VERSION 4.4

Introduction toWave Optics Module

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Part number. CM023502

I n t r o d u c t i o n t o t h e W a v e O p t i c s M o d u l e 19982013 COMSOL

Protected by U.S. Patents 7,519,518; 7,596,474; 7,623,991; and 8,457,932. Patents pending.

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Version: November 2013 COMSOL 4.4

Contents

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

The Use of the Wave Optics Module . . . . . . . . . . . . . . . . . . . . . . . . 2

The Wave Optics Module Physics Interfaces . . . . . . . . . . . . . . . . 9

The P

The Mo

Tutoria

Intro

Mode

Resu

Refer | i

hysics Interface List by Space Dimension and Study Type . 13

del Libraries Window . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

l Example: Directional Coupler . . . . . . . . . . . . . . . . . . . . 15

duction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

l Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

lts and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

ence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

ii |

Introduction

The Wave Optics Module is used by engineers and scientists to understand, predict, and design electromagnetic wave propagation and resonance effects in optical applications. Simulations of this kind result in more powerful and efficient products and engineering methods. It allows its users to quickly and accurately predict electromagnetic field distributions, transmission and reflection coefficients, and power dissipation in a proposed design. Compared to traditional prototyping, itpredict entitieexploration ofhazardous.This module cthree-dimensioequations togemodeling capareferred to as WelectromagnetelectromagneteigenfrequencUnder the hooform of MaxwThe equationsedge element preconditioninare presented upower flow, anexpressions of values obtaineThe work flowdefine the geodefine boundasolver, and visDesktop. The default settingThe Wave Opdifferent featuformulations. nonlinear optimodels for verIntroduction | 1

offers the benefits of lower cost and the ability to evaluate and s that are not directly measurable in experiments. It also allows the operating conditions that would destroy a real prototype or be

overs electromagnetic fields and waves in two-dimensional and nal spaces. All modeling formulations are based on Maxwells ther with material laws for propagation in various media. The bilities are accessed via predefined physics interfaces, collectively

ave Optics interfaces, which allow you to set up and solve ic models. The Wave Optics interfaces cover the modeling of ic fields and waves in frequency domain, time domain, y, and mode analysis.d, the Wave Optics interfaces formulate and solve the differential

ells equations together with the initial and boundary conditions. are solved using the finite element method with numerically stable discretization in combination with state-of-the-art algorithms for g and solution of the resulting sparse equation systems. The results sing predefined plots of electric and magnetic fields, S-parameters, d dissipation. You can also display your results as plots of

the physical quantities that you define freely, or as tabulated derived d from the simulation. is straightforward and can be described by the following steps: metry, select materials, select a suitable Wave Optics interface, ry and initial conditions, define the finite element mesh, select a ualize the results. All these steps are accessed from the COMSOL solver selection step is usually carried out automatically using s, which are tuned for each specific Wave Optics interface.tics Modules model library describes the interfaces and their res through tutorial and benchmark examples for the different The library includes models addressing gratings and metamaterials, cs, optical scattering, waveguides and couplers, and benchmark ification and validation of the Wave Optics interfaces.

2 | Introduction

This introduction is intended to give you a jump start in your modeling work. It has examples of the typical use of the Wave Optics Module, a list of the physics interfaces with a short description, and a tutorial example that introduces the modeling workflow.

The Use of the Wave Optics ModuleThe Wave Optics interfaces are used to model electromagnetic fields and waves in optical applications. Typical wavelengths for optical applications are in the nanometer to micrometer range, corresponding to frequencies of the order of hundreds of Tare frequentlyWave Optics siproperties for field distributiwaveguide. Thpropagation. TGaAs.

Figure 1: Electric fwhereas the crossmodel DirectionalHz. A characteristic of optical applications is also that the structures much larger than the wavelength.mulations are often used for determining propagation and coupling different types of waveguide structures. Figure 1 shows the electric on in a directional coupler. A wave is launched into the left e wave couples over to the right waveguide after 2 mm he waveguides consist of ion-bombarded GaAs, surrounded by

ield distribution in a directional coupler. Notice that the propagation length is 2 mm, -sectional area is 12 m by 18 m. From the Wave Optics Module model library Coupler.

Figure 2 shows an example of transient propagation in a nonlinear crystal. The figure shows the total electric field, the sum of the incoming fundamental wave and the generated second harmonic wave, when the wave is located in the middle of the crystal.

Figure 2: The z-coWave Optics Mod

Transient simuinvolving shoris the effect ofIntroduction | 3

mponent of the electric field after 61 fs propagation in a nonlinear crystal. From the ule model library model Second Harmonic Generation.

lations are useful for modeling nonlinear optical processes t optical pulses. Another example of nonlinear optical propagation self-focusing, as shown in Figure 3. Here, the Gaussian beam and

4 | Introduction

the intensity-dependent refractive index forms a self-induced lens in the material that counteracts the spreading effect of diffraction.

