tech background and tutorials

7/23/2019 Tech Background and Tutorials

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OptiBPMTechnical Background and Tutorials

Waveguide Optics Modeling Software System

Version 12.1for Windows XP, Vista, Windows 7


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OptiBPMTechnical Background and TutorialsWaveguide Optics Modeling Software Systems

Copyrigh t 2013 OptiwaveAll rights reserved.

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DisclaimerOptiwave makes no representation or warranty with respect to the adequacy of this documentation or the programs which itdescribes for any particular purpose or with respect to its adequacy to produce any particular result. In no event shall Optiwave, itsemployees, 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.


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Contact Information

Technical Support

Tel (613) 224-4700 E-mail [email protected]

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General Enquiries

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Table of contents

Technical Background..............................................................................................11

Notes: .....................................................................................................................................12

Beam Propagation Method (BPM) ...........................................................................13

Slowly Varying Envelope Approx imation ...........................................................................13

Differential Equations of BPM..............................................................................................15

Semi-Vector and Scalar BPM ...............................................................................................19

Crank-Nicholson Method and Scheme Parameter .............................................................20

ADI ..........................................................................................................................................22

Boundary Condi tions for the Beam Propagation Method .................................................23

Perfectly Matched Layer (PML) ............................................................................................26

Wide Angle Beam Propagation Method ..............................................................................30

Wide-Angle Beam Propagation Method based on Pad Approximant Operators ..........48

References .............................................................................................................................65

Conformal Mapping Regions ...................................................................................67

References .............................................................................................................................69

Notes: .....................................................................................................................................70

Diffusion in Lithium Niobate ....................................................................................71


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Titanium Diffusion in Lithium Niobate ................................................................................71

Magnesium Diffusion in Lithium Niobate............................................................................73

Proton Exchange Process in Lithium Niobate ...................................................................74

Electroopt ic Effect ....................................................................................................77

Index elli psoid .......................................................................................................................78

References .............................................................................................................................80

Scatter ing Data..........................................................................................................81

Introduct ion ...........................................................................................................................81

Modeling of the Opt ical Components - Survey of Methods ..............................................81

Circui t Complexi ty - Class if ication ......................................................................................83

The Tool - Scattering Data Approach ..................................................................................86

Implementation with OptiSystem.........................................................................................90

Solu tions ................................................................................................................................90

References: ..........................................................................................................................104

Non-linear BPM algorithm ......................................................................................105

Appendix A: Technical background and manual calculat ion method ...............107

Vectoral Beam Propagation for Anisotropic Waveguides ..................................117

Introduct ion .........................................................................................................................117

Mathematical Formulations................................................................................................118

Vectoral Modal Analysis for Anisotropic Waveguide..........................................129

Introduct ion .........................................................................................................................129

Appendix I ............................................................................................................................131

Appendix II ...........................................................................................................................137

References ...........................................................................................................................138


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Fiber Mode Solvers .................................................................................................139

Introduct ion .........................................................................................................................139

Real-valued formulation .....................................................................................................140

Debye Potential ...................................................................................................................140

Separation of Variab les ......................................................................................................142

Solu tion of the linear system .............................................................................................143

Dispersion equat ion ............................................................................................................144

LP Modes .............................................................................................................................146

References ...........................................................................................................................148

Finite Difference Mode Solver................................................................................149

Introduct ion .........................................................................................................................149

Magnetic Formulation .........................................................................................................150

Magnetic Finite Difference Equat ions ...............................................................................151

Impli cit ly Restarted Arnold i Method (IRAM).....................................................................152

Transparent Boundary Cond it ion (TBC) ...........................................................................158

References ...........................................................................................................................161

Tutorials ...................................................................................................................165

Lesson 1: Getting Started ......................................................................................167

Install ing OptiBPM ..............................................................................................................167

Hardware and software requirements ...............................................................................167

Protecti on key......................................................................................................................167

OptiBPM directory ...............................................................................................................168

Installation ...........................................................................................................................168

Installing OptiBPM on Windows XP, Vista or Windows 7 ......................................168

Introduction to OptiBPM.....................................................................................................169


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What is OptiBPM? ...............................................................................................................169

Introduction to optical waveguides ...................................................................................172

What are optical waveguides? ...........................................................................................172

Quick start............................................................................................................................173

Starting OptiBPM ...................................................................................................173

GUI main parts .....................................................................................................................174

How to use OptiBPM ...........................................................................................................176

Defining materials ..................................................................................................176

Defining 2D and 3D channel profiles......................................................................181

Defining the layout settings ....................................................................................185

Creating a basic project .........................................................................................189

Inserting the input plane.........................................................................................190

Running the simulation...........................................................................................195

Lesson 2: Create a simple MMI coupler................................................................207

Defining materials for the MMI coupler ..................................................................207


Creating an MMI coupler........................................................................................211



Viewing the simulation results in OptiBPM_Analyzer.............................................225

Lesson 3: Create a single-bend device.................................................................227

Defining materials for the single-bend device ........................................................227


Creating an arc waveguide ....................................................................................230


Selecting output data files......................................................................................239



Notes: ...................................................................................................................................248


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Lesson 4: Create an MMI star coupler ..................................................................249

Defining materials for the MMI star coupler ...........................................................250


Creating an MMI star coupler.................................................................................251



Viewing the maxima...............................................................................................256

Drawing the output waveguide...............................................................................260

Assigning a path to the output waveguide .............................................................261


Adding output waveguides and viewing the new simulation results.......................265

Viewing the new simulation results in OptiBPM_Analyzer .....................................267

Notes: ...................................................................................................................................269

Lesson 5: Wavelength scripting with VB Script...................................................271

Defining materials for the waveguide .....................................................................271


Creating the waveguides........................................................................................272

Modifying the input plane .......................................................................................273

Assigning paths to the waveguides........................................................................274


Viewing the simulation results in OptiBPM_Designer ............................................278

Using VB Scripting for the simulation.....................................................................280


Notes ....................................................................................................................................287

Lesson 6: Design a 3dB coupler using VB Script................................................289

Defining the materials for the 3dB coupler .............................................................289


Drawing and positioning the waveguides...............................................................295

Generating the layout script ...................................................................................302

Inserting and editing the input plane ......................................................................303



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Modifying the layout script......................................................................................309

Viewing simulation results in OptiBPM_Analyzer...................................................312

Lesson 7: Applying predefined dif fus ion processes...........................................315

Creating a linear waveguide formed by the Titanium Diffusion in Lithium Niobate 315


Designing the waveguide.......................................................................................319

Setting the simulation parameters..........................................................................321


Creating a buried waveguide formed by Titanium Diffusion in Lithium Niobate.....325

Saving the template under a new name.................................................................325

Adding a new profile...............................................................................................325

Creating the top linear waveguide..........................................................................326

