wave optics model library manual

VERSION 4.4

Wave Optics ModuleModel Library Manual

C o n t a c t I n f o r m a t i o n

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Part number: CM023503

W a v e O p t i c s M o d u l e M o d e l L i b r a r y M a n u a l 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

Solved with COMSOL Multiphysics 4.4

Beam Sp l i t t e r

Introduction

A beam splitter is used for splitting a beam of light in two. One way of making a splitter is to deposit a thin layer of metal between two glass prisms. The beam is slightly attenuated within the layer and then split into two paths. This example models the thin mre

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etal layer using a transition boundary condition, which reduces the memory quirements. Losses in the metal layer are also computed.

gure 1: A beam splitter composed of two prisms with a thin layer of metal between them.

odel Definition

odel the beam splitter in the 2D plane, as shown in Figure 1, under the assumption at the electric field is polarized perpendicular to the plane. A Gaussian beam of avelength 700 nm propagates in the x direction through the glass prism of refractive dex n 1.5. A 13 nm thin layer of silver sandwiched between the two prisms splits e beams.

he model geometry is a square region around the region where the Gaussian beam osses the silver layer. The focus of the beam is at the left boundary, so the expression r the beam intensity at the focal plane can be used as the excitation. The expression r the relative electric field intensity at the focal plane of a Gaussian is

Thin metal layer

nput

Output

Output


2 | B E A M S P L I T

(1)

where w = 3500 nm is the beam waist, and the y = 0 line is the centerline of the beam. Use this expression in a Port boundary condition on the left side to model the incident beam. Model all the other domain boundaries using Scattering Boundary Conditions. These conditions are appropriate when they are placed several wavelengths away from any scattering objects and the wave is known to be traveling at normal or almost n

Twimtoinpdelin

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ormal incidence.

he thin silver layer is modeled using a Transition Boundary Condition. At a free-space avelength of 700 nm, the dielectric of silver is about r 16.5 1.06i, where the aginary part accounts for the losses. Thus, you can set the conductivity of the metal zero. This boundary condition allows for a discontinuity in the fields across the terface by splitting the mesh at the boundary. It can introduce both losses and a hase shift across the interface. It does not require a mesh of the thickness of the omain, and thus saves significant memory. Mesh the two domains with triangular ements, with the maximum size set such that there are six elements per wavelength the glass.

esults and Discussion

igure 2 shows the electric field intensity in the modeling domain. The beam is split to two beams, one propagating in the x direction and the other one in the

direction. The splitting can be evaluated by computing the flux crossing the coming boundary and the two outgoing boundaries. Figure 3 plots the power ossing these boundaries as well as the losses at the mirror.


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gure 2: The electric field intensity shows that the incoming beam is split into two beams approximately equal intensity.

gure 3: The power flux crossing the input boundary and the two output boundaries as ell as the losses at the silver surface.



Model Library path: Wave_Optics_Module/Optical_Scattering/beam_splitter

Modeling Instructions

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rom the File menu, choose New.

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In the New window, click the Model Wizard button.

O D E L W I Z A R D

In the Model Wizard window, click the 2D button.

In the Select physics tree, select Optics>Wave Optics>Electromagnetic Waves, Frequency Domain (ewfd).

Click the Add button.

Click the Study button.

In the tree, select Preset Studies>Frequency Domain.

Click the Done button.

L O B A L D E F I N I T I O N S

rametersOn the Home toolbar, click Parameters.

In the Parameters settings window, locate the Parameters section.

In the table, enter the following settings:

ere, c_const is a predefined COMSOL constant for the speed of light in vacuum.

ame Expression Value Description

da0 700[nm] 7.000E-7 m Wavelength

0 c_const/lda0 4.283E14 1/s Frequency

_max 0.2*lda0 1.400E-7 m Maximum mesh size

ps_Ag -16.5-1.06*i -16.5 - 1.06i Relative dielectric constant, Silver


G E O M E T R Y 1

1 In the Model Builder window, under Component 1 click Geometry 1.

2 In the Geometry settings window, locate the Units section.

3 From the Length unit list, choose m.

Create a triangle using Polygon for one prism.

Polygon 11 Right-click Component 1>Geometry 1 and choose Polygon.

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In the Polygon settings window, locate the Coordinates section.

In the x edit field, type -10 -10 10.

In the y edit field, type -10 10 10.

Click the Build Selected button.

otate the triangle to create the other prism.

otate 1On the Geometry toolbar, click Rotate.

Select the object pol1 only.

In the Rotate settings window, locate the Input section.

Select the Keep input objects check box.

Locate the Rotation Angle section. In the Rotation edit field, type 180.



6 Click the Build All Objects button.

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L E C T R O M A G N E T I C WA V E S , F R E Q U E N C Y D O M A I N ( E W F D )

ow set up the physics.

cattering Boundary Condition 1On the Physics toolbar, click Boundaries and choose Scattering Boundary Condition.


2 Select Boundaries 2, 4, and 5 only.

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rt 1On the Physics toolbar, click Boundaries and choose Port.

Select Boundary 1 only.

In the Port settings window, locate the Port Properties section.



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

5 Locate the Port Mode Settings section. Specify the E0 vector as

6 In the edit field, type ewfd.k.Tr1

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exp(-(y/3500[nm])^2) zT E R

ansition Boundary Condition 1On the Physics toolbar, click Boundaries and choose Transition Boundary Condition.


In the Transition Boundary Condition settings window, locate the Transition Boundary Condition section.

From the Electric displacement field model list, choose Relative permittivity.

From the r list, choose User defined. In the associated edit field, type eps_Ag.From the r list, choose User defined. Leave the default value of 1.From the list, choose User defined. Leave the default value of 0.In the d edit field, type 13[nm].


M A T E R I A L S

Next, assign material properties. Use Glass (quartz) for all domains.

1 On the Home toolbar, click Add Material.

A D D M A T E R I A L

1 Go to the Add Material window.

2 In the tree, select Built-In>Glass (quartz).

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In the Add material window, click Add to Component.

Close the Add material window.

E S H 1

hoose the maximum mesh size in the air domain smaller than 0.2 wavelengths using e parameter h_max that you defined earlier. Scale the mesh size by the inverse of the fractive index.

ze 1In the Model Builder window, under Component 1 right-click Mesh 1 and choose Size.

In the Size settings window, locate the Element Size section.

Click the Custom button.

Locate the Element Size Parameters section. Select the Maximum element size check box.

In the associated edit field, type h_max/1.5.

ee Triangular 1In the Model Builder window, right-click Mesh 1 and choose Free Triangular.

Right-click Free Triangular 1 and choose Build All.

T U D Y 1

ep 1: Frequency DomainIn the Model Builder window, expand the Study 1 node, then click Step 1: Frequency Domain.

In the Frequency Domain settings window, locate the Study Settings section.

In the Frequencies edit field, type f0.

On the Home toolbar, click Compute.


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R E S U L T S

Electric Field (ewfd)The default plot shows the E-field norm. Compare the plot with Figure 2.

Follow the steps below to reproduce the plot in Figure 3.

1D Plot Group 21 On the Home toolbar, click Add Plot Group and choose 1D Plot Group.

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On the 1D Plot Group 2 toolbar, click Line Graph.


In the Line Graph settings window, locate the y-Axis Data section.

In the Expression edit field, type -ewfd.nPoav.

Locate the x-Axis Data section. From the Parameter list, choose Expression.

In the Expression edit field, type y.




In the Expression edit field, type ewfd.nPoav.


In the Expression edit field, type y.




In the Expression edit field, type ewfd.nPoav.


