parameter Insertion loss, fnc is the filter center frequency defined by the parameter Frequency for each channel n, B is the parameter Bandwidth, and f is the frequency.

Hn f( ) αk f fc–( ) B⁄( )sin

k f fnc–( ) B⁄( )---------------------------------------=

424

MACH-ZEHNDER INTERFEROMETER

Mach-Zehnder Interferometer

Ports

Parameters

Main








Delay

Time delay of the first path

0 s s, ms, ns [1e-9,+INF[

Coupling coefficient

Cross coupling coefficients

0.5 — — [0,1]

Additional loss

Loss applied to the signal at the output

0 dB — [0,+INF[

425

MACH-ZEHNDER INTERFEROMETER

Technical backgroundThe Mach-Zehnder filter is tunable and consists of two couplers, which are connected by two waveguides. The filter transfer function for such a case is defined by:

where H(f) is the filter transfer function and f is the frequency.

with:

where α is the parameter Coupling coefficient, and

where t is the parameter time Delay.

H f( ) Hcoupler f( )HτHcoupler f( )=

Hcoupler f( )1 α–

j α=

j α

1 α–

Hτ f( ) e j2πfτ–

0= 0

1

426

INVERTED OPTICAL IIR FILTER

Inverted Optical IIR filter

Inverted infinite impulse response filter (IIR) for optical signals.

Ports

Parameters

Main


Input Input Optical




Frequency



Filter sample rate



[1e-9,+INF[

Additional loss


0 dB — [0,+INF[




427




Simulation




3 — [1,+INF[










3 — [1,+INF[









Enabled



Resample



Sample rate



428


Noise

Technical backgroundThe transfer function is of the form:

where H(f) is the filter transfer function, α is the parameter Insertion loss, HIIR(f) is the IIR filter transfer function (see Optical IIR filter), and f is the frequency.


Units Value range

Noise threshold



Noise dynamic


3 dB ]-INF,+INF[

H f( ) α 1 HIIR f( )– 2=

429

INVERTED RECTANGLE OPTICAL FILTER

Inverted Rectangle Optical filter

Optical filter with an inverted rectangle frequency transfer function.

Ports

Parameters

Main


Input Input Optical




Frequency



Bandwidth



]0,+INF[

Insertion loss


0 dB — [0,+INF[

Depth


100 dB — [0,+INF[

430

INVERTED RECTANGLE OPTICAL FILTER

Simulation

Noise

Technical backgroundThe transfer function is of the form:

where H(f) is the filter transfer function, α is the parameter Insertion loss, HRect(f) is the rectangle filter transfer function (see Rectangle Optical filter), and f is the frequency.



Enabled



Resample



Sample rate




Units Value range

Noise threshold



Noise dynamic


3 dB ]-INF,+INF[

H f( ) α 1 HRect f( ) 2–=

431

INVERTED TRAPEZOIDAL OPTICAL FILTER

Inverted Trapezoidal Optical filter

Optical filter with an inverted trapezoidal frequency transfer function.

Ports

Parameters

Main


Input Input Optical




Frequency


193.1 THz Hz, THz, nm [30,3e5]

Zero dB bandwidth 10 GHz Hz, GHz, THz, nm

[1e-9,+INF[

Bandwidth



[1e-9,+INF[

Cutoff magnitude 3 dB — [0,+INF[

Insertion loss


0 dB — [0,+INF[

Depth


100 dB — [0,+INF[

432

INVERTED TRAPEZOIDAL OPTICAL FILTER

Simulation

Noise



Enabled



Resample



Sample rate




Units Value range

Noise threshold


–100 dB ]-INF,0[

Noise dynamic


3 dB [0,+INF[

433

INVERTED GAUSSIAN OPTICAL FILTER

Inverted Gaussian Optical filter

Optical filter with an inverted gaussian frequency transfer function.

Ports

Parameters

Main


Input Input Optical




Frequency



Bandwidth



]0,+INF[

Insertion loss


0 dB — [0,+INF[

Depth


100 dB — [0,+INF[

Order


1 — — [1,100]

434

INVERTED GAUSSIAN OPTICAL FILTER

Simulation

Noise

Technical backgroundThe transfer function is:

where H(f) is the filter transfer function, α is the parameter Insertion loss, HGauss(f) is the filter transfer function (see Gaussian Optical filter), and f is the frequency.



Enabled



Resample



Sample rate




Units Value range

Noise threshold


–100 dB ]-INF,0[

Noise dynamic


3 dB [0,+INF[

H f( ) α 1 HGauss f( )– 2=

435

INVERTED BUTTERWORTH OPTICAL FILTER

Inverted Butterworth Optical filter

Optical filter with an inverted Butterworth frequency transfer function.

Ports

Parameters

Main


Input Input Optical




Frequency



Bandwidth



]0,+INF[

Insertion loss


0 dB — [0,+INF[

Depth


100 dB — [0,+INF[

Order


1 — — [1,100]

436

INVERTED BUTTERWORTH OPTICAL FILTER

Simulation

Noise


where H(f) is the filter transfer function, α is the parameter Insertion loss, HB(f) is the filter transfer function (see Butterworth Optical filter), and f is the frequency.



Enabled



Resample



Sample rate




Units Value range

Noise threshold


–100 dB ]-INF,0[

Noise dynamic


3 dB [0,+INF[

H f( ) α 1 HB f( )– 2=

437

INVERTED BESSEL OPTICAL FILTER

Inverted Bessel Optical filter

Optical filter with an inverted Bessel frequency transfer function.

Ports

Parameters

Main


Input Input Optical




Frequency



Bandwidth



]0,+INF[

Insertion loss


0 dB — [0,+INF[

Depth


100 dB — [0,+INF[

Order


1 — — [1,100]

438

INVERTED BESSEL OPTICAL FILTER

Simulation

Noise


where H(f) is the filter transfer function, α is the parameter Insertion loss, HB(f) is the filter transfer function (see Bessel Optical filter), and f is the frequency.



Enabled



Resample



Sample rate




Units Value range

Noise threshold –100 dB ]-INF,0[

Noise dynamic 3 dB [0,+INF[

H f( ) α 1 HB f( )– 2=

439

FIBER BRAGG GRATING (FBG)

Fiber Bragg Grating (FBG)

Simulates an FBG.

Ports

Parameters

Main


Input Input Optical





Frequency

Optical frequency of the center of the Fiber Bragg Grating reflection spectrum

193.1 THz Hz, THz, nm [30,3e5]

Effective index

Modal index of the optical fiber grating

1.45 — — [1,10]

Length

Length of the optical fiber grating

2 nm — [1e-6,1e3]

440


Apodization

Chirp


Apodization function

Modulates the grating intensity over the grating length.

Uniform — Uniform, Gaussian, Tanh, user-defined

Gauss parameter

Apodization is defined by a Gaussian function using the S parameter. See Technical Background for the definition.

0.5 — [0.01,100]

Tanh parameter

Apodization is defined by an hyperbolic tangent function using the S parameter. See Technical Background for the definition.

0.5 — [0.01,100]

Apodization filename

You supply a file for the apodization. The ith element of this file is applied as the local apodization for the ith segment of the grating.

Apodization.dat — —

Modulation AC

Index modulation when the apodization is unity. The product of this number with the apodization function determines the local index modulation.

0.00001 — ]0,1e3]

Modulation DC

Modifies the modal index of the fiber in proportion to the apodization function.

0 — ]0,1e3]


Chirp function

Period that the grating can be changed over the length of the fiber.

None — None, Linear, Quadratic, Square root, Cubic root, user-defined

Linear parameter

Period varies in a linear way, as defined in the Technical Background.

0.00001 µm [0.01,100]

Quadratic parameter

Period varies in a quadratic way, as defined in the Technical Background.

0.00001 µm [0.01,100]

Square root parameter

Period varies as defined in the Technical Background.

0.00001 µm [0.01,100]

441


Calculation

Simulation

Cubic root parameter

Period varies as defined in the Technical Background.

0.00001 µm [0.01,100]

Chirp filename

In the user-defined file, the ith entry is used as the period for the ith segment of the chirped grating.

ChirpPeriod.dat — —


Units Value range

Number of segments

The non-uniform grating will be divided into this number of equal length uniform segments to calculate the spectrum

101 — [1,1e9]

Max. number of spectral points

Maximum nuber of points for the transmission and reflection complex spectrum

1000 — [100,1e6]



Enabled



Resample



Sample rate




442


Noise

Technical backgroundThe non-uniform (chirped and apodized) grating [1] is divided into Number of Segments uniform gratings. The coupled mode theory is used to calculate the scattering matrix of each uniform segment, and the spectral response of the whole grating is found by connecting the uniform segments using the transfer matrix theory. The apodization functions Gaussian and Hyperbolic tangent are defined with the following parameters:

Gaussian

Hyperbolic tangent



Noise threshold


–100 dB — ]-INF,0[

Noise dynamic


3 dB — [0,+INF[

Noise calculation bandwidth

Calculation bandwidth — outside of this range, calculation is replaced by the attenuation


[0, 1e+100]

A z( ) 2 2 z L 2⁄–( )⋅s L⋅

-------------------------------2

⋅ln–

exp=

A z( ) s z L⁄⋅( ) s 1 z L⁄–( )⋅[ ] 1 h s 2⁄( )2tan–+tanh⋅tanh=

443


When the parameter Apodization function is user-defined, you provide a file with the data describing the apodization. The input file is formatted containing two items per line — the length in mm and the apodization value.

The chirp functions depend on a parameter, ∆, which is used as follows:

Linear

Quadratic

Square Root

0 7.99437714249507e-007

0.2 2.39785072153609e-006

0.4 3.99496320824255e-006

0.6 5.58995679966756e-006

0.8 7.18201727067935e-006

1.0 8.770334716246e-006

1.2 1.03541096905246e-005

.

.

.

Λ z( ) Λ0z L 2⁄–

L------------------∆–= ∆ Λ0«

Λ z( ) Λ0zL---

2 1

4---+ ∆–= ∆ Λ0«

Λ z( ) Λ0zL--- 1

2-------– ∆–= ∆ Λ0«

444


Cubic Root

When the parameter Chirp function is user-defined, you provide a file with the data describing the chirp. The input file is formatted containing two items per line — the length in mm and the chirp value in µm.

References[1] Erdogan, R., “Fiber Grating Spectra”, J. Light. Technol., 15, 1277-1294, (1997).

0 0.53368353843689

0.2 0.53369003534317

0.4 0.533694565296173

0.6 0.533698260784149

0.8 0.533701419830322

1.0 0.533704221248627

1.2 0.533706843852997

.

.

.

Λ z( ) Λ0zL---3

123

-------– ∆–= ∆ Λ0«

445

UNIFORM FIBER BRAGG GRATING

Uniform Fiber Bragg Grating

Simulates a Uniform FBG.

Ports

Parameters

Main


Input Input Optical





Frequency

Optical frequency of the center of the Fiber Bragg Grating reflection spectrum

193.1 THz Hz, THz, nm [30,3e5]

Bandwidth

Width of the reflection stop band of the Fiber Bragg Grating


[0,+INF[

Reflectivity

Desired maximum reflectivity of the grating (maximum is at the centre wavelength)

0.99 — — [ 1e-100, 1]

446

UNIFORM FIBER BRAGG GRATING

Simulation

Noise

Technical backgroundThe solution to the coupled mode equations for a uniform grating is used. The unknown parameters in the grating (grating period, grating modulation intensity) are found by employing the information about maximum reflectivity and bandwidth. The result is a module for the calculation of the reflection and transmission spectra [1].

References[1] Agrawal, G.P., Fiber-Optic Communication Systems. John Wiley & Sons, New York, (1997).



Enabled



Resample



Sample rate





Noise threshold


–100 dB — ]-INF,0[

Noise dynamic


3 dB — [0,+INF[

Noise calculation bandwidth

Calculation bandwidth, outside of this range calculation is replaced by the attenuation


[0, 1e+100]

447

IDEAL DISPERSION COMPENSATION FBG

Ideal Dispersion Compensation FBG

Approximation of an ideal chirped FBG designed for dispersion compensation.

Ports

Parameters

Main


Input Input Optical





Frequency



Bandwidth



]0,+INF[

Insertion loss


0 dB — [0,+INF[

Depth


100 dB — [0,+INF[

Dispersion

Group delay slope

800 ps/nm ps/nm s/m ] -INF, +INF[

448


Simulation

Noise

Technical backgroundThis model is a filter with user-defined group delay. The filter transfer function is:

where f is the frequency dependence phase of the filter.

The group delay is defined by [1]:

Writing Equation 2 as a function of wavelength:

where c is the speed of light.



Enabled



Resample



Sample rate




Units Value range

Noise threshold



Noise dynamic


3 dB ]-INF,+INF[

(1)

(2)

(3)

H f( ) ejφ f( )=

τ f( ) 12π------dφ

df------–=

τ λ( ) λ2

2πc---------dφ

dλ------–=

449


You define by entering the center wavelength , bandwidth , and the group delay slope D in s/m:

This generates the following group delay curve:

Figure 1 Group delay

Calculate the phase from this curve to calculate the filter transfer function.

Phase calculationThe phase is calculated from Equation 3 and Equation 4:

(4)

:

(5)

τ λc ∆λ

τ λ( )

τ0

D.λτλc ∆λ 2⁄+

=λ λc ∆λ 2⁄–≤

λc ∆λ 2 λ λc ∆λ 2⁄+≤<⁄–λ λc ∆λ 2⁄+>

φ 2πc τ λ( ) 1λ2----- λd∫–=

λ λc ∆λ 2⁄–≤

φ 2πcτ01λ2----- λd

λ1

λ

∫– 2πcτ01λ--- 1

λ1-----–

= =

λ1 ∞ τ0,– 0= =φ 0=

450


:

(6)

:

(7)

λc ∆λ 2 λ λc≤<⁄– ∆λ 2⁄+

φ 2πcD λ λ1–( )λ2

------------------- λ φλc ∆λ 2⁄–+dλ1

λ

∫ 2πcD λ( ) 2πcDλ1

λ-----– φλc ∆λ 2⁄–+ln= =

φλc ∆λ 2⁄– 2πcD λ1( ) 2πcD λ1,–ln λc ∆λ 2⁄–( )= =

φ 2πcD λ( ) 2πcD λc ∆λ 2⁄–( )λ

----------------------------– 2πcD λc ∆λ 2⁄–( ) 2πcD–ln+ln=

λ λc ∆λ 2⁄+>

φ 2πcτλc ∆λ 2⁄–1λ2----- λ φλc ∆λ 2⁄–+d

λ1

λ

∫– 2πcτλc ∆λ 2⁄–1λ--- 1

λ1-----–

φλc ∆λ 2⁄–+= =

λ1 λc ∆λ 2⁄+( ) λ2, λ τλc ∆λ 2⁄–, D ∆λ( )–= = =

φλc ∆λ 2⁄– 2πcD λc ∆λ 2⁄+( ) 2πcD λc ∆λ 2⁄–( )λc ∆λ 2⁄+( )

-----------------------------– 2πcD λc ∆λ 2⁄–( ) 2πcD–ln+ln=

φ 2πcD∆λ– 1λ--- 1

λc ∆λ 2⁄+( )-----------------------------–

+= =

2πcD λc ∆λ 2⁄+( ) 2πcD λc ∆λ 2⁄–( )λc ∆λ 2⁄+( )

-----------------------------– 2πcD λc ∆λ 2⁄–( ) 2πcD–ln+ln

451


This generates the following typical phase curve (for :

Figure 2 Cumulative phase

References[1] Madsen, C. K. and Zhao, J H., Optical Filter Design and Analysis: A Signal Processing

Approach. John Wiley & Sons, New York, (1999).

D 0.8s m⁄–=

452

LOW PASS IIR FILTER (ELECTRICAL)

Low Pass IIR filter (Electrical)

Infinite impulse response filter (IIR) for electrical signals.

Ports

Parameters

Main






Filter sample rate


10 GHz Hz, GHz [1e-9,+INF[

Additional loss


0 dB dB [0,+INF[




453




Simulation


Units Value range



3 — [1,+INF[








Units Value range



3 — [1,+INF[








Units Value range

Enabled



454



where H(z) is the filter transfer function in the Z domain, α is the parameter related to Additional loss, N is the parameter number of Numerator coefficients, an are the coefficients for the numerator, M is the parameter number of Denominator coefficients, and bm are the coefficients for the denominator.

Also

where fs is the parameter Filter sample rate, and f is the frequency.

According to the parameter Filter coefficients type, the filter transfer function can be given in the z (z domain) or in the frequency domain. In the second case, the filter is determined by the numerator and the denominator polynomial, which can be expressed by their roots (Poles and zeros) or by the polynomial coefficients (in Frequency domain).

H z( ) α

anz n–

n 0=

N

∑

bmz m–

m 0=

M

∑-----------------------=

z j2πf fs⁄( )exp=

455

LOW PASS RECTANGLE FILTER (ELECTRICAL)

Low Pass Rectangle filter (Electrical)


Ports

Parameters

Main

Simulation






Cutoff frequency

3 dB cutoff frequency of the filter

0.75 * bit rate Hz Hz, MHz, GHz [0,+INF[

Insertion loss


0 dB — [0,+INF[

Depth


100 dB — [0,+INF[


Units Value range

Enabled



456

LOW PASS RECTANGLE FILTER (ELECTRICAL)


where H(f) is the filter transfer function,α is the parameter Insertion loss, d is the parameter Depth, fc is the cutoff frequency, and f is the frequency.

H f( )α,d,

=0 f fc< <

otherwise

457

LOW PASS GAUSSIAN FILTER (ELECTRICAL)

Low Pass Gaussian filter (Electrical)


Ports

Parameters

Main

Simulation






Cutoff frequency



Insertion loss


0 dB — [0,+INF[

Depth


100 dB — [0,+INF[

Order


1 — — [1, 100]


Units Value range

Enabled



458

LOW PASS GAUSSIAN FILTER (ELECTRICAL)


where H(f) is the filter transfer function, α is the parameter Insertion loss, fc is the filter cutoff frequency, N is the parameter Order, and f is the frequency.

H f( ) αe2 f

2N

fc--------

ln–

=

459

LOW PASS BUTTERWORTH FILTER (ELECTRICAL)

Low Pass Butterworth filter (Electrical)


Ports

Parameters

Main

Simulation






Cutoff frequency



Insertion loss


0 dB — [0,+INF[

Depth


100 dB — [0,+INF[

Order


1 — — [1, 100]


Units Value range

Enabled



460

LOW PASS BUTTERWORTH FILTER (ELECTRICAL)

Technical backgroundButterworth filters are a class of all-pole filters with maximally flat frequency response. In this case. the filter transfer function is:

where

where H(f) is the filter transfer function, α is the parameter Insertion loss, fc is the filter cutoff frequency, N is the parameter Order, and f is the frequency.

H f( ) αfc( )N

j f( ) pk–( )k 0=

N 1–

∏--------------------------------=

pk fc ejπ2-- 1 2k 1+

N--------------+

⋅=

461

LOW PASS BESSEL FILTER (ELECTRICAL)

Low Pass Bessel filter (Electrical)

Filter with a Bessel frequency transfer function.

Ports

Parameters

Main

Simulation






Cutoff frequency



Insertion loss


0 dB — [0,+INF[

Depth


100 dB — [0,+INF[

Order


4 — — [1, 100]


Units Value range

Enabled



462

LOW PASS BESSEL FILTER (ELECTRICAL)

Technical backgroundBessel filters have the following transfer function:

where α is the parameter Insertion loss, N is the parameter Order, and

being a normalizing constant and BN(s) an nth-order Bessel polynomial of the form:

where

and

where fc is the filter cutoff frequency defined by the parameter Frequency and Wb denotes the normalized 3 dB bandwidth and can be approximated by:

for N≥ 3

H s( ) αd0

BN s( )-------------=

d02N( )!

2N N!⋅----------------=

BN s( ) dksk

k 0=

N

∑=

dk2N k–( )!

2N k– k! N k–( )!⋅---------------------------------------=

s j f wb⋅fc

------------ =

wb 2N 1–( ) 2ln⋅≈

463

LOW PASS CHEBYSHEV FILTER (ELECTRICAL)

Low Pass Chebyshev filter (Electrical)

Filter with a Chebyshev frequency transfer function.

Ports

Parameters

Main

Simulation






Cutoff frequency



Insertion loss


0 dB — [0,+INF[

Depth


100 dB — [0,+INF[

Order


1 — — [1, 100]

Ripple factor

Ripple parameters

0.5 — — [0, 1]


Units Value range

Enabled



464

LOW PASS CHEBYSHEV FILTER (ELECTRICAL)

Technical backgroundChebychev of order N filters have the following transfer function:

where α is the parameter Insertion loss and N is the parameter Order.

Also

and

where fc is the filter cutoff frequency.

The parameters:

and

where

where rp is the parameter ripple factor.

H s( ) α

sk

k 0=

N 1–

∏

s sk–( )k 0=

N 1–

∏-------------------------⋅–=

s jf=

sk fc δ βk j δ βksin⋅cosh⋅+cos⋅sinh( )⋅=

δ 1N----ar r 1–( )sinh=

βkπ 2 k 1+( ) N 1–+( )

2N------------------------------------------------=

r 11 rp–------------- 1–=

465

LOW PASS RC FILTER (ELECTRICAL)

Low Pass RC filter (Electrical)

Filter with an RC frequency transfer function.

Ports

Parameters

Main

Simulation






Cutoff frequency



Insertion loss


0 dB — [0,+INF[

Depth


100 dB — [0,+INF[


Units Value range

Enabled



466

LOW PASS RC FILTER (ELECTRICAL)

Technical backgroundRC filter has the following transfer function:

where α is the parameter Insertion loss and fc is the filter cutoff frequency.

H f( ) α 1

1 j ffc---+

---------------⋅=

467

LOW PASS RAISED COSINE FILTER (ELECTRICAL)

Low Pass Raised Cosine filter (Electrical)

Filter with a raised cosine frequency transfer function.

Ports

Parameters

Main

Simulation






Cutoff frequency



Insertion loss


0 dB — [0,+INF[

Depth


100 dB — [0,+INF[

Roll-off factor 0.5 — — [0, 1]


Units Value range

Enabled



468

LOW PASS RAISED COSINE FILTER (ELECTRICAL)

Technical backgroundRaised cosine filter has the following transfer function:

where

where α is the parameter Insertion loss, fc is the filter cutoff frequency, and rp is the parameter Roll off factor.

H f( )

α

α cos2 π2rp∆f------------- f( )

1 rp–( )2

------------------∆f–⋅

0

=

f 1 rp–( )2

------------------∆f<

1 rp–( )2

------------------∆f f 1 rp+( )2

------------------∆f<≤

1 rp+( )2

------------------∆f f≤

∆f 2fc1

1 rp– 4 π rp arc 24cos⋅ ⋅⁄+--------------------------------------------------------------------⋅=

469

LOW PASS COSINE ROLL OFF FILTER (ELECTRICAL)

Low Pass Cosine Roll Off filter (Electrical)

Filter with a cosine roll off frequency transfer function.

Ports

Parameters

Main

Simulation






Cutoff frequency



Insertion loss


0 dB — [0,+INF[

Depth


100 dB — [0,+INF[

Roll off factor 0.5 — — [0, 1]


Units Value range

Enabled



470

LOW PASS COSINE ROLL OFF FILTER (ELECTRICAL)

Technical backgroundCosine Roll Off Filter has the following transfer function:=

where a is the parameter Insertion loss, fc is the filter cutoff frequency, and rp is the parameter Roll off factor.

The parameters f1 and f2 are:

and

H f( )

α

0.5 α2 1 f f1–rp ∆fFWHM⋅------------------------- π⋅

cos+⋅ ⋅

0

=

f f1<

f1 f f2<≤

f2 f≤

f1 1 rp–( )fc= 0 rp 1≤ ≤

f2 1 rp+( )fc= 0 rp 1≤ ≤

471

LOW PASS SQUARED COSINE ROLL OFF FILTER (ELECTRICAL)

Low Pass Squared Cosine Roll Off filter (Electrical)

Filter with a square cosine roll off frequency transfer function.

Ports

Parameters

Main

Simulation






Cutoff frequency



Insertion loss


0 dB — [0,+INF[

Depth


100 dB — [0,+INF[



Units Value range

Enabled



472

LOW PASS SQUARED COSINE ROLL OFF FILTER (ELECTRICAL)

Technical backgroundSquare cosine roll off filter has the following transfer function:

where α is the parameter Insertion loss and rp is the roll off factor.

The parameter is related to the filter frequency cutoff by:

where fc is the filter cutoff frequency.

H f( )

α

0.5 α 1 f f1–rp ∆f⋅-------------- π⋅

cos+⋅ ⋅

0

=

f f1<

f1 f f2<≤

f2 f≤

∆f

∆f 2fc

1 2π--- arc 2 1–( ) 1–cos⋅ rp⋅+

-------------------------------------------------------------------------------=

473

BAND PASS IIR FILTER (ELECTRICAL)

Band Pass IIR filter (Electrical)

Infinite impulse response filter (IIR) for electrical signals.

Ports

Parameters

Main






Frequency


10 GHz Hz, MHz, GHz [0,+INF[

Filter sample rate


10 GHz Hz, GHz [1e-9,+INF[

Additional loss


0 dB — [0,+INF[




474




Simulation


Units Value range



3 — [1,+INF[








Units Value range



3 — [1,+INF[






Denominator[2].imag3 0 — ]-INF,+INF[


Units Value range

Enabled



475



where H(z) is the filter transfer function in the Z domain, α is the parameter related to Additional loss, N is the parameter number of Numerator coefficients, an are the coefficients for the numerator, M is the parameter number of Denominator coefficients, and bm are the coefficients for the denominator.

Also

where fc is the filter center frequency defined by the parameter Frequency, fs is the parameter Filter sample rate, and f is the frequency.

According to the parameter Filter coefficients type, the filter transfer function can be given in the z (Z domain) or in the frequency domain. In the second case, the filter is determined by the numerator and the denominator polynomial, which can be expressed by their roots (Poles and zeros) or by the polynomial coefficients (Frequency domain).

H z( ) α

anz n–

n 0=

N

∑

bmz m–

m 0=

M

∑----------------------=

z j2π f fc–( ) fs⁄( )exp=

476

MEASURED FILTER (ELECTRICAL)

Measured filter (Electrical)

Filter based on measurements.

Ports

Parameters

Main

Measurements









Frequency




Units Value range

File frequency unit


Hz — Hz, THz

File format


Power — Power, Power Phase, Real Imag, phase

Linear scale

Determines whether or not the measured data is in linear scale

True — ]-INF,+INF[

477


Numerical

Simulation

Technical backgroundThe input file is formatted containing two items per line, the frequency and filter measurement. The parameter File frequency unit determines the frequency or wavelength unit of the first item; It can be in Hz or THz.

According to the parameter File format the second item can be one value (Power or Phase) or two values (Power and Phase or Real and Imag):

Filename


Filter.dat — —


Units Value range

Interpolation




Units Value range

Enabled




Units Value range

478


Power (Phase is set to zero, assuming frequency unit is THz)

Power Phase

Real Imag

193.10 0

193.11 0.5

193.12 0.5

193.13 0

...

193.10 0 0

193.11 0.5 3.14

193.12 0.5 3.14

193.13 0 0

...

193.10 0

193.11 –0.5 7.9e-4

193.12 –0.5 7.9e-4

193.13 0 0

...

479



The parameter User defined frequency determines if you can enter the center frequency.

From the measured data,

where is the center frequency of the loaded file, is the maximum frequency of the file, and is the minimum frequency of the file. If the option 'User Defined Frequency' is selected, then the center frequency of the loaded file becomes centered at the user defined frequency.

193.10 0

193.11 3.14

193.12 3.14

193.13 0

...

