Phase-Multiplied Photoelastic and Series Interferometer Arrangement for Full-Field Stress Measurement in Single Crystals

Abstract

A compact, phase-multiplied, circular polariscope and series interferometer arrangement is developed for high-resolution, full-field stress measurement in single crystals with weak piezo-optical coefficients. We present a general stress-optic law, derived from anisotropic piezo-optical constitutive relations, which provides the theoretical framework for obtaining stress field components from measured optical isoclinic, isochromatic and isopachic phase maps. A new phase image processing technique is also developed, which combines data obtained from different interference configurations for the successful removal of low-modulation zones within isoclinic and isopachic phase maps. The validity and accuracy of the proposed interferometer arrangement and stress measurement methodology are demonstrated through a compression test of a c-cut single crystal sapphire plate loaded by a cylindrical indenter.

Appendix: Jones Calculus of Bright- and Dark-field Interferometry

In the optical arrangement as shown in Fig. 2(a), the input light beam becomes right-circularly polarized after it passes through the horizontal linear polarizer and the λ/4 plate whose fast axis is aligned at 45^o with respect to the positive x-axis. The normalized Jones vector of the right-circular light is given by

$$ a = \frac{1}{{\sqrt {2} }}\left[ {\begin{array}{*{20}{c}} 1 \\ { - i} \\ \end{array} } \right]. $$

(A1)

The Jones matrix of the birefringent specimen plate is written as

$$ R = \left[ {\begin{array}{*{20}{c}} {{e^{{ - i{\delta_1}}}}{{\cos }^2}\theta + {e^{{ - i{\delta_2}}}}{{\sin }^2}\theta } & { - \left( {{e^{{ - i{\delta_1}}}} - {e^{{ - i{\delta_2}}}}} \right)\sin \theta \cos \theta } \\ { - \left( {{e^{{ - i{\delta_1}}}} - {e^{{ - i{\delta_2}}}}} \right)\sin \theta \cos \theta } & {{e^{{ - i{\delta_1}}}}{{\sin }^2}\theta + {e^{{ - i{\delta_2}}}}{{\cos }^2}\theta } \\ \end{array} } \right], $$

(A2)

in which δ ₁ and δ ₂ are the two principal retardations and θ is the orientation of δ ₁ with respect to the positive x-axis.

According to the Jones calculus, the Jones vector of the first-order reference beam emerging from the specimen is a ₁ = Ra. For the (2m+3)-th order measurement beam, it travels through the plate for (2m+3) times. Therefore, its Jones vector is given by ${\mathbf{a}}_{{2{\text{m}} + 3}} = {\mathbf{R}}^{{2{\text{m}} + 3}} {\mathbf{a}}$. The Jones vector of coherent combination of the reference and measurement beams is expressed by

$$ {\mathbf{a}}\prime = {\mathbf{a}}_{1} + {\mathbf{a}}_{{2{\text{m}} + 3}} = {\left( {{\mathbf{R}} + {\mathbf{R}}^{{2{\text{m}} + 3}} } \right)}{\mathbf{a}}. $$

(A3)

For the configurations of bright- and dark-field interferometry, the Jones vectors of the light beams that emerge from the output λ/4 plate and analyzer are given by

$$ a\prime \prime = {P_{{3\pi /4}}}{Q_{{\pi /2}}}a\prime = \frac{1}{2}\left[ {\begin{array}{*{20}{c}} 1 & 1 \\ 1 & 1 \\ \end{array} } \right]\left[ {\begin{array}{*{20}{c}} 1 & 0 \\ 0 & i \\ \end{array} } \right]a\prime, $$

(A4)

and

$$ a\prime \prime \prime = {P_{{\pi /4}}}{Q_{{\pi /2}}}a\prime = \frac{1}{2}\left[ {\begin{array}{*{20}{c}} 1 & { - 1} \\ { - 1} & 1 \\ \end{array} } \right]\left[ {\begin{array}{*{20}{c}} 1 & 0 \\ 0 & i \\ \end{array} } \right]a\prime, $$

(A5)

respectively. The light intensities of the two beams are calculated as ${I_{\text{Bright}}} = \widetilde{a}\prime \prime a\prime \prime$ and ${I_{\text{Dark}}} = \widetilde{a}\prime \prime \prime a\prime \prime \prime$, where the tilde notation denotes the transpose complex conjugate operation. Substitution of equations (A1)–(A5) into the two intensity equations and subsequent simplification lead to

$$ {I_{\text{Bright}}} = 1 + \frac{1}{2}\cos \left( {{\delta_{\text{diff}}}} \right) + \frac{1}{2}\cos \left[ {\left( {2m + 3} \right){\delta_{\text{diff}}}} \right] + \left\{ {\cos \left[ {\left( {m + 1} \right){\delta_{\text{diff}}}} \right] + \cos \left[ {\left( {m + 2} \right){\delta_{\text{diff}}}} \right]} \right\}\cos \left[ {\left( {m + 1} \right){\delta_{\text{sum}}}} \right], $$

(A6)

and

$$ {I_{\text{Dark}}} = 1 - \frac{1}{2}\cos \left( {{\delta_{\text{diff}}}} \right) - \frac{1}{2}\cos \left[ {\left( {2m + 3} \right){\delta_{\text{diff}}}} \right] + \left\{ {\cos \left[ {\left( {m + 1} \right){\delta_{\text{diff}}}} \right] - \cos \left[ {\left( {m + 2} \right){\delta_{\text{diff}}}} \right]} \right\}\cos \left[ {\left( {m + 1} \right){\delta_{\text{sum}}}} \right]. $$

(A7)