In the Expression edit field, type x.

On the 1D plot group toolbar, click Line Graph.



In the Expression edit field, type ewfd.Qsrh.


In the Expression edit field, type x.


26 On the 1D Plot Group 2 toolbar, click Plot. The plot describes the power flux crossing the input boundary and the two output boundaries together with the losses at the silver surface. Compare with Figure 3. 11 | B E A M S P L I T T E R


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D i e l e c t r i c S l a b Wav e gu i d e

Introduction

A planar dielectric slab waveguide demonstrates the principles behind any kind of dielectric waveguide such as a ridge waveguide or a step index fiber, and has a known analytic solution. This model solves for the effective index of a dielectric slab w

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aveguide as well as for the fields, and compares to analytic results.

gure 1: The guided modes in a dielectric slab waveguide have a known analytic solution.

odel Definition

dielectric slab of thickness hslab = 1 m and refractive index ncore = 1.5 forms the re of the waveguide, and sits in free space with ncladding = 1. Light polarized out of e plane of propagation, of wavelength = 1550 nm, is perfectly guided along the is of the waveguide structure, as shown in Figure 1. Here, only the TE0 mode can opagate. The structure varies only in the y direction, and it is infinite and invariant the other two directions.


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The analytic solution is found by assuming that the electric field along the direction of propagation varies as EzE(y)exp(-ikxx), where E(y)C1cos(kyy) inside the dielectric slab, and E(y)C0exp(yhslab2 in the cladding. Because the electric and magnetic fields must be continuous at the interface, the guidance condition is

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2----------- tan=C S L A B W A V E G U I D E

here ky and satisfy

ith kcore = 2ncore and kcladding = 2ncladding. It is possible to find the solution the above two equations via the Newton-Raphson method, which is used whenever OMSOL Multiphysics detects a system of nonlinear equations, the only requirement eing that of an adequate initial guess.

his model considers a section of a dielectric slab waveguide that is finite in the x and directions. Because the fields drop off exponentially outside the waveguide, the fields n be assumed to be zero at some distance away. This is convenient as it makes the

oundary conditions in the y direction irrelevant, assuming that they are imposed fficiently far away.

se Numerical Port boundary conditions in the x direction to model the guided wave ropagating in the positive x direction. These boundary conditions require first lving an eigenvalue problem that solves for the fields and propagation constants at e boundaries.


igure 2 shows the results. The numerical port boundary condition at the left side cites a mode that propagates in the x direction and is perfectly absorbed by the

umerical port on the right side. The analytic and numerically computed propagation nstants agree.

ky2 kcore

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gure 2: The electric field in a dielectric slab waveguide.

odel Library path: Wave_Optics_Module/Verification_Models/electric_slab_waveguide

odeling Instructions


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4 Click the Study button.

5 In the tree, select Custom Studies>Empty Study.

6 Click the Done button.

G L O B A L D E F I N I T I O N S

Parameters1 On the Home toolbar, click Parameters.

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E O M E T R Y 1

In the Model Builder window, under Component 1 click Geometry 1.

In the Geometry settings window, locate the Units section.

From the Length unit list, choose m.

ectangle 1Right-click Component 1>Geometry 1 and choose Rectangle.

In the Rectangle settings window, locate the Size section.

In the Width edit field, type w_slab.

In the Height edit field, type h_core.

Locate the Position section. From the Base list, choose Center.

e Expression Value Description

da0 1550[nm] 1.550E-6 m Wavelength

re 1.5 1.500 Refractive index, core

adding 1 1.000 Refractive index, cladding

re 1[um] 1.000E-6 m Thickness, core

adding 7[um] 7.000E-6 m Thickness, cladding

ab 5[um] 5.000E-6 m Slab width

re 2*pi[rad]*n_core/lambda0

6.081E6 rad/m Wave number, core

adding 2*pi[rad]*n_cladding/lambda0

4.054E6 rad/m Wave number, cladding

c_const/lambda0 1.934E14 1/s Frequency


6 Click the Build Selected button.

Rectangle 21 In the Model Builder window, right-click Geometry 1 and choose Rectangle.

2 In the Rectangle settings window, locate the Size section.

3 In the Width edit field, type w_slab.

4 In the Height edit field, type h_cladding.

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Click the Build All Objects button.

Click the Zoom Extents button on the Graphics toolbar.

L E C T R O M A G N E T I C WAV E S , F R E Q U E N C Y D O M A I N

he wave is excited at the port on the left side.


Select Boundaries 1, 3, and 5 only.


From the Type of port list, choose Numeric.

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

ow, add the exit port.


Select Boundaries 810 only (the boundaries on the right side).



A T E R I A L S

aterial 1In the Model Builder window, under Component 1 right-click Materials and choose New Material.

In the Material settings window, locate the Material Contents section.



3 In the table, enter the following settings:

4 Right-click Component 1>Materials>Material 1 and choose Rename.

5 Go to the Rename Material dialog box and type Cladding in the New name edit field.

6 Click OK.

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Refractive index n n_cladding 1 Refractive index

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y default, the first material you add applies on all domains. Add a core material.

aterial 2Right-click Materials and choose New Material.

Select Domain 2 only.



Right-click Component 1>Materials>Material 2 and choose Rename.

Go to the Rename Material dialog box and type Core in the New name edit field.

Click OK.

E S H 1

izeIn the Model Builder window, under Component 1 right-click Mesh 1 and choose Free Triangular.



Locate the Element Size Parameters section. In the Maximum element size edit field, type lambda0/n_cladding/8.

ize 1In the Model Builder window, under Component 1>Mesh 1 right-click Free Triangular 1 and choose Size.

In the Size settings window, locate the Geometric Entity Selection section.

From the Geometric entity level list, choose Domain.

roperty Name Value Unit Property group

efractive index n n_core 1 Refractive index


4 Select Domain 2 only.

5 Locate the Element Size section. Click the Custom button.

6 Locate the Element Size Parameters section. Select the Maximum element size check box.

7 In the associated edit field, type lambda0/n_core/8.

8 Click the Build All button.

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T U D Y 1

ep 1: Boundary Mode AnalysisOn the Study toolbar, click Study Steps and choose Other>Boundary Mode Analysis.

In the Boundary Mode Analysis settings window, locate the Study Settings section.

In the Search for modes around edit field, type n_core. This value should be in the vicinity of the value that you expect the fundamental mode to have.

In the Mode analysis frequency edit field, type f0.

dd another boundary mode analysis, for the second port.

ep 2: Boundary Mode Analysis 2On the Study toolbar, click Study Steps and choose Other>Boundary Mode Analysis.


In the Search for modes around edit field, type n_core.



4 In the Port name edit field, type 2.

5 In the Mode analysis frequency edit field, type f0.

Finally, add the study step for the propagating wave in the waveguide.

Step 3: Frequency Domain1 On the Study toolbar, click Study Steps and choose Frequency Domain>Frequency

Domain.

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On the Study toolbar, click Compute.

E S U L T S

lectric Field (ewfd)he default plot shows the norm of the electric field. Modify the plot to shows the component (compare with Figure 2).

In the Model Builder window, expand the Electric Field (ewfd) node, then click Surface 1.

In the Surface settings window, click Replace Expression in the upper-right corner of the Expression section. From the menu, choose Electromagnetic Waves, Frequency Domain>Electric>Electric field>Electric field, z component (ewfd.Ez).

Locate the Coloring and Style section. From the Color table list, choose WaveLight.

On the Electric Field (ewfd) toolbar, click Plot.

inish by comparing the simulation results to the analytic solution. To compute the tter, add a Global ODEs and DAEs interface and then set up and solve the relevant uations.

O M P O N E N T 1

n the Home toolbar, click Add Physics.

D D P H Y S I C S

Go to the Add Physics window.

In the Add physics tree, select Mathematics>ODE and DAE Interfaces>Global ODEs and DAEs (ge).


3 Find the Physics in study subsection. In the table, enter the following settings:

4 In the Add physics window, click Add to Component.

5 Close the Add physics window.

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Studies Solve

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n the Home toolbar, click Add Study.

D D S T U D Y

Go to the Add Study window.

Find the Studies subsection. In the tree, select Custom Studies>Preset Studies for Some Physics>Stationary.

Find the Physics in study subsection. In the table, enter the following settings:

In the Add study window, click Add Study.

Close the Add study window.

L O B A L O D E S A N D D A E S

lobal Equations 1In the Model Builder window, under Component 1>Global ODEs and DAEs click Global Equations 1.

In the Global Equations settings window, locate the Global Equations section.


T U D Y 2

n the Study toolbar, click Compute.

hysics Solve

lectromagnetic Waves, Frequency Domain (ewfd)

ame f(u,ut,utt,t) (1) Initial value (u_0) (1)

Initial value (u_t0) (1/s)

Description

lpha alpha-k_y*tan(k_y*h_core/2)

k_core/2 0

_y k_y^2-(k_core^2-k_cladding^2-alpha^2)

k_core/2 0


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R E S U L T S

Derived ValuesFinally, compare analytical and computed propagation constants.

1 On the Results toolbar, click Global Evaluation.

2 In the Global Evaluation settings window, click Replace Expression in the upper-right corner of the Expression section. From the menu, choose Electromagnetic Waves, Frequency Domain>Ports>Propagation constant (ewfd.beta_1).

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Locate the Expression section. Select the Description check box.

In the associated edit field, type Propagation constant,beta_1.

Click the Evaluate button.

On the Results toolbar, click Global Evaluation.

In the Global Evaluation settings window, click Replace Expression in the upper-right corner of the Expression section. From the menu, choose Electromagnetic Waves, Frequency Domain>Ports>Propagation constant (ewfd.beta_2).

Locate the Expression section. Select the Description check box.

In the associated edit field, type Propagation constant, beta_2.

Right-click Results>Derived Values>Global Evaluation 4 and choose Evaluate>Table 1 - Global Evaluation 3 (ewfd.beta_1).

On the Results toolbar, click Global Evaluation.

In the Global Evaluation settings window, locate the Data section.

From the Data set list, choose Solution 4.

Locate the Expression section. In the Expression edit field, type sqrt(k_core^2-k_y^2).

Select the Description check box.

In the associated edit field, type Propagation constant, computed.



D i r e c t i o n a l C oup l e r

Introduction

Directional 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 th

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e waveguides.

gure 1: Schematic drawing of the waveguide structure. The structure consists of the two aveguide cores and the surrounding cladding. Port 1 and 2 are used for exciting the aveguides and Port 3 and 4 absorb the waves.

ight that propagates through a dielectric waveguide has most of the power ncentrated within the central core of the waveguide. Outside the waveguide core, in e cladding, the electric field decays exponentially with the distance from the core. owever, if you put another waveguide core close to the first waveguide (see

igure 1), that second waveguide will perturb the mode of the first waveguide (and ce versa). Thus, instead of having two modes with the same effective index, one calized in the first waveguide and the second mode in the second waveguide, the odes and their respective effective indexes split and you get a symmetric supermode ee Figure 2 and Figure 4 below), with an effective index that is slightly larger than

Port 1 andPort 2

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2 | D I R E C T I O N

the effective index of the unperturbed waveguide mode, and an antisymmetric supermode (see Figure 3 and Figure 5), 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 flthcodth

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uctuates back and forth between the two waveguides, as the waves propagate through e waveguide structure. You can adjust the length of the waveguide structure to get upling from one waveguide to the other waveguide. By adjusting the phase

ifference between the fields of the two supermodes, you can decide which waveguide at will initially be excited.

odel Definition

he directional coupler, as shown in Figure 1, consists of two waveguide cores bedded in a cladding material. The cladding material is GaAs, with ion-implanted

aAs for the waveguide cores. The structure is modeled after Ref. 1.

he core cross-section is square, with a side length of 3 m. The two waveguides are parated 3 m. The length of the waveguide structure is 2 mm. Thus, given the tiny oss-section, compared to the length, it is advantageous to use a view that doesnt reserve the aspect ratio for the geometry.

or this kind of problem, where the propagation length is much longer than the avelength, the Electromagnetic Waves, Beam Envelopes interface is particularly itable, as the mesh does not need to resolve the wave on a wavelength scale, but ther the beating between the two waves.

he model is setup to factor out the fast phase variation that occurs in synchronism ith the first mode. Mathematically, we write the total electric field as the sum of the ectric fields of the two modes,

he expression within the square parentheses is what will be solved for. It will have a eat length L defined by

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

=

=

2 1 L 2=


or

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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.

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he model uses two numeric ports per input and exit boundary (see Figure 1). The o ports define the lowest symmetric and antisymmetric modes of the waveguide

ructure.


igure 2 to Figure 5 shows the results of the initial boundary mode analysis. The first o modes (those with the largest effective mode index) are both symmetric. Figure 2 ows the first mode. This mode has the transverse polarization component along the direction. The second mode, shown in Figure 4, has transverse polarization along e y-direction.

otice that your plots may look different from the plots below, as the plots show the al part of the boundary mode electric fields. The computed complex electric fields n have different phase factors than for the plots below. Thus, the color legends can ve different scales and the fields can either show minima (a blue color) or maxima (a d color) at the locations for the waveguide cores. However, for a symmetric mode,



it will have the same field value for both waveguide cores and for an antisymmetric mode, it will have opposite field values for the two waveguide cores.

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igure 2: The symmetric mode for z-polarization. Notice that the returned solution can lso show the electric field as positive values in the peaks at the cores.

igure 3: The antisymmetric mode for z-polarization.


Figure 3 and Figure 5 show the antisymmetric modes. Those have effective indexes that are slightly smaller than those of the symmetric modes. Figure 3 shows the mode for z-polarization and Figure 5 shows the mode for y-polarization.

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gure 4: The symmetric mode for y-polarization. Notice that the returned solution can so show the electric field as positive values in the peaks at the cores.

gure 5: The antisymmetric mode for y-polarization.



Figure 6 shows how the electric field increases in the receiving waveguide and decreases in the exciting waveguide. In a longer waveguide, the waves would oscillate back and forth between the waveguides.

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igure 6: Excitation of the symmetric and the antisymmetric modes. The wave couples om the input waveguide to the output waveguide. Notice your result may show that the ave is excited in the other waveguide core, if your mode fields have different signs than hat is displayed in Figure 2 to Figure 5.

igure 7 shows the result, when there is a phase difference between the fields of the citing ports. In this case, the superposition of the two modes results in excitation of e other waveguides (as compared to the case in Figure 6).