Figure 3: The elecintensity-dependen

In Figure 4 anshows the scatis setup using entered as a baMatched Layetric field distribution for a Gaussian beam propagating in a medium with an t refractive index. From the Wave Optics Module model library model Self-Focusing.

d Figure 5, a model from the Wave Optics Module model library tering of an incoming plane wave by a small gold sphere. The model the scattered field formulation, where the incoming plane wave is ckground field. The scattered wave is absorbed by a Perfectly r (PML). Figure 4 shows the volume resistive losses in the gold

sphere.

Figure 4: The volumFrom the Wave OIntroduction | 5

e resistive losses in a small gold sphere, when excited by an incoming plane wave. ptics Module model library model Scattering Nanosphere

6 | Introduction

A Far-Field Domain is used in the model to calculate the far-field pattern of the scattered waves, as shown in Figure 5.

Figure 5: The far-f700 nm.

The Wave Opmodeling incluield radiation pattern in the E-plane (blue) and H-plane (green) when wavelength is

tics Module also offers a comprehensive set of features for 2D ding both source driven wave propagation and mode analysis.

Figure 6 shows mode analysis of a step-index profile optical fiber.

Figure 6: The surfathe Wave Optics M

Both in 2D anexample of a p

Figure 7: Electric fiPlasmonic Wire G

The Wave Opresults, for exaS-parameters cIntroduction | 7

ce plot visualizes the longitudinal component of the electric field in the fiber core. From odule model library model Step Index Fiber Bend.

d 3D, the analysis of periodic structures is popular. Figure 7 is an lane wave incident on a wire grating with a dielectric substrate.

eld norm for TE incidence at /5. From the Wave Optics Module model library model rating.

tics Module has a vast range of tools to evaluate and export the mple, evaluation of far-field and scattering matrices (S-parameters). an be exported in the Touchstone file format.

8 | The Wave O

The Wave Optics Module Physics Interfaces

The Wave Optics physics interfaces are based upon Maxwells equations together with material laws. In the module, these laws of physics are translated by the Wave Optics interfaces to sets of partial differential equations with corresponding initial and boundary conditions.The Wave Optics physics define a number of features. Each feature represents a term or condition in the underlying Maxwell-based formulation and may be defined in a ge3D models), oFigure 8 uses model library tselected Wavenode adds thein a selected gFurthermore, Materials featuand/or refractdielectric. Thethe modeled pNotice that th3 by the WaveElectric Displarelative permitThe simulatiodefault boundis the Perfect EElectric CondPeriodic Condabsorbing wavptics Module Physics Interfaces

ometric entity of the model, such as a domain, boundary, edge (for r point.

the Plasmonic Wire Grating model from the Wave Optics Module o show the Model Builder window and the settings window for the Equation, Electric 1 feature node. The Wave Equation, Electric 1 terms representing Electromagnetic Waves to the model equations eometrical domain in the model.the Wave Equation, Electric 1 feature node may link to the re node to obtain physical properties such as relative permittivity ive indexin this case the refractive index of a user-defined properties, defined by the Dielectric material, can be functions of hysical quantities, such as temperature.e Wave Equation, Electric 1 feature node is overridden for domain Equation, Electric 2 feature. That feature node uses another cement Field model, the Relative Permittivity model, that gets the tivity from Gold, defined below the Materials node.n domain is delimited by boundary condition feature nodes. The ary condition feature for the interfaces in the Wave Optics Module lectric Conductor feature. For the example in Figure 8 the Perfect

uctor 1 feature node is overridden by two Port feature nodes and a ition feature node. The Port features are used for exciting and es and the Periodic Condition relates boundaries with periodicity

conditions (for instance continuity, antiperiodicity, and general Floquet phase relationships).

Figure 8: The Modsection shows the model equations. between the DieleThe Wave Optics Module Physics Interfaces | 9

el Builder (left), and the Wave Equation, Electric settings window (right). The Equation model equations and the terms added by the Wave Equation, Electric 1 node to the The added terms are underlined with a dotted line. The text also explains the link ctric node and the values for the refractive index.

10 | The Wave

Figure 9 shows the Wave Optics physics interfaces as displayed in the Model Wizard for this module.

Figure 9: The Wave Optics Module physics interfaces as displayed in the Model Wizard.

This module imodeling andHeating multiPhysics Interfaoverview of th

ELECTROMAGThe Electromfrequency-domform of point It is used primand structuresthe eigenfrequeigenvalue protransmission liare waveguide

ELECTROMAGThe Electromfrequency-domfield is represerapidly varyingslower spatial vElectromagneoptically largeThe sources caelectric or magpropagation presonant strucare waveguidelaser beam proOptics Module Physics Interfaces

ncludes Wave Optics physics interfaces ( ) for frequency-domain time-domain modeling, respectively. It also includes the Laser physics interface, available under Heat Transfer. Also see The ce List by Space Dimension and Study Type on page 13. A brief e Wave Optics physics interfaces follows.

