br Here denote the components of the electric vector c

Here denote the components of the electric vector, c is the light speed, ω is the frequency, χ is permittivity of the stress free media, are components of the permittivity tensor induced by the residual stresses. The matriciant of Eq. (1) (Jones matrix) Ω(γ, α0, α*) can be expressed via its characteristic parameters: γ is the characteristic phase difference, α0 is the initial characteristic direction, α* is the secondary characteristic direction. In the case of a slow rotation of quasi-principal directions along the light propagation direction these MLN0128 Supplier parameters are related to the components of the dielectric tensor through the relationships
Equations of photoelasticity of a single cubic crystal [14] are the following: they contain the fourth-order elastic–optical tensor The essential difference of a problem under study from a problem for an isotropic medium is the presence of the second addend in the Eq. (2). This fact considerably complicates the solution of the problem. At first, an algorithm of determining the residual stress in long cubic crystal (the assumption of plane deformation) will be presented. Then, the application of the parametric tomography to the reconstruction of stresses in the common case will be considered.
Reconstruction of stresses in case of plane strain deformation
A cylindrical crystal without axial stress gradient is illuminated in the plane z = const. The components and are equal to zero and there is no rotation of the quasi-principal directions of the permittivity tensor. Thus, the problem of the optical tomography is simplified and only characteristic phase differences are measured on the ray. The optical relation (3) is complemented by equations of the elasticity theory: equations of state, equilibrium equations, compatibility equations. The residual stresses are considered to be of thermal character:
Here are the coefficients of the elasticity, α is the coefficient of thermal expansion, T is fictitious temperature, and are the components of the strains tensor. In the case of the plane strain deformation  = 0, and from Eqs. (4) stromatolite follows
Inserting the components of strain into the equation of compatibility and expressing the components of stress through the Airy function one can obtain
This equation can be written in more suitable form [20]: where and k is a solution of the equation
This equation is based on the fact that the Airy function F and its normal derivative ∂F/∂n are equal to zero in the load free conditions on the lateral surface. If the distribution of is described only by harmonic functions, the other stress components do not develop in a cylinder. The same situation takes place in the case of an isotropic cylinder [21]. General solution of Eq. (5) can be written as the sum of functions F = F1 + F2 which must satisfy the equation where . We can write Eq. (6) as a system of the following equations:
The boundary conditions for give the integral equation for the determination of the Here G12 are arbitrary solution of corresponding equations . At last the line integral (3) can be simplified by using the Airy function:
Thus, residual stresses can be determined from the partial solution of the Eq. (5) and the ray integral equation is
Application of the MPE method to the full determination of stresses
In the case of an arbitrary distribution of internal stresses in a sample, the number of variables increases and the reconstruction problem becomes significantly more complicated. In this case it is impossible to achieve a complete reconstruction of stresses by using only the equations of the elasticity theory. We apply the additional homogeneous magnetic field along the light propagation direction and measure the angle of rotation of the polarization plane due to the Faraday effect. The following two integrals can be determined using the polarization measurements [6,7]: