http://www.wiretechworld.com/the-future-of-optical-fibres/

EE 443/CS 543 Optical Fiber Communications

Dr. Donald EstreichFall Semester

1

Lecture 5

Optical FibersII

2

Highlights from Lecture 4 – I

1. Silica is silicon dioxide and exists in many forms or structures2. The index of refraction n of silica is weakly dependent upon

wavelength (common values range from n = 1.4 to 1.6)3. The equation governs wave travel along a transmission line with a

superposition of waves of the form where kis the wave number (2/) and is the radian frequency

4. Waves have a phase velocity vphase ( = /k) and the energy of the wave travels at the group velocity vgroup ( = d/dk)

5. Modes exist in transmission lines and in general each mode travels at a different group velocity

6. There are two primary categories of optical fibers: multi-mode fibers and single-mode fibers – the single-mode fiber uses a very small diameter core whereas the multi-mode fiber has a larger core diameter

( ) −exp ( )amplitudeA j kz t

3

Highlights from Lecture 4 – II

7. Optical fibers can be either step index (constant index n) or graded index (core n is shaped)

8. The index of refraction of the core is always greater than the cladding index of refraction (n1 > n2)

9. Single-mode fiber is best for long-haul applications such as for WAN, data center and submarine cable links; multi-mode fiber is best for shorter distance applications such as LAN and MAN networks

10. The numerical aperature NA of a fiber is11. The acceptance angle ac is related to the NA by12. Linearly polarized modes exist in optical fibers for multi-mode fiber,

but not of single-mode fiber13. Four generations of fiber systems: (1st generation) 45 Mbps & 850 nm,

(2nd generation) 0.1 to 1.7 Gbps & 1310 nm, (3rd generation) 10 Gbps & 1550 nm, and (4th generation) 10 Tbps & DWDM 1450 to 1620 nm

221 2NA n n= −

sin( )acNA n =

4

https://www.slideshare.net/tossus/waveguiding-in-optical-fibers

Linearly Polarized (LP) ModesReview – From Lecture 4:

5

LP01 Mode Distribution LP11 Mode Distribution

https://www.newport.com/t/fiber-optic-basics

When light is launched into a fiber, modes are excited to varying degrees depending on

the conditions of the launch — input cone angle, spot size, axial centration, etc. The

distribution of energy among the modes evolves with distance as energy is exchanged

between them. Energy can be coupled from guided to radiation modes by

perturbations such as microbending and twisting of the fiber.

LP01 and LP11 Modes in the Core of an Optical Fiber

6

The V-Number (or Normalized Frequency)

https://www.rp-photonics.com/v_number.html

The V-number is an often-used dimensionless parameter in the context of

MMF step-index fibers. V is given by these three expressions:

2 2

1 2 1

2 2 22

− = =

a a aV n n NA n

1

2

a

n

n

NA

Radius of fiber core within the claddingIndex of refraction of fiber coreIndex of refraction of claddingWavelength of optical signal in vacuumNumerical AperatureIndex difference between n1 and n2 is

− =

2 2

1 2

2

12

n n

n

See Section 2.4.1 in Senior, 3rd ed., Eqs. (2.69 and 2.70)

7http://www.invocom.et.put.poznan.pl/~invocom/C/P1-9/swiatlowody_en/p1-1_1_3.htm

Propagation Constant vs. V-number for Different Modes

V = 2.405

Single moderange

LP01

LP11

LP21

0.00

1.00

( ) −

−

2 22

2 21 2

/ k nb

n n

From the solutions of Maxwell’s equations

No

rmal

ized

Pro

pag

atio

n C

on

stan

tb

See Section 2.4.1 in Senior, 3rd ed.

8

The V-Number (or Normalized Frequency)

The V-number is a dimensionless parameter used in the context of step-index

fibers. It is defined as

where λ is the vacuum wavelength, a is the radius of the fiber core, and NA is the

numerical aperture. The V-number can be viewed as a kind of normalized optical frequency. (It is proportional to the optical frequency but rescaled using a waveguide’s properties.) Its useful properties for optical fibers are:

2 2

1 2

2 2

− =

a aV n n NA

(1) For V values less than 2.405, a fiber supports only one mode per polarization

direction (→ single-mode fibers).

(2) Multimode fibers can have much higher V numbers. For large values, the

number of supported modes of a step-index fiber (including polarization

multiplicity) can be calculated approximately as

2

; for 2.4052

V

M V

9

Example: Number of Modes in a MMF given a V-Number Value

Consider a multimode fiber (MMF) with the following

parameters; core radius = ½ 50 m (radius a = 25 m),

numerical aperature NA = 0.39, and wavelength = 1500 nm.