NETIC WAVES, FREQUENCY DOMAINagnetic Waves, Frequency Domain interface ( ) solves a

ain wave equation for the electric field. The sources can be in the dipoles, line currents, or incident fields on boundaries or domains. arily to model electromagnetic wave propagation in different media . Variants of the formulation solves an eigenvalue problem to find encies of a structure or, at a prescribed frequency, solves an blem to find the propagating modes in waveguides and nes. Some typical applications that are simulated with the interface s, gratings, and scattering from small particles.

NETIC WAVES, BEAM ENVELOPESagnetic Waves, Beam Envelopes interface ( ) solves one or two

ain wave equations for the electric field envelope(s). The electric nted as the product of the solved for electric field envelope and a prescribed phase function. As the electric field envelopes has a ariation than the electric field, a coarse mesh can be used. Thus, the

tic Waves, Beam Envelopes interface is suitable for simulations of structures (structures that are much larger than the wavelength). n be in the form of incident fields on boundaries, surface current, netic fields on boundaries. The interface can be used for

roblems at a fixed frequency and for finding eigenfrequencies in a ture. Some typical applications that are simulated with the interface structures, like directional couplers and fiber Bragg gratings, and pagation.

ELECTROMAGNETIC WAVES, TIME EXPLICITThe Electromagnetic Waves, Time Explicit interface ( ) solves a system of two first-order partial differential equations (Faradays law and Maxwell-Ampres law) for the electric and magnetic fields using the Time Explicit Discontinuous Galerkin method. The sources can be in the form of volumetric electric or magnetic currents or electric surface currents or fields on boundaries. It is used primarily to model electromagnetic wave propagation in linear media. Typical applications involve the transient propagation of electromagnetic pulses.

ELECTROMAGThe Electromaequation for thcurrents, or incelectromagnettime-domain sfor nonlinear melectromagnetmedia.

LASER HEATINThe Laser Heasystems and dewavelength scaBeam Envelopmultiphysics cwaves as a heatthe temperatuelectromagnetCombinationsBeam Envelopinterface, calleare supported The Wave Optics Module Physics Interfaces | 11

NETIC WAVES, TRANSIENTgnetic Waves, Transient interface ( ) solves a time-domain wave e electric field. The sources can be in the form of point dipoles, line ident fields on boundaries or domains. It is used primarily to model ic wave propagation in different media and structures when a olution is requiredfor example, for non-sinusoidal waveforms or edia. Typical applications involve the propagation of

ic pulses and the generation of harmonics in nonlinear optical

G

ting interface ( ) is used to model electromagnetic heating for vices where the electric field amplitude varies slowly on a le. This multiphysics interface adds an Electromagnetic Waves, es interface and a Heat Transfer in Solids interface. The ouplings add the electromagnetic losses from the electromagnetic source, and the electromagnetic material properties can depend on re. The modeling approach is based on the assumption that the ic cycle time is short compared to the thermal time scale. of frequency-domain modeling for the Electromagnetic Waves, es interface and stationary modeling for the Heat Transfer in Solids d frequency-stationary and, similarly, frequency-transient modeling, in 2D and 3D.

12 | The Wave

The Physics Interface List by Space Dimension and Study TypeThe table below list the interfaces available specifically with this module in addition to the COMSOL Multiphysics basic license.

INTERFACE ICON TAG SPACE DIMENSION

PRESET STUDY TYPE

Heat Transfer

Electromagnetic Heating

Laser Heating*

Joule Heating*

Wave Opt

Electromagnetic WFrequency Doma

Electromagnetic WBeam Envelopes

Electromagnetic WExplicit

Electromagnetic WTransient

* This interface isinterfaces and coOptics Module Physics Interfaces

3D, 2D, 2D axisymmetric

frequency-stationary; frequency-transient

all dimensions stationary; time dependent; frequency-stationary; frequency-transient

ics

aves, in

ewfd 3D, 2D, 2D axisymmetric

eigenfrequency; frequency domain; frequency-domain modal; boundary mode analysis; mode analysis

aves, ewbe 3D, 2D, 2D axisymmetric

eigenfrequency; frequency domain; frequency-domain modal; boundary mode analysis

aves, Time teew 3D, 2D, 2D axisymmetric

time dependent

aves, ewt 3D, 2D, 2D axisymmetric

eigenfrequency; time dependent; time dependent modal

a predefined multiphysics coupling that automatically adds all the physics upling features required.

