Fractional Fourier transform applied to spatial filtering in the Fresnel domain

The fractional Fourier transform can be optically defined through a phase-space coordinate rotation of the Wigner distribution function associated with the input signal. This operation can be achieved by performing three successive shearing processes, which are reduced to a free-space Fresnel diffraction originated by a scaled version of the input object illuminated with a spherical wave. This result is applied to describe the behavior of spatial filtering devices based on the self-imaging phenomenon (Fresnel spatial filters). Light propagation in a medium of position-varying refractive index can be described by analyzing the effect of an operator acting on the input field amplitude. For the case of free-space propagation (or constant refractive index), the Kirchhoff integral provides an adequate operator which can be approximated depending on the involved distances by the Fresnel and Fraunhofer integrals. Thus, the diffraction process leads to a Fourier transform relationship, operation which is the basis for developing spatial filtering devices. On the other hand, for a quadratic graded index (GRIN) medium the amplitude distribution at different locations along the propagation axis can be obtained from a fractional Fourier transform operation applied to the input signal, an approach introduced in optics by Mendlovic and Ozaktas [ l-31. When light propagates a certain finite distance inside such media, an ordinary Fourier transform is accomplished same as in the far field free-space diffraction. However, the intermediate states of the field amplitude, where the spatial and the spectral information is mixed, are different for the free-space and the GRIN media. By using an alternative definition based on considerations about the Wigner distribution function (WDF) [4-61, Lohmann suggested how to obtain the fractional Fourier transform through a proper combination of free-space propagation and lens action [ 71. The equivalence between both optical definitions was later established [ 81. For a certain kind of input objects that can be expressed as a linear superposition of binary grating structures, free-space diffraction in the Fresnel region was used for developing spatial filtering devices based on the self-imaging phenomenon [9-l 11. In this paper, we propose a description of these Fresnel spatial filters by establishing the relationship between the Talbot conditions that are found at the self-image planes and the fractional order of the Fourier transform associated with the grating component which is cancelled as result of the interaction filter-diffracted field. L Also: Facultad de Ciencias Exactas, Universidad National de La Plata, Argentina. 0030-4018/95/$09.50 @ 1995 Elsevier Science B.V. All rights reserved sSDr0030-4018(95)00348-7 276 S. Gmnieri et al. /Optics Communications 119 (1995) 275-278