Generalized Far-Field Distance of Antennas and the Concept of Classical Photons

A generalized far-field (Rayleigh) distance is derived for a frequency-domain or time-domain antenna in terms of the radius of the significant reactive power of the antenna, where this radius is given in terms of the maximum value N of the degree number n of the spherical waves needed to accurately represent the far-field of the antenna. The maximum possible gain of the antenna is given in terms of the radius of the significant reactive power, which enables supergain to be defined in terms of the physical radius of the antenna. Although the energy in pulses from finite-energy, finite-extent sources, including the energy in “electromagnetic missiles,” must eventually decay, it is shown that wavelength-size wavepackets with a well-defined center frequency can remain localized in free space for a limited amount of travel time and distance. These quasi-monochromatic “classical photons” are used, along with Planck’s constant, to determine the electromagnetic energy density below which quantum scattering theory, rather than the classical Maxwell equations, may be required to determine electromagnetic scattering.

I. Introduction

The approximate distance at which the time-harmonic (single frequency ) fields of an antenna form a far-field angular pattern with radial dependence (, with c the speed of light and the wavelength), sometimes referred to as the Rayleigh distance, is commonly given as , where is the spherical circumscribing diameter of the source region of the antenna, and the radial distance r is measured from the center of the circumscribing sphere [1, sec. 7.7]. However, this is not a generally valid formula, not only because it does not apply if but, more importantly, because the sources of any given antenna with nonzero can be replaced by sources in a smaller region of space that have practically the same fields outside the circumscribing diameter of the original sources [2], [3]. Thus, this article begins with a derivation using spherical-wave representations to obtain a generalized far-field distance for arbitrary single-frequency antennas with sources confined to a volume of finite extent.