Freeform Optics and Supporting Quadric Method : Introduction

This chapter along with Chapters 8 and 9 are devoted to optical design with freeform reflecting surfaces. We begin by summarizing some of the notations and terminology used in these chapters.

Notation

R², R³: The two- and three-dimensional vector spaces, respectively.

§²: The sphere of unit radius with center at the origin О of a

Cartesian coordinate system in R³.

A: An arbitrary subset of R².

C^k(A): The set of к times continuously differentiable functions

defined on a set A; к is a nonnegative integer.

L(A): The set of functions on a set A such that |/(x)|dx < oo .

jy

MU ^:= f | f(x)dx : The norm off on L(A).

C^k(A) n C'(B): Functions which are in class C*(A) and in C'(B).

CfA)xC'(B): Set of pairs of functions/,# such that / e C'(A), g e C'(B),

A,J3<=R².

Radiance: (= radiance pattern) /; radiant flux per unit source area,

W/m².

Irradiance: (= irradiance pattern) L: radiant flux per unit target area,

W/m².

For a point source the input aperture is always taken on the unit sphere §² with center at the source. The source radiance (= intensity or radiance intensity) in this

case is W/sr.

In the far-field problem the reflected directions are identified with points on the unit sphere §². The target irradiance (often referred to as intensity or irrradiance intensity) in this case is the flux per unit area of the aperture on §² representing the directions of reflected rays, W/sr, see Figure 7.1.

FIGURE 7.1 Far-field reflector design problem.

The optical systems considered in this book are required to redirect the light and reshape the radiance pattern of a source so that the total output irradiance is distributed on a given target and has a pattern prescribed in advance. Following Winston et al. (2005) we refer to problems of this kind as prescribed irradiance problems. The goals of the design, in addition to the redirection and reshaping of the flow of light, may also include efficiency, compactness of the device, and optimized performance according to some criteria. Consider, for example, the problem of designing a reflector system shown schematically in Figure 7.1.

Here and in the rest of the book the framework for formulating prescribed irradiance problems is geometrical optics (GO). Afterwards, the designed optics can be tested computationally and/or experimentally by other techniques, including physical optics.

Let О denote a nonisotropic point source of light placed at the origin (also denoted by O) of a Cartesian coordinate system in 3D-space R³. The source О emits light rays in a set of directions defined by the aperture D (= D U dD. where dD is the boundary of D) given as a closed set on a sphere §² of unit radius centered at O. Points on §² are treated also as unit vectors in R³ with initial points at O. Usually, we assume that D is a convex domain on §².

We denote by I(m) the radiance of the source О in direction msD and assume that / is a nonnegative and integrable function on D. Suppose that a light ray emitted by the source О in direction m e 5 is incident on a smooth perfectly reflecting surface R at point r(m) with unit normal и = u(m) and reflects off it in direction у = y(m), I y(m)I = 1. In this case, the aim is to design a system with one reflecting surface R (to be determined!) intercepting the light beam from the source О and redistributing the light with prescribed irradiance over a given region T in the far field. The input and output irradiances are not shown in Figure 7.1. This example illustrates the type of design problems which can be analyzed by methods described in Chapters 8 and 9.

Many applications require optical systems capable of controling output irradiance. Such systems are needed in nonimaging optics (concerned with light concentration, illumination, and imaging applications), laser optics (concerned with reshaping laser beams), and the design of reflector antennas with prescribed directivity properties, to name just a few (Winston et al., 2005, Schruben, 1974; Malyak, 1992; Burkhard & Shealy, 1987; Olikeret al., 1994; Kinber, 1962; Kinber, 1984; Galindo-Israel et al., 1979; Galindo-Israel et al., 1987; Galindo-Israel et al., 1991a; Galindo-Israel et al., 1991b; Westcott, 1983; Westcott & Norris, 1975; Oliker, 2006a; Rhodes & Shealy, 1980; Dickey, 2014; Dickey et al., 2005).

Traditional methods for designing optics for prescribed irradiance problems are often based on an a priori assumption that the sources, targets, and light patterns have rotational or some other symmetry. However, designers of optics often encounter practical tasks in which optical surfaces are expected to accommodate light sources, targets, and illumination patterns without special symmetries such as rotational or rectangular. In these cases, traditional designs constrained by an a priori set-particular symmetry may lead to highly energy inefficient devices.

