Fredholm Integral Equation for Finite 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. We seek the function that its total power is maximized in finite Fresnel transform plane, 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 nonnegative and real number. By discretizing the kernel, the problem depends on the eigenvalue problem of Hermitian conjugate matrix in finite dimensional vector space. By using the Jacobi method, we compute the eigenvalues and eigenvectors of the matrix. We applied it to the problem of approximating a function and evaluated the error.