<title>Two Extensions Of Fourier Optical Processors</title>

Abstract

We discuss two generalizations of the continuous Fourier transform performed by coherent optical systems. The first concerns the introduction of an appropriate exponential damping factor in the input plane, which leads to a processor that evaluates a two-dimensional slice through the four-dimensional complex Laplace transform domain. By performing Laplace filtering, rather than Fourier filtering, one can in principle trade off dynamic range in the filter plane for dynamic range in the input plane. Using a Laplace transform, it is also possible to find the complex roots of polynomials. The second generalization concerns modification of the continuous Fourier transform to behave as a discrete Fourier transform. With such a modification, it is in principle possible to find (in a single step, without iterations) the eigenvalues of any circulant matrix, or any circulant approximation to a Toeplitz matrix (including correlation matrices) using a coherent optical processor. Furthermore, if a light valve having a suitable nonlinear relation between amplitude transmittance and exposure is available, it is possible to obtain the inverse of any matrix in the class described above in a single pass through a coherent optical processor.