Generalized Huygens principle with pulsed-beam wavelets

TL;DR

This paper extends Huygens' principle by complexifying the integration surface, replacing spherical wavelets with focused pulsed beams, which improves back-propagation and computational efficiency in radiation field representations.

Contribution

It introduces a novel complexification of the Huygens surface, leading to a complete and efficient pulsed-beam representation of radiation fields.

Findings

Replaces spherical wavelets with focused pulsed beams

Provides a complete representation of radiation fields

Enhances computational efficiency by ignoring irrelevant beams

Abstract

Huygens' principle has a well-known problem with back-propagation due to the spherical nature of the secondary wavelets. We solve this by analytically continuing the surface of integration. If the surface is a sphere of radius $R$, this is done by complexifying $R$ to $R+ia$. The resulting complex sphere is shown to be a real bundle of disks with radius $a$ tangent to the sphere. Huygens' "secondary source points" are thus replaced by disks, and his spherical wavelets by well-focused pulsed beams propagating outward. This solves the back-propagation problem. The extended Huygens principle is a completeness relation for pulsed beams, giving a representation of a general radiation field as a superposition of such beams. Furthermore, it naturally yields a very efficient way to compute radiation fields because all pulsed beams missing a given observer can be ignored. Increasing $a$ sharpens the focus of the pulsed beams, which in turn raises the compression of the representation.