Band-limited angular-spectrum approximation to a spherical scalar wave field

Abstract

The spherically symmetric scalar field eikr/r is studied in a band-limited angular-spectrum approximation. The field is represented by Weyl’s integral as an expansion into an angular spectrum of plane waves, the evanescent portion of the spectrum is discarded, and the remaining homogeneous part of Weyl’s integral is evaluated over a region within a few wavelengths around the source, by numerical integration. The modified field differs markedly from the complete spherical wave field over this region. The contribution of the evanescent plane waves to the spherical field does not decay away from the origin along the Z axis. Thus, a frequent approximation that involves neglect of evanescent plane waves in an angular-spectrum representation over the Z ≥ 0 half-space cannot be used near the Z axis with the spherical field. The close relation between this field and a well-known image field due to a well-corrected lens will be described.