The Model Libraries Window

To open a Wave Optics Module model library model, click Blank Model in the New screen. Then on the Home or Main toolbar click Model Libraries . In the Model Libraries window that opens, expand the Wave Optics Module folder and browse or search the contents. Click Open Model to open the model in COMSOL Multiphysics or click Open PDF Document to read background about the model including the step-by-step inlibrary can hav Full MPH-f

window theexceeds 25Myou positio

Compact Mand solutiosolutions foand to meshversionswupdate youwindow witthe Model LMPH-file isincludes a D

To check all avLibrary ( ) f(Mac and LinuA model fromExample: DireThe Model Libraries Window | 13

structions to build it. The MPH-files in the COMSOL model e two formatsFull MPH-files or Compact MPH-files.iles, including all meshes and solutions. In the Model Libraries se models appear with the icon. If the MPH-files size B, a tip with the text Large file and the file size appears when

n the cursor at the models node in the Model Libraries tree.PH-files with all settings for the model but without built meshes

n data to save space on the DVD (a few MPH-files have no r other reasons). You can open these models to study the settings and re-solve the models. It is also possible to download the full ith meshes and solutionsof most of these models when you

r model library. These models appear in the Model Libraries h the icon. If you position the cursor at a compact model in ibraries window, a No solutions stored message appears. If a full

available for download, the corresponding nodes context menu ownload Full Model item ( ).

ailable Model Libraries updates, select Update COMSOL Model rom the File>Help menu (Windows users) or from the Help menu x users).

the model library is used as a tutorial in this guide. See Tutorial ctional Coupler, which starts on the next page.

14 | Tutorial Ex

Tutorial Example: Directional Coupler

IntroductionDirectional couplers are used for coupling a light wave from one waveguide to another waveguide. By controlling the refractive index in the two waveguides, for instance by heating or current injection, it is possible to control the amount of coupling between the waveguides.

Figure 10: Schemacores and the surrabsorb the waves.waveguides is arouThe waveguide se

Light that proconcentrated wcore, in the clathe core. How(see Figure 10waveguide (aneffective indexsecond waveguget a symmetriindex that is sl

Port 1 and

Port 2

Coreample: Directional Coupler

tic drawing of the waveguide structure. The structure consists of the two waveguide ounding cladding. Port 1 and 2 are used for exciting the waveguides and Port 3 and 4 Notice that the waveguide structure is not drawn to scale. The length of the nd 2 mm, whereas the waveguide cross-section is square with a side length of 3 m.

paration is 3 m.pagates through a dielectric waveguide has most of the power ithin the central core of the waveguide. Outside the waveguide

dding, the electric field decays exponentially with the distance from ever, if you put another waveguide core close to the first waveguide ), that second waveguide will perturb the mode of the first d vice versa). Thus, instead of having two modes with the same , one localized in the first waveguide and the second mode in the ide, the modes and their respective effective indexes split and you

c supermode (see Figure 11 and Figure 13 below), with an effective ightly larger than the effective index of the unperturbed waveguide

Port 3 and

Port 4

s

Cladding

mode, and an antisymmetric supermode (see Figure 12 and Figure 14), with an effective index that is slightly lower than the effective index of the unperturbed waveguide mode.Since the supermodes are the solution to the wave equation, if you excite one of them, it will propagate unperturbed through the waveguide. However, if you excite both the symmetric and the antisymmetric mode, that have different propagation constants, there will be a beating between these two waves. Thus, you will see that the power fluctuates back and forth between the two waveguides, as the waves propagate through the waveguide structure. You can adjust the length of the waveguwaveguide. Bysupermodes, y

Model DefinThe directionaembedded in aion-implanted1.The core crossare separated 3the narrow crothat dont preFor this kind owavelength, thsuitable, as therather the beatThe model is swith the first mthe electric fie

The expressiona beat length L

orTutorial Example: Directional Coupler | 15

ide structure to get coupling from one waveguide to the other adjusting the phase difference between the fields of the two ou can decide which waveguide that initially will be excited.

itionl coupler, as shown in Figure 10, consists of two waveguide cores cladding material. The cladding material is GaAs, with GaAs for the waveguide cores. The structure is modeled after Ref.

-section is square, with a side length of 3 m. The two waveguides m. The length of the waveguide structure is 2 mm. Thus, given ss-section, compared to the length, it is advantageous to use a view serve the aspect ratio for the geometry.f problem, where the propagation length is much longer than the e Electromagnetic Waves, Beam Envelopes interface is particularly mesh does not need to resolve the wave on a wavelength scale, but ing between the two waves.

etup to factor out the fast phase variation that occurs in synchronism ode. Mathematically, we write the total electric field as the sum of

lds of the two modes,

within the square parentheses is what will be solved for. It will have defined by

.

E r E1 j1x exp E2 j2x exp+E1 E2 j 2 1 x exp+ j1x exp

=

=

2 1 L 2=

L 22 1------------------=

16 | Tutorial Ex

In the simulation, this beat length must be well resolved. Since the waveguide length is half of the beat length and the waveguide length is discretized into 20 subdivisions, the beat length will be very well resolved in the model.The model uses two numeric ports per input and exit boundary (see Figure 10). The two ports define the lowest symmetric and antisymmetric modes of the waveguide structure.

