Abstract

In this paper a rigorous mathematical approach to the near-field diffraction of 1D periodic structures at quarter-Talbot distances is presented. At the beginning of the paper, we present a mathematical investigation on the behavior of periodic functions under squaring by the aid of Fourier analysis. We determine sufficient conditions for a periodic function in which its fundamental period is halved by squaring. We show that if Fourier expansion of a given periodic function includes only coefficients with odd indexes, and minimum difference of the indexes is 2, then its fundamental period halves by squaring. This result is clarified by considering some typical periodic functions. Based on the presented mathematical results, we explain the formation of halved-period sub-images at quarter-Talbot distances from the 1D periodic structures in the near-field diffraction. In this work, 1D periodic structures are categorized into two classes in terms of their Fourier expansion forms. We show that Fourier expansion of a given 1D periodic structure may include Fourier coefficients either with both even and odd indexes or only with odd indexes plus the DC term. For clarifying the matter, the near-field diffractions from four sets of structures are considered, including parabolic and triangular gratings; sawtooth and Ronchi gratings; two- and three-level binary gratings; and sin-sinusoidal and cos-sinusoidal gratings. It is shown that, for all of these sets, by excluding even-order Fourier coefficients from the Fourier expansions, except the DC term, the value of intensity contrasts decreases at quarter-Talbot distances. We show that, in addition to the consistency of the presented approach for analyzing quarter-Talbot images with the conventional fractional Talbot formulation, it also reveals new details on the subject. As a direct address to the potential applications of the work, we show that high-contrast sub-images of a binary grating can be produced at quarter-Talbot distances only by choosing suitable opening numbers and that it can be used in various domains such as lithography.

© 2017 Optical Society of America

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