Notes ....................................................................................................................................329

Lesson 8: 3D OptiMode solver - COST project Waveguide ................................331

Defin ing the materials for the 3D Mode solver .................................................................331

Defin ing the waveguide ......................................................................................................334

Parameterize the Problem ..................................................................................................336

Add Substrate Layer ...........................................................................................................336

Setting Simulati on Parameters ..........................................................................................338

Modi fy ing the Film Thickness ............................................................................................340

Automat icall y modifying a parameter by scr ip t ...............................................................340

Lesson 9: Create a chip-to-fiber but t coupler ......................................................343

Defining materials and waveguides for the chip-to-fiber butt coupler ....................345


Creating a chip-to-fiber butt coupler.......................................................................347

Editing the Input plane ...........................................................................................349




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References ...........................................................................................................................355

Notes: ...................................................................................................................................356

Lesson 10: Electro-Optic Modulator ....................................................................357

Define Dielectric Materials .....................................................................................359

...............................................................................................................................361

Define Electrode Materials .....................................................................................361

Define Profiles........................................................................................................363

Draw the waveguide...............................................................................................366

Draw the electrodes ...............................................................................................372

Electrostatic Simulation..........................................................................................376

Electro-Optic Simulation.........................................................................................380

References ...........................................................................................................................383

Lesson 11: Integrated optical circuit simulation using OptiBPM and OptiSystem Scattering data export .........................................................................................385

Theoretical Background .....................................................................................................385

Part I OptiBPM.................................................................................................................387

Generating Scattering Data script..........................................................................390

Exporting scattering data .......................................................................................394Creating the arms...................................................................................................395

Part II OptiSystem...........................................................................................................397

Loading the *.s file in OptiSystem ..........................................................................397

Loading the files for both Arms ..............................................................................401

Completing the layout in OptiSystem.....................................................................403

Connecting the components ..................................................................................405

Running the calculation..........................................................................................407

Creating the graph to view the results....................................................................407

Reference:............................................................................................................................409

Notes ....................................................................................................................................410

Lesson 12: Scan the refractive index (RI).............................................................411


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Defining the materials and the channel..................................................................411


Drawing the Linear Waveguide..............................................................................417

Inserting the Input Plane ........................................................................................418

Creating the script..................................................................................................419



Lesson 13: Applying User-defined dif fus ion profiles..........................................425

Developing the user defined prof ile ..................................................................................426

Defining the materials ............................................................................................426

Creating the reference profile.................................................................................427


User defined prof ile.............................................................................................................428

Defining variables...................................................................................................430

Defining functions...................................................................................................430

Defining the user defined profile ............................................................................431

Checking the user defined profile against the reference profile .............................432

Notes ....................................................................................................................................434

Lesson 14: Mach-Zehnder Interferometer switch ................................................435

CAD design of the circuit ...................................................................................................436


Creating the Ti-diffused profile...............................................................................436

Defining the Wafer .................................................................................................438

Creating the device ................................................................................................439

Checking the RI profile of the x-y slice...................................................................442

Defining the electrode region .................................................................................443

Defining the input plane and simulation parameters.............................................. 447


Creating a script.....................................................................................................450

Lesson 15: Step index fiber modes .......................................................................453


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Step index fiber ...................................................................................................................454


Creating the SMF-28 fiber profile...........................................................................455


Creating the linear waveguide................................................................................458


Viewing the refractive index distribution (X-Y cut)..................................................461

Calculating the mode .............................................................................................463

Lesson 16: Graded index fiber modes ..................................................................467

Graded Index - Core fiber ..................................................................................................467

Creating the material..............................................................................................470

Defining user variables...........................................................................................470

Defining user functions...........................................................................................471

Defining the User defined profile............................................................................473


Creating the linear waveguide fiber........................................................................476


Viewing the refractive index distribution (X-Y cut)..................................................477


Notes ....................................................................................................................................482

Lesson 17: Modal analysis of an anisotropic buried waveguide .......................483


Creating the profile.................................................................................................486


Creating the linear waveguide................................................................................490


Viewing the elements of the permittivity (XY cut)...................................................493


Computing the magnetic field.................................................................................500

References: ..........................................................................................................................500


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Lesson 18: BPM analysis of an anisotropic bur ied waveguide..........................501

Before you start th is lesson ...............................................................................................501

Setting the simulat ion parameters ....................................................................................502

Viewing the elements of the permi tti vity (XY cut) ............................................................503

Setting the field in the input plane.....................................................................................505

References: ..........................................................................................................................509


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11

Technical Background

The following section contains information on the following topics:

Beam Propagation Method (BPM)

Conformal Mapping Regions

Diffusion in Lithium Niobate

Scattering Data

Non-linear BPM algorithm

Vectoral Beam Propagation for Anisotropic Waveguides

Vectoral Modal Analysis for Anisotropic Waveguide


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Notes:


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BEAMPROPAGATIONMETHOD(BPM)

13

Beam Propagation Method (BPM)

The finite difference beam propagating method (BPM) is one of the most powerfultechniques to investigate linear and nonlinear lightwave propagation phenomena inaxially varying waveguides such as curvilinear directional couplers, branching andcombining waveguides, S-shaped bent waveguides, and tapered waveguides. BPMis also quite important for the analysis of ultra short light pulse propagation in opticalfibers [9].

Like the Finite Difference Time Domain method, finite difference BPM solvesMaxwell's equations by using finite differences in place of partial derivatives. In thissense BPM is computational intensive, and able to accurately model a very widerange of devices. It differs from a full and direct solution of the equations as found inthe Finite Difference Time Domain Method in two ways. The first is that BPM is doneentirely in the frequency domain, and as such only weak non-linearities can be

modelled. The second is in the use of a slowly varying envelope approximation in theparaxial direction. In BPM, it is assumed the device has an optical axis, and that mostof the light travels in this direction, or at least approximately in this direction (paraxialapproximation). In OptiBPM, as with the majority of literature on the subject, this axisis taken to be , the third space co-ordinate. Many practical optical devices arenaturally aligned close to a single direction, so once this is associated with the axis,the slowly varying approximation can be applied.

Slowly Varying Envelope Approximation

Suppose is an electric or magnetic component of the optical electromagnetic field.This component is a periodic (harmonic) function of position, it changes most rapidlyalong the optical axis, , and has a period that is on the order of the optical

wavelength. The slowly varying approximation involves replacing the quickly varyingcomponent, , with a slowly varying one,

where , and is known as the Reference Index. If the light is travellingmostly parallel to the axis (paraxial approximation), and is monochromatic(wavelength ), then it should be possible to select a reference index whichmakes a slowly varying function in all three directions, , , and . If is slowlyvarying, the requirements on the mesh to represent derivatives by finite differencesare relaxed. It is possible to choose fewer mesh points to improve the speed of thecalculation without compromising the accuracy too much. This is the main reason whyBPM can do accurate calculations of light propagation using step sizes many times

larger than the optical wavelength, especially in the direction.

