Generalized Huygens principle with pulsed-beam wavelets

Huygens’ principle states that the solution of the wave equation radiated by a bounded source can be represented outside the source region as a superposition of spherical ‘Huygens wavelets’ radiated by secondary point-sources on a surface enclosing the primary source. This was originally proposed as a geometrical explanation of wave propagation, but as such it is conceptually problematic because the spherical wavelets propagate equally in all directions, thus implying that the wave propagates backwards (toward the source) as well as forwards. We propose a solution to this problem by generalizing the idea of Huygens wavelets. Choosing the surface to be the sphere SR of radius R, we show that the Huygens representation of the exterior wave can be continued analytically to a complex radius α = R + ia. For any real unit vector n̂, the complex vector αn̂ is shown to represent a real disk of radius a tangent to SR at the point Rn̂. The complex sphere Sα consisting of all such vectors αn̂ is therefore equivalent to a real tangent disk bundle with base SR. Just as the points Rn̂ ∈ SR radiate spherical wavelets, so do the tangent disks αn̂ ∈ Sα radiate well-focused pulsed-beam wavelets propagating in the outward direction n̂. The analytically continued Huygens formula can be given the following real interpretation: the interior wave radiated by the source is intercepted by the set of tangent disks αn̂, which then re-radiate it as a set of outgoing pulsed beams. The original wave is thus represented in the exterior as a superposition of pulsed beams emanating from disks tangent to the sphere SR, and the coefficients are interpreted as local reception amplitudes by the disks. The generalized principle is a completeness relation for pulsed-beam wavelets, enabling a pulsedbeam representation of radiation fields. Since the new wavelets can be 1 ar X iv :0 90 4. 06 83 v1 [ m at hph ] 4 A pr 2 00 9 focused by increasing the disk radius a, our construction solves the directionality problem of Huygens’ original construction. Furthermore, it leads to substantial gains in the efficiency of computing radiation fields. Only pulsed beams propagating toward the observer need to be included and the rest can be ignored while incurring little error. This leads to a significantly compressed representation of radiation fields, with the compression controlled by the disk radius a. We confirm these results by numerical simulations.