The structure of quadratic Gauss sums in Talbot effect

TL;DR

This paper introduces a new method for computing quadratic Gauss sums related to the fractional Talbot effect, simplifying analysis and design of optical systems using this phenomenon.

Contribution

It presents a novel integer-based approach to compute Gauss sums efficiently, reducing complexity and providing explicit formulas for different cases.

Findings

Simplified computation of Gauss sums using a new integer parameter

Reduction of Gauss sums to two cases based on parity of q

Application to design of Talbot array illuminators

Abstract

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.