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F 7 | D I R E C T I O N A L C O U P L E R

gure 7: The same excitation conditions as in Figure 6, except that there is a phase fference between the two ports of radians. Notice your result may show that the wave is cited in the other waveguide core, if your mode fields have different signs than what is splayed in Figure 2 to Figure 5.

eference

S. Somekh, E. Garmire, A. Yariv, H.L. Garvin, and R.G. Hunsperger, Channel ptical Waveguides and Directional Couplers in GaAs-lmbedded and Ridged, pplied Optics, vol. 13, no. 2, pp. 32730, 1974.

odel Library path: Wave_Optics_Module/Waveguides_and_Couplers/rectional_coupler





N E W

1 In the New window, click the Model Wizard button.

M O D E L W I Z A R D

1 In the Model Wizard window, click the 3D button.

2 In the Select physics tree, select Optics>Wave Optics>Electromagnetic Waves, Beam Envelopes (ewbe).

3 Click the Add button.

4

5

6

G

Fp

Pa1

2

3

G

C

Nam

wl

f0

a

d

len

widt

heig

ncl

dn

ncoA L C O U P L E R


In the tree, select Preset Studies>Boundary Mode Analysis.



irst, define a set of parameters for creating the geometry and defining the material arameters.




E O M E T R Y 1

reate the calculation domain.

e Expression Value Description

1.15[um] 1.150E-6 m Wavelength

c_const/wl 2.607E14 1/s Frequency

3[um] 3.000E-6 m Side of waveguide cross-section

3[um] 3.000E-6 m Distance between the waveguides

2.1[mm] 0.002100 m Waveguide length

h 6*a 1.800E-5 m Width of calculation domain

ht 4*a 1.200E-5 m Height of calculation domain

3.47 3.470 Refractive index of GaAs

0.005 0.005000 Refractive index increase in waveguide core

ncl+dn 3.475 Refractive index in waveguide core


Block 11 On the Geometry toolbar, click Block.

2 In the Block settings window, locate the Size section.

3 In the Width edit field, type len.

4 In the Depth edit field, type width.

5 In the Height edit field, type height.

6 Locate the Position section. From the Base list, choose Center.

N

B1

2

3

4

5

6

7

Apo

B1

2

3

4

D

Si

1

C1

2

3 9 | D I R E C T I O N A L C O U P L E R

ow add the first embedded waveguide.

lock 2On the Geometry toolbar, click Block.

In the Block settings window, locate the Size section.

In the Width edit field, type len.

In the Depth edit field, type a.

In the Height edit field, type a.


In the y edit field, type -d.

dd the second waveguide, by duplicating the first waveguide and modifying the sition.

lock 3Right-click Component 1>Geometry 1>Block 2 and choose Duplicate.

In the Block settings window, locate the Position section.

In the y edit field, type d.


E F I N I T I O N S

nce the geometry is so long and narrow, don't preserve the aspect ratio in the view.

In the Model Builder window, expand the Component 1>Definitions node.

ameraIn the Model Builder window, expand the Component 1>Definitions>View 1 node, then click Camera.

In the Camera settings window, locate the Camera section.

Clear the Preserve aspect ratio check box.


10 | D I R E C T I O

4 Click the Apply button.

5 Click the Zoom Extents button on the Graphics toolbar.

M

N

M1

2

3

4

5

6

M1

P

RN A L C O U P L E R

A T E R I A L S

ow, add materials for the cladding and the core of the waveguides.

aterial 1In the Home toolbar, click New Material.




Go to the Rename Material dialog box and type GaAs cladding in the New name edit field.

Click OK.

aterial 2In the Home toolbar, click New Material.


efractive index n ncl 1 Refractive index


2 Select Domains 2 and 3 only.

3 In the Material settings window, locate the Material Contents section.


5 Right-click Component 1>Materials>Material 2 and choose Rename.

6

7

E

Sipr

1

2

3

A

Po1

2

3

4

5

N


Refractive index n nco 1 Refractive index

e

0


Go to the Rename Material dialog box and type Implanted GaAs core in the New name edit field.

Click OK.

L E C T R O M A G N E T I C WAV E S , B E A M E N V E L O P E S

nce there will be no reflected waves in this model, it is best to select unidirectional opagation.

In the Electromagnetic Waves, Beam Envelopes settings window, locate the Wave Vectors section.

From the Number of directions list, choose Unidirectional.

Specify the k1 vector as

This sets the wave vector to be that of the lowest waveguide mode.

dd two numeric ports per port boundary. The first two ports excite the waveguides.






ow duplicate the first port and rename it.

wbe.beta_1 x

y

z



Port 21 Right-click Component 1>Electromagnetic Waves, Beam Envelopes>Port 1 and choose

Duplicate.

2 In the Port settings window, locate the Port Properties section.

3 In the Port name edit field, type 2.

Next create the ports at the other end of the waveguides.

Po1

2

3

4

D

Po1

2

3

M

Dw

Fr1

2

S1

2

3

4N A L C O U P L E R


Select Boundaries 1618 only.



uplicate this port and give it a new unique name.

rt 4Right-click Component 1>Electromagnetic Waves, Beam Envelopes>Port 3 and choose Duplicate.


In the Port name edit field, type 4.

E S H 1

efine a triangular mesh on the input boundary and then sweep that mesh along the aveguides.

ee Triangular 1In the Mesh toolbar, click Boundary and choose Free Triangular.


ize 1Right-click Component 1>Mesh 1>Free Triangular 1 and choose Size.

Set the maximum mesh element size to be one wavelength, which will be enough to resolve the modes.





5 In the associated edit field, type wl.

6 Select the Minimum element size check box.

7 In the associated edit field, type wl/2.

Sweep the mesh along the waveguides. Twenty elements along the waveguide will be sufficient to resolve the mode-coupling that will occur.

Swept1

2

3

4

5

6

S

D

1


In the Model Builder window, right-click Mesh 1 and choose Swept.

In the Model Builder window, click Mesh I>Size.



Locate the Element Size Parameters section. In the Maximum element size edit field, type len/20.

Click the Build All button.

T U D Y 1

on't generate the default plots.

In the Model Builder window, click Study 1.

In the Study settings window, locate the Study Settings section.



3 Clear the Generate default plots check box.

Step 1: Boundary Mode AnalysisNow analyze the four lowest modes. The first two modes will be symmetric. Since the waveguide cross-section is square, there will be one mode polarized in the z-direction and one mode polarized in the y-direction. Mode three and four will be antisymmetric, one polarized in the z-direction and the other in the y-direction.

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

2

3

4

5

6

R

C

31

2

3

4

5

6

7



In the Desired number of modes edit field, type 4.

Search for the modes with effective index close to that of the waveguide cores.

In the Search for modes around edit field, type nco.

In the Mode analysis frequency edit field, type f0.

Compute only the boundary mode analysis step.

Right-click Study 1>Step 1: Boundary Mode Analysis and choose Compute Selected Step.

E S U L T S

reate a 3D surface plot to view the different modes.

D Plot Group 1On the Home toolbar, click Add Plot Group and choose 3D Plot Group.

On the 3D Plot Group 1 toolbar, click Surface.

First look at the modes polarized in the z-direction.

In the Surface settings window, click Replace Expression in the upper-right corner of the Expression section. From the menu, choose Electromagnetic Waves, Beam Envelopes>Boundary mode analysis>Tangential boundary mode electric

field>Tangential boundary mode electric field, z component (ewbe.tEbm1z), by double-clicking it or selecting it and pressing Enter.

In the Model Builder window, click 3D Plot Group 1.

In the 3D Plot Group settings window, locate the Data section.

From the Effective mode index list, choose the largest effective index.

On the 3D plot group toolbar, click Plot. This plot shows the symmetric mode polarized in the z-direction. Compare with Figure 2.

From the Effective mode index list, choose the third largest effective index.


9 On the 3D plot group toolbar, click Plot. This plot shows the anti-symmetric mode polarized in the z-direction. Compare with Figure 3.

