Research of the nonuniform layers by a reflectometry method near the Brewster’s angle

The mathematical model of the electromagnetic wave reflection with p-polarization from the non-sharp boundary between two transparent media is built and studied. The feature of the model consisted in the fact that the thickness of the transition layer was much less than the electromagnetic wave length. The analytical ratio expressing the reflection coefficient through the coordinate dependence of permittivity is deduced. It is shown that the logarithmic derivative from the reflection coefficient with respect to angle can be used as the inhomogeneity indicator. The interval of the angles located near the Brewster’s angle is the most informative in this case. The developed theoretical representations are illustrated by calculations executed for the terahertz range wave reflection from an implanted layer.


Introduction
It is known that the angular dependence of a reflection coefficient of the p-polarised electromagnetic wave from a homogeneous medium has the zero value minimum. The corresponding angle is named as Brewster , s angle. The presence of thin transitive layer near a partition boundary results in the fact that the reflection coefficient, as before, has the minimum in angular dependence, which, however, doesn't reach zero. It is convenient in this case to use a logarithmic derivative instead of angular dependence of the reflection coefficient   θ p R [1]: The function   θ η unlike the function has not only the minimum but the maximum near the Brewster , s angle. Besides that the experimental dependence   θ η can be built in the values of the reflection coefficient expressed by the relative units. This circumstance simplifies execution of the necessary measurements because it is possible to ignore the change of the sounding radiation intensity as well as reflections from the input and the exit windows of the measuring system.
It is possible to deduce the analytical expression for electromagnetic wave reflection coefficient from a non-homogeneous medium only in the exclusive cases [2,3]. The simplest example is the Drude model in which the permittivity changes by the jump at some depth. The reflection coefficient in this case can be expressed through the amplitude reflection coefficients from the semi-infinite homogeneous media with the sharp boundaries of partition (Fresnel , s coefficients). The numerical methods for arbitrary profile of the permittivity go out on the first plan. The method in which the nonhomogeneous part of a medium breaks into the thin homogeneous layers is the most often used [4].
The function   z h was presented in the expression form: The values   are the amplitudes of the incident, passed and reflected waves respectively.
Since two functions were injected instead of one required, it is possible to impose an additional condition on them. The condition was expressed by the equation: For calculation of the reflection coefficient it is convenient to use a function Then a wave reflection coefficient is connected with   z V by equality From the Eqs. (2), (4), and (6) the equation follows: The boundary condition for   z V can be deduced from the physical restriction: . Then (9) Thus, the reflection coefficient from the media partition boundary with the non-uniform transition layer can be calculated from the differential Eq. (8) with the boundary condition (9) and from the formula (7).
The Eq. (8) generally hasn't an analytical solution. However approximate methods can be used [5]. The equation has the feature. Its right part approaches to zero at (8), presented in the row form, leads to finite values of each term of the row even when the limits of integration will be infinite. Such feature of the equation arises due to the dz dp derivative [6]. It allows to replace in final expressions the integration interval d zΔ on integration with the infinite limits. Therefore the dependences   z ε aiming to asymptotic values at least exponentially were considered.
Decomposition of the function   z V into the row of the perturbations theory was used: (12) was as a first approximation used, and the function   z V 1 is defined by the equation: In the second approximation it was assumed that , and the function  