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We implemented the linear programming approach proposed by Oliker and by Wang to solve the single reflector problem for a point source and a far-field target. The algorithm was shown to produce solutions that aim the input rays at the intersections between neighboring reflectors. This feature makes it possible to obtain the same reflector with a low number of rays – of the order of the number of targets – as with a high number of rays, greatly reducing the computation complexity of the problem.
©2012 Optical Society of America
1. Introduction
Deriving the optics to direct light from a given source to a desired target illumination distribution is one of the main problems addressed by illumination design. If it is only desired to control either the irradiance or the intensity distribution at the target, one surface is sufficient. We can classify the single reflector design process for a point source as consisting of two main steps: first, a reflector must be obtained that collects light from the source and directs it to the target with a rough matching to the prescribed target illumination distribution. We refer to this solution as a starting point solution. Next, the reflector is adjusted to obtain the desired target irradiance or intensity. We refer to this solution as the point source solution. While real sources are extended sources, it is of great interest to derive a point source solution, because some real sources can be approximated as point sources, or the extended source solution can then be derived from the point source solution. If the problem has rotational or translational symmetry, it is straightforward to derive the surface that produces the required illumination [1], but in general non-rotationally symmetric surfaces are needed. The shape of the reflector is in general governed by a nonlinear second-order partial differential equation of the Monge-Ampère kind for which there is no analytic solution and which are not usually solved with standard numerical integration techniques.
Four approaches have been proposed to generate a non-rotationally symmetric reflector surface to produce the desired target illumination from a point source. Parkyn and Wang subdivide the problem in equi-flux regions and assign a mapping to direct the flux from the source to the target in the desired distribution, generating faceted reflectors [2,3]. Ries and Muschaweck apply a tailoring method to solve a set of partial differential equations that govern the reflector shape [4]; the solution is determined iteratively with a multigrid approach. Oliker exploits the imaging property of ellipses to design a reflector made of ellipsoid patches in an iterative process; the flux is initially collected entirely by one ellipsoid, then the ellipsoids are scaled until the desired target distribution is produced [5,6]. The Oliker algorithm is guaranteed to converge to the solution and, as the number of ellipsoids approaches infinity, the reflector surface becomes continuously smooth. Finally, Wang and Oliker use a variational approach, formulating the problem as a linear program to obtain a reflector consisting of patches of paraboloids [7–9]. To the authors’ knowledge, there are no detailed examples in the literature describing the properties of the latter method.
This paper explores the practical implementation of the linear programming in two dimensions and shows initial results in three dimensions. The linear programming algorithm appears to be especially suited to obtain a starting point solution.
2. Linear programming formulation of the reflector problem
A variational formulation of the single reflector design for a point source and a far-field target was proposed independently by Wang and by Oliker in 2003 [7–9]. Assuming a point source located at the origin, by sampling the source and target intensity distributions in a discrete set of NS source ray directions s and NT target directions t, it is possible to find a reflector solution consisting of NT patches of paraboloids, with each paraboloid directing light from the source into one target direction.
The reflector problem is formulated as a maximization problem
with linear constraintwhere IS(s) and IT(t) are the normalized source and target intensities in directions s and t respectively, 〈·,·〉 denotes the inner product, and the solutions u(s) and v(t) are related to the focal parameters C(t) and the polar radii ρ(s) of the paraboloids asFor energy conservation, it is necessary to ensure that the sum of the source flux collected by the reflector is equal to the sum of the target flux. Equations (1)-(2) can be formulated in matrix form as
where cT is the transpose of c, and the vectors c, x and b are defined asThe matrix A has size NSNT × (NS + NT) and is defined according to Eq. (1) to have only two non-zero elements per row. The size of A is the main limitation of the implementation of the linear programming algorithm, making it more suited to solve the starting point problem. As the number of rays and targets increases, the size of A quickly becomes very large. For NS = 100 and NT = 100, for example, the size of A is 10,000 × 200.
Since the linear programming algorithm solves a far-field problem, there is an entire family of solutions that provide the desired target illumination. To select one set of reflectors, it is possible to require the solution to meet certain requirements. For example, it is possible to set the distance to the reflector along the first ray to 1 by adding the constraint
where Aeq is a matrix of size NSNT × (NS + NT) whose only non-zero element is the first one and beq is a zero vector of length NSNT.The reflector solution obtained by solving the linear programming as expressed in Eqs. (1)-(2) and Eqs. (5)-(6) has a crossing geometry. If a non-crossing geometry is desired, the maximization in Eqs. (1) and (5) becomes a minimization and the direction of the inequality in Eqs. (2) and (6) must be inverted.
3. 2D reflector design
To illustrate the main features of the linear programming algorithm, let us consider the simple example of a 2D reflector with a source distribution with rays uniformly spaced from 0 to 90° and four target directions (NT = 4) uniformly distributed between 0 and −10°. Figures 1(a) -1(c) shows the reflectors obtained as a solution of the linear programming method with 5, 21 and 4 rays, respectively.
Fig. 1 2D reflectors obtained as a linear programming solution for four uniform targets directions from 0 to −10° and an isotropic source distribution from 0 to 90°, with crossing geometry. The number of rays is NS = NT + 1 in (a), NS = 5NT + 1 in (b) and NS = NT in (c). While a number of rays equal to NS = nNT + 1, with n integer, yields a valid solution (the same for any n), other combinations can produce reflectors that receive only one ray, as shown in the inset.