In contrast, freeform lenses and mirrors are expected to be designed without any a priori assumed symmetry and therefore significantly more degrees of freedom are available to the designer to create optical systems with desired characteristics (Winston et al., 2005; Ries & Muschaweck, 2002; Berry, 2017; Oliker, 2017; Oliker, 2003; Oliker, 2005a; Oliker, 2007; Oliker, 2005b; Oliker, 2006b). A freeform mirror shown in Figure 7.2 provides an example of the complexity of a shape representable by a freeform optical surface.

FIGURE 7.2 Freeform reflector transforming a sinusoidal shaped radiation pattern from a point source into a uniform planar far-field irradiance; see Section 9.4 for more details. Designed by V. Oliker using supporting quadric method; I/O data provided by J. C. Minano and P. Benitez (Benitez et al., 2006).

Unfortunately, in tasks involving freeform surfaces, traditional design methods in most cases can not be used and the development of new design methods capable of utilizing freeform surfaces is necessary. This area of research and development is presently at an early stage.

Theoretical and computational methods for designing freeform mirrors for applications which require the generation of prescribed irradiances is the main subject of this work. Freeform lenses will be discussed in a separate publication. For mirrors specifically we present here rigorous approaches to optical design producing effective tools for development, analysis, and description of required freeform optics. Such description may be a collection of formulas or functions defining the designed optical surfaces or an algorithm clearly indicating the steps leading to quantitative definitions of required surfacefs).

We outline now the issues involved in the development of methods for design witli freeform surfaces. In the GO approximation the derivation of equations of freeform optical surface(s) transforming a given input radiation into a prescribed irradiance pattern is based on a systematic application of the ray-tracing equations and energy conservation law. Physically, the resulting equations express the fact that the radiance of the source and the required output irradiance are related by a factor which is the determinant of the Jacobian of the ray-tracing map (defined by the reflecting/ refracting surface[s] to be determined!) The required optical surfaces are determined by solving these equations. Depending on the approach to derivation of the Jacobian equation, this step requires careful analysis of the theoretical and numerical solvability of either a nonlinear system of first order partial differential equations (PDEs) (Galindo-Israel et al., 1979; Galindo-Israel et al„ 1987; Galindo-Israel et al„ 1991a; Galindo-Israel et al., 1991b) or a nonlinear second order PDE of Monge-Ampere type for the function(s) describing the desired optical surface(s) (Winston et al., 2005; Kinber, 1962; Westcott, 1983; Westcott & Norris. 1975; Bosel & Gross, 2017; Oliker, 2014; Oliker, 2003). The resulting equations are quite complex even when it is assumed that the source is small or a collimator.

Solving numerically such equations is a very challenging problem. Because of strong nonlinearities the traditional discretization schemes based on finite differences in general fail to converge (Oliker & Prussner, 1988) and, for the same reason, the discretization schemes based on projection methods are not applicable. This is true for problems formulated as systems of PDEs (Galindo-Israel et al., 1987; Galindo-Israel et al., 1991a; Galindo-Israel et al., 1991b) and as Monge-Ampere equations (Westcott, 1983; Oliker, 2003). The engineering literature contains numerous papers in which attempts are made to solve the equations of beam-shaping numerically with ad hoc approaches. Flowever, the authors almost always rely on specific numerical examples to justify their approaches and results (Galindo-Israel et al., 1979; Galindo-Israel et al., 1987; Galindo-Israel et al., 1991a; Galindo-Israel et al., 1991b; Westcott, 1983). This is a serious drawback making the reliability of such approaches problematic.

Alternative formulations of some of such design problems based on variational considerations are sometimes possible. For example, it has been shown that, when the desired irradiance pattern is in the far field or when two refracting/reflecting surfaces are used, the design problems can be formulated as problems of linear programming (Glimm & Oliker, 2003; Glimm & Oliker, 2004; Oliker, 2011). In the near-field case for a single lens, however, this formulation is impossible (Graf & Oliker, 2012).

In recent years a very general geometric framework for solving several classes of equations arising in beam-shaping problems has been developed and applied by the author of this survey and his collaborators in Oliker et al. (1994); Oliker and Prussner (1994); Kochengin and Oliker (1998); Oliker (1989); Newman and Oliker (1994); Kochengin and Oliker (1997); Kochengin et al. (1998); Caffarelli and Oliker (2008); Caffarelli et al. (1999); and Kochengin and Oliker (2003). The developed “Supporting Quadric Method” (SQM) is based on new geometric ideas for constructing the so-called “weak” solutions of Monge-Ampere equations. It provides a rigorous and unified framework for establishing existence of solutions and calculating them numerically. However, weak solutions representing sag functions, while continuous, may not be differentiable at all points and the resulting optical surface may have straight and curved edges and corner points. It is important to note that the set of all points at which nondifferentiability occurs has zero area and with some additional efforts weak solutions can be smoothed out.