Fc Max Min+( ) 2⁄=

Fc MaxMin

480

BAND PASS RECTANGLE FILTER (ELECTRICAL)

Band Pass Rectangle filter (Electrical)


Ports

Parameters

Main

Simulation






Frequency



Bandwidth



Insertion loss


0 dB — [0,+INF[

Depth


100 dB — [0,+INF[


Units Value range

Enabled



481

BAND PASS RECTANGLE FILTER (ELECTRICAL)


where H(f) is the filter transfer function, α is the parameter Insertion loss, d is the parameter Depth, fc is the filter center frequency defined by the parameter Frequency, B is the parameter Bandwidth, and f is the frequency.

H f( )α,d,

=

fc B 2 f fc B 2⁄+< <⁄–otherwise

482

BAND PASS GAUSSIAN FILTER (ELECTRICAL)

Band Pass Gaussian filter (Electrical)


Ports

Parameters

Main

Simulation






Frequency



Bandwidth



Insertion loss


0 dB — [0,+INF[

Depth


100 dB — [0,+INF[

Order


1 — — [1, 100]


Units Value range

Enabled



483

BAND PASS GAUSSIAN FILTER (ELECTRICAL)



H f( ) αe2 2

f fc–( )2N

B-----------------------

ln–

=

484

BAND PASS BUTTERWORTH FILTER (ELECTRICAL)

Band Pass Butterworth filter (Electrical)


Ports

Parameters

Main

Simulation






Frequency



Bandwidth



Insertion loss


0 dB — [0,+INF[

Depth


100 dB — [0,+INF[

Order


1 — — [1, 100]


Units Value range

Enabled



485

BAND PASS BUTTERWORTH FILTER (ELECTRICAL)

Technical backgroundButterworth filters are a class of all-pole filters with maximally flat frequency response. The filter transfer function is:

where


H f( ) α B 2⁄( )N

j f fc–( ) pk–( )k 0=

N 1–

∏

------------------------------------------=

pkB2--- e

jπ2--- 1 2k 1+

N---------------+

⋅=

486

BAND PASS BESSEL FILTER (ELECTRICAL)

Band Pass Bessel filter (Electrical)

Filter with a Bessel frequency transfer function.

Ports

Parameters

Main

Simulation






Frequency



Bandwidth



Insertion loss


0 dB — [0,+INF[

Depth


100 dB — [0,+INF[

Order


4 — — [1, 100]


Units Value range

Enabled



487

BAND PASS BESSEL FILTER (ELECTRICAL)

Technical backgroundBessel filters have the following transfer function:

where α is the parameter Insertion loss, N is the parameter Order, and

is a normalizing constant and BN(s) is an nth-order Bessel polynomial of the form

where

and

where fc is the filter center frequency defined by the parameter Frequency, B is the parameter Bandwidth, and Wb denotes the normalized 3 dB bandwidth and can be approximated by:

for N≥ 3

H s( ) αd0

BN s( )-------------=

d02N( )!

2N N!⋅----------------=

BN s( ) dksk

k 0=

N

∑=

dk2N k–( )!

2N k– k! N k–( )!⋅---------------------------------------=

s j 2 f fc–( ) wb⋅B

-------------------------- =

wb 2N 1–( ) 2ln⋅≈

488

BAND PASS CHEBYSHEV FILTER (ELECTRICAL)

Band Pass Chebyshev filter (Electrical)

Filter with a Chebyshev frequency transfer function.

Ports

Parameters

Main






Frequency



Bandwidth



Insertion loss


0 dB — [0,+INF[

Depth


100 dB — [0,+INF[

Order


1 — — [1, 100]

Ripple factor

Bandpass ripple parameter

0.01 — — [0, 1]

489


Simulation

Technical backgroundChebychev of order N filters have the following transfer function:

where is the parameter Insertion loss and N is the parameter Order.

with

where fc is the filter center frequency defined by the parameter Frequency.

Here, Sk are the poles of the filter defined by:

where B is the parameter Bandwidth.

and

where rp is the parameter ripple factor.


Units Value range

Enabled



H s( ) α

sk

k 0=

N 1–

∏

s sk–( )k 0=

N 1–

∏-------------------------⋅=

α

s j f fc–( )=

skB2--- δ βk j δ βksin⋅cosh⋅+cos⋅sinh( )⋅=

r 11 rp–------------- 1–=

δ 1N----ar r 1–( )sinh=

490


and

βkπ 2 k 1+( ) N 1–+( )

2N------------------------------------------------=

491

BAND PASS RC FILTER (ELECTRICAL)

Band Pass RC filter (Electrical)

Filter with an RC frequency transfer function.

Ports

Parameters

Main

Simulation






Frequency



Bandwidth



Insertion loss


0 dB — [0,+INF[

Depth


100 dB — [0,+INF[


Units Value range

Enabled



492

BAND PASS RC FILTER (ELECTRICAL)

Technical backgroundRC filter has the following transfer function:

where α is the parameter Insertion loss, fc is the filter center frequency defined by the parameter Frequency, and B is the parameter Bandwidth.

H f( ) α 1

1 j2f fc–B

-----------+--------------------------⋅=

493

BAND PASS RAISED COSINE FILTER (ELECTRICAL)

Band Pass Raised Cosine filter (Electrical)

Filter with a raised cosine frequency transfer function.

Ports

Parameters

Main

Simulation






Frequency



Bandwidth



Insertion loss


0 dB — [0,+INF[

Depth


100 dB — [0,+INF[



Units Value range

Enabled



494

BAND PASS RAISED COSINE FILTER (ELECTRICAL)

Technical backgroundRaised cosine filter has the following transfer function:

where

where α is the parameter Insertion loss, fc is the filter center frequency defined by the parameter Frequency, B is the parameter Bandwidth, and rp is the parameter Roll off factor.

H f( )

α

α cos2 π2rp∆f------------- f fc–( )

1 rp–( )2

------------------∆f–⋅

0

=

f fc– 1 rp–( )2

------------------∆f<

1 rp–( )2

------------------∆f f fc– 1 rp+( )2

------------------∆f<≤

1 rp+( )2

------------------∆f f fc–≤

∆f B 11 rp– 4 π rp arc 24cos⋅ ⋅⁄+--------------------------------------------------------------------⋅=

495

BAND PASS COSINE ROLL OFF FILTER (ELECTRICAL)

Band Pass Cosine Roll Off filter (Electrical)

Filter with a cosine roll off frequency transfer function.

Ports

Parameters

Main

Simulation






Frequency



Bandwidth



Insertion loss


0 dB — [0,+INF[

Depth


100 dB — [0,+INF[



Units Value range

Enabled



496

BAND PASS COSINE ROLL OFF FILTER (ELECTRICAL)

Technical backgroundCosine Roll Off Filter has the following transfer function:

where α is the parameter Insertion loss, fc is the filter center frequency defined by the parameter Frequency, B is the parameter Bandwidth, and rp is the parameter Roll off factor.

The parameters f1 and f2 are:

and

H f( )

α

0.5 α2 1 f fc– f1–rp ∆fFWHM⋅------------------------- π⋅

cos+⋅ ⋅

0

=

f fc– f1<

f1 f fc– f2<≤

f2 f fc–≤

f11 rp–

2-------------B= 0 rp 1≤ ≤

f11 rp+

2-------------B= 0 rp 1≤ ≤

497

BAND PASS SQUARE COSINE ROLL OFF FILTER (ELECTRICAL)

Band Pass Square Cosine Roll Off filter (Electrical)

Filter with a square cosine roll off frequency transfer function.

Ports

Parameters

Main






Frequency



Bandwidth



Insertion loss


0 dB — [0,+INF[

Depth


100 dB — [0,+INF[


498

BAND PASS SQUARE COSINE ROLL OFF FILTER (ELECTRICAL)

Simulation

Technical backgroundSquare cosine roll off filter has the following transfer function:

where α is the parameter Insertion loss, fc is the filter center frequency defined by the parameter Frequency, and rp is the roll off factor.

The parameter is related to the filter bandwidth by:

(2)

where B is the parameter Bandwidth.


Units Value range

Enabled



H f( ) 0.5 α 1

α

f fc– f1–rp ∆f⋅

----------------------- π⋅

0

cos+f fc– f1<

f1 f fc– f2<≤

f2 f fc–≤

⋅ ⋅

=

∆f

∆f B

1 2π--- arc 2 1–( ) 1–cos⋅ rp⋅+

-------------------------------------------------------------------------------=

499

S PARAMETERS MEASURED FILTER (ELECTRICAL)

S Parameters Measured filter (Electrical)

Loads files with S Parameter measurements. You can load files directly from measurements by using the Touchstone (.s2p) format.

Ports

Parameters

Main

Measurements



Transmission Output Electrical

Reflection Output Electrical






Frequency


0 GHz Hz, MHz, GHz [0, 1e+100]


Units Value range

Filename (.s2p)


Device.s2p — —

500


Numerical

Simulation

Graphs

Technical backgroundThe Touchstone Format is a common standard for S Parameter data. The model expects the .s2p file to be in the following general format (lines starting with the comment symbol '!' and blank lines are ignored):

# freq_unit param_type data_form term_type term_val

f1 s11a s11b s21a s21b s12a s12b s22a s22b

f2 s11a s11b s21a s21b s12a s12b s22a s22b

.

.

.

fn s11a s11b s21a s21b s12a s12b s22a s22b

where: • freq_unit: Specifies the frequency units — can be Hz, kHz, MHz, or GHz.• param_type: Usually set to S to indicate S Parameter file.


Units Value range

Interpolation




Units Value range

Enabled




Reflection - real part Frequency (Hz) Amplitude (a.u.)

Reflection - imag part Frequency (Hz) Amplitude (a.u.)

Transmission - real part Frequency (Hz) Amplitude (a.u.)

Transmission - imag part Frequency (Hz) Amplitude (a.u.)

501


• data_form: Either RI (for real imaginary), MA (for magnitude & angle) or DB (for magnitude in dB scale & angle). Indicates how the component should treat the pair of S Parameter values.

• term_type: Termination type (R for real or Z for terminating impedance). Usually R.

• term_val: Termination value (if R, then the value in Ohms, else a pair representing the impedance).

The header is followed by the data. Each line has nine values — the frequency and the eight values representing four S Parameters. This model loads only the S11 and S21 (direct reflection and transmission).

The following example was generated by a network analyzer. The units are in Hz and the data is in real and imaginary values.

The parameter User defined frequency determines if you can enter the center frequency. This means that the filter data is shifted from the measured center frequency to the user center frequency that you define by the parameter Frequency.

! Network Analyzer

! Model 1

! 16 Dec 1999 15:02:50

!Frequency S11 S21 S12 S22

# HZ S RI R 50

3000 2.17788E-1 0.24215E-1 -5.69091E0 4.64843E-1 3.02257E-2 0.33741E-2 -6.33483E-1 0.40252E-1

30029850 1.72088E-1 -1.57524E-1 -5.98193E0 -1.68359E0 4.33025E-2 1.31721E-2 -4.84573E-1 1.45126E-1

60029700 0.49133E-1 -2.12097E-1 -7.35302E0 -2.20703E0 5.24978E-2 1.82323E-2 -3.78585E-1 1.96167E-1

90029550 -4.32815E-2 -2.02163E-1 -8.36279E0 -2.04736E0 5.92289E-2 1.87740E-2 -2.99804E-1 1.91909E-1

120029400 -9.79766E-2 -1.74827E-1 -8.99023E0 -1.67724E0 6.32743E-2 1.8013E-2 -2.49618E-1 1.72729E-1

.

.

.

502


Notes:

503

OPTICAL FILTER ANALYZER

Optical Filter analyzer

Extracts the frequency response of an optical component by comparing a reference optical signal before and after the calculation.

Ports

Parameters

Main


From DUT output Input Optical

To DUT input Output Optical



Frequency

Signal center frequency

193.1 THz Hz, THz, nm [0,3e5[

Bandwidth

Signal bandwidth or sample rate


[0,+INF[

Frequency unit

Unit scale for the graphs


Linear scale

Determines whether or not the graph is in linear scale


Minimum value

Minimum value when using log scale

–100 dBm — ]-INF,+INF[

504

OPTICAL FILTER ANALYZER

Results

Export

Graphs

Results


Units Value range

Cutoff magnitude

Reference cutoff magnitude for bandwidth result calculation

3 dB [0,100]


Save to file

Determines if the filter transmission will be saved as a file


Filename

Filename with the filter data

Filter.dat — —


Transmission function Frequency (Hz) Power (dB)

Transmission Phase X

Phase at the polarization X

Frequency (Hz) Phase (rad)

Name and description Unit

Frequency at Max. Transmission Hz

Filter Bandwidth (BandPass) at Cut off Hz

Frequency at Max. Transmission nm

Filter Bandwidth (BandPass) at Cut off nm

Frequency at Min. Transmission Hz

Filter Bandwidth (Band-Reject) at Cut off Hz

Frequency at Min. Transmission nm

Filter Bandwidth (Band-Reject) at Cut off nm

505

ELECTRICAL FILTER ANALYZER

Electrical Filter analyzer

Extracts the frequency response of an electrical component by comparing a reference electrical signal before and after the calculation.

Ports

Parameters

Main

Results


From DUT output Input Electrical

To DUT input Output Electrical



Bandwidth

Signal bandwidth or sample rate

100 GHz Hz, GHz, THz [0,+INF[

Frequency unit

Unit scale for the graphs

Hz — — Hz, THz

Linear scale

Determines whether or not the graph is in linear scale


Minimum value

Minimum value when using log scale

–100 dBm — ]-INF,+INF[


Units Value range

Cutoff magnitude

Reference cutoff magnitude for bandwidth result calculation

3 dB [0,100]

506

ELECTRICAL FILTER ANALYZER

Export

Graphs

Results


Save to file

Determines if the filter transmission will be save as a file


Filename

Filename with the filter data

Filter.dat — —


Transmission function Frequency (Hz) Power (dB)


Frequency at Max. Transmission Hz

Filter Bandwidth (BandPass) at Cut off Hz

Frequency at Min. Transmission Hz

Filter Bandwidth (Band-Reject) at Cut off Hz

507

WDM Multiplexers Library

This section contains information on the following WDM Multiplexers.

Add and Drop

• WDM Add• WDM Drop• WDM Add and Drop

Demultiplexers

• WDM Demux 1x2• WDM Demux 1x4• WDM Demux 1x8• WDM Demux• WDM Demux ES• Ideal Demux

Multiplexers

• WDM Mux 2x1• WDM Mux 4x1• WDM Mux 8x1• WDM Mux• WDM Mux ES• Ideal Mux

507

Notes:

508

WDM ADD

WDM AddAdds a WDM channel and a WDM signal.

Ports

Parameters

Main






Frequency 193.1 THz Hz, THz, nm [30, 300000]

Bandwidth3 dB filter bandwidth


[0,+INF[

Insertion lossInsertion loss of the demux

0 dB — [0,+INF[

DepthMaximum attenuation value for the filter

100 dB — [0,+INF[

Filter typeInternal filter type

Bessel — — Rectangle, Gaussian, Bessel

Filter orderOrder of the function when using Gaussian or Bessel filter type

2 — — [1,1000]

509

WDM ADD

Simulation

Noise


The input signals are filtered by an optical filter and are combined in one signal. The first signal is filtered by an inverse filter. The optical filters can be a Rectangle, Gaussian, or Bessel optical filter. The subsystem is illustrated in Figure 1.

Figure 1 WDM Add subsystem

Defining crosstalk

The level of crosstalk for both MUX and DEMUX components, is defined by bandwidth, ripple, and depth of the filter. These 3 factors will determine how much power, from neighboring channels, will act as crosstalk terms when calculating the performance of a specific channel. The most important parameter is depth, as it will play the most significant role in determining the power levels of the neighboring channels.




ResampleDetermines if the filter will down sample the signal bandwidth to the filter sample rate


Sample rateNew output signal sample rate






3 dB ]-INF,+INF[

510

WDM DROP

WDM DropDrops a WDM channel from a WDM signal.

Ports

Parameters

Main


Input Input Optical







[0,+INF[


0 dB — [0,+INF[


100 dB — [0,+INF[




2 — — [1,1000]

511

WDM DROP

Simulation

Noise


The input signal is split into two signals. Each signal is filtered by an optical filter. The first signal is filtered by an inverse filter. The optical filters can be a Rectangle, Gaussian, or Bessel optical filter. The subsystem is illustrated in Figure 1.

Figure 1 WDM Drop subsystem

Defining crosstalk













3 dB ]-INF,+INF[

512

WDM ADD AND DROP

WDM Add and DropWDM Add and Drop multiplexer. Equivalent to a subsystem based on the WDM Add and WDM Drop components.

Ports

Parameters

Main


Input Input Optical

Output (Drop) Output Optical

Input (Add) Input Optical






[0,+INF[


0 dB — [0,+INF[

Depth Maximum attenuation value for the filter

100 dB — [0,+INF[




2 — — [1,1000]

513

WDM ADD AND DROP

Simulation

Noise


In the drop section, the input signal is divided in two signals. Each signal is filtered by an optical filter. An inverse filter filters the first signal.

In the add section, the input signals are filtered by an optical filter and are combined in one signal. An inverse filter filters the first signal.

The optical filters can be a Rectangle, Gaussian, or Bessel optical filter. The subsystem is illustrated in Figure 1.












3 dB ]-INF,+INF[

514

WDM ADD AND DROP

Figure 1 WDM Add and drop subsystem

Defining crosstalk


515

WDM ADD AND DROP

Notes:

516

WDM DEMUX 1X2

WDM Demux 1x2Demultiplexes two WDM signal channels.

Ports

Parameters

Main

Channels


Input Input Optical






[0,+INF[


0 dB — [0,+INF[


100 dB — [0,+INF[




2 — — [1,1000]


Frequency[0]Filter center frequency for channel 0

193.1 THz Hz, THz, nm [30,3e5]


193.2 THz Hz, THz, nm [30,3e5]

517

WDM DEMUX 1X2

Ripple

Simulation

Noise


Ripple[0]Additional loss of the filter for channel 0

0 dB ]-INF,+INF[

Ripple[1] Additional loss of the filter for channel 1

0 dB ]-INF,+INF[












3 dB ]-INF,+INF[

518

WDM DEMUX 1X2

Technical backgroundThe input signal is split into two signals that are filtered by an optical filter. The optical filter can be a Rectangle, Gaussian, or Bessel optical filter. The subsystem is illustrated in Figure 1.

Figure 1 Demultiplexer subsystem

Defining crosstalk


519

WDM DEMUX 1X2

Notes:

520

WDM DEMUX 1X4

WDM Demux 1x4Demultiplexes four WDM signal channels.

Ports

Parameters

Main


Input Input Optical








[0,+INF[


0 dB — [0,+INF[


100 dB — [0,+INF[




2 — — [1,1000]

521

WDM DEMUX 1X4

Channels

Ripple

Simulation

Noise



193.1 THz Hz, THz, nm [30,3e5]


193.2 THz Hz, THz, nm [30,3e5]


193.3 THz Hz, THz, nm [30,3e5]


193.4 THz Hz, THz, nm [30,3e5]



0 dB ]-INF,+INF[


0 dB ]-INF,+INF[


0 dB ]-INF,+INF[


0 dB ]-INF,+INF[












3 dB ]-INF,+INF[

522

WDM DEMUX 1X4

Technical backgroundThe input signal is split into four signals that are filtered by an optical filter. The optical filter can be a Rectangle, Gaussian, or Bessel optical filter. The subsystem is illustrated in Figure 1.


Defining crosstalk


523

WDM DEMUX 1X4

Notes:

524

WDM DEMUX 1X8

WDM Demux 1x8Demultiplexes eight WDM signal channels.

Ports

Parameters

Main


Input Input Optical












[0,+INF[


0 dB — [0,+INF[


100 dB — [0,+INF[




2 — — [1,1000]

525

WDM DEMUX 1X8

Channels

Ripple



193.1 THz Hz, THz, nm [30,3e5]


193.2 THz Hz, THz, nm [30,3e5]


193.3 THz Hz, THz, nm [30,3e5]


193.4 THz Hz, THz, nm [30,3e5]


193.5 THz Hz, THz, nm [30,3e5]


193.6 THz Hz, THz, nm [30,3e5]


193.7 THz Hz, THz, nm [30,3e5]


193.8 THz Hz, THz, nm [30,3e5]



0 dB ]-INF,+INF[


0 dB ]-INF,+INF[


0 dB ]-INF,+INF[


0 dB ]-INF,+INF[


0 dB ]-INF,+INF[


0 dB ]-INF,+INF[


0 dB ]-INF,+INF[


0 dB ]-INF,+INF[

526

WDM DEMUX 1X8

Simulation

Noise


The input signal is split into eight signals that are filtered by an optical filter. The optical filter can be a Rectangle, Gaussian, or Bessel optical filter. The subsystem is illustrated in Figure 1.













3 dB ]-INF,+INF[

527

WDM DEMUX 1X8

Defining crosstalk


528

WDM DEMUX

WDM DemuxDemultiplexes a user-defined number of WDM signal channels.

Ports

Parameters

Main


Input Input Optical










Number of output ports 8 — — [2, 1000]



[0,+INF[


0 dB — [0,+INF[


100 dB — [0,+INF[




2 — — [1,1000]

529

WDM DEMUX

Channels

Ripple



193.1 THz Hz, THz, nm [30,3e5]


193.2 THz Hz, THz, nm [30,3e5]


193.3 THz Hz, THz, nm [30,3e5]


193.4 THz Hz, THz, nm [30,3e5]


193.5 THz Hz, THz, nm [30,3e5]


193.6 THz Hz, THz, nm [30,3e5]


193.7 THz Hz, THz, nm [30,3e5]


193.8 THz Hz, THz, nm [30,3e5]



0 dB ]-INF,+INF[


0 dB ]-INF,+INF[


0 dB ]-INF,+INF[


0 dB ]-INF,+INF[


0 dB ]-INF,+INF[


0 dB ]-INF,+INF[


0 dB ]-INF,+INF[


0 dB ]-INF,+INF[

530

WDM DEMUX

Simulation

Noise


The input signal is split into N signals, where N is the number of output ports. The Signals are filtered by an optical filter. The optical filter can be a Rectangle, Gaussian, or Bessel optical filter. The subsystem is illustrated in Figure 1.













3 dB ]-INF,+INF[

531

WDM DEMUX

Defining crosstalk


532

WDM DEMUX ES

WDM Demux ESDemultiplexes a user-defined number of WDM signal channels. The center frequencies of the internal filters are equally spaced (ES).

Ports

Parameters

Main


Input Input Optical











FrequencyCenter frequency of the first filter

193.1 THz, Hz, nm [30,+INF[

Frequency spacingFrequency spacing between adjacent filters

100 GHz, THz, Hz, nm

]-INF,+INF[



[0,+INF[


0 dB — [0,+INF[


100 dB — [0,+INF[

533

WDM DEMUX ES

Simulation

Noise


The WDM Demux ES is equivalent to the conventional WDM Demux component. However, the WDM Demux ES is easier to set up for WDM systems, since it requires only the filter center frequency and the spacing.




2 — — [1,1000]












3 dB ]-INF,+INF[


534

IDEAL DEMUX

Ideal DemuxDemultiplexes a user-defined number of output WDM signal channels. This model is equivalent to an ideal splitter, since there is no power splitting and filtering.

Ports

Parameters

Main


The input signal is duplicated and attenuated. The subsystem is illustrated in Figure 1.

Figure 1 Subsystem — duplicated and attenuated input signal


Input Input Optical






0 dB [0,+INF[

535

WDM MUX 2X1

WDM Mux 2x1Multiplexes two WDM signal channels.

Ports

Parameters

Main

Channels








[0,+INF[


0 dB — [0,+INF[

Depth Maximum attenuation value for the filter

100 dB — [0,+INF[




2 — — [1,1000]



193.1 THz Hz, THz, nm [30,3e5]


193.2 THz Hz, THz, nm [30,3e5]

536

WDM MUX 2X1

Ripple

Simulation

Noise



0 dB ]-INF,+INF[


0 dB ]-INF,+INF[












3 dB ]-INF,+INF[

537

WDM MUX 2X1

Technical backgroundThe two input signals are filtered by an optical filter and are combined in one signal. The optical filter can be a Rectangle, Gaussian, or Bessel optical filter. The subsystem is illustrated in Figure 1.

Figure 1 Multiplexer subsystem

Defining crosstalk


538

WDM MUX 4X1

WDM Mux 4x1Multiplexes four WDM signal channels.

Ports

Parameters

Main










[0,+INF[


0 dB — [0,+INF[


100 dB — [0,+INF[




2 — — [1,1000]

539

WDM MUX 4X1

Channels

Ripple

Simulation

Noise



193.1 THz Hz, THz, nm [30,3e5]


193.2 THz Hz, THz, nm [30,3e5]


193.3 THz Hz, THz, nm [30,3e5]


193.4 THz Hz, THz, nm [30,3e5]



0 dB ]-INF,+INF[


0 dB ]-INF,+INF[


0 dB ]-INF,+INF[


0 dB ]-INF,+INF[












3 dB ]-INF,+INF[

540

WDM MUX 4X1

Technical backgroundThe four input signals are filtered by an optical filter and are combined in one signal. The optical filter can be a Rectangle, Gaussian, or Bessel optical filter. The subsystem is illustrated in Figure 1.


Defining crosstalk


541

WDM MUX 4X1

Notes:

542

WDM MUX 8X1

WDM Mux 8x1Multiplexes eight WDM signal channels.

Ports

Parameters

Main














[0,+INF[


0 dB — [0,+INF[


100 dB — [0,+INF[




2 — — [1,1000]

543

WDM MUX 8X1

Channels

Ripple



193.1 THz Hz, THz, nm [30,3e5]


193.2 THz Hz, THz, nm [30,3e5]


193.3 THz Hz, THz, nm [30,3e5]


193.4 THz Hz, THz, nm [30,3e5]


193.5 THz Hz, THz, nm [30,3e5]


193.6 THz Hz, THz, nm [30,3e5]


193.7 THz Hz, THz, nm [30,3e5]


193.8 THz Hz, THz, nm [30,3e5]



0 dB ]-INF,+INF[


0 dB ]-INF,+INF[


0 dB ]-INF,+INF[


0 dB ]-INF,+INF[


0 dB ]-INF,+INF[


0 dB ]-INF,+INF[


0 dB ]-INF,+INF[


0 dB ]-INF,+INF[

544

WDM MUX 8X1

Simulation

Noise


The eight input signals are filtered by an optical filter and are combined in one signal. The optical filter can be a Rectangle, Gaussian, or Bessel optical filter. The subsystem is illustrated in Figure 1.













3 dB ]-INF,+INF[

545

WDM MUX 8X1

Defining crosstalk


546

WDM MUX

WDM MuxMultiplexes a user-defined number of input WDM signal channels.

Ports

Parameters

Main












Number of input ports 8 — — [2,1000]



[0,+INF[


0 dB — [0,+INF[


100 dB — [0,+INF[




2 — — [1,1000]

547

WDM MUX

Channels

Ripple



193.1 THz Hz, THz, nm [30,3e5]


193.2 THz Hz, THz, nm [30,3e5]


193.3 THz Hz, THz, nm [30,3e5]


193.4 THz Hz, THz, nm [30,3e5]


193.5 THz Hz, THz, nm [30,3e5]


193.6 THz Hz, THz, nm [30,3e5]


193.7 THz Hz, THz, nm [30,3e5]


193.8 THz Hz, THz, nm [30,3e5]



0 dB ]-INF,+INF[


0 dB ]-INF,+INF[


0 dB ]-INF,+INF[


0 dB ]-INF,+INF[


0 dB ]-INF,+INF[


0 dB ]-INF,+INF[

Ripple[6]Additional loss of the filter for hannel 6

0 dB ]-INF,+INF[


0 dB ]-INF,+INF[

548

WDM MUX

Simulation

Noise


The input signals are filtered by an optical filter and combined in one signal. The optical filter can be a Rectangle, Gaussian, or Bessel optical filter. The subsystem is illustrated in Figure 1.













