Optically Anisotropic Waveguide

Optical waveguides consist of a core with a high refractive index, surrounded by a cladding with a low refractive index. Thereby, due to total internal reflection, the mode is guided by the high-index core. The smaller the refractive index difference is between the core and the cladding, the more the mode extends into the cladding. For example, the tutorial called Directional Coupler demonstrates how to model a GaAs-based waveguide structure consisting of two closely spaced waveguide cores surrounded by a cladding.

For most waveguide structures, the optical properties for the core and the cladding are different. However, within each domain, the material properties are uniform and isotropic. This model demonstrates what happens when the material in the core exhibit anisotropy. Specifically, in this case, the electric displacement field is no longer parallel to the electric field. Instead, the electric displacement field D relates to the electric field E by a matrix-vector expression

(1)

where Dx, Dy, and Dz are the components of the electric displacement field D, ε0 is the vacuum permittivity, εrij is the relative permittivity tensor element for row index i and column index j, and Ex, Ey, and Ez are the components of the electric field E.

However, for all materials, you can find a coordinate system in which you only have nonzero diagonal elements in the permittivity tensor, whereas the off-diagonal elements are all zero. The three coordinate axes in this rotated coordinate system are the principal axes of the material and, correspondingly, the three values for the diagonal elements in the permittivity tensor are called the principal permittivities of the material.

There are basically two kinds of anisotropic crystals: uniaxial and biaxial crystals. With a suitable choice of coordinate system, where only the diagonal elements of the permittivity tensor are nonzero, uniaxial crystals are characterized by an extraordinary refractive index ne for waves polarized along the crystal’s optical axis and an ordinary refractive index no for waves polarized orthogonally to the crystal’s optical axis. So the diagonal elements of the relative permittivity in the rotated coordinate system are given by εrxx = εryy = (no)2, εrzz = (ne)2. However, for a biaxial crystal, all three principal permittivities are different.

Model Definition

In this model a uniaxial material will be used. Two cases will be considered — diagonal transverse anisotropy, where the crystal’s optical axis is oriented in an orthogonal Cartesian direction to the propagation direction of the waveguide, and off-diagonal longitudinal anisotropy, where the optical axis has a components both in the propagation direction and in the transverse direction. Here, for the diagonal transverse anisotropy, the extraordinary refractive index is defined in the x direction. For the off-diagonal longitudinal anisotropy case, the optical axis is defined in a direction 45 degrees from the z-axis in the yz-plane.

The ordinary refractive index is 1.52 and the extraordinary refractive index is 1.48.

A mode analysis is performed where the width and the thickness is parametrically swept from 0.5 μm to 4 μm. For each geometry, the effective index is solved for and plotted in a dispersion diagram. The results agree well with the results reported in Ref. 1.

Results and Discussion

Figure 1 shows the power in the dominant mode Ey11 for diagonal transverse anisotropy. Here, the superscript denotes the dominant polarization direction and the first and second subscripts denote the number of peaks in the x and y direction, respectively.

Figure 1: The power in the propagation direction (out-of-plane) for the dominant mode, Ey11. The arrows show that the mode is polarized in the y direction.

Figure 2 shows the dispersion curves for the lowest order modes for the case of diagonal transverse anisotropy. The dispersion curves show good agreement with those in Fig. 3 of Ref. 1.

Figure 2: Dispersion curves for the lowest order modes for the diagonal transverse anisotropic case.

Figure 3 shows the power in the dominant mode Ex11 for off-diagonal longitudinal anisotropy. As seen from the plot, the polarization is not uniform for this type of anisotropy.

Figure 4 shows the dispersion curves for off-diagonal longitudinal anisotropy. These results show good agreement with Fig. 8 in Ref. 1.

Figure 3: The lowest order mode, Ex11, for off-diagonal longitudinal anisotropy.

Figure 4: Dispersion curves for the lowest order modes for off-diagonal longitudinal anisotropy.

Reference

1. M. Koshiba, K. Hayata, and M. Suzuki, “Finite-element solution of anisotropic waveguides with arbitrary tensor permittivity,” J. Lightwave Technol., vol. LT-4, p. 121, 1986.

Wave_Optics_Module/Verification_Examples/optically_anisotropic_waveguide

Modeling Instructions

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New

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Model Wizard

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Global Definitions

Parameters 1

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Browse to the model’s Application Libraries folder and double-click the file optically_anisotropic_waveguide_parameters.txt.

Geometry 1

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Square 1 (sq1)

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In the text field, type 20[um].

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Rectangle 1 (r1)

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In the text field, type height.

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Electromagnetic Waves, Frequency Domain (ewfd)

Wave Equation, Electric 1

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Materials

Cladding

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In the window for , type Cladding in the text field.

Locate the section. In the table, enter the following settings:


Property	Variable	Value	Unit	Property group
Relative permittivity	epsilonr_iso ; epsilonrii = epsilonr_iso, epsilonrij = 0	eps_clad	1	Basic
Relative permeability	mur_iso ; murii = mur_iso, murij = 0	1	1	Basic
Electrical conductivity	sigma_iso ; sigmaii = sigma_iso, sigmaij = 0	0	S/m	Basic

Core

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In the window for , type Core in the text field.

Select Domain 2 only.

Locate the section. In the table, enter the following settings:


Property	Variable	Value	Unit	Property group
Relative permittivity	{epsilonr11, epsilonr22, epsilonr33} ; epsilonrij = 0	{epsr_eo, epsr_o, epsr_o}	1	Basic
Relative permeability	mur_iso ; murii = mur_iso, murij = 0	1	1	Basic
Electrical conductivity	sigma_iso ; sigmaii = sigma_iso, sigmaij = 0	0	S/m	Basic

Mesh 1

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Size

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Size 1

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Study - Diagonal Transverse Anisotropy

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Parametric Sweep

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In the table, enter the following settings:


Parameter name	Parameter value list	Parameter unit
height (Height of waveguide)	range(500[nm],250[nm],4000[nm])	nm

Step 1: Mode Analysis

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In the text field, type f0.

