Vortex axicons for hypergeometric beams formation
Abstract A method for obtaining multi-ring distributions by using combinations of diffractive optical elements (DOE) for the formation of hypergeometric (HG) vortices is described. The calculated vortex axicons form a superposition of two HG vortices due to a combination of two logarithmic axicons in the phase function. Experimental results on the optical capture and transport of polystyrene microparticles with a diameter of 5 μm using calculated DOEs are presented.
We study multiple orthogonal polynomials of Meixner-Pollaczek type with respect to a symmetric system of two orthogonality measures. Our main result is that the limiting distribution of the zeros of these polynomials is one component of the solution to a constrained vector equilibrium problem. We also provide a Rodrigues formula and closed expressions for the recurrence coefficients. The proof of the main result follows from a connection with the eigenvalues of block Toeplitz matrices, for which we provide some general results of independent interest. ::: The motivation for this paper is the study of a model in statistical mechanics, the so-called six-vertex model with domain wall boundary conditions, in a particular regime known as the free fermion line. We show how the multiple Meixner-Pollaczek polynomials arise in an inhomogeneous version of this model.