3 dB ]-INF,+INF[

549

WDM MUX

Defining crosstalk


550

WDM MUX ES

WDM Mux ESThis component multiplexes a user-defined number of WDM signal channels. The center frequencies of the internal filters are equally spaced (ES).

Ports

Parameters

Main












Number of input ports 8 — — [2,1000]

FrequencyCenter frequency of the first filter

193.1 — THz, HZ, nm [30,+INF[

Frequency spacingFrequency spacing between adjacent filters

100 — GHz, THz, Hz, nm

]-INF,+INF[



[0,+INF[


0 dB — [0,+INF[


100 dB — [0,+INF[

551

WDM MUX ES

Simulation

Noise


The WDM Mux ES is equivalent to the conventional WDM Mux component. However, the WDM Mux ES is easier to set up for WDM systems, since it only requires the filter center frequency and the spacing.




2 — — [1,1000]












3 dB ]-INF,+INF[


552

IDEAL MUX

Ideal MuxMultiplexes a user-defined number of input WDM signal channels. This model is equivalent to an ideal adder, since there is no power splitting and filtering.

Ports

Parameters

Main






Number of input ports 2 — [2,1000]

LossInsertion loss of the demux

0 dB [0,+INF[

553

IDEAL MUX

Technical backgroundThe input signals are added and attenuated. The subsystem is illustrated in Figure 1.

Figure 1 Ideal Multiplexer subsystem

554

Network Library

This section contains information on the following network components.

Optical Switches

• Dynamic Y Select Nx1 Measured• Dynamic Y Switch 1xN Measured• Dynamic Y Switch 1xN• Dynamic Y Select Nx1• Dynamic Space Switch Matrix NxM Measured• Dynamic Space Switch Matrix NxM• Optical Switch• Digital Optical Switch• Optical Y Switch• Optical Y Select• Ideal Switch 2x2• Ideal Y Switch• Ideal Y Select• Ideal Y Switch 1x4• Ideal Y Select 4x1• Ideal Y Switch 1x8• Ideal Y Select 8x1• Ideal Y Select Nx1• Ideal Y Switch 1xN

555

Frequency Conversion

• Ideal Frequency Converter

556

DYNAMIC Y SELECT NX1 MEASURED

Dynamic Y Select Nx1 MeasuredY select with a user-defined mapping table for different switching events.

Ports

Parameters

Main












Number of input ports 8 — — [2, 1000]

Time constantSwitching time constant

50 ns s, ms, ns [0,+INF[

Switching event timeTime instant when the switching event occurs


Repeat eventsDetermines if the events will be repeated for each event time


Mapping table filenameFilename with the measured data

Table.dat — — —

557



Static solution

The switch model allows for the selection of the number of input ports N.

For the input ports i = 1…N, you can select the complex values of a mapping table:

where

If the light electric field complex amplitude entering the input port number 'i' is Ei, then the electric field complex amplitude at the output port due to Ei is:

(1)

When all input ports of the switch are used, the output complex amplitude at the output port is:

(2)

This sum includes all different wavelength contributions.

i = 1

i = 2

i = 3

.

.

.

i = N

n1 j α1×+

n2 j α2×+

n3 j α3×+

nN j αN×+

j 1–( )=

EOutput EiInpute

j ni jαi+( )=

EOutput EiInpute

j ni jαi+( )

i 1=

N

∑=

558


Transients

This type of switch is characterized by switching time with a time constant.

Mathematically, a switching event is a replacement of one mapping table, T1, with a different one, T2.

To a first order approximation, the change from to resembles a charging process of a linear capacitor through a linear resistor. It has an exponential time behavior, with a time constant . The parameter time constant is universal and is shared by all transient events.

For a switching event that takes place at time t0, the real part of a mapping matrix element will change as:

(3)

t0 is the parameter Switching event time.

The parameter Repeat events allows you to generate multiple switching events. For example, changing the map table from 1 to 2 and vice versa.

File format

The file format for the data with the map table is:

where the first index is the input port (row) and the second index is the table number (1 or 2).

Assuming a component with 3 input ports, and transient from port 1 to 3:

n1,1 n1,2

n2,1 n2,2

.

.

nN,1 nN,2

0 0 0 10

0 10 0 10

0 10 0 0

ni j αi×+{ }T1 ni j αi×+{ }T2

τ τ

ni t( ) ni T1t t0–( ) τ⁄–( ) ni T2

1 t t0–( ) τ⁄–( )exp–{ }×+exp×=

α1 1, α1 2,

α2 1, α2 2,

αN 1, αN 2,

559

DYNAMIC Y SWITCH 1XN MEASURED

Dynamic Y Switch 1xN MeasuredY switch with user-defined mapping table for different switching events.

Ports

Parameters

Main


Input Input Optical



















560



Static solution

The switch model allows for the selection of the number of output ports N.

For the output ports i = 1…N, you can select the complex values of a mapping table:

where

If the light electric field complex amplitude at the output port number 'i' is Ei, calculated from the electric field complex amplitude at the input port, Ei is:

(1)

Transients





(2)


i = 1

i = 2

i = 3

.

.

.

i = N

n1 j α1×+

n2 j α2×+

n3 j α3×+

nN j αN×+

j 1–( )=

EOutput EiInpute

j ni jαi+( )=


τ τ

ni t( ) ni T1t t0–( ) τ⁄–( ) ni T2

1 t t0–( ) τ⁄–( )exp–{ }×+exp×=

561



File format


where the first index is the output port (row) and the second index is the table number (1 or 2).

Assuming a component with 3 output ports, and transient from port 3 to 1:

n1,1 n1,2

n2,1 n2,2

.

.

nN,1 nN,2

0 0 0 10

0 10 0 10

0 10 0 0

α1 1, α1 2,

α2 1, α2 2,

αN 1, αN 2,

562

DYNAMIC Y SWITCH 1XN

Dynamic Y Switch 1xNY switch that allows you to control the different values for attenuation and phase values with transient effects when switching from different input ports.

Ports

Parameters

Main


Input Input Optical





Port before eventPort number to use before the event

1 — — [1, 1000]

Port after eventPort number to use after the event

2 — — [1, 1000]







563


Table


Static solution

The switch model allows for the selection of the number of input ports N.


where


(1)


Real coeff. at selected portReal coeff. equivalent to the phase at the selected port

1e-006 — ]-INF,+INF[

Imag coeff. at selected portImag coeff. equivalent to the attenuation at the selected port

1 — ]-INF,+INF[

Real coeff. at other portsReal coeff. equivalent to the phase at other ports

1e-006 — ]-INF,+INF[

Imag coeff. at other portsImag coeff. equivalent to the attenuation at other ports

1e-006 — ]-INF,+INF[

i = 1

i = 2

i = 3

.

.

.

i = N

n1 j α1×+

n2 j α2×+

n3 j α3×+

nN j αN×+

j 1–( )=

EOutput EiInpute

j ni jαi+( )=

564


When all input ports of the switch are used, the output complex amplitude at the output port is:

(2)


Transients





(3)



Mapping table

The mapping table is generated based on the values for the selected and unselected ports. You can select the values of the real and imag coefficients for the selected port and for the unselected ports. The models assumes that all unselected ports have the same phase and attenuation. For arbitrary values for these coefficients, use the equivalent measured component.

EOutput EiInpute

j ni jαi+( )

i 1=

N

∑=


τ τ

ni t( ) ni T1t t0–( ) τ⁄–( ) ni T2

1 t t0–( ) τ⁄–( )exp–{ }×+exp×=

565


Notes:

566

DYNAMIC Y SELECT NX1

Dynamic Y Select Nx1Y select that allows you to control the different values for attenuation and phase values with transient effects when switching from different output ports.

Ports

Parameters

Main







Port before eventPort number to use before the event

1 — — [1, 1000]

Port after eventPort number to use after the event

2 — — [1, 1000]







567


Table


Static solution

The switch model allows for the selection of the number of output ports N.


where

If the light electric field complex amplitude at the output port number 'i' is Ei, calculated from the electric field complex amplitude at the input port, Ei is:

(1)


Real coeff. at selected portReal coeff. equivalent to the phase at the selected port

1e-006 — ]-INF,+INF[

Imag coeff. at selected portImag coeff. equivalent to the attenuation at the selected port

1 — ]-INF,+INF[


1e-006 — ]-INF,+INF[


1e-006 — ]-INF,+INF[

i = 1

i = 2

i = 3

.

.

.

i = N

n1 j α1×+

n2 j α2×+

n3 j α3×+

nN j αN×+

j 1–( )=

EOutput EiInpute

j ni jαi+( )=

568


Transients





(2)



Mapping table

The mapping table is generated based on the values for the selected and unselected ports. You can select the values of the real and imag coefficients for the selected port and for the unselected ports. The models assume that all unselected ports have the same phase and attenuation. For arbitrary values for these coefficients, use the equivalent measured component.


τ τ

ni t( ) ni T1t t0–( ) τ⁄–( ) ni T2

1 t t0–( ) τ⁄–( )exp–{ }×+exp×=

569


Notes:

570

DYNAMIC SPACE SWITCH MATRIX NXM MEASURED

Dynamic Space Switch Matrix NxM MeasuredSpace switch matrix with a user-defined mapping table for different switching events.

Ports


















571


Parameters

Main












572



Static solution

The switch model allows for the selection of the number of input ports N and output ports M.


where


(1)

When all input ports of the switch are used, the output complex amplitude at each output port is:

(2)


i = 1

i = 2

i = 3

.

.

.

i = N

n1 j α1×+

n2 j α2×+

n3 j α3×+

nN j αN×+

j 1–( )=

EOutput EiInpute

j ni jαi+( )=

EOutput EiInpute

j ni jαi+( )

i 1=

N

∑=

573


Transients





(3)



File format


where the first index is the input port (row), the second index is the table number (1 or 2), and the third index is the output port. This means that there is one row for each input port and 4 columns for each output port.

Assuming a component with 3 input and output ports, and transient from port 1 to 3:

n1,1 n1,2 ... n1,1,M n1,2,M

n2,1 n2,2 ... n2,1,M n2,2,M

..

..

nN,1 nN,2 ... nN,1,M nN,2,M

0 0 0 10 0 10 0 10 0 10 0 10

0 10 0 10 0 10 0 10 0 10 0 10

0 10 0 10 0 10 0 10 0 10 0 0


τ τ

ni t( ) ni T1t t0–( ) τ⁄–( ) ni T2

1 t t0–( ) τ⁄–( )exp–{ }×+exp×=

α1 1, α1 2, α1 1 M, , α1 2 M, ,

α2 1, α2 2, α2 1 M, , α2 2 M, ,

αN 1, αN 2, αN 1 M, , αN 2 M, ,

574

DYNAMIC SPACE SWITCH MATRIX NXM

Dynamic Space Switch Matrix NxMSpace switch matrix that allows you to control the different values for attenuation and phase values with transient effects when switching from different input and output ports.

Ports


















575


Parameters

Main

Table




Input port before eventPort number to use before the event

1 — — [1, 1000]

Input port after eventPort number to use after the event

1 — — [1, 1000]

Output port before eventPort number to use before the event

2 — — [1, 1000]

Output port after eventPort number to use after the event

2 — — [1, 1000]








Real coeff. at selected portReal coeff. equivalent to the phase at selected port

1e-006 — ]-INF,+INF[

Imag coeff. at selected portImag coeff. equivalent to the attenuation at selected port

1 — ]-INF,+INF[


1e-006 — ]-INF,+INF[


1e-006 — ]-INF,+INF[

576



Static solution

The switch model allows for the selection of the number of input ports N and output ports M.


where


(1)

When all input ports of the switch are used, the output complex amplitude at each output port is:

(2)


i = 1

i = 2

i = 3

.

.

.

i = N

n1 j α1×+

n2 j α2×+

n3 j α3×+

nN j αN×+

j 1–( )=

EOutput EiInpute

j ni jαi+( )=

EOutput EiInpute

j ni jαi+( )

i 1=

N

∑=

577


Transients





(3)



Mapping table

The mapping table is generated based on the values for the selected and unselected ports. You can select the values of the real and imag coefficients for the selected port and for the unselected ports. The models assume that all unselected ports have the same phase and attenuation. For arbitrary values for these coefficients, use the equivalent measured component.


τ τ

ni t( ) ni T1t t0–( ) τ⁄–( ) ni T2

1 t t0–( ) τ⁄–( )exp–{ }×+exp×=

578

OPTICAL SWITCH

Optical SwitchSimulates a non-ideal switch 2x2.

Ports

Parameters

Main


The optical switch routes the optical signals at input port 1 and 2 to the two output ports, according to the parameter phase shift described as follows:• If the phase shift is 0, then the optical signal at input 1 is passed to output 2 and the optical

signal at input 2 is passed to output 1 (see Figure 1). • If the phase shift is , then the optical signal at input 2 is passed to output 2 and the optical

signal at input 1 is passed to output 1 (Figure 1).







Phase shift Zero — Zero, pi

Additional loss 0 dB [0, 1e100]

π

579

OPTICAL SWITCH

Figure 1 Switch behavior

The following equations describe the switch behavior:

where E1in and E2in are the input signals at input port 1 and 2 respectively.

(4)

(5)

(6)

(7)

where the coupling coefficient, cc, is 0.5, is the phase shift parameter, and is the additional loss.

E1out

E2out

αm11

m21

⋅=m12

m22

E1in

E2in

⋅

m11 1 cc–( ) j φ⋅( ) cc–exp⋅=

m12 1 cc– j cc j φ⋅( ) 1+exp( )⋅ ⋅ ⋅=

m21 1 cc– j cc j φ⋅( ) 1+exp( )⋅ ⋅ ⋅=

m22 1 cc–( ) cc j φ⋅( )exp⋅–⋅=

φ α

580

DIGITAL OPTICAL SWITCH

Digital Optical SwitchSimulates a non-ideal switch 2x2 with a control signal.

Ports

Parameters

Main


Control Input Binary






Additional loss 0 dB [0, 1e100]

581

DIGITAL OPTICAL SWITCH

Technical backgroundThe digital optical switch routes the optical signals at input port 1 and 2 to the two output ports, according to the control signal described as follows:• If the control signal is 0, then the optical signal at input 1 is passed to output 1 and the

optical signal at input 2 is passed to output 2.• If the control signal is 1, then the optical signal at input 2 is passed to output 1 and the

optical signal at input 1 is passed to output 2.

The working behavior of this component is similar to the optical switch component. When the control signal is 0, internally the phase shift is set at , and when the control signal is 1, the phase shift is set at 0.

π

582

OPTICAL Y SWITCH

Optical Y SwitchSimulates a non-ideal optical switch 1x2.

Ports

Parameters

Main



Input Input Optical




Insertion loss 0 dB [0, 1e100]

Crosstalk 1 30 dB [0, 1e100]


Phase shift 1 90 deg [-1e50, 1e50]


583

OPTICAL Y SWITCH

Technical backgroundThe digital optical 1x2 switch routes the input signal to one of two output ports, including crosstalk and phase shift between the two input signals. The parameters responsible for crosstalk between the two output signals are crosstalk 1 and crosstalk 2. The phase shift is specified by phase shift 1 and phase shift 2.

This model has two modes of operation:• If the control is 0, then the optical signal at input is routed to output 1 (see Figure 1).• If the control is 1, then the optical signal at input is routed to output 2 (see Figure 1).


584

OPTICAL Y SELECT

Optical Y SelectSimulates a non-ideal optical switch 2x1.

Ports

Parameters

Main







Insertion loss 0 dB [0, 1e100]





585

OPTICAL Y SELECT

Technical backgroundThe digital optical 2x1 switch selects one of the two input signals and the route to the output port, including crosstalk and phase shift between the two input signals. The parameters responsible for crosstalk between the input signals are crosstalk 1 and crosstalk 2. The phase shift is specified by phase shift 1 and phase shift 2.

This model has two modes of operation:• If the control is 0, then the optical signal at input 1 is passed to the output (see Figure 1).• If the control is 1, then the optical signal at input 2 is passed to the output (see Figure 1).


586

IDEAL SWITCH 2X2

Ideal Switch 2x2Simulates an ideal switch 2x2.

Ports

Parameters

Simulation










587

IDEAL SWITCH 2X2

Technical backgroundThe ideal optical 2x2-switch routes the optical signals at input port 1 and 2 to the two output ports according with the control signal.

The ideal 2x2 switch has two modes of operation:• If the control is 0, then the optical signal at input 1 is passed to output 1 and the optical

signal at input 2 is passed to output 2 (see Figure 1).• If the control is 1, then the optical signal at input 2 is passed to output 1 and the optical

signal at input 1 is passed to output 2 (see Figure 1).


588

IDEAL Y SWITCH

Ideal Y SwitchSimulates an ideal optical 1x2 switch.

Ports

Parameters

Simulation



Input Input Optical






589

IDEAL Y SWITCH

Technical backgroundThe ideal optical 1x2 switch routes a signal in the input port to one of two output ports.

The ideal 2x1 switch has two modes of operation as follows:• If the control is 0, then the optical signal at input 1 is passed to output 1 (see Figure 1).• If the control is 1, then the optical signal at input 2 is passed to output 2 (see Figure 1).


590

IDEAL Y SELECT

Ideal Y SelectSimulates an ideal optical select switch.

Ports

Parameters

Simulation









591

IDEAL Y SELECT

Technical backgroundThe ideal Y select switch has two modes of operation:• If the control is 0, then the optical signal at input 1 is passed to the output (see Figure 1).• If the control is 1, then the optical signal at input 2 is passed to the output (see Figure 1).


592

IDEAL Y SWITCH 1X4

Ideal Y Switch 1x4Simulates an ideal optical 1x4 switch.

Ports

Parameters

Simulation



Input Input Optical








593

IDEAL Y SWITCH 1X4

Technical backgroundThe ideal optical 1x4 switch routes a signal in the input port to one of four output ports.

The ideal 1x4 switch has four states of operation, as follows:• If the control is 00, then the optical signal at input is passed to output 1 (see Figure 1).• If the control is 01, then the optical signal at input is passed to output 2 (see Figure 1).• If the control is 10, then the optical signal at input is passed to output 3 (see Figure 1).• If the control is 11, then the optical signal at input is passed to output 4 (see Figure 1).

Figure 1 Two possible working states of the 4x1 switch

594

IDEAL Y SELECT 4X1

Ideal Y Select 4x1Simulates an ideal optical switch 4x1.

Ports

Parameters

Simulation











595

IDEAL Y SELECT 4X1

Technical backgroundThe ideal Y select 4x1 switch has four states of operation:• If the control is 00, then the optical signal at input 1 is passed to out (see Figure 1).• If the control is 01, then the optical signal at input 2 is passed to out (see Figure 1).• If the control is 10, then the optical signal at input 3 is passed to out (see Figure 1).• If the control is 11, then the optical signal at input 4 is passed to out (see Figure 1).

Figure 1 Two possible working states of the 4x1 switch

596

IDEAL Y SWITCH 1X8

Ideal Y Switch 1x8Simulates an ideal optical 1x8 switch.

Ports

Parameters

Simulation



Input Input Optical












597

IDEAL Y SWITCH 1X8

Technical backgroundThe ideal optical 1x8-switch routes a signal in the input port to one of eight output ports (see Figure 1).

Figure 1 One possible working state of the 1x8 switch

Table 1 displays the switching states for the eight output ports.

Table 1 Switching states — output ports

Control Output 1 Output 2 Output 3 Output 4 Output 5 Output 6 Output 7 Output 8

000 X — — — — — — —

001 — X — — — — — —

010 — — X — — — — —

011 — — — X — — — —

100 — — — — X — — —

101 — — — — — X — —

110 — — — — — — X —

111 — — — — — — — X

598

IDEAL Y SELECT 8X1

Ideal Y Select 8x1Simulates an ideal optical switch 8x1.

Ports

Parameters

Simulation















599

IDEAL Y SELECT 8X1

Technical backgroundThe ideal optical 8x1-switch routes one of the 8 input signals to the output port.

Figure 1 One possible working state of the 8x1 switch

Table 2 displays the switching states for the eight input ports.

Table 2 Switching states — input ports

Control Input 1 Input 2 Input 3 Input 4 Input 5 Input 6 Input 7 Input 8

000 X — — — — — — —

001 — X — — — — — —

010 — — X — — — — —

011 — — — X — — — —

100 — — — — X — — —

101 — — — — — X — —

110 — — — — — — X —

111 — — — — — — — X

600

IDEAL Y SELECT NX1

Ideal Y Select Nx1Simulates an ideal optical switch with a variable number of input ports.

Ports

Parameters

Simulation







Number of input ports 2 — [2, 1000]



601

IDEAL Y SELECT NX1

Technical backgroundThe number of input ports for the Nx1 switch is given by the number of input ports parameter. The bit sequence length of control signals must be enough for the correct use of the switch. The minimum number of bits is:

where nb is the number of bits and Nin is the number of input ports.

The control signal specifies which input port will have the optical signal routed to the output port.

nb log2 Nin( )=

602

IDEAL Y SWITCH 1XN

Ideal Y Switch 1xNSimulates an ideal optical 1xN switch with a variable number of output ports.

Ports

Parameters

Simulation



Input Input Optical







603

IDEAL Y SWITCH 1XN

Technical backgroundThe control signal must be long enough for the correct use of the switch. The minimum number of bits is:

where nb is the number of bits and Nout is the number of output ports.

The control signal specifies which output port will have the optical signal roouted at the input port.

nb 2 Nout( )log=

604

IDEAL FREQUENCY CONVERTER

Ideal Frequency ConverterSimulates an ideal frequency converter.

Ports

Parameters

Main

Simulation


Optical Input Optical

Optical Output Optical


Frequency offset 100 GHz [-1e6, 1e6 ]

Shift band True — True, False




605

IDEAL FREQUENCY CONVERTER

Technical backgroundThe ideal frequency converter shifts the optical signal spectrum by the amount (frequency offset). There are two modes of operation:• If the shift band parameter is true, then the center frequency is changed and the complex

amplitude of the sampled eletrical field remains unchanged (see Figure 1).• If shift band parameter is false, a cyclic shift is performed (see Figure 1). The complex

amplitudes are changed according to:

For parameterized and noise bins signals, there is only one mode of operation — shift band true.

Figure 1 Ideal frequency converter behavior: (a) input signal, (b) output signal – shift band false and (c) output signal – shift band true

∆f

Eout t( ) Ein t( ) 2 π ∆f t⋅⋅ ⋅( )exp⋅=

606

Passives LibraryThis section contains information on the following passives.

Electrical• Electrical Signal Time Delay

Optical

• Optical Attenuator• Phase Shift• PMD Emulator• Time Delay

Couplers

• X Coupler• Pump Coupler Co-Propagating• Pump Coupler Counter-Propogating

Power Splitters

• Power Splitter 1x2• Power Splitter 1x4• Power Splitter 1x8• Power Splitter

Power Combiners

• Power Combiner 2x1• Power Combiner 4x1• Power Combiner 8x1• Power Combiner

607

Polarization

• Linear Polarizer• Circular Polarizer• Polarization Attenuator• Polarization Combiner• Polarization Controller• Polarization Rotator• Polarization Splitter

Isolators

• Isolator• Ideal Isolator

Circulators

• Circulator• Ideal Circulator

608

ELECTRICAL SIGNAL TIME DELAY

Electrical Signal Time Delay

Adds a time delay to the electrical signal input.

Ports

Parameters

Main

Simulation






DelayDelay to apply to the signal input

0 s s, ms, ns [0,+INF[


Units Value range



609

ELECTRICAL SIGNAL TIME DELAY

Notes:

610

OPTICAL ATTENUATOR

Optical Attenuator

Attenuates the optical signal power.

Ports

Parameters

Main

Simulation


Input Input Optical



AttenuationPower attenuation

0 dB [0,+INF[




611

OPTICAL ATTENUATOR

Technical backgroundThe signal input electrical field for both polarizations is attenuated as:

(1)

where α is the power attenuation.

EOutX Y, t( ) EInX Y,t( )10

α–20-------

=

612

PHASE SHIFT

Phase Shift

Adds a time phase advance/delay to the optical signal input.

Ports

Parameters

Main

Simulation


Input Input Optical



Phase shiftPhase shift to apply to the signal

0 deg ]-INF,+INF[




613

PHASE SHIFT

Notes:

614

PMD EMULATOR

PMD Emulator

This component simulates the effects of first- and second-order PMD in a linear fiber.

Ports

Parameters

Main




Name and description Symbol Default value

Units Value range

Length

Fiber length

L 50 km ]0,+INF[

Attenuation

Fiber attenuation coefficient

0.2 dB/km [0,+INF[

Dispersion

Dispersion at the frequency reference

D 17 ps / (nm - km)

]-INF, +INF[

Dispersion slope

Slope of the dispersion at the frequency reference

S 0.075 ps / (nm2 - km)

]-INF, +INF[

Frequency reference

Frequency of reference for the specified parameters

f 193.1 THz ]-INF, +INF[

Differential group delay 71 ps [0, +INF[

Polarization chromatic dispersion 1.3 ps/GHz ]-INF, +INF[

Depolarization rate 2k 10.8 Deg/GHz ]-INF, +INF[

α

∆τ0

∆τ'

615

PMD EMULATOR

Simulation

Technical backgroundAs bit rates increase to 10 Gbps and 40 Gbps, Polarization Mode Dispersion (PMD) becomes one of the leading causes of signal degradation in data transmission. A physical phenomenon in optical fiber that is statistical in nature, PMD causes dispersion, or spreading of pulses in time and distance, causing adjacent signal pulses to overlap and produce bit errors. The PMD emulator component consists of the PMD channel transfer function considering the first (frequency independent) and second order (frequency dependent) PMD effects.

A linear dispersive fiber can be represented by a 2x2 transfer matrix of the form [1]:

where is the fiber attenuation, is the mean propagation constant, and is the unitary matrix that can be written as:

takes into account the rotation of the principal states of polarization (PSP):

where the coefficient k is defined by the depolarization rate .

The parameter s is the direction of one of the two orthogonal eigenvectors.

D takes into account the different propagation speeds on the two PSPs, with the expressions:




(1)

(2)

T ω( ) α– j β ω( )⋅–( ) z⋅( ) M ω( )⋅exp=

α β M

M ω( ) R 1– ω( ) D ω( ) R ω( )⋅⋅=

R

R ω( ) k ω⋅( ) k ω⋅( )sincosk ω⋅( ) k ω⋅( )cossin–

=

2k ∂s∂ω-------=

D ω( ) j ∆τ ω 2⁄⋅ ⋅( )exp 0 0 j– ∆τ ω 2⁄⋅ ⋅( )exp

=

616

PMD EMULATOR

In the second-order approximation the time difference between the two polarizations is given by:

Where is frequency independent differential group delay, and the differential group delay frequency dependency is represented by the depolarization rate .

∆τ ∆τ0 ∆τ'ω+=

∆τ0∆τ'

617

PMD EMULATOR

References:[1] Cristian Francia, Frank Bruyere, Denis Penninckx, and Michel Chbat. " PMD Second-Order

Effects on Pulse Propagation in Single-Model Optical Fibers". IEEE Photonics Technology Letters, December 1998.

[2] L. E. Nelson, R. M. Jopson, H. Kogelnik, and G. J. Foschini. "Measurement of Depolarization and Scaling Associated with Second Order Polarization Mode Dispersion in Optical Fibers". IEEE Photonics Technology Letters, December 1999.

618

TIME DELAY

Time Delay

Adds a time delay to the optical signal input.

Ports

Parameters

Main

Simulation


Input Input Optical





0 s s, ms, ns [0,+INF[


Units Value range



619

TIME DELAY

Notes:

620

X COUPLER

X Coupler

Cross coupler for combining or splitting optical signals.

Ports

ParametersMain







Coupling coefficientCoupling factor from port 1 to port 2

0.5 — [0,1]

Additional lossLoss applied to the signal after coupling

0 dB [0,+INF[

621

X COUPLER

Technical backgroundThe transmission matrix for the cross:

(1)

where c is coupling coefficient and and α is the additional loss.