Regarding numerical calculations required for explicit representation of the sag functions of the designed optics, the SQM, in essence, includes a built-in numerical procedure. In particular, for problems with one reflecting surface, specific computational algorithms based on SQM were proposed in Kochengin and Oliker (1998) and Caffarelli et al. (1999). The algorithms described in these papers are motivated by the physical and geometric properties of weak solutions to these problems (see Section 2.9 for more details) and are proven to converge. However, the rate of convergence of the corresponding numerical algorithms may become low when the number of discretization nodes is high. Such behavior is typical for strongly nonlinear problems, including equations of the Monge-Ampere type. Presently, computing reliable numerical freeform solutions to many such problems with high accuracy is possible but can be hard in terms of the complexity of the algorithms and computing time requirements. Because of the underlying strong nonlinearities, a reasonably accurate discretization of such design problems requires fine meshes for representing the light source and its radiation pattern, the target set, the required irradiance on the target, and the solution. Consequently, the algorithms and their implementations into codes may be quite complex as they have to solve a large number of equations relating the data on the source with the data on the target. Thus, many theoretical and computational issues important for applications still remain open. In particular, analysis and enhancements of computational algorithms for solving Monge-Ampere equations numerically remains an important direction of research. This direction of mathematical and optical research has become very active, especially during the last two decades (Oliker & Prussner, 1988; Oliker, 1989; Caffarelli et al., 1999; Glowinski et al., 2018; Nochetto & Zhang, 2018; de Castro et al.. 2016).

The selection of problems considered in this chapter was aimed at providing illustrations of applications and design methods. Consequently, many applications which may benefit from the use of freeform surfaces and developed design methods are not discussed here. In order to focus our presentation and due to constraints on the size of this discussion we discuss only problems involving reflecting optical system. The refractive systems will be discussed in a subsequent publication. Chapter 8 is partially based on our earlier survey (Oliker, 2003). In order to maintain a reasonable level of rigor, we had to use some mathematical terminology. For the convenience of the reader, a summary of notations and terminology is provided at the beginning of this chapter. Beginning graduate textbooks (Lieb & Loss, 1997; Taylor, 2006) can be used as general references. Finally, we note the paper of Fang et al. (2013) which includes an extensive list of references to papers where the manufacturing of freeform optics is discussed.

References

Benitez, P., Winston, R„ and Minano, J. C. (2006). Nonimaging Optics. Short Course # SC388, 76-78.

Berry, M. V. (2017). Laplacian magic windows. J. Opt. 19, 1-5.

Bosel, C., and Gross, H. (2017). Single freeform surface design for prescribed input wavefront and target irradiance. J. Opt. Soc. Am. A 34(9), 1490-1499.

Burkhard, D. G., and Shealy, D. L. (1987). A different approach to lighting and imaging: Formulas for flux density, exact lens and mirror equations and caustic surfaces in terms of the differential geometry of surfaces. In SPIE Vol. 692 Materials and Optics for Solar Energy Conversion and Advanced Lighting Technology, 248-272.

Caffarelli, L. A., and Oliker, V. (2008). Weak solutions of one inverse problem in geometric optics. Preprint, 1994. J. Math. Sci. 154(1), 37-46.

Caffarelli, L. A., Kochengin, S., and Oliker, V. (1999). On the numerical solution of the problem of reflector design with given far-field scattering data. Contemp. Math. 226, 13-32. de Castro, P. M. M., Merigot, Q., and Thibert, B. (2016). Far-field reflector problem and intersection of paraboloids. Numer. Math. 134, 389-411.

Dickey, F. M. (Ed.) (2014). Laser Beam Shaping: Theory and Techniques,

Dickey, F. M., Holswade, S., and Shealy, D. L. (2005). Laser Beam Shaping Applications.

Fang, F. Z., Zhang, X. D., Weckenmann, A., Zhang, G. X., and Evans, C. (2013). Manufacturing and measurement of freeform optics. CIRP Ann.-Manuf. Technol. 62, 823-846. Galindo-Israel, V., Imbriale, W. A., and Mittra, R. (1987). On the theory of the synthesis of single and dual offset shaped reflector antennas. IEEE Trans. Antennas Propag. AP- 35(8), 887-896.