Photoelastic effect (stress-optic effect)

The photoelastic effect is the change in refractive index that accompanies mechanical stress or strain in a transparent material. Because a stress usually acts more strongly along one direction than another, it changes the index differently for the two polarizations, and an otherwise isotropic glass becomes birefringent. The effect is small: in N-BK7 a uniaxial stress of 1 MPa produces an index difference of about 2.8 × 10⁻⁶, and in fused silica about 3.5 × 10⁻⁶. Over centimeter paths this is tens of nanometers of retardance, enough to spoil polarization extinction in a precision system and enough to be seen by eye between crossed polarizers.

Stress-optic law and strain-optic tensor

For a plate under plane stress with principal stresses $\sigma_1$ and $\sigma_2$, the index difference between light polarized along the two principal directions is proportional to their difference,

$$\Delta n = C\,(\sigma_1 - \sigma_2),$$

where $C$ is the stress-optic coefficient, usually quoted in units of 10⁻¹² Pa⁻¹ (the brewster). Across a thickness $d$ the retardance is $\Gamma = C(\sigma_1 - \sigma_2)\,d$. Glass catalogs list $C$ for each glass; for N-BK7 it is about 2.77 × 10⁻¹² Pa⁻¹.

The microscopic description works with strain $S$ and the impermeability tensor $1/n^2$. To first order $\Delta(1/n^2)_i = p_{ij}S_j$, with $p_{ij}$ the strain-optic (Pockels photoelastic) coefficients, so a strain changes the index by roughly

$$\Delta n \approx -\tfrac{1}{2}\,n^3\,p\,S.$$

For silica, $n = 1.444$, $p_{11} = 0.121$ and $p_{12} = 0.270$. A strain of 10⁻⁴ acting through $p_{12}$ changes the index by about 4.1 × 10⁻⁵. In an isotropic solid only the difference $p_{11} - p_{12}$ produces birefringence; the average part shifts both indices together.

Worked example

A 10 mm thick N-BK7 window clamped with a uniaxial stress of 1 MPa acquires a retardance of 2.77 × 10⁻¹² × 10⁶ × 0.01 m = 27.7 nm. Viewed between crossed polarizers at 633 nm with the stress axis at 45°, the leakage is $\sin^2(\pi\Gamma/\lambda) = 1.9\%$, so the extinction of that path cannot exceed about 53:1 (17 dB). Holding the retardance to 5 nm raises the limit to about 1600:1 (32 dB).

How it is observed

The classic observation is photoelasticity in the mechanical-engineering sense: a model part cast from a polymer with a large stress-optic coefficient, loaded and viewed in a polariscope between crossed or circular polarizers, shows colored isochromatic fringes, each of which marks one more wave of retardance and so a fixed increment of $\sigma_1 - \sigma_2$. In optics labs the same arrangement, a light table between crossed polarizers, reveals residual stress in lens blanks, windows and molded parts. Quantitative maps come from imaging polarimeters that measure the full Stokes parameters or the retardance and its axis at each pixel, and glass suppliers specify residual stress birefringence in nm/cm.

Where it matters

• Fiber. Bending a fiber makes it birefringent, which is the operating principle of the paddle polarization controller, and frozen-in stress from boron-doped regions creates the high birefringence of polarization-maintaining fiber. Axial strain shifts the Bragg wavelength of a fiber Bragg grating by about 1.2 pm per microstrain at 1550 nm, of which the photoelastic part, with an effective coefficient of about 0.21 in silica, offsets roughly a fifth of the pure length change.
• Acousto-optics and scattering. A sound wave is a traveling strain grating, and through the photoelastic coefficients it diffracts light in an acousto-optic modulator, whose figure of merit contains $p^2$. The same coupling underlies Brillouin scattering.
• High-power lasers. Temperature gradients in a pumped rod create thermal stress, and the resulting birefringence depolarizes the beam alongside thermal lensing.
• Modulators. A photoelastic modulator drives a fused silica bar at a mechanical resonance, typically tens of kilohertz, to produce a periodically varying retardance for polarimetry and circular dichroism.

Pitfalls

Mounting stress is the usual cause of unexplained depolarization: a lens squeezed by a retaining ring or bonded with a shrinking adhesive can add several nanometers of retardance near its edge, and the effect varies with temperature as the mount and glass expand differently. Integrated waveguides inherit stress from cladding films, which shifts the TE and TM effective indices differently. In all these cases the error is a change of polarization state, which a power meter alone does not detect.

Common questions

What is the difference between the photoelastic and the electro-optic effect?

Both change the index tensor. The electro-optic effect responds to an electric field and needs a crystal without a center of symmetry for its linear form; the photoelastic effect responds to strain and occurs in every material, including glasses.

Why do some glasses show almost no stress birefringence?

In lead silicate glasses the stress-optic coefficient falls as the lead content rises, passes through zero at a high lead fraction and changes sign beyond it. Glasses formulated near the zero crossing, such as SF57, keep their polarization performance under mounting and thermal stress, at the cost of high density and strong dispersion.

References: A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley, 1984); B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 3rd ed. (Wiley, 2019).