References

[1] Gerd Keiser, “Optical Fiber Communications,” Third Edition, McGraw-Hill, Higher Education, 2000.

E1OutX Y,

E2OutX Y,

α1 c– j c

j c 1 c– E1InX Y,

E2InX Y,

=

622

PUMP COUPLER CO-PROPAGATING

Pump Coupler Co-Propagating

Equivalent to a pump coupler subsystem where you can control the attenuation of the signal and pump independently.

Ports

Parameters

Main


Signal Input Input Optical

Pump Input Input Optical



Signal attenuationSignal power attenuation

0 dB [0,+INF[

Pump attenuationPump power attenuation

0 dB [0,+INF[

623

PUMP COUPLER CO-PROPAGATING

Technical backgroundThe input signals are attenuated and combined. The subsystem is illustrated in Figure 1.

Figure 1 Pump coupler co-propogating subsystem

624

PUMP COUPLER COUNTER-PROPOGATING

Pump Coupler Counter-Propogating

Equivalent to a subsystem where you can control the attenuation of the signal and pump independently.

Ports

Parameters

Main


Signal Input Input Optical

Pump Input Input Optical

Pump Output Output Optical



Signal attenuationSignal power attenuation

0 dB [0,+INF[

Pump attenuationPump power attenuation

0 dB [0,+INF[

625

PUMP COUPLER COUNTER-PROPOGATING

Technical backgroundThe input signals are attenuated independently. The subsystem is illustrated in Figure 1.

Figure 1 Pump coupler counter-propogating subsystem

626

POWER SPLITTER 1X2

Power Splitter 1x2

Ideal power splitter — splits an optical input signal into two ouput signals.

Ports

Parameters

Main


Input Input Optical




LossLoss applied to the signal after splitting

0 dB [0,+INF[

627

POWER SPLITTER 1X2

Technical backgroundThe signal output for each port is attenuated by:

(1)

where α is the power attenuation and N is the number of ouput ports (N=2).

EOutX Y, t( )EInX Y,

t( )10α–

20-------

N-------------------------------=

628

POWER SPLITTER 1X4

Power Splitter 1x4

Ideal power splitter — splits an optical input signal in four ouput signals.

Ports

Parameters

Main


Input Input Optical







0 dB [0,+INF[

629

POWER SPLITTER 1X4


(1)



t( )10α–

20-------

N-------------------------------=

630

POWER SPLITTER 1X8

Power Splitter 1x8

Ideal power splitter — splits an optical input signal in eight output signals.

Ports

Parameters

Main


Input Input Optical











0 dB [0,+INF[

631

POWER SPLITTER 1X8


(1)



t( )10α–

20-------

N-------------------------------=

632

POWER SPLITTER

Power Splitter

Ideal power splitter — splits an optical input signal into a user-defined number of ouput signals.

Ports

Parameters

Main


Input Input Optical




Number of output ports 2 — [2,1000]


0 dB [0,+INF[

633

POWER SPLITTER


(1)

where α is the power attenuation and N is the number of ouput ports.


t( )10α–

20-------

N-------------------------------=

634

POWER COMBINER 2X1

Power Combiner 2x1

Ideal power combiner — combines two optical input signals.

Ports

Parameters

Main







0 dB [0,+INF[

635

POWER COMBINER 2X1


(1)

where α is the power attenuation and N is the number of input ports (N=2).

EOutX Y, t( ) 10α–

20-------

N----------- EIn NX Y,, t( )

1

N

∑=

636

POWER COMBINER 4X1

Power Combiner 4x1

Ideal power combiner — combines four optical input signals.

Ports

Parameters

Main









0 dB [0,+INF[

637

POWER COMBINER 4X1


(1)



20-------

N----------- EIn NX Y,, t( )

1

N

∑=

638

POWER COMBINER 8X1

Power Combiner 8x1

Ideal power combiner — combines eigth optical input signals.

Ports

Parameters

Main













0 dB [0,+INF[

639

POWER COMBINER 8X1


(1)



20-------

N----------- EIn NX Y,, t( )

1

N

∑=

640

POWER COMBINER

Power Combiner

Ideal power combiner — combines a user-defined number of input signals.

Ports

Parameters

Main






Number of input ports 2 — [2,1000]


0 dB [0,+INF[

641

POWER COMBINER


(1)

where α is the power attenuation and N is the number of input ports.


20-------

N----------- EIn NX Y,, t( )

1

N

∑=

642

LINEAR POLARIZER

Linear Polarizer

Simulates an ideal linear polarizer.

Ports

Parameters

Main

Simulation


Input Input Optical



Device angle 0 deg [ 1e-50, 1e-50 ]




643

LINEAR POLARIZER

Technical backgroundThe ideal linear polarizer transmits the linear polarization component that coincides with the transmission axis of the polarizer (given by device angle). The orthogonal component removed.

The following Jones (1) and Mueller (2) matrices describe the polarization transfer function of this model with an arbitrary device angle :

(1)

(2)

The Jones matrix is used in sampled signals. The Mueller matrix is used for parameterized and noise bins signals.

θ

cos2 θ( )θ( ) θ( )sin⋅cos

θ( ) θ( )sin⋅cos

sin2 θ( )

12---

12θ( )cos2θ( )sin

0

2θ( )cos

cos2 2θ( )2θ( ) sin 2θ( )⋅cos

0

2θ( )sin2θ( ) 2θ( )sin⋅cos

sin2 2θ( )0

0000

644

CIRCULAR POLARIZER

Circular Polarizer

Simulates an ideal circular polarizer.

Ports

Parameters

Main

Simulation


Input Input Optical



Circular type Right — Left, Right




645

CIRCULAR POLARIZER

Technical backgroundThe ideal circular polarizer transmits the circular polarization component of the input signal. The input signal type coincides with the polarizer type (given by circular type). The orthogonal circular polarized component is removed.

The following Jones and Mueller matrices describe the model:

(1)

(2)

The Jones matrices are used in sampled signals. The Mueller matrices are used for parameterized and noise bins signals.

Right Left

Right Left

1 2⁄j 2⁄

j 2⁄–0

1 2⁄j 2⁄–

j 2⁄0

1 2⁄00

1 2⁄

0000

0000

1 2⁄00

1 2⁄

1 2⁄00

1 2⁄–

0000

0000

1 2⁄–00

1 2⁄

646

POLARIZATION ATTENUATOR

Polarization Attenuator

Simulates a polarization attenuator.

Ports

Parameters

Main

Simulation


Input Input Optical



Attenuation X 0 dB [0, 1e+050]

Attenuation Y 0 dB [0, 1e+050]




647

POLARIZATION ATTENUATOR

Technical backgroundThe polarization attenuator allows the attenuation of each polarization component by multiplication with constant values. The parameters attenuation x and attenuation y define the amount of attenuation the x polarization and the y polarization components will be multiplied by.

The following Jones and Mueller matrices describe the transmission of the signal:

(1)

(2)


10αx 20⁄–

0

0

10αy 20⁄–

12---

10αx 10⁄–

10αy 10⁄–

+

10αx 10⁄–

10αy 10⁄–

–00

10αx 10⁄–

10αy 10⁄–

–

10αx 10⁄–

10αy 10⁄–

+00

00

2 10⋅αx 20⁄–

10αy 20⁄–

⋅0

000

2 10⋅αx 20⁄–

10αy 20⁄–

⋅

⋅

648

POLARIZATION COMBINER

Polarization Combiner

Simulates a polarization combiner.

Ports

Parameters

Main

Simulation






Device angle 0 deg [-1e-50, +1e-50 ]




649

POLARIZATION COMBINER

Technical backgroundThis model combines the two input signals to one output port. The polarization combiner selects the appropriate polarization component of each signal at the input ports and adds the selected polarization components. Figure 1 shows how this model is implemented. There is a linear polarizer at each input port. The angle of each polarizer is given by device angle. An angle of 90° is added to the device angle of the polarizer at input port 2.

Figure 1 Polarizer Combiner

650

POLARIZATION CONTROLLER

Polarization Controller

Simulates a polarization controller.

Ports

Parameters

Main

Simulation


Input Input Optical



Azimuth 0 deg [ -90, 90 ]

Ellipticity 0 deg [ -45, 45 ]

Symmetry factor 0 — [ -1e100, 1e100 ]




651

POLARIZATION CONTROLLER

Technical backgroundThe polarization controller sets the input signal in an arbitrary polarization state. The azimuth and ellipticity parameters define the polarization state of the output signal. In this case, the output polarization is independent of the input signal polarization.

Considering Einx and Einy as the polarization components of the input signal, the output signal is:

(1)

where k is the power splitting ratio parameter and is the phase difference between the x and y components.

The splitting ratio is:

(2)

and the phase difference is

(3)

The x and y phase components are derived from:

(4)

where sf is the symmetry factor.

For sampled signals, Equation 1, Equation 2, and Equation 3 describe the output signal. The following Stokes representation describes parameterized and noise bins signals:

(5)

Eout t( )1 k– j δx t( )⋅( )exp⋅

k j δx t( )⋅( )exp⋅

Einx2 Einy

2+⋅=

δyx t( )

k 1 2 η⋅( ) 2 ε⋅( )cos⋅cos–( ) 2⁄=

δyx arc 2 ε⋅( )sin2 k 1 k–( )⋅⋅-----------------------------------

sin=

sfδx δinx–δy δiny–--------------------=

Sout Einx2 Einy

2+( )

12 ε⋅( ) 2 η⋅( )cos⋅cos2 ε⋅( ) 2 η⋅( )sin⋅cos

2 ε⋅( )sin

=

652

POLARIZATION ROTATOR

Polarization Rotator

Simulates a rotation of coordinate axes.

Ports

Parameters

Main

Simulation


Input Input Optical



Rotation angle 0 deg [ 1e-50, 1e-50 ]




653

POLARIZATION ROTATOR

Technical backgroundThe polarization rotator performs a rotation of the coordinate axes system. The parameter Device angle rotates the angle by counterclockwise.

The rotation is:

(1)

(2)

Based on Equation 1 and Equation 2, the corresponding Jones matrix is defined as:

(3)

The corresponding Mueller matrix is:

(4)


φ

xo x φ( ) y φ( )sin⋅+cos⋅=

yo x φ( ) y φ( )cos⋅+sin⋅–=

φ( )cosφ( )sin–

φ( )sinφ( )cos

1000

02 φ⋅( )cos2 φ⋅( )sin–

0

02 φ⋅( )sin2 φ⋅( )cos0

0001

654

POLARIZATION SPLITTER

Polarization Splitter

Simulates a polarization splitter.

Ports

Parameters

Main

Simulation


Input Input Optical




Device angle 0 deg [-1e-50, +1e-50 ]




655

POLARIZATION SPLITTER

Technical backgroundThis model splits the input signal to two output ports. The polarization splitter selects the appropriate polarization component of the signal at the input port and each polarization component for one of two output ports. Figure 1 shows how this model is implemented.

Figure 1 Polarizer Splitter

656

ISOLATOR

Isolator

Optical isolator. You can control insertion loss, return loss, and isolation.

Ports

Parameters

Main


Input 1 (forward) Input Optical

Output 2 (forward) Output Optical

Input 2 (backward) Input Optical

Output 1 (backward) Output Optical


Insertion lossAttenuation between Input 1 – Output 2

0 dB [0,+INF[

Return lossReturn loss between Input 1 – Output 1, and Input 2 – Output 2

60 dB [0,+INF[

IsolationIsolation between Input 2 – Output 1

60 dB [0,+INF[

657

ISOLATOR

Technical backgroundThe subsystem is illustrated in Figure 1.

Figure 1 Isolator subsystem

658

IDEAL ISOLATOR

Ideal Isolator

Ideal optical isolator. You can control insertion loss — there is no return loss or ideal isolation.

Ports

Parameters

Main


Input 1 (forward) Input Optical

Output 1 (forward) Output Optical

Input 2 (backward) Input Optical


Insertion lossAttenuation between Input 1 – Output 1

0 dB [0,+INF[

659

IDEAL ISOLATOR


Figure 1 Ideal Isolator subsystem

660

CIRCULATOR

Circulator

Optical circulator. You can control insertion loss, return loss, and isolation.

Ports

Parameters

Main









Insertion lossAttenuation between Input 1 – Output 2, Input 2 – Output 3, and Input 3 – Output 1

0 dB [0,+INF[

Return lossReturn loss between Input 1 – Output 1, Input 2 – Output 2, and Input 3 – Output 3

60 dB [0,+INF[

IsolationIsolation between Input 3 – Output 2, Input 1 – Output 3, and Input 2 – Output 1

60 dB [0,+INF[

661

CIRCULATOR


Figure 1 Circulator subsystem

662

IDEAL CIRCULATOR

Ideal Circulator

Ideal optical isolator. Yyou can control insertion loss — there is no return loss or ideal isolation.

Ports

Parameters

Main









Insertion lossAttenuation between Input 1 – Output 2, Input 2 – Output 3, and Input 3 – Output 1

0 dB [0,+INF[

663

IDEAL CIRCULATOR


Figure 1 Ideal Circulator subsystem

664

Signal Processing Library

This section contains information on the following signal processors.

Arithmetic

Electrical

• Electrical Gain• Electrical Adder• Electrical Subtractor• Electrical Multiplier• Electrical Bias• Electrical Norm• Electrical Differentiator• Electrical Integrator• Electrical Limiter

Optical

• Optical Gain• Optical Adder• Optical Subtractor• Optical Bias• Optical Multiplier

Tools

Optical

• Merge Optical Signal Bands• Convert to Parameterized• Convert to Noise Bins

665

Logic

Binary

• Binary NOT• Binary AND• Binary OR• Binary XOR• Binary NAND• Binary NOR• Binary XNOR• Binary Delay• Duobinary precoder

666

ELECTRICAL GAIN

Electrical GainIdeal gain element.

Ports

Parameters

Main

Simulation



Output = Input * Gain Output Electrical


GainGain to apply to the signal port

1 — ]-INF,+INF[




667

ELECTRICAL ADDER

Electrical AdderAdds the input electrical signals.

Ports




Output = (Input 1) + (Input 2) Output Electrical

668

ELECTRICAL SUBTRACTOR

Electrical SubtractorSubtracts the input electrical signals.

Ports




Output = (Input 1) – (Input 2) Output Electrical

669

ELECTRICAL MULTIPLIER

Electrical MultiplierMultiplies the input electrical signals.

Ports




Output = (Input 1) * (Input 2) Output Electrical

670

ELECTRICAL BIAS

Electrical BiasAdds a constant value to the input signal.

Ports

Parameters

Main

Simulation



Output = Input + Bias Output Electrical


BiasConstant value to add to the input signal

0 — ]-INF,+INF[




671

ELECTRICAL NORM

Electrical NormCalculates the equivalent power value of the electrical signal.

Ports

Parameters

Simulation



Output = Input * Input Output Electrical




672

ELECTRICAL DIFFERENTIATOR

Electrical DifferentiatorCalculates the time derivative of the input signal. It can be used in frequency demodulators.

Ports

Parameters

Simulation


This component calculates the derivative of the input electrical signal according to:

, where is the input electrical signal.



Output = d(Input) / dt Output Electrical




vout t( )dvin t( )

dt----------------= vin

673

ELECTRICAL INTEGRATOR

Electrical IntegratorCalculates the time integral of the input signal. It can be used in phase demodulators.

Ports

Parameters

Simulation


Calculates the integral of the input electrical signal according to:




Output = Integ(Input) Output Electrical




vout t( ) vin t( ) td∫= vin

674

ELECTRICAL LIMITER

Electrical LimiterScales the minimum and maximum values of the input signal to user-defined minimum and maximum values.

Ports

Parameters

Main

Simulation





Minimum amplitudeThe minimum value of the signal output

0 a.u. ]-INF,+INF[

Maximum amplitudeThe maximum value of the signal output

1 a.u. ]-INF,+INF[




675

ELECTRICAL LIMITER

Technical BackgroundThis model estimates the minimum and maximum values of the input signal and calculates the scale and bias factor according to the user defined values. The output signal is scaled according to:


and are calculated according to:

where and are the maximum and minimum values for the signal input and output.

vout t( ) avin t( ) b+= vin

a b

aMinout Maxout–Minin Maxin–

----------------------------------------=

bMaxoutMinin MaxinMinout–

Minin Maxin–-----------------------------------------------------------------------=

Max Min

676

OPTICAL GAIN

Optical GainIdeal gain element.

Ports

Parameters

Main

Simulation


Input Input Optical

Output = Input * Gain Output Optical


GainGain to apply to the signal port

0 — ]-INF,+INF[




677

OPTICAL ADDER

Optical AdderAdds the input optical signals.

Ports




Output = (Input 1) + (Input 2) Output Optical

678

OPTICAL SUBTRACTOR

Optical SubtractorSubtracts the input optical signals.

Ports




Output = (Input 1) – (Input 2) Output Optical

679

OPTICAL BIAS

Optical BiasAdds a constant value to the input signal.

Ports

Parameters

Main

Simulation


Input Input Optical

Output = Input + Bias Output Optical


BiasConstant value to add to the input signal

0 — ]-INF,+INF[




680

OPTICAL MULTIPLIER

Optical MultiplierMultiplies the input optical signals.

Ports




Output = (Input 1) * (Input 2) Output Optical

681

MERGE OPTICAL SIGNAL BANDS

Merge Optical Signal BandsMerges multiple sampled signals into one signal.

Ports

Parameters

Simulation


Input Input Optical





682

CONVERT TO PARAMETERIZED

Convert to ParameterizedThis component converts sampled signals and noise bins to parameterized signals.

Ports

Parameters

Main

Simulation


This component converts sampled signals and noise bins to parameterized signals. The user selects the type of signals to be converted by using the parameter Signal type. By default only sampled signals will be converted to parameterized signals.

Sampled signal channels are converted to parameterized in the frequency domain. The total power for each channel per polarization is translated to a parameterized signal. The same approach is used for noise bins, where the total power for each noise bin is translated to a parameterized signal.


Input Input Optical



Signal typeDefines the input signal type to be converted to parameterized signals

Sampled signals — Sampled signals, Noise bins, Sampled signals and Noise bins


Enabled



683

CONVERT TO NOISE BINS

Convert to Noise BinsThis component converts sampled signals and parameterized signals to noise bins.

Ports

Parameters

Main

Simulation


This component converts sampled signals and parameterized signals to noise bins. The user selects the type of signals to be converted by using the parameter Signal type. By default only sampled signals will be converted to noise bins.

Sampled signals are first converted to parameterized signals. The total power for each channel per polarization is translated to a parameterized signal.

Parameterized signals are then converted to noise bins. The power spectral density depends on the noise bin bandwidth. If there is only one channel available, the bandwidth is 100 GHz. If more then one channel is available, the bandwidth is the frequency separation between adjacent channels.


Input Input Optical



Signal typeDefines the input signal type to be converted to noise bins

Sampled signals — Sampled signals, Parameterized signals, Sampled and Parameterized signals


Enabled



684

BINARY NOT

Binary NOTLogic NOT operator.

Ports

Parameters

Simulation


Input Input Binary

Output Output Binary




685

BINARY AND

Binary ANDLogic AND operator.

Ports


Input 1 Input Binary


Output = (Input 1) AND (Input 2) Output Binary

686

BINARY OR

Binary ORLogic OR operator

Ports




Output = (Input 1) OR (Input 2) Output Binary

687

BINARY XOR

Binary XORLogic XOR operator.

Ports




Output = (Input 1) XOR (Input 2) Output Binary

688

BINARY NAND

Binary NANDLogic NAND operator.

Ports




Output = (Input 1) NAND (Input 2) Output Binary

689

BINARY NOR

Binary NORLogic NOR operator.

Ports




Output = (Input 1) NOR (Input 2) Output Binary

690

BINARY XNOR

Binary XNORLogic XNOR operator.

Ports




Output = (Input 1) XNOR (Input 2) Output Binary

691

BINARY DELAY

Binary DelayAdds a time delay in number of bits to the binary signal input.

Ports

Parameters

Main

Simulation


Input Input Binary




1 — [0, 1e+009]




692

DUOBINARY PRECODER

Duobinary precoderThis component simulates a precoder generally utilized in a duobinary modulation.

Ports

Parameters

Main

Simulation


Normally, an optical duobinary system requires a precoder in order to avoid recursive decoding in the receiver, error propagation and reduce hardware complexity. The precoder is composed on an exclusive-or gate with a delayed feedback path. The precoding rule for this is:

where is the transmitted binary data sequence, is the precoded binary sequence, and represents the logic instruction exclusive-or “XOR”. Due to the use of the precoder in a

transmitter, decoding in the receiver is simple.

Figure 1 shows a diagram detailing the precoder. You can specify the number of bits delayed in the feedback path.


Input1 Input Binary

Output1 Output Binary



1 bits [1, +INF]




bk dk bk 1–⊕=

dk bk ⊕

693

DUOBINARY PRECODER

Figure 1 Duobinary precoder

694

Tools LibraryThis section contains information on the following tools.• Switch• Select• Loop Control• Ground• Buffer Selector• Fork 1xN• Binary Null• Optical Null• Electrical Null• Binary Delay• Optical Delay• Electrical Delay• Optical Ring Controller• Electrical Ring Controller• Duplicator• Limiter• Initializer• Save to file• Load from file• Command Line Application

695

Notes:

696

SWITCH

Switch

This component is a switch. The signal entering the input port will be send to one of the output ports.

Ports

Main


Input Input All types

Output 1 Output All types





SelectionSelects the signal output port

1 — — [1 - number of output ports]

697

SWITCH

Technical backgroundThe signal entering the input port will go to the selected output port. This component is used to sweep components. The user can define the signal path by changing the parameter Selection.

The following block diagram shows an example where 4 types of filters are used with a system:

In this project, the user can sweep the Selection parameter from the Switch and the Select components from 1 to 4. By changing these parameters, a different filter will be used for each sweep iteration.

698

SELECT

Select

This component is a select switch. One of the signals entering the input ports will be sent to the output port.

Ports

Main


Input 1 Input All types




Output Output All types



SelectionSelects the signal output port

1 — — [1 - number of output ports]

699

SELECT

Technical BackgroundOne of the signals entering the input ports will go to the output port. This component is used to sweep components. The user can define the input signal and the signal path by changing the parameter Selection.

The following block diagram shows an example where 4 types of filters are used with a system:

In this project, the user can sweep the Selection parameter from the Switch and the Select components from 1 to 4. By changing these parameters, a different filter will be used for each sweep iteration.

700

FORK 1X2

Fork 1x2

Copies the input signal into two output signals. This tool allows you to duplicate component output ports.

Ports


Input Input Any type

Output 1 Output Any type


701

LOOP CONTROL

Loop Control

Allows you to build systems using loop structures.

Ports

Parameters

Main

Simulation

Technical backgroundThe loop topology starts at the Loop output port and terminates at the Loop input port. The signal enters the Input port and circulates in the loop N times, where N is defined by the parameter Number of Loops.



Loop Input Input Any type

Output Output Any type

Loop Output Output Any type


Number of loops

Number of iterations in the loop

0 — [0,+INF[


Enabled



702

GROUND

Ground

Ports



703

BUFFER SELECTOR

Buffer Selector

Allows you to select one of the signals from the input buffer.

Ports

Parameters

Main

Simulation





Selection

Index of the signal buffer

0 — [0,+INF[


Enabled



Iterations

Maximum number of signals in the input buffer

Iterations — [1,1e9]

704

FORK 1XN

Fork 1xN

Copies the input signal into a user-defined number of ouptut signals. Allows you to duplicate component ouput ports.

Ports

Parameters

Main









705

BINARY NULL

Binary Null

Generates a zero-value binary signal.

Ports

Parameters

Simulation




Iterations

Number of times to repeat the calculation


706

OPTICAL NULL

Optical Null

Generates a zero-value optical signal.

Ports

Parameters

Simulation




Iterations



707

ELECTRICAL NULL

Electrical Null

Generates a zero-value electrical signal.

Ports

Parameters

Simulation




Iterations



708

BINARY DELAY

Binary Delay

Generates binary signal delays. The delay is added by sending a NULL signal to the output port.

Ports

Parameters

Main

Simulation


Input Input Binary



Delay

Number of delay signals

1 — [0,+INF[


Enabled



709

OPTICAL DELAY

Optical Delay

Generates optical signal delays. The delay is added by sending NULL signals to the output port.

Ports

Parameters

Main

Simulation


Input Input Optical



Delay


1 — [0,+INF[


Enabled



710

ELECTRICAL DELAY

Electrical Delay

Generates electrical signal delays. The delay is added by sending NULL signals to the output port.

Ports

Parameters

Main

Simulation





Delay


1 — [0,+INF[


Enabled



711

ELECTRICAL DELAY

Notes:

712

OPTICAL RING CONTROLLER

Optical Ring Controller

This component allows the user to build systems using ring structures with optical signals.

Ports

Parameters

Main

Simulation


Input Input Optical



Number of loops


0 — [0, 1e+009]


Enabled



713

OPTICAL RING CONTROLLER

Technical backgroundThe signal enters the Input port and circulates through the ring N times, where N is defined by the parameter Number of Loops. The ring is initialized by a null optical signal.

The block diagram of this component is:

714

ELECTRICAL RING CONTROLLER

Electrical Ring Controller

This component allows the user to build systems using ring structures with electrical signals.

Ports

Parameters

Main

Simulation





Number of loops


0 — [0, 1e+009]


Enabled



715


Technical backgroundThe signal enters the Input port and circulates through the ring N times, where N is defined by the parameter Number of Loops. The ring is initialized by a null electrical signal.

The block diagram of this component is:

716


Duplicator

Ports

Simulation


Input Input Binary



Enabled



Iterations



717


Notes:

718

LIMITER

Limiter

This component controls the number of signals passing from the input to the output port. It can be used as a ring controller module.

Ports

Parameters

Main

Simulation

(same as delays, parameter ‘enable only)


Input Input All types



Number of loops

Number of iterations in the loop ring

1 — [0,+INF[


Enabled



719

LIMITER

Technical BackgroundThis component counts the number of signals passing from the input to the output port. It will interrupt the signal propagation when the number of signals is equal to the parameter Number of loops.

The Limiter is used for ring controlling, since a signal can enter the input port and circulate through the ring the number of times dictated by the parameter Number of loops.

720

INITIALIZER

Initializer

This component is a select switch. The signal entering the first input port is sent to the output a user-defined number of times.

Ports

Parameters

Main






Number of signals

Number of signals from the first input port to be sent to the output port

1 — [0,+INF[

721

INITIALIZER

Technical BackgroundThe signal entering the first input port goes to the output port times, where is defined by the parameter Number of signals. After that, signals from the second input port go to the output port.

The Initializer is used to initialize ring structures, as it allows the user to specify the initial signal to circulate in the ring.

The block diagram below shows an optical ring controller using the Initializer component instead of the Optical Delay tool.

N N

722

INITIALIZER

Save to file

Saves the input signal to a file.

Ports

Parameters

Main


Input Input Binary


Save signal mode

Select if you want to save all signals, only the first signal, or the last signal from the signal buffer

All signals — First signal, All signals, Last signal

Filename

Filename for the saved data

— — —

723

LOAD FROM FILE

Load from file

Loads the input signal from a file.

Ports

Parameters

Main

Simulation




Filename

Filename to load the data

Signal.ods — —

Number of signals to skip

Number of signal to skip when loading a file with multiple signals

0 — [0, 1e+009]


Iterations



724

COMMAND LINE APPLICATION

Command Line Application

This component can create a process with user defined command line parameters. It can be used to call any Windows application. It requires a signal at the input port to trigger the calculation.

Ports

Parameters

Main

Simulation





Application

The name of the program to be called

Command line arguments

List of command line arguments available


Enabled



725


Technical BackgroundThis component allows the user to run another program or application during the simulation process. Command Line Application will run the application with the user defined command line arguments. The component only runs if there is a signal of any type in the input port. After closing the application the component will generate a trigger signal at the output. The trigger signal will be the same signal at the input port.

