Caustics, multiply reconstructed by Talbot interference

In planar geometrical optics, the rays normal to a periodically undulating wavefront curve W generate caustic lines that begin with cusps and recede t o infinity in pairs; therefore these caustics are not periodic in the propa gation distance z. On the other hand, in paraxial wave optics the phase dif fraction grating corresponding to W gives a pattern that is periodic in z, the period for wavelength lambda and grating period a being the Talbot dist ance, z(T) = a(2)/lambda, that becomes infinite in the geometrical limit. A model where W is sinusoidal gives a one-parameter family of diffraction fi elds, which we explore with numerical simulations, and analytically, to see how this clash of limits (that wave optics is periodic but ray optics is n ot) is resolved. The geometrical cusps are reconstructed by interference, n ot 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 w ithin each grating period, down to a resolution scale set by lambda. In add ition to caustics, the patterns show dark lanes, explained in detail by an averaging argument involving interference.