1. INTRODUCTION ABSTRACT

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1 Copyright 2008, Society of Photo-Optical Instrumentation Engineers (SPIE). This paper was published in the proceedings of the August 2008 SPIE Annual Meeting and is made available as an electronic preprint with permission of SPIE. One print or electronic copy may be made for personal use only. Systematic or multiple reproduction, distribution to multiple locations via electronic or other means, duplication of any material in this paper for a fee or for commercial purposes, or modification of the content of the paper are prohibited. Optimization using rational Bézier control points and weighting factors Mary G. Turner & Kevin J. Garcia Breault Research Organization, 6400 East Grant Road, Tucson, AZ ABSTRACT Optimization of traditional optical systems involves defining a merit function and a corresponding set of variables. The variables are changed to configure the optical system for optimum performance as required from the merit function. The typical variables used in lens design codes for designing classical optical systems are surface curvature, conic constant, aspheric coefficients, thickness, and refractive indices. The surface curvature, conic constant and aspheric coefficients are related directly to a polynomial equation representation of the surface. However, sometimes these variables are not the best choice for optimization especially in illumination system design where the required optical prescription is in the form of computer aided design (CAD) geometrical representation. As an alternative to these traditional variables, optimization using rational Bézier control points and weighting factors as variables is proposed and explored in this paper. Non-uniform rational B-splines (NURBs) using the Bézier basis are natural, graphical design curves exhibiting end-point interpolation whose interior control points and weighting factors are ideal variables for optical system optimization. Furthermore, optical designs created with NURBS are already in the language of CAD and numerical control machining environments and do not require the troublesome process of converting polynomial surfaces to their parametric representations. Keywords: Bezier polynomials, optimization, downhill simplex, simulated annealing, Brent s method, illumination design 1. INTRODUCTION Optical system complexity has increased over the years and so too has the need to model light interactions in a nonsequential ray tracing environment in order to simulate realistic system behavior. Most of these systems can benefit from automated optimization algorithms integrated with the non-sequential ray trace engine to improve system performance. There are two distinct requirements for the optimization process: a merit function which defines the design requirements and an optimization method which will drive the process. A range of general optimization methods combined with the use of smart merit functions provides an effective toolset for optimizing non-sequential systems. Many of these systems use extended sources which require large ray sets to achieve suitable radiometric or photometric simulation accuracy. Optimization routines coupled with these systems will require a large amount of time to find a solution because of the ray trace overhead. Although optimization with large ray sets does have its applicability, an alternative approach is to reduce the ray trace overhead by choosing appropriate ray sets that simulate a smaller subset of the problem. This paper examines using such ray sets to optimize non-sequential optical systems in the context of damp least squares, downhill simplex, and simulated annealing optimization routines. A key need for all forms of optical analysis programs is an ability to optimize the performance of the optical system. Sequential ray tracing programs are often called lens design programs because their most important capability is the ability to optimize imaging optical systems such as photographic lenses, microscopes and telescopes. Optimization refers to a generalized process which systematically varies the initial parameters of a system in an automated fashion to design a system which best meets a specified performance metric. Optimization consists of three separate components: a list of design goals or targets, a set of variables and an appropriate optimization algorithm. The optimization algorithm is used to systematically adjust any defined variables so that the system performance best matches the design goals. A merit function is a set of rules or targets and constraints which are related to the system s defined parameter values and is used to specify a performance goal. The goal of optimization is to determine a set of parameter values which

