Download Edge diffraction in Monte Carlo ray tracing

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
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.
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.
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,
Copyright  1999 Lambda Research Corporation
Published in Optical Design and Analysis Software, Proceedings of SPIE, Volume 3780, Denver, 1999
but is not suitable for predicting wide-angle diffraction. It does, however, have much in common with the other methods, as
we shall see.
2.5. Quantum electrodynamics
Diffraction patterns can also be predicted using quantum electrodynamics (or QED). Analogous to the classical treatment,
one computes a probability distribution for “photon paths” by a superposition of all possible paths of the photon between the
source and the observation point. One can see parallels with Fermat’s principle, Huygens’ principle, and the integral
methods, and, as expected, the resulting probability amplitude for photons is equal to the diffracted field as computed by
classical methods. Again, this requires that one know the incident field a priori, the same objection as with the other methods.
The quantum interpretation of diffraction lends new insight to the problem at hand. The conventional quantum interpretation
of the Young double slit experiment is that each photon that passes through the experiment has a probability density
determined by the incident field and the geometry of the slits. If one records photons that pass through the apparatus, each
photon is absorbed by a discrete atom or molecule in the observation plane and so has a very localized position. Averaging
over many photons, however, produces the expected diffraction pattern. Since photons do not interact with each other, this
means that each photon “knows” the resulting diffraction pattern and so must pass through both slits at once. Furthermore,
each photon must know the characteristics of the source. A photon is a quantization of the energy of the electromagnetic
field, and this quantization has no implications for the spatial extent of the photon. Taken to an extreme, a photon may have
size comparable to the size of the universe, if emitted from a star, for example. The photon is localized only when it interacts
with matter, i.e., when it is absorbed or passes through an aperture or near an edge. Rays are not good representations of
photons!
3. HEISENBERG UNCERTAINTY RAY BENDING
In the Heisenberg Uncertainty Principle ray bending method (or HURB), the underlying premise for bending rays is that
when a ray passes close to the edge of an aperture, the uncertainty in its location can be no larger than the closest distance to
the edge of the aperture. This is in contradiction to the quantum interpretation, in which photons are not localized spatially
unless they pass through an aperture or are absorbed. Using the Heisenberg Uncertainty Principle and equating a ray to a
photon nonetheless, we assert that the product of the uncertainty in the position and the uncertainty in the momentum of the
photon is
∆x∆p x ≥ 12 h
(1)
where h = h / 2π and h is Planck’s constant. Because position and momentum are vectors, this uncertainty relation applies
only to the component of the vector in the direction defined by the line connecting the ray with the edge at its point of closest
approach, as shown in Figure 1. The uncertainty in the ray position in an orthogonal direction may also be finite. This can be
determined by constructing a projection plane orthogonal to the closest-direction vector and measuring the shortest distance
to the aperture edge. The uncertainty in the momentum can be interpreted as an uncertainty in the direction of propagation.
Specifically, the ray or photon may be deviated from its path by an angle, and the uncertainty in the momentum is interpreted
as an uncertainty in the direction. The angular uncertainty in direction can be found by geometry,
tan σ x =
∆p x
r ,
p
(2)
and the momentum of a photon is
r
r
p = hk
where
(3)
r
k is the propagation vector of length k=2π/λ. This yields for the angular uncertainty
 1 
 .
σ x = arctan
 2 ∆x k 
(4)
Copyright  1999 Lambda Research Corporation
Published in Optical Design and Analysis Software, Proceedings of SPIE, Volume 3780, Denver, 1999
Figure 1 The diffraction uncertainty ellipse near the corner of a rectangular aperture.
In the orthogonal direction we likewise have an angular uncertainty
 1 
 .
σ y = arctan
 2 ∆y k 
(5)
The two orthogonal uncertainties form an uncertainty ellipse as shown in Figure 1. We postulate that the probability
distribution governing the angular deviation of the ray in each direction is a Gaussian distribution, and the mechanisms for
deviation in the two directions are independent and separable. Then the probability distribution for the angular deviation of
the ray is an elliptical Gaussian distribution,
p(θ x ,θ y ) =
− (θ 2 / 2σ x 2 +θ y 2 / 2σ y 2 )
1
e x
.
2πσ xσ y
(6)
In practice, when a ray passes through an aperture, the distance from the ray to the nearest point on the edge of the aperture is
measured, the orthogonal distance is measured, and a probability distribution constructed. Then a random direction is chosen
for the ray, weighted by the probability distribution. The closer the ray is to the aperture edge, the wider the probability
distribution in that direction. Rays that pass close to an edge are spread out in a plane containing the ray and the normal to the
edge.
It is instructive to look at the problem in reverse. At a particular observation point behind the aperture in the shadow region,
the probability of a ray being bent toward the observation point is vanishingly small except for rays that are both close to the
edge of the aperture and lying in a plane containing the normal to the aperture edge and the observation point. For apertures
with corners, there is also a contribution from the corners, because the probability distribution is spread in both directions
near a corner.
Another interpretation of this method is as a superposition of Gaussian wavelets. The angular size or obliquity factor of the
wavelets is not stationary, that is, it changes shape depending on the position in the aperture. We can, nevertheless, write
down the integral corresponding to this algorithm. It is easier to write it down, however, than to solve it, because the angles
θx and θy and the uncertainties σx and σy are themselves complicated functions of the position in the aperture. In general, the
equivalent integral for the far-field intensity is
Copyright  1999 Lambda Research Corporation
Published in Optical Design and Analysis Software, Proceedings of SPIE, Volume 3780, Denver, 1999
I (θ x ,θ y ) =
∞ ∞
∫ ∫ A( x, y) E0 ( x, y ) p(θ x ,θ y ) dx dy
(7)
−∞ −∞
where E0 is the incident irradiance and A is the aperture function. This integral is difficult to solve even for the simplest
aperture shape, a rectangle, for which


