Abstract

We show that the Fresnel field at a fraction of the Talbot distance behind a complex transmittance grating is conveniently described by a matrix operator. We devote special attention to a discrete-type grating, whose basic cell (of length d) is formed with a finite number (Q) of intervals of length d/Q, each with a constant complex transmittance. Ignoring the physical units of the optical field, we note that the transmittance of the discrete grating and its Fresnel field belong to a common Q-dimensional complex linear space (VQ). In this context the Fresnel transform is recognized as a linear operator that is represented by a Q × Q matrix. Several properties of this matrix operator are derived here and employed in a discussion of different issues related to the fractional Talbot effect. First, we review in a simple manner the field symmetries in the Talbot cell. Second, we discuss novel Talbot array illuminators. Third, we recognize the eigenvectors of the matrix operator as discrete gratings that exhibit self-images at fractions of the Talbot distance. And fourth, we present a novel representation of the Fresnel field in terms of the eigenvectors of the matrix operator.

© 1996 Optical Society of America

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