## 1 Introduction, Purpose and Statement of the Problem  
At present, the development and use of contactless optical measuring instruments based on laser interferometers is one of the most relevant and promising scientific and technical areas. This is due to the need to improve the information content and accuracy of measurements of displacements of control surfaces. Such studies allow us to establish the features of the formation of defects in advanced materials and to perform diagnostics of the state of structural materials and goods. The measuring instruments [1; 2; 3; 4], created on the base of a two-way laser interferometer with combined branches, are the most applicable and preferred for the solution of the pointed problems. The scheme of the laser interferometer is shown in Figure 1, where the beam splitter 1 with known amplitude transmission is located in a plane \(S\), related to a associated Cartesian coordinate system \(Oxyz\). The reflector 2 is in a plane \(S_{r}\) that is rotated relative to the plane \(S\) by an angle \(\alpha\). The reflector 2 is rigidly fixed on the surface of the control object and can perform linear (\(\Delta h\)) and angular (\(\Delta\alpha\)) displacements simultaneously with it. The beam splitter 1 receives coherent radiation from a point source of radiation in the form of an electric dipole located at the point O\({}^{\prime}\) (\(x_{0}\), \(y_{0}\), \(z_{0}\)). The photodetectors for analyzing the interference field are located in the direction of reflection from the beam splitter of the source radiation, at a certain distance from the beam splitter corresponding to the Fresnel zone. This device allows us to determine the coordinates of the source of radiation from the interference pattern created, or if fixed, to find the movement of the interferometer reflector in space.  
To determine the spatial displacement of the radiation source or reflector, it is necessary to define the change in the distribution of the electromagnetic field scattered by the device under consideration in the zone of observation of the interference pattern [5], i.e., in half-space \(y>0\).  
Using the tensor Green's function [6] with mixed Neumann-Dirichlet boundary conditions [7], and also taking into account the fact that the electromagnetic field has a transverse character and corresponds to the Maxwell equations, for its determination at the observation points it is sufficient to know only the tangential components of the electric field on the surface where the sources are located, i.e., on the beam splitter.  
To determine the tangential components of the electric field intensity vector \(E\), the image method [8] is used, in accordance with which the total field is a superposition of the fields from the source at the point \(O^{\prime}\) and the imaginary mirror source located at the point \(O^{\prime}\) taking into account the transmission of the beam splitter. Thus, the problem of determining the distribution of the intensity of the optical field in the area of the photodetector devices is solved successively in two stages, and initially, the field is determined on the beam splitter surface, and further, the field is determined in the area of the photodetector devices. At a given distribution of the intensity field in the area of photodetectors, it is possible to obtain displacements of the surface of the test object, uniquely associated with this area.  
The aim of the work is to obtain the constitutive relations for determining the electromagnetic field on the surface of the beam splitter of the laser interferometer under consideration for the cases of perpendicular and parallel polarizations of the source radiation. These relations can be used for numerical simulation of the process of measuring the displacements of control object surfaces.  
Figure 1: Circuits of interferometer under consideration.