Monte Carlo simulations are widely applied in the fields such as imaging evaluations, graphical rendering, scattering analysis, and illumination design. Light source modeling, which directly determines the accuracy of the simulation results, is crucial in Monte Carlo simulation. However, light source modeling, especially surface light source modeling, is difficult and rarely discussed publicly. Surface light sources including extended filaments and curved fluorescent tubes are still commonly employed in general and special lighting. Additionally, the external radiation of the non-transparent components of the mechanical structure can also be considered as surface light sources in stray light analysis of far-infrared optical systems. We provide a Monte Carlo modeling method for surface light sources. In this method, we introduce a statistical model of the surface light source and two ray sampling strategies. Results show that the proposed modeling method has high precision. The influence of different sampling strategies and different random numbers on the modeling accuracy and speed is also discussed to guide balancing the modeling accuracy and speed.

Based on the homogeneity assumption, we analyze the spatial and orientational properties of the surface light source separately. We clarify the stochastic ray parameters including the starting point coordinates, direction vectors, energy weights, and their physical implications in the Monte Carlo modeling. Based on the radiation properties of the source, the desired probability density functions for different parameters of the ray are derived. In addition, we describe how to sample the parameters following an arbitrary two-dimensional probability density function based on inverse transform sampling. We introduce two ray sampling strategies of uniform sampling with equal weights and uniform sampling in parameter space. The former strategy samples the rays strictly according to the probability density functions, with equal energy weights. The latter strategy assigns the corresponding weights to the rays and ensures that the weights are proportional to the desired probability density functions, which can considerably improve the computational speed by avoiding numerical integration and interpolation operations. The proposed method can model light sources with arbitrary surfaces, with strong versatility. To verify the accuracy of the modeling results, the integral formula of the irradiance distribution formed by the surface light source on the receiver is derived as the theoretical illuminance distribution (Fig. 1). The accuracy of the modeling method is measured by comparing the relative deviation of the simulated irradiance distribution of the sampled rays from the theoretical value.

Monte Carlo modeling results and precision analysis are implemented for two different surface light sources, which are expressed by XY-polynomial (Fig. 2) and non-uniform rational B-spline (NURBS) (Fig. 6) respectively. The sampled starting points, ray directions, and rays (Figs. 3 and 7) are provided respectively to show the differences between the two sampling strategies. The calculated theoretical irradiance distributions formed by the two surface light sources at the specified receiver have an extremely high spatial resolution, which can be regarded as continuous (Figs. 4 and 8). The maximum relative deviation between the simulated value and the theoretical value is within 1% for 224(≈1.6×107) sampling rays, demonstrating a high modeling accuracy (Figs. 5 and 9). The uniform sampling strategy with equal weights leads to slightly higher modeling accuracy than that of uniform sampling in parameter space. For the NURBS surface light source, we analyze the differences in modeling accuracy and speed between the two sampling strategies under different numbers of rays and the influence of different random numbers on modeling accuracy (Fig. 10). This shows that the average modeling error gradually decreases while the modeling time increases with the rising number of rays. In contrast to pseudorandom numbers, the utilization of quasi-random numbers can improve the modeling accuracy. The strategy of uniform sampling in parameter space is faster than that of uniform sampling with equal weights since the latter employs the computationally expensive inverse transform sampling.

We propose a Monte Carlo modeling method for surface light sources. Based on the homogeneity assumption, the spatial and orientational radiation characteristics are analyzed separately. Probability density functions and sampling strategies are presented for different parameters of the rays, and a way to verify the accuracy of the modeling results is also proposed. For the two modeling examples of surface light sources, the maximum relative deviation of the simulated irradiance distribution from the theoretical value at the specified receiver is less than 1% when the number of sampled rays is at the order of 10⁷, demonstrating high modeling accuracy. In addition, the effect of different sampling strategies on modeling accuracy and speed is analyzed under different numbers of rays. The uniform sampling strategy with equal weights leads to higher modeling accuracy. In contrast, the uniform sampling strategy in parameter space is considerably faster. Comparisons through different random numbers show that quasi-random numbers can improve modeling accuracy.