On the generality of the Talbot condition for inducing self-imaging effects on periodic objects

Integer and fractional self-imaging effects can be induced on periodic waveforms across the time, frequency, space, or angular frequency domains by imposing a quadratic phase profile along the corresponding Fourier dual domain. This phase must satisfy the well-known "Talbot condition." The resulting period-divided fractional self-images exhibit deterministic pulse-to-pulse phase variations that arise from the solution of a Gauss sum. In turn, these self-images can be regarded as inducing a Talbot effect in the Fourier dual domain. This suggests the possibility of observing self-imaging effects by imposing phase profiles that are not defined by the Talbot condition. In this Letter, we show otherwise that the phase profiles retrieved from a Gauss sum also satisfy the Talbot condition, which implies that this condition may encompass all possible quadratic phase patterns for inducing self-imaging effects. We establish here the precise relationships between the solutions of Gauss sums and the corresponding Talbot phases, and derive additional properties of Talbot phase patterns of fundamental and practical interest.