25Stress-Induced Birefringence

25.1Introduction to Stress Birefringence

Stressis the distribution of forces inside a body that the neighboring parts of its materials exert on each other. Strain is the resulting deformation of the material, the atoms moving closer together or further apart in response to the forces, with a corresponding change in dimensions and volume. Such changes of volume change the refractive index and birefringence of the material, a change labeled stress-induced birefringence. This chapter presents algorithms for polarization ray tracing through lenses with stress birefringence and has examples showing the impact of stress on image quality. For such analyses, the stress distributions are usually described by arrays of stress tensors in files generated by computer-aided design (CAD) programs performing finite element analyses of stress and strain.

Consider some examples where stress is generated in optical elements. Each optical element has a distribution of forces throughout its volume. For example, gravity pulls a lens downward against its mount with the forces between the lens and mount squeezing the lens and generating internal stresses. Tightening screws in the lens mount further increases the forces against the lens, thus increasing the stresses inside. Heating the lens and the mount causes the lens and mount materials to expand at different rates, based on their coefficients of thermal expansion, modifying the stress distribution. Many lens coatings are applied in vacuum to hot lenses because the coating material will be deposited more uniformly and densely so that the resulting coatings are stronger. When the lens is removed from the chamber and the lens and coating cool and contract at different rates, the surface of the lens often ends up in compression, with a coating-induced stress near the lens surfaces.

Stress-induced birefringence is common and often unavoidable in optical systems. Optical materials undergo strain at the molecular level due to various environmental conditions, such as external forces and pressures, vibration, or temperature change. The microscopic strains induce birefringence and affect the wavefront and point spread function of optical systems. The associated stress-induced retardance is generally undesirable, changing the wavefront aberration and polarization aberration in complex patterns. Thus, when dealing with optical elements with stress birefringence, it is useful to be able to ray trace the effects of stress birefringence to assess its impact. Further, the optical system can be ray traced with different levels of stress birefringence to tolerance the maximum amount of stress that is acceptable based on the system’s image quality specifications.

Scottish physicist David Brewster1 discovered stress-induced birefringence in isotropic substances in 1816; this phenomenon is also called mechanical birefringence, photoelasticity, and stress birefringence. Stress presents in optical systems in two different forms, mechanically-induced or residual. Mechanical stress results from physical pressure, vibration, or thermal expansion and contraction. It often arises from mounts squeezing or applying force to elements. Such a property of an optical material is referred to as photoelasticity.2Residual stress is a permanent stress inside an element, independent of external forces. It commonly occurs during fabrication of injection-molded lenses or when glass is poorly annealed. As the material cools from liquid to solid, stress can easily become frozen into the material, particularly when the outer surface solidifies before the inner material. Most optical glass is annealed, heated to its annealing temperature, a temperature where the molecules have enough thermal energy to rearrange themselves slightly to reduce stress, but where the glass is not hot enough to deform. The glass is then cooled slowly to avoid the introduction of additional stress. Either mechanical stress or residual stress alters the material’s molecular structure slightly, some molecules are closer than equilibrium, others further apart, changing the optical properties in the direction of the stress and inducing birefringence.

Isotropic materials with stress-induced birefringence behave as spatially varying weakly uniaxial or biaxial materials. This induced birefringence is readily observed with interferometers3,4 and polariscopes,5,6 as shown in Figure 25.1. The colorful patterns of materials with stress birefringence between crossed polarizers is due to the wavelength-dependent variation of the retardance, with more birefringence for blue light and less for red light.

Figure 25.1The colors in this plastic cup, plastic tape dispenser, and prescription eyeglasses placed between crossed polarizers indicate large amounts of spatially varying stress.

Transparent isotropic optical elements can exhibit temporary birefringence when subjected to stress and revert to isotropic when the stress is released. This stress can be induced by overtightening a knob of an optical mount as shown in Figure 25.2.

Figure 25.2(a) An isotropic glass plate placed in an optical mount and viewed between crossed polarizers. (b) The same glass plate as a screw is tightened against the glass on the bottom. Because of the stress created, induced birefringence has caused significant light leakage between the crossed polarizers.

Stress birefringence can be intentionally induced in tempered glass for strength. In photoelastic modulators, an enormous sound wave is resonantly built up in the crystal to create a sinusoidally varying retardance for use as a polarization modulator. In general, stress birefringence is undesirable. For example, the stress that is common in injection-molded lenses induces wavefront aberrations and causes polarization aberrations, thus increasing the size of the point spread function and degrading the image.

25.2Stress Birefringence in Optical Systems

In most optical designs, stress must be nearly eliminated to ensure that the image quality of a high-performance optical system is maintained. A quick analysis can be performed by understanding the allowable induced birefringence. Some typical birefringence tolerances on lenses and glass blanks are 2 nm/cm for critical applications such as photolithography systems, polarimeters, and interferometers; 5 nm/cm in precision optics; 10 nm/cm for microscope objectives; and 20 nm/cm for eyepieces, viewfinder, and magnifying glasses.7,8 Many lenses are fabricated by injection molding plastics these days due to its low cost and ease of fabrication. However, plastic optics, particularly the harder plastics such as polycarbonate, suffer from significant stress birefringence resulting from the manufacturing process, including molding, cooling, and mounting. Molding parameters such as the pressure applied to the resin in a mold, the time spent in the mold, and the rate the molded lens cool are often adjusted to minimize stress birefringence, but completely eliminating it is difficult.

When significant amount of stress presents in an optical system, it can be important to incorporate the stress birefringence into a polarization ray trace to simulate its effect on image formation, fringe visibility, and other optical metrics. This chapter presents the mathematical description of stress and its relationship to optical birefringence. Standard methods to translate the non-uniform stress distribution in optical components to retardance are presented and algorithms for polarization ray tracing through components with stress are shown. The finite element model of example plastic lenses will be analyzed using retardance maps, Jones pupil images, and polarization point spread matrices to visually demonstrate the induced birefringence and its effects.

25.3Theory of Stress-Induced Birefringence

The stress birefringence can be described mathematically. When stress is applied to an object, the material deforms slightly as atoms reposition themselves in response to applied forces, as shown in Figure 25.3. The magnitudes of stress discussed in this chapter only induces small changes in atomic and molecular positions but causes negligible change in the material’s physical shape. This applied stress, contact stress, or principal stress is characterized by the force applied in kg·m·s−2 or Newton (N) to the relevant cross-sectional area; the resulting stress is given in units of N/m2 or Pascals (Pa). For a material such as glass or clear plastic, compression increases the refractive index in the compressive stress direction; atoms move closer together in this direction but further apart in the perpendicular plane. Conversely, tension decreases refractive index in the tensile stress direction as atoms expand along this axis. For small stress, after releasing the force, the glass returns to an isotropic state. This is referred to as elastic deformation. Above some threshold, irreversible changes to the molecular arrangements occur, and the object cannot return to its original shape. For example, glass may shatter or a plastic lens may become dented.

