Guided complex waves. Part 1: Fields at an interface

Abstract

The variety of waves which can be supported by a plane homogeneous interface includes surface waves of the forward and backward type, and several kinds of complex wave, the latter being characterized by wave numbers which are complex even though the media involved are not necessarily lossy. The present study views all these waves as contributions due to poles in several alternative integral representations of a source-excited field, and places particular stress on the steepest-descent representation. The pole locations, field distributions and power-transport properties are explored in detail for all the wave types. Distinctions are made between proper (spectral, modal) and improper waves, and between lossy and lossless structures; complex waves along lossless structures are shown to appear always in degenerate pairs consisting of a forward and a backward wave, with interesting power-flow characteristics. The different wave types are grouped into the general category of guided complex waves which propagate without attenuation as inhomogeneous slow plane waves at some angle to the interface. Power-transport considerations via the steepest-descent representation show that these waves either carry power to compensate for losses in the system or account for a transfer of energy into the radiation field. kList of symbols c -Velocity of light in free space f(«0 -Complex amplitude function for the /c-mode G(x, z) -•-Total solution for the field in Cartesian co-ordinates G P (A', Z) = Pole contribution to G(x, z) G(r, 6) = Total solution for the field in polar co-ordinates G p (r, 8) = Pole contribution to G(r, 8) G s (r, 8) -Space-wave contribution to G(r, 8) G 0 ((f) p ) -Field contribution at the origin due to a p o l e a t <j> = <f> p t(ju. o e o ) l/2 = Wave number of plane wave in free space. P x , P z --Components of power flow r •-Radial field co-ordinate v g -Group velocity v p = Phase velocity x, y, z = Cartesian field co-ordinates ZJ(K) = Impedance at the interface ZO(K) -Characteristic impedance of transmission line representing free space a,. ~ Radial attenuation factor P,--Radial phase factor = Absolute permittivity of free space = Wave number in the z-direction -Location of a pole in the complex £-plane r) = Imaginary component of (f> p -Imaginary component of <f> p 8 -Angular field co-ordinate ' c = Angle of definition for pole contributions = Wave number in the jc-direction +JK p i -Location of a pole in the complex /c-plane /x 0 = Absolute permeability of free space g = Real component of <j> £ p = Real component of <j> p <fi = The steepest-descent complex variable cf) p -Location of a pole in the complex 0-plane co = Angular frequency

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