V. Arrizon et al., MATRIX FORMULATION OF THE FRESNEL TRANSFORM OF COMPLEX TRANSMITTANCE GRATINGS, Journal of the Optical Society of America. A, Optics, image science,and vision., 13(12), 1996, pp. 2414-2422
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.