Stress-Optical Effects with Generalized Plane Strain

The assumptions made for plane strain in the previous analysis of the waveguide structure (see the Application Library model Stress-Optical Effects in a Photonic Waveguide, the model name is stress_optical) do not hold in a situation where the silicon-silica laminate is free to expand in the z direction. Instead, it is necessary to use a generalized plane strain model that allows for free expansion in the z direction. The boundary conditions in the xy-plane already allow the structure to expand freely in all directions in the plane. When the different materials in a laminate expand with different expansion coefficients, the laminate bends. In this model, the silica-silicon laminate bends in both the x and z directions.

This application requires the Wave Optics Module and the Structural Mechanics Module.

Model Definition

Generalized Plane Strain

One possible extension of the plane strain formulation is to assume that the strains are independent of the out-of -plane coordinate z, that is

Under the small strain assumption, the above equations have the following 3D solution:

where e0, e1, and e2 are constants. Thus, at the cross-section

, one has

and

The above conditions differ from the plane strain state only by the fact that the normal out-of -plane strain component can vary linearly throughout the cross section. This approximation is expected to be good when the structure is free to expand in the out-of-plane direction, and possible bending curvature is small with respect to the extents of the structure in the xy-plane. It corresponds to a small rotation that is representative of each cross section of the structure along the z-axis.

In COMSOL Multiphysics, the Generalized Plane Strain option is available under Structural Mechanics interface in 2D. Coefficients e0, e1, and e2 in the expression for the εz strain are modeled as extra degrees of freedom that are constant throughout the model (global variables). See the Structural Mechanics Module User’s Guide for more details.

Results and Discussion

Figure 1 shows the von Mises stress distribution together with the deformed shape of the waveguide.

Figure 1: von Mises equivalent stress computed with (upper) and without (lower) the generalized plane strain assumption.

A 3D representation of the generalized plane strain solution is shown in this figure:

For symmetry reasons, the strain components εx and εz should be equal. The plot in Figure 2 visualizes the area where the relative difference between εx and εz is within 5%. The model is most accurate in the regions close to the core, far from the boundaries on the far left and right.

Figure 2: In the red-colored region, the relative difference between the x and z strain components is within 5%.

Figure 3 compares the effective mode indices for the first four propagating modes using the generalized plane strain equations with those obtained from the analysis in the previous model. As the plot shows, there is a systematic shift in the propagation constants when the strain in the z direction is taken into account.

Figure 3: Effective mode indices assuming plane strain and generalized plane strain.

Figure 4 shows the stress-induced birefringence along the symmetry line within the waveguide. Nonzero out-of-plane strain leads to an increase in the birefringence effect.

Figure 4: Birefringence along the vertical symmetry line within the waveguide for plane strain (lower curve) and generalized plane strain (upper curve).

Figure 5 gives the details of the eigenmode with the lowest effective mode index. It presents the visualization of the power flow, also called the optical intensity or the Poynting vector, in the out-of-plane direction.

Figure 5: Eigenmode with lowest mode index, computed with the stress-optical effect under generalized plane strain assumption.

Structural_Mechanics_Module/Stress_Optical_Effects/stress_optical_generalized

Modeling Instructions

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New

In the window, click

Model Wizard

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Click .

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Click

In the tree, select .

Click

Global Definitions

Parameters 1

In the window, under click .

In the window for , locate the section.

Click

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

Geometry 1

In the window, under click .

In the window for , locate the section.

From the list, choose .

Rectangle 1 (r1)

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In the window for , locate the section.

In the text field, type 320.

In the text field, type 83.

Locate the section. In the text field, type -160.

In the text field, type -100.

Rectangle 2 (r2)

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In the window for , locate the section.

In the text field, type 320.

In the text field, type 14.

Locate the section. In the text field, type -160.

In the text field, type -17.

Rectangle 3 (r3)

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In the window for , locate the section.

In the text field, type 320.

In the text field, type 16.

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In the text field, type -3.

Rectangle 4 (r4)

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In the window for , locate the section.

In the text field, type 6.

Locate the section. In the text field, type -3.

In the text field, type -3.

Rectangle 5 (r5)

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In the window for , locate the section.

In the text field, type 20.

Locate the section. In the text field, type -10.

In the text field, type -10.

Click

The last rectangular region encloses the optical computational domain. It can be enlarged if needed for validating the results. The region should be chosen large enough so that the computed propagation constants do not change significantly if the region is enlarged.

Solid Mechanics (solid)

In the window, under click .

In the window for , locate the section.

In the d text field, type 2*d.

Linear Elastic Material 1

In the window, under click .

Thermal Expansion 1

In the toolbar, click

and choose .

