Travelling and evanescent parts of the optical near field

Abstract

The optical near field of a localized source has been studied by means of the angular spectrum representation of the electromagnetic Green's tensor. This Green's tensor can be expressed in terms of four auxiliary functions, which depend on the field point through the dimensionless radial distance q to the source, or origin of coordinates, and the polar angle ρ with the z axis. Each function separates into a part containing travelling (radiative) waves and a part which is a superposition of evanescent (decaying) waves. We have derived series expansions in q of the four functions, both for the travelling and for the evanescent parts. It is shown that the travelling waves are finite at the origin of coordinates, and that all singular behaviour of the radiation field is governed by the evanescent waves. It is illustrated numerically that the series expansions are applicable up to about five wavelengths from the origin. In order to extend the range to also cover larger values of q, we have derived series expansions of the auxiliary functions which converge rapidly near the x-y plane, and a full asymptotic expansion with the z coordinate as the large variable. Finally, from the properties of the Taylor coefficients we have derived simple new integral representations for the auxiliary functions.