The component will only finish the calculation when the application is closed, this means that if you run Notepad, for example, only when you close notepad the calculation of other components will continue.

Typically this component is used for cosimulation with EDA tools, together with triggered load and save modules from the EDA cosimulation library (see Figure 1 and Figure 2).

Figure 1 Cosimulation using command line application

726


Figure 2 Command line arguments for the application

727


Notes:

728

Optiwave Software Tools

This section contains information on the following Optiwave software tools.

• OptiAmplifier• IFO_Gratings• WDM_Phasar Demux 1xN• WDM_Phasar Mux Nx1• OptiBPM Component NxM

729

Notes:

730

OPTIAMPLIFIER

OptiAmplifierOptiSystem can call Optiwave’s OptiAmplifier software to design optical fiber amplifiers and lasers.

Since the amplifier performance depends on the input signal, OptiSystem calls the OptiAmplifier engine to simulate the amplifer (see Figure 1).

Figure 1 OptiAmplifier

731

OPTIAMPLIFIER

Ports

Parameters

Main

Polarization

Simulation


Input Input Optical



OptiAmplifier ProjectFilename with the OptiAmplifier project

— — —

Show project after calculationDetermines if the amplifier layout will appear after the calculation ends



Polarization filterDetermines the polarization filter type





732

OPTIAMPLIFIER

Noise

Random numbers


From the OptiAmplifier icon, you can create a new project file, open an existing project file, show the layout of the current project, or access the component properties and parameters (see Figure 2).

Figure 2 OptiAmplifier open menu

You can enter component parameters in the Properties dialog box (see Figure 3).



–100 dB ]-INF,0[


3 dB [0,+INF]


Convert noise bins

— True, False





0 — [0,4999]

733

OPTIAMPLIFIER

Figure 3 OptiAmplifier component properties dialog box

A new OptiAmplifier file opens in OptiAmplifier Main Layout. You can use OptiAmplifier to design the amplifier (see Figure 4).

734

OPTIAMPLIFIER

Figure 4 OptiAmplifier software

OptiAmplifier receives signals from OptiSystem using the input and output ports. OptiAmplifier calculates the layout and sends the output signal back to OptiSystem (see Figure 5).

735

OPTIAMPLIFIER

Figure 5 OptiAmplifier and OptiSystem integration

You can close OptiAmplifier and open it later by selecting Open Layout from the OptiAmplifier Component dialog box. You can also load an existing OptiAmplifier project by selecting Open OptiAmplifier File from the OptiAmplifier component dialog box (see Figure 6).

736

OPTIAMPLIFIER

Figure 6 Loading OptiAmplifier files

737

OPTIAMPLIFIER

Notes:

738

IFO_GRATINGS

IFO_GratingsLoads Optiwave’s IFO_Gratings complex spectrum files. IFO_Gratings is an Integrated and Fiber Optical Gratings Design Software which can export the results to OptiSystem. This component can also be used to load measured data from files.

Ports

Parameters

Main

Simulation


Input Input Optical




User-defined frequency Determines whether you can define the filter center frequency or use the value from the measurements


Frequency User-defined filter center frequency


FBG filename FBG.txt — — —








739

IFO_GRATINGS

Noise

Graphs





3 dB ]-INF,+INF[


Filter reflection - real part Frequency (Hz) Amplitude (a.u.)

Filter reflection - imag part Frequency (Hz) Amplitude (a.u.)

Filter transmission - real part Frequency (Hz) Amplitude (a.u.)

Filter transmission - imag part Frequency (Hz) Amplitude (a.u.)

740

IFO_GRATINGS

Technical backgroundThe input file is formatted containing three items per line — the wavelngth in microns, a complex value transmission, and a complex value for the reflection.

1.5465000 -0.95054476 0.31058130 0.0019538914 0.00030748692

1.5465047 -0.83933357 -0.54361247 -0.00053900650 0.0020880603

1.5465093 -0.10532500 -0.99443599 -0.0017730023 -0.00074527952

1.5465140 0.70683493- 0.70737604 0.0012775158 -0.0013700226

1.5465187 0.99451620 0.10456029 0.0012534218 0.0017631583

1.5465233 0.54426584 0.83891013 -0.0018252456 0.0010779574

1.5465280 -0.30982698 0.95079110 -0.00049861026- 0.0018114039

.

.

.

The parameter User defined frequency determines if you can enter the center frequency. This means that the filter data is shifted from the grating center frequency to the user center frequency that you define in the parameter Frequency.

This file can be generated by IFO_Grating using the Export Complex Spectrum tool (see Figure 1).

741

IFO_GRATINGS

Figure 1 Exporting results from IFO_Grating to OptiSystem

742

WDM_PHASAR DEMUX 1XN

WDM_Phasar Demux 1xNLoads Optiwave’s WDM_Phasar PIW files. WDM_Phasar is Phased Array WDM Device Design Software which can export the results to OptiSystem. This components can also be used to load measured data from files.

Ports

Parameters

Main

Simulation

Noise


Input Input Optical





FilenameFilename with the .piw data

WDM_Phasar.piw — —






-100 dB ]-INF,+INF[


3 dB ]-INF,+INF[

743

WDM_PHASAR DEMUX 1XN

Graphs


The PIW file contains the file header, number of wavelength points, and number of wavelength channels.

BCF2DMC

121 6

[wl ][ -4.564250E+001] [ -3.258100E+001] [ -1.954000E+001] [ -6.511500E+000] [ 6.510500E+000] [ 1.953900E+001] [ 3.258000E+001] [ 4.564150E+001]

1.546000E+000 9.689700E-005 8.340647E-005 7.325889E-006 5.303238E-005 3.161631E-005 7.236055E-005

1.546075E+000 1.293506E-004 8.015659E-005 2.596500E-005 1.780735E-005 4.972572E-005 2.968608E-005

1.546150E+000 1.690047E-004 7.332815E-005 3.134233E-005 1.449511E-005 4.502246E-005 3.533981E-006

1.546225E+000 1.238766E-004 7.758708E-005 9.527608E-006 2.024385E-005 1.532031E-005 3.790632E-006

1.546300E+000 6.985230E-005 7.710248E-005 1.613273E-006 1.720816E-005 4.933527E-007 1.170849E-005

OptiSystem skips the first four lines of the file. The resulting file is:

1.546000E+000 9.689700E-005 8.340647E-005 7.325889E-006 5.303238E-005 3.161631E-005 7.236055E-005

1.546075E+000 1.293506E-004 8.015659E-005 2.596500E-005 1.780735E-005 4.972572E-005 2.968608E-005

1.546150E+000 1.690047E-004 7.332815E-005 3.134233E-005 1.449511E-005 4.502246E-005 3.533981E-006

1.546225E+000 1.238766E-004 7.758708E-005 9.527608E-006 2.024385E-005 1.532031E-005 3.790632E-006

1.546300E+000 6.985230E-005 7.710248E-005 1.613273E-006 1.720816E-005 4.933527E-007 1.170849E-005

The first column is the transmission wavelength in microns. The other columns are the transmission power for each channel. OptiSystem recognizes the number of columns and associates each one to an internal filter and an ouput port.


Filter transmission Frequency (Hz) Amplitude (a.u.)

744

WDM_PHASAR MUX NX1

WDM_Phasar Mux Nx1Loads Optiwave’s WDM_Phasar PIW files. WDM_Phasar is Phased Array WDM Device Design Software which can export the results to OptiSystem. This component can also be used to load measured data from files.

Ports

Parameters

Main

Simulation






Number of input portsDetermines whether or not the component is enabled

2 — [2,1000]

FilenameFilename with the .piw data

WDM_Phasar.piw — —




745

WDM_PHASAR MUX NX1

Noise

Graphs


The PIW file contains the file header, number of wavelength points, and number of wavelength channels.

BCF2DMC

121 6

[wl ][ -4.564250E+001] [ -3.258100E+001] [ -1.954000E+001] [ -6.511500E+000] [ 6.510500E+000] [ 1.953900E+001] [ 3.258000E+001] [ 4.564150E+001]

1.546000E+000 9.689700E-005 8.340647E-005 7.325889E-006 5.303238E-005 3.161631E-005 7.236055E-005

1.546075E+000 1.293506E-004 8.015659E-005 2.596500E-005 1.780735E-005 4.972572E-005 2.968608E-005

1.546150E+000 1.690047E-004 7.332815E-005 3.134233E-005 1.449511E-005 4.502246E-005 3.533981E-006

1.546225E+000 1.238766E-004 7.758708E-005 9.527608E-006 2.024385E-005 1.532031E-005 3.790632E-006

1.546300E+000 6.985230E-005 7.710248E-005 1.613273E-006 1.720816E-005 4.933527E-007 1.170849E-005

OptiSystem skips the first four lines of the file. The resulting file is:

1.546000E+000 9.689700E-005 8.340647E-005 7.325889E-006 5.303238E-005 3.161631E-005 7.236055E-005

1.546075E+000 1.293506E-004 8.015659E-005 2.596500E-005 1.780735E-005 4.972572E-005 2.968608E-005

1.546150E+000 1.690047E-004 7.332815E-005 3.134233E-005 1.449511E-005 4.502246E-005 3.533981E-006

1.546225E+000 1.238766E-004 7.758708E-005 9.527608E-006 2.024385E-005 1.532031E-005 3.790632E-006

1.546300E+000 6.985230E-005 7.710248E-005 1.613273E-006 1.720816E-005 4.933527E-007 1.170849E-005

.

.

.

The first column is the transmission wavelength in microns. The other columns are the transmission power for each channel. OptiSystem recognizes the number of columns and associates each one to an internal filter and an ouput port.





3 dB ]-INF,+INF[


Filter transmission Frequency (Hz) Amplitude (a.u.)

746

OPTIBPM COMPONENT NXM

OptiBPM Component NxM

Description

This component loads Optiwave's OptiBPM 's' files. OptiBPM is a software suite for the design of a variety of integrated and fiber optic guided problems, which can export the results to OptiSystem. This component can also be used to load measured data from files.

Ports


















747


Parameters

Main

Enhanced

Simulation

Noise


File formatDefines the format of the file with the ‘s’ data.

Real Imag — Real Imag, Amplitude Phase

Filename(s)Filename with the ‘s’ data

OptiBPM.s — —

Vertical flipDefines whether or not the vertical flip is enabled.



Central wavelength approximationDefines whether or not the central wavelength approximation is enabled.


LengthWaveguide length

100 ]0,+INF[

Reference index (real)Real part of the complex reference index.

1 — ]0,+INF[

Reference index (imag)Imaginary part of the complex reference index.

0 — ]-INF,+INF[





Noise thresholdMinimum value for adaptation of noise bins.


Noise dynamicThreshold ratio for adaptation of noise bins.

3 dB ]-INF,+INF[

µm

748


Technical BackgroundThe first line of the 's' file contains the number of inputs and outputs, the consecutive lines have the wavelength in microns and the complex values of the amplitude transmission between each input x output port. The parameter File format defines whether the complex values are defined by the real and imaginary parts, or amplitude and phase.

Ninputs Noutputs

l(1) S11(1) S12(1) … S1M(1) S21(1) S22(1) S2M(1) … SN1(1) SN2(1) … SNM(1)

(2) S11(2) S12(2) … S1M(2) S21(2) S22(2) S2M(2) … SN1(2) SN2(2) … SNM(2)

.

.

.

(L) S11(L) S12(L) … S1M(L) S21(L) S22(L) S2M(L) … SN1(L) SN2(L) … SNM(L)

Where Sij(k) is a complex number, i is the input port index, j is the output port index and k is the row index. The complex number is represented by two real numbers: real and imaginary.

The total number of elements in each row is the number of inputs times the number of outputs times 2 (real/amplitude and imaginary/phase parts) plus one (wavelength).

Example with 2 inputs and 2 outputs:

Example with 1 input and 2 outputs:

λ

λ

λ

749


Example with 3 inputs and 1 output:

Central wavelength approximation

The approximation uses the following expression:

where is the central wavelength, is the reference index and L is the length.

The simulation was done with only one wavelength, which is considered to be the central one.

S'11 λ( ) S11 λ0( ) j2πn0L 1λ0-----

1λ---

–exp⋅=

λ0 n0

750

MATLAB Library

This section contains information on the following MATLAB components.

• MATLAB Filter Component• MATLAB Optical Filter Component• MATLAB Component

751

MATLAB LIBRARY

Notes:

752

MATLAB FILTER COMPONENT

MATLAB Filter ComponentSimulates an electrical filter using MATLAB.

Ports

Parameters

Main




Name and description

Default value Default unit

Units Value range

Convert units True — — True, False

FrequencyFilter center frequency

0 GHz Hz, MHz, GHz

[30,+INF[

BandwidthFilter bandwidth

3.5 GHz Hz, MHz, GHz

[0,+INF[

Run commandMATLAB command to execute during the calculation

Order = Factor =(( Frequency - CenterFrequency )/( Bandwidth / 2.0 ) ).^(2.0 * Order)TransferFunction =exp( -0.5 * 0.693147180559945309417 * Factor )

— — —

753


MATLAB

Simulation


MATLAB search pathPath to add to the MATLAB search path

— — —




754


Technical backgroundWhen the MATLAB filter component is active, it opens the MATLAB program. The MATLAB filter component performs a co-simulation with the MATLAB.

At the beginning of the simulation, the MATLAB adds the path (defined by the matlab search path parameter) to the MATLAB search path. This is where you put the created MATLAB files that will be used in the simulation.

Next, OptiSystem puts the following frequencies into the MATLAB workspace:• Vector Frequency with whole frequencies of the electrical signal spectrum• Variable CenterFrequency with center frequency of the MATLAB filter defined by the

Frequency parameter• Variable Bandwidth with the 3 dB bandwidth of filter defined by Bandwidth parameter

(see Figure 1).

Figure 1 MATLAB workspace in OptiSystem

OptiSystem executes the command defined by the Run command in MATLAB. This parameter can contain a command, a file name, or a sequence of commands, such as the default run command.

755


Note: Your program must be able to handle the frequency vector, center frequency, and 3 dB frequency bandwidth to successfully perform the transfer function of the electrical filter (see Figure 2).

Figure 2 MATLAB workspace after generation of the transfer function for OptiSystem

756

MATLAB OPTICAL FILTER COMPONENT

MATLAB Optical Filter ComponentSimulates an optical filter using MATLAB.

Ports

Parameters

Main


Input Input Optical


Name and description

Default value Default unit

Units Value range

FrequencyFilter center frequency

193.1 THz Hz THz nm

[30,+INF[

BandwidthFilter bandwidth

10 GHz Hz GHz THz nm

[0,+INF[

Run commandMATLAB command to execute during the calculation

Order = Factor =(( Frequency - CenterFrequency )/( Bandwidth / 2.0 ) ).^(2.0 * Order)TransferFunction =exp( -0.5 * 0.693147180559945309417 * Factor )

— — —

757


MATLAB

Simulation

Noise



— — —










–100 dB — ]-INF,+INF[



Noise calculation bandwidthCalculation bandwidth, outside of this range calculation is replaced by the attenuation

0.5 THz Hz, GHz, THz, nm

]0,+INF[

758


Technical backgroundWhen the MATLAB optical filter component is active, it opens the MATLAB program. The MATLAB optical filter component performs a co-simulation with the MATLAB.

At the beginning of the simulation, the MATLAB adds the path (defined by the matlab search path parameter) to the MATLAB search path. This is where you put the created MATLAB files that will be used in the simulation.

Next, OptiSystem puts the following frequencies into the MATLAB workspace:• Vector Frequency with whole frequencies of the electrical signal spectrum• Variable CenterFrequency with center frequency of the filter defined by the Frequency

parameter• Variable Bandwidth with the 3 dB bandwidth of filter defined by Bandwidth parameter (see

Figure 1).

Figure 1 MATLAB workspace in OptiSystem

OptiSystem executes the command defined by the Run command in MATLAB. This parameter can contain a command, a file name, or a sequence of commands, such as the default run command.

759


Note: Your program must be able to handle the frequency vector, center frequency, and 3 dB frequency bandwidth to successfully perform the transfer function of the optical filter (see Figure 2).

Figure 2 MATLAB workspace after generation of the transfer function for OptiSystem

760

MATLAB COMPONENT

MATLAB ComponentEnables the utilization of components created in MATLAB.

Ports

Parameters

Main

Simulation

Input





Load MATLABDefines whether MATLAB should be loaded before the calculation starts and kept open after the simulation is completed.

False — True or False

Run commandMATLAB command to execute during calculation

OutputPort1 = InputPort1

— —


— — —

Sampled signal domainSignal domain when transferring signal to MATLAB

Frequency — Frequency, Time





Number of input portsSpecify the number of input ports

1 — [1, 50]

761

MATLAB COMPONENT

Output

User parameters

Simulation

Random numbers

Signal type (Input 1)Specify signal type for Input Port 1)* Optical signal = 0, Electrical signal = 1, Binary signal = 2

0* — [0, 2]*


Number of input portsSpecify the number of output ports

1 — [1, 50]

Signal type (Output 1)Specify signal type for Output Port 1)* Optical signal = 0, Electrical signal = 1, Binary signal = 2

0* — [0, 2]*


Number of parametersSpecify the number of user parameters

1 — [1, 50]

Parameter0Specify the value of the parameter

0 — [-1e+100, 1e+100]








0 — [0,4999]


762

MATLAB COMPONENT

Technical backgroundBy selecting Load MATLAB, the MATLAB software is loaded before the calculation starts. It will stay open after the calculation is complete. Use this option when you want to display graphs from MATLAB. By default, MATLAB is loaded and closed during the calculations.

In this model, you define the number of input ports and output ports and the signal type (optical, electrical, or binary) of each port. For each input port, OptiSystem assembles a structure with the signal and puts this structure into the MATLAB workspace.

Optical signal

For an optical signal at InputPort1, the following structure is launched in the workspace:

Structure 1.0

The field Channel is a double array containing the wavelengths of each channel at InputPort1. As the optical signal can be presented in three formats (sampled, parameterized, and noise bins), the structure for the optical signal contains fields for each format. Figure 1 shows examples of optical signals launched in the workspace.

Figure 1 Examples of optical signals launched in the MATLAB workspace

InputPort1

TypeSignal Optical

Sampled [ struct ]

Parameterized [ struct ]

Noise [ struct ]

Channels [ channels array ]

763

MATLAB COMPONENT

The field Sampled indicates if the input signal at InputPort1 has optical sampled signals. The structure is defined by:

Structure 1.1

If Signal contains an electrical field, it can be a vector 1XN (one polarization component) or a matrix 2XN (two polarization components) of complex numbers. N is the number of samples in the signal. Figure 2 shows examples of structures for the sampled signal.

Figure 2 Structures of optical sampled signals in the MATLAB workspace

The field CentralFrequency indicates the central frequency of the frequency window for the signal. Depending on the parameter Sampled domain, the sampled signal will be in the time or frequency domain. The default value of the Sampled domain parameter is frequency domain. Figure 3 shows an example of a signal in time domain.

Figure 3 Structure of optical sampled signal (time domain) in the MATLAB workspace

The field Parameterized in Structure 1.0 indicates if the InputPort1 signal has optical parameterized signals. The structure is defined by:

Structure 1.2

The field Power indicates the total power of each parameterized signal in the input port.

InputPort1.Sampled

Signal [ Ex; Ey ]

CentralFrequency Central frequency

Frequency | Time [ frequency array ] | [ time array ]

InputPort1.Parameterized

Power [ power ]

Frequency [ frequency ]

SplittingRatio [ sr ]

Phase [ phase ]

764

MATLAB COMPONENT

SplittingRatio gives the ratio between the power of the polarization components and the total power. Phase shows the phase difference between the x and y components. The Frequency field indicates the frequency of each parameterized channel. Figure 4 shows two examples of a parameterized signal structure.

Figure 4 Structure of optical parameterized signals in the MATLAB workspace

The field Noise in Structure 1.0 indicates if the InputPort1 signal has optical noise bins. The structure is defined by:

Structure 1.3

The optical noise structure represents a set of noise bins. The field Power contains a vector or matrix with the power of each noise bin for each polarization state. The LowerFrequency and UpperFrequency fields define the range of each noise bin. The central frequency of each noise bin can be defined as:

CentralFrequency = (InputPort1.Noise.UpperFrequency + InputPort1.Noise.LowerFrequency)/2

The frequency slot of each noise bin can be defined as:

FrequencySlot = InputPort1.Noise.UpperFrequency - InputPort1.Noise.LowerFrequency

Figure 5 shows an example of optical noise structure.

Figure 5 Structure of an optical noise signal in the MATLAB workspace

InputPort1.Noise

Power [ powerX, powerY ]

LowerFrequency [ lowfrequency ]

UpperFrequency [ upfrequency ]

Phase [ phase ]

765

MATLAB COMPONENT

Electrical signalThe structure for electrical signals is simpler than for optical signals. For an electrical signal at InputPort1, the following structure is launched in the workspace:

Structure 2.0

This signal has two other structures inside, one for the signal (Sampled) and the other for noise (Noise). Figure 6 shows an example of an electrical signal launched in the workspace.

Figure 6 Structure of an electrical sampled signal in the MATLAB workspace

The field Sampled indicates if the InputPort1 signal has electrical sampled signals. It is defined by:

Structure 2.1

As with the optical sampled signal, the user can select the domain (time domain or frequency domain) of the electrical sampled signal through the parameter Sampled domain. Figure 7 shows two examples of electrical signals.

Figure 7 Structures of an electrical sampled signal in the MATLAB workspace

InputPort1

TypeSignal Electrical

Sampled [ struct ]

Noise [ struct ]

InputPort1.Sampled

Signal [ E ]

Frequency | Time Frequency | time

766

MATLAB COMPONENT

The field Noise in Structure 2.0 indicates if the InputPort1 signal has electrical noise. The structure is defined by:

Structure 2.2

The structure found in Structure 2.2 is equal to the sampled signal and has the same time domain options. Figure 8 shows two examples of noise signals in different signal domains.

Figure 8 Structures of an electrical noise signal in the MATLAB workspace

InputPort1.Noise

Signal [ E ]

Frequency | Time Frequency | time

767

MATLAB COMPONENT

Binary signalsThe structure for a Binary signal is characterized by the bit sequence and the bit rate. For a binary signal at InputPort1, the following structure is launched in the workspace:

Structure 3.0

Figure 9 is an example of a binary structure.

Figure 9 Structure of a binary signal in the MATLAB workspace

After OptiSystem puts the input signals in the MATLAB workspace, OptiSystem executes the command defined by the Run command parameter in MATLAB. This parameter can contain a command, a file name, or a sequence of commands.

Note: Your program must be able to handle the structures for each input port to successfully obtain the output signals. For each output port, you must create a structure according to the signal type.

For an optical signal to the OutputPort1:

Structure 4.0

The field Sampled in Structure 4.0 is defined by a structure similar to Structure 1.1.

The field Parameterized in Structure 4.0 is similar to Structure 1.2, and the field Noise in Structure 4.0 is similar to Structure 1.3.

InputPort1

TypeSignal Binary

Sequence [ Sequence of bits ]

BitRate bitrate

OutputPort1

TypeSignal Optical

Sampled [ struct ]

Parameterized [ struct ]

Noise [ struct ]

768

MATLAB COMPONENT

For an electrical signal to OutputPort1:

Structure 5.0

The field Sampled in Structure 5.0 is similar to Structure 2.1.

For a binary signal to OutputPort1, the structure is similar to Structure 3.0.

OptiSystem loads the output signal for the appropriate output port and continues the simulation.

Example

Designing an optical amplitude modulator

To design an optical amplitude modulator using a MATLAB component, the component has to be able to handle an optical and electrical signal input to generate an optical signal output. The input and output tab parameters are shown in Figure 10.

Figure 10 Input and Output parameters

As we use some MATLAB files to model the amplitude modulator, all files are located in the ‘c:\temp’ folder, and the parameter Matlab search path has to point to it (see Figure 11).

The file that contains the program utilized by OptiSystem is AmplitudeModulatorComponent, and is specified by the parameter Run command. Because this example modulates an optical signal in time, the parameter Sampled domain is set to Time (see Figure 11).

OutputPort1

TypeSignal Electrical

Sampled [ struct ]

769

MATLAB COMPONENT

Figure 11 MATLAB component Main tab for amplitude modulator model

The equation describing the behaviour of this model is similiar to that found in the Amplitude modulator - Transmitters library. An amplitude modulator parameter Modulation index is necessary to make the MATLAB model work in a manner similiar to the original amplitude modulator. Parameter0 on the User Parameters tab is defined as our Modulation index (see Figure 12).

Figure 12 MATLAB component User Parameters tab for amplitude modulator model

With the parameters of the MATLAB components already defined, the MATLAB code that handles the OptiSystem signals must be generated. In accordance with the Run command parameter, the file AmplitudeModulatorComponent is executed (the code can be seen in Figure 13).

The functions of this file are to:• count the number of sampled signals• count the number of parameterized signals• count the number of noise signals• call the corresponding routine to calculate the output signals for each existing signal

(sample, parameterized, and noise)• verify the other polarization component, in the case of sampled signals

This example shows how generic code can be made to handle all the possibilities of the OptiSystem signals. Users have to be aware of all these possibilities.

770

MATLAB COMPONENT

Figure 13 Code of the AmplitudeModulatorComponent file

The following MATLAB files are used to handle different kinds of signals:• Figure 14 is for optical sampled signals• Figure 15 is for parameterized signals• Figure 16 is for noise bin signals

771

MATLAB COMPONENT

Figure 14 Optical sampled signals

Figure 15 Parameterized signals

Figure 16 Noise bin signals

772

MATLAB COMPONENT

The system shown in Figure 17 demonstrates how this component works. The MATLAB component modulates the two optical signals in accordance with the electrical modulation signal. One of the optical signals is parameterized, and the other is sampled. Both have noise bins. The modulation index (Parameter0) is defined as 1.

Figure 17 Amplitude modulator system

773

MATLAB COMPONENT

Notes:

774

EDA Cosimulation Library

This section contains information on the following EDA Cosimulation components.

• Save ADS File• Load ADS File• Save Spice Stimulus File• Load Spice CSDF File• Triggered Save Spice Stimulus File• Triggered Load Spice CSDF File

775

EDA COSIMULATION LIBRARY

Notes:

776

SAVE ADS FILE

Save ADS FileThis component can save files in the 'TIM' format. The .tim and files are signal data files in Agilent EEsoft ADS, MDIF format. They contain time-domain waveform data for defining the signals associated with certain sources.

Ports

Parameters

Main

Random numbers




Output dataThe signal type to save to the file

Signal and noise SignalSignal and NoiseNoise

File typeType of file to be saved

TIM MDIF

Filename (.tim)File name with the electrical signal in time domain with TIM format

Signal_2.tim

Reference resistanceThe resistance to be added to the output file

50 Ohm [0,+INF[


Generate random seedDetermines the interpolation algorithm for the measured data

YES Yes, No

Random seed indexUser-defined seed index for signal generation

0 — [0,4999]

777

SAVE ADS FILE

Technical backgroundThe .tim files are signal data files in Agilent EEsoft ADS, MDIF format. They contain time-domain waveform data for defining the signals associated with certain sources.

The general .tim file format is:

An exclamation point (!) at the beginning of a line makes it a comment line. Characters following the ! are ignored by the program. The TIMEDATA data block is required. The option line format is the following:

# T ( [ SEC / MSEC / USEC / NSEC / PSEC ] V/MV R xx )

where

The Format line is: % t v

where

By design of the program, the syntax t and v in the Format line are completely arbitrary. These values can be whatever you prefer. For example, an option line such as: % time voltage.