Equation 1can also be interpreted to give some indication about what will be requiredfrom the mesh, it must be sufficiently fine to approximate spatial derivatives ofaccurately by using finite differences. Since varies much more slowly than ,significant savings in calculation time can be obtained. On the other hand, if theproblem cannot be reduced so simply, then the will need to vary more quickly, and

(1)

zz

z

x y z, ,( ) x y z, ,( ) jkn0z( )exp=

k 2 = n0z

n0 x y z

z


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14

the changes must be taken up by making the mesh finer (smaller , , and shorterpropagation step ). For many practical waveguides, the refractive index contrastin the transverse plane ( plane) is small, which means the variation of the field

in the transverse plane is slow, compared to the wavelength. On the other hand,sometimes there are large index contrasts in the transverse plane, and a finer mesh

in , , or both axes might be needed. Another case in which the effectiveness ofEquation 1is compromised is when some of the light deviates from the direction ofthe axis. This is another case in which faster variation in is unavoidable, andsometimes a finer mesh or Wide Angle methods are necessary.

In any case, it is recommended to experiment with several different meshes appliedto the same problem, and to compare the results. Usually, the same results are foundin all cases, but failing this, the results can be analyzed for trends. Eventually asuitable range of mesh parameters can be found which gives consistent results. InOptiBPM, it is very easy to apply many different meshes to the same problem andcompare the results.

Sometimes a single value of reference index will not reduce the variation everywhere,

but will follow the optical field well in one region of the device. Another region mayfollow a different Reference Index. For example, in a star coupler, a waveguideconnects to a free propagation region. The reference index for the waveguide is thewaveguide's modal index, but the reference index for the coupler would be the slabwaveguide's modal index. The best solution in this case is to define one referenceindex for one region and a different one for another region. This is easily done inOptiBPM.

x yz

x y

x y

z


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15

Differential Equations of BPM

In this section we show the derivation for the differential equations found in BPM. Ofcourse, more complete accounts of this material can be found elsewhere (seeReferences [1]- [8]). This section exists for the convenience of the user of OptiBPM,

to define important terms and illustrate the nature of the different levels ofapproximation.

The derivation starts with Maxwell's equations for electromagnetic fields in acontinuous medium in the frequency domain. For optical materials it is usuallypermittivity which differs from the vacuum level. The permeability is unchanged, andso the electromagnetic field is influenced by the device geometry because therefractive index is a function of position. The refractive index distributionenters Maxwell's equations as a coefficient

and no charges are present

The magnetic field can be eliminated by taking the curl of Equation 2and usingEquation 3.

where .

By applying the vector identity

Equation 6becomes

BPM has a bias towards the axis, so it is natural to treat the component of bothand differently than the transverse components, and .

(2)

(3)

(4)

(5)

(6)

(7)

(8)

n x y z, ,( )

E j0H=

H j0n2

x y z, ,( )E=

n2E( ) 0=

H 0 .=

E k2n2E=

k 00=

( . ) 2= .

2E k2n2E+ E( ).=

z zE x y


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16

Writing

the transverse components of Equation 8can be written

The divergence Equation 4can be used to eliminate the longitudinal term above.

Equations (9)and (10)can be used to separate the transverse and longitudinal termsin Equation 4

If the refractive index is changing slowly along , then the second term can beneglected. Then the second term on the right hand side of Equation 11can bereplaced with the divergence term of Equation 12.

Using the chain rule on the second divergence term on the right hand side gives aterm that cancels the first divergence term, leaving a single term on the right handside. Also, on the left side, the derivative can be written explicitly

(9)

(10)

(11)

(12)

(13)

(14)

E Et zEz+=

t zz-----+=

2Et k2n

2Et+ t t Et

Ezz--------+

.=

t n2Et( )

n2

z--------Ez n

2Ezz--------+ + 0 .=

z

2Et k2n

2Et+ t t Et

1

n2

----t n( 2 Et)=

z

t2

Et2Etz2---------- k

2

n2

Et+ + t1

n2---- ( tn

2

) Et .=


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17

At this point the slowly varying envelope approximation is used, the field componentsExandEyare varying quickly in the direction, but are probably varying slowly inand . Therefore the substitution

is likely to result in a function which is slowly varying in all directions,compared to the wavelength of the light. Of course, a suitable value of ReferenceIndex needs to be used, and often OptiBPM can find by itself an appropriatenumber. However, sometimes OptiBPM cannot find a good number, and theReference Index needs to be adjusted manually. It is easy to see if the ReferenceIndex has been set correctly after the simulation is finished. A good value will resultin slow variation of the phase of the solution with respect to .

Substituting Equation 15into Equation 14

If the reference index has been set correctly then the first term will be much smallerthan the second, and it is safe to neglect the first term. The remaining equation is offirst order in . It is customary to collect the transverse differential operators on theright hand side,

where the components of the operator are

(15)

(16)

(17)

(18)

z xy

Et x y z, ,( ) et x y z, ,( ) jn 0kz( )exp=etx y z, ,( )

n0

et x y z, ,( ) z

2etz2--------- 2jkn

0

etz------- k

2

n( 2

n02

)et t2

et t1

n2---- tn

2

( ) et+ + + 0 .=

z

2jkn0z-----

ex

ey

Pxx

Pyx

Pxy

Pyy

ex

ey=

P

Pxx

x-----

1

n2

----x-----n

2. 2

y2-------+= k

2n( 2 n0

2 )+

Pxy

x-----

1

n2

----y-----n

2. 2

xy------------=

Pyx

y-----

1

n

2----

x-----n

2.=2

yx------------

Pyy

y-----

1

n2

----y-----n

2. 2

x2-------+= k

2n( 2 n0

2 ).+


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19

Semi-Vector and Scalar BPM

The above system of equations for BPM is called the Full-Vector form, as it includesboth transverse components of the field. Often it is not necessary to have both fieldcomponents in the simulation. If it is known that the device does not change the

polarization of light, then it is sufficient to model one polarization at a time (or onlymodel the expected polarization). In these cases the contribution of to and viceversa can be ignored, and the governing equation is

for Semi-Vector TE and

for Semi-Vector TM. In Semi-Vector BPM, one can notice a difference in thepropagation of light depending on whether the initial light is polarized parallel orperpendicular to the device substrate. However, once polarized horizontally, forexample, the model does not allow the horizontal light to transfer into a verticalpolarization.