2 result in the values of all of the targets being met. In general, variations in the value of the system parameters impact the merit function in a non-linear fashion. For most systems, the topography of solution space consists of many peaks, valleys or even flat regions. The lowest point on each valley is referred to as a local minimum. The location in solution space where the values of the merit function is it s absolute lowest is referred to as the global minimum. Almost all optical systems contain both imaging as well as non-imaging or illumination paths. A typical example of this is a projection system. The illumination system provides an appropriate distribution of light to the modulator, or object. The image of the object is the projected to the screen. Each part of the system requires a different type of optimization technique as well as significantly different merit functions. For optimization of imaging systems, several techniques are used where both the defined weighted targets are provided with an appropriate optimization algorithm. The targets usually consist of a set of rays that are traced for proscribed points in the object space of the system (across the field of view) which travel through specifically selected locations within the system s entrance pupil, or aperture stop through to the image surface. The optimization algorithms, often based on damped least squares (DLS) or singular value decomposition (SVD) techniques attempt to form a point image for each point in object space. Nonsequential systems, especially illumination systems, provide significant challenges for optimization. In illumination systems, there is usually no defined pupil through which all the energy passes, multiple sources may be used which are not located along a common surface, all of the energy does not interact with all (or even any) of the optical surfaces, and the common goal is to generate a specific spread of energy, rather than to perform point to point imaging. Complicating the task further is that many surfaces used in illumination designs are defined parametrically, as in a CAD program, rather than represented as implicit or explicit polynomials similar to common optical surfaces. Although predefined sets of targets combined with a selected optimization algorithm may be appropriate for very specific problems, the real need in illumination or general nonsequential optimization is the ability to define any combination of targets and to apply an optimization technique that can best work within the topography of the solution space of the system. This requires a variety of optimization algorithms as well as the capability to define complex targets based on a wide range of goals including illumination uniformity (in spatial, angular or direction cosine space), separation of energy paths based on selected surface interaction requirements, or other ray history data including effects of scatter, coherence, Fresnel effects, etc. In ASAP, several optimization algorithms are available for both imaging as well as non-imaging design. A wide range of ray interrogation tools are available to define any necessary optimization target. Optimization can be performed on any surface, including Bézier-based surfaces. 2.1 Local damped optimization 2. OPTIMIZATION TECHNIQUES Damped least squares (DLS) is referred to as a downhill optimizer. The optimizer will search a downhill path through the solution space to find a nearby minimum. DLS assumes a merit function of the form Φ = ϕ + ϕ + +ϕ m The are m targets. The contribution of each target to the total merit function is ϕ wi ( v t ) i = i i (2) where v is the actual value, t is the target value and w is a weighting or importance factor. The intent of optimization is to minimize Φ in a least squares sense, which implies minimizing all of theϕ i s. This is done by setting the first derivatives to zero. In this process an assumption is made over the range of variable values, the behavior of the functional derivatives is linear. A damping factor is used to help force this condition. Singular value decomposition (SVD) is a modification of the damped least squares technique. In this case the damping factor is based on the second derivative of the merit function. Using double-sided derivatives enhances the prediction of an appropriate damping factor by considering the departures from linearity of the derivative. These optimization techniques are generally applied to optimization of imaging systems, to minimize the RMS or transverse ray or wavefront aberrations. A simple example is to change the bending factor of a singlet lens. The figures below demonstrate the shape of solution space for this example. (1)

3 TRANSVERSE RAY ABERRATION MERIT FUNCTION VS LENS BENDING MERIT B BENDING PARAMETER/SHAPE FACTOR TRANSVERSE RAY ABERRATION MERIT FUNCTION IS SUM OF SQUARES OF TRANSVERSE RAY ABERRATION IN MM ASAP RMS SPOT SIZE AS A FUNCTION OF LENS BENDING RMS BENDING PARAMETER/SHAPE FACTOR RMS SPOT SIZE AS A FUNCTION OF LENS BENDING RMS SPOT SIZE IS IN MM ASAP B Figure 1 RMS spot size as a function of lens bending 2.2 Brent s method Figure 2 Transverse ray aberration as a function of lens bending Brent s method is a technique for finding the minimum value of a function in a 1-dimensional solution space (i.e., only one variable). This technique combines several root-finding algorithms to form a very fast and robust optimization technique as long as the solution space is not discontinuous. 2.3 Simulated annealing Simulated annealing is an optimization technique that attempts to mimic the physical process of annealing materials such as metals or glass. Annealing is used to form materials with improved physical properties. There are three steps in the annealing process: heating the material to the annealing temperature, holding the material at this temperature until it is uniformly heated, and cooling the material at a predefined rate to allow for the optimal orientation of atoms and molecules. The optimal orientation is the distribution which requires the least energy to maintain. In applying this process to optimization, the various systems (variables and their associated values as well as the resulting merit function value) are analogous to the equivalent states of the annealing material. The merit function defines the energy state of a particular solution. A control parameter, the temperature, is used to limit the acceptable range of solutions. If the temperature is relatively large, solutions which result in a higher energy state may be accepted as they may indicate a path to a more optimal state. As the temperature is reduced, the allowed range of positive departure in the merit function is reduced and as the temperature approaches zero, only solutions resulting in a lower merit function are accepted. As a local minimum is achieved, the system is perturbed to a random starting point and the process runs again, looking for a different, possibly better local minimum. 2.4 Downhill simplex The Nelder-Mead method uses a simplex to find the minima of an N-dimensional function, where N is the number of variables. The simplex is defined by N+1 vertex points. In 2-dimensions, the simplex has the form of a triangle formed on a plane. In 3-dimensions, the simplex is tetrahedral. The downhill simplex techniques solves for the minimum solution of a continuous solution space by evaluating the function at the vertex locations and then adjusting the orientation, size or shape of the simplex based on these values. After each modification of the simplex, the merit function is reevaluated at the new vertex locations, resulting in a further change to the simplex. This technique is useful for finding a local minimum solution for situations where the merit function is smooth and continuous. Modifying the simplex near the local minimum has the potential to move the solution away from a local minimum in an attempt to find the location of a better minimum. 3. BÉZIER FUNCTIONS Bézier functions are parametric interpolating functions that were developed to describe automobile components in computer-aided design (CAD) programs. Bézier polynomials are the most stable of all of the polynomial bases used in