The choice of the number of rays is crucial for obtaining a viable solution. We found that, provided that the rays sample the source in equi-flux regions, if nNT + 1 rays are used, with n integer, then each reflector collects n + 1 rays, with NT -1 rays falling at the intersections between neighboring reflectors. Additionally, the NT paraboloids generated with nNT + 1 rays are identical even for different n, i.e., they have the same focal parameters and polar radii, as shown in Figs. 1(a) and 1(b) for an isotropic source with 5 and 21 rays respectively, making it possible to obtain the solution with a low number of rays (NT + 1). If a different number of rays is used, NS = NT for example, the solution is generally not feasible, as reflectors that only collect one ray are generated, as shown in Fig. 1(c) for 4 rays.
Figure 2 illustrates that both crossing and non-crossing geometries can be obtained with the linear programming algorithm.
Fig. 2 2D reflectors obtained as a linear programming solution for four uniform targets directions from 0 to −10° and an isotropic source distribution from 0 to 90°, with crossing (top) or non-crossing geometry (bottom).
Since the linear programming algorithm produces solutions in which neighboring reflectors share a source ray, to design a reflector for a non-isotropic source distribution, instead of modifying the weight of the rays in Eq. (7), it is better to sample the angular distribution of the source according to the desired distribution. Examples of a reflector for isotropic, Gaussian (with a sigma of 1/(2π)1/2) and Lambertian source distributions sampled in equi-flux regions are shown in Fig. 3 .
Fig. 3 2D reflectors with crossing geometry obtained as a linear programming solution for four uniform target directions from 0 to −10° and an (a) isotropic, (b) Gaussian and (c) Lambertian source distribution from 0 to 90°.
In general, if a nonuniform target distribution is desired, the discrete target directions should be selected to sample the target distribution into equi-flux regions and the same property of being able to obtain the solution with a low number of rays (NT + 1) will apply.
4. 3D reflector design
To extend the linear programming method to 3D, we consider a number of rays NS = (NT,x + 1)(NT,y + 1), where NT,x and NT,y are the number of target points in the x and y directions (z being the optical axis).
To illustrate how the linear programming algorithm applies to a three-dimensional case, let us consider a point source emitting over a tangent plane square with a 90° side and a 4 × 4 uniform target grid covering directions ± 5.6°. Figure 4 shows the starting point solutions generated with the iterative ellipsoid algorithm proposed by Oliker [5,6] with 64 rays and with the linear programming algorithm with 25 rays. To be able to compare the two reflectors, since the linear programming is for a far-field target, we placed the target far away from the origin.
Fig. 4 3D reflectors obtained (a) with the linear programming algorithm and (b) with the ellipsoid algorithm. The dots indicate where on the reflectors the rays used to run the algorithms hit. The number of rays used was (NT,x + 1)(NT,y + 1) = 25 for the linear programming and 4NT,xNT,y = 64 for the ellipsoid algorithm. (c) Normalized flux at the 16 target points evaluated tracing 16,000 rays for the ellipsoid algorithm (solid line) and linear programming algorithm (dashed line).
The solution produced by the ellipsoid algorithm with a low number of rays (4 per reflector) is not symmetric because of the nature of the algorithm itself; the flux is initially all collected by one reflector, and then the flux is redistributed, in an iterative process, between all the other reflectors to produce the desired illumination. It was shown that the ellipsoid algorithm has to be run with several rays per reflector to overcome this initial bias [10,11]. On the contrary, the linear programming algorithm was run with a low number of rays (1-2 rays per reflector) to yield the result of Fig. 4(a).
We can define a merit function for a uniform target illumination as
where Ti represents the flux directed to the i-th target. Tracing 16,000 rays to evaluate the flux collected by the reflectors of Fig. 4, the merit function obtained with the linear programming reflector was 0.18, while the merit function for the ellipsoid algorithm was 0.66.5. Discussion
The linear programming algorithm finds a numerical solution to the nonlinear second-order partial differential equation of the Monge-Ampère kind that governs the shape of the reflector, directing light from a point source to a discrete far-field target. It produces a reflector solution consisting of paraboloid patches; as the number of targets increases, the reflector surface approaches a smooth surface, as desired for fabrication [12]. However, while we have shown the linear programming to be a good candidate for obtaining a starting point solution with a low number of rays, the computational complexity introduced by the increasing size of the matrix A with number of rays and targets does not make it currently feasible to exploit the algorithm to get a point source solution. Instead, the starting point obtained with the linear programming can be directly optimized to produce the point source solution.
We have shown that the linear programming method aims the input rays to the intersections between neighboring paraboloids. Future work will address whether this property makes it possible to obtain a point source solution from the linear programming algorithm run with a low number of input rays.
6. Conclusion
We have implemented the linear programming algorithm in two dimensions and shown that if the number of rays used to run the problem is chosen as the number of targets plus one in each direction, the result is a set of paraboloids that intersect at the ray locations. We have also shown that it is possible to obtain a starting point solution with the linear programming algorithm in three dimensions using a low number of rays (of the order of the number of targets).
While immediately applicable to deriving a starting point solution, the linear programming currently appears limited by the computation complexity and its use to obtain a full point source solution remains to be explored.
Acknowledgments
The authors acknowledge Synopsys, Inc. for the educational license of LightTools®. This work was funded under a fellowship from Synopsys, Inc., the National Science Foundation (EECS-1002179), and the NYSTAR Foundation (C050070).
References and links
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