On the other hand, for structures with a very low index contrast, it won't make muchdifference in the order of application of operators like and . In this case, theoperators commute, and both P

xxand P

yyreduce to the scalar operator:

(19)

(20)

(21)

ey ex

2jkn0exz------- Pxx ex=

2jkn0eyz------- Pyy ey=

x n2

P 2

x2-------

2

y2------- k

2n

2n0

2( ).+ +=


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21

often realizes the most stable method. In Equation 24, it is possible to show (beforethe introduction of lossy boundary conditions) that the norm of the operator on the leftis the same as the operator on the right. Therefore the application of many stepsshould lead to no change in the norm of the solution vector . This guarantees thestability of the method, at least in the sense of conservation of energy.

et


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27

conductivities of PML, respectively. The modified differential operator used inEquation 33and Equation 34is defined as

with

where , and are the unit vectors in the , ,and directions, respectively,and the values of are summarized in Table 1.

Figure 2 Optical waveguide sur rounded by PML

(38)

(39)

(40)

'

' 't z'z+=

't xsx

x----- ysy

y-----+=

'z sz-----=

x y, z x y zsx


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29

The parameter is written as

where are the free-space wavelength, the PML thickness, theposition of the PML surface, and the theoretical reflection coefficient, respectively.Here, the PML is terminated with the perfect electric or magnetic conductor for TE orTM mode, respectively. Usually, a parabolic is assumed for the conductivity, .

(45)

s

s

1 in non PML region

1 j 1+( )

4dn---------------------

d---

1

Rt---- ln in PML region

=

2c d x0 , , ,=

m 2=


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35

By substituting Equation 66, Equation 67, and Equation 68into Equation 64, we get:

Thus, using FD scheme into Equation 50for TM modes, we get:

with

(69)

(70)

(71)

(72)

(73)

(74)

rs----

x-----

1

sr------

Hyx

--------- 4

x2

--------r i( )s i( )-----------

Hy i 1( )s i 1( ) s i( )r i( ) r i 1( )+ +-----------------------------------------------------------------------_

1

s x h+( )rx h+( )-----------------------------------------

+1

s x h( )rx h( )---------------------------------------- Hy x( )

Hy x 2h )+(s x h+( )r x h+( )-----------------------------------------

+

+Hy i 1+( )

s i( ) s i 1+( )+( ) r i( ) + r i 1+( )( )----------------------------------------------------------------------------------.

2k0nre fHyz

--------- 2Hy

z2----------- PHy+=

PHyB

x2

--------Hy i 1( ) C

x2

--------Hy i( ) F

x2

--------Hy i 1+( )+ +

B 4r i( )

s i( )--------------

1

s i 1( ) s i( )+( ) r i( ) r i 1( )+( )-------------------------------------------------------------------------------=

C 4r i( )

s i( )--------------

1

s i( ) s i 1+( )+( ) r i( ) + r i 1+( )( )----------------------------------------------------------------------------------=

+1

s i( ) s i 1( )+( ) ( r i( ) r i 1( ) )+---------------------------------------------------------------------------------- + k0

2r i( ) nre f

2( )

F 4r i( )

s i( )--------------

1

s i( ) s i 1+( )+( ) r i( ) + r i 1+( )( )---------------------------------------------------------------------------------- .=


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41

a) Discretization of the weak di fferential equation

The discretization of Equation 87to a linear set of equation is done by introducing anexpansion for and then making appropriate choices for the weighting functions

. We chose the linear representation

where are the unknown coefficients of the expansion. When , wedefine linear elements. In this case, the basis functions for element are defined as

Figure 5 Nodal expansion function for ethfunctions considering linear approximation

The basis functions have unit magnitude at one node and vanish at all others withlinear variation between the nodes. When ,we have quadratic elements thatare also known as second order elements. Each element has three nodes, one of twoendpoints, and the third is usually placed at the center of the element. Within eachelement, the unknown function is approximated as a quadratic function

(88)

(89)

(90)

(91)

x( )Wm x( )

jex( ) Nj

ex( )j

e

j 1=

eo

=

jex( ) eo 2=

e

N1ex( )

x2e

x

x2e

x1e

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

N2e

x( ) x x1

e

x2e

x1e

---------------.=

eo 3=

ex( ) ae bex cex2.+ +=


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42

Enforcing Equation 91at the three nodes of the element yields:

Solving for ae, be, and ceand substituting them back into Equation 91, we obtain

where the interpolation or expansion functions are given by

(92)

(93)

(94)

1e

ae

bex1

ec

ex1

e( )2

+ +=

2e

ae

bex2

ec

ex2

e( )2

+ +=

3e

ae

bex3

ec

ex3

e( )2.+ +=

e

x( ) Njex( )j

e

j 1=

eo 3=

=

N1e x x 2

e( ) x x 3e( )

x1e

x 2e

( ) x( 1e

x3e )

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

N2e x x 1

e( ) x x 3e( )

x2e

x 1e

( ) x( 2e

x3e )

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


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43

Figure 6 The nodal expansion function for ethfunctions considering quadratic approximation

(95)N3e x x 1

e( ) x x 2e( )

x3e

x 1e

( ) x( 3e

x2e )

-----------------------------------------.=


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44

When the expansion Equation 92is substituted into Equation 87we get

The terms in the brackets are due to contributions from endpoints of the domain andtheir evaluation is subject to the specific boundary conditions. This equation nowexplicitly shows how the boundary conditions enter into the construction of the linearsystem. Hereon, we will refer to their contributions as endpoints since we have not yetspecified the type of boundary condition to be imposed.

We are now ready to make different choices for the weighting function to generate asystem of linear equations for the solution of . As stated earlier, this step is alsoreferred to as testing and Galerkin's method is usually employed in the finite elementmethod. Specifically, we choose and for each of these testing orweighting functions a single linear equation is generated.

(96)

k02

qs nre f2

( )Wm x( )Nie

x( )

p

s---

dWm x( )dx------------------

dNiex( )

dx----------------- xie

dx1

e

x2e

i 1=

eo

e 1=

Ne

+ psNiex( )Wm x( )dx

2ie

z2---------- 2jk0nre f i

epsNi

ex( )Wm x( ) x

ie

z--------d

x1e

x2e

j 1=

eo

x

1e

x2e

j 1=

eo

pWm x( )d

dx------

x xf

=

pWm x( )d

dx------

x x0=

= 0 .

n{ }

Wm x( ) Njex( )=


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47

Here .

b) Boundary Condition

The endpoint contributions appear only when or and vanish when theNeumann

or Dirichlet conditions are imposed.

To set the TBC as defined in Transparent Boundary Condition on page 23, wedefine:

Here,

where the real parts of and must be restricted to be positive to ensure onlyradiation outflow.

For Dirichlet, Neumann, and TBC boundary conditions, we set . For PML, weintroduce the parameter as defined by Equation 45.

(105)

(106)

(107)

(108)

I2Ni

ed

xd--------

Nje

d

xd-------- x

1

3le

------7 8 1

8 16 8

1 8 7

.=d

e

=

le x2e

x1e

=

e 1= e m=

d

dx------

x 0=

d

dx------

x xf

=

0==

x 0= x xf

==

ge

jk1 ife 1=

jkfp ife Ne=

0 otherwise.

=

k1 i, j 1

x2 i, x1 i,--------------------

1 i,

2 i,-------

ln=

k1 i, j 1

xf i, xf 1 i,------------------------

f i,f i,-------

ln=

k1 i, km i,

s 1=s


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48

Wide-Angle Beam Propagation Method based on Pad Approximant

Operators

Here, we follow the Pad approximant approach to get the wide angle beam

propagation [31].It is worth it to point out that expansion via Pad is more accurate than Taylorexpansion for the same order of terms. When Pad is employed, larger angles, higherindex contrast and more complex mode interference can be analyzed in both guidedwave and free space as the Pad order increases. See below the guidelines for usingthe Pad technique:

We may formally rewrite Equation 50in the form

Equation 109can be reduced to the following expression:

where and are both polynomials of the operator .

Differentiating Equation 110based on FD scheme, we get

The parameter is introduced to control the finite difference scheme. The range ofis between 0and 1. We can have standard implicit scheme with , explicitscheme with and Crank-Nicolson scheme with . For Crank-Nicolson scheme the propagating field does not diverge as the beam propagates.

The numerator of the term on the right-hand side of Equation 111can be factorized as

where the coefficients can be obtained by solving the algebraicequations

(109)

(110)

(111)

(112)

(113)

z------ j

P 2k0 nre f1 j 2k0 nre f( ) z( )+-----------------------------------------------------.=

z------ j

N

D----,=

N D P

l 1+ D j z 1 ( )N

D j zN+----------------------------------------

l.=

0= 1= 0.5=

AnPN

BnPN 1

CnPN 2 + + +( ) 1 cnP+( ) 1 c2P+( ) 1 c1P+( )=

c1 c2 cn, , ,

D j z 1 ( )N( ) An

PN

Bn

PN 1

Cn

PN 2 + + +( ) 0 .= =


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49

The denominator of the term on the right-hand side of Equation 111can be factorizedas

where the coefficients can be obtained by solving the algebraic equation

Here,

Thus, the unknown field at is related to the known field asfollows

or

In general, determination of the 's and 's requires the one-time solution of an th-order complex algebraic equation.

The explicit expressions for the corresponding Wide-angle (WA) orders using Padapproximant operator are shown below.

The term

in Equation 109can be taken into account quite effectively by using the recursion

relation:

(114)

(115)

(116)

(117)

(118)

AdPN

BdPN 1

CdPN 2 + +( ) 1 dnP+( ) 1 d2P+( ) 1 d1P+( )=

d1 d2 dn, ,

D j zN+ di Pi

i 0=

N

0 .= =

c0 d0 P 0 1= = =

l 1+

z z+ lat z

l 1+ AnP

NBnP

N 1CnP

N 2 + + +

AdPN

BdPN 1

CdPN 2 + + +

------------------------------------------------------------------------l

=

l 1+ 1 cnP+( ) 1 c2P+( ) 1 c1P+( )

1 dnP+( ) 1 d2P+( ) 1 d1P+( )----------------------------------------------------------------------------

l.=

c d n

2z2--------

z-----

n

j P 2k0nre f

1 j 2 k0nre f( ) z( )n 1+------------------------------------------------------------- .=


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50

If Equation 118is now used to replace the derivative in the denominator ofEquation 118we can get the following explicit expressions for several wide-angle(WA) orders.

Fresnel Approximation (Pad 0thorder)

For paraxial approximant, we get the following equation:

here ,

From Equation 119, we get

and therefore

where

(119)

(120)

(121)

(122)

(123)

(124)

(125)

z

z-----

0

j P a

1 j

a---

z-----

1

+

------------------------ jP

a---= =

a 2k0nre f=z-----

1

0=

D 1,=

N P a=

D j z 1 ( )N 1 j z 1 ( )P

a--- 1 AnP+= =

D j zN+ 1 j zP

a---+ 1 AnP+= =

An j z 1 ( )

a------------------------=

Ad j za------.=


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54

Wide Angle (WA), Pad(2,2)

Using recurrence formula Equation 118for Pad(2,2) we get:

Here .

Using Equation 144into Equation 143, we get:

From Equation 145, we get:

Thus,

(143)

(144)

(145)

(146)

(147)

(148)

z----- 3j

P a

1 j a( ) z( ) 2+--------------------------------------------=

z-----

2

j

P

a---

P2

a3

-----+

1 2P

a2

-------+

---------------- .=

a 2k0nre f=

z-----

3

j

P

a---

2

a3

----P2

+

I 3P

a2

------- P

2

a4

-----+ +

---------------------------.=

N P

a---

2

a3

----P2

+=

D 1 3P

a2

------- P

2

a4

-----.+ +=

D j z 1 ( )N 1 3a

2---- j z

1 ( )a

----------------- P+=

+1

a4

---- 2 z1

a3

----------------------j P2 1= BnP AnP

2+ +


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55

and,

where:

Thus, the unknown field is related to the known field asfollows

Multistep Method

In order to solve Equation 154, we applied the multistep method developed byHadley.

First, we rewrite Equation 154as

(149)

(150)

(151)

(152)

(153)

(154)

(155)

D j zN 1 3

a2

---- j z

a---+

P + +=+

1

a4

---- 2 z

a3

-------------j+ P2 1= BdP AdP

2+ +

An1

a4

---- 2 z 1 ( )

a3

---------------------------j ,=

Bn 3a

2---- z 1 ( ) a

------------------------j ,=

Ad1

a4

---- 2 z

a3

-------------j+ ,=

Bd3

a2

---- z

a

----------j.+=

l 1+

atz z+ latz

l 1+ 1 BnP AnP

2+ +

1 BdP AdP2

+ +-------------------------------------

l 1 c2P+( ) 1 c1P+( )1 d2P+( ) 1 d1P+( )

----------------------------------------------l.= =

1 d2P+

1 c2P+------------------

l 1+ 1 c1P+

1 d1P+------------------

l.=


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56

Then, defining the field as

We rewrite Equation 154as

Since is known, we can obtain by solving:

If we consider FEM, we get,

Thus, we can rewrite Equation 158as

here

Using , we rewrite Equation 156as

If we are using FEM, we get,

Solving Equation 162or Equation 162, we can obtain the unknown field . It isapparent from the form of Equation 154that an th-order Pad propagator may bedecomposed into an -step algorithm for which the th partial step takes the form

when .

