The Angular Spectrum Representation of Electromagnetic Fields in Crystals. I. Uniaxial Crystals

Abstract

The electromagnetic field in a linear, nonmagnetic, nonabsorbing uniaxial crystal which fills the entire half-space z ≥ 0 and whose optic axis is perpendicular to the plane z = 0 is represented as an angular spectrum of plane waves. The angular spectrum representation consists of a superposition of plane electromagnetic waves expressed as the sum of two integrals. In general both homogeneous and evanescent plane waves are required in each integral. Each plane wave of the spectrum satisfies the identical equations obeyed by the entire field. The homogeneous plane waves of the first integral are all ordinary waves and those of the second integral all extraordinary waves. The spectral amplitudes of the field are explicitly expressed in terms of the Fourier transform of the field in the place z = 0. The method of stationary phase is applied to the integral representation and it is thereby shown that in the far zone the field may be expressed as the sum of an outgoing (nonuniform) spherical wave and an outgoing (nonuniform) ellipsoidal wave. The amplitude of these waves, at each point on the wave surface, is expressed in terms of the Fourier transform of the field in the plane z = 0.