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Results

Power and Electric Field - Diagonal Transverse Anisotropy

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In the window for , type Power and Electric Field - Diagonal Transverse Anisotropy in the text field.

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Surface 1

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Arrow Surface 1

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Contour 1

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Power and Electric Field - Diagonal Transverse Anisotropy

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In the text area, type Height=eval(height,nm) nm Effective mode index=eval(ewfd.neff) Surface and Countour: Power Arrows: Electric field.

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button in the toolbar twice.

Dispersion Curves - Diagonal Transverse Anisotropy

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Global 1

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In the table, enter the following settings:


Expression	Unit	Description
ewfd.neff^2	1

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In the text field, type ewfd.k0*height.

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In the text field, type 100.

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Dispersion Curves - Diagonal Transverse Anisotropy

Remove solution 3, which represents the Ey22 mode, as this mode was not included in Ref. 1.

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In the text field, type 1 2 4 5 6 7.

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In the associated text field, type k*thickness.

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In the associated text field, type (Effective index)<sup>2</sub>.

Adjust the axis limits to be the same as in Ref. 1.

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In the text field, type 0.

In the text field, type 15.

In the text field, type 2.05.

In the text field, type 2.31.

Annotation 1

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In the text field, type \[E^{y}_{11}\].

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In the text field, type 2.225.

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Annotation 2

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In the text field, type \[E^{x}_{11}\].

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Annotation 3

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In the text field, type \[E^{y}_{21}\].

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In the text field, type 2.165.

Annotation 4

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In the text field, type \[E^{y}_{12}\].

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Annotation 5

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In the text field, type \[E^{x}_{21}\].

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In the text field, type 2.09.

Annotation 6

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In the text field, type \[E^{x}_{12}\].

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Dispersion Curves - Diagonal Transverse Anisotropy

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In the window for , click to expand the section.

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Electromagnetic Waves, Frequency Domain (ewfd)

Now, redo the simulation for off-diagonal longitudinal anisotropy. That is, in this case the optical axis is directed at 45-degree angle from the y-axis in the yz-plane.

Wave Equation, Off-Diagonal Longitudinal Anisotropy

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Select Domain 2 only.

Locate the section. From the list, choose .

From the εr list, choose . From the list, choose .

In the εr table, enter the following settings:


epsr_o	0	0
0	epsr_eo(cos(phi))^2+epsr_o(sin(phi))^2	epsr_eosin(phi)cos(phi)-epsr_osin(phi)cos(phi)
0	epsr_eosin(phi)cos(phi)-epsr_osin(phi)cos(phi)	epsr_o(cos(phi))^2+epsr_eo(sin(phi))^2

Add Study

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to open the window.

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Study - Off-Diagonal Longitudinal Anisotropy

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In the window for , type Study - Off-Diagonal Longitudinal Anisotropy in the text field.

Study - Diagonal Transverse Anisotropy

Parametric Sweep

In the window, under right-click and choose .

Study - Off-Diagonal Longitudinal Anisotropy

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Step 1: Mode Analysis

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In the text field, type f0.

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In the associated text field, type n_core_o.

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to close the window.

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Results

Power and Electric Field - Off-Diagonal Longitudinal Anisotropy

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Arrow Surface 1, Contour 1

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Arrow Surface 1

In the window, right-click and choose .

Surface 1

In the window for , click in the upper-right corner of the section. From the menu, choose .

Power and Electric Field - Off-Diagonal Longitudinal Anisotropy

In the window, click .

In the window for , locate the section.

From the list, choose .

Select the check box.

From the list, choose .

In the text field, type 5.

In the text area, type Height=eval(height,nm) nm Effective mode index=eval(ewfd.neff) Surface and Countour: Power Arrows: Electric field.

Clear the text field.

Dispersion Curves - Off-Diagonal Longitudinal Anisotropy

In the window, right-click and choose .

In the window for , type Dispersion Curves - Off-Diagonal Longitudinal Anisotropy in the text field.

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From the list, choose .

Annotation 1

In the window, expand the node, then click .

In the window for , locate the section.

In the text field, type \[E^{x}_{11}\].

Annotation 2

In the window, click .

In the window for , locate the section.

In the text field, type \[E^{y}_{11}\].

Locate the section. In the text field, type 2.175.

Annotation 3

In the window, click .

In the window for , locate the section.

In the text field, type \[E^{x}_{12}\].

Locate the section. In the text field, type 12.

In the text field, type 2.145.

Annotation 4

In the window, click .

In the window for , locate the section.

In the text field, type \[E^{x}_{21}\].

Locate the section. In the text field, type 12.5.

In the text field, type 2.13.

Annotation 5

In the window, click .

In the window for , locate the section.

In the text field, type \[E^{y}_{21}\].

Locate the section. In the text field, type 12.5.

In the text field, type 2.111.

Annotation 6

In the window, click .

In the window for , locate the section.

In the text field, type \[E^{y}_{12}\].

Locate the section. In the text field, type 13.

In the text field, type 2.095.

In the toolbar, click

epsr_o	0	0
0	epsr_eo(cos(phi))^2+epsr_o(sin(phi))^2	epsr_eosin(phi)cos(phi)-epsr_osin(phi)cos(phi)
0	epsr_eosin(phi)cos(phi)-epsr_osin(phi)cos(phi)	epsr_o(cos(phi))^2+epsr_eo(sin(phi))^2