The run time for an th-order propagator is obviously times the paraxial run time.Therefore, for Pad (2,2) we follow the steps:

(156)

(157)

(158)

(159)

(160)

(161)

(162)

(163)

(164)

l 1 2+

l 1 2+ 1 d2P+

1 c2P+------------------

l 1+.=

l 1 2+ 1 c1P+

1 d1P+------------------

l.=

l

l 1 2+

1 d1P+( )l 1 2+

1 c1P+( )l.=

P Me

[ ] 1

Ke

[ ] k02

nre f2

Me

[ ]( ).=

Me[ ] d1 L

e[ ]+( )l 1 2+ Me[ ] c1 Le[ ]+( )l=

Le[ ] Ke[ ] k0

2nre f

2M

e[ ].=

l 1 2+

1 d2P+( )l 1+ 1 c2P+( )l 1 2+ .=

Me[ ] d2 L

e[ ]+ )l 1+ Me[ ] c2 Le[ ]+( )l 1 2+ .=

l 1+

nn i

l i n+ 1 ciP+

1 diP+

----------------- l i 1( ) n+

=

i 1 2 n, , ,=

n n


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59

Thus, the unknown field is related to the known field asfollows

Multistep method

In order to solve Equation 183, we use the Multistep method, that is, the unknown fieldcan be obtained from the known field by successively solving Equation 164

for . Therefore for Pad(3,3) we follow the steps:

1 Using , compute considering the linear system:

In case of using FEM solve,

2 Using , compute considering the linear system:

In case of using FEM compute,

(180)

(181)

(182)

(183)

(184)

(185)

(186)

(187)

Ad1

a6

---- 3j z

a5

---------------+=

Bd6

a4

---- 4j z

a3

---------------+=

Cd5

a2

---- j za

----------+ .=

l 1+

atz z+ latz

l 1+ 1 CnP BnP

2AnP

3+ + +

1 CdP BdP2

AdP3

+ + +-------------------------------------------------------

l 1 c3P+( ) 1 c2P+( ) 1 c1P+( )1 d3P+( ) 1 d2P+( ) 1 d1P+( )

----------------------------------------------------------------------l.= =

l 1+

l

i 1 2 3, ,=

l

l 1 3+

1 d1P+( )l 1 3+

1 c1P+( )l.=

Me[ ] d1 L

e[ ]+( )l 1 3+ Me[ ] c1 Le[ ]+( )l.=

l 1 3+

l 2 3+

1 d2P+( )l 2 3+

1 c2P+( )l 1 3+

.=

Me[ ] d2 L

e[ ]+( )l 2 3+ Me[ ] c2 Le[ ]+( )l 1 3+ .=


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60

3 Finally, using , compute the following linear system to obtain theunknown field :


(188)

(189)

l 2 3+

l 1+

atz z+

1 d3P+( )l 1+

1 c3P+( )l 2 3+

.=

Me[ ] d3 L

e[ ]+( )l 1+ Me[ ] c3 Le[ ]+( )l 2 3+ .=


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62

Thus,

and

here

(195)

(196)

(197)

(198)

(199)

(200)

D j z 1 ( )N 1 7a

2---- j

z 1 ( )a

------------------------ P+=

+15

a4

------ j6 z 1 ( )

a3

--------------------------- P2+ 10

a6

------ j10 z 1 ( )

a5

------------------------------ P3

+1

a8

---- j4 z 1 ( )

a7

--------------------------- P4=1 DnP CnP

2BnP

3AnP

4+ + + +

D j zN 1 7

a2

---- j za

----------+ P 15

a4

------ j6 z

a3

-------------+ P2+ +=+

+10

a6

------ j10 z

a5

----------------+ P3 1

a8

---- j4 z

a7

-------------+ P4+

= 1 DdP CdP2

BdP3

AdP4

+ + + +

An1

a8

---- j4 z 1 ( )

a7

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

Bn10

a6

------ 10j z 1 ( )

a5

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

Cn15

a4

------ 6j z 1 ( )

a3

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

Dn7

a2

---- j z 1 ( )a

--------------------------.=


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63

Thus, the unknown field is related to the unknown field asfollows:

Multistep Method

In order to solve Equation 201, we use the multistep method, that is, the unknown fieldis obtained with the following the steps:

1 Compute solving the linear system:






(201)

(202)

(203)

(204)

(205)

(206)

(207)

l 1+

atz z+ latz

l 1+ 1 DdP CdP

2BdP

3AdP

4+ + + +

1 DnP CnP

2

BnP

3

AnP

4

+ + + +

--------------------------------------------------------------------------l

=

=1 c4P+( ) 1 c3P+( ) 1 c2P+( ) 1 c1P+( )1 d4P+( ) 1 d3P+( ) 1 d2P+( ) 1 d1P+( )

----------------------------------------------------------------------------------------------l.

l 1+

l 1 4+

1 d1P+( )l 1 4+ 1 c1P+( )l.=

Me[ ] d1 L

e[ ]+( )l 1 4+ Me[ ] c1 Le[ ]+( )l.=

l 1 2+

1 d2P+( )l 1 2+

1 c2P+( )l 1 4+

.=

Me[ ] d2 L

e[ ]+( )l 1 2+ Me[ ] c2 Le[ ]+( )l 1 4+ .=

l 3 4+

1 d3P+( )l 3 4+

1 c3P+( )l 1 2+

.=

Me[ ] d3 L

e[ ]+( )l 3 4+ Me[ ] c3 Le[ ]+( )l 1 2+ .=


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64

4 Finally, knowing , compute the known field solvinglinear system:


The advantage of the multistep is that the matrix equation to be solved in each stepis the same size as the Fresnel equation and for 2D problems is tridiagonal when weconsider Finite Difference method (FD) or Finite Element Method (FEM) consideringlinear element.

(208)

(209)

l 3 4+

l 1+

atz zl+

1 d4P+( )l 1+

1 c4P+( )l 3 4+

.=

Me[ ] d4 L

e[ ]+( )l 1+ Me[ ] c4 Le[ ]+( )l 3 4+=


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DIFFUSIONINLITHIUMNIOBATE

72

The diffusion-induced index change is described by the product of the dispersionfactor and the distribution function [1],

where the dispersion factor is different for the ordinary and extraordinary cases( in m)

In turn, the diffusion induced distribution function is a function of the distributionconstant , the dopant concentration profile and the distribution powerfactor ,

where the distribution constants

and

and the distribution power factors

and

are different for the ordinary and extraordinary cases. They are fitted from publishedexperimental data [1].

The concentration profile can be derived following the classical diffusion theory[2], [3]. The profile has a bell-shaped form

In the lateral direction, that is horizontal to the crystal surface, the profile ischaracterized by the combination of error functions. In the crystal in-depth direction,that is vertical to the crystal surface, it has a Gaussian shape. The profile's parametersinclude the profile constant , the dopant stripe width before diffusion , thehorizontal (lateral) diffusion length , and the vertical (in depth) diffusion length .

