On the fresnel approximation

The purpose of this paper is to show the power of the Sommerfeld expansion in computing Fresnel and near fields of antennas, a matter which has become of increasing importance in high resolution antennas. A connection is shown between the Fresnel and Fraunhofer approximations for radiation fields which is derived by using Sommerfeld's expansion of the field in inverse powers of radial distance. This expansion permits an estimate of the error incurred in using the Fresnel approximation. Higher-order corrections to the phase and amplitude portion of the Fresnel approximation are also exhibited. By way of illustrating the power of the Sommerfeld expansion of the fields in the Fresnel (intermediate) region of a radiation source, numerical calculations of amplitude, phase, and power patterns have been made for a finite line source of length
D
with an equiphase cosine-on-a-pedestal current distribution. It is found that the first five terms of the series are sufficient to obtain accurate results when
r\geqD^{2}/2\lambda
, as compared with the Fraunhofer approximation which is usually considered valid for
r\geq2D^{2}/\lambda
. Non-Fraunhofer zone effects on the power pattern and phase front are discussed as a function of distance
r
and the type of current distribution.