10 In the Model Builder window, under Results>3D Plot Group 1 click Surface 1.

11 In the Surface settings window, locate the Expression section.

12 In the Expression edit field, type ewbe.tEbm1y.

13 In the Model Builder window, click 3D Plot Group 1.

14 In the 3D Plot Group settings window, locate the Data section.

15

16

17

18

DYobo

1

2

T

Cte

1

S

St1

2


From the Effective mode index list, choose the second largest effective index.

On the 3D plot group toolbar, click Plot. This plot shows the symmetric mode polarized in the y-direction. Compare with Figure 4.

From the Effective mode index list, choose the smallest effective index.

On the 3D plot group toolbar, click Plot. This plot shows the anti-symmetric mode polarized in the y-direction. Compare with Figure 5.

erived Valuesu will need to copy the effective indexes for the different modes and use them in the undary mode analyses for the different ports.

In the Global Evaluation settings window, click Replace Expression in the upper-right corner of the Expression section. From the menu, choose Electromagnetic Waves, Beam Envelopes>Ports>Propagation constant (ewbe.beta_1), by double-clicking it or selecting it and pressing Enter.


A B L E

opy all information in the table to the clipboard. Then paste that information in a xt editor, so you easily can enter the values later in the boundary mode analysis steps.

In the Table window, click Full Precision, then click Copy Table and Headers to Clipboard.

T U D Y 1

ep 1: Boundary Mode AnalysisIn the Model Builder window, under Study 1 click Step 1: Boundary Mode Analysis.


In the Desired number of modes edit field, type 1.



4 In the Search for modes around edit field, type 3.4716717443092047, by selecting the value in you text editor and then copying and pasting it here. This should be the largest effective index. The last figures could be different from what is written here.

Step 3: Boundary Mode Analysis 11 Right-click Study 1>Step 1: Boundary Mode Analysis and choose Duplicate.

2 In the Model Builder window, under Study 1 click Step 3: Boundary Mode Analysis 1.

3 In the Boundary Mode Analysis settings window, locate the Study Settings section.

4

5

S1

2

3

4

5

S1

2

3

S1

2

3

4


In the Search for modes around edit field, type 3.4714219480792172., by selecting the value in you text editor and then copying and pasting it here. This should be the third largest effective index. The last figures could be different from what is written here.


tep 4: Boundary Mode Analysis 2Select the two boundary mode analyses, Step 1: Boundary Mode Analysis and Step 3: Boundary Mode Analysis 1.

In the Model Builder window, right-click Step 1: Boundary Mode Analysis and choose Duplicate.

In the Model Builder window, under Study 1 click Step 4: Boundary Mode Analysis 2.



tep 5: Boundary Mode Analysis 3In the Model Builder window, under Study 1 click Step 5: Boundary Mode Analysis 3.



tep 2: Frequency DomainIn the Model Builder window, under Study 1 click Step 2: Frequency Domain.



Finally, move Step2: Frequency Domain to be the last study step.

Right-click Study 1>Step 2: Frequency Domain and choose Move Down. Repeat this command twice.



R E S U L T S

3D Plot Group 1Remove the surface plot and replace it with a slice plot of the norm of the electric field.

1 In the Model Builder window, under Results>3D Plot Group 1 right-click Surface 1 and choose Delete. Click Yes to confirm.

2 Right-click 3D Plot Group 1 and choose Slice.

3

4

5

6

7

8

9

10

11

E

PoT

1

2

3

S

O

R

3Nco 17 | D I R E C T I O N A L C O U P L E R

In the Slice settings window, locate the Plane Data section.

From the Plane list, choose xy-planes.

In the Planes edit field, type 1.

Right-click Results>3D Plot Group 1>Slice 1 and choose Deformation.

In the Deformation settings window, locate the Expression section.

In the z component edit field, type ewbe.normE.

On the 3D plot group toolbar, click Plot.

Click the Go to View 1 button on the Graphics toolbar.

Click the Zoom Extents button on the Graphics toolbar. The plot shows how the light couples from the excited waveguide to the unexcited one; compare with Figure 6.

L E C T R O M A G N E T I C WAV E S , B E A M E N V E L O P E S

rt 2o excite the other waveguide, set the phase difference between the exciting ports to .In the Model Builder window, under Component 1>Electromagnetic Waves, Beam Envelopes click Port 2.


In the in edit field, type pi.

T U D Y 1

n the Home toolbar, click Compute.

E S U L T S

D Plot Group 1ow the other waveguide is excited and the coupling occurs in reverse direction, mpared to the previous case. Compare your result with that in Figure 7.


18 | D I R E C T I O N A L C O U P L E R


F ab r y - P e r o t C a v i t y

Introduction

A Fabry-Perot cavity is a slab of material of higher refractive index than its surroundings, as shown in Figure 1. Such a structure can act as a resonator at certain frequencies. Although such solutions can be found analytically, this model de

FiwW

M

Tmof 1 | F A B R Y - P E R O T C A V I T Y

monstrates how to find the resonant frequencies and the Q-factor.

gure 1: A Fabry-Perot cavity. An electromagnetic wave, traveling at normal incidence, ill be partially reflected and transmitted at each interface between differing dielectrics. hen the length, L, is an integer fraction of the wavelength, this will act as a resonator.

odel Definition

he geometry is a slab of a material with refractive index higher than the surrounding edium. It is assumed that the mode of interest is polarized with the electric field out the plane, and that the wave vector of the mode of interest is parallel to the x-axis.

L

R1

T2

T1

x

n2 n1


2 | F A B R Y - P E R

Because the mode of interest propagates in the x direction, the models y-dimension is arbitrary. The model space is composed of three types of domains:

a central domain of unit width and refractive index n4 domains of n1 on both sides of the central domain two outer perfectly-matched-layer (PML) domains

The PMLs absorb without reflection any incoming evanescent or propagating wave. The boundary condition on the top and bottom edges is perfect magnetic conductor (PAbpar

E

Fnseteeiso

F

Tmso2wanatb

ArefrpTreO T C A V I T Y

MC), which implies that the solution will be mirror symmetric about those planes. scattering boundary condition (SBC) applies at the left and right sides. This oundary condition is only perfectly transparent to an incoming plane wave, and will artially reflect any other component. Using a PML backed by an SBC reduces any tificial reflections due to the boundary conditions.

I G E N F R E Q U E N C Y M O D E L

irst, solve the model as an eigenvalue problem, which requires that you specify the umber of eigenfrequencies to solve for and the frequency range around which to arch. The PML and the SBC make this problem nonlinear, by introducing a damping rm that depends upon the frequency. This, in turn, requires that you specify an genvalue transform point, which only needs to be within an order of magnitude or of the expected resonant frequency.

R E Q U E N C Y - D O M A I N M O D E L

he approach described above has several drawbacks. First, the results must be anually examined to identify the spurious, nonphysical, modes. Second, it requires lving a nonlinear eigenvalue problem using a memory-intensive direct solver. For a

D model, this is not a computational hurdle, but for structurally complex 3D cases, here far more mesh elements are required, it can be a concern. The convergence rate d solution time of the eigenvalue solver also depend on the choice of starting guess the resonant frequency, the number of modes requested as output, and the spacing etween these modes.

n alternative approach to determining the resonant frequency and Q-factor is to cast this as a frequency-domain model, and to excite the structure over a range of equencies. The excitation should be as isotropic as possible, so that it can excite all ossible modes. The present example uses a line current condition applied to a point. he model is run in the frequency domain over a range covering the expected sonances.


