Exact formulation of Huygens' principle in terms of generalized spatiotemporal-dipole secondary sources

TL;DR

This work addresses an exact formulation of Huygens' principle by representing the field inside a region as generated by generalized spatiotemporal dipole (GSTD) secondary sources placed on a surface S. The approach derives geometry-dependent delay and attenuation parameters, yielding a GSTD operator that reproduces the Kirchhoff integral for a single monopole primary source, while suppressing backward secondary waves through a circle-of-cancellation geometry. The key results provide explicit matching conditions $\tau_h = \frac{h}{c}\cos(n,r)$ and $\alpha_h = \frac{h}{r}\cos(n,r)$ and show how an arbitrary integration surface can be used to achieve near-field exact cancellation, generalizing Huygens’ principle beyond primary wavefronts. Overall, the framework unifies wave-field reconstruction with a clear geometrical-optical interpretation and extends the applicability of Huygens’ principle to more general surfaces and near-field regimes.

Abstract

A "spatiotemporal dipole" wave source, as defined by D.A.B. Miller (1991), differs from an ordinary ("spatial") dipole source in that the inverted monopole is delayed relative to the uninverted monopole, the delay being equal to the propagation time from one monopole to the other. A "generalized" spatiotemporal dipole (GSTD), as defined here, is generalized in two ways: first, the delay may be smaller in absolute value (but not larger) than the propagation time, so that the radiated waves cancel at a certain angle from the axis of the dipole; second, one monopole may be attenuated relative to the other, so that the cancellation is exact at a finite distance - on a circle coaxial with the dipole. I show that the Kirchhoff integral theorem, for a single monopole primary source, gives the same wave function as a certain distribution of GSTD secondary sources on the surface of integration. In the GSTDs, the "generalized" delay allows the surface of integration to be general (not necessarily a primary wavefront), whereas the attenuation allows an exact match of the wave function even in the near field of the primary source. At each point on the surface of integration, the circle of cancellation of the GSTD secondary source passes through the primary source, which therefore receives no backward secondary waves, while the direction of specular reflection of the primary wave passes through the same circle, giving a geometrical-optical explanation of the suppression of backward secondary waves at any field point.