4 CAD programs. Bézier curves have several properties which make them very useful for designing curves and surfaces related to optical and illumination system design. Bézier curves are described by a minimum of three points: two endpoints and at least on control point. The total number of control points is determined by the order of the polynomial function. An n th order polynomial requires n-1 control points. A key feature of Bézier curves is that they use end-point interpolation. That is, the curve passes through each of the endpoints. The curve does not pass through any of the control points. The control points instead provide a force which pulls on the curve, affecting the shape of the curve between the endpoints. 3.1 Interpolation Polynomial interpolating functions are most often written in parametric form. That is, the coordinates of the curve or surface are specified as a function of an auxiliary variable. In the case of an explicit function, the dependent variable can be defined explicitly in terms of the independent variable. Consider a two-dimensional explicit parabolic equation defined in a Cartesian coordinate system: Now let t = x. With y = t 2, the points on the curve are given by 2 y = f( x) = x (3) 2 P(t) { x( t), y( t)} { t, t } The domain of the function is t and the range is P(t). = = (4) Consider a linear interpolation between two control points p 0 and p 1 as shown in Figure 1. These control points are used to define, or control the shape of the surface or curve. Interpolating between the control points is simply a matter of finding the parametric equation of the line between them. The direction vector between the points is (p 1 - p 0 ). The coordinates of any other location along the line can be determined by moving the required distance along the direction vector. In this case This may be rewritten as Figure 3 Control points and their relationship to auxiliary variables p() t = p + t( p p ),0 t 1 (6) 0 1 0

5 p() t = (1 t) p + t p,0 t pt () = αp + βp, α + β = 1 (7) α and β are the projective or barycentric coordinates. This leads to another important characteristic of Bézier curves: geometries can be shifted, rotated, scaled or skewed by directly manipulating the control points using affine mapping. An affine map is any transformation that leaves the barycentric properties of an interpolator invariant. This holds for Bézier polynomials because the Bézier interpolation uses linear interpolants. Since three points along the line in Figure 1 are collinear, we can expand Equation (7) to show t ti ti+ 1 ti =. (8) pt () p p p i That is, the ratio of the distances between any three points on a line is related to the ratio of the barycentric coordinates of those points: Ratios are preserved under affine mapping. i+ 1 i dist( pi, p( t)) α Ratio( pi, p( t), pi+ 1) = =. (9) dist( p( t), p ) β The convex hull property of Bézier curves is also important in geometric modeling. This means in addition to the barycentric coordinates summing to 1, these coordinates are also non-negative. This means that Bézier curve is completely contained within the region defined by the control points. i+ 1 Figure 4 Convex and non-convex sets

