Method for calculating multiwave scattering by layered anisotropic media

Abstract

The method of transfer matrices of orders n=2 and n=4 is known from the theory of the propagation of light and elastic waves in layered media. The development of this matrix method is presented for cases n>4. The transfer matrix T of waves through a homogeneous layer of thickness d is defined as the matrix exponential T=exp(Wd). The elements of the matrix W depend on the physical properties of the layer and are determined from the differential equations describing the waves. The matrix T is calculated using the scaling and squaring method: T=[exp(Wd∕(2j))]2j. A new way is proposed for choosing the scaling parameter m=2j, which provides the required accuracy in computing the elements of the matrix T. Two procedures for calculating the exponential of the scaled matrix A=Wd∕(2j) are considered: 1) truncation of the Taylor series, 2) polynomial representation of the matrix exponential. Both of these methods do not require finding the eigenvalues of the matrix A. The effectiveness of these methods is compared. It is shown that when calculating the matrix exponential with double precision, the polynomial method is preferable for matrices of order n<10. An alternative to multiple squaring for the cases j>n is proposed. The application of the scaling method is shown by the example of calculations of six-beam scattering of elastic waves by an anisotropic layer.