Authored by: Julia Zhang, Chris Normanshire

Introduction

Stress birefringence is a phenomenon where a material becomes optically anisotropic when subjected to stress from various sources—including mechanical loads, thermal induced gradients, or manufacturing processes. This leads to system performance changes including:

• Phase retardation between orthogonal polarization components
• Local variations in refractive index causing wavefront aberrations
• Depolarization when stressed optical components alter the polarization state of light, resulting in loss of polarization information, reduced image quality, and unpredictable performance.

These effects produce observable optical defects including double imaging, reduced contrast, color fringing, and distorted wavefronts that lead to focus shifts or aberrations.

In this article, we will introduce basic concepts of the index ellipsoid, stress tensors, and how they are connected through stress-optical coefficients to analyze stress birefringence. We will use a polariscope example to demonstrate how to load stress data and evaluate performance affected by stress birefringence. Please also refer to the article Introduction to STAR Stress Birefringence for more information on this topic.

 

Index ellipsoid

The index ellipsoid, also known as the optical indicatrix, is a 3D ellipsoid that helps us understand how light moves through materials by visualizing the refractive index "n" for all polarization directions. The shape is defined by using three mutually perpendicular axes (principal index axes) with the magnitude of refractive indices (n₁, n₂, n₃). These indices are determined by the material's crystal structure or stress state. In isotropic materials, all three indices are the same (n₁ = n₂ = n₃), and light travels the same way in all directions with no special optical axes. In birefringent materials:

• Uniaxial crystals have two indices equal (n₁ = n₂ ≠ n₃), with one principal optical axis that lines up with the unique index (n₃)
• Biaxial crystals have all three indices different (n₁ ≠ n₂ ≠ n₃), and they have two principal optical axes in the same plane as the largest and smallest indices[1].

The principal optical axes (where light travels without splitting into two beams and which are determined by crystal symmetry) are different from principal index axes, which show directions of maximum and minimum refractive indices.

 

 

Fig1:Index ellipsoid of isotropic, uniaxial, biaxial crystals

Electric displacement vector and refraction index

In electromagnetism, the following fundamental quantities that describe electromagnetic fields include the electric field E, magnetic field H,  electric displacement D ,magnetic induction B . These quantities are interconnected through Maxwell's equations and material equations.

The Poynting vector S quantifies the directional energy flow of the electromagnetic field. In anisotropic materials, the direction of the wavefront (k) differs from the direction of energy propagation.

D=εE

H∝k X E

S = E X H

Due to the properties of cross products, H is perpendicular to both k and E, S  is perpendicular to both E  and H,

For linear, homogeneous, isotropic materials, the permittivity is a scalar, and the directions of D and E  are identical. However, in anisotropic media, this is not the case, as the permittivity becomes a second-rank tensor (3x3 matrix). In tensor form, the relationship of the refractive index and permittivity can be written as:

For an isotropic material under no stress:

• The diagonal elements are equal: ε₁₁ = ε₂₂ = ε₃₃ = n²
• Off-diagonal elements are zero: εᵢⱼ = 0 (where i ≠ j)

By applying the tensor effect in anisotropic materials,  and  can point in different directions.

By utilizing the index ellipsoid for an anisotropic material, we can determine the allowed electric displacement vector  and its associated index value. For any given wave vector , there are two allowed polarization directions for the electric displacement vector . These directions are perpendicular to each other, and the  vectors correspond to the major and minor axes of the ellipse formed when the k-perpendicular plane intersects the index ellipsoid. The two vectors denoted as  and  with the following relationships:

This leads to different propagation speeds for different polarization directions, and the direction of energy flow for the two polarizations can differ from the direction of the wavefront.

Fig2: Sketch showing the use of the index ellipsoid to determine n1​ and n2​

 

Stress tensor and stress optical coefficients

As discussed above, a material's refractive index changes due to anisotropy, and mechanical stress can induce anisotropic behavior in optical materials, altering their refractive properties. This relationship between mechanical stress and optical behavior is fundamental for understanding light propagation through materials. By knowing the stress distribution within a material, we can calculate the resulting refractive index changes. The stress-optical law provides the mathematical framework connecting mechanical stress states to optical properties, showing how stress information translates into refractive index modifications.

Stress tensor

The Cauchy Stress Tensor[2] is a mathematical representation that describes the state of stress at a point in a material. It is represented by a 3×3 matrix:

where:

• σᵢᵢ (i=1,2,3) are normal stress components
• σᵢⱼ (i≠j) are shear stress components

 

Due to moment equilibrium:

The number of independent components reduces from 9 to 6. There are:

• 3 Normal Stress Components (σᵢᵢ), represented by diagonal elements in the stress matrix (σ₁₁, σ₂₂, σ₃₃). These act perpendicular to a surface, and cause compression or tension in the material
• And 3 Shear Stress Components (σᵢⱼ where i≠j), represented by off-diagonal elements in the matrix, these act parallel to a surface, they cause deformation through sliding

Fig3: Components of stress in three dimensions

In OpticStudio, the allowed FEA/CFD data format are nine columns in a tab-delimited format [X position, Y position, Z position, SX, SY, SZ, SXY, SYZ, SXZ]. SX, SY, SZ are Normal Stress Components and SXY, SYZ, SXZ are Shear Stress Components.

