Stochastic ray tracing for Fresnel diffraction

We propose stochastic ray tracing for laser beam propagation in Fresnel diffraction to find the duality between wave and ray representations. We transform from the Maxwell equations to the Schrödinger equation for a monochromatic laser beam in the slowly varying envelope approximation. The stochastic ray tracing method interprets this Schrödinger equation as a stochastic process, of an analogy of Nelson's stochastic mechanics. It can illustrate the stochastic paths and the wavefront of an optical beam. This ray tracing method includes Fresnel diffraction effects naturally. We show its general theoretical construction and numerical tests for a Gaussian laser beam with diffraction, that stochasticity realizes the beam waist around the Rayleigh range.