Results and DiscussionFigure 11 to Ffirst two modeFigure 11 shocomponent aloample: Directional Coupler

igure 14 show the results of the initial boundary mode analysis. The s (those with the largest effective mode index) are both symmetric. ws the first mode. This mode has the transverse polarization ng the z-direction. The second mode, shown in Figure 13, has

transverse polarization along the y-direction.

Figure 11: The symelectric field as po

Figure 12: The ant

Figure 12 andindexes that arshows the modTutorial Example: Directional Coupler | 17

metric mode for z-polarization. Notice that the returned solution can also show the sitive values in the peaks at the cores.

isymmetric mode for z-polarization.

Figure 14 show the antisymmetric modes. Those have effective e slightly smaller than those of the symmetric modes. Figure 12 e for z-polarization and Figure 14 shows the mode for

18 | Tutorial Ex

y-polarization.

Figure 13: The symelectric field as po

Figure 14: The anample: Directional Coupler

metric mode for y-polarization. Notice that the returned solution can also show the sitive values in the peaks at the cores.

tisymmetric mode for y-polarization.

Figure 15 shows how the electric field increases in the receiving waveguide and decreases in the exciting waveguide. If the waveguide had been longer, the waves would switch back and forth between the waveguides.

Figure 15: Excitatiwaveguide to the other waveguide cFigure 14.

Figure 16 shoof the excitingexcitation of thTutorial Example: Directional Coupler | 19

on of the symmetric and the antisymmetric modes. The wave couples from the input output waveguide. Notice that your result may show that the wave is excited in the ore, if your mode fields have different signs than what is displayed in Figure 11 to

ws the result, when there is a phase difference between the fields ports. In this case, the superposition of the two modes results in e other waveguides (as compared to the case in Figure 15).

20 | Tutorial Ex

Figure 16: The sambetween the two other waveguide cFigure 14.

Reference1. S. Somekh, EWaveguides andvol. 13, no. 2, p

Model Wi

These step-by-directional couThen the lowethe antisymmestructure. Finaexcite the otheample: Directional Coupler

e excitation conditions as in Figure 15, except that there is a phase difference ports of radians. Notice that your result may show that the wave is excited in the ore, if your mode fields have different signs than what is displayed in Figure 11 to

. Garmire, A. Yariv, H.L. Garvin, and R.G. Hunsperger, Channel Optical Directional Couplers in GaAs-lmbedded and Ridged, Applied Optics, p. 32730, 1974.

zard

step instructions guide you through the design and modeling of the pler in 3D. First the simple geometry and the materials are defined. st order modes are determined. Those modes (the symmetric and tric modes combined) are then used to excite the waveguide lly, it is shown how you can change the sign of one of the modes to r waveguide.

Note: These instructions are for the user interface on Windows but apply, with minor differences, also to Linux and Mac.

1 To start the software, double-click the COMSOL icon on the desktop. When the software opens, you can choose to use the Model Wizard to create a new COMSOL model or Blank Model to create one manually. For this tutorial, click the Model Wizard button.If COMSOL is already open, you can start the Model Wizard by selecting New from the File menu and then click Model Wizard .

The Model Wnext window

2 In the Selec3 In the Selec

Waves, Beam4 Click Add a5 In the Studi6 Click Done

Global De

First, define a sparameters.1 On the Hom

If COMSOLaccessible on

Note: On Linunear the top oTutorial Example: Directional Coupler | 21

izard guides you through the first steps of setting up a model. The lets you select the dimension of the modeling space.

t Space Dimension window click 3D .t Physics tree under Optics>Wave Optics, select Electromagnetic Envelopes (ewbe) .

nd then click Study .es tree, select Preset Studies>Boundary Mode Analysis .

.

f init ions - Parameters

et of parameters for creating the geometry and defining the material

e toolbar, click Definitions and choose Parameters . Multiphysics is running in full-screen mode, Parameters is the Home toolbar without first clicking Definitions .x and Mac, the Home toolbar refers to the specific set of controls

f the Desktop.

22 | Tutorial Ex

2 In the Parameters settings window under Parameters, enter these settings in the Parameters table

Geometry

In section Glwere defined. different dimeTo build the gtwo embeddedample: Directional Coupler

1

obal Definitions - Parameters the parameters for the geometry Using a parametrized geometry it is easy to experiment with nsions for your waveguide structure.eometry, start by defining the simulation domain and then add the waveguides.

Block 11 On the Geometry toolbar, click Block .2 Go to the Block settings window. Under the

Size and Shape section in the:- Width field enter len.- Depth field enter width.- Height field enter height.

3 Under Position choose Center from the Base list.

Block 2Now add the f1 On the Geo2 Go to the Bl

Size and Sh- Width fie- Depth fie- Height fie

3 Under PositBase list.

4 In the y field

Block 3Add the seconfirst waveguide1 Right-click

Duplicate 2 Go to the B

Position secTutorial Example: Directional Coupler | 23

irst embedded waveguide.metry toolbar, click Block .ock settings window. Under the ape section in the:ld enter len.ld enter a.ld enter a.ion choose Center from the

enter -d.

d waveguide, by duplicating the and modifying the position.