Find the (1) value of the V-number and (2) estimate the number of modes M supported by this value.

2 2

2 2 (25 μm)(0.39) 40.8

1.500 μm

(40.8)832 modes

2 2

aV NA

VM

= = =

= =

10

where Vlm corresponds to the (l,m)-mode. The cut-off wavelength may also be defined as the wavelength for which multimode transmission begins. Thus,

1 1 21 2

1

2 22 2

2

2 22 ,

2

−= = −

C

lm lm

an n n an n

V n V

Optical Fiber Cut-Off Wavelength

2(single-mode)

2.405

=

C

aNA

The cut-off frequency of an electromagnetic waveguide is the lowest frequency

for which a mode will propagate in it. In fiber optics, it is more common to

consider the cut-off wavelength, C , the maximum wavelength that will

propagate in an optical fiber or waveguide. The cut-off wavelength is given by

https://en.wikipedia.org/wiki/Cutoff_frequency

Section 2.5.1 to 3.12 in Senior, 3rd ed. (pp. 59 to 60 )

11

Example 2.8: Optical Fiber Cut-Off Wavelength

Determine the cut-off wavelength for a step-index fiber to exhibit single-mode

operation when the core refractive index and core radius are n1 = 1.46 and a =

4.5 microns (m), respectively, with a relative index difference being 0.25%.

Solution: Using equation (2.98) on page 59 of Senior, 3rd edition, we find

12 2 2 (4.5 μm) 1.46 2 (0.00025)

2.405 2.405

1.214 μm or 1,214 nm

C

C

an

= =

=

12

https://physics.stackexchange.com/questions/106477/graded-index-fiber?rq=1

Single Mode Optical Fiber Field Distribution

Step-Index SMF Fiber

13

http://en.optipedia.info/lsource-index/fiberlaser-index/fiber/parameters/mode-field/

Mode Field Diameter (MFD) in Single-Mode Fiber

For SMF, the diameter of the circular region for the light intensity of 1/e2 = 0.1356 of

the core center light intensity is defined as mode field diameter (provided that the

light intensity distribution in SMF is approximated by the Gaussian function).

=MFD 2wThe MFD is sometimesassumed to be definedby the 1/e (= 0.368) amplitude points.

Senior, 3rd ed., pp. 60-61

14

Mode-Field Diameter is Dependent Upon Wavelength

https://www.slideshare.net/tossus/waveguiding-in-optical-fibers

Wavelength (nm)

Mo

de-

Fiel

d D

iam

eter

(

m)

15

Next Topic: Dispersion in Optical Fibers

Sections 3.8 to 3.12 in Senior, 3rd ed. (pages 105 to 140)Also refer to Section 3.13.2 (pages 144 to 147).

Initial shape

Later shape

16

Pulse Spreading Caused by Dispersion

From: Senior, 3rd ed.– Figure 3.7 on page 106 in Chapter 3.

Composite pattern

1 0 1 1

Attenuation of pulsesnot included in figure

17

A Demonstration of Wave Dispersion

https://en.wikipedia.org/wiki/Wave_packet

A wave packet with dispersionA wave packet without dispersion

▪ Single pulse▪ Pulse maintains its shape▪ Phase and group velocities equal

▪ Single pulse▪ Pulse spreads out as it travels▪ Phase and group velocities unequal

Lossless travel along a transmission path

Group velocity is the speed of the centroid of the pulse.It is the velocity of energy transport.

18

And Pulses Trains Consist of Many Frequency Components

http://mriquestions.com/fourier-transform-ft.html

Periodic Waveforms use Fourier series to express the frequency components.

19

Pulses Are the Sum of Many Frequencies

http://mriquestions.com/fourier-transform-ft.html

t

Infinite durationFourier TransformPair

20

Parameter: Digital Bit Rate

For no overlapping light pulses in an optical fiber, thedigital bit rate (BT) must be less than the reciprocalof the pulse duration (2). Hence,

We must distinguish between digital bit rate and the band-width of the medium. The next slide addresses this.

1(approximately)

2TB

21

Digital Bit Rate (BT) and Bandwidth (BW)

From: Figure 3.8 (page 107) in Senior, 3rd ed.

