Second-order optical differentiation of a 3D light beam at oblique incidence using a multilayer metal-dielectric structure

We investigate the optical implementation of a second-order differentiation operation using a metal-dielectric layered structure in the oblique incidence geometry. It is shown that the transformation of the profile of a three-dimensional light beam occurring upon its reflection from a layered structure can be described using the theory of linear systems. The transfer function of the corresponding linear system is obtained, and it is shown that if a layered structure has a reflection zero of the second order with respect to the spatial frequency for one of the polarizations, the transformation performed by the structure corresponds to the weighted sum of the second derivatives of the incident beam profile with respect to the spatial coordinates. Using the presented theoretical description, layered metal-dielectric structures for computing the second derivative with respect to one of the spatial coordinates and for computing the Laplace operator of the profile of a three-dimensional linearly polarized light beam are calculated. The presented numerical simulation results demonstrate high-quality computation of these operators.