(4)

(5)

(6)

(7)

di ( ) hi x y,( )

ni x y, ,( ) di ( )hi x y,( ),= i o e.,=

do ( ) 0.67

2

2

0.13---------------------= de ( )

0.8392

2

0.0645---------------------------=

Fi c x y,( )i

hi x y,( ) Fic x y,( )[ ]

i= i o e,=

Fo 1.3 10 25

cm3=

Fe 1.2 10 23

cm3=

o 0.55=

e 1.00=

c x y,( ) c0 erf w

2D

H

---------- 1 2x

w

------+ erf w

2D

H

---------- 1 2x

w

------ +

y2

DV2

------ exp=

c0 wDH DV


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DIFFUSIONINLITHIUMNIOBATE

76

The distribution function in the vertical direction is defined by the hyper-Gaussiandependence [4], [5]

where is the power of the hyper-Gaussian distr ibuti on, usually ranging between10 and 23, and is the diffusion depth after annealingor effective guide depth.The diffusion depth as well as the maximum refractive index difference afterannealing might have much different values then the ones before annealing. Itis up to the user to define these parameters which can be case dependent. As anexample, we quote empirical relationships to calculate and for annealing at200 and 400C taken from [4], [5]:

where is the initial PE depth, is the initial change of the refractive indexafter proton exchange and are the fitting parameters (Table 3). Therefractive index distribution is strongly dependent on the post-exchange time andthe post-exchange temperature .

In a case when the predefined profiles are found not sufficient, user has the option todefine his own refractive index profiles by using the User Function Profiles in Profile

designer.

Table 2

Table 3

(16)

(17)

Taken from [6] Toluic Acid Benzoic Acid

Temperature range [oC] 109-263 122-249

Diffusion constant [ m2/hr] 7.02 107 7.36 109

Activation energy [kJ/mol] 75.58 94

[oC] b [hrc] c [] p [hrq] q []

300 0.7031 0.0754 0.2325 0.1033

400 1.2884 0.7577 0.3749 0.6574

f y( ) y DV( )a

[ ]exp=

aDV

DVnes

nes DV

DV

DV0( )--------- 1 bta

c

= 1

nes

nes0( )----------- pta

q

=

V

0( )nes

0( )b c p q, , ,

taTa

D0

Q

Ta


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ELECTROOPTICEFFECT

80

References

[1] E. Strake, G.P. Bava, and I. Montrosset: Modes of channel waveguides: A novel quasi-analyticaltechnique in comparison with scalar finite-element method, J. Light. Technol.6, (1988): 1126-1135.

[2] J. Crank: The mathematics of diffusion. Oxford University Press, New York, (1975).

[3] G.B. Hocker and W.K. Burns: Mode dispersion in diffused channel waveguides by the effectiveindex method,Appl. Opt. 16, (1977): 113-118.

[4] J. Nikolopoulos and G.L. Yip: Accurate modeling of the index profile in annealed proton-exchanged LiNbO3, Proc. SPIE1583, Integrated Optical Circuits, (1991): 71-82.

[5] J. Nikolopoulos and G.L. Yip: Theoretical modeling and characterization of annealed proton-exchanged planar waveguides in z-cut LiNbo3, J. Light. Technol.9, (1991): 864-870.

[6] E.Y.B. Pun, K.K. Loi, and P.S. Chung: Experimental studies of proton-exchanged waveguidesin Lithium Niobate using toluic acid, Proc. SPIE1583, Integrated Optical Circuits, (1991): 64-70.

[7] J.L. Jackel: Proton exchange: past, present, and future, Proc. SPIE1583, Integrated Optical

Circuits, (1991): 54-63.

[8] H. Jin, M. Belanger, and Z. Jakubczyk: General analysis of electrodes in integrated-optics andelectrooptic devices, J. Quant. Electron.27, (1991): 243-251.

[9] H. Jin, R. Vahldieck, M. Belanger, and Z. Jakubczyk: A mode projecting method for the quasi-static analysis of electrooptic device electrodes considering finite metallization thickness andanisotropic substrate, J. Quant. Electron.27, (1991): 2306-2314.

[10] R. Syms and J. Cozens: Optical guided waves and devices, Section 3.7, London, McGraw-Hill,(1992).

[11] A. Yariv: Optical Electronics in Modern Communications. 5th edition, Oxford University Press,(1996).


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SCATTERINGDATA

84

Figure 3 Basic four -channel Mach-Zehnder multi /demultiplexer

The device can be fully analyzed by the BPM, however we can show it is notnecessary. The distances among the input ports (named with the respect to

multiplexed wavelengths) are typically 0.25 mm and the device is several millimeterslong, the circuit area is quite large. Note, the speed of the BPM simulator isapproximately proportional to the layout area. Further, we can see that the layout is,say, somehow coarse, i.e. we can declare some places where the field exists in theevanescent form and the field is negligible. Running the simulation through theseplaces is inefficient, however we cannot exclude these dead places from therectangular simulation window. Our new approach proposes an elegant solution (seethe pertinent paragraph below).

Multidirectional BPM device

The second category is very close to the previous one. The structure on the figurebelow is another arrangement of the simple MZI. There are two crucial items on this

example. First, the path difference is relatively big, what was designed by a circularloop instead of a primitive arc as done in the first example. The second item appearingrather crucial when BPM would be considered as the simulation tool, is - the input andoutput ports are mutually in perpendicular directions. As the final remark, the so-called dead space again occupies the majority of space in the simulation window(see Figure 4).


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SCATTERINGDATA

85

Figure 4 Multidirectional BPM device

Devices consisting of the combination of BPM & gratings (Add/drop)

In this category, we shall study the devices, where a grating is present, BPM isunsuitable (see Figure 5). Of course, as it was mentioned we can use CMT, i.e. theOptiGrating product. At this point, we may recognize the final item of our new method.

To analyze some advanced structures by means of several independent techniques,we will have to connect all the particular results using some common way. Moreover,we will need a common environment to do so.

Concerning the functionality of the circuit, two identical Bragg gratings are designedto have the reflection maximum, say, at . For example, the principalparameters of the sine groove grating are as follows: .

Figure 5 Mach-Zehnder interferometer

0 1550nm=L 5mm , n 0.0007==


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SCATTERINGDATA

88

reduced form of it, hence it is called S-data matrixthrough the text). The four-portcoupler is then characterized by the following relation:

Equation 1can be easily described in an oral form. The element is responsiblefor a response from the first input waveguide into the first output waveguide. Similarly,the matrix element relates to the response in the second output waveguide whenthe light was introduced into the first input waveguide. The meaning of the other matrixelements is defined in analogous way. Note, the elements are complex numbers andthey consist of real and imaginary parts. Equation 1has to be repeatedly determinedfor each wavelength run. The relations among the input and output ports takes thegeneral form (for one particular wavelength)

We use the following relations to obtain the desired S-data elements. We aresearching for a response on a unitary input modal field (unitary means the unitamplitude and zero phase additional advance of the modal filed). We have to use thefollowing approximation, because it would be somehow cumbersome to transport acomplete information carrying the output field distribution yielded from the BPMsimulation. Concerning the relative power presence at the output waveguide, we thus

compare the actual final BPM field obtained after the simulation at each outputwaveguide with that one being theoretically present if the single straight waveguidelight propagation considered for the pertinent output waveguide. This can be easilydone by well-known power overlap integral

where one field is the BPM one, while the second is the modal field distribution of

output waveguide (the asterisk means the complex conjugated field). Thedenominator, consisting of the product of two integrals, ensures the normalization ofthe overlap integral from 0 to 1. determines the integration region.

(1)

(2)

(3)

b1

b2

S11S12

S21S22

a1

a2=

S11

S12

b1b2

bM

S11 S12 S1NS21 S22 S2N SM1SM2 SMN

a1a2

aN

=

P

E1E2 xd

2

E12

x E22

xd

d

---------------------------------------- ,=


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SCATTERINGDATA

89

The response with respect to the cumulated phase delay is then given by (both realand imaginary part)

where , is the reference index, is the propagation length. Finally,is an additional angle related to the BPM propagation (this can be observed from

the analyzer). Sometimes in one BPM calculation more than one reference index willbe used. There may be regions having different reference indices. If more then oneregion appears in the layout, we have to characterize these regions separately byparticular lengths and reference indices and the phase delay term in Equation 4must be replaced accordingly as shown schematically in Equation 5

with being the number of regions in the layout. In other words Equation 4simplyadds the phase shift due to the optical length of the wafer. Since the responses weresought for the unit input modal fields, the collected values of for each combination(input/output/wavelength) created according to Equation 2the S-data matrixelements. The exported file is a text file with an '.s ' extension. The first line containstwo integer values: the number of inputs and the number of outputs. All thesubsequent lines represent an individual scattering data matrix for a single

wavelength. They contain the wavelength in microns and all the complex values of theamplitude transmission between each input and output port. The total number ofelements in each row is the number of the inputs times the number of the outputstimes 2 (real and imaginary parts) plus one (wavelength). These lines appear suchthat the wavelength is in increasing order.

Central wavelength approximation

We can, however, employ a more approximate solution which is much faster tocompute. Concerning the wavelength interval, we are mostly dealing with a very smallvicinity of some central wavelength, say. To save more simulation time, we can runthe simulation just for this wavelength to obtain the one set of matrix elements. Let uscall the matrix (the *.sfile will then be just one line with the headings). In most

practical applications, the weighted power distribution coming from the power overlapintegral in an output port varies by a negligible amount. The phase change may becrucial, on the other hand. We can then approximately derive the phase change withrespect to the wavelength. We can approximately write in the vicinity of a centralwavelength

(4)

(5)

(6)

b P j kn0L wg+( )[ ],exp=

k 2 = n0 Lwg

kn0L

n0

L nr

Lr,

r

r

bSij

Sc

c

S k( ) Sc i kc k( )n0L{ },exp


94/515

SCATTERINGDATA

90

with , while where is the actual wavelength. The valuesare the S-data obtained after the ordinary simulation for the central wavelength. In

conclusion, we may need just one wavelength simulation to describe the deviceoptical response. The limitation of the central wavelength approximation is obvious.

The accuracy is decreased with increasing propagation length as well as with thebroader wavelength interval. The approximation will be also suitable for lower valuesof the reference refractive index.

Implementation with OptiSystem

We are now ready to embark into the main part of the paper. Namely, we have alreadyshown we could divide an advanced circuit into smaller parts that are suitable for theBPM analysis. We have also the S-data tool to provide the effective mathematicaldescription of all those particular sub-elements. The last part missing in our accountis to find an environment connecting the sub-elements into the entire circuit. To reachour goal, we used OptiSystem, which provides a very flexible environment forschematic designing and simulation of advanced optical communication systems.

Any complex circuit can be assumed to be an optical system, and OptiSystem can bevery effectively used as the motherboard in the presented study.

The practical implementation is trivial from a customer point of view. At this point, keepin mind that we have the sub-elements characterized in the form of small files.OptiSystem introduces a general component OptiBPM Component NxMthat canload these small files easily.

The component on the left is shown with its default value of 8x8input/output ports. Apparently, we do not need much additionaldescription saying that the loaded component by the *.s filereads firstly the headings, i.e. the number of both ports andautomatically adapts the icon to have the correct number of

ports (visually can be seen in the solved examples below). TheOptiSystem environment also follows the unidirectional of theBPM, this can be seen from the icon by the orientation of the

port arrows (from left to right). We are now able to fully exploit a number of variousOptiSystem facilities to analyze a given circuit. This brings an immense flexibility intothe circuit simulations. Moreover, we can decrease the simulation time comparing thenew approach with a BPM run. We will demonstrate these improvements by solvingthe four examples previously mentioned.

Solutions

The concluding part of our discourse will show both the complexity and the flexibilityof the presented modeling improvements. The OptiSystem schematic layout brings anew vision of the optical circuit behavior. Any advanced photonic circuit may befurther studied as a set consisting of separated fundamental components. As we willalso see, the use of the OptiSystem platform extends the simulation possibilitiesdramatically, because of a possible entering of various photonic and even electronicsimulation tools.

kc 2 c= k 2 = Sc


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Four Channel Mach-Zehnder Multi/Demultiplexer

The realistic layout of the mux/demux example in OptiBPM looks as shown inFigure 8(note the size ratio of 1:10). What first attracts our attention is the presenceof a number of relatively primitive, i.e. straight and curved, waveguides. The layout is

very coarse compared to the entire wafer area. Moreover, the light interactionsactually occur on just a small fraction of the layout. In addition, all six four-portcouplers are identical (see Figure 9).

Figure 8 Mach-Zehnder MUX/DEMUX example in OptiBPMSize ratio 1:10

Let us mark the couplers as shown in Figure 9. At this point we encounter the keypoint of the approach - its division criteria with the respect to the sub-circuitfunctionality. We have to split the circuit into the particular cells. The cells must be selfexisting fully functional units, although the splitting of the whole device must notdisturb its functionality in any way when later composed together by means of theOptiSystem.


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First the most accurate way is to simulate them by BPM similarly to the coupler - letus call it total analysis with respect to this approach. However, the detailed study isnot needed for all waveguides. We would prefer to do so for the waveguides playinga role between the couplers, since we are primarily interested in the phase differencebetween the arms. The waveguides connecting the ports with the core of the device

(i.e. the outermost waveguides) are simple isolated waveguides engaged in modalpropagation in the fundamental mode. After some preliminary inspection we can evenneglect the losses or we can prescribe them analytically. In other words, a user'sexperience and intuitive estimat

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