In the window for , locate the section.

From the T list, choose . In the associated text field, type T1.

Click

for .

Global Definitions

Default Model Inputs

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In the window for , locate the section.

Find the subsection. In the text field, type T0.

Definitions

Variables 1

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and choose .

In the window for , locate the section.

In the table, enter the following settings:


Name	Expression	Unit	Description
N	nCore		Refractive index for stress-free material

Locate the section. From the list, choose .

Select Domain 6 only.

Variables 2

In the toolbar, click

and choose .

In the window for , locate the section.

In the table, enter the following settings:


Name	Expression	Unit	Description
N	nSiO2		Refractive index for stress-free material

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Select Domains 4 and 5 only.

Variables 3

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and choose .

In the window for , locate the section.

In the table, enter the following settings:


Name	Expression	Unit	Description
Nx	N-(B1solid.sx+B2(solid.sy+solid.sz))		Refractive index, x-component
Ny	N-(B1solid.sy+B2(solid.sx+solid.sz))		Refractive index, y-component
Nz	N-(B1solid.sz+B2(solid.sx+solid.sy))		Refractive index, z-component

Locate the section. From the list, choose .

Select Domains 4–6 only.

Solid Mechanics (solid)

Rigid Motion Suppression 1

In the toolbar, click

and choose .

Select Domain 1 only.

Electromagnetic Waves, Frequency Domain (ewfd)

The computational domain is reduced significantly for the optical mode analysis.

In the window, under click .

Select Domains 4–6 only.

Wave Equation, Electric 1

In the window, under click .

In the window for , locate the section.

From the n list, choose . From the list, choose .

In the n table, enter the following settings:


Nx	0	0
0	Ny	0
0	0	Nz

From the k list, choose .

Materials

In the window, under right-click and choose .

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

Select Domain 1 only.

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


Property	Variable	Value	Unit	Property group
Young’s modulus	E	ESi	Pa	Young’s modulus and Poisson’s ratio
Poisson’s ratio	nu	nuSi	1	Young’s modulus and Poisson’s ratio
Density	rho	rhoSi	kg/m³	Basic
Coefficient of thermal expansion	alpha_iso ; alphaii = alpha_iso, alphaij = 0	alphaSi	1/K	Basic

SiO2

Right-click and choose .

In the window for , type SiO2 in the text field.

Select Domains 2–6 only.

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


Property	Variable	Value	Unit	Property group
Young’s modulus	E	ESiO2	Pa	Young’s modulus and Poisson’s ratio
Poisson’s ratio	nu	nuSiO2	1	Young’s modulus and Poisson’s ratio
Density	rho	rhoSiO2	kg/m³	Basic
Coefficient of thermal expansion	alpha_iso ; alphaii = alpha_iso, alphaij = 0	alphaSiO2	1/K	Basic

Mesh 1

Free Triangular 1

In the toolbar, click

Size 1

Right-click and choose .

In the window for , locate the section.

From the list, choose .

Select Domains 4–6 only.

Locate the section. Click the button.

Locate the section.

Select the check box. In the associated text field, type 0.2.

Size

In the window, under click .

In the window for , locate the section.

From the list, choose .

Click

Study 1

Mode Analysis

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and choose .

In the window for , locate the section.

From the list, choose .

In the text field, type 1.46.

Select the check box. In the associated text field, type 4.

In the text field, type c_const/lambda0_ewfd.

These settings make the eigenmode solver search for the 4 eigenmodes with effective mode indices closest to the value 1.46. This value is an estimate of the effective mode index for the fundamental mode.

Exclude Solid Mechanics from the Mode Analysis step.

Locate the section. In the table, clear the check box for .

Step 1: Stationary

In the window, click .

In the window for , locate the section.

In the table, clear the check box for .

In the toolbar, click

Results

Stress (solid)

In the window for , locate the section.

From the list, choose .

In the toolbar, click

Click the

button in the toolbar.

Click the

button in the toolbar.

In the toolbar, click

Study 1/Solution 1 - Plane Strain

In the window, expand the node, then click .

In the window for , type Study 1/Solution 1 - Plane Strain in the text field.

Solid Mechanics (solid)

In the window, under click .

In the window for , locate the section.

From the list, choose .

Definitions

Variables 4

In the window, under right-click and choose .

In the window for , locate the section.

In the table, enter the following settings:


Name	Expression	Unit	Description
u1	u-solid.wxZ/2*Z^2	m	Displacement, x-component
v1	v-solid.wyZ/2*Z^2	m	Displacement, y-component
w1	(solid.wxZX+solid.wyZY+solid.w0Z)*Z	m	Displacement, z-component