Derivation of the Angular Dispersion Error Distribution of Mirror Surfaces for Monte Carlo Ray-Tracing Applications

Of paramount importance to the optical design of solar concentrators is the accurate characterization of the specular dispersion errors of the reflecting surfaces. An alternative derivation of the distribution of the azimuthal angular dispersion error is analytically derived and shown to be equivalent to the well-known Rayleigh distribution obtained by transforming the bivariate circular Gaussian distribution into polar coordinates. The corresponding inverse cumulative distribution function applied in Monte Carlo ray-tracing simulations, which gives the dispersion angle as a function of a random number sampled from a uniform distribution on the interval (0,1), does not depend on the inverse error function, thus simplifying and expediting Monte Carlo computations. Using a Monte Carlo ray-tracing example, it is verified that the Rayleigh and bivariate circular Gaussian distribution yield the same results. In the given example, the Rayleigh method is found to be ∼40% faster than the Gaussian method.