From this stress tensor data, we can calculate eigenvalues[3] to determine the principal stresses (σ₁, σ₂, σ₃) in the material in engineering.

Stress optical coefficients

The analysis of stress-induced birefringence can be systematically approached through dimensional reduction to two-dimensional plane stress problems, followed by the alignment of principal stress directions with principal optical index axes, wherein stress-optic coefficients (K, K₁₁, K₁₂) establish a linear relationship between principal stresses and principal index changes of the material:

• K: relates the difference of principal stresses to birefringence
• K₁₁: relates principal stress σ₁ to principal refractive index n₁ changes
• K₁₂: principal stress σ₂ to refractive index n₂ changes

After identifying principal stress directions through stress transformation, we calculate the principal stress values and apply them to the stress-optic law to determine the principal index values:

where:

• nᵢ is the refractive index along principal axis i in the index ellipsoid
• n₀ is the unstressed refractive index
• σᵢᵢ, σⱼⱼ, σₖₖ are normal stress components when shear stresses are 0

A use coefficient K can be used to relate the difference in principal stresses to stress optical birefringence

Where , ,are the principal stresses[4] (eigenvalues of the stress tensor), , are the principal indices.

 

Example

Building on these stress birefringence concepts, once we’ve computed the stress induced refractive index changes we can investigate their impact on the performance of optical systems. To demonstrate this, we'll examine how stress patterns can be measured and analyzed using a polariscope, a specialized optical instrument designed for visualizing stress distributions in transparent materials.

 

Two common configurations are used in photoelastic analysis: plane polariscope and circular polariscope[5]. Each setup has its unique advantages and applications in stress measurement. We will demonstrate an example using OpticStudio to model the measurement of a planar optics with prepared stress data. Using the generated sample data, we will demonstrate how these stress patterns appear in the optical performance results when applied through OpticStudio's STAR feature.

For engineering simulation, stress tensor data can be generated using several softwares packages:

• Structural analysis software like Ansys Mechanical, ABAQUS, and NASTRAN use finite element analysis (FEA) to generate complete stress tensor fields
• Injection molding software such as Moldflow, Moldex3D, and SIGMA can simulate molding processes and calculate residual stress tensors in optical components
• Specialized optical stress analysis tools that calculate component stress patterns and export tensor data

Optical setup

The polariscope optical setup consists of a monochromatic light source (such as a laser at 632.8 nm), followed by a Y-oriented linear polarizer that establishes vertically polarized light. This light then passes through a quarter-wave plate, with its fast axis at 45 degrees (with respect to Y), to create circular polarization before encountering the planar testing plate.

After passing through the testing plate, a second quarter-wave plate (fast axis at 135 degrees relative to Y) and X-oriented analyzer work together to analyze stress-induced changes. In a stress-free testing plate, the light would be converted back to Y-polarized and completely blocked by the X-analyzer. However, any stress in the testing plate creates birefringence that allows some light to pass through the analyzer, creating patterns that reveal stress distribution.

Fig4: Circular polariscope optical setup

 

A plane polariscope includes a light source, followed by a Y-oriented linear polarizer, the testing plate, and an X-oriented analyzer. This basic arrangement differs from the circular polariscope by removing the two quarter-wave plates. In this configuration, when the principal stress directions in the testing plate align with either the polarizer or analyzer axes, dark bands (isoclinics) appear. The dark bands from the results indicate stress direction in the material. By rotating the polarizer and analyzer together, we can observe where dark bands form—the angle at which a dark band appears directly reveals the direction of stress at that location.

 

Fig5: Plane polariscope optical setup

To demonstrate how a circular polariscope measures stress-induced birefringence while eliminating isoclinic effects, we can replace the testing plate with a quarter-wave plate and observe how the interference patterns' peak irradiance changes as we rotate the replacement plate's fast axis.

Fig6: Comparison of peak irradiance in circular and plane Polariscopes due to quarter-wave plate rotation

The red curve maintains a constant value regardless of the fast axis rotation, while dark bands emerge as the axis rotates for the plane setup (blue curve). This comparison shows that the circular setup provides a clear view of stress magnitude distribution without confusion from polarization alignment effects. In contrast, the planar setup reveals additional information about the alignment relationship between the polarizer/analyzer and the sandwiched (testing) waveplate.