# = Delimiter that tells the program you are specifying these parameters

T = Time

SEC / MSEC / USEC / NSEC / PSEC

= Your choice of Seconds, Milliseconds, Microseconds, Nanoseconds, or Picoseconds

V/MV = Your choice of Volts or Millivolts

R = Reference resistance, default is 50.0

xx = User-specified value for reference resistance

% = Delimiter that tells the program you are specifying

t = time

v = voltage

778

LOAD ADS FILE

Load ADS FileThis component can load files in the 'TIM' format. The .tim and files are signal data files in Agilent EEsoft ADS, MDIF format. They contain time-domain waveform data for defining the signals associated with certain sources.

Ports

Parameters

Main

Numerical




File typeType of file to be loaded

TIM MDIF

Filename (.tim)File name with the electrical signal in time domain with TIM format

Signal.tim

Reload fileDefines whether the component should reload the signal for each run.

no — True, false




779

LOAD ADS FILE

Simulation

The .tim and files are signal data files in Agilent EEsoft ADS, MDIF format. They contain time-domain waveform data for defining the signals associated with certain sources.





Iterations — —

Sample rateOutput signal sample rate

Hz Hz, GHz, THz

780

LOAD ADS FILE

The general .tim file format is:

he ! are ignored by the program. The TIMEDATA data block is required. The option line format is the following:

# T ( [ SEC / MSEC / USEC / NSEC / PSEC ] V/MV R xx )

where

The Format line is the following: % t v

where

By design of the program, the syntax t and v in the Format line are completely arbitrary. These values can be whatever you prefer. For example, an option line such as:

% time voltage

# = Delimiter that tells the program you are specifying these parameters

T = Time

SEC / MSEC / USEC / NSEC / PSEC

= Your choice of Seconds, Milliseconds, Microseconds, Nanoseconds, or Picoseconds

V/MV = Your choice of Volts or Millivolts

R = Reference resistance, default is 50.0

xx = User-specified value for reference resistance

% = Delimiter that tells the program you are specifying

t = time

v = voltage

781

LOAD ADS FILE

Notes:

782

SAVE SPICE STIMULUS FILE

Save Spice Stimulus FileThis component can save ASCII files in a user define format. By default the file has the PSpice Stimulus data format .stl. The .stl files are signal data files used in PSpice. They contain time-domain waveform data, based on a piece-wise linear algorithm, for defining the signals associated with certain sources and nodes.

Ports

Parameters

Main

File format





Signal and noise SignalSignal and noiseNoise

Filename (.stl)File name with the electrical signal in time domain

Signal.stl


File commentCharacter string for comments in the file

*

File beginCharacter string to be added to the first line of the file

.STIMULUS SIGNAL PWL

File endCharacter string to be added to the last line of the file

783


Random numbers


This component can save the data in a user defined file format. The parameters in the File format tab define the additional format information to be added to the signal data.

The general file format is:

COMMENT Written by OptiSystem 3.0

COMMENT EDA Cosimulation Library

COMMENT Save Spice Stimulus File Component

BEGIN

LINE BEGIN time1 DELIMITER amplitude1 LINE END

LINE BEGIN time2 DELIMITER amplitude2 LINE END

LINE BEGIN time… DELIMITER amplitude… LINE END

…

END

File line beginCharacter string to be added to the beginning of each line of data

+(

File line endCharacter string to be added to the end of each line

)

File delimiterCharacter string to be used in each line of data, to separate multiple values

,



YES Yes, No

Random seed indexUser-defined seed index for signal generation

0 — [0,4999]


784


By default, the signal will be saved using the .stl format, e.g. after a source:

* Written by OptiSystem 3.0

* EDA Cosimulation Library

* Save Spice Stimulus File Component


+ (0,1)

+ (6.25e-012, 1)

+ (1.25e-011, 1)

+ (1.875e-011, 1)

+ (2.5e-011, 1)

+ (3.125e-011, 1)

+ (5.119375e-008, 1)

In this case, the parameter Comment is "*", the Begin is ".STIMULUS SIGNAL PWL", the Line Begin is "+(", the Delimiter is ",", the Line End is ")", there is no format for the End, and the value is "".

Typically this component is used after an electrical pulse generator in the transmitter stage, or after the photodetectors, in the receiver stage. The electrical signals are exported to a file and processed by a circuit simulator such as PSpice.

Figure 1 Exporting data using Save Spice Stimulus File

785


Figure 2 Formatting the file using Stimulus format

After the simulation, a text file ‘NRZ.stl’ is generated by OptiSystem. Figure 3 shows the file data using the Stimulus format.

786


Figure 3 NRZ.stl STIMULUS file

The signal can be loaded into a Stimulus editor, and then compared with the original signal from OptiSystem. Figure 4 shows the same signals in OptiSystem and in the Stimulus editor:

Figure 4 OptiSystem and Stimulus editor signals

787


Notes:

788

LOAD SPICE CSDF FILE

Load Spice CSDF FileThis component can load Common Simulation Data Format (CSDF) files from EDA tools that can export PROBE results into CSDF file format. The .csd files are signal data files exported from circuit simulators such as PSpice.

Ports

Parameters

Main

Simulation




Node nameName of the circuit node. The signal at this node will be extracted from the file.

V (0)

Filename (.stl)File name with the data signal in time domain with CDSF format

Signal.csd

Reload fileDefines whether the component should reload the signal for each run

no — True, false





Iterations — —


Hz Hz, GHz, THz

789


Technical BackgroundThe .csd files are signal data files exported from circuit simulators such as PSpice. They are used for post-processing and waveform analysis. They contain time-domain waveform data for defining the signals associated with certain nodes in the circuit. Usually the data is voltage, current, or digital levels (1 or 0).

Circuit simulators can export PROBE data using CSDF option. In the following example, a RLC filter is used to filter a 2.5 GB/s signal. The simulation circuit file and the result is displayed in Figure 1 and Figure 2.

Figure 1 Spice circuit file will generate the PROBE data using CSDF option

790


Figure 2 Simulation results after filtering a NRZ signal using a RLC filter

The following figure (Figure 3) shows OptiSystem loading the file using Load Spice CSDF File component, after the simulation the results will be displayed in the Oscilloscope Visualizer (Figure 4).

Figure 3 OptiSystem project that loads a .csd file

791


Figure 4 Signal from circuit simulator in OptiSystem visualizer

792

TRIGGERED SAVE SPICE STIMULUS FILE

Triggered Save Spice Stimulus FileThis component has the same engine as the Save Spice Stimulus File component. It will copy the input signal to the output signal. The output port can be connected to other component to be used as a signal trigger.

Ports

Parameters

Main

File format








Signal.stl



*




793


Random numbers


Refer to the Technical Background of the Save Spice Stimulus File component for additional information. Additionally this component can generate a signal trigger after saving the signal. It can be used together with the Command Line Application component, from the Tools component library. This module can save the signal into a file, then triggers the command line component to open another application that can load the saved file.

Typically this component is used for cosimulation with EDA tools, together with triggered load component from the EDA cosimulation library. Figure 1 shows one example of application, the file will be saved with a 2.5 GB/s signal, the file format is Spice PWL, with a source named Vsupply. The file name will be NRZ25.stl, and it will be loaded as a voltage source into the circuit simulation (Figure 2).

The file data after the simulation is presented in Figure 3. After saving the file a signal trigger will be send to the Command Line Application component, that will perform its own calculation.


+(


)


,



YES Yes, No


0 — [0,4999]


794


Figure 1 Cosimulation using Triggered Save Spice Stimulus File component

Figure 2 Triggered Save Spice Stimulus file parameter for the file header

795


Figure 3 NRZ25.stl file generated from OptiSystem

796

TRIGGERED LOAD SPICE CSDF FILE

Triggered Load Spice CSDF FileThis component has the same engine as the Load Spice CSDF File component. It will run only if there is a signal at the input port. The signal can be off any type and it will work as a trigger.

Ports

Parameters

Main

File format







Signal.stl



*




797


Simulation


Refer to the Technical Background of the Load Spice CSDF File component for additional information. Additionally, this component can generate a signal trigger after loading the signal. It can be used together with the Command Line Application component, from the Tools component library. This module can load the simulation results into OptiSystem after receiving a signal trigger at the input port.

Typically this component is used for cosimulation with EDA tools, together with triggered save component from the EDA cosimulation library. Figure 1 shows one example of application, the file will be saved with a 2.5 GB/s signal, the file format is Spice PWL, with a source named Vsupply. The file name will be NRZ25.stl, and it will be loaded as a voltage source into the circuit simulation (Figure 2).

The Command Line Application component will call the application, in this case, PSpice, that will perform the simulation and generate a .csd file. This component will load the signal at node V(2) into OptiSystem.


+(


)


,





Hz Hz, GHz, THz


798


Figure 1 Cosimulation using Triggered Load Spice CSDF file component

Figure 2 Triggered Load Spice CSDF file parameters: selected CSDF file data

799


Notes:

800

Cable Access Library

This section contains information on the following Cable Access components:

Carrier generators

• Carrier Generator• Carrier Generator Measured

Transmitters

Modulators

• Electrical Amplitude Modulator (AM)• Electrical Frequency Modulator (FM)• Electrical Phase Modulator• Quadrature Modulator• PAM Modulator• QAM Modulator• PSK Modulator• DPSK Modulator• OQPSK Modulator• MSK Modulator• FSK Modulator• CPFSK Modulator

Pulse generators

• M-ary Pulse Generator• PAM Pulse Generator• QAM Pulse Generator• PSK Pulse Generator• DPSK Pulse Generator

801

CABLE ACCESS LIBRARY

• OQPSK Pulse Generator• MSK Pulse Generator

Sequence generators

• PAM Sequence Generator• QAM Sequence Generator• PSK Sequence Generator• DPSK Sequence Generator

Receivers

Demodulators

• Electrical Amplitude Demodulator• Electrical Phase Demodulator• Electrical Frequency Demodulator• Quadrature Demodulator

Decoders

• PAM Sequence Decoder• QAM Sequence Decoder• PSK Sequence Decoder• DPSK Sequence Decoder

Detectors

• M-ary Threshold Detector

802

CARRIER GENERATOR

Carrier Generator

This component generates a user-defined number of carriers. The output is a sum of sinusoidal electrical signals with constant amplitude. The phase can be constant or random.

Ports

Parameters

Main




Number of channelsNumber of output signal carriers

2 — [1,+INF[

FrequencyFrequency of the first carrier

50 Hz, MHz, GHz

[0,+INF[

Frequency spacingSpacing between adjacent carriers

3.5 Hz, MHz, GHz

[0,+INF[

AmplitudeOutput signal amplitude of each carrier

1 a.u. ]-INF,+INF[

BiasDC bias

0 a.u. ]-INF,+INF[

Random phaseDefines whether the phase of the output carriers will be random or user defined

Yes — True, false

PhaseConstant phase

0 deg ]-INF,+INF[

803

CARRIER GENERATOR

Simulation

Random numbers

Technical backgroundThis component generates a sum of sinusoidal carriers with the same zero peak amplitude according to:

, where is the signal for each carrier, is the parameter,

Number of channels and Vbias is the parameter Bias.

Each carrier is defined by:

, where is the frequency of each carrier.

The phase can be defined as random, or user-defined. The user-defined phase is the same for all the carriers.










Units Value range




0 — [0,4999]

vout vi t( ) vbias+i 1=

N

∑= vi N

vi A 2πft φi+( )sin= fi

804

CARRIER GENERATOR MEASURED

Carrier Generator Measured

This component loads a file with the list of frequency, amplitude and phase of each carrier, and generates a sum of sinusoidal electrical signals.

Ports

Parameters

Main




Frequency amplitude phase (Hz a.u. deg)Table with the carrier data

50e6 1 0

58e6 1 0

Hz, a.u., deg

Amplitude and phase file nameFile name with the list of carriers.

1 Hz, MHz, GHz

[0,+INF[

Frequency spacingSpacing between adjacent carriers

8 Hz, MHz, GHz

0,+INF[

BiasDC bias

0 a.u. ]-INF,+INF[

805

CARRIER GENERATOR MEASURED

Simulation

Technical backgroundThis component generates a sum of sinusoidal carriers according to:

, where is the signal for each carrier, is the number of channels,

and Vbias is the parameter Bias.

Each carrier is defined by:

, where are the amplitude, frequency and phase of each carrier.

The user can provide the measurements in the parameter Frequency amplitude phase (Hz a.u. deg); alternatively the measurements can be loaded from a file using the parameter Amplitude and phase file name. The amplitude and phase curves must be provided in the file containing three columns, where the first one refers to the frequency specified in [Hz] units; the second one gives the amplitude curve in [a.u.] units, and the last one gives the phase in [deg] units.

As an example of input file we have:









vout vi t( ) vbias+i 1=

N

∑= vi N

vi A 2πfit φi+( )sin= Ai, fi, and φl

806

ELECTRICAL AMPLITUDE MODULATOR (AM)

Electrical Amplitude Modulator (AM)

Ports

Parameters

Main

Frequency [Hz] Amplitude [a.u.] Phase [deg]

50e6 1.00 0.00

58e6 1.00 0.00






Frequency

Frequency of the input signal carrier

50 MHz, Hz, GHz, Thz

[0,+INF[


1 a.u. ]-INF,+INF[

Gain

Linear gain to be applied to the signal input

1 ]-INF,+INF[

PhasePhase of the input signal carrier

0 deg,rad ]-INF,+INF[

807

ELECTRICAL AMPLITUDE MODULATOR (AM)

Simulation

Technical BackgroundThe Electrical Amplitude Modulator implements an analog amplitude modulator. The output signal is modulated according to:

where is the input electrical signal, is the parameter gain, is the bias, is the carrier frequency, and is the phase of the carrier.

Figure 1 shows the block diagram of this component.

Figure 1 Electrical Amplitude Modulator block diagram




vout t( ) Gvin t( ) 2πfct φc+( ) b+cos=

vin G b fcφc

808

ELECTRICAL FREQUENCY MODULATOR (FM)

Electrical Frequency Modulator (FM)

Ports

Parameters

Main

Simulation






Frequency



[0,+INF[


1 a.u. ]-INF,+INF[


1 a.u. ]-INF,+INF[



Modulation constantFrequency change relative to the input signal amplitude

1 Hz, kHz, Mhz, GHz

]-INF,+INF[




809

ELECTRICAL FREQUENCY MODULATOR (FM)

Technical BackgroundThe Electrical Frequency Modulator implements an analog frequency modulator. The output signal is modulated according to:

where is the input electrical signal, is the modulation constant, is the parameter amplitude, is the bias, is the carrier frequency, and is the phase of the carrier.

vout t( ) A 2( πfct 2π m vin t( )∫ dt φc )++ b+cos=

vin m Ab fc φc

810

ELECTRICAL PHASE MODULATOR

Electrical Phase Modulator

Ports

Parameters

Main

Simulation






Frequency



[0,+INF[


1 a.u. ]-INF,+INF[


1 a.u. ]-INF,+INF[



Modulation constantPhase change relative to the input signal amplitude

1 deg, rad ]-INF,+INF[




811

ELECTRICAL PHASE MODULATOR

Technical BackgroundThe Electrical Phase Modulator implements an analog phase modulator. The output signal is modulated according to:

where is the input electrical signal, is the modulation constant, is the parameter amplitude, is the bias, is the carrier frequency, and is the phase of the carrier.

vout t( ) A 2πfct mvin t( ) φ+ c+( ) b+cos=

vin M Ab fc φc

812

QUADRATURE MODULATOR

Quadrature Modulator

Ports

Parameters

Main

Simulation


Input-I Input Electrical

Input-Q Input Electrical




Frequency



[0,+INF[


1 a.u. ]-INF,+INF[

GainLinear gain to be applied to the signal input

1 ]-INF,+INF[






813

QUADRATURE MODULATOR

Technical BackgroundThe Quadrature Modulator implements a quadrature analog amplitude modulator. The output signal is modulated according to:

where and are the input electrical signals, is the parameter gain, is the bias, is the carrier frequency, and is the phase of the carrier.


Figure 1 Quadrature Modulator block diagram

Reference:[1] Benedetto, S., Biglieri, E., Castellani, V., Digital Transmission Theory. Prentice-Hall, N.Y.,

(1987).

vout t( ) G I t( ) 2πfct φc+( ) Q t( ) 2πfct φc+( )sin–cos[ ] b+=

I Q G bfc φc

814

PAM MODULATOR

PAM Modulator

Encodes and modulates binary signal to an electrical signal using pulse amplitude modulation (PAM).

Ports

Parameters

Main


Bit Sequence Input Binary




Frequency



[0,+INF[

Amplitude 1 a.u. ]-INF,+INF[


1 a.u. ]-INF,+INF[



Bits per symbolNumber of bits per symbol used in the coding

2 [0,100]


0.5 bit [0,1]

Position 0 bit

Gray codeDefines whether or not to use Gray code

False True, False

815

PAM MODULATOR

Simulation

Technical BackgroundThe PAM Modulator implements a PAM modulator [1].

Figure 1 shows a block diagram of the component.

Figure 1 PAM Modulator block diagram


(1987).


Default units

Unit Value range





816

QAM MODULATOR

QAM Modulator

Encodes and modulates a binary signal to an electrical signal using quadrature amplitude modulation (QAM).

Ports

Parameters

Main






Frequency



[0,+INF[

Amplitude


0 a.u. ]-INF,+INF[




2 [0,100]


0.5 bit [0,1]

Position 0 bit


False True, False

817

QAM MODULATOR

Simulation

Technical BackgroundThe QAM Modulator implements a QAM Modulator [1].

Figure 1 shows a block diagram of this component.

Figure 1 QAM Modulator block diagram


(1987).


Default units

Unit Value range





818

PSK MODULATOR

PSK Modulator

Encodes and modulates a binary signal to an electrical signal using phase shift keying modulation (PSK).

Ports

Parameters

Main






Frequency



[0,+INF[

Amplitude


0 a.u. ]-INF,+INF[




2 [0,100]

Phase offsetDefines whether to use Gray coding or not



False True, False

819

PSK MODULATOR

Simulation

Technical BackgroundThe PSK Modulator implements a PSK modulator [1].


Figure 1 PSK Modulator block diagram


(1987).


Default units

Unit Value range





820

DPSK MODULATOR

DPSK Modulator

Encodes and modulates a binary signal to an electrical signal using differential phase shift keying modulation (DPSK).

Ports

Parameters

Main






Frequency



[0,+INF[

Amplitude


0 a.u. ]-INF,+INF[




2 [0,100]




False True, False

821

DPSK MODULATOR

Simulation

Technical BackgroundThe DPSK Modulator implements a DPSK modulator [1].


Figure 1 DPSK Modulator block diagram


(1987).


Default units

Unit Value range





822

OQPSK MODULATOR

OQPSK Modulator

Encodes and modulates a binary signal to an electrical signal using offset quadrature phase shift keying modulation (OQPSK).

Ports

Parameters

Main






Frequency



[0,+INF[

Amplitude


0 a.u. ]-INF,+INF[






False True, False

823

OQPSK MODULATOR

Simulation

Technical BackgroundThe OQPSK Modulator implements an OQDPSK modulator [1].


Figure 1 OQPSK Modulator block diagram


(1987).


Default units

Unit Value range





824

MSK MODULATOR

MSK Modulator

Encodes and modulates a binary signal to an electrical signal using minimum shift keying modulation (MSK).

Ports

Parameters

Main






Frequency



[0,+INF[

Amplitude


0 a.u. ]-INF,+INF[






False True, False

825

MSK MODULATOR

Simulation

Technical BackgroundThe MSK Modulator implements a MSK modulator [1].


Figure 1 MSK Modulator block diagram


(1987).


Default units

Unit Value range





826

FSK MODULATOR

FSK Modulator

Encodes and modulates a binary signal to an electrical signal using frquency shift keying modulation (FSK).

Ports

Parameters

Main


Input Input Binary




Frequency



[0,+INF[

Amplitude


0 a.u. ]-INF,+INF[




2 [0,100]

Frequency separationFrequency separation between symbols

1 Hz, MHz, GHz, THz

[0,+INF[


False True, False

827

FSK MODULATOR

Simulation

Technical BackgroundThe FSK Modulator implements a FSK modulator [1].

When transmitting information, we can vary the frequency of a signal according to the source symbols. The frequency values takes information from the set of amplitudes [1]:

where is the frequency separation, is the number of possible sequences of binary digits, calculated according to:

where is the number of bits per symbol.

where is the parameter amplitude, is the bias, is the carrier frequency, and is the phase of the carrier.


(1987).


Default units

Unit Value range





ai fs 2i 1– M–( ) i, 1 2 ...M, ,= =

fs M

M 2h=

h

vout t( ) A 2πfct 2πai φc+ +( ) b+cos=

A b fcφc

828

CPFSK MODULATOR

CPFSK Modulator

Encodes and modulates a binary signal to an electrical signal using continuous phase frequency shift keying modulation (CPFSK).

Ports

Parameters

Main


Input Input Binary




Frequency



[0,+INF[

Amplitude


0 a.u. ]-INF,+INF[




2 [0,100]

Frequency separationFrequency separation between symbols

1 Hz, MHz, GHz, THz

[0,+INF[


False True, False

829

CPFSK MODULATOR

Simulation

Technical BackgroundThe CPFSK Modulator implements a CPFSK modulator [].

When transmitting information, we can vary the frequency of a signal according to the source symbols. The frequency values takes information from the set of amplitudes []:

where is the frequency separation, is the number of possible sequences of binary digits, calculated according to:


where is the parameter amplitude, is the bias, is the carrier frequency, and is the phase of the carrier.

In this model, because the phase transitions are constant, a single oscillator with a modulated frequency modulated is used. The absence of abrupt phase transitions results in a narrower spectrum.

Reference:Benedetto, S., Biglieri, E., Castellani, V., Digital Transmission Theory. Prentice-Hall, N.Y., (1987)


Default units

Unit Value range





ai fs 2i 1– M–( ) i, 1 2 ...M, ,= =

fs M

M 2h=

h

vout t( ) A 2πfct 2πai φc+ +( ) b+cos=

A b fcφc

830

M-ARY PULSE GENERATOR

M-ary Pulse Generator

Generates multilevel pulses according to the M-ary signal input.

Ports

Parameters

Main

Simulation


Input Input M-ary



Units Value range


0 ]-INF,+INF[


1 a.u. ]-INF,+INF[


1 bit [0,1]

Position 0 bit


Default units

Unit Value range





831

M-ARY PULSE GENERATOR

Technical BackgroundThis model generates pulses according to:

where is the input M-ary signal, is the linear gain, and is the parameter Bias.

is the bit period, is the duty cycle, and is the pulse position.

vout t( )b 0 t t1<≤,

avin t( ) b t1 t t1 tc+<≤,+b t1 tc t T<≤+,

=

vin a b

T tc t1

832

PAM PULSE GENERATOR

PAM Pulse Generator

Generates a M-ary electrical signal from binary signals using pulse amplitude modulation (PAM).

Ports

Parameters

Main

Simulation



PAM Pulses Output Electrical


Units Value range


2 [0,100]


0.5 bit [0,1]

Position 0 bit


False True, False


Default units

Unit Value range





833

PAM PULSE GENERATOR

Technical BackgroundWhen transmitting information, we can vary the amplitude of a signal according to the source symbols. The amplitude values are taken from the set of amplitudes [1]:

where is the number of possible sequences of binary digits, calculated according to:


Using Gray code, the adjacent signal amplitudes that correspond to the binary sequences will differ by only one digit.

This model generates pulses according to:

where is the amplitude of the signal , is the bit period, is the duty cycle, and is the pulse position.


Figure 1 PAM Pulse Generator block diagram

Reference:

[1] Benedetto, S., Biglieri, E., Castellani, V., Digital Transmission Theory. Prentice-Hall, N.Y., (1987).

ai 2i 1– M–( ) i, 1 2 ...M, ,= =

M

M 2h=

h

vk out–

0 0 t t1<≤,

ak t1 t t1 tc+<≤,

0 t1 tc+ t T<≤,

=

ak k T tct1

834

QAM PULSE GENERATOR

QAM Pulse Generator

Generates two parallel M-ary electrical signals from binary signals using quadrature amplitude modulation (QAM).

Ports

Parameters

Main



Output - I Output Electrical

Output - Q Output Electrical


Units Value range


2 [0,100]


0.5 bit [0,1]

Position 0 bit


False True, False

835

QAM PULSE GENERATOR

Simulation

Technical BackgroundWith the QAM sequence generator, the bit sequence is split into two parallel subsequences, each transmitted in two quadrature carriers when building a QAM modulator. This is done by using a serial to parallel converter.

When transmitting information, we can vary the amplitude of a signal according to the source symbols.

For each output port, the value of the amplitude takes value from the set of amplitudes [1]

where is the number of possible sequence of binary digits, calculated according to:

where is the number of bits per symbol. The equivalent QAM set is given by the square of .

This means:

If , , then we have a 4-QAM. If , , then we have a 16-QAM.




Default units

Unit Value range





a1 2i 1– M–( ) i, 1 2 ..., M, ,= =

M

M 2h 2⁄=

hM

h 2= M 2= h 4= M 4=

h 6= M 8= h 8= M 16=

836

QAM PULSE GENERATOR


where is the amplitude of the signal , is the bit period, is the duty cycle, and is the pulse position.

Figure 1 represents the block diagram of this component.

Figure 1 QAM Pulse Generator block diagram

vk out– t( )0 0 t t1<≤,

ak t1 t t1 tc+<≤,

0 t1 tc+ t T<≤,

=

ak k T tct1

837

QAM PULSE GENERATOR

Reference:


838

PSK PULSE GENERATOR

PSK Pulse Generator

Generates two parallel M-ary electrical signals from binary signals using phase shift keying modulation (PSK).

Ports

Parameters

Main

Simulation






Units Value range


2 [0,100]

Phase offsetInitial phase offset



False True, False


Default units

Unit Value range





839

PSK PULSE GENERATOR

Technical BackgroundWhen transmitting information, we can vary the phase of a signal according to the source symbols. The phase values are taken from the set of angles [1]:


where is the number of bits per symbol, and is the phase offset. The in-phase and the quadrature-channel will have amplitudes according to:



where and are the amplitudes of the output signals and is the bit period.


Figure 1 PSK Pulse Generator block diagram

Reference:


ϕi2πM------ i 1–( ) φ+

i, 1 2 ...M, ,= =

M

M 2h=

h φ

Ii ϕi( ) i,cos 1 2 ...M, ,= =

Qi ϕi( ) i,sin 1 2 ...M, ,= =

Ik out– t( ) Ik 0 t T<≤,=

Qk out– t( ) Qk 0 t T<≤,=

Ik Qk k T

840

DPSK PULSE GENERATOR

DPSK Pulse Generator

Generates two parallel M-ary electrical signals from binary signals using differential phase shift keying modulation (DPSK).

Ports

Parameters

Main

Simulation






Units Value range


2 [0,100]




False True, False


Default units

Unit Value range





841

DPSK PULSE GENERATOR

Technical BackgroundWhen transmitting information, we can vary the phase of a signal according to the source symbols. The phase values are taken from the set of angles [1], [2]:

where is the phase value for the current symbol, and is the phase value for the previous symbol. is the number of possible sequence of binary digits, calculated according to:



where and are the amplitudes of the output signals and is the bit period.


Figure 1 DPSK Pulse Generator block diagram

Reference:


[2] Pawula, R.F., “On M-ary DPSK Transmission Over Terrestrial and Satellite Channels”, IEEE Trans. on Commun. COM-32, 752-761, (July 1984).

ϕki ϕk 1–2πM------ i 1–( ) φ+

+ i, 1 2 ...M, ,= =

ϕki ϕk 1–M

M 2h=

h φ

Iki ϕki( ) i,cos 1 2 ...M, ,= =

Qki ϕki( ) i,sin 1 2 ...M, ,= =


Qk out– t( ) Qk 0 t T<≤,=

Ik Qk k T

842

OQPSK PULSE GENERATOR

OQPSK Pulse Generator

Ports

Parameters

Main

Simulation









False True, False

Name and description Default value Default units Unit Value range





843

OQPSK PULSE GENERATOR

Technical BackgroundWhen transmitting information, we can vary the phase of a signal according to the source symbols. The phase values take the values in the set of angles [1]:

where is the number of possible sequence of binary digits, when using quadrature phase shift keying (QPSK), this number is equal to 4, and is the phase offset. A reduction of the signal fluctuations is possible by delaying the Q channel by one bit period. The bit period is calculated from the input binary signal.