Figure 25.3The forces on an object generate compressive stress and tensile stress. (Left) The original shape and atomic position of an optic. (Right) The changes when stress is applied, the shape of the object slightly deformed; for the object shown, the change is exaggerated to visualize the effect of compressive and tensile stress.

Stress typically induces small changes in a material’s refractive index with Δn less than 1. Typically, a 15-MPa stress yields a Δn in the order of 0.0001 for glass, as shown in Figure 25.4. When an external force is applied along one direction to an isotropic piece of glass, it becomes a uniaxial material with its optic axis along the direction of external force. Application of a second force along a different direction causes the glass to become a biaxial material. These stresses alter the dielectric tensor material. Therefore, the stressed optical element is simulated as an anisotropic material. A real optical component generally has varying stress throughout its volume, such as Figures 25.1 and 25.2; hence, stress birefringence should be simulated as a spatially varying birefringent material.

Figure 25.4The change of refractive index Δn plotted as a function of material stress-optic coefficient C for 15 MPa applied stress, where Δn = C·stress. Glass such as N-BK7 and SF4 have C < 5. Polymers such as polymethylmethacrylate (PMMA), polystyrene (PS), and polycarbonate (PC) have larger C, which also vary significantly with temperature.

The typical stress-induced birefringence causes a very small amount of ray splitting. Figure 25.5 shows the very small change in refractive index (Δn) due to typical stress-induced birefringence for N-BK7 and polycarbonate (PC), which results in a negligibly small amount of ray doubling (Δθ°); thus, the two split modes propagate very close to each other, typically within a few micrometers. Since a few wavelengths of ray separation in a ray trace is insignificant and would usually not affect the accuracy of many calculations, this ray doubling can be safely ignored. The induced birefringence of the two orthogonally polarized modes can be modeled as retardance along a single ray path. In the ray trace analysis, a polarization ray tracing P matrix will be constructed to keep track of such stress-induced retardance. The following sections present the algorithm to translate the stress of an optical material into the resultant retardance, and the refraction is typically modeled based on Snell’s law using the unstressed refractive index.

Figure 25.5Refractive index change Δn and its corresponding ray split Δθas a function of applied stress for glass N-BK7 (left) and polymer polycarbonate (right). The ray split is calculated with algorithms presented in Chapter 19.

25.4Ray Tracing in Stress Birefringent Components

Algorithms will now be developed for simulating stress birefringence in optical systems by polarization ray tracing to allow its effects on image formation, fringe visibility and other optical metrics to be calculated. The applied stress and its induced retardance have a linear relationship. Therefore, the stress-induced birefringence is represented as a spatially varying linear retarder. P matrices keep track of stress-induced retardance along ray paths. The algorithms take the stress of an optical material as input and calculate the resultant retardance experienced by light rays. The calculation of the stress distributions involves methods from mechanical engineering that are not treated here.9

When multiple stresses are applied to an isotropic material in different directions, the stressed component becomes biaxial. The applied stress is represented as a 3 × 3 strain tensorS, defined in terms of normal stresses σ and shear stresses τ in an (x, y, z) coordinate system,

$$S=\left(\begin{array}{lll}{\sigma }_{xx}& {\tau }_{xy}& {\tau }_{xz}\\ {\tau }_{xy}& {\sigma }_{yy}& {\tau }_{yz}\\ {\tau }_{xz}& {\tau }_{yz}& {\sigma }_{zz}\end{array}\right).$$25.1

25.1

S contains six stress tensor coefficients σxx, σyy, σzz, τxy, τxz, and τyz,10–14 which represent the force per unit area from different directions in 3D. The amount of deformation in the optic relative to the original shape due to S is characterized by the strain tensor Γ,

$$\mathbf{\Gamma }=\left(\begin{array}{lll}{\gamma }_{xx}& {\gamma }_{xy}& {\gamma }_{xz}\\ {\gamma }_{xy}& {\gamma }_{yy}& {\gamma }_{yz}\\ {\gamma }_{xz}& {\gamma }_{yz}& {\gamma }_{zz}\end{array}\right),$$25.2

25.2

where γ’s are the strain tensor coefficients. The material refractive index is represented as a 3 × 3 dielectric tensor ε. Isotropic materials have a refractive index n0, a diagonal ε0 ($n_0^2$ × identity matrix), which changes and becomes non-diagonal ε when external stress is applied in arbitrary directions,

$${\mathbf{\epsilon }}_{\text{0}}=\left(\begin{array}{lll}{n}_{\text{0}}^2& 0& 0\\ 0& n_{\text{0}}^2& 0\\ 0& 0& n_{\text{0}}^2\end{array}\right)\to \mathbf{\epsilon }=\left(\begin{array}{lll}{n}_{xx}^2& n_{xy}^2& n_{xz}^2\\ n_{xy}^2& n_{yy}^2& n_{yz}^2\\ n_{xz}^2& n_{yz}^2& n_{zz}^2\end{array}\right),$$25.3

25.3

where n0 is the unstressed refractive index. Similarly, the impermeability tensor, the inverse of ε0, also changes from a diagonal tensor to a non-diagonal tensor with external stress,

$${\mathbf{\eta }}_0={\mathbf{\epsilon }}_0^{-1}=\left(\begin{array}{lll}1/n_0^2& 0& 0\\ 0& 1/n_0^2& 0\\ 0& 0& 1/n_0^2\end{array}\right)=\left(\begin{array}{lll}{\eta }_0& 0& 0\\ 0& {\eta }_0& 0\\ 0& 0& {\eta }_0\end{array}\right)\to \mathbf{\eta }=\left(\begin{array}{lll}{\eta }_{xx}& {\eta }_{xy}& {\eta }_{xz}\\ {\eta }_{xy}& {\eta }_{yy}& {\eta }_{yz}\\ {\eta }_{xz}& {\eta }_{yz}& {\eta }_{zz}\end{array}\right).$$25.4

25.4

Thus, the stress alters the index ellipsoid from spherical to ellipsoidal,

$$\begin{array}{l}\frac{x^2}{n_{\text{0}}^2}+\frac{y^2}{n_{\text{0}}^2}+\frac{z^2}{n_{\text{0}}^2}=x^2{\eta }_{\text{0}}+y^2{\eta }_{\text{0}}+z^2{\eta }_{\text{0}}=1\\ \downarrow \\ {\eta }_{xx}{x}^2+{\eta }_{yy}{y}^2+{\eta }_{zz}{z}^2+2{\eta }_{xy}xy+2{\eta }_{xz}xz+2{\eta }_{yz}yz=1,\end{array}$$25.5

25.5

where $x=\left(\frac{D_x}{\sqrt{2u}}\right)$, $y=\left(\frac{D_y}{\sqrt{2u}}\right)$, and $z=\left(\frac{D_z}{\sqrt{2u}}\right)$, and $u=\frac{1}{2}E\cdot D=\frac{1}{2}\left(\frac{D_x^2}{n_x^2}+\frac{D_y^2}{n_y^2}+\frac{D_z^2}{n_z^2}\right)$ is the energy density of the light field.