Nonreturn-to-Zero

Return-to-Zero

Maximum rateof change ina bit pattern

(b) 1 1 1 1

(a) 1 0 1 0

time

time

Finding the maximum digital bit rate BT (bps) from bandwidth BW (Hz) of transmission line depends upon the digital coding:

For Nonreturn-to-Zero: BT(max) = 2BWFor Return-to-Zero: BT(max) = BW

Two bits per second per Hz

One bit per second per Hz

NRZ

RZ

22

Bandwidth-Length Product is a Useful Metric

The information carrying capability of an optical fiber is limited bythe amount of pulse distortion at the receiving end of the fiber.Pulse broadening is proportional to the length of the fiber cable and thus bandwidth is inversely proportional to distance.

The Bandwidth-Length parameter is a very useful parameter statingthe information capability of a transmission line (optical fiber).

A more accurate estimate for maximum bit rate BT(max) is obtained ifwe have a Gaussian pulse with standard deviation ,

TB L

0.2(max) [bits/sec]TB

=

See Appendix B (pp. 1052-1053)of Senior, 3rd ed.

~

23

Example 3.5 (Page 109) of Senior, 3rd Edition

A multi-mode graded index fiber exhibits a pulse broadening of 0.1 microseconds (0.1 s) over a length of 15 km.

(a) Estimate the maximum possible bandwidth of this fiber link assuming not ISI.

The maximum possible optical bandwidth which is equivalent to the maximumPossible bit rate (for to return-to-zero pulses) without ISI is

(b) The dispersion per unit length may be estimated by dividing the total dispersion by the total length of the fiber link.

−= = = =

6second)

1 15 MHz

2 2 (0.1 10opt TB B

6second0.1 10

Dispersion 6.667 ns/km15 km

−= =

24

Example 3.5 (continued)

(c) The bandwidth-length product can be determined in two ways. First, multiply the maximum bandwidth by its length.

Second, use the dispersion per unit length and use BT = 1/2.

5 MHz 15 km 75 MHz kmoptB L = =

9

1 175 MHz km

2 2 6.667 10 [sec/km]optB L

−= = =

25

https://fiberopticnetwork.wordpress.com/tag/fibre-optic-bandwidth/

Fiber

Size (m) Wavelength 850 nm 1300 nm 1550 nm

9/125 Single-mode 2000 20,000 + 4000 – 20,000 +

50/125 Multimode 200 - 500 400 - 1500 300 - 1500

62.5/125 Multimode 100 - 400 200 - 1000 150 - 500

Bandwidth Distance Product (MHz km)

Bandwidth-Distance Product (MHz-km)Fiber

Some commonly cited numbers:

26

https://www.intechopen.com/books/current-developments-in-optical-fiber-technology/multimode-graded-index-optical-fibers-for-next-generation-broadband-access

Dispersion Mechanisms in Optical Fibers

MMF = Multi-Mode Fiber; SMF = Single Mode Fiber

Optical Fiber Dispersion

Profile dispersion(MMFs)

27

Dispersion Mechanisms in Optical Fibers – I

Modal Dispersion is a distortion mechanism occurring in multimode fibers, where

signals spread in time because the propagation velocity of the signals are not the

same for different modes.

Modal Dispersion (also called Intermodal Dispersion)

Chromatic Dispersion is the result of the different colors, or wavelengths, in a light

beam arriving at their destination at slightly different times. It results from the

refractive index being a function of wavelength.

Chromatic Dispersion

Waveguide Dispersion is the result of wavelength-dependence of the propagation

constant of the optical waveguide. It is important in single-mode waveguides. The

larger the wavelength, the more the fundamental mode will spread from the core into

the cladding. Related to Goos-Häenchen shift.

Waveguide Dispersion

28

Dispersion Mechanisms in Optical Fibers – II

Polarization Mode Dispersion

Polarization Mode Dispersion (PMD) is a form of modal dispersion where two

different polarizations of light, which normally travel at the same speed, now travel

at different speeds due to their different polarizations.

Material Dispersion

Material Dispersion is one kind of Chromatic Dispersion where the material

refractive index varies with wavelength and therefore causes the group

velocity to vary. It is called “material” dispersion to distinguish it from “waveguide”

dispersion, which is a consequence of how light is guided in a dielectric fiber.

29

Review: Group Velocity

A packet of waves (signal) moves at the group velocity given by

where

The group velocity is the velocity of the energy transport of theoptical signal propagating on the fiber.

The phase velocity of a single frequency is

= = = =1

g gr

d cv v

d dd

d dk

p phv v

= =

= = =1 1 1

2n n k n

c

Ref. Section 2.3.3,pp. 28 to 30 in Senior, 3rd ed.

30

Review: Group Delay

❑ Group delay gr is the time taken by a modulated signal to travel (such as a length L of optical fiber).