The in-phase and the quadrature-channel will have amplitudes according to:



where is the amplitude of the signal , is the bit period, and is the input bit period.


Figure 1 OQPSK Pulse Generator block diagram

Reference:


ϕi2πM------ i 1–( ) φ+

i, 1 2 ...M, ,= =

Mφ

Ii ϕi( ) i,cos 1 2 ...M, ,= =

Qi ϕi( ) i,sin 1 2 ...M, ,= =


Qk out– t( ) Qk Ts t T Ts+<≤,=

k I T Ts

844

MSK PULSE GENERATOR

MSK Pulse GeneratorGenerates two parallel M-ary symbol sequences from binary signals using minimum shift keying modulation (MSK).

Ports

Parameters

Main

Simulation









False True, False

Name and description Default value Default units Unit Value range





845

MSK PULSE GENERATOR

Technical BackgroundWhen transmitting information, we can vary the phase of a signal according to the source symbols. The phase values take the values in the set of angles [1]:

where is the number of possible sequence of binary digits, when using quadrature phase shift keying (QPSK), this number is equal to 4, and is the phase offset. A reduction of the signal fluctuations is possible by delaying the Q channel by one bit period. The bit period is calculated from the input binary signal. The MSK is a special case of OQPSK in which a sinusoidal pulse replaces the rectangular waveform.

The in-phase and the quadrature-channel will have amplitudes according to:



where is the amplitude of the signal , is the bit period, and is the input bit period.


ϕi2πM------ i 1–( ) φ+

i, 1 2 ...M, ,= =

Mφ

Ii ϕi( ) i,cos 1 2 ...M, ,= =

Qi ϕi( ) i,sin 1 2 ...M, ,= =

Ik out– t( ) Ikπt

2Ts---------

sin 0 t T<≤,=

Qk out– t( ) Qkπt

2Ts---------

cos Ts t T Ts+<≤,=

k I T Ts

846

MSK PULSE GENERATOR

Figure 1 MSK Pulse Generator block diagram

847

MSK PULSE GENERATOR

Reference:[1] Benedetto, S., Biglieri, E., Castellani, V., Digital Transmission Theory. Prentice-Hall, N.Y., (1987).

848

PAM SEQUENCE GENERATOR

PAM Sequence Generator

Generates a M-ary symbol sequence from binary signals using pulse amplitude modulation (PAM).

Ports

Parameters

Main

Simulation



PAM sequence Output M-ary



2 [0,100]

Gray codeDefines whether to use Gray coding or not

False True, False





849


Technical BackgroundWhen transmitting information, we can vary the amplitude of a signal according to the source symbols. The value of the amplitude takes value from the set of amplitudes [1]:



If bits per symbol ( ) equals 2, is equal to 8, and values of and will be:

If bits per symbol ( ) equals 3, is equal to 8, and values of and will be:


Bit sequence i ai

00 1 -3

01 2 -1

10 3 1

11 4 3

Bit sequence i ai

000 1 -7

001 2 -5

010 3 -3

011 4 -1

100 5 1

101 6 3

110 7 5

111 8 7

a1 2i 1– M–( ) i, 1 2 ..., M, ,= =

M

M 2h=

h

h M a i

h M a i

850


In the case of bits per symbol ( ) equals 3, is equal to 8, with Gray code, and the values of will be:

Bit sequence ai

000 -7

001 -5

101 -3

100 -1

110 1

111 3

011 5

010 7

h Ma

851


Reference:


852

QAM SEQUENCE GENERATOR

QAM Sequence Generator

Generates two parallel M-ary symbol sequences from binary signals using quadrature amplitude modulation (QAM).

Ports

Parameters

Main

Simulation



Output - I Output M-ary

Output - Q Output M-ary



2 Hz, THz, nm [0,100]


False True, False





853


Technical BackgroundWith the QAM sequence generator, the bit sequence is split into two parallel subsequences, each can be transmitted in two quadrature carriers when building a QAM modulator. This is achieved by using a serial to parallel converter.


For each output port, the amplitude takes one of the values from the set of amplitudes [1]:



This means:



If bits per symbol ( ) are equal to 4, we have a 16-QAM that requires 2 consecutive bits from the input sequence for each subsequence:

Sequence Subsequence I/i a Subsequence Q / i

a

0000 00 / 1 -3 00 / 1 -3

0001 00 / 1 -3 01 / 2 -1

0010 00 / 1 -3 10 / 3 1

0011 00 / 1 -3 11 / 4 3

0100 01 / 2 -1 00 / 1 -3

0101 01 / 2 -1 01 / 2 -1

0110 01 / 2 -1 10 / 3 1

0111 01 / 2 -1 11 / 4 3

1000 10 / 3 1 00 / 1 -3

1001 10 / 3 1 01 / 2 -1

1010 10 / 3 1 10 / 3 1

a1 2i 1– M–( ) i, 1 2 ..., M, ,= =

M

M 2h 2⁄=

hM

h 2= M 2= h 4= M 4=

h 6= M 8= h 8= M 16=

h

854



1011 10 / 3 1 11 / 4 3

1100 11 / 4 3 00 / 1 -3

1101 11 / 4 3 01 / 2 -1

1110 11 / 4 3 10 / 3 1

1111 11 / 4 3 11 / 4 3


a

855


Reference:


856

PSK SEQUENCE GENERATOR

PSK Sequence Generator

Generates two parallel M-ary symbol sequences from binary signals using phase shift keying modulation (PSK).

Ports

Parameters

Main

Simulation







2 [0,100]


45 deg, rad ]-INF, +INF[


False True, False





857





Assuming , if bits per symbol ( ) equals 2, equals 4, the values of and will be:


Bit sequence I Q

00 1 0

01 0 1

10 -1 0

11 0 -1

Bit sequence I Q

000 1 0

001

010 0 1

ϕi2πM------ i 1–( ) φ+

i, 1 2 ...M, ,= =

M

M 2h=

h φ

Ii ϕi( ) i,cos 1 2 ...M, ,= =

Qi ϕi( ) i,sin 1 2 ...M, ,= =

φ 0= h M IQ

φ 0= h M IQ

22

------- 22

-------

858



011

100 -1 0

101

110 0 -1

111

Bit sequence I Q

22

-------– 22

-------

22

-------– 22

-------–

22

------- 22

-------–

859


Reference:


860

DPSK SEQUENCE GENERATOR

DPSK Sequence Generator

Generates two parallel M-ary symbol sequences from binary signals using differential phase shift keying modulation (DPSK).

Ports

Parameters

Main

Simulation







2 [0,100]




False True, False





861



where is the phase value for the current symbol, and is phase value for the previous symbol. is the number of possible sequence of binary digits, calculated according to:




k Bit sequence I Q

0 00 1 0

1 01 0 1

2 10 -1 0

3 11 0 -1

k Bit sequence I Q

0 000 1 0

1 001

ϕki ϕk 1–2πM------ i 1–( ) φ+

+ i, 1 2 ...M, ,= =

ϕki ϕk 1–M

M 2h=

h φ

Iki ϕki( ) i,cos 1 2 ...M, ,= =

Qki ϕki( ) i,sin 1 2 ...M, ,= =

φ 0= h M IQ

φ 0= h M IQ

22

------- 22

-------

862



2 010 0 1

3 011

4 100 -1 0

5 101

6 110 0 -1

7 111

k Bit sequence I Q

22

-------– 22

-------

22

-------– 22

-------–

22

------- 22

-------–

863


Reference:



864

ELECTRICAL AMPLITUDE DEMODULATOR

Electrical Amplitude Demodulator

A coherent amplitude demodulator.

Ports

Parameters

Main

Low Pass Filter





Units Value range

FrequencyFrequency of the input signal carriert

50 MHz, Hz, GHz, Thz [0,+INF[




1 ]-INF,+INF[


Units Value range

Cut off frequency3 dB cut off frequency of the filter



Cosine Roll Off

Rectangular, Cosine Roll Off, Squared Cosine Roll Off

Roll Off factor 0.2 [0.1]

865

ELECTRICAL AMPLITUDE DEMODULATOR

Simulation

Technical BackgroundThe component implements an analog demodulator for amplitude-modulated signals. The output signal is demodulated according to:

,

where is the input electrical signal, is the parameter gain, is the carrier frequency,

is the phase of the carrier, and is the time response of the low pass filter.

The filter type is described according to filter components in the Electrical Filters library:

• rectangle• cosine roll off• squared cosine roll off


Figure 1 Electrical amplitude demodulator block diagram




vout t( ) Gvin t( ) 2πfct φc+( )cos[ ]∗hlow t( )=

vin G fc

φc hlow

866

ELECTRICAL PHASE DEMODULATOR

Electrical Phase Demodulator

A coherent phase demodulator.

Ports

Parameters

Main

Low Pass Filter





Units Value range





Peak to peak amplitudePeak to peak output signal

1 ]-INF,+INF[


Units Value range




Cosine Roll Off



867

ELECTRICAL PHASE DEMODULATOR

Simulation

Technical BackgroundThis component implements an analog demodulator for phase-modulated signals. The output signal is demodulated using a frequency discriminator followed by an integrator according to:

where is the input electrical signal, is the carrier frequency, is the phase of the carrier, and is the time response of the low pass filter. The signal is then scaled to the user-defined peak-to-peak amplitude.

The filter type is described according to filter components in the Electrical Filters library:• rectangle• cosine roll off• squared cosine roll off


Figure 1 Electrical phase demodulator block diagram




vd t( ) ddt-----vin t( )=

v t( ) vd∫ t( ) td=

vout t( ) v t( ) 2πfct φc+( )cos[ ]∗hlow t( )=

vin fc φchlow

868

ELECTRICAL FREQUENCY DEMODULATOR

Electrical Frequency Demodulator

Frequency demodulator based on a frequency discriminator.

Ports

Parameters

Main

Low Pass Filter





Units Value range





Peak to peak amplitude

Peak to peak output signal

1 ]-INF,+INF[


Units Value range




Cosine Roll Off



869

ELECTRICAL FREQUENCY DEMODULATOR

Simulation

Technical BackgroundThis component implements an analog demodulator for frequency-modulated signals. The output signal is demodulated using a frequency discriminator according to:

.

where is the input electrical signal, is the carrier frequency, is the phase of the

carrier, and is the time response of the low pass filter. The signal is then scaled to the user-defined peak-to-peak amplitude.

The filter type is described according to filter components in the Electric Filters library:• rectangle• cosine roll off• squared cosine roll off


Figure 1 Electrical frequency demodulator block diagram




vd t( ) ddt-----vin t( )=

vout t( ) vd t( ) 2πfct φc+( )cos[ ]∗hlow t( )=

vin fc φc

hlow

870

QUADRATURE DEMODULATOR

Quadrature Demodulator

A coherent amplitude demodulator for quadrature components (I and Q).

Ports

Parameters

Main

Low Pass Filter



Output-I Output Electrical

Output-Q Output Electrical


Units Value range






1 ]-INF,+INF[


Units Value range




Cosine Roll Off



871

QUADRATURE DEMODULATOR

Simulation

Technical BackgroundThis component implements an alalog demodulator using a carrier generator for Q and I quadrature components. The output signal is demodulated according to:

where is the input electrical signal, is the parameter gain, is the carrier frequency,

is the phase of the carrier, and is the time response of the low pass filter.

The filter type is described according to filter components in the Electric Filters library:• rectangle• cosine roll off• squared cosine roll off


Figure 1 Quadrature demodulator block diagram




vI t( ) Gvin t( ) 2πfct φc+( )cos[ ]∗hlow t( )=

vQ t( ) G– vin t( ) 2πfct φc+( )sin[ ]∗hlow t( )=

vin G fc

φc hlow

872

PAM SEQUENCE DECODER

PAM Sequence Decoder

Decodes a PAM M-ary symbol sequence to a binary signal.

Ports

Parameters

Main

Simulation


PAM sequence Input M-ary




2 [0,100]


False True, False





873


Technical BackgroundWhen transmitting information, we can vary the amplitude of a signal according to the source symbols. The amplitude values are taken from the set of amplitudes [1]:

is the number of possible sequence of binary digits, calculated according to:

where is the number of bits per symbol. . The PAM decoder will calculate the value of for each amplitude of the signal :

and convert the values of to the equivalent binary sequence.

If bits per symbol ( ) equals 2, equals 4, the values of and will be:

If bits per symbol ( ) equals 3, equals 8, the values of and will be:

Bit sequence i ai

00 1 -3

01 2 -1

10 3 1

11 4 3

Bit sequence i ai

000 1 -7

001 2 -5

010 3 -3

011 4 -1

100 5 1

101 6 3

110 7 5

111 8 7

ai 2i 1– M–( ) i, 1 2 ...M, ,= =

MM 2h=

h φi k

i ak 1 M+ +( ) 2⁄= i

h M a i

h M a i

874


Using Gray code, the adjacent signal amplitudes that correspond to the binary sequences will differ by only one digit. In the case where bits per symbol ( ) equals 3, equals 8, with Gray code, the values of will be:

Bit sequence ai

000 -7

001 -5

101 -3

100 -1

110 1

111 3

011 5

010 7

hM a

875


Reference:


876

QAM SEQUENCE DECODER

QAM Sequence Decoder

Decodes two parallel QAM M-ary symbol sequences to a binary signal.

Ports

Parameters

Main

Simulation


Input - I Input M-ary

Input - Q Input M-ary




2 [0,100]


False True, False





877


Technical BackgroundIn the QAM sequence decoder, the bit sequence is split into two parallel subsequences, each can be transmitted in two quadrature carriers when building a QAM modulator. This is achieved by using a serial to parallel converter.


For each output port, the value of the amplitude takes value from the set of amplitudes [1]



This means:



The QAM decoder calculates the value of for the amplitude of each signal input :


If bits per symbol ( ) equals 4, we have a 16-QAM that requires 2 consecutive bits from the input sequence for each subsequence:


a

0000 00 / 1 -3 00 / 1 -3

0001 00 / 1 -3 01 / 2 -1

0010 00 / 1 -3 10 / 3 1

0011 00 / 1 -3 11 / 4 3

0100 01 / 2 -1 00 / 1 -3

0101 01 / 2 -1 01 / 2 -1

0110 01 / 2 -1 10 / 3 1

0111 01 / 2 -1 11 / 4 3

a1 2i 1– M–( ) i, 1 2 ..., M, ,= =

M

M 2h 2⁄=

hM

h 2= M 2= h 4= M 4=

h 6= M 8= h 8= M 16=

i k

i ak 1 M+ +( ) 2⁄=

i

h

878


Using Gray code, adjacent signal amplitudes that correspond to binary sequences will differ by only one digit.

1000 10 / 3 1 00 / 1 -3

1001 10 / 3 1 01 / 2 -1

1010 10 / 3 1 10 / 3 1

1011 10 / 3 1 11 / 4 3

1100 11 / 4 3 00 / 1 -3

1101 11 / 4 3 01 / 2 -1

1110 11 / 4 3 10 / 3 1

1111 11 / 4 3 11 / 4 3


a

879


Reference:


880

PSK SEQUENCE DECODER

PSK Sequence Decoder

Ports

Parameters

Main

Simulation







2 [0,100]




False True, False





881




where is the number of bits per symbol, and is the phase offset. The in-phase and the quadrature channel will have amplitudes according to:

The PSK decoder will calculate the value of for the phase of each signal input :


Assuming , if bits per symbol ( ) equals 2, and , then the values for and will be:

Bit sequence I Q

00 1 0

01 0 1

10 -1 0

11 0 -1

ϕi2πM------ i 1–( ) φ+

i, 1 2 ...M, ,= =

M

M 2h=

h φ

Ii ϕi( ) i,cos 1 2 ...M, ,= =

Qi ϕi( ) i,sin 1 2 ...M, ,= =

i k

ϕk arc Qk Ik⁄( )tan=

iϕk ϕ–( )M

2π------------------------- 1+=

i

ϕ 0= h M 4=I Q

882


Assuming , if bits per symbol ( ) equals 3, and , then the values for and will be


Bit sequence I Q

000 1 0

001

010 0 1

011

100 -1 0

101

110 0 -1

111

ϕ 0= h M 8=I Q

22

------- 22

-------

22

-------– 22

-------

22

-------– 22

-------–

22

------- 22

-------–

883


Reference:


884

DPSK SEQUENCE DECODER

DPSK Sequence Decoder

Decodes two parallel DPSK M-ary symbol sequences to binary signals.

Ports

Parameters

Main

Simulation







2 [0,100]




False True, False





885



where is the phase value for the current symbol, and is phase value for the previous symbol.

is the number of possible sequence of binary digits, calculated according to:


The DPSK decoder will calculate the value of from the phase difference between consecutive signals and :


k Bit sequence I Q

0 00 1 0

1 01 0 1

2 10 -1 0

3 11 0 -1

ϕki ϕk 1–2πM------ i 1–( ) φ+

+ i, 1 2 ...M, ,= =

ϕki ϕk 1–

M

M 2h=

h φ

Iki ϕki( ) i,cos 1 2 ...M, ,= =

Qki ϕki( ) i,sin 1 2 ...M, ,= =

ik k 1–

ϕk arc Qk Ik⁄( )tan=

iϕk ϕk 1–– φ–( )M

2π------------------------------------------- 1+=

φ 0= h M IQ

886




k Bit sequence I Q

0 000 1 0

1 001

2 010 0 1

3 011

4 100 -1 0

5 101

6 110 0 -1

7 111

φ 0= h M IQ

22

------- 22

-------

22

-------– 22

-------

22

-------– 22

-------–

22

------- 22

-------–

887


Reference:



888

M-ARY THRESHOLD DETECTOR

M-ary Threshold Detector

Decodes multilevel pulses to a M-ary signal output.

Ports

Parameters

Main



Output Output M-ary



Bit rate Bits/s Bits/s

MBits/s

GBits/s

[0,+INF[



Threshold amplitudesList of threshold levels for decision

-5, -3.5, -1.5, 0

1.5, 3.5, 5

a.u. ]-INF,+INF[

Decision instantValue for the decision instant to use when recovering the bit sequence

0.5 Bit — [0,1]

Output amplitudesList of multilevel symbols for the output M-ary sequence

-7, -5, -3, -1, 1, 3, 5, 7

a.u. ]-INF,+INF[

889

M-ARY THRESHOLD DETECTOR

Simulation

Random numbers

Technical BackgroundThis model compares the electrical signal at a user-defined decision instant with a list of threshold levels. The comparison generates an index used to generate the output amplitude.

For example, if the signal input has a value of -3.3, the output level will be -3, since -3.3 is between -3.5 and -1.5.

The delay compensation parameter allows the user to compensate delay occurred during the signal propagation. The number of output levels must be greater than the number of threshold levels.

By selecting ‘parameter enable’ to false, the module will generate the levels at the decision instant without comparison and decision based on the output levels. This means the user can access the values at decision instant before the quantization.


Units Value range




Units Value range




0 — [0,4999]

890

Visualizer LibraryThis book contains information on the following visualizers.

Optical

• Optical Spectrum Analyzer (OSA)• Optical Time Domain Visualizer (OTDV)• Optical Power Meter Visualizer• WDM Analyzer (WDMA)• Dual Port WDM Analyzer (DPWDMA)

Electrical

• Oscilloscope Visualizer• RF Spectrum Analyzer (RFSA)• Eye Diagram Analyzer• BER Analyzer• Electrical Power Meter• Electrical Carrier Analyzer (ECAN)• Electrical Constellation Visualizer

891

Notes:

892

OPTICAL SPECTRUM ANALYZER (OSA)

Optical Spectrum Analyzer (OSA)

Ports

Parameters

Resolution bandwidth

Graphs






Determines whether or not the resolution filter is enabled

Off — On, Off

Filter type

Determines the type of resolution filter

Rectangle — Rectangle, Gaussian, Butterworth

Bandwidth

Resolution filter bandwidth

0.01 nm [0, 1e+100]



Power unit dBm — dBm, W

Minimum value –100 dBm [-1e+100, 1e+100]

Frequency unit m — m, Hz

Limit number of points True — True, False

Max. number of points 128000 — [100, 1e+008]

893


Simulation

Graphs

Sampled signals

Parameterized signals

Noise bins



Enabled




Sampled signal spectrum Wavelength (m) Power (dBm)

Sampled signal spectrum X Wavelength (m) Power (dBm)

Sampled signal spectrum Y Wavelength (m) Power (dBm)


Parameterized signal spectrum Wavelength (m) Power (dBm)

Parameterized signal spectrum X Wavelength (m) Power (dBm)

Parameterized signal spectrum Y Wavelength (m) Power (dBm)


Noise bins signal spectrum Wavelength (m) Power (dBm)

Noise bins signal spectrum X Wavelength (m) Power (dBm)

Noise bins signal spectrum Y Wavelength (m) Power (dBm)

894


Technical backgroundAfter you run a simulation, the visualizers in the project generate graphs and results based on the signal input. You can access the graphs and results from the Project Browser (see Figure 1), or by double-clicking a visualizer in the Main Layout.

Figure 1 Project browser

Access the Optical Spectrum Analyzer (OSA) parameters, graphs, and results from the simulation (see Figure 2).

Figure 2 OSA display

895


Use the signal index to select the signal to display from the signal buffer.

Use the tabs on the left side of the graph to select the representation that you want to view (see Figure 3).• Signal• Noise• Signal + Noise• All

Figure 3 Multiple signal types display

Use the tabs at the bottom of the graph to access the optical signal polarization (see Figure 4).• Power: Total power• Power X: Power from polarization X• Power Y: Power from polarization Y

Figure 4 Signal polarization display

896

OPTICAL TIME DOMAIN VISUALIZER (OTDV)

Optical Time Domain Visualizer (OTDV)

Ports

Parameters

Graphs





Time unit

Time unit for the horizontal axis

s — — s, bits

Reference bit rate

Reference bit rate to use when the time unit is Bit period

Bit rate Bits/s Bits/s

MBits/s

GBits/s

[0, 1e+012]

Phase unit

Phase unit for the vertical axis

deg — — deg, rad

Unwrap phase

Determines whether or not to remove the phase discontinuity


Power unit

Power unit for the vertical axis

W — — W, dBm

Minimum value

Minimum value for power when using units in dBm

–100 dBm — [-1e+100, 1e+100]

Limit number of points

Determines if you can enter the maximum number of points to display


897


Downsampling

Simulation

Random numbers

Max. number of points

Maximum number of points displayed per graph

128,000 — — [100, 1e+008]


Default unit Default unit Value range

Centered at max power

Determines whether the internal filter will be centered at the maximum amplitude of the signal or if it will be user-defined


Center frequency

User-defined center frequency for the internal filter

193.1 THz Hz, THz, nm [30,3e5]

Sample rate 5*(Sample rate) THz Hz, GHz, THz, nm

[1, 1e+100]



Enabled








Random seed index


0 — [0,4999]



898


Graphs

Signal

Noise

Signal + Noise


Signal power Time (s) Power (W)

Signal power X Time (s) Power (W)

Signal power Y Time (s) Power (W)

Signal phase X Time (s) Phase (deg)

Signal phase Y Time (s) Phase (deg)

Signal chirp X Time (s) Frequency (Hz)

Signal chirp Y Time (s) Frequency (Hz)


Noise power Time (s) Power (W)

Noise power X Time (s) Power (W)

Noise power Y Time (s) Power (W)

Noise phase X Time (s) Phase (deg)

Noise phase Y Time (s) Phase (deg)

Noise chirp X Time (s) Frequency (Hz)

Noise chirp Y Time (s) Frequency (Hz)


Signal + Noise power Time (s) Power (W)

Signal + Noise power X Time (s) Power (W)

Signal + Noise power Y Time (s) Power (W)

Signal + Noise phase X Time (s) Phase (deg)

Signal + Noise phase Y Time (s) Phase (deg)

Signal + Noise chirp X Time (s) Frequency (Hz)

Signal + Noise chirp Y Time (s) Frequency (Hz)

899




900


The Optical Time Domain Visualizer (OTDV) is an Oscilloscope for optical signals. Access the OTDV parameters, graphs, and results from the simulation (see Figure 2).

Figure 2 OTDV display.




901


Use the tabs at the bottom of the graph to access the optical signal polarization (see Figure 4).• Power: Total power• Power X: Power from polarization X• Power Y: Power from polarization Y

Figure 4 Signal polarization display

When you select Power X or Power Y, you can access the signal phase and chirp by selecting the Analysis option.

902

OPTICAL POWER METER VISUALIZER

Optical Power Meter Visualizer

Ports

Parameters

Main

Simulation

Results





Minimum value


–100 dBm [-1e+100, 1e+100]



Enabled




Total power dBm

Total power W

Signal power dBm

Signal power W

Sampled signal power dBm

Sampled signal power W

903

OPTICAL POWER METER VISUALIZER



Access the Optical Power Meter Visualizer (OTDV) parameters, graphs, and results from the simulation (see Figure 2).

Figure 2 OPMV display

You can select the total signal power to display for each signal type. When you select the signal power, the result is the sum of the sampled and parameterized signals.

Parameterized signal power dBm

Parameterized signal power W

Noise power dBm

Noise power W


904

WDM ANALYZER (WDMA)

WDM Analyzer (WDMA)

Ports

Parameters

Main

Interpolation







0.1 nm [0,+INF[

Minimum value


–100 dBm ]INF,+INF[

Lower frequency limit

Defines the lower frequency limit for the calculatino bandwidth

185 Hz, THz, and nm

[30,+INF[

Upper frequency limit

Defines the upper frequency limit for the calculation bandwidth

200 Hz, THz, and nm

[30,+INF[



Noise interpolation

Determines if the noise will be estimated by using the signal

Auto — On, Off, Auto

Interpolation offset

Spacing between the signal maximum and the signal value used as noise value

0.5 nm [0,+INF[

905

WDM ANALYZER (WDMA)

Graphs

Simulation

Graphs

Results

Signal



Frequency unit

Frequency unit for the horizontal axis




Enabled




Signal spectrum Wavelength (nm) Power (dBm)

Noise spectrum Wavelength (nm) Power (dBm)


Min. signal power dBm

Min. signal power W

Frequency at min. signal power Hz

Wavelength at min. signal power nm

Max. signal power dBm

Max. signal power W

Frequency at max. signal power Hz

Wavelength at max. signal power nm

Total signal power dBm

Total signal power W

Ratio max/min signal power dB

Ratio max/min signal power —

906

WDM ANALYZER (WDMA)

Noise

OSNR


Min. noise power dBm

Min. noise power W

Frequency at min. noise power Hz

Wavelength at min. noise power nm

Max. noise power dBm

Max. noise power W

Frequency at max. noise power Hz

Wavelength at max. noise power nm

Total noise power dBm

Total noise power W

Ratio max/min noise power dB

Ratio max/min noise power —


Min. OSNR dB

Frequency at min. OSNR Hz

Wavelength at min. OSNR nm

Max. OSNR dB

Frequency at max. OSNR Hz

Wavelength at max. OSNR nm

Ratio max/min OSNR dB

907

WDM ANALYZER (WDMA)



908

WDM ANALYZER (WDMA)

The WDMA estimates the signal and the noise power for each optical signal channel based on the resolution bandwidth. Click the Analysis tab to view results such as frequency, power, noise, and OSNR.(see Figure 2).

Figure 2 WDMA analysis tab

Click the Details tab to view the detailed analysis of the results, such as the minimum and maximum values for the signals (see Figure 3).

