For the first time, an algorithm for reconstructing an arbitrary distribution of residual stresses by the polarization tomography method for cylindrical rod structures with a radial distribution of the refractive index has been presented. The reconstruction took into account the ray refraction. The algorithm is based on the expansion of the tensor stress field in angular harmonics (singular value expansion). The case of an axisymmetric tensor field with an arbitrary stress gradient along the cylinder axis was considered. Numerical calculations were carried out for an axially symmetric stress distribution in a grad an for the case of a plane deformation state. The reconstruction was based on the expansion of the stress tensor in eigenfunctions of the boundary value problem. The regularized solution of the resolving equation (of Abelian type) used the expansion in Zernike polynomials. The results of the reconstruction are given with taking into account the additional term due to the deflection of the transmission rays as well as without this doing.

В большинстве случаев цилиндрические структуры с градиентным показателем преломления (граданы) формируются на основе диффузионной технологии.Показатель преломления и остаточные напряжения в них носят аксиально симметрический характер.Дисторсия, обусловленная диффузией, и температурные внутренние напряжения в них моделируются фиктивной температурой остаточных напряжений.Применение интегральной фотоупругости для реконструкции остаточных напряжений в граданах обычно ограничивается

In most cases, cylindrical structures with a gradient refractive index (GRIN) are formed on the basis of diffusion technology. The refractive index and the residual stresses in them are axially symmetrical. The distortion determined by diffusion and the thermal internal stresses in them are simulated by the fictive temperature of residual stresses. The use of the integral photoelasticity for the reconstruction of residual stresses in GRIN lenses is usually limited to the case of a planar deformed state. In this study, this algorithm is extended to the case of the axial change of residual stresses. It is assumed that the illumination procedure is carried out in the sample cross section. The optical problem is solved within the quasi-isotropic approximation. The reconstruction is based on measuring the characteristic parameters of polarized light passed through the sample and on the combined solution of the thermoelasticity problem.

Предложено обобщение алгоритма Кормака обращения

A generalization of the Cormac algorithm for the inversion of the Radon transformation in an optical medium with an axisymmetric index of refraction has been proposed. The distribution of rays is such that only one ray passes through any two points in the circle. A “parallel scanning scheme” of tomography has been considered: a cylindrical object is illuminated by a parallel beam of light, the rays of which are deflected into the cylinder. There is no refraction on the surface of the cylinder. The algorithm assumes the possibility of attenuation on a ray, which also has an axisymmetric character. Such a type of ray deflection occurs in problematic issues of the tomography of GRADANs, light guides, and plasma.

An algorithm for determining the thermal stress in round cylindrical monocrystals by means of the integral photoelasticity method is analyzed. Transillumination in a cross section of a sample is assumed. The optical problem is solved within the framework of the paraxial approximation assuming weak birefringence. Reconstruction is based on the measurement of characteristic parameters of the polarized light propagated through the sample.

A way to determine residual stresses in cylindrical trigonal single crystals the optical axis of which is directed along the crystal axis is proposed. It is assumed that the residual deformation tensor is of thermal character and is characterized by fictive temperature. The measurements are performed in the middle part of a single crystal the length of which is much larger than its diameter; therefore, the stresses in this part do not vary along the single crystal axis. The reconstruction of stresses is based on determining characteristic parameters of polarized light by use of the tomographic method in the plane orthogonal to the single crystal axis.

Algorithms for determining residual stresses in glass round waveguides by the method of integrated photoelasticity are considered. It is assumed that the residual deformation tensor is of thermal character and is characterized by fictitious temperature varying over the sample. The reconstruction is based on measuring characteristic parameters of polarized light and is carried out by the tomographic method in the plane orthogonal to the waveguide axis.

A method for determining residual stresses in elongated transparent blanks of hexagonal single crystals the optical axis of which is directed along the crystal is proposed. It is assumed that the residual deformation tensor is of thermal character and is characterized by a fictitious temperature. Characteristic parameters of polarized light have been measured by the tomographic method in the plane orthogonal to the single crystal axis.

Optical polarization tomography of the tokamak plasma magnetic field is considered in the context accepted for large magnetic fields. Two models of the interaction between the magnetic field and radiation are studied: (1) the Faraday rotation (FR) of the polarization plane is imposed on birefringence caused by the toroidal component and (2) the Cotton-Mouton effect (CME) is imposed on the FR. Previously, these problems were solved with allowance for only one of the optical effects. New integrals caused by the amplification of the magnetic field are added to ray integrals used earlier. The retrieval of the magnetic field is based on the vector Radon transform (VRT) of imaginary absorption. Formulas of the inverse VRT are derived using generalized conditions from R. Novikov on the Radon transform with absorption.

As initial data, optical polarization tomography of tensor and vector fields uses ray integrals the values of which are determined from polarization measurements. Most algorithms of integrated photoelasticity are based on the linear approximation of the solution of the birefringence equations. In this work, the approximation of smooth rotation of quasi-principal axes of the polarization tensor is proposed for use in approximating values of ray integrals.

The generalization of the Cormack algorithm for inversion of the Radon transform with axially symmetric absorption is proposed. The Radon transform is one of the main mathematical tools of optical tomography. In many technical applications, the absorption is axially symmetric. This allows one to perform the expansion of the original and its image in terms of angular harmonics, which simplifies the inversion of the transform.

An approximate method is proposed for determining residual stresses in elonagted transparent articles featuring weak variation of the stress field along the axis. The proposed method is a generalization of the well-known method of determining internal stresses in optic fibers based on the integrated photoelasticity measurements. Complete determination of the stress tensor components is performed within the framework of the concept of temperature-dependent residual stresses.

A tomographic method for reconstruction of the stress field that is based on the magnetophotoelasticity effect is proposed. The input data are provided by the difference between path lengths and isocline parameters measured in the presence and the absence of a magnetic field. The reconstruction employs the imaginary-parameter exponential Radon transformation. Formulas are derived that generalize Cormack transforms to the case of the exponential Radon transformation.