Results and Discussion

E I G E N F R E Q U E N C Y M O D E L

The results of the eigenvalue analysis are plotted in Figure 2, and the Q-factor is reported in Table 1. Some of these results are clearly nonphysical, and in fact represent numerical modesthat is, solutions to the numerical eigenvalue problem that have no physically meaningful interpretation. These nonphysical eigenmodes can be identified in two ways:

Fiei 3 | F A B R Y - P E R O T C A V I T Y

A visual examination of the field solutions can reveal that some modes exist purely in the PML regions. This is, however, a manual task, and it is not always obvious that a mode is indeed physical.

Alternatively, it is possible to examine the Q-factor for each mode. A nonphysical mode has a Q-factor less than 12.

gure 2: The electric field across the entire modeling domain for various solutions to the genvalue problem. Only physical modes are visualized.


4 | F A B R Y - P E R

F

Fmthisthcoelenre

TABLE 1: COMPUTED RESONANT FREQUENCY AND Q-FACTORS, PHYSICAL MODES ARE HIGHLIGHTED

Resonant frequency (MHz) Q-factor Note

~0 ~0 Nonphysical mode

37.5 3.08

74.9 6.15

112.4 9.22

1O T C A V I T Y

R E Q U E N C Y - D O M A I N M O D E L

igure 3 plots the results of the frequency-domain analysis. The total energy density is onitored at a point inside the cavity region. The peaks in this graph correspond to e resonant frequencies, f0, and the Q-factor can be computed as Qf0f, where f

the full width at half maximum. This method is an alternative approach to finding e resonant frequencies and Q-factors that requires less memory, an important ncern for 3D models. This approach entirely avoids the problem of finding and

iminating spurious modes. The only limitations are that some care must be taken to sure that the desired modes are indeed excited and that evaluation of the Q-factor quires manual postprocessing of the data.

49.9 12.3


Fibe

Mfa

M

F

N

1

M

1

2 5 | F A B R Y - P E R O T C A V I T Y

gure 3: Plot of energy density within the cavity over a range of frequencies. This plot can used to find the resonant frequencies and Q-factors.

odel Library path: Wave_Optics_Module/Verification_Models/bry_perot



E W


O D E L W I Z A R D




6 | F A B R Y - P E R

3 Click the Add button.

4 Click the Study button.

5 In the tree, select Preset Studies>Eigenfrequency.

6 Click the Done button.

G L O B A L D E F I N I T I O N S

Parameters1

2

3

G

R1

2

3

4

5

6

A1

2

3

4

5

6

7

N

f

fO T C A V I T Y

On the Home toolbar, click Parameters.



E O M E T R Y 1

ectangle 1In the Model Builder window, under Component 1 right-click Geometry 1 and choose Rectangle.

In the Rectangle settings window, locate the Size section.

In the Height edit field, type 0.2.


In the x edit field, type -2.

Click the Build Selected button.

rray 1On the Geometry toolbar, click Array.

Select the object r1 only.

In the Array settings window, locate the Size section.

From the Array type list, choose Linear.

In the Size edit field, type 5.

Locate the Displacement section. In the x edit field, type 1.


ame Expression Value Description

_min 20[MHz] 2.000E7 Hz Minimum frequency in sweep

_max 100[MHz] 1.000E8 Hz Maximum frequency in sweep



E

Nso

1

2

3

A

Pe1 7 | F A B R Y - P E R O T C A V I T Y


ow set up the physics. The model is based on differences in refractive indices and you lve for the E-field's out-of-plane component.

In the Model Builder window, under Component 1 click Electromagnetic Waves, Frequency Domain.

In the Electromagnetic Waves, Frequency Domain settings window, locate the Components section.

From the Electric field components solved for list, choose Out-of-plane vector.

ssign a PMC condition on the top and bottom edges.

rfect Magnetic Conductor 1On the Physics toolbar, click Boundaries and choose Perfect Magnetic Conductor.


8 | F A B R Y - P E R

2 Select Boundaries 2, 3, 5, 6, 8, 9, 11, 12, 14, and 15 only.

D

Pe1

2O T C A V I T Y

E F I N I T I O N S

rfectly Matched Layer 1On the Definitions toolbar, click Perfectly Matched Layer.

Select Domains 1 and 5 only.


E L E C T R O M A G N E T I C WAV E S , F R E Q U E N C Y D O M A I N

Scattering Boundary Condition 11 On the Physics toolbar, click Boundaries and choose Scattering Boundary Condition.

2 Select Boundaries 1 and 16 only.

Li1 9 | F A B R Y - P E R O T C A V I T Y

ne Current (Out-of-Plane) 1On the Physics toolbar, click Points and choose Line Current (Out-of-Plane).


10 | F A B R Y - P E

2 Select Point 5 only.

3

4

M

N

M1R O T C A V I T Y

In the Line Current (Out-of-Plane) settings window, locate the Line Current (Out-of-Plane) section.

In the I0 edit field, type 1.

A T E R I A L S

ow, specify the material properties. First, define the medium surrounding the slab.



2 Select Domains 1, 2, 4, and 5 only.

3

4

5

6

7

T

M1

P

R 11 | F A B R Y - P E R O T C A V I T Y




Go to the Rename Material dialog box and type n=1 in the New name edit field.

Click OK.

he refractive index of the slab is 4.

aterial 2Right-click Materials and choose New Material.


efractive index n 1 1 Refractive index


12 | F A B R Y - P E


3

4

5

6

7

M

S1

2

3

P

RR O T C A V I T Y




Go to the Rename Material dialog box and type n=4 in the New name edit field.

Click OK.

E S H 1

izeIn the Model Builder window, under Component 1 right-click Mesh 1 and choose Free Triangular.


From the Predefined list, choose Extremely fine.


efractive index n 4 1 Refractive index


4 Click the Build All button.

S

St1

2

3

4

Bad

So1

2

3

4


T U D Y 1

ep 1: EigenfrequencyIn the Model Builder window, under Study 1 click Step 1: Eigenfrequency.

In the Eigenfrequency settings window, locate the Study Settings section.

In the Desired number of eigenfrequencies edit field, type 8.

In the Search for eigenfrequencies around edit field, type 1e7.

ecause of the PMLs and SBCs, this is a nonlinear eigenvalue problem and an ditional setting is required to handle the associated damping term.

lver 1On the Study toolbar, click Show Default Solver.

In the Model Builder window, expand the Solver 1 node, then click Eigenvalue Solver 1.

In the Eigenvalue Solver settings window, locate the Values of Linearization Point section.

Find the Value of eigenvalue linearization point subsection. In the Point edit field, type 1e7.

In the Model Builder window, click Study 1.


14 | F A B R Y - P E

6 In the Study settings window, locate the Study Settings section.

7 Clear the Generate default plots check box.

8 On the Home toolbar, click Compute.

R E S U L T S

Derived Values1 On the Results toolbar, click Global Evaluation.

2

3

T

Rco

R

11

2

3

4

5

6

7

8

9

10R O T C A V I T Y

In the Global Evaluation settings window, click Replace Expression in the upper-right corner of the Expression section. From the menu, choose Electromagnetic Waves, Frequency Domain>Global>Quality factor (ewfd.Qfactor), by double-clicking it or selecting it and pressing Enter.


A B L E

eview the evaluated eigenfrequencies and Q-factors; Q-factors less than 0.5 rrespond to nonphysical eigenmodes.

E S U L T S


In the 1D Plot Group settings window, click to expand the Legend section.

From the Position list, choose Lower right.

Locate the Data section. From the Eigenfrequency selection list, choose Manual.

In the Eigenfrequency indices (1-5) edit field, type 2 3 4 5.


In the Line Graph settings window, locate the Selection section.

Click Paste Selection.

Go to the Paste Selection dialog box.

In the Selection edit field, type 3, 6, 9, 12, 15.


11 Click the OK button.

12

13

14

15

16

17

R

A

1

L

3

7

1



In the Expression edit field, type Ez.

Click to expand the Legends section. Select the Show legends check box.

From the Legends list, choose Manual.


On the 1D plot group toolbar, click Plot. Compare the plot with that shown in Figure 2.

O O T

dd a new study as an alternative approach to examine the Q-factor for each mode.

On the Home toolbar, click Add Study.

egends

7.5 MHz

4.9 MHz

12.4 MHz

49.9 MHz


16 | F A B R Y - P E

A D D S T U D Y

1 Go to the Add Study window.

2 Find the Studies subsection. In the tree, select Preset Studies>Frequency Domain.

3 In the Add study window, click Add Study.

S T U D Y 2

Step 1: Frequency Domain1

2

3

4

R

E1

2

3R O T C A V I T Y

In the Model Builder window, under Study 2 click Step 1: Frequency Domain.


In the Frequencies edit field, type range(f_min,0.25[MHz],f_max).


E S U L T S

lectric Field (ewfd)In the 2D Plot Group settings window, locate the Data section.

From the Parameter value (freq) list, choose 3.75e7.

On the Electric Field (ewfd) toolbar, click Plot.



T

F

11

2

3

4

5

6

7

8

E

e 17 | F A B R Y - P E R O T C A V I T Y

his is the E-field norm at 37.5 MHz.

inally, reproduce the plot in Figure 3.


In the 1D Plot Group settings window, locate the Data section.

From the Data set list, choose Solution 2.

On the 1D Plot Group 3 toolbar, click Global.

In the Global settings window, locate the y-Axis Data section.


Click to expand the Legends section. Clear the Show legends check box.

On the 1D plot group toolbar, click Plot. Using the definition Q = f0/f you can use this plot to evaluate the Q-factor at each resonance.

xpression Unit Description

wfd.intWe+ewfd.intWm J Total energy


18 | F A B R Y - P E R O T C A V I T Y


F r e s n e l Equa t i o n s

Introduction

A plane electromagnetic wave propagating through free space is incident at an angle upon an infinite dielectric medium. This model computes the reflection and transmission coefficients and compares the results to the Fresnel equations.

M

Auptrthre

Fidi

Amapalco 1 | F R E S N E L E Q U A T I O N S

odel Definition

plane wave propagating through free space (n1) as shown in Figure 1 is incident on an infinite dielectric medium (n1.5) and is partially reflected and partially

ansmitted. If the electric field is p-polarizedthat is, if the electric field vector is in e same plane as the Poynting vector and the surface normalthen there will be no flections at an incident angle of roughly 56, known as the Brewster angle.

gure 1: A plane wave propagating through free space incident upon an infinite electric medium.

lthough, by assumption, space extends to infinity in all directions, it is sufficient to odel a small unit cell, as shown in Figure 1; a Floquet-periodic boundary condition plies on the top and bottom unit-cell boundaries because the solution is periodic

ong the interface. This model uses a 3D unit cell, and applies perfect electric nductor and perfect magnetic conductor boundary conditions as appropriate to

Unit cell

Reflected

Incident

Transmitted

n1 n2


2 | F R E S N E L E Q

model out-of-plane symmetry. The angle of incidence ranges between 090 for both polarizations.

For comparison, Ref. 1 and Ref. 2 provide analytic expressions for the reflectance and transmittance. Reflection and transmission coefficients for s-polarization and p-polarization are defined respectively as

(1)

R

T

rsn1 incidentcos n2 transmittedcosn1 incidentcos n2 transmittedcos+------------------------------------------------------------------------------------=U A T I O N S

(2)

(3)

(4)

eflectance and transmittance are defined as

(5)

(6)

he Brewster angle at which rp0 is defined as

(7)

ts2n1 incidentcos

n1 incidentcos n2 transmittedcos+------------------------------------------------------------------------------------=

rpn2 incidentcos n1 transmittedcosn1 transmittedcos n2 incidentcos+------------------------------------------------------------------------------------=

tp2n1 incidentcos

n1 transmittedcos n2 incidentcos+------------------------------------------------------------------------------------=

R r 2=

Tn2 transmittedcosn1 incidentcos

------------------------------------------ t2

=

Bn2n1------atan=


Results and Discussion

Figure 2 is a combined plot of the y component of the electric-field distribution and the power flow visualized as an arrow plot for the TE case.

Fith 3 | F R E S N E L E Q U A T I O N S

gure 2: Electric field, Ey (slice) and power flow (arrows) for TE incidence at 70 inside e unit cell.



For the TM case, Figure 3 visualizes the y component of the magnetic-field distribution instead, again in combination with the power flow.

Fin

NshthU A T I O N S

igure 3: Magnetic field, Hy (slice) and power flow (arrows) for TM incidence at 70 side the unit cell.

ote that the sum of reflectance and transmittance in Figure 4 and Figure 5 equals 1, owing conservation of power. Figure 5 also shows that the reflectance around 56e Brewster angle in the TM caseis close to zero.


Fiso

Fiso 5 | F R E S N E L E Q U A T I O N S

gure 4: The reflectance and transmittance for TE incidence agree well with the analytic lutions.

gure 5: The reflectance and transmittance for TM incidence agree well with the analytic lutions. The Brewster angle is also observed at the expected location.



References

1. C.A. Balanis, Advanced Engineering Electromagnetics, Wiley, 1989.

2. B.E.A. Saleh and M.C. Teich, Fundamentals of Photonics, Wiley, 1991.

Model Library path: Wave_Optics_Module/Verification_Models/fresnel_equations

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E W


O D E L W I Z A R D





In the tree, select Preset Studies>Frequency Domain.



efine some parameters that are useful when setting up the mesh and the study.





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Name Expression Value Description

n_air 1 1.000 Refractive index, air

n_slab 1.5 1.500 Refractive index, slab

lda0 1[m] 1.000 m Wavelength

f

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t 7 | F R E S N E L E Q U A T I O N S

he angle of incidence is updated while running the parametric sweep. The refraction ransmitted) angle is defined by Snell's law with the updated angle of incidence. The rewster angle exists only for TM incidence, p-polarization, and parallel polarization.

E F I N I T I O N S

riables 1In the Model Builder window, under Component 1 right-click Definitions and choose Variables.

0 c_const/lda0 2.998E8 1/s Frequency

lpha 70[deg] 1.222 rad Angle of incidence

eta asin(n_air*sin(alpha)/n_slab)

0.6770 rad Refraction angle

_max lda0/6 0.1667 m Maximum element size, air

lpha_brewster atan(n_slab/n_air) 0.9828 rad Brewster angle, TM only

_s (n_air*cos(alpha)-n_slab*cos(beta))/(n_air*cos(alpha)+n_slab*cos(beta))

-0.5474 Reflection coefficient, TE

_p (n_slab*cos(alpha)-n_air*cos(beta))/(n_air*cos(beta)+n_slab*cos(alpha))

-0.2061 Reflection coefficient, TM

_s (2*n_air*cos(alpha))/(n_air*cos(alpha)+n_slab*cos(beta))

0.4526 Transmission coefficient, TE

_p (2*n_air*cos(alpha))/(n_air*cos(beta)+n_slab*cos(alpha))

0.5292 Transmission coefficient, TM



2 In the Variables settings window, locate the Variables section.


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Name Expression Unit Description

ka ewfd.k0 rad/m Propagation constant, air

kax ka*sin(alpha) rad/m kx for incident wave

kay 0 ky for incident wave

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LU A T I O N S

E O M E T R Y 1

irst, create a block composed of two domains. Use layers to split the block.

lock 1On the Geometry toolbar, click Block.