The link between the stress/strain and their optical effect on isotropic, non-magnetic, and non-absorbing material are the 6 × 6 stress optic tensorC and strain optic tensorΩ,

$$C=\left(\begin{array}{llllll}{C}_1& C_2& C_2& 0& 0& 0\\ C_2& C_1& C_2& 0& 0& 0\\ C_2& C_2& C_1& 0& 0& 0\\ 0& 0& 0& C_3& 0& 0\\ 0& 0& 0& 0& C_3& 0\\ 0& 0& 0& 0& 0& C_3\end{array}\right)\text{ and }\mathbf{\Omega }=\left(\begin{array}{llllll}{p}_1& p_2& p_2& 0& 0& 0\\ p_2& p_1& p_2& 0& 0& 0\\ p_2& p_2& p_1& 0& 0& 0\\ 0& 0& 0& p_3& 0& 0\\ 0& 0& 0& 0& p_3& 0\\ 0& 0& 0& 0& 0& p_3\end{array}\right)$$25.6

25.6

for isotropic materials and polymers with unidirectional symmetric structure. C is a function of the stress optic coefficients, C1 and C3 = C1 − C2 with units of inverse Pascals (1/Pa). Ω is a function of the strain optic coefficients, p1 and p2 with p3 = (p1 − p2)/2. These stress/strain optic coefficients are directly related to each other, the Young’s modulus (E) and Poisson’s ratio (ν) as

$$C_1=\frac{n_o^3}{2E}(p_1-2\nu p_2)\text{and}{C}_2=\frac{n_o^3}{2E}\left[p_2-\nu (p_1+p_2)\right].$$25.7

25.7

Then, the matrices Ω and C relate the strain and stress to the material’s refractive index as1

$$∆\mathbf{\eta }=\left(\begin{array}{l}1\text{/}{n}_{xx}^2-1\text{/}{n}_0^2\\ 1\text{/}{n}_{yy}^2-1\text{/}{n}_0^2\\ 1\text{/}{n}_{zz}^2-1\text{/}{n}_0^2\\ 1\text{/}{n}_{xy}^2\\ 1\text{/}{n}_{yz}^2\\ 1\text{/}{n}_{xz}^2\end{array}\right)=\mathbf{\Omega }\left(\begin{array}{l}{\gamma }_{\mathrm{xx}}\\ {\gamma }_{\mathrm{yy}}\\ {\gamma }_{\mathrm{zz}}\\ {\gamma }_{xy}\\ {\gamma }_{yz}\\ {\gamma }_{xz}\end{array}\right)\text{ and }∆\mathbf{\eta }=C_0\left(\begin{array}{l}{\sigma }_{xx}\\ {\sigma }_{yy}\\ {\sigma }_{zz}\\ {\tau }_{xy}\\ {\tau }_{yz}\\ {\tau }_{xz}\end{array}\right)\text{or}\left(\begin{array}{l}{n}_{xx}-n_{\text{0}}\\ n_{yy}-n_{\text{0}}\\ n_{zz}-n_{\text{0}}\\ n_{xy}\\ n_{yz}\\ n_{xz}\end{array}\right)\approx -C\left(\begin{array}{l}{\sigma }_{xx}\\ {\sigma }_{yy}\\ {\sigma }_{zz}\\ {\tau }_{xy}\\ {\tau }_{yz}\\ {\tau }_{xz}\end{array}\right).$$25.8

25.8

Table 25.1 contains a set of strain and stress optic tensor coefficients C1 and C2. The stress optic tensor coefficients for plastic are typically larger than glass, which means an equivalent stress applied on plastic yields a larger change of refractive index than glass. More compressible materials have larger response in general. PC is a very strong polymer, but generally has high stress-induced birefringence. Thus, PC is avoided in application where stress is a problem, but is popular in eyeglasses for its strength and the eye protection provided.

Table 25.1Refractive Index n0, Strain Optic Coefficients (p1 and p2), Young’s Modulus E, Poisson’s Ratio ν, and Stress Optic Tensor Coefficients (C1 and C2) for Glasses and Plastics at 633 nm

Materials	no	p1	p2	E (GPa)	ν	C1 (10−12/Pa)	C2 (10−12/Pa)
Corning 7940 fused silica12,15	1.46	0.121	0.270	70.4	0.17	0.65	4.50
Corning 7070 glass15	1.469	0.113	0.23	51.0	0.22	0.37	4.80
Corning 8363 glass15	1.97	0.196	0.185	62.7	0.29	5.41	4.54
Al2O312	1.76	−0.23	−0.03	367	0.22	−1.61	0.202
As2S315	2.60	0.24	0.22	16.3	0.24	72.5	59.1
Polystyrene16	1.57	0.30	0.31	3.2	0.34	53.9	62.0
Lucite17	1.491	0.30	0.28	4.35	0.37	35.4	24.9
Lexan17	1.582	0.252	0.321	2.2	0.37	13.0	98.1

Example 25.1Uniaxial Stress Optic Effect

An isotropic material under a stress in one direction becomes a uniaxial material with its optic axis along the direction of the applied stress. From Equation 25.8, when there is only a stress acting along the x-direction,18–20

$$\begin{array}{l}{C}_{\text{0}}\sigma =∆\eta =\frac{1}{n_{xx}^2}-\frac{1}{n_{\text{0}}^2}\\ \phantom{C_{\text{0}}\sigma }=\frac{n_{\text{0}}^2-n_{xx}^2}{n_{\text{0}}^2{n}_{xx}^2}=\frac{(n_{\text{0}}-n_{xx})(n_{\text{0}}+n_{xx})}{n_{\text{o}}^2{n}_{xx}^2},\text{ where }{n}_{\text{0}}\approx n_{xx}\\ \approx \frac{(n_{\text{0}}-n_{xx})(2n_{\text{0}})}{n_{\text{0}}^4}\\ \approx -\frac{2}{n_{\text{0}}^3}∆n.\end{array}$$

Thus,

$$∆n\approx -\frac{n_0^3}{2}{C}_0\sigma =-C\sigma .$$25.9

25.9

For common polymers, such as polymethyl methacrylate (PMMA), C0 is in the order of 10−12 Pa−1, which is equivalent to a Brewster. Several stress optic coefficients C of glass and polymers are listed in Table 25.2.

