#### 1. Introduction

Studies of the light beam propagation in anisotropic optical windows are of greater interest because of a wide range of possible applications of these elements in optical devices as well as new fundamental physical mechanisms that are present in such samples. The measurement of birefringence of anisotropic samples becomes attractive in the case of using of non-uniform anisotropic materials, for example, of gradient-doped laser crystals or tempered glasses. In this point of view the polarimetry techniques, based on the photoelastic phenomena, are the certain interest for investigation of the mechanical stress in tempered glasses [1 - 7]. Such glasses are anisotropic and measurements of the intensity of polarized light transmitted through the sample under study allow one to estimate the stress-induced birefringence of optical glass plates. By measuring the light intensity distribution it can be analyzed and evaluated the birefringence for a single point or a small glass area, but this is still not good enough for a big anisotropic sample. The polarization-optical technique proposed in [8] allows visualizing the stress distribution in an extended area of the analyzed optical windows. However, usually the magnitude of residual stresses, and hence birefringence, has a non-uniform distribution in the sample thickness. Determination of surface stress requires the preliminary measuring of a central stress, which can be measured using, for example, analysis of light scattering by inhomogeneities in the distribution of the glass refractive index [9,10].

In this work, using the example of a study of tempered glass with a non-uniform thickness-distribution of the birefringence value, a technique for measuring the birefringence-induced phase difference profile of the electric vectors of linearly polarized components of optical radiation passing through a birefringent sample is presented.

#### 2. Experimental details

This approach is based on recording of the spatial intensity distribution of the light scattered by inhomogeneities in the distribution of the glass refractive index. A typical experimental situation is simulated with the tempered glass parallelepiped stated along the vertical axis, making an angle of

In a practical experimental setup we expected the value of residual-stress over the glass plate thickness is not constant, but a stress distribution in the plate thickness

(1)
where σ
*d* is the thickness of the tempered glass plate.

Due to the non-uniform dependence of the residual stresses in the glass plate thickness the refractive index of tempered glass is also a function of the coordinates. It can be shown that the birefringence-induced phase difference in this case is given by

(2) or

(3)
where
*n*
*n*

In order to determine σ
*n*
*z*) and *n*
*z*) the measurement of the spatial distribution of the scattered light intensity is performed using scan of glass plate by a narrow beam in its cross-section along Z - axis, and the subsequent processing of the distribution of the light intensity scattered by the inhomogeneities of the refractive index in glass is made [12]. To achieve this, we propose to use the experimental setup that allows visualizing propagation of light beam in anisotropic sample. In this case analyzing intensity distributions of scattered light it is possible to study almost locally anisotropy of the glass refraction index. The light beam goes through polarizer, lens and a normal-incidence face of glass plate 6. The laser light at 533 nm with intensity of 1 mW is produced by the solid-state laser 1. The light is collimated by lens 3 to a beam width of about 180

It is known that if the observation direction is perpendicular to the direction of polarization of the input light then the intensity of the scattered light is defined as [9]

(4)
where

In experimental setup the intensity distribution of scattered light is recorded in a plane, which is perpendicular to the direction of its propagation, by a photorecording device 5 (CCD camera).

#### 3. Results and discussions

Because ordinary and extraordinary waves have various phase velocity in anisotropic material, between them there is a phase delay. The interference of ordinary and extraordinary waves changes a condition of polarization states of light beam at its distribution along the sample. Spatial modulation of scattered light surely is recorded in the direction which is perpendicular to the direction of propagation of the probing radiation in a anisotropic sample (Figure 2,*a*). In this case the phase difference δ = π arises on the length *x*, equaled to the period of the regular distribution intensity of scattered light.

As distribution of mechanical stress in glass in direction of Z- axis
*x*
*x* coordinate.

The imposed pictures of spatial modulation obtained at various coordinates of the input point of radiation in glass plate z
*I*(*z*, *x*) (see Figure 2,*b*), obtained when scanning the sample by the light beam in direction of Z - axis showed in Figure 1, allows to receive estimates of

According to [11] the stress profile in the tempered glass is described by a parabolic function. Only in the case of full compliance with the conditions of the quenching technology, it has the form (1). In practice, this is usually not performed. Therefore we represent the stress profile for glasses by function

(5)

The values of the coefficients *A
*(

*i*= 1, 2,...,

*p*) can be found by analyzing the magnitude of the beam deflection when it propagates through the glass plate parallel to the X-axis. Analysis of the recorded distributions (Figure 2

*b*), constructed by scanning the sample with a light beam along the Z-axis, allows to analyze the magnitude of the ray deflection when it propagates parallel to the X-axis (Figure 3) and estimate the function

(6)
where

(7)
and taking into account the boundary conditions

(8)
where

The integral on the left-hand side of equation (8) is an improper integral of type 2, which permits regularization. Obtained solution allows us to determine the unknown coefficients *A
* by the least squares method based on the results of recording the trajectory of the light ray in the glass. The minimized functional has the form

(9)
where

In the simplest case of the dependence
*A
* with an error of less than 10%.

The estimate of the stress σ
*T* of a function

(10)

In accordance with the results of data processing presented in Figures 2 and 3, the intensity distribution has period *T* = 6.1 mm at *z
* = 0.4 mm. The estimated value of the σ

#### 4. Conclusions

This work demonstrates that the using the analysis of the recorded intensity distribution of scattered light it is possible to estimate mechanical stress in the tempered glass and to investigate their asymmetric distribution in a cross-section of glass plate and to make estimates of values of the stretching and squeezing stress in glass and also the surface stress. It is shown that distribution of residual stresses in the tempered glasses is significantly the non-uniform. Obtained results show a good agreement with the theoretical calculation dates for central area of the glass plate. When moving away from the central area of the glass plate to its surface we observe the deviation of the refractive index profile from parabolic form. The presented results indicate that spatial distribution of birefringence-induced phase difference in non-uniform anisotropic materials can be correctly measured using the offered approach.