Block 2 and choose .

lock settings window. In the tion in the y field enter d.

24 | Tutorial Ex

3 Click the Build All Objects button .

Definit ion

Since the geomratio in the vie1 In the Mod

node , th2 Go to the C

Camera sectratio check ample: Directional Coupler

s

etry is so long and narrow, it is better to not preserve the aspect w.el Builder window, expand the Component 1>Definitions>View 1 en click Camera .amera settings window. In the ion, clear the Preserve aspect box.

3 From the Graphics toolbar, click the little down arrow next to the View button and select the Go to View 1 button from the drop-down menu. Then click the Zoom Extents button to produce the figure below.

Materials

Now, add matwaveguide claddomains. In thFor this wavegGaAs is used a

Material 11 From the HTutorial Example: Directional Coupler | 25

erials for the cladding and the core of the waveguides. First add the ding material. By default, the first material will be assigned to all e second step, you will define the waveguide core material.uide structure, GaAs is used as cladding material and ion-implanted s core material.

ome toolbar, select New Material .

26 | Tutorial Ex

2 Go to the Material settings window. Under the Material Contents section enter the following settings in the table:

3 Right-click Material 1, choose Rename and type GaAs cladding in the New name field.

4 Click OK.

Material 21 From the H2 Select only

3 Go the Matthe followin

4 Right-click New name

5 Click OK.ample: Directional Coupler

ome toolbar, select New Material .the waveguide cores, domains 2 and 3.

erial settings window. Under the Material Contents section enter g settings in the table:

Material 2, choose Rename and type Implanted GaAs core in the field.

Electromagnetic Waves, Beam Envelopes

For many simulation problems you have some boundaries between different materials, where reflected waves propagating in the backward direction is created.Thus, the default setting for the Electromagnetic Waves, Beam Envelopes interface is to do bidirectional simulations.However, for the directional coupler problem, we dont expect any reflected waves. So it is best to select unidirectional propagation in this case.1 From the H

ElectromagnIn the settinElectromagninterface, unchoose Uniddirections li

2 Enter ewbe.table, as shoThis sets thelowest waveelectric fieldmismatchedModel Def

Port 1To excite the wboundary, twowaveguides.Each numeric Boundary Mofields and the Analysis steps Study 1 on 1 On the PhyTutorial Example: Directional Coupler | 27

ome toolbar, select etic Waves, Beam Envelopes. gs window for the etic Waves, Beam Envelopes der the Wave Vectors section, irectional from the Number of

st.beta_1 in the x field of the k1 wn in the figure to the right. wave vector to be that of the guide mode. With this wave vector setting, the phase factor of the will perfectly match the lowest mode, but will be slightly to the also excited higher-order mode, as discussed in section inition on page 16.

ave at the input boundary and to absorb the wave at the output numeric ports per boundary are used. The first two ports excite the

port has a Boundary Mode Analysis study step associated to it. The de Analysis study performs an Eigenvalue study to find the mode propagation constant associated to the port. The Boundary Mode are defined in the section Study 1 on page 32 and in section page 37.sics toolbar, click Boundaries and choose Port .

28 | Tutorial Ex

2 Select Boundaries 1, 5, and 10, as shown in the figure to the right.

3 Go to the Port settings window. In the Port Properties section choose Numeric from the Type of port list.

4 From the Wave excitation at this port list, choose On.

Now duplicaterename it.

Port 21 Right-click

choose Dup2 Go to the Po

in the Port

Next, create th

Port 31 On the Phy2 Select Boun

shown in th3 Go to the P

In the Port choose Numof port list.

Duplicate thisnew unique na

Port 41 Right-click

Duplicate 2 Go to the P

section in thwindow andample: Directional Coupler

the first port and

Port 1 and licate .rt Properties section

settings window and enter 2 in the Port name field.

e absorbing ports at the other end of the waveguide structure.

sics toolbar, click Boundaries and choose Port .daries 1618 only, as e figure to the right.ort settings window. Properties section eric from the Type

port and give it a me.

Port 3 and choose .

ort Properties e Port settings enter 4 in the Port name field.

Mesh

We set up the mesh by defining a triangular mesh on the input boundary. That mesh will then be swept along the waveguide structure.

Free Triangular 11 On the Mesh toolbar, click Boundary and select Free Triangular .2 Select Boundaries 1, 5, and

10 only, as sfigure to the

Size 11 Right-click

Triangular 1choose Sizemaximum mto be one wwill be enoumodes.

