Calculation of output photometric characteristics describing the energy of radiation is an integral part of a lighting device optical system designing. The most common way to solve this problem today is the Monte-Carlo method. However, its significant disadvantage is the low rate of convergence, which often does not allow obtaining the result in an acceptable time even on modern computing systems. Therefore, the search for ways to improve this method does not lose its relevance and is the subject of research by numerous authors. One of these ways is the use of the histogram method known from the theory of statistical modelling, which is reduced to the expansion of the sought-for value in a series of orthogonal functions forming an orthonormal basis. This approach paves the way to eliminate the averaging of the sought-for value and organize the calculation in such a way that any beam passing through the optical system contributes to the distribution function of the entire sought-for response of a receiver. Thus, we achieve an increase in the rate of convergence of the method. Of greatest interest here is the expansion of the sought-for quantity distribution in a series in terms of Legendre polynomials and spherical functions related to them. To date, the described modification of the Monte-Carlo method has been implemented only for the case of lighting devices optical systems with axisymmetric light distribution. In this article, we propose a method that allows to apply the method modification to lighting devices optical systems with arbitrary light distribution. Resulting computational time reduction while solving direct problems allows new ways for solving reverse problems, such as designing of optical systems for a given light distribution.
1. Sobol, I.M. Numerical Monte-Carlo Methods [Chislennye metody Monte-Karlo] / M.: Nauka, 1973, 312 p.
2. Rozenberg, G.V. The light ray (contribution to the theory of the light field) // Sov. Phys. Usp., 1977, Vol. 20, # 1, pp. 55–80.
3. Korobko, A.A., Kushch, O.K. Using the Monte-Carlo method in lighting calculations [Ispol’zovanie metoda Monte-Karlo v svetotekhnicheskih raschyotah] // Svetotekhnika, 1986, # 10, pp. 14–17.
4. Kushch, O.K. Optical calculation of lighting and irradiating devices on a computer [Opticheskij raschyot svetovyh i obluchatel’nyh priborov na EVM] / M.: Energoatomizdat, 1991, 152 p.
5. Feder, D.P. Optical Calculations with Automatic Computing Machinery // J. Opt. Soc. Am., 1951, Vol. 41, # 9, pp. 630–635.
6. Bartsev, A.A., Budak, V.P. Calculation of the photometric characteristics of optical systems by the Monte-Carlo method in the direct path of rays [Raschyot fotometricheskih harakteristik opticheskih sistem metodom Monte-Karlo v pryamom hode luchej] // Svetotekhnika, 1993, # 4, pp. 4–8.
7. Bartsev, A.A. Development of methods for mathematical modelling of optical systems as an element of automation of the design of lighting devices [Razrabotka metodov matematicheskogo modelirovaniya opticheskih sistem kak elementa avtomatizacii proektirovaniya svetovyh priborov] / Dissertation … candidate of technical sciences: 05.09.07. Moscow, 1994. 202 p.
8. Chencov, N.N. Statistical decision rules and optimal inference / NY: ACM, 2000, 499 p.
9. Gelfand, I.M., Minlos, R.A., Cummins, G. Representations of the rotation group and the Lorentz group, their applications / Eastford, CT, USA: Martino Fine Books, 2012, 386 p.
10. Korn, G., Korn, T. Mathematical Handbook for Scientists and Engineers / Mineola, NY, USA: Dover Publications, 2000, 1152 p.
11. Karyakin, N.A. Light fixtures of searchlight and projector types [Svetovye pribory prozhektornogo i proektornogo tipov] / M.: Higher school, 1966, 412 p.
12. Wang, K. et al. Freeform Optics for LED Packages and Applications / Hoboken, NJ, USA: Wiley, 2017, 372 p.

• Previous article
• Next article

• optical system
• Monte-Carlo
• ray tracing
• spherical functions