1. THE PROBLEM
Analysis of optical systems by Monte Carlo ray tracing, once an unusual technique, is now commonplace. Ray tracing can be used to simulate reflection, transmission, scattering, and absorption of light. For many optical systems, the effects of diffraction by edges are also important. This is especially true when stray light must be calculated, and the ratio of the wavelength to aperture size is relatively large. It is desirable, then, to simulate the effects of diffraction by edges within the framework of a Monte Carlo ray-tracing program. The opto-mechanical system to be analyzed may be complex, and the distributions of light illuminating edges may not be known. Traditional methods involving the solution of integrals, often by Fourier transform, are not suitable because of their strict sampling requirements. Furthermore, it is often desired to get a partial solution to edge diffraction by tracing a few rays, and have this solution be representative of the full solution. This is analogous to a measurement of an irradiance distribution in which only a few photons are measured. We know from experiment and from the quantum theory of electrodynamics that such a measurement, while noisy, is a predictor of the irradiance resulting from a more complete measurement. We present a method for predicting diffraction by edges that, while not new, is not well known. We also provide some new insight into the relationship of this method to other methods, and explain why we have chosen to use it in a production Monte Carlo ray-tracing program.
2. SURVEY OF EDGE DIFFRACTION CALCULATIONS
The study of diffraction has a long history – an overview can be found in Born and Wolf.1 As early as 1801, Thomas Young2 sought to explain the diffraction of light when passing through an aperture as an interaction of light with the edge of the aperture. Later solutions of the aperture diffraction problem employed integral techniques in which the entire aperture is considered as a whole. In this century, the idea of diffraction of light by edges was revived through the Maggi-Rubinowicz theory and generalized by Miyamoto and Wolf.3 Lyot4 took advantage of the experimental observation that an illuminated aperture edge appears bright when viewed from the shadow region, in the design of his coronagraph. Keller,5 working in the microwave field, developed a Geometrical Theory of Diffraction to aid in antenna design. In the 1960s, ray-tracing for Gaussian beams using a waist ray and divergence ray was developed.6,7 This was later employed by Greynolds8 in his Gaussian beam superposition method of modeling the propagation of electromagnetic waves. Meanwhile, the bending of rays near edges by employing the Heisenberg uncertainty principle was first suggested by Carlin9 and later employed in Monte Carlo ray-tracing algorithms for stray light analysis.10,11 The theory of quantum electrodynamics can also be used to predict diffraction by edges. The theory has similarities to the classical theory, but is fraught with many of the same problems for numerical computations as the integral formulations. 2.1. Integral methods The well-known integral methods of predicting diffraction by apertures, pioneered by Fresnel and Kirchhoff and later refined by Rayleigh and Sommerfeld require performing an integral of the incident field over the entire aperture weighted by a propagator or Green’s function. These methods accurately predict diffraction patterns complete with the high-frequency ripples caused by constructive and destructive interference. Unfortunately, they suffer from many practical difficulties.
ABSTRACT
Monte Carlo ray tracing programs are now being used to solve many optical analysis problems in which the entire optomechanical system must be considered. In many analyses, it is desired to consider the effects of diffraction by mechanical edges. Smoothly melding the effects of diffraction, a wave phenomenon, into a ray-tracing program is a significant technical challenge. This paper discusses the suitability of several methods of calculating diffraction for use in ray tracing programs. A method based on the Heisenberg Uncertainty Principle was chosen for use in TracePro, a commercial Monte Carlo ray tracing program, and is discussed in detail. Keywords: Ray tracing, diffraction, Monte Carlo, stray light analysis.