2 In the Size slocate the Esection and radio button

3 Locate the EParameters element sizethe associate

4 Finally, seleccheck box aassociated fi

Swept 1Sweep the meTwenty elemebe sufficient tothat will occur1 On the Mes

Size1 In the ModTutorial Example: Directional Coupler | 29

hown in the right.

the Free node and

. Set the esh element size

avelength, which gh to resolve the

ettings window, lement Size click the Custom .

lement Size section. Select the Maximum check box and enter wl in d field.t the Minimum element size nd enter wl/2 in the eld.

sh along the waveguides. nts along the waveguide will resolve the mode-coupling .h toolbar, click Swept .

el Builder window, under the Mesh 1 node click Size .

30 | Tutorial Ex

2 In the Size settings window, locate the Element Size section and click the Custom radio button.

3 Locate the Element Size Parameters section and enter len/20 in the Maximum element size field.

4 Click the Build All button .ample: Directional Coupler

Study 1

For this model, we will generate special plots, so start by clearing the check box for the generation of the default plots.1 In the Model Builder window, click

Study 1 .2 In the Study settings window, locate the

Study SettinGenerate de

Step 1: BoundNow analyze tthe waveguidez-direction andbe antisymmet1 In the Mode

Analysis 2 In the Boun

section and modes.

3 In the Searcaround fieldsearch for theffective indthe wavegui

4 In the Modfrequency fi

5 To computeboundary mright-click SMode Analychoose Com

Results

Create a 3D suTutorial Example: Directional Coupler | 31

gs section and clear the fault plots check box.

ary Mode Analysishe four lowest modes. The first two modes will be symmetric. Since cross-section is square, there will be one mode polarized in the one mode polarized in the y-direction. Mode three and four will

ric, one polarized in the z-direction and the other in the y-direction.l Builder window, under Study 1 click Step 1: Boundary Mode .dary Mode Analysis settings window, locate the Study Settings enter 4 in the Desired number of modes field to find the four lowest

h for modes enter nco to e modes with ex close to that of de cores.e analysis eld enter f0. only the ode analysis step, tep 1: Boundary sis and pute Selected Step .

rface plot to view the different modes.

32 | Tutorial Ex

3D Plot Group 11 On the Home toolbar, point to Add Plot Group and select 3D Plot

Group .2 On the 3D Plot Group 1 toolbar, select Surface .

Now, first look at the modes polarized in the z-direction.3 In the Surface settings window, click Replace Expression in the

upper-right corner of the Expression section. From the menu, navigate to and select the variable Electromagnetic Waves, Beam Envelopes>Boundary mode analysis>Tanelectric field

4 In the Mod5 In the 3D P6 From the E7 Click the Pl

in the z-dire

Notice that the real partelectric fieldfield can eitample: Directional Coupler

gential boundary mode electric field>Tangential boundary mode , z component (ewbe.tEbm1z).el Builder window, click 3D Plot Group 1 .lot Group settings window, locate the Data section.ffective mode index list, choose the largest effective index.ot button . The plot below shows the symmetric mode polarized ction (same as Figure 11 on page 18 in Results and Discussion).

your plot may look different from the plot above, as the plot shows of the boundary mode electric field. The computed complex can have a different phase factor than for the plot above. Thus, the her show minima (a blue color) or maxima (a red color) at the

locations for the waveguide cores. However, since this is a symmetric mode, it will have the same field values for both waveguide cores.Another consequence of the arbitrary phase factor is that the magnitude for the displayed real part of the electric field in your plot can be different from what is shown in the plot above.

8 From the Effective mode index list, choose the third largest effective index.9 Click the Plot button. The plot below shows the anti-symmetric mode

polarized in the z-direction (same as Figure 12 on page 18 in Results and Discussion

10In the Mod11From the Suewbe.tEbm1y-direction.

12In the Mod13In the 3D P14From the ETutorial Example: Directional Coupler | 33

).

el Builder window, under 3D Plot Group 1 click Surface 1 .rface settings window, locate the Expression section and enter y in the Expression field, to plot the mode field polarized in the

el Builder window, click 3D Plot Group 1 .lot Group settings window, locate the Data section.ffective mode index list, choose the second largest effective index.

34 | Tutorial Ex

15Click the Plot button . The plot below shows the symmetric mode polarized in the y-direction (same as Figure 13 on page 19 in Results and Discussion).

16From the Eample: Directional Coupler

ffective mode index list, choose the smallest effective index.

17Click the Plot button . The plot below shows the anti-symmetric mode polarized in the y-direction (same as Figure 14 on page 19 in Results and Discussion).

Derived ValueYou will need in the bounda1 On the Resu2 Locate the E

enter ewbe.propagation

3 Click the EvNow, copy ainformationin the bounTutorial Example: Directional Coupler | 35

sto copy the effective indexes for the different modes and use them ry mode analyses for the different ports.lts toolbar select Global Evaluation .xpression section, in the Global Evaluation settings window, and beta_1 in the Expression field. The variable ewbe.beta_1 is the constant related to the first port.aluate button .ll information in the table to the clipboard. Then paste that into your favorite text editor, so you easily can enter the values later dary mode analysis steps.