❑ The collection of all frequencies constitutes the signal’s waveform and the energy of the signal.

❑ Each spectral component travels independently of the others.❑ Group velocity vgr is the speed of the waveform (i.e., group of

frequencies) as it travels through a component.

Ref. Section 2.3.3,pp. 28 to 30 in Senior, 3rd ed.

1gr

grv =

31

A Formula for Group Velocity

−

−

− = = =

− = −

= =

−

1

1

1

1 1

2

11

2( )

( ) ( )1

2

gr

gr

d d d dv n

d d d d

dn n

d

c c

dn Nn

d

where Ngr is known as the “group index of the fiber.”For comparison,

Ref. Section 2.3.3,pp. 28 to 30 in Senior, 3rd ed.

=1

phase

cv

n

32

Example: Wave Packet Formed By Two Sinusoids

http://hyperphysics.phy-astr.gsu.edu/hbase/Sound/beat.html

The formation of a wave packet from combining two sinusoidsof nearly equal frequencies.

The envelope of the wave packet moves at the group velocity.

=gr

dv

d

envelope

33

Wave Packet Formed By Many Sinusoids

https://www.slideshare.net/srkrishna341/isi-and-nyquist-criterion

FrequencyDomain

Time Domain

34

Pulse Dispersion Illustrated

http://umich.edu/~elements/5e/18chap/summary.html

Time evolution: t5 > t4 > t3 > t2 > t1

Why does dispersion occur?

Because the Fourier spectral components travel at different velocities causing the pulse to spatially spread out.

35

Derivation of Phase and Group Velocity

Neglecting attenuation, the electric field of a single frequency sinusoid traveling in the z-direction is

The phase front is the plane where the optical phase is constant.

Differentiating both sides gives the phase velocity,

Next, modulate the signal by a sinusoid of frequency .

The modulation splits the signal into two frequency components.

( )( )

= − − = = =0

2( , ) exp ( , whereE z t E j t z n nk n

c

= =phase

dzv

dt

− = constantt z

( )( ) ( ) = − − 0( , ) exp ( cosE z t E j t z

36

Derivation of Phase and Group Velocity (continued)

At the input (z = 0), we have

because

Now we have two frequencies, + and - , so this givestwo propagation constants: + and - , where = n/c.Thus,

( ) ( ) ( ) − − + − −= = +00

( ) ( )(0, ) cos2

j t j t j tEE t E e e e

( ) ( ) − = +( ) ( )1cos .

2

j je e

( ) ( )( )( )( ) ( )

( ) ( )

+ − + − − −

−

− −

−

= +

= −

=

( ) ( ) ( ) ( )0

0

0

( , )2

( , ) cos

( , ) optical carrier envelope of carrier

t z t z

t z

j j

j

EE z t e e

E z t E e t z

E z t E

37

Derivation of Phase and Group Velocity (continued)

The envelope of the carrier represents the information that is modulated onto the optical carrier signal. The propagation speed is called the group velocity and is found by differentiating

This gives the relationship for group velocity,

Letting → d and → d ,

When the index of refraction is constant with frequency and geometry, then

− = constantt z

= =

group

dzv

dt

=group

dv

d

=group phasev v

38

Group Velocity Dispersion

Consider two sinusoids of frequencies of (+½) and

(-½) modulated onto carrier frequency .

Each modulating frequency has its own group velocity.

2

Optical carrier

Frequency

- - - + + - + +

From: Rongqing Hui, Introduction to Fiber-Optic Communications, Academic Press, London, 2020; pp. 48 to 51. © Elsevier.

39

Group Velocity Dispersion (continued)

The propagation group delay group over a one-meter length is defined as the reciprocal of the group velocity.

The group delay difference between + ½ and - ½ is

Obviously, the group delay difference is introduced by the frequency dependence of the propagation constant ().

1 1group

group

d

dv d

d

= = =

2

2

group

group

d d d d

d d d d

= = =

From: Rongqing Hui, Introduction to Fiber-Optic Communications, Academic Press, London, 2020; pp. 48 to 51. © Elsevier.

40

Group Velocity Dispersion (continued)

In general, a frequency dependent propagation constant () can be written as a Taylor’s series,

where

0 0

0 0 0 02

0 0 1 0 2 0

22

2

1( ) ( ) ( ) ( )

2

1( ) ( ) ( ) ( )

2

d d

d d

= =

= + − + − +

= + − + − +

From: Rongqing Hui, Introduction to Fiber-Optic Communications, Academic Press, London, 2020; pp. 48 to 51. © Elsevier.

