Caustics, multiply reconstructed by Talbot interference

Abstract

In planar geometrical optics, the rays normal to a periodically undulating wavefront curve W generate caustic lines that begin with cusps and recede to infinity in pairs; therefore these caustics are not periodic in the propagation distance z. On the other hand, in paraxial wave optics the phase diffraction grating corresponding to W gives a pattern that is periodic in z, the period for wavelength γ and grating period a being the Talbot distance, z _T = a ²/γ, that becomes infinite in the geometrical limit. A model where W is sinusoidal gives a one-parameter family of diffraction fields, which we explore with numerical simulations, and analytically, to see how this clash of limits (that wave optics is periodic but ray optics is not) is resolved. The geometrical cusps are reconstructed by interference, not only at integer multiples of z _T but also, according to the fractional Talbot effect, at rational multiples of z = z _T p/q, in groups of q cusps within each grating period, down to a resolution scale set by γ. In addition to caustics, the patterns show dark lanes, explained in detail by an averaging argument involving interference.