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Stress tensor data

Three stress data files were prepared to demonstrate patterns using the polariscope setup. The first data file generates symmetric patterns for basic evaluation using a circular polariscope setup, which we'll examine in this section. The second data file shows isochromatic lines that reveal areas of equal principal stress differences, also using a circular polariscope (and broadband wavelengths). The third data file demonstrates isoclinic effects using a plane polariscope. We'll explore the second and third stress data files in the section of extended simulation.

The first testing multiphysics data follows a quadratic stress distribution according to the equation below:

 

The symmetry in the stress distribution allows for straightforward comparison between theoretical predictions and observed fringe patterns. This stress data was selected specifically to showcase the feature's functionality without needing to explain its physical meaning.

Fig7: Constructed symmetric stress distribution

 

Data load and validation

In the optical configuration 1 lens file, navigate to the STAR tab. In the Multiphysics Data section, click on Multiphysics Data Loader to load the stress data. Since this is volume data, apply it to the first surface of the testing plate and select "Local" to transform the data to the lens local coordinate system.

 

Fig8: Stress data loading through Multiphysics Data Loader

Go to STAR...Analyses...System Viewer to check the Lens Retardance (the accumulated retardance as the wavefront travels through an optical component) or index change data overlaid on the lens within the ray footprint region.

Fig9: Showing Change in Index and Retardance from System Viewer

After applying the data, perform comprehensive analyses in OpticStudio, including Wavefront Map analysis, Huygens PSF calculations, Polarization Pupil Map data and Physical Optics Propagation (POP) simulations to validate the results. These analyses can be cross-referenced to verify consistency and accuracy, and demonstrate how stress-induced birefringence affects optical performance through detailed visualization of the test results after the testing plate.

Fig10: Huygens PSF results

 

Fig11: Wavefront Map results

Fig12: Physical Optics Propagation results

Fig13: Intensity plot using Polarization Pupil Map data

 

There are two methods for calculating stress-induced effects: the GRIN Ray trace model and Linear Ray trace model. The GRIN model discretizes the stress-induced birefringence into multiple steps, computing both the accumulated retardance and ray path deviations as light propagates through each step of the stressed material. The Linear model uses a simplified approximation for cases with minimal stress variations—rays follow straight paths inside the material while still accounting for retardance through the material. For detailed information about these two methods, please refer to the corresponding discussion in the help file.

Fig14: GRIN Ray trace model and Linear Ray trace model

 

Important settings to note:

• Uncheck Afocal Image Space under System Explorer to compare results in spatial space rather than angular space, even though this is an afocal system.
• Enable Force Planar when computing the Huygens Integral because this is an afocal system.
• Use the GRIN Ray Trace Model for calculating stress birefringence effects with Wavefront Map and Huygens PSF. Note that the Linear model will not affect POP results for this example.
• Since this example uses polarization components and stress birefringence effects are sensitive to polarization, enable the Use Polarization checkbox for all three analyses.
• Especially the "Unpolarized" setting in System Explorer should be turned off for accurate Huygens PSF calculation.

Extended simulation

In stress analysis using white light, isochromatic lines appear at locations where the difference between principal stresses (σ₁ - σ₂) are equal. For the optical configuration 1 circular polariscope setup, we analyze the white light response by applying the second stress data using three main wavelengths and combining their results.

The second stress data file uses asymmetric stress patterns to enhance color separation across wavelengths. In this case, regions with equal principal stress differences are indicated by lines of the same color, resulting in more distinct color separation.

Fig15: Three main wavelengths Physical Optics Propagation(POP) results

Fig16: Combined POP results showing isochromatic effects

 

In the optical configuration 2 plane polariscope setup, we load the third stress data that shows how stress direction changes across the lens when we rotate the (X and Y) polarizers or the testing plate. When we look at the POP simulation results, we can see dark areas that show where the stress direction matches the angle we set up. In a stress-free lens, crossing the polarizers would block all light, but stress in the material allows some light to pass through.

 

Fig17: Isoclinic effects indicating stress directions

 

Although the stress data used for demonstration purposes is fabricated in this article, accurate stress data for real-world applications requires proper stress simulation.

 

Summary

This article analyzes stress birefringence in optical system, covering index ellipsoid and stress tensor concepts and reviewing the stress optical law. It details applications using polariscope setups and demonstrate optical performance after applying stress data.

 

[1] https://lotusgemology.com/component/content/article?id=296:pleochroism...

[2] https://en.wikipedia.org/wiki/Cauchy_stress_tensor

[3] https://en.wikipedia.org/wiki/Eigenvalues_and_eigenvectors 

[4] https://docs.mat3ra.com/properties-directory/non-scalar/stress-tensor/

[5] https://en.wikipedia.org/wiki/Photoelasticity#Plane_polariscope_setup