Figure 3 WDMA details tab

909

WDM ANALYZER (WDMA)

Notes:

910

DUAL PORT WDM ANALYZER (DPWDMA)

Dual Port WDM Analyzer (DPWDMA)

Ports

Parameters

Main

Interpolation








0.1 nm [0,+INF[

Minimum value


–100 dBm ]–INF,+INF[

Lower frequency limit

Defines the lower frequency limit for the calculatino bandwidth

185 Hz, THz, and nm

[30,+INF[

Upper frequency limit

Defines the upper frequency limit for the calculation bandwidth

200 Hz, THz, and nm

[30,+INF[



Noise interpolation

Determines if the noise will be estimated by using the signal

Off — On, Off, Auto

Interpolation offset

Spacing between the signal maximum and the signal value used as noise value

0.1 nm [0,+INF[

911


Graphs

Simulation

Graphs

Results

Input signal



Frequency unit

Frequency unit for the horizontal axis




Enabled




Input signal spectrum Wavelength (nm) Power (dBm)

Input noise spectrum Wavelength (nm) Power (dBm)

Output signal spectrum Wavelength (nm) Power (dBm)

Output noise spectrum Wavelength (nm) Power (dBm)

Gain Wavelength (nm) Gain (dB)

Noise figure Wavelength (nm) NF (dB)


Input: Min. signal power dBm

Input: Min. signal power W

Input: Frequency at min. signal power Hz

Input: Wavelength at min. signal power nm

Input: Max. signal power dBm

Input: Max. signal power W

Input: Frequency at max. signal power Hz

Input: Wavelength at max. signal power nm

912


Input noise

Input OSNR

Input: Total signal power dBm

Input: Total signal power W

Input: Ratio max/min signal power dB

Input: Ratio max/min signal power —


Input: Min. noise power dBm

Input: Min. noise power W

Input: Frequency at min. noise power Hz

Input: Wavelength at min. noise power nm

Input: Max. noise power dBm

Input: Max. noise power W

Input: Frequency at max. noise power Hz

Input: Wavelength at max. noise power nm

Input: Total noise power dBm

Input: Total noise power W

Input: Ratio max/min noise power dB

Input: Ratio max/min noise power —


Input: Min. OSNR dB

Input: Frequency at max. OSNR Hz

Input: Wavelength at max. OSNR nm

Input: Max. OSNR dB

Input: Frequency at min. OSNR Hz

Input: Wavelength at min. OSNR nm

Input: Ratio max/min OSNR dB


913


Output signal

Output noise


Output: Min. signal power dBm

Output: Min. signal power W

Output: Frequency at min. signal power Hz

Output: Wavelength at min. signal power nm

Output: Max. signal power dBm

Output: Max. signal power W

Output: Frequency at max. signal power Hz

Output: Wavelength at max. signal power nm

Output: Total signal power dBm

Output: Total signal power W

Output: Ratio max/min signal power dB

Output: Ratio max/min signal power —


Output: Min. noise power dBm

Output: Min. noise power W

Output: Frequency at min. noise power Hz

Output: Wavelength at min. noise power nm

Output: Max. noise power dBm

Output: Max. noise power W

Output: Frequency at max. noise power Hz

Output: Wavelength at max. noise power nm

Output: Total noise power dBm

Output: Total noise power W

Output: Ratio max/min noise power dB

Output: Ratio max/min noise power —

914


Output OSNR

Details

Gain

Noise figure


Output: Min. OSNR dB

Output: Frequency at min. OSNR Hz

Output: Wavelength at min. OSNR nm

Output: Max. OSNR dB

Output: Frequency at max. OSNR Hz

Output: Wavelength at max. OSNR nm

Output: Ratio max/min OSNR dB


Min. gain dB

Frequency at min. gain Hz

Wavelength at min. gain nm

Max. gain dB

Frequency at max. gain Hz

Wavelength at max. gain nm

Total gain dB

Ratio max/min gain dB


Min. noise figure dB

Frequency at min. noise figure Hz

Wavelength at min. noise figure nm

Max. noise figure dB

Frequency at max. noise figure Hz

Wavelength at max. noise figure nm

Ratio max/min noise figure dB

915




The Dual Port WDM Analyzer (DPWDMA) estimates the signal and the noise power for each optical signal channel based on the resolution bandwidth for each input port.

916


Click the Analysis tab to view the results (such as gain and noise figure) comparing the signal from the two input ports (see Figure 2).

Figure 2 DPWDMA analysis tab

Click the Details tab to view the detailed analysis for the results, such as the minimum and maximum values for the signals (see Figure 3).

Figure 3 DPWDMA details tab

917


Notes:

918

OSCILLOSCOPE VISUALIZER

Oscilloscope Visualizer

Ports

Parameters

Main

Simulation


Electrical Input Electrical



Time unit

Time unit for the horizontal axis

s — — s, bits

Reference bit rate

Reference bit rate to use when the time unit is Bit period

Bit rate Bits/s Bits/s, MBits/s, GBits/s

[0,+INF[






128000 — — [100, 1e+008]



Enabled



919


Random numbers

Graphs






Random seed index


0 — [0,4999]


Signal amplitude Time (s) Amplitude (a.u.)

Noise amplitude Time (s) Amplitude (a.u.)

Signal + noise amplitude Time (s) Amplitude (a.u.)

920




Access the Oscilloscope parameters, graphs, and results from the simulation (see Figure 2).

Figure 2 Oscilloscope display

921





922

RF SPECTRUM ANALYZER (RFSA)

RF Spectrum Analyzer (RFSA)

Ports

Parameters

Main



Electrical Input Electrical



Power unit

Power unit for the vertical axis

dB — dBm, W

Minimum value


–100 dBm [-1e+100, 1e+100]

Negative frequencies False — True, False






128000 — [100, 1e+008]





Off — On, Off

Filter type

Determines the type for the resolution filter

Rectangle — Rectangle, Gaussian, Butterworth

923


Simulation

Random numbers

Graphs

Bandwidth

Resolution filter bandwidth

10 MHz [0,+INF[



Enabled








Random seed index


0 — [0,4999]


Signal spectrum Frequency (GHz) Power (dBm)

Noise spectrum Frequency (GHz) Power (dBm)

Signal + noise spectrum Frequency (GHz) Power (dBm)



924




925


Access the RF Spectrum Analyzer (RFSA) parameters, graphs, and results from the simulation (see Figure 2).

Figure 2 RFSA display




926

EYE DIAGRAM ANALYZER

Eye Diagram Analyzer

Ports

Parameters

Main

Clock







Time window

Time window for the eye diagram display

1.5 bit [1,3]

Ignore start bits

Number of start bits to be ignored in the eye diagram

1 bits [0,+INF[

Ignore end bits

Number of end bits to be ignored in the eye diagram

1 bits [0,+INF[



Clock recovery

Determines if the delay compensation between the reference and the received signal will be applied

On — On, Off

927


Threshold

Graphs

Simulation

Noise



Threshold mode

Determines the value mode for the user-defined threshold

Relative — Relative, Absolute

Absolute threshold

Amplitude value for the threshold

0 (a.u.) ]–INF,+INF[

Relative threshold

Relative value for the threshold, relative to the average values of 1s and 0s

50 % [0,100]

Decision instant

The user-defined decision instant for the eye analysis

0.5 Bit [0,1]



Time unit Bit period — s, Bit period

Ratio unit dB — none, dB, %






128000 — [100, 1e+008]



Enabled





Add noise to signal True — True, False

928


Random numbers

Graphs

Results






Random seed index


0 — [0,49999]


Eye diagram Time (s) Amplitude (a.u.)

Min. BER Time (s) log (BER)

Q-factor Time (s) Q

Threshold at min. BER Time (s) Amplitude (a.u.)

Eye Height Time (s) Amplitude (a.u.)

Eye Amplitude Time (s) Amplitude (a.u.)

Eye Closure Time (s) Amplitude (a.u.)

Eye Opening Factor Time (s) Ratio (dB)

Eye Extinction Ratio Time (s) Ratio (dB)


Total Power dBm

Total Power W

Signal Power dBm

Signal Power W

Noise Power dBm

Noise Power W

Signal Delay s

Signal Delay samples

Bit Rate Bits/s

Max. Q Factor —

929


Min. BER —

Min. log of BER —

Max. Eye Height a.u.

Threshold at Min. BER a.u.

Decision Instant at Min. BER Bit period

Max. Eye Amplitude a.u.

Max. Eye Closure a.u.

Max. Eye Opening Factor dB

Max. Eye Opening Factor —

Max. Eye Opening Factor %

Extinction Ratio at Min. BER dB

Extinction Ratio at Min. BER —

Extinction Ratio at Min. BER %

Q Factor at User Defined Decision Instant —

Eye Height at User Defined Decision Instant a.u.

Min. BER at User Defined Decision Instant —

Min. log of BER at User Defined Decision Instant —

BER at User Defined Threshold —

BER at User Defined Decision Instant and Threshold —

log of BER at User Defined Threshold —

log of BER at User Defined Decision Instant and Threshold —

Eye Amplitude at User Defined Decision Instant a.u.

Eye Closure at User Defined Decision Instant a.u

Eye Opening Factor at User Defined Decision Instant dB

Eye Opening Factor at User Defined Decision Instant —

Eye Opening Factor at User Defined Decision Instant %

Extinction Ratio at User Defined Decision Instant dB

Extinction Ratio at User Defined Decision Instant —

Extinction Ratio at User Defined Decision Instant %


930




931


The Eye Diagram Analyzer generates eye diagrams and BER analysis. Double-click the Eye Diagram Analyzer to access the parameters, graphs, and results from the simulation (see Figure 2).

Figure 2 Eye diagram display


The available results are:• Max Q-factor: Maximum value for the Q-factor in the eye time window.• Min BER: Minimum value for the bit error rate in the eye time window.• Eye height: Maximum value for the eye height in the eye time window.• Threshold: Value of the threshold at the decision instant for the maximum Q-

factor / minimum BER.• Decision inst: Value of the decision instant for the maximum Q-factor/minimum

BER.

932


Figure 3 Eye diagram analysis

933


BER and Q-Factor EstimationThe BER estimation method in this visualizer compare the bits generated by a binary signal and the signal received.

Assuming Gaussian noise with the standard deviations and the BER is [1]

(1)

where P0 and P1 are the probabilities of the symbols, M is the number of samples for the logical 0, and N is the number of samples for the logical 1.

Pe0 and Pe1 are:

(2)

(3)

where , , , and are average values and variances of the sampled values respectively, and S is the threshold value.

The Q-factor is calculated by:

(4)

The eye height is calculated by [2]

(5)

σ0 σ1

PeM

N M+----------------Pe0

NN M+----------------Pe1+=

Pe012---erfc

S µ0–

2σ0

---------------

,=

Pe112---erfc

µ1 S–

2σ1

---------------

=

µ0 µ1 σ0 σ1

Qµ1 µ0–σ1 σ0+--------------------=

EH µ1 3σ1–( ) µ0 3σ0+( )–=

934


The eye amplitude is calculated by:

(6)

The eye closure is calculated by:

(7)

where min(V1) is the minimum value of the amplitude for the marks and max(V0) is the maximum value for the amplitude of the spaces.

The eye-opening factor is calculated by:

(8)

The extinction ratio is:

(9)

EA µ1 µ0–=

Ec min V1( ) max V0( )–=

E0µ1 σ1–( ) µ0 σ0–( )–

µ1 µ0–( )---------------------------------------------------=

ERµ1µ0-----=

935


References[1] G.P. Agrawal, "Fiber Optic Communication Systems," John Wiley & Sons, New York, 1997.

[2] D. Derickson, "Fiber Optic Test and Measurement," Prentice Hall, New Jersey, 1998.

936

BER ANALYZER

BER Analyzer

Ports

Parameters

Main







Algorithm

Determines the algorithm used to estimate the BER

Gaussian — Gaussian, Average Gaussian, Gaussian Worse Case

Time window

Time window for the eye diagram display

1.5 bit [1, 3]

Ignore start bits

Number of start bits to be ignored in the eye diagram

1 bits [0,+INF[

Ignore end bits

Number of end bits to be ignored in the eye diagram

1 bits [0,+INF[

Lower calculation limit

Defines the lower calculation limit for the time window

0 Bit period [0,1.5]

Upper calculation limit

Defines the upper calculation limit for the time window

1 Bit period [0,1.5]

937

BER ANALYZER

Clock

FEC

Threshold



Clock recovery

Determines if the delay compensation between the reference and the received signal will be applied

On — On, Off



Enabled FEC gain estimation False — True, False



Threshold mode

Determines the value mode for the user-defined threshold

Relative — Relative, Absolute

Absolute threshold

Amplitude value for the threshold

0 (a.u.) ]–INF,+INF[

Relative threshold

Relative value for the threshold, relative to the average values of 1s and 0s

50 % [0,100]

Load threshold from file

Defines whether the threshold will be loaded from a file or not


Measured threshold filename

Threshold file name

Threshold.dat — —

Reload before calculation

Defines whether the file should be reloaded when the calculation starts


Decision instant

The user-defined decision instant for the eye analysis

0.5 Bit period [0,100]

938

BER ANALYZER

Graphs

Graphs

BER patterns



Time unit Bit period — s, Bit period

Ratio unit None — none, dB, %






128000 — [100, 1e+008]


Measured Threshold Time (Bit period) Amplitude (a.u.)

BER at Measured Threshold Time (s) log of BER



Calculate patterns

Determines whether or not the component will generate BER patterns


Number of points

Number of vertical points for the patterns

16 — [10, 1e+008]

BER for pattern 1 1e-012 — [0,1]





Calculate 3D graph

Determines whether or not the component generates a 3D graph with the BER


939

BER ANALYZER

Penalty calculations


Reference values setup User-defined — User defined, First sweep iteration, Current sweep iteration

Total power –1000 dBm [-1e+100, 1e+100]

Signal power –1000 dBm [-1e+100, 1e+100]

Noise power –1000 dBm [-1e+100, 1e+100]

Min. BER 1 — [0, 1]

Q factor from min. BER 0 — [0, 1000]

Max. Q factor 0 — [0, 1000]

Max. eye height 0 a.u. [-1e+100, 1e+100]

Max. eye amplitude 0 a.u. [-1e+100, 1e+100]

Max. eye closure 0 a.u. [-1e+100, 1e+100]

Max. eye opening factor 0 dB [-1e+100, 1e+100]

Extinction ratio at min. BER 0 dB [-1e+100, 1e+100]

Min. BER at user defined decision instant

1 — [0, 1]

Q factor from min. BER at user defined decision instant

0 — [0, 1000]

Q factor at user defined decision instant

0 — [0, 1000]

BER at user-defined threshold 1 — [0, 1]

Q factor from BER at user defined threshold

0 — [0, 1000]

BER at user defined decision instant and threshold

1 — [0, 1]

Q factor from BER at user defined decision instant and threshold

0 — [0, 1000]

Eye height at user defined decision instant

0 a.u. [-1e+100, 1e+100]

Eye amplitude at user defined decision instant

0 a.u. [-1e+100, 1e+100]

Eye closure at user defined decision instant

0 a.u. [-1e+100, 1e+100]

Eye opening factor at user defined decision instant

0 dB [-1e+100, 1e+100]

940

BER ANALYZER

Simulation

Noise

Random numbers

Graphs

Extinction ratio at user defined decision instant

0 dB [-1e+100, 1e+100]



Enabled





Add noise to signal True — True, False






Random seed index


0 — [0,4999]


Eye diagram Time (s) Amplitude (a.u.)

Min. BER Time (s) log of BER

Q-factor Time (s) Q

Threshold at min. BER Time (s) Amplitude (a.u.)

Eye height Time (s) Amplitude (a.u.)

Eye Amplitude Time (s) Amplitude (a.u.)

Eye Closure Time (s) Amplitude (a.u.)

Eye Opening Factor Time (s) Ratio (dB)


941

BER ANALYZER

Results

Eye Extinction Ratio Time (s) Ratio (dB)

BER pattern 1 Time (s) Amplitude (a.u.)





BER pattern 3D graph Amplitude (a.u.) Time (s)


Total Power dBm

Total Power W

Signal Power dBm

Signal Power W

Noise Power dBm

Noise Power W

Signal Delay s

Signal Delay samples

Bit Rate Bits/s

Max. Q Factor —

Q Factor from Min. BER —

Min. BER —

Min. log of BER —

Max. Eye Height a.u.

Threshold at Min. BER a.u.

Decision Instant at Min. BER Bit period

Max. Eye Amplitude a.u.

Max. Eye Closure a.u.

Max. Eye Opening Factor dB

Max. Eye Opening Factor —

Max. Eye Opening Factor %


942

BER ANALYZER

Extinction Ratio at Min. BER dB

Extinction Ratio at Min. BER —

Extinction Ratio at Min. BER %

Q Factor at User Defined Decision Instant —

Eye Height at User Defined Decision Instant a.u.

Min. BER at User Defined Decision Instant —

Q Factor from Min. BER at User Defined Decision Instant —

Min. log of BER at User Defined Decision Instant —

BER at User Defined Threshold —

BER at User Defined Decision Instant and Threshold —

Q Factor from BER at User Defined Threshold —

Q Factor from BER at User Defined Decision Instant and Threshold —

log of BER at User Defined Threshold —

log of BER at User Defined Decision Instant and Threshold —

Eye Amplitude at User Defined Decision Instant a.u.

Eye Closure at User Defined Decision Instant a.u.

Eye Opening Factor at User Defined Decision Instant dB

Eye Opening Factor at User Defined Decision Instant —

Eye Opening Factor at User Defined Decision Instant %

Extinction Ratio at User Defined Decision Instant dB

Extinction Ratio at User Defined Decision Instant —

Extinction Ratio at User Defined Decision Instant %

Penalty: Total Power dB

Penalty: Signal Power dB

Penalty: Noise Power dB

Penalty: Max. Q Factor dB

Penalty: Q Factor from Min. BER dB

Penalty: Min. BER dB

Penalty: Max. Eye Height dB

Penalty: Max. Eye Amplitude dB

Penalty: Max. Eye Closure dB


943

BER ANALYZER

Penalty: Max. Eye Opening Factor dB

Penalty: Extinction Ratio at Min. BER dB

Penalty: Q Factor at User Defined Decision Instant dB

Penalty: Eye Height at User Defined Decision Instant dB

Penalty: Min. BER at User Defined Decision Instant dB

Penalty: Q Factor from Min. BER at User Defined Decision Instant dB

Penalty: BER at User Defined Threshold dB

Penalty: BER at User Defined Decision Instant and Threshold dB

Penalty: Q Factor from BER at User Defined Threshold dB

Penalty: Q Factor from BER at User Defined Decision Instant and Threshold

dB

Penalty: Eye Amplitude at User Defined Decision Instant dB

Penalty: Eye Closure at User Defined Decision Instant dB

Penalty: Eye Opening Factor at User Defined Decision Instant dB

Penalty: Extinction Ratio at User Defined Decision Instant dB

Min. BER after FEC —

Min. log of BER after FEC —

Min. BER after FEC at User Defined Decision Instant —

Min. log of BER after FEC at User Defined Decision Instant —

BER after FEC at User Defined Threshold —

BER after FEC at User Defined Decision Instant and Threshold —

log of BER after FEC at User Defined Threshold —

log of BER after FEC at User Defined Decision Instant and Threshold —


944

BER ANALYZER



The BER Analyzer estimates and analyzes the BER of the signal received. Double-click the BER Analyzer to access the parameters, graphs, and results from the simulation (see Figure 2).

945

BER ANALYZER

Figure 2 BER Analyzer display

Use the signal index to select the signal to display from the signal buffer (see Figure 3).

The available results are:• Max Q-factor: Maximum value for the Q-factor in the eye time window.• Min BER: Minimum value for the bit error rate in the eye time window.• Eye height: Maximum value for the eye height in the eye time window.• Threshold: Value of the threshold at the decision instant for the maximum Q-

factor / minimum BER.• Decision inst: Value of the decision instant for the maximum Q-factor/minimum

BER.

946

BER ANALYZER

Figure 3 BER analysis

When the parameter Calculate 3D graph is enabled, you can visualize a 3D graph that shows the values of BER versus the decision instant and threshold (see Figure 4).

Figure 4 3D BER graph

947

BER ANALYZER

BER and Q-factor estimationThe parameter Algorithm defines the numerical method to use for the BER estimation.

GaussianAssuming Gaussian noise with the standard deviations and , the BER is [1]:

(1)

where P0 and P1 are the probabilities of the symbols, M is the number of samples for the logical 0, and N is the number of samples for the logical 1.

Also, Pe0 and Pe1 are:

(2)

(3)

where , , , and are average values and variances of the sampled values respectively, and S is the threshold value.

Average GaussianAn enhancement of the simple Gaussian approximation can be achieved by averaging the separately estimated BERs for different sampled symbols [2]. For M sampled values for the logical 0 and N sampled values for the logical 1, the corresponding error rates are:

(4)

(5)

σ0 σ1

PeM

N M+----------------Pe0

NN M+----------------Pe1+=

Pe012---erfc

S µ0–

2σ0

---------------

,=

Pe112---erfc

µ1 S–

2σ1

---------------

=

µ0 µ1 σ0 σ1

Pe01

2M-------- erfc

S µ0i–

2σ0i

----------------

i 1=

M

∑=

Pe11

2N------- erfc

µ1i S–

2σ1i

----------------

i 1=

N

∑=

948

BER ANALYZER

If the signal is mixed with the noise, the Average Gaussian method is modified to calculate the average error patterns. The detailed description is [4]:

where NP is the number of one occurrence of any pattern, N is the total number of patterns, and are average values and variances of the sampled values for each pattern respectively, and S is the threshold value.

Worst-case GaussianSince the Average Gaussian method can estimate the BER per bit or per pattern, the Worst-case Gaussian searches for the min BER for each bit or pattern instead of calculating the average values.

Calculating results

There are two modes to calculate the Q-Factor:

The Q-Factor from BER is calculated numerically by:

(6)

where the Q-factor is calculated

(7)

The eye height is calculated by [2]:

(8)

The eye amplitude is calculated by:

(9)

The eye closure is calculated by:

(10)

where min(V1) is the minimum value of the amplitude for the marks and max(V0) is the maximum value for the amplitude of the spaces.

PeNP

N------erfc

µi S–

2σi

--------------

i 1=

8

∑=

µi σi

Pe12---erfc Q

2-------

=

Qµ1 µ0–σ1 σ0+--------------------=

EH µ1 3σ1–( ) µ0 3σ0+( )–=

EA µ1 µ0–=

Ec min V1( ) max V0( )–=

949

BER ANALYZER

The eye-opening factor is calculated by:

(11)

The extinction ratio is calculated by:

(12)

For the user defined threshold, the input file, given by the parameter Measured threshold filename, is formatted with two items per line, the time and threshold amplitude. Time is given in ratio of the bit period, and amplitude is given in arbitrary units (voltage or current)

As an example of input file, we have:

References[1] G.P. Agrawal, "Fiber Optic Communication Systems," John Wiley & Sons, New York, 1997.

[2] J.C. Cartledge, G.S. Burley, "The Effect of Laser Chirping on Lightwave System Performance," Journal of Lightwave Technology, Vol. 7, Nr. 3, 1989, S. 568-573.

[3] D. Derickson, "Fiber Optic Test and Measurement," Prentice Hall, New Jersey, 1998.

[4] C.J. Anderson, J.A. Lyle, “Technique for evaluation of systems performance using Q in numerical simulation exhibiting intersymbol interference,” Electronic Letters, Vol. 30, No. 1, 1994, S. 71-72.

0 0.5

0.1 0.5

0.2 0.5

...

0.9 0.5

E0µ1 σ1–( ) µ0 σ0–( )–

µ1 µ0–( )---------------------------------------------------=

ERµ1µ0-----=

950

ELECTRICAL POWER METER

Electrical Power Meter

Ports

Parameters

Main

Simulation

Results





Minimum value

Minimum value for power when using units in dBm.

-100 dBm —



Enabled




Total Power dBm

Total Power W

Signal Power dBm

Signal Power W

Noise Power dBm

Noise Power W

951

ELECTRICAL POWER METER

Technical BackgroundVisualizers are similar to components, however they generate graphs and results based on the signal input. You can access the results generated by the visualizer with the Project Browser.

Figure 1 Project Browser

The graphs and results can be accessed by the project browser or by using Displays.

Displays is the visualizer interface; in the EPMV, you can access the parameters and results from the simulation using the EPMV display.

Figure 2 EPMV Display

952

ELECTRICAL CARRIER ANALYZER (ECAN)

Electrical Carrier Analyzer (ECAN)

The Electrical Carrier Analyzer (ECAN) measures and compares different results in two different frequencies. It can also calculate carrier to noise ratio.

Ports

Parameters

Main





Frequency 1

Center frequency of the first filter.

50 MHz, Hz, kHz, THz

[0,+INF[

Bandwidth 1

Bandwidth of the first filter.

10 MHz, Hz, kHz, GHz

[0,+INF[

Frequency 2

Center frequency of the second filter.

50 MHz, Hz, kHz, THz

[0,+INF[

Bandwidth 2

Bandwidth of the second filter.

10 MHz, Hz, kHz, GHz

[0,+INF[

Filter type Gaussian — Gaussian, Rectangle

Filter order

Order of the Gaussian filter.

1 — [1,+INF[

Minimum value

Minimum value for power when using units in dBm.

-100 dBm ]-INF,+INF[

953


Simulation

Results

Frequency 1

Frequency 2



Enabled




Total Power1 dBm

Total Power1 W

Signal Power1 dBm

Signal Power1 W

Noise Power1 dBm

Noise Power1 W

SNR1 dB


Total Power2 dBm

Total Power2 W

Signal Power2 dBm

Signal Power2 W

Noise Power2 dBm

Noise Power2 W

SNR2 dB

954


Details

Total Power

Signal

Noise


Min. Total Power dBm

Min. Total Power W

Frequency at Max. Total Power Hz

Max. Total Power dBm

Max. Total Power W

Frequency at Min. Signal Power Hz

Ratio Max/Min Signal Power dB

Ratio Max/Min Signal Power —



Min. Total Power W



Max. Total Power W






Min. Total Power W



Max. Total Power W


955


SNR




Min. SNR dB

Frequency at Min. SNR Hz

Max. SNR dB

Frequency at Max. SNR Hz

Ratio Max/Min SNR dB


956





Displays is the visualizer interface; in the ECAN, you can access the parameters and results from the simulationusing the ECAN display.

957


The ECAN will estimate the signal and the noise power for each electrical signal channel based on the central frequency of the internal filters. The analysis tab displays results such as frequency, power, noise, and SNR.

Figure 2 Analysis tab

The Details tab displays the detailed analysis for the results shown in the Analysis tab, including the minimum and maximum values for the signals.

Figure 3 Details tab

958

ELECTRICAL CONSTELLATION VISUALIZER

Electrical Constellation Visualizer

Displays the In-Phase and Quadrature-Phase electrical signals in a constellation diagram.

Ports

Parameters

Graphs

Simulation


Electrical - I Input Electrical

Electrical - Q Input Electrical




Defines whether you can enter the maximum number of points to be displayed.

True — [0,+INF[

Maximum number of points

Maximum number of points that can be displayed in a graph.

128,000 —



Enabled



959


Random numbers

Graphs






Defines whether the seed is automatically defined and unique.


Random seed index

User defined seed index for noise generation.

0 — [0,4999]


Signal Amplitude Amplitude - I (a.u.) Amplitude - Q (a.u.)

Noise Amplitude Amplitude - I (a.u.) Amplitude - Q (a.u.)

Signal + Noise Amplitude Amplitude - I (a.u.) Amplitude - Q (a.u.)

960



Displays is the visualizer interface; in the constellation visualizer, you can access the parameters and results from the simulation using the constellation display.

Figure 2 Constellation display

You can select the signal to be displayed from the signal buffer by selecting the signal index. The vertical tab gives access to the signal types:• Signal• Noise• Signal and Noise


961


Notes:

962

Optiwave Corporation7 Capella CourtOttawa, Ontario, K2E 8A7, Canada

Tel.: 1.613.224.4700Fax: 1.613.224.4706

E-mail: [email protected]: www.optiwave.com

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