OUTLINE OF A 1D-PHOTOELASTICITY EXPERIMENT
Abstract
We take a transparent material (in this case a slab of acrylic), apply differing amounts of stress to it, and measure the resulting changes in the optical characteristics of the material. Once we understand our measurements well, we can then apply the technique to measure a known material having an unknown stress; the measured optical characteristics of the sample can then tell us the stress distribution in the sample.
To simplify things, we start out by performing 1D stresses – stresses applied along a single axis to an optically isotropic material. Such stresses alter an isotropic material into a uniaxial one, where the induced birefringence axis coincides with that of the applied stress (uniaxial crystals may be familiar as the crystalline form used to build waveplates). In the real world, no material is completely isotropic, but some materials are close enough; acrylic doesn’t adhere to this approximation very well because it contains large residual stresses frozen into the material during the manufacturing process. Thus, the 1D model is appropriate for acrylic only at very large applied stresses (where the residual stress is small in relative terms). In any case, if we stress the sample along one axis and use the 1D photoelasticity model, then we can treat the sample as a linear retarder of unknown retardation amplitude and unknown orientation.
To perform the measurement, we first install a broad-spectrum source which outputs circularly-polarized collimated light. Into this beam we place the transparent sample (our piece of acrylic) at normal incidence. After the sample we then place a spectropolarimeter to measure the polarization state of the beam across the entire wavelength region of interest (in this case 400nm – 700 nm). During the experiment, we apply different linear stresses to the sample, taking data at each step. Each data point then represents the polarization state of the beam exiting from the sample, averaged across the beam diameter. The center of the beam, having greater intensity, will of course have a greater weight in the overall average.
Once we have the Stokes spectra (polarization state at each wavelength) for each given step, we can then plot the Stokes spectra on the equator of the Poincaré Sphere, which amounts to plotting the data points on the (
s1,s2) plane (ignoring s3). A linear retarder acts on an input polarization vector by rotating that vector by an angle δ (the magnitude of the retardation) where the axis of rotation is twice the orientation angle φ of the waveplate: tan(2φ) = s1/s2. More concretely, we plot the data points on the (s1,s2) plane and fit a line to the points. The angle of that fitted line to the s2-axis gives the axis of rotation. Half that angle gives the orientation angle of the waveplate wrt the optical system. Next, to find the magnitude of the retardation, we can first rotate the Poincaré Sphere in 3D such that the new coordinate system (s'1, s'2, s'3) has its s01-axis coincident with the axis of rotation (2φ) of the stokes vectors. In this new coordinate system, we can then plot the data points on the (s'2, s'3) plane, producing an arc which may be part of a circle, or perhaps a circle which overlaps itself. The arctangent of the (s'2, s'3) vectors will then give us the angle (in radians) of the given data point, modulo 2π.In order to determine the absolute retardance of our linear retarder model, we can first note that whereas the relation of retardance to wavelength is
δ = 2πdΔn/λ, we can invert the equation to be in terms of wavenumber, such that δ = 2πdΔnσ, where d is the physical thickness of the retarder and Δn is its birefringence. Here we have a linear equation in sigma, where δ = 0 for σ = 0. Thus if we fit a line to our data in the wavenumber domain, extrapolate that line out to σ = 0 and find out where that extrapolated line intersects the δ-axis, we can determine how many 2π we need to add to our relative δ’s in order to get our absolute δ’s. Having our retardance values, we can then measure the physical thickness of our sample to derive the birefringence, Δn = λδ/2πd, or we can use the "Photoelastic Law" to derive the differential stress experienced by the sample, Δσ = δ/Cd, where Δσ is the differential stress (expressed in units of force per unit area) and C is the stress-optic coefficient for the sample material. Using a known coefficient for the type of material being used, we can then use this result to get a measure of the accuracy of our instrument. If we are using a material with a poorly-known coefficient (which is generally the case for plastics, which vary widely in optical characteristics with different manufacturing processes and even between batches using the same process), then we can use the resulting measurement to determine a value of C for the material.
Stressed Plastic
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