6 3.2 Interpolating basis function The convex hull property of Bézier curves allows a function to be evaluated using only the control points. Effectively, any polynomial or part of a polynomial is a Bézier polynomial as long as the control point data is properly selected. The equation of a line connecting control points p 1 and p 0 can be written 1 p () t = (1 t) p + t p,0 t 1. (10) Equations for the lines between the control points p p 1 0, p 1 1, p 0 and 1 can be written Combining yields a quadratic function in the variable t: p () t = (1 t) p + t p,0 t 1 and p () t = (1 t) p + t p,0 t 1 p () t = (1 t) p + 2 t p + t p,0 t (11) (12) Figure 5 Bézier control polygon and interpolated curve Extending this relationship to higher orders results in the decasteljau recursion r 2 r 1 r 1 r = 1,... n pj() t = (1 t) pj () t + t pj+ 1(), t j = 0,... n r 3.3 Rational Béziers Rational curves described on a Bézier basis can be described in a modified form, using a normalized mass: n n p jwb j j() t n j p () t = n n wb() t j j j (13) (14) where BBj n is a Bernstein polynomial and the w j s are weighting factors. As shown previously, the quadratic form of the non-rational Bézier ia a parabola. It has three control points. Assume the end control points are located in the same z-plane and have a weighting factor of 1. Consider the effect of varying the

7 weighting factor for the interior control point. If the control point is also in the same z-plane and also has a weighting factor of 1, then we have a parabola in that z-plane. As the weighting factor changes, the projection of the parabola rotates around the exterior control points. Look at the conic sag equation. Figure 6 shows a parabola projected onto the projection plane defined by w=1. The weighting factor on both the end control points is one. This is the standard form for the Bézier. The point m is the bisector of a line between the two endpoints. m p + p = (15) Figure 6 Projected parabola Point s indicates the vertex of the curve. The distance from the vertex to control point p 1 is d and the sag at the end control point is z. For a rational Bézier in standard form, m, s and p 1 are collinear and the ratio between these points defines the weighting. For this case z w1 = (16) d Substituting the sag equation as well as the equation for the line from p 1 to p 2 yields ρ w1 = 1 ( 1 + k) R The interior control point weighting factor is equal to the radical term of the conic sag equation. The distance d can be easily computed after the weighting factor is known. 4. OPTIMIZATION OF BÉZIER SURFACES Optimization of surfaces and curves defined in the Bézier space has significant advantages in illumination design. First, these polynomials represent the shape as defined in the CAD environment. Commonly, non-imaging optics are first designed in a CAD environment. If the optimization is performed using the same basis as the original design form, then 2 (17)

8 there will not be any loss of accuracy when the surface is transferred between the CAD program and the optical program and then back. Also, commonly used optical surfaces, such as aspheres, are not exactly represented in the CAD environment and these surfaces also are generally not the most effective form for illumination applications, other than simple conics. 4.1 Projection illumination Often the requirement for an illumination system is to provide an energy distribution that may be defined in terms of distribution as well as shape. An example of this is to provide a square, uniform distribution of light on a projection screen at a specified location and of a specified size. For a smoothly varying solution space, the downhill simplex optimization technique is often a good approach. For this design, a Bézier patch was defined having the necessary size and shape for the reflecting optic. Optimization can be done on any of the parameters defining the patch, which include the coordinates of each of the control points as well as any of the weighting factors associated with each of the control points. If appropriate, symmetry should be exploited to improve the efficiency of the optimization. The figures below show the surface profiles and the energy distributions before and after optimization. Figure 7 Surface before optimization Figure 8 Energy distribution before optimization

9 Figure 9 Surface after optimization Figure 10 Energy distribution after optimization In this example, the weighting factors can be adjusted to maximize the squareness of the distribution. Similar techniques based on non-bézier polynomial techniques have significantly less design freedom. 4.2 Angle mapper A typical task for an illumination engineer is to design a system, such as a CPC to meet a required illumination condition. Although CPC are ideal 2D angle mappers, the required length or other parameters of the optimal CPC may not be appropriate in all applications. By defining a CPC-like surface and allowing the parameters, including weighting factors as well as control points to vary, a more acceptable energy distribution can be obtained. Although the change in the shape of the cone is almost imperceptible, a noticeable change can be seen in to output. Figure 11 Initial cone Figure 12 Initial output energy

10 Figure 13 Optimized collector Figure 14 Final energy distribution 5. CONCLUSION The combination of robust optimization algorithms appropriate for optimizing illumination and other non-imaging systems is a necessary capability of optical analysis programs. The ability to combine such optimization techniques with Bézier surface representation opens a wide range of opportunities to the optical engineer which have not been available previously. REFERENCES [1] Garcia, K.J. Non-rational and rational parametric descriptions of the geometric propagation of light in an optical system UMI Microform (1999)