1

2

2

2

is the group delay

is the group delay dispersion

d

d

d

d

=

=

41

Dispersion Coefficient

The relative time delay between two frequency components in an optical fiber of length L is

It is more convenient to use the wavelength separation instead of the frequency separation, then the relative delay is

where D is the group delay dispersion coefficient. It is related to the parameter 2 as follows:

From: Rongqing Hui, Introduction to Fiber-Optic Communications, Academic Press, London, 2020; pp. 48 to 51. © Elsevier.

2group groupL L = =

group

group

dD

d

= =

22

2group groupd dd cD

d d d

= = =

42

Dispersion Coefficient

The relative time delay between two wavelength components separated by is

In practical fiber systems the delay is measured in picoseconds (ps), wavelengths in nanometers (nm) and fiber distances in kilometers (km). The units for the dispersion coefficient D is [ps/nm-km].

When more than one dispersion mechanisms are present in the fiber the dispersion coefficients are added together.

From: Rongqing Hui, Introduction to Fiber-Optic Communications, Academic Press, London, 2020; pp. 48 to 51. © Elsevier.

group D L =

iitotalD D=

Examples: Modal, chromatic, material, waveguide and polarization dispersion.

43

https://apps.lumerical.com/pic_circuits_optical_fiber.html

Modal Dispersion

Modes in fiber

Each mode has a uniquevelocity of propagation

Pulseshape

Dispersion by path length

Sum of the threecomponents

44

Modal Dispersion

1min

1

distance

velocity ( / )

LnLT

c n c= = =

1max

distance

velocity cos( )

LnT

c = =

Ref. Section 3.10.1,pp. 114 to 119 in

Senior, 3rd ed.

1

2

max

2

LnT

c n=

The minimum time Tmin for a signal to travel distance L within the fiber is

The maximum time Tmax for a signal to travel distance L within the fiber is

Using Snell’s Law we can write

45

Modal Dispersion (continued)

The time delay difference between meridional ray and the axial ray by subtraction,

and by definition delta is

And

What we want to know is the rms pulse broadening MD

1 1 11 1 2max min

2 2 2 2

2 2 2Ln Ln LnLn n nT T T

c n c c n n c n

−= − = − = =

− − =

2 2

1 2 1 2

2

1 1

, 12

n n n n

n n

2

1 1 2 1

2 1

( )or

2

Ln n n Ln L NAT T

c n c n c

−= = =

( )2

1

12 3 4 3MD s

L NALn

c n c

= = =

~

~

~That was a homework problem

46From: Figure 3.9 (page 108) in Senior, 3rd ed.

Pulse Broadening From Intermodal Dispersion

20 MHz-km

1 GHz-km

100 GHz-km

Multimode step index fiber

Multimode graded index fiber

Single-mode step index fiber

Bandwidth Distance Product

47

Example 3.8 (Modal Dispersion)

Senior, 3rd ed.Pages 117-118

Example 3.8. A six-kilometer (6 km) optical link with multi-mode step-index fiber withcore index of refraction = 1.500 and relative refractive index difference of 1%. Find (a) The delay difference between the lowest and fastest modes of the fiber link.

Solution: The delay difference is given by

(b) Find the rms pulse broadening from modal dispersion over the fiber link of 6 km.

Solution:

(c) Next, find the maximum bit rate without errors on the fiber link.

3

1

8

(6 10 m) 1.500 0.01300 ns

3 10 m/sec

LnT

c

= = =

3

1

8

1 (6 10 m) 1.500 0.01 300ns 86.6 ns

3 10 m/sec 3.46412 3 2 3MD

Ln

c

= = = =

48

Example 3.8 (continued)

Example 3.8. (c) Next, find the maximum bit rate without errors on the fiber link. (Assume only

modal dispersion is present)

Solution: The maximum bit rate can be estimated in two ways. First, we may use the relationship between BT and the reciprocal of the delay , namely,

Or we can estimate it from the calculated rms pulse broadening from part (b).

(d) Find the bandwidth-length product corresponding to (c) above. Assume return-to-zero pulses.

9maximum

1 1 1( ) 1.67 Mbps

2 2 2 300 10 secTB

T −= = = =

9maximum

0.2 0.2( ) 2.31Mbps

86.6 10 secT

MD

B −

= = =

Bestestimate

2.31MHz 6 km 13.86 MHz kmoptB L = =

~

Senior, 3rd ed.Pages 117-118

49

Questions are requested.

50

Pulse Spreading Caused by Dispersion

https://www.researchgate.net/figure/Pulse-Spread-and-Attenuation-due-to-Dispersion_fig1_277014078

51

1 0 1