36 | Tutorial Ex

4 In the Table window, click Full Precision .

5 In the Table window, click Copy Table and Headers to Clipboard .

6 Paste the contents of the clipboard into your text editor, for later reference.

Study 1

Step 1: Bound1 In the Mod

Analysis 2 Go to the B

section and 3 In the Searc

the value inbe the largewritten here

Step 3: Bound1 Right-click 2 In the Boun

section.3 In the Searc

the value inbe the thirdis written he

4 Enter 2 in t

Step 4: Bound1 Select the tw

and Step 3: 2 In the Mod

and choose ample: Directional Coupler

ary Mode Analysisel Builder window, under Study 1 click Step 1: Boundary Mode .oundary Mode Analysis settings window, locate the Study Settings enter 1 in the Desired number of modes field.h for modes around field enter 3.4716717443092047, by selecting you text editor and then copying and pasting it here. This should st effective index. The last figures could be different from what is .

ary Mode Analysis 1Step 1: Boundary Mode Analysis and choose Duplicate .dary Mode Analysis settings window, locate the Study Settings

h for modes around field enter 3.4714219480792177, by selecting you text editor and then copying and pasting it here. This should largest effective index. The last figures could be different from what re.he Port name field.

ary Mode Analysis 2o boundary mode analyses, Step 1: Boundary Mode Analysis

Boundary Mode Analysis 3 .el Builder window, right-click Step 1: Boundary Mode Analysis Duplicate .

3 For the node Step 4: Boundary Mode Analysis 2, go to the Boundary Mode Analysis settings window, locate the Study Settings section and enter 3 in the Port name field.

Step 5: Boundary Mode Analysis 31 In the Model Builder window, under Study 1 click Step 5: Boundary Mode

Analysis 3 .2 Go to the Boundary Mode Analysis settings window, locate the Study Settings

section and enter 4 in the Port name field.

Step 2: Frequ1 In the Mod

Domain 2 Go to the Fr

and enter f03 Finally, mov

be the last sFrequency DDown (Frequency DAfter the molike the sequ

4 Right-click

Results

3D Plot GrouRemove the sufield.1 In the Mod

Surface 1 2 On the 3D 3 From the Sl

Plane list, ch4 In the Plane5 Right-click Tutorial Example: Directional Coupler | 37

ency Domainel Builder window, under Study 1 click Step 2: Frequency .equency Domain settings window, locate the Study Settings section in the Frequencies field.e Step2: Frequency Domain to tudy step, by right-clicking Step 2: omain and choose Move

three times) or by dragging Step2: omain to the last study step. ve, the study sequence should look ence to the right.

Study 1 and choose Compute .

p 1rface plot and replace it with a slice plot of the norm of the electric

el Builder window, under 3D Plot Group 1 right-click and choose Delete . Click Yes to confirm.

Plot Group 1 toolbar, select Slice .ice settings window, locate the Plane Data section and from the oose xy-planes.s field enter 1.Slice 1 and choose Deformation .

38 | Tutorial Ex

6 In the Deformation settings window, locate the Expression section and in the z component field enter ewbe.normE.

7 Click the Plot button .8 Click the little down arrow next to the

View button on the Graphics toolbar and select the Go to View 1 button from the drop-down menu.

9 Click the Zoom Extents button on the Graphicshows how (same as Fig

Electroma

Port 2To excite the oto .ample: Directional Coupler

s toolbar. The plot below the light couples from the excited waveguide to the unexcited one ure 15 on page 20 in Results and Discussion).

gnetic Waves, Beam Envelopes

ther waveguide, set the phase difference between the exciting ports

1 In the Model Builder window, under Electromagnetic Waves, Beam Envelopes click Port 2 .

2 In the Port settings window, locate the Port Properties section and enter pi in the in field.

Study 1

On the Home

Results

3D Plot GrouNow the othercompared to t21 in ResultsTutorial Example: Directional Coupler | 39

toolbar, select Compute .

p 1 waveguide is excited and the coupling occurs in reverse direction, he previous case. The figure below reproduces Figure 16 on page and Discussion.

40 | Tutorial Ex

This concludes this introduction to the Wave Optics Module. For further reading including theory sections, see the Wave Optics Module User's Guide.ample: Directional Coupler

IntroductionThe Use of the Wave Optics Module

The Wave Optics Module Physics InterfacesThe Physics Interface List by Space Dimension and Study Type

The Model Libraries WindowTutorial Example: Directional CouplerIntroductionModel DefinitionResults and DiscussionReferenceModel WizardGlobal Definitions - ParametersGeometry 1DefinitionsMaterialsElectromagnetic Waves, Beam EnvelopesMeshStudy 1ResultsStudy 1ResultsElectromagnetic Waves, Beam EnvelopesStudy 1Results