Huygens' Principle for Complex Spheres

Huygens' relations that express wave fields of primary sources in terms of Huygens' sources on a spherical surface remain valid when the sphere radius is a complex number in a certain bounded domain. Such Huygens relations are derived for frequency and time-domain acoustic and electromagnetic fields. Interior and exterior source configurations are combined to obtain dual-sphere Huygens' relations, in which the primary sources are enclosed by one sphere and the observation points by another. The Huygens sources are complex point sources that exhibit directivity, which for certain parameter ranges makes it possible to achieve high accuracy with only a small fraction of the Huygens surface included in the integration. In general, for a dual-sphere configuration with fixed physical dimensions, the fraction of the spheres required to achieve a given accuracy diminishes with frequency, up to a certain frequency limit. Beyond this upper frequency limit the size of the required regions on the two spheres remain roughly constant. The upper frequency limit is increased when the imaginary part of the complex sphere radius is increased. A similar result holds in the time domain with respect to diminishing pulse width.