CALCULATION OF OPTICAL WAVES PROPAGATION THROUGH GYROTROPIC ANISOTROPIC MEDIA: LR- AND SP-POLARIZATION

Abstract

A new method of electrodynamic analysis of gyrotropic (isotropic and anisotropic) media is developed. This method is based on the scalar representation of Maxwell's equations corresponding to 4 × 4-matrix formulation and coupling equations for gyrotropic medium in the Drude's form. It is utilized by solving the wave equations of second and fourth order, followed by cross-linking the fields at the boundary. The obtained results are experimentally verified by their good matching with the popular benchmark data, such as quartz rotatory power and in comparison with a known standard parameter of an optical element, such as λ/4-plate. This method simply summarizes the polarimetric and ellipsometric calculations. In 1972, Berreman finally formulated a theoretical approach to the electrodynamics of condensed matter (1), first introduced by Teitler and Henvis (2), called 4 × 4-matrix technique or 4 × 4-matrix formulation. According to this approach, the Maxwell's equations, along with the constitutive equations (also referred to as coupling equations) for a particular optical media, shall be converted into a scalar system of four first order differential equations in four unknowns, which are the field components Ex, Ey, Hx, Hy in the Cartesian coordinates system. These components should be chosen in the following way. Let us consider a plane wave propagating from vacuum (air) in the xz plane of incidence in the direction of the z axis, perpendicular to the plane xy, which is a boundary of the medium under investigation. The field components along the x axis are assumed to be ∼ exp(ikxx), where kx is a coordinate along the axis of the wave vector k0 of the reflected wave and the wave vector k of the refracted wave. The y coordinate is expected to be ∂/∂y = 0. This makes it possible to eliminate components Ez and Hz in a scalar system of the six first order differential equations equivalent to the Maxwell's vector equations. The Berreman's method is a commonly used algebraic method for solving the matrix wave equation. It is reduced to the calculation of the eigenvalue problem for a 4 × 4-matrix. However, the amount of calculation in the corresponding numerical method is too high, which makes its practical implementation difficult. Nevertheless, it is difficult to overestimate the importance of the 4 × 4-matrix representation due to the initial presentation of Maxwell equations as four scalar differential equations. This paper presents a new method for solving this problem. Its main advantage when compared to the Berreman's method is computational efficiency. It is interesting that it is not necessary to use the flow of matrix transformations, but to use only the basic theory of ordinary differential equations. For simplicity, but without loss of generality, we will reduce our considerations here to some extension of the general case of an anisotropic gyrotropic medium, combining two cases discussed in (1), namely orthorhombic crystal and certain optically active medium. We will benefit from an important fact that a 4 × 4-matrix of scalar equations is a sparse matrix, because more than a half of the coefficients of the system (the matrix elements) are equal to zero. As will be shown below, this makes it possible to substitute a system of