A fast numerical algorithm for calculating electromagnetic waves in an inhomogeneous slab

Abstract

AbstractConsider a plane monochromatic electromagnetic wave normally incident on an absorbing dielectric slab, bounded by z = 0 and z = L, and whose electrical parameters ε, μ and σ are arbitrary functions of z. The media on either side of the slab have constant but generally different electrical properties. This paper describes an efficient numerical algorithm for calculating the electromagnetic field within the slab, as well as its absolute reflection and transmission coefficients. The waveform within the slab is represented by a new complex dependent variable satisfying a modified (complex) Helmholtz equation to be solved subject to mixed inhomogeneous conditions at the boundaries of the slab. If derivatives are replaced by nth‐order centred differences, the difference equation set is linear, sparse and efficiently solved by direct recursion. In this paper, complex arithmetic is avoided by replacing the complex Helmholtz equation with two coupled, real, second‐order ordinary differential equations. An algorithm of second‐order accuracy is developed which leads to a bitridiagonal difference equation set. Calculations are carried out for a variable‐property slab problem which can also be solved analytically. Numerical comparisons indicate that the method is accurate and efficient enough that a desk‐top computer is a practical computational machine.