Vectorial differentiation of a light-beam profile with respect to two spatial coordinates upon reflection from an interface at Brewster’s angle

Subject of study. This study investigates vectorial differentiation of an optical beam, involving the computation of derivatives with respect to two spatial coordinates of the nonzero transverse electric field component of a linearly polarized incident beam in the two transverse field components of the beam reflected from an interface at Brewster’s angle. Aim of study. The aim of this study is to develop a theoretical framework and provide experimental verification for the optical implementation of vectorial differentiation of a linearly polarized optical beam incident on an interface at Brewster’s angle. Method. The theoretical description of the vectorial differentiation is formulated using a vectorial transfer function that characterizes the transformation of a linearly polarized incident beam upon reflection from an interface at Brewster’s angle. To validate the theoretical results, electromagnetic numerical simulations and an optical experiment demonstrating the application of vectorial differentiation for the optical edge detection of phase objects were performed. Main results. The feasibility of vectorial differentiation for a beam incident at Brewster’s angle is theoretically investigated and experimentally confirmed for the first time. Isotropic vectorial differentiation can be achieved by introducing a quarter-wave plate and polarizer into the optical path of the reflected beam, such that the intensity of the reflected beam is proportional to the squared magnitude of the gradient of the electric field of the incident beam. Practical significance. The obtained results may facilitate the development of novel optical image-processing systems.