1

σ x ( x) = arctan
 2 ( | x | −a) k 
(8)
and


1
 .
σ y ( y ) = arctan
 2 ( | y | −b ) k 
(9)
This rectangular aperture has x-width equal to 2a and y-width equal to 2b. The integral for a rectangular aperture becomes
I (θ x ,θ y ) =
b a
∫ ∫ E 0 ( x, y )
2
2



 



1
1

  
  +θ y 2 / 2 arctan
− θ x 2 / 2 arctan



 2 ( | y | −b) k   
 2 ( | x | − a ) k  

e 
−b − a




1
1
 arctan

2π arctan
−
2
(
|
x
|
a
)
k
2
(
|
y
|
−
b
)
k




dx dy
(10)
where the aperture function is represented by the integration limits. Although we have not attempted an analytical solution of
this integral, the transcendental functions suggest that a numerical solution is required. Furthermore, in a Monte Carlo
simulation, the incident field is not known a priori.
3.1. Connections to other methods
The bending of rays by a Gaussian probability distribution in the HURB method has many parallels with the other methods.
The BWD and the GTD are based on waves or rays emanating from the edge, while the HURB method, in a similar way,
bends or emanates rays that pass near the edge. We argue that the HURB method is more intuitive. The GTD requires a ray
to actually strike an edge, but a ray and an edge both have zero width, so the probability of a ray actually hitting an edge is
zero in the Monte Carlo method. Also, the cross-sectional area of the edge is zero, so the fraction of the incident field that is
modified by the edge in the BWD method should seemingly also be zero. It seems that the energy assigned to the diffracted
portion of the field at the edge in both the BWD and GTD is a mathematical artifice. The discontinuities in the field in the
BWD are also artificial.
The BWD, GTD, and HURB have a notable feature in common: the rays in the GTD and HURB and the stationary phase
points in the BWD require that the optical path or time of flight is a minimum, as in Fermat’s principle. This is a strict
requirement in the GTD, and a method for simplifying a line integral in the BWD. In the HURB, flux can only propagate in
directions satisfying this variational principle, because the Gaussian falls in amplitude very rapidly away from this direction.
This feature was not designed into the method intentionally, but is a side-benefit of the geometrical construction of the
HURB method.
There is another similarity between the BWD and HURB methods. From the theory of distributions, we know that a delta
function can be represented as a Gaussian distribution in the limit as σ → 0,
δ ( x) =
lim
σ →0
1
2π σ
2
2
e − x / 2σ .
(11)
Copyright  1999 Lambda Research Corporation
Published in Optical Design and Analysis Software, Proceedings of SPIE, Volume 3780, Denver, 1999
Then the integral in equation 10 can be seen to approximate the geometric wave from the BWD near the center of the
aperture, since the Gaussian distributions in the integrand are very narrow at this position and approximate a delta function.
The sifting property of the delta function will ensure that the incident field (diffracted ray in the HURB method) is
propagated almost geometrically. Near the edge of the aperture the Gaussian distributions are wider, approximating the
boundary wave from the BWD method. In this way, the HURB can be seen to be similar to the BWD, but providing a smooth
transition from the geometric wave to the boundary wave. No discontinuous or singular fields are required. The integral has
no phase term, so we do not expect our result to predict interference fringes. The larger the number of rays that are traced, the
