TracePro 中文教程

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 dLeabharlann Baiduffraction 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.
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.
Copyright © 1999 Lambda Research Corporation
Published in Optical Design and Analysis Software, Proceedings of SPIE, Volume 3780, Denver, 1999
Solutions are available in closed form for only a few simple aperture shapes and incident fields. In addition, the incident field must be known a priori in order to carry out the calculation. This is a serious disadvantage in a Monte Carlo simulation, in which the incident light distribution is in general not known at the beginning of the ray-trace – indeed, the purpose of the raytrace is to determine the light distribution. Even when the incident field is known, the sampling requirements for performing the integrations numerically are prodigious. By extending the Shannon sampling theorem12 one can see that the integrand must be sampled with a spatial interval of λ/2 or smaller, where λ is the wavelength of light. For example, with a one-meter aperture and λ = 0.5µm, on the order of 1013 samples are needed for each point in the diffracted field. Because the resultant field is a coherent summation of the field over the aperture, and much of the resultant field is small or even zero due to interference, at least 128-bit arithmetic is necessary to avoid round-off errors. At the current state of computer hardware and software, this calculation is not feasible. 2.2. Boundary wave theory The Maggi-Rubinowicz theory of boundary wave diffraction (or BWD) is based on the fact that the Kirchhoff diffraction integral can be decomposed into two terms. One of the terms represents a wave originating from every point on the boundary of the aperture (the boundary diffraction wave), and the other represents a wave propagated according to geometrical optics (the geometrical wave).3 This was done by constructing a new vector potential that has the property that the normal component of its curl (on a surface enclosing the observation point) is equal to the integrand of the Kirchhoff integral. The total field consists of an integral over a surface that includes singular points in the aperture (the geometrical wave), and a line integral around the edge of the aperture (the boundary diffraction wave). Since the geometrical wave has a discontinuity at the edge of the aperture, the boundary diffraction wave has a discontinuity that exactly cancels it in order to produce a continuous total wave. This technique can be further exploited to predict the diffracted intensity in the shadow region by reducing the line integral to a discrete sampling of points on the edge using the method of stationary phase. Stationary phase points occur in the integrand where the path length from the incident field point to the aperture point to the observation point is stationary, i.e., the derivative of path length with respect to position on the edge is zero. The argument goes that because the phase is rapidly varying over the remainder of the line integrand, the integrand is sinusoidal and so “cancels out” because its average value is zero, so that only the stationary phase points need to be evaluated in order to evaluate the integral. This is a very powerful method for calculating the diffracted intensity in the shadow region when the incident light distribution is known a priori. Again, it does not fit with our Monte Carlo way of tracing rays, in which we know the distribution of incident light only on a ray by ray basis. The discontinuities and singular points are somewhat troubling as well. 2.3. Geometrical theory of diffraction The geometrical theory of diffraction (or GTD) was developed in order to aid in the design of microwave antennae, and is still used for that purpose. It is based on the hypothesis of a Fermat’s principle for edge diffraction. That is, just as the path of a ray through a medium between two points is an extremum in the time of flight or the path length (Fermat’s principle in brief), the path of a ray between two points diffracted by an edge is an extremum, with a discontinuity in the path occurring at a point on the edge. This is in concert with the idea that diffraction is an edge phenomenon. It has parallels with the boundary wave theory, with the Fermat’s principle for edge diffraction describing the same points on the edge as the stationary phase points. Unfortunately, this technique requires that rays actually intersect exactly with edges, in turn requiring a deterministic distribution of rays. In our Monte Carlo way of doing things, rays occur in random locations and with random directions, making this concept unsuitable. 2.4. Gaussian beam decomposition In the Gaussian beam decomposition method (or GBD), advantage is taken of the fact an arbitrary function can be represented as a superposition of Gaussian functions. This, combined with the fact that a wavefront can be propagated with a Fourier transform and a propagation kernel, and the Fourier transform of a Gaussian is another Gaussian, means that the initial representation of the wavefront by the superposition of Gaussian beams is all that is needed. Propagating the wavefront can be accomplished by propagating the individual Gaussian beams. This can be done simply and elegantly by propagating the central ray of the Gaussian beam, called the base ray, and two additional rays representing the waist of the beam and the divergence angle of the beam, called parabasal rays. An elliptical Gaussian beam requires two additional parabasal rays, and symmetry is maintained by adding four more parabasal rays. The electric field can be propagated in this way, subject to some limitations. First, the incident wave must be represented in a computer program by a finite number of discrete Gaussian beams rather than a continuum of distributions. This can cause fuzzy edges and spurious ripples if there is insufficient sampling, i.e., not enough Gaussian beams. Aliasing can also occur if there is a moiré effect between the sample distribution and the aperture shape. Second, the Gaussian shape solves the homogeneous scalar wave equation only in the paraxial region, meaning that results for wide-angle diffraction are suspect. This is a clever and elegant method for propagating wavefronts,
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.
Published in Optical Design and Analysis Software, Proceedings of SPIE, Volume 3780, Denver, 1999
Edge diffraction in Monte Carlo ray tracing
Edward R. Freniere, G. Groot Gregory, and Richard A. Hassler Lambda Research Corporation, 80 Taylor Street, Littleton, MA 01460-4400