Methods of choosing sample rays in ray-tracing computations

Ray-tracing computations for calculating the fraction of radiant energy (intercept factor I) transmitted by an optical system are often performed by using the Monte Carlo method, i.e., selection of sample rays at random. However, this is not economical regarding computing time since the error E in I is proportional to N(-(1/2)), where N is the number of rays traced. It is found that, in example computations, the selection of sample rays according to a regular grid gives better results. An equation for estimating E in this case is derived on a theoretical basis and agrees with the computations. This predicts E alpha N(-(3/4)) for systems where three ray coordinates are needed (e.g., with rotational symmetry) and E alpha N(-?) where four are required, which is the general case.