In the Block settings window, locate the Size section.

In the Width edit field, type 0.2.

In the Depth edit field, type 0.2.

In the Height edit field, type 0.8.

Click to expand the Layers section. Find the Layer position subsection. In the table, enter the following settings:


az ka*cos(alpha) rad/m kz for incident wave

b n_slab*ewfd.k0 rad/m Propagation constant, slab

bx kb*sin(beta) rad/m kx for refracted wave

by 0 ky for refracted wave

bz kb*cos(beta) rad/m kz for refracted wave

ayer name Thickness (m)

ayer 1 0.4



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Choose wireframe rendering to get a better view of each boundary.

Click the Wireframe Rendering button on the Graphics toolbar.


t up the physics based on the direction of propagation and the E-field polarization. irst, assume a TE-polarized wave which is equivalent to s-polarization and rpendicular polarization. Ex and Ez are zero while Ey is dominant.

he wave is excited from the port on the top.





Locate the Port Mode Settings section. Specify the E0 vector as

In the edit field, type abs(kaz).

x

xp(-i*kax*x)[V/m] y

z


10 | F R E S N E L E

Port 21 On the Physics toolbar, click Boundaries and choose Port.

2 Select Boundary 3 only.

3 In the Port settings window, locate the Port Mode Settings section.

4 Specify the E0 vector as

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In the edit field, type abs(kbz).he bottom surface is an observation port. The S21-parameter from Port 1 and Port provides the transmission characteristics.

he E-field polarization has Ey only and the boundaries are always either parallel or erpendicular to the E-field polarization. Apply periodic boundary conditions on the oundaries parallel to the E-field except those you already assigned to the ports.

riodic Condition 1On the Physics toolbar, click Boundaries and choose Periodic Condition.

Select Boundaries 1, 4, 10, and 11 only.

In the Periodic Condition settings window, locate the Periodicity Settings section.

From the Type of periodicity list, choose Floquet periodicity.

xp(-i*kbx*x)[V/m] y

z


5 Specify the kF vector as

AE

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kax x

0 y

0 z 11 | F R E S N E L E Q U A T I O N S

pply a perfect electric conductor condition on the boundaries perpendicular to the -field. This condition creates a virtually infinite modeling space.

rfect Electric Conductor 2On the Physics toolbar, click Boundaries and choose Perfect Electric Conductor.



2 Select Boundaries 2, 5, 8, and 9 only.

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ow set up the material properties based on refractive index. The top half is filled with r.






Go to the Rename Material dialog box and type Air in the New name edit field.

Click OK.

he bottom half is glass.


efractive index n n_air 1 Refractive index


Material 21 Right-click Materials and choose New Material.


3 In the Material settings window, locate the Material Contents section.


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Go to the Rename Material dialog box and type Glass in the New name edit field.

Click OK.

E S H 1

he periodic boundary condition performs better if the mesh is identical on the riodicity boundaries. This is especially important when dealing with vector degrees freedom, as will be the case in the TM version of this model. The maximum element ze is smaller than 0.2 times the wavelength. The bottom half domain is scaled versely by the refractive index of the material.

zeIn the Model Builder window, under Component 1 right-click Mesh 1 and choose Size.



Locate the Element Size Parameters section. In the Maximum element size edit field, type h_max.

ze 1In the Model Builder window, under Component 1>Mesh 1 click Size 1.

In the Size settings window, locate the Geometric Entity Selection section.

From the Geometric entity level list, choose Domain.


Locate the Element Size section. Click the Custom button.


In the associated edit field, type h_max/n_slab.

efractive index n n_slab 1 Refractive index



Free Triangular 11 In the Model Builder window, right-click Mesh 1 and choose Free Triangular.


Copy Face 11 Right-click Mesh 1 and choose Copy Face.


3 In the Copy Face settings window, locate the Destination Boundaries section.

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Select the Destination group focus toggle button.

Select Boundaries 10 and 11 only.

ee Tetrahedral 1Right-click Mesh 1 and choose Free Tetrahedral.

Right-click Mesh 1 and choose Build All.

T U D Y 1

tep 1: Frequency DomainIn the Model Builder window, under Study 1 click Step 1: Frequency Domain.




Parametric Sweep1 On the Study toolbar, click Parametric Sweep.

2 In the Parametric Sweep settings window, locate the Study Settings section.

3 Click Add.


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Parameter names Parameter value list

a 15 | F R E S N E L E Q U A T I O N S

se a direct solver instead of an iterative one for faster convergence.

lver 1On the Study toolbar, click Show Default Solver.

In the Model Builder window, under Study 1>Solver Configurations>Solver 1>Stationary Solver 1 right-click Direct and choose Enable.

Right-click Study 1>Solver Configurations>Solver 1>Stationary Solver 1>Direct and choose Compute.

E S U L T S

lectric Field (ewfd)he default plot is the E-field norm for the last solution, which corresponds to ngential incidence. Replace the expression with Ey, add an arrow plot of the power w (Poynting vector), and choose a more interesting angle of incidence for the plot.

In the Model Builder window, under Results>Electric Field (ewfd) click Multislice 1.

In the Multislice settings window, click Replace Expression in the upper-right corner of the Expression section. From the menu, choose Electromagnetic Waves, Frequency Domain>Electric>Electric field>Electric field, y component (ewfd.Ey).

Locate the Multiplane Data section. Find the x-planes settings and in the Planes edit field, type 0.

Find the z-planes settings and in the Planes edit field, type 0.

Locate the Coloring and Style section. From the Color table list, choose Wave.

In the Model Builder window, right-click Electric Field (ewfd) and choose Arrow Volume.

In the Arrow Volume settings window, click Replace Expression in the upper-right corner of the Expression section. From the menu, choose Electromagnetic Waves,

lpha range(0,2[deg],90[deg])



Frequency Domain>Energy and power>Power flow, time average

(ewfd.Poavx,...,ewfd.Poavz).

8 Locate the Arrow Positioning section. Find the y-grid points and in the Points edit field, type 1.

9 Locate the Coloring and Style section. From the Color list, choose Green.

10 In the Model Builder window, click Electric Field (ewfd).

11 In the 3D Plot Group settings window, locate the Data section.

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From the Parameter value (alpha) list, choose 1.22173.

On the 3D plot group toolbar, click Plot.

Click the Zoom Extents button on the Graphics toolbar. The plot should look like that in Figure 2.

dd a 1D plot to see the reflection and transmission versus the angle of incidence.


In the 1D Plot Group settings window, locate the Plot Settings section.

Select the x-axis label check box.

In the associated edit field, type Angle of Incidence.

Select the y-axis label check box.

In the associated edit field, type Reflectance and Transmittance.

Click to expand the Legend section. From the Position list, choose Upper left.

On the 1D plot group toolbar, click Global.

In the Global settings window, locate the y-Axis Data section.


Click to expand the Coloring and style section. Locate the Coloring and Style section. Find the Line markers subsection. From the Line list, choose None.

From the Marker list, choose Cycle.

From the Line list, choose None.

From the Marker list, choose Cycle.

xpression Unit Description

bs(ewfd.S11)^2 1 Reflectance

bs(ewfd.S21)^2 1 Transmittance


15 On the 1D plot group toolbar, click Global.

16 In t

wave optics model library manual

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