Table 25.2Stress Optic Coefficients C

Materials	C (10−12/Pa)
N-BK7 (Schott glass)21	2.77
F2 (Schott glass)21	2.81
SF4 (Schott glass)21	1.36
Poly methyl methacrylate (PMMA)22–24	~3.3, 4.5a
Polycarbonate25,27	30 to 72a
LBC3N (HOYA)27	0.43
FF5 (HOYA)27	3.31

Schott glasses were measured at 589.3 nm,21 and polymers and HOYA glasses were measured at 632.8 nm.27

a Parameter varies with modified material formulations for different applications.

From Equation 25.8, the stressed impermeability tensor is $\mathbf{\eta }={\mathbf{\eta }}_0+∆\mathbf{\eta }$, which can be converted to a dielectric tensor in the principal coordinate system. When η is expressed in its principal coordinates, it is a diagonal matrix with its eigenvalues (L1, L2, L3) along the diagonal,

$${\mathbf{\eta }}_{\mathrm{principle}}=\left(\begin{array}{lll}{L}_1& 0& 0\\ 0& L_2& 0\\ 0& 0& L_3\end{array}\right)\text{, so }{\mathbf{\epsilon }}_{\mathrm{principle}}=\left(\begin{array}{lll}1\text{/}{L}_1& 0& 0\\ 0& 1\text{/}{L}_2& 0\\ 0& 0& 1/L_3\end{array}\right).$$25.10

25.10

The dielectric tensor in principal coordinates is rotated to the global coordinate system as

$$\mathbf{\epsilon }={\mathbf{R}}^{-1}\cdot {\epsilon }_{\mathrm{principle}}\cdot \mathbf{R},$$25.11

25.11

where R is (v1v2v3) whose columns are the eigenvectors of η.

Stress applied to uniaxial or biaxial crystals requires additional stress optic coefficients to the coefficients shown in Equation 25.6; these are related to other strain coefficients. Their stress optic tensor require more non-zero components to account for interaction between off-diagonal ε components in both shear and normal stress.10,28,29

Using the results in Equations 25.8, 25.10, and 25.11, we obtain the stressed dielectric tensor that alters the polarization properties of a ray in a way similar to a biaxial dielectric tensor. As described in Chapter 19, when a ray passes through a stressed biaxial material, it splits into two modes with orthogonal electric field vectors E1 and E2, which carry two different refractive indices n1 and n2. The following assumptions are made to simplify the calculation of the stressed P matrix while retaining the necessary accuracy for useful optics calculations (generally better than λ/100 for phases):

1. The applied stress only causes small changes to the refractive index. Hence, refraction into and out of an element is accurately calculated by Snell’s law using the unstressed refractive index.

2. Ray doubling is ignored since two modes exiting the stress birefringent material are essentially on top of each other, as shown in Figure 25.5, and the path is well modeled by a single ray segment traced from the entrance to exiting face. Therefore, $\widehat{k}\approx \widehat{S}$ and $\widehat{D}\approx \widehat{E}$.

Using these assumptions, the refracted propagation vector is k,

$$n_i\sin {\theta }_i=n_0\sin {\theta }_0\text{ and }k=\frac{n_i}{n_0}{k}_i+\left(\frac{n_i}{n_0}\cos {\theta }_i-\cos {\theta }_0\right)\widehat{\mathbf{\eta }},$$25.12

25.12

where ni and n0 are the incident and unstressed refractive indices, θi and θ0 are the incident and refraction angles, ki is the incident propagation vector, and $\widehat{\mathbf{\eta }}$ is the surface normal. The refractive index and electric field vector for the two modes are calculated with the method of Section 19.5. By combining the anisotropic constitutive relation in Equation 19.7 and the Maxwell equations, the eigenvalue equation for E is

$$\left[\mathbf{\epsilon }+{(nK)}^2\right]E=0,$$25.13

25.13

where $K=\left(\begin{array}{ccc}0& -k_z& k_y\\ k_z& 0& -k_x\\ -k_y& k_x& 0\end{array}\right)$ and $\widehat{k}=(k_x,k_y,k_z)$ is the propagation direction within the stressed material. For non-zero E,

$$\left|\mathbf{\epsilon }+{(nK)}^2\right|=0$$25.14

25.14

with two refractive index solutions, n1 and n2 correspond to the two eigenmodes. Then, E1 and E2 are obtained through the singular value decomposition of [ε + (nK)2] with n1 and n2. Their Poynting vectors are the cross product of the electric and magnetic vectors,

$$S=\frac{ℛe\left[E\times H^{*}\right]}{\left|ℛe\left[E\times H^{*}\right]\right|},$$25.15

25.15

where

$$H=nK\cdot E.$$25.16

25.16

The P matrix of the ray within the stressed material is a linear retarder. It maps vectors $\left({\widehat{E}}_1,{\widehat{E}}_2,\widehat{S}\right)$ to $\left({\widehat{E}}_1{e}^{i\frac{2\pi }{\lambda }{n}_{1}d},{\widehat{E}}_2{e}^{i\frac{2\pi }{\lambda }{n}_{2}d},\widehat{S}\right)$. Thus,

$$P_{\mathrm{stressed}}=\left(\begin{array}{lll}{\widehat{E}}_1{e}^{i\frac{2\pi }{\lambda }{n}_{1}d}& {\widehat{E}}_2{e}^{i\frac{2\pi }{\lambda }{n}_{2}d}& \widehat{S}\end{array}\right){\left(\begin{array}{lll}{\widehat{E}}_1& {\widehat{E}}_2& \widehat{S}\end{array}\right)}^T,$$25.17

25.17

where λ is the wavelength of the light and d is the distance the light travels inside the material.

Example 25.2Uniaxial Stress of Compression and Tension

A compressive stress increases refractive index and produces a positive uniaxial effect; an isotropic medium changes into a weakly positive uniaxial medium. The direction of compression is the slow axis. The compressive stress σ has a negative sign.

A tensile stress or tension produces a negative uniaxial effect altering an isotropic material into a negative uniaxial material. Tension decreases refractive index; hence, the direction of such a stress denotes the fast axis. The tensile stress σ has a positive sign.

The phase of the light polarized along the fast axis is ahead of the mode polarized along the slow axis. For compression stress along y, retardance $\delta =2\pi \frac{∆n}{\lambda }d=\frac{2\pi }{\lambda }(n_y-n_x)d>0$. For tensile stress along y, retardance $\delta =2\pi \frac{∆n}{\lambda }d=\frac{2\pi }{\lambda }(n_y-n_x)d<0$. The linear retarder P matrix with retardance δ for this ray propagating in the z-direction is

$$\left(\begin{array}{lll}{e}^{i\frac{2\pi }{\lambda }{n}_{x}d}& 0& 0\\ 0& e^{i\frac{2\pi }{\lambda }{n}_{y}d}& 0\\ 0& 0& 1\end{array}\right)\text{or}\left(\begin{array}{lll}{e}^{-i\delta /2}& 0& 0\\ 0& e^{i\delta /2}& 0\\ 0& 0& 1\end{array}\right),$$25.18

25.18

and the two eigenmodes are e−iδ/2Ex =e−iδ/2 (1, 0, 0) and eiδ/2Ey = eiδ/2 (0, 1, 0).

25.5Ray Tracing through Stress Birefringence Components with Spatially Varying Stress

In optical elements with stress birefringence, the stress varies in a complex way. For example, when plastic optical elements are manufactured by injection molding, stress from lens mounts and vacuum windows create complex spatially varying stress birefringence, like Figure 25.2. Mechanical engineering software packages, such as SigFit,30 calculate the stress distributions within mechanical parts due to forces such as bolting items together, welding, and gravity. This finite element modeling (FEM) of stresses and many other parameters such as vibration characteristics are part of a mechanical design process. The resultant stress distribution is calculated at a discrete set of data points, thus the designation finite element. For injection molding, other software packages such as Moldflow and Timon3D31–33 analyze the complex physical process of injection molding by simulating the flow of the viscous melted plastic resin into a heated mold, the non-uniform cooling of the mold and part, the solidification under high compression, and the separation of the lens from the mold. Modeling has improved efficiency and quality in plastic molding to incorporate new polymers and meet the high demand in high-quality electronics, consumer products, and automobile industry.34 In both cases, mechanical stresses and molding stresses, the distribution of stress inside a 3D object is expressed as an array of 3 × 3 stress tensors. Figure 25.6 shows the 3 × 3 symmetric stress tensor distribution on the surface of an injection-molded lens, where high stress variations are observed in the diagonal elements.

Figure 25.6The 3 × 3 stress tensor distribution of an injection-molded lens. Red indicates tension, blue indicates compression, and gray indicates zero stress. The plastic entered the lens mold from the bottom at the region called the gate. The cylinder of plastic extending from the bottom of the lens is for handling the lens, which is moved by robot to a saw, where the cylinder is removed and the lens falls into packaging for delivery.

Simulating polarization change through an optical element with spatially varying stress involves four general steps listed below. These steps calculate OPL, retardance, and P matrices for a finite element stress model:

1. Extract the element’s optical shape from the finite element program’s files.

2. Calculate the optical path of a ray inside the element, and divide it into short segments.

3. Extract the stress information along that optical path.

4. Convert the stress optical distribution into retarder P matrices for the segments.

5. Multiply the P matrices to obtain an overall P matrix from entrance to exit.

25.5.1Storage of System Shape

Although the storage details of a 3D object vary between CAD systems, the idea is similar; the goal is to discretize a continuous region to a finite number of sub-domains. In general, an object stored in a CAD file is using numerous simple building blocks or finite elements, such as cuboids or tetrahedrons. Each element is specified by its vertices or nodes. A cuboid requires eight nodes, and a tetrahedron requires four nodes as shown in Figure 25.7b and c.

Figure 25.7(a) A tetrahedron element has four vertices (1, 2, 3, and 4) and four surfaces (I, II, III, and IV). (b) and (c) are surface plots of two different injection-molded lens structures. The surfaces are represented by surface triangles.

25.5.2Refraction and Reflections

Refraction and reflection are performed using the unstressed refractive indices and the surface triangles at the object surface extracted from a CAD file. Each surface triangle has its surface normal specified in global coordinate systems. For a given triangle, two vectors are calculated by taking the differences between the triangle vertices. The cross product of these two vectors approximates the surface normal for the triangular face. For a given ray, a ray intercept is calculated using the algorithm in Ref. 35. Figure 25.8 shows a ray grid refracting through an injection-molded lens. Using the ray intercept routine and Snell’s law, a set of incident collimated rays converge to an image after refracting through the lens surfaces.

Figure 25.8Refraction through an injection-molded lens uses surface triangles and Snell’s law to calculate ray intercepts. The figure shows a simulation of collimated rays converging through the lens.

25.5.3Stress Data Format

Stress information is contained in CAD files as an array of stress tensors, each with six stress coefficients, σxx, σyy, σzz, τxy, τxz, and τyz in Equation 25.8. A stress tensor is assigned to each element building block. When a ray propagates through a 3D object, the ray passes through multiple elements and experiences varying stress tensors along the ray segment. Usually, each stress tensor is associated with the center of one element block as shown in Figure 25.9. Thus, for an object composed of N element blocks, the stress file provides N stress data points distributed throughout the object’s volume. The light ray propagates through this cloud of points and interpolation can estimate the stress at arbitrary locations along the ray.

Figure 25.9A tetrahedron element defined by four corners (top row, left column) is reduced to one data point, shown as a red point at the center of the element. Ten tetrahedrons shown in the middle row are reduced to 11 data locations. The last row shows that all 95,656 tetrahedrons of an object are reduced to 95,656 data locations, each of them are associated to a stress tensor.

The stress component σxx of an example injection-molded lens is shown in Figure 25.10. It is common that most of the stress is concentrated in a thin layer at the surface. A stress tensor map with all nine components of another example lens is shown in Figure 25.11 with the magnitude of the stress represented by the color scale. Higher stress is observed around the gate (where the plastic melt flows into the mold) and the flange (an annular structure around the outside of the lens for mounting) of the plastic lens.

Figure 25.10Two views of the stress tensor coefficient σxx from the CAD file of an injection-molded lens.

Figure 25.11Stress tensor maps for the nine components are plotted across an object cross section in color scale.

25.5.4Polarization Ray Tracing Matrix for Spatially Varying Biaxial Stress

When a ray propagates through material with spatially varying stress, it experiences retardance change throughout the ray path. This is similar to a ray propagating through a spatially varying birefringent material; hence, the ray path is modeled as a stack of constant biaxial materials as shown in Figure 25.12. Figure 25.12 (left) shows a spatially varying birefringent material with a rotating optic axis and changing retardance magnitude. Such spatially varying behavior is simulated by dividing the material into thin slabs along the ray path as shown in Figure 25.12 (right). Each of the slabs represents a constant biaxial material with its unique optic axis orientation and retardance magnitude.

Figure 25.12For a single ray, a spatially varying biaxial material (left) can be simulated as a stack of constant biaxial slabs (right). A different stack is calculated for each ray.

The concept of slicing a ray path into segments through a material is applied to the stress data shown in Figure 25.13. (a) Along a ray path, a ray passes many stress data points. (b) The ray path is then divided into steps. (c) The midpoint of a step is unlikely to land exactly on top of a data point. (d) The stress tensor at the center of each step is interpolated from the data; for example, the weighted average of the three closest data points can be used. (e) A P matrix is calculated for the step from the stress tensor. The data point closest to the step contributes more than the data point data further away from the step. The number of steps should be chosen to be sufficiently large to model spatial variation of the stress birefringence and ensure accurate results.

Figure 25.13The ray path inside the spatially varying biaxial material is divided into steps. Interpolation is used to obtain stress information at each step. (a) A ray passes near many stress data points shown in blue along its ray path. (b) The ray path is then divided into steps shown as orange parallel lines along the ray path. (c) It is unlikely that the step exactly intersects any data point. (d) The N closest data points, three are highlighted in purple, are used to interpolate the stress for the step. (e) A P matrix is calculated for the step from the stress tensor, the ray direction, and the step length.

The stress grid is intended to sample the spatial stress variation densely enough that the change between nearby modes is small. In this limit, the stress can be interpolated. However, for more rapidly varying stresses, higher-order fitting equations could be needed, and interpolation would be performed on the retardance magnitude and retardance orientation separately.

One straightforward method to calculate the weighted average stress is similar to the interpolation algorithm shown in Chapter 20. Consider a set of 3 × 3 stress tensors (S1, S2, … SN) at Ndata locations (rs1, rs2, …, rsn, … rsN), describing the stress distribution of an object. A ray propagates through the object and its ray path within the stressed object is evenly divided into M steps (r1, r2, …rm, … rM). The distance between each step is d. For step m, the Q closest data points are (rm1, rm2, …rmq, …rmQ) with stress tensors (Sm1, Sm2, …,Smq …, SmQ). Q is chosen to produce a reasonable stress estimation at a location other than the N data points. Q = 3 is sufficient for the examples shown in this chapter. These data points are distance $\left(\left|{\overline{r}}_{m1}\right|,\left|{\overline{r}}_{m2}\right|,\mathrm{...},\left|{\overline{r}}_{mQ}\right|\right)$ = (|rm1 − rm|, |rm2 − rm|, …, |rmQ − rm|) away from step m. The stress at step m is interpolated from this data. Using

$$S_m={\displaystyle \sum _{q=1}^Q{A}_q{S}_q},$$25.19

25.19

where

$$A_q=\frac{{\left(\left|{\overline{r}}_{mq}\right|+{\epsilon }^{-17}\right)}^{-2}}{{\displaystyle \sum _{q=1}^Q\left[{\left(\left|{\overline{r}}_{mq}\right|+{\epsilon }^{-17}\right)}^{-2}\right]}}$$25.20

25.20

is a scaling factor based on distance $\left|{\overline{r}}_{mq}\right|$, which has a maximum of 1 and ε ≈ 10−17. Aq approaches Sq when it is near point rm and deemphasizes points further away. If rm is exactly on top of rmq, $\left|{\overline{r}}_{mq}\right|=0$ and Aq ≈ 1. Between data points, Aq averages effects from the closest Q data points.

Each step behaves as a linear retarder whose effect is represented by a P matrix. The stressed Pm matrix corresponding to step m is calculated using methods described in Section 25.3 and Equation 25.17 with average Sm in Equation 25.19. The net effect of the sequences of retardances along the ray path through the stressed element is obtained by multiplying the P matrices,

$$P_{\mathrm{stress}}={\displaystyle \prod _{m=1}^M{P}_{M-m+1}}.$$25.21

25.21

To represent a ray refracting into the stress material, propagating through the stressed material, and refracting out of the material, the total P matrix is

$$P_{\mathrm{out}}\cdot P_{\mathrm{stress}}\cdot P_{\mathrm{in}},$$25.22

25.22

where Pin and Pout are isotropic P matrices calculated as described in Chapter 9 with the unstressed refractive index for the incident and exiting surfaces.

25.5.5Examples of Spatially Varying Stress Function

Consider the example of light propagating through tempered glass, such as in an automobile windshield. As illustrated in Example 25.3, the parabolic stress distribution in a piece of tempered glass provides strength and stability through equalizing tension and compression. An imbalance in stress can cause unexpected weakness leading to spontaneous breakage.

Example 25.3Parabolic Stress in Tempered Glass

Tempered glass uses a parabolic stress distribution produced in a glass plate by controlled cooling. The example stress distribution is shown in Figure 25.14, where positive stress is tension and negative stress is compression applied along the x-direction. Having compression on the surface results in stronger molecular bonds and a stronger material. The tension in the center (a less dense region) balances the forces across the glass thickness.

Figure 25.14Parabolic stress across to a glass plate thickness.

Consider 500 nm light normal incident onto the plate with index 1.52. The ray experiences the retardance induced by compression stress from A to B in Figure 25.14. With the method described in Section 25.5.4, the ray path is divided into 30 slices of linear retarders, and the resultant PAB is

$$\left(\begin{array}{lll}0.707+0.707i& 0& 0\\ 0& 0.707-0.707i& 0\\ 0& 0& 1\end{array}\right),$$

which has 90° retardance magnitude with a y fast axis. Then, it travels from B to C with an increasing tension stress. The resultant PBC for that segment is

$$\left(\begin{array}{lll}0.707-0.707i& 0& 0\\ 0& 0.707+0.707i& 0\\ 0& 0& 1\end{array}\right),$$

which has 90° retardance magnitude with an x fast axis. The overall P from A to E is

$$\left(\begin{array}{lll}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right),$$

where retardance is cancelled to zero by the balance between tension and compression. The stress is purposely set up to distribute tensions and compressions through the component. Each segment does not need to have a quarter wave of retardance, but the important point is to balance the resultant retardance to zero.

Another example of stress tensor as a function of location on the component cross section is shown in Example 25.4. A 2 mm × 2 mm glass plate under a stress with the stress tensor function of Equation 25.23 becomes a rotationally symmetric retarder. Figure 25.16 shows the simulation results of the stressed plate under a linear polariscope. About 10 waves of retardance are observed across the 2 mm width. The orientation of the zero intensity cross rotates with the polariscope, since the stress axis of the plate is radially orientated. The simulated 2 × 2 Jones pupil (Figure 25.17) shows the coupling between the vertical (y) and horizontal (x) polarization through the stress plate. The coupling of xx and yy has the same pattern, which is the opposite pattern of the cross-coupling leakage of xy and yx. In the phase components, the xx and yy coupling shows the radial cycles from −π to π, while the cross couplings have jumps from −π/2 to π/2. When the amplitude and the phase images are considered together, the cross-coupling Jones pupils are real valued and have a saddle shape varying from +1 to −1.

Example 25.4Simulation of a Rotationally Stressed Plate

A stressed plane parallel plate under polariscope is simulated using P matrix multiplication. Consider of 0.1844 mm thick plate made of N-BK7 with a stress tensor:

$$S(x,y)=\left(\begin{array}{lll}{x}^2& x\times y& 0\\ x\times y& y^2& 0\\ 0& 0& 0\end{array}\right)\cdot 2000\text{MPa}\text{.}$$25.23

25.23

The stress magnitude and orientation is shown in Figure 25.15; the magnitude is zero along (x, y) = (0, 0) axis and increases quadratically from the center.

Figure 25.15Stress distribution of Equation 25.23. The stress is applied along the line and the stress magnitude is proportional to the length of the line. The larger the stress, the darker the line is, so larger stress is located toward the end of the plate.

Consider placing the stressed plate in a polariscope with two crossed polarizers. The stressed plate is a spatially varying A-plate. The polariscope’s P matrix is obtained from the 0° and 90° linear polarizer P matrices as P0° · Pstress · P90°, and the resultant intensity image is shown in Figure 25.16.

Figure 25.16Linear polariscope images of a stressed plate; the stress plate is in between horizontal and vertical polarizers shown on the left and in between 45° and 135° polarizers shown on the right. Black denotes zero intensity and white denotes high intensity.

The Jones pupil corresponding to Pstress is plotted in Figure 25.17.

Figure 25.17Jones pupil amplitude (left) and phase (right) of the stress waveplate.

25.6Effects of Stress Birefringence on Optical System Performance

There are many ways to analyze ray tracing results for systems containing stress birefringence, such as polarization state change or birefringence change. The most common figure of merit is retardance. Retardance magnitude and fast axis orientation across the pupil give insight into the stress location, orientation, and magnitude. In most optical systems, a quarter of a wave of retardance yields a significant amount of aberration, causing noticeable image degradation in the point spread function. In this section, simulations of various injection-molded lenses and images taken by polariscope for stress-induced plastic and glass elements are presented.

25.6.1Observing Stress Birefringence Using Polariscope

A quick way to observe birefringence is to use the polariscope. Since the polariscope measures the integrated retardance along the ray path, it is commonly used to analyze stress birefringence of transparent samples. Different configurations of polariscope reveal the magnitude and orientation of induced birefringent. Polariscopes have been used for quality inspections during glass and clear plastic lens manufacturing processes to identify defects and stress. Polariscopes are discussed in Section 7.7.1.

The polariscope images of some plastic samples (Figures 25.18 through 25.21) are taken from different polariscope configurations. The light leakage through the sample is a sign of stress birefringence in polariscope with cross polarizers. A plastic tape dispenser in Figure 25.18 and the CD substrate in Figure 25.19 show large variation of birefringence near the edges where the shape changes abruptly. Since the tape dispenser and the CD substrate are not typically used as optical elements in polarization-sensitive optical systems, large stress birefringence is not a problem. A pair of prescription eyeglasses, however, shows unexpectedly high birefringence. Since human eyes are insensitive to polarization, the birefringence has little effect on the performance of the eyeglasses.

Figure 25.18A plastic tape dispenser is placed under (a) crossed linear polarizers, (b) crossed circular polarizers, and (c) sensitive tint plate polariscope.

Figure 25.19A CD substrate is placed under (a) parallel linear polarizers, (b) crossed linear polarizers, (c) crossed circular polarizers, (d) sensitive tint plate polariscope, and (e) crossed polarizers with a quarter waveplate.

Figure 25.20A pair of plastic glasses is placed under (a) crossed linear polarizers, (b) crossed circular polarizers, and (c) sensitive tint plate polariscope.

Figure 25.21A plastic lens is placed under (a) parallel linear polarizers, (b) crossed linear polarizers, (c) crossed circular polarizers, (d) sensitive tint plate polariscope, and (e) crossed polarizers with a quarter waveplate. The plastic lens is oriented to show stress. The gate for plastic injection appears different for the two orientations, except in circular polariscope, which produces intensity independent of fast axis orientation.

The injection-molded convex lens in Figure 25.21 shows the typical Maltese cross pattern of lenses between crossed polarizers. The Maltese cross mostly comes from thin compressive layers on the surfaces of the lens as shown in Figure 25.10. The gate of the injection mold, where the plastic flows into the mold, is located at the side of the lens and typically has high stress birefringence relative to the rest of the lens.36 The two sets of images in Figure 25.21 show the polariscope images of the lens in five configurations of polariscope; the bottom row is rotated by 45°. By changing the orientation of the lens with different polariscope configurations, orientation of the stressed induced retardance can be qualitatively calculated. This top row of Figure 25.21 provides information on the linear vertical/horizontal and circular retardance; the bottom row provides information on the linear 45°/135° and circular retardance. In between linear cross polarizers, the main pattern is invariant with the lens orientation because the stress is predominantly radially oriented, similar to Figure 25.15. From the circular polariscope configuration, no pattern change is detected by rotating the lens; the retardance magnitude is revealed in one snapshot independent from the lens orientation. The color shift between blue and yellow when rotating the lens for 45° in the sensitive tint plate polariscope reveals that the birefringent axis is 45° with respect to two orthogonal directions of the Maltese cross pattern.

Mounting optical components can apply external forces that induce birefringence. Figures 25.22 through 25.24 show polariscope images of a rectangular plane parallel glass plate with increasing forces applied by a screw in the middle of one edge. The plate is supported by two pins on the opposite edge. It shows that unnecessary stress applied to optical elements causes stress that adds to the wavefront aberration. As shown later, this undesired stress induces polarization aberrations.

Figure 25.22A plane parallel glass plate in a linear polariscope with increasing force from (a) to (f).

Figure 25.23Linear polariscope images as the plane parallel glass plate of Figure 25.22f is rotated. The dark band indicates when its fast axis is aligned with the polariscope axes.

Figure 25.24With the addition of a tint plate retarder, the resultant color pattern changes. The pinkish background represents near zero retardance or nearly isotropic conditions.

A glass sample formed with intentionally high stress, as observed under a polariscope in Figure 25.25. The view of the stress pattern changes in different polariscope configurations. Testing is usually performed at 0° and 45° to observe both components of stress, associated with Stokes parameters S1 and S2, with different glass orientations providing kaleidoscope-like images.

Figure 25.25Glas Spannungenwith intrinsic stress is placed under (a) parallel linear polarizers, (b) crossed linear polarizers, (c) crossed circular polarizers, (d) sensitive tint plate polariscope, and (e) crossed polarizers with a quarter waveplate. The stressed glass is viewed in 0° and 45° orientations in the two rows that reveal all the stress. The birefringence pattern appears different for the two orientations, except in circular polariscope, which produces intensity independent of fast axis orientation.

25.6.2Simulations of Injection-Molded Lens

The simulation methods explained in Section 25.5 are applied to calculate the retardance induced by an injection-molded lens shown in Figure 25.26 with stress shown in Figure 25.27. The complete 3 × 3 stress tensor of this lens is also shown in Figure 25.11.

Figure 25.26The CAD image of an injection-molded lens in three different views.

Figure 25.27The stress images of the injection-molded lens (the same lens as in Figure 25.26) with a cutaway through the CAD variation file.

A collimated grid of rays is traced through this stressed lens. The retardance variation across the lens is calculated from the ray tracing P matrices and plotted in Figure 25.28. Rapid variation of retardance appears on the flange and the gate of the lens. The retardance is lower at the center and slightly higher toward the edge. This lens is also simulated in a polariscope with cross polarizers. The light leakage through the polariscope is due to the shape of the lens as well as the stress-induced retardance. Different stress optic coefficients result in different induced retardance across the object. The set of simulated figures with the higher stress coefficients gives higher retardance magnitude and induces more leakage through the polariscope.

Figure 25.28Retardance magnitude, retardance orientation, polariscope images, and birefringence color calculated from a polarization ray trace of an injection-molded lens with two sets of stress optic coefficients. (Top row) {C1, C2} is {5 × 10−14 Pa−1, 5 × 10−13 Pa−1}. (Bottom row) {C1, C2} is {6.5 × 10−13 Pa−1, 4.5 × 10−12 Pa−1}. The gate of the lens is located at the top of the lens shown.

25.6.3Simulation of a Plastic DVD Lens

DVD and CD systems are sensitive to stress-induced polarization from injection-molded lenses since the DVD signal is routed twice through a polarization beam splitter. The DVD signal is degraded by too much stress-induced aberrations. Too much degradation causes the bit error rate to increase to unacceptable levels. Without stress, the uncoated lens has zero retardance and performs as a conventional ray trace would predict for isotropic lenses; with stress, the lens has stress-induced retardance. The stress-induced retardance of an example DVD pickup lens (Figure 25.29) across the pupil is shown in Figure 25.30.

Figure 25.29(Left) Optical layout of an optical pickups system in a CD player.37 (Right) A model of an injection-molded lens focuses on a collimated incident beam.

Figure 25.30Induced retardance magnitude (left) and retardance orientation (right) of a stress lens across the pupil.

The DVD pickup lens configuration in Figure 25.29 (left) is illuminated through and then images through a polarizing beam splitter. The image quality is evaluated by its amplitude response matrix ARM, as shown in Figure 25.31. The main ARM components xx and yy are nearly Airy disks for the unstressed lens (Figure 25.31, left) with a small amount of light coupled into the off-diagonal elements xy and yx. When the polarization ray trace analysis is repeated with the stress data, the main xx and yy components are only slightly affected, but much more light is now coupled into the orthogonal polarization states, as shown by the off-diagonal elements xy and yx. The corresponding polarization modulation transfer functions (MTFs) are shown in Figure 25.32 with an overall lower modulation.

Figure 25.31The ARM of an unstressed (left) and stressed (right) injection-molded lens represented in log scale.

Figure 25.32The corresponding MTFs of an unstressed (left) and stressed (right) injection-molded lens for Figure 25.31.

As illustrated in Figure 25.33, the cross-coupling components in the resultant ARM depart more and more from an Airy disk as the stress increases. At extreme levels of stress (Figure 25.33e), the lens loses its ability to form an image.

Figure 25.33Point spread functions of polarized light refracting through a lens with (a) no stress, (b) some modeled stress, (c) 10 times the stress than (b), (d) 100 times the stress, and (e) 1000 times the stress.

25.7Conclusion

Stress in optical elements affects performance. In this chapter, complex stress distributions are modeled as a varying anisotropic material with spatially varying dielectric tensor. Stress in isotropic materials was treated here. Anisotropic material requires more complex stress optic tensors and strain optic tensors to describe its directional-dependent change induced by external stress. Similar algorithms with more elements than Equation 25.6 can still model the optical performance. This ray tracing method, modeling optical components with spatially varying optical properties, can be applied to simulate gradient index optics, liquid crystal, and other similar optical elements.

Being able to analyze and simulate the effect of stress in optical elements is important for industrial inspection, product control, and tolerance analysis for precision optical system. Stress birefringence can be measured by Mueller matrix polarimeters, polariscopes, the sensitive tint plate method, and the Senarmont method. Other birefringence measurement methods are also developed for higher accuracy and shorter processing time using photoelastic modulators, optical heterodyne interferometry,38 phase shifting technique,39 and near-field optical microscopy.40 By comparing measurement and simulation, improvements can be made in manufacturing procedures for molded lenses and other elements. The stress simulation has been used to predict the product performance and the trade-off between the processing time and the product quality in injection molding and to reduce the expensive cost of trial and error.

25.8Problem Sets

1. What are the differences between ray tracing through a sample with stress birefringence and a uniaxial sample?

2. Images in Figures 25.1 and 25.2 are both taken with crossed polarizers. Why does Figure 25.2 contain only black and white? And why does Figure 25.1 contain color?

3. 1. Is the polarizer in polarizing sunglasses on the outer surface, the inner surface, or spread throughout?

   2. If the stress birefringence of a pair of polarizing sunglasses was located in the middle, but not in the outer surfaces, what would be the effect in the three cases in (a)?

   3. Perform measurements with a polariscope or linear polarizers. What do your measurements show?

4. For the change in refractive index Δn of 0.0001, what is the corresponding stress for the materials in Table 25.2?

5. In the weak stress limit for isotropic materials, the change in refractive index Δn is related to the principal stress difference σxx − σyy through the stress-optic coefficients C1 and C2:

   $$∆n=C({\sigma }_{xx}-{\sigma }_{yy})=\frac{1}{2}{n}_0^3(C_1-C_2)\sigma ,$$

   

   2

   where no is the refractive index of unstressed material. At a wavelength of 633 nm:

   Lucite	n0 = 1.491	C1 = 35.4 × 10−12/Pa	C2 = 24.9 × 10−12/Pa
   Fused silica	n0 = 1.46	C1 = 0.65 × 10−12/Pa	C2 = 4.5 × 10−12/Pa

   1. For principal stress difference σxx − σyy = 100 MPa, what is the change in refractive index for lucite plastic? And for fused silica glass?

   2. Given the above stress, what is the retardance magnitude (expressed in waves) for a 1-mm-thick piece of lucite? And for a 1-mm-thick piece of fused silica?

   3. Assume 1-mm thickness for both fused silica and lucite. What is the principal stress difference needed in the glass to get the same retardance as 100 MPa stress in the plastic?

6. Using Figures 25.23 and 25.24, locate the birefringent axis of the stressed glass.

7. Explain what information can be concluded about the sample in Figure 25.25.

Acknowledgments

This chapter incorporates the work of Greg Smith, who developed the spatially varying stress algorithms used in many of the examples. Some of this work resulted from collaborations with Nalux Co. Ltd and Emhart Glass.

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1 See Equation 25.9 for the relationship between C0 and C.

2 A negative sign might apply for another convention of compression and tension.