The IODC illumination design problem: Rays from a square Lambertian source must be directed toward this crossshaped target located in the farfield, with parameters as shown.
The winning solution: Squaretocross converter comprising a freeform solid TIR light pipe immersed in a rotational symmetric projection lens. The design is of the multiparameteroptimization type. The insight of the designer was crucial to provide a good starting design for the optimization. Design by Bill Cassarly.
The runnerup: Chaves, Grabovickic and Garcia used light pipes comprising a combination of light shifters, angle transformers and angle rotators for the squaretocross transformation, used as input for the gradedindex Morgan projection lens. This solution did not use optimization methods.

Considerably more development is needed for SMS methods, both at the theoretical and practical levels. These methods will be extended for more surfaces and more ray congruences. Such extensions are not trivial. One of the most exciting developments is imaging applications of SMS3D. The design procedure in 2D involves the simultaneous calculation of N axisymmetric surfaces, with the condition that N oneparameter ray bundles (and their symmetric counterparts) are perfectly imaged. The design strategy selects ray bundles that are evenly distributed in the phase space (i.e., the spaceangle domain).
By continuity, good image quality for those rays will guarantee sufficient image quality of all other rays through the system. This strategy was first used in the design of the RX, an ultrahigh numerical aperture imaging device showing higher imaging quality than conventional aplanatic designs, for which the system is optimized only for firstorder deviation from the onaxis point.
Emerging illumination design models
These three methods for freeform surface designs—multiparameter optimization, partial differential equations and SMS—are very different. The question of which one works best depends on the specific problem.
A good example of a challenging design problem was proposed at the 2006 International Optical Design Conference (IODC). This conference can be traced back to 1905, and has a long tradition of organizing interesting imaging lens design contests. For the 100^{th} anniversary meeting, the conference included its firstever contest on illumination design.
Participants were asked to come up with the best solution to a fluxcoupling problem. The source was a flat, square Lambertian emitter, and the angular contour of the target is shown as a cross in the top figure on the facing page. The angles H and V are formed by a direction vector with the Cartesian coordinate planes x = 0 and y = 0. The exit aperture of the optical system is to be a circular diaphragm. A quantitative merit function evaluates the performance of different solutions, a function depending on both fluxtransfer efficiency and diaphragm diameter, to bias the solutions to be close to the minimum diameter for which it is theoretically possible to have 100 percent raytransfer efficiency. Optical losses must be computed as well (Fresnel reflections, nonideal mirror reflectivity, absorption).
A total of eight valid solutions were submitted, representing a variety of approaches, including rotational reflective optics, freeform surfaces, squaretocross light pipe converters, and projection lenses (from singlets to those with a spherical symmetric nonhomogeneous refractive index).
The two solutions that performed the best are shown on the right. The firstplace winner, in the middle, was submitted by Bill Cassarly (ORA, USA). It was a squaretocross converter comprising a solid freeform TIR light pipe (parameterized so the “speed” of conversion can be adjusted), coupled to a threelens projection optic. The exit of the light pipe was optimized for a ±38º emission cone.
The projection lens has the double function of focusing the 38º rays onto the rim of the exit lens (i.e., to make the exit lens rim coincide with the exit pupil in the imaging optics terminology) and to make the edge of the cross be projected toward 20 degrees. Despite the absorption in the materials (the system was over 17 mm long), the combination of graded interfaces made this improved projection lens reach a merit function of 0.750.
The secondplace winner was submitted by Julio Chaves (LPILLC, USA) and Dejan Grabovickic and Fernando Garcia (Universidad Politécnica de Madrid, Spain). Their design also had two stages: a squaretocross converter and a projection lens. The projection lens was a spherically symmetric graded index lens, as designed by Morgan in 1958, with refractive index profile going from n=1.56 at the center to n=1.33 at the sphere’s edge. This Morgan lens collimates perfectly in 3D geometry and within the geometricaloptics approximation, with all rays emitted by a point source placed in air very close to its surface. (The Morgan lens is a generalization of the Luneburg lens.)
This lens is therefore an ideal projection lens, to which the squaretocross converter is to be coupled. The difference, however, between this projection lens and Cassarly’s is the input ray bundle it accepts, which is formed by all the rays hitting a virtual sphere of radius R/1.33, where R is the Morgan lens radius (this virtual sphere is the input pupil in the imaging terminology).
The first stage makes the squaretocross conversion by using light pipes that includes a combination of light shifters, angle transformers and angle rotators. Besides its sophistication, this system had a merit function of 0.706. Its main performance limitation resulted from the miscoupling of the light pipes and the projection lens: The intensity out of the guides did not fully illuminate the previously mentioned virtual sphere.
The winning solution used an optimizer integrated into the design. Therefore, it can be classified as a multiparameter optimization design. On the other hand, the secondplace approach used the nonimaging optics edgeray theorem, but not numerically optimized. This underscored the importance of optimization methods—which, if they had been applied to the design, could have improved its miscoupling problem.
At the same time, however, the optimization procedures themselves do not yet provide good solutions. (Other multiparameter optimization designs were submitted, with poorer performance). Thus, the designