# Techniques for the determination of absolute retardation in photoelasticity

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## Abstract

The purpose of this paper is to give an interpretation to the fringes observed in holographic interferometry when plane-polarized light or circularly polarized light is utilized. It is shown that, when plane-polarized light is utilized and both the loaded and the unloaded states are considered, the obtained patterns are formed by the superposition of three families of fringes: the two families of absolute optical retardation and the family of relative retardation. The intensity distribution is also a function of the orientation of the plane of polarization, and along the points where the plane of polarization is parallel to one of the principal directions, only one of the families of absolute retardation is observed. By utilizing circularly polarized light, the dependence on the orientation of the principal axis is eliminated and patterns consisting of the superposition of the three above-mentioned families are obtained. If only the loaded state is considered, the holographic interferometer behaves as an ordinary polariscope with the reference beam playing the role of the analyzer. The relationships between the observed families are discussed. Examples of application to the disk and ring under diametral compression are also given.

### Keywords

Mechanical Engineer Fluid Dynamics Loaded State Intensity Distribution Principal Axis### List of Symbols

*a*stress-optical constant

- \(\bar a\)
\(stress - optical constant = \frac{a}{{(a^2 - b^2 )h}}\)

**A**_{R}complex representation of the reconstructed first order wavefront (scalar wave)

*b*stress-optical constant

- \(\bar b\)
\(stress - optical constant = \frac{b}{{(a^2 - b^2 )h}}\)

*c*cos β

*E*_{1}exposure=

*I·t*, first exposure*E*_{2}same, second exposure

*h*thickness of the unloaded model

*h*′thickness of the loaded model

*I*_{1}light intensity, first exposure

*I*_{R}light intensity of the first order in the reconstruction

*I*_{PL}light intensity in the reconstructed image, plane-polarized light, bright background

*I*_{PD}light intensity in the reconstructed image, plane-polarized light, dark background

*I*_{CL}light intensity in the reconstructed image, circularly polarized light, bright background

*I*_{CD}light intensity in the reconstructed image, circularly polarized light, dark background

*k*\(\frac{{2\pi }}{\lambda }\)

*K*\(ratio of the exposure times = \frac{{t_1 }}{{t_2 }}\)

*k*_{1},*k*_{2}parameters

*M*constant of proportionality

*n*_{o}index of refraction of the model in the unloaded condition

*n*_{1},*n*_{2}principal indexes of refraction

**R**complex representation of the reconstruction wavefront

*R*amplitude modulus of the reconstruction wavefront

*s*sin β

*S*\(R^2 \in 1^2 \in R^2\)

*t*_{1},*t*_{2}exposure times, first and second exposures respectively

*T*transmission function of the holographic plate

- γ
slope of the straight-line portion of the plot density vs. the logarithm of the exposure

- \(\delta _1\)
absolute variation of the optical path of the polarized component vibrating parallel to the principal direction 1

- \(\delta _2\)
same, with respect to the direction 2

- \(\delta _3\)
difference between the absolute variation of the optical paths 1 and 2

- \(\varepsilon 1\)
object wavefront during the first exposure (vector wave) primed →

*x′*,*y′*,*z′*unprimed →*x*,*y*,*z;*- \(\varepsilon 2\)
same, during the second exposure

- \(\varepsilon R\)
reference wavefront

- \(\varepsilon 1T,\varepsilon 2T\)
total light vectors, first and second exposure respectively

- \(\bar \varepsilon\)
- complex amplitude vectors primed →
*x′, y′, z′*unprimed →*x, y, z* - \(\varepsilon _x ,\varepsilon _y\)
scalar components of the complex amplitude vectors

- \(\varepsilon R, \varepsilon 1, \varepsilon 2\)
modulus of the reference wavefront, the object wavefront (first exposure), the object wavefront (second exposure).\(\varepsilon = \sqrt {\varepsilon _{x^2 } + \varepsilon _{y^2 } }\)

- \(\psi x, \psi y\)
phases of the components of the amplitude vector

- θ
angle of inclination of the reference and object wavefronts with respect to the plane of the hologram

- λ
wavelength of the light

- \(\sigma _1 ,\sigma _2\)
principal stresses

- ψ
phase change introduced by the unloaded model

- \(\psi _1\)
change of phase introduced by the loaded model in the principal direction 1

- \(\psi _2\)
same, in the principal direction 2

- ω
angular frequency of the light

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