MATRIX FORMULATION OF THE FRESNEL TRANSFORM OF COMPLEX TRANSMITTANCE GRATINGS

We show that the Fresnel field at a fraction of the Talbot distance be hind a complex transmittance grating is conveniently described by a ma trix 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. I gnoring the physical units of the optical field, we note that the tran smittance of the discrete grating and its Fresnel held belong to a com mon Q-dimensional complex linear space (V-Q). In this context the Fres nel transform is recognized as a linear operator that is represented b y a Q X Q matrix. Several propel-ties of this matrix operator are deri ved here and employed in a discussion of different issues related to t he fractional Talbot effect. First, we review in a simple manner the f ield symmetries in the Talbot cell. Second, we discuss novel Talbot ar ray illuminators. Third, we recognize the eigenvectors of the matrix o perator 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. (C) 1996 Optical Society of America.