The structure of quadratic Gauss sums in Talbot effect

The field diffracted from a one-dimensional, coherently illuminated periodic structure at fractional Talbot distances can be described as a coherent sum of shifted units cells weighted by a set of phases given by quadratic Gauss sums. We report on the computation of these sums by use of the properties of a recently introduced integer $s$, which is constructed here directly from the two coprime numbers $p$ that $q$ that define the fractional Talbot plane. Using integer $s$, the computation is reduced, up to a global phase, to the trivial completion of the exponential of the square of a sum. In addition, it is shown that the Gauss sums can be reduced to two cases, depending only on the parity of integer $q$. Explicit and simpler expressions for the two forms of integer $s$ are also provided. The Gauss sums are presented as a Discrete Fourier Transform pair between periodic sequences of length $q$ showing perfect periodic autocorrelation. The relationship with one-dimensional multilevel phase structures is exemplified by the study of Talbot array illuminators. These results represent a simple means for the design and analysis of systems employing the fractional Talbot effect.