The bidirectional reflectance distribution function (BRDF) is commonly used to accurately characterize the scattering property of the surface of opto-mechanical structures in stray light analysis. Software for stray light analysis based on the Monte Carlo method (MCM) can construct probability models for scattered ray tracing on the basis of BRDF models. However, the types of BRDF models allowing surface property setting in the software are limited. Although the inverse transform technique can be used to construct probability models, the BRDF of most scattering models is modulated by multiple variables with complex forms, and the analytical solution of the cumulative distribution function is absent. Consequently, this method becomes invalid, and it also limits the application of BRDF models to some extent. As scattered ray tracing is limited by the difficulty in obtaining an analytical solution, a probability model for scattered ray tracing is constructed by the rejection sampling method. The proposed method circumvents the integral solution process by setting test conditions and then screens out the effective samples to achieve scattered ray tracing, whereby it gains the advantage of wide applicability.
The rejection sampling method is applied to construct the probability model for MCM-based scattered ray tracing in the present study. Specifically, the BRDF describing the scattering model is converted into a probability density function, and random sampling based on uniform distribution is performed. Then, a reasonable squeezing function is used, and the effective samples are screened out under the test conditions. Finally, the effective samples are taken as the direction of the scattered ray, and scattered ray tracing based on the BRDF model is thus achieved. For the shift-invariant BRDF model, a symmetric sampling scheme is further proposed to sample the half-space after determining the sampling interval. The angular coordinates are converted into direction cosines, and the effective samples are selected by the rejection sampling method. The effective samples in the half-space are then used to obtain those in the full-space by applying mirror symmetry about the axis of symmetry. Simulation programs are prepared in Matlab according to the proposed method, and the simulation results in Matlab are compared with those in LightTools from the aspects of repeatability and accuracy. The same simulation parameters of surface property, incidence angle, and number of traced rays are set to simulate the BRDF models commonly used in engineering for scattered ray tracing. Since scattered energy distribution is the direct reflection of the simulated tracing results, the universal quality index (UQI) is used to quantify the different energy distributions on the analyzed surface at different times of simulation. The repeatability and accuracy of the simulation are described by the UQI.
The ABg model of the oxidatively blackened mechanical component for scattered ray tracing is simulated, and the obtained UQI values of the simulation results based on the proposed method and those of the results in LightTools are all higher than 0.9985 (Fig. 5). The simulation results based on the rejection sampling method are comparable to those in LightTools in terms of repeatability and accuracy. The ABg model is used to model the two scattering surfaces of shiny aluminum alloy and standard lens glass for scattered ray tracing, and the Harvey model is used to model an optical surface for the same purpose. The UQI values of the simulation results based on the proposed method and those of the results in LightTools are all higher than 0.9994 (Fig. 6). The scattered energy distribution based on the simulation programs is highly consistent with the result delivered by LightTools, which verifies the rationality and validity of the proposed method. The Phong model and the K-correlation model that are not included in LightTools are also simulated for scattered ray tracing, the UQI values obtained which are used to describe the repeatability of the simulation are all higher than 0.9970 (Fig. 7). This result further verifies the universality of the proposed method.
To address the limited applicability of the existing scattered ray tracing methods based on probability models, this study proposes the probability model by the rejection sampling method. Specifically, the BRDF is converted into the probability density function, and the probability model is thereby constructed for random sampling. Then, the effective samples that meet test conditions are used as the direction of the scattered ray. Finally, the spatially continuous distribution of scattered energy is converted into the probability distribution of a discrete ray, and scattered ray tracing is thus achieved. For the shift-invariant BRDF model, a symmetric sampling method is further proposed to enhance the sampling rate by halving the sampling area and then mirroring it. In the case of BRDF models with different materials, ray tracing programs are constructed to achieve scattered ray tracing in Matlab. To verify the simulation results based on the proposed method and those delivered by LightTools in terms of repeatability and accuracy, this study sets the same simulation parameters in Matlab and LightTools. The simulation results based on the rejection sampling method in the present study are almost the same as those in LightTools, and scattered ray tracing based on BRDF models that are not included in LightTools is also achieved.
