Band-limited Orthogonal Functional Systems for Optical Fresnel Transform

Abstract

The fundamental formula in an optical system is Rayleigh diffraction integral. In practice, we deal with Fresnel diffraction integral as approximate diffraction formula. By optical instruments, an optical wave is subject to a band limited. To reveal the band-limited effect in Fresnel transform plane, we seek the function that its total power in finite Fresnel transform plane is maximized, on condition that an input signal is zero outside the bounded region. This problem is a variational one with an accessory condition. This leads to the eigenvalue problems of Fredholm integral equation of the first kind. The kernel of the integral equation is Hermitian conjugate and positive definite. Therefore, eigenvalues are real non-negative numbers. Moreover, we also prove that the eigenfunctions corresponding to distinct eigenvalues have dual orthogonal property. By discretizing the kernel and integral calculus range, the eigenvalue problems of the integral equation depend on a one of the Hermitian matrix in finite dimensional vector space. We use the Jacobi method to compute all eigenvalues and eigenvectors of the matrix. We consider the application of the eigenvectors to the problem of approximating a function and showed the validity of the eigenvectors in computer simulation.