better the approximation to the integral in equation 10. In this sense we are using the Monte Carlo method to evaluate the
integral directly, a popular use for Monte Carlo.
The HURB and GBD methods are similar in that the field is represented by a superposition of Gaussian distributions in both
methods. In the HURB method, only the angular spread is Gaussian, while in the GBD a Gaussian distribution encapsulates
the spatial-angular representation of each beam. In the GBD method, a distribution of Gaussian beams is defined at the
beginning of the ray-trace to represent the incident field, while in the HURB method, the width of the Gaussian distribution is
determined when the ray passes near an edge. The HURB method is, in this sense, adaptive in that when a ray passes close to
an edge, a very wide Gaussian distribution is generated, and in the center very narrow distributions are generated. These
distributions are independent of each other and do not depend on how many rays are used in the simulation. This means that
partial results from a ray-trace using the HURB method are useful, i.e. they are indicative of the correct result, but contain
noise due to the stochastic nature of the Monte Carlo method.
4. EXAMPLE
As stated in section 1, the HURB method of simulating diffraction is not new. Previous authors10,11,13 have reported on the
accuracy of this method for predicting near-field and far-field diffraction. Here we present an example, diffraction by a
circular aperture in the far field, for which analytical results are well known. At small angles the diffracted intensity was
computed by reference to the well-known Airy pattern, and for large angles an asymptotic form of the Airy pattern was used,
I (θ ) ≈
E i aλ
2π 2 sin 3 θ
,
(12)
where Ei is the incident irradiance, a is the aperture radius, and θ is the polar angle from the peak of the Airy pattern. Figure 2
shows the result of a Monte Carlo simulation of the diffracted intensity compared with the analytical calculations. The HURB
method does not produce the rings in the Airy pattern, but has good agreement with the analytical results over many orders of
magnitude.
5. CONCLUSION
We have reviewed several methods for calculating diffraction by apertures and edges in the search for a method that is
suitable for integration into a Monte Carlo ray-tracing program. We compared the boundary wave diffraction theory,
geometrical theory of diffraction, Gaussian beam decomposition, and bending of rays by the Heisenberg uncertainty
principle. All the methods correctly predict diffraction patterns within their own limitations, and we have found similarities
and connections between the methods. We have found that the Heisenberg uncertainty ray bending method is uniquely suited
for integration into a Monte Carlo program, and have chosen it for use in TracePro, a commercial Monte Carlo ray tracing
program.
Copyright  1999 Lambda Research Corporation
Published in Optical Design and Analysis Software, Proceedings of SPIE, Volume 3780, Denver, 1999
Diffracted flux
TracePro Diffraction - Circular Aperture
1.0E+03
1.0E+02
1.0E+01
1.0E+00
1.0E-01
1.0E-02
1.0E-03
1.0E-04
1.0E-05
1.0E-06
1.0E-07
1.0E-08
1.0E-09
1.0E-10
1.0E-11
1.0E-12
Asymptotic
TracePro
Airy Pattern
0.1
1
10
100
1000
10000
Airy radii
Figure 2 Diffraction pattern for a circular aperture produced by HURB ray-tracing compared with analytical results.
ACKNOWLEDGMENTS
This work reported in this paper was supported in part by NASA Goddard Space Flight Center under contract number NAS597010.
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Copyright  1999 Lambda Research Corporation