Cortés Luis Romero, Guillet de Chatellus Hugues, Azaña José
Opt Lett. 2016 Jan 15;41(2):340-3. doi: 10.1364/OL.41.000340.
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.
通过在相应的傅里叶对偶域施加二次相位分布，整数和分数自成像效应可以在时间、频率、空间或角频率域的周期波形上被诱导出来。这个相位必须满足著名的“塔尔博特条件”。所得到的周期划分分数自像表现出由高斯和的解产生的确定性脉冲间相位变化。反过来，这些自像可以被视为在傅里叶对偶域中诱导出塔尔博特效应。这表明通过施加不由塔尔博特条件定义的相位分布来观察自成像效应的可能性。在本信函中，我们以其他方式表明从高斯和中检索到的相位分布也满足塔尔博特条件，这意味着该条件可能涵盖用于诱导自成像效应的所有可能的二次相位模式。我们在此建立高斯和的解与相应塔尔博特相位之间的精确关系，并推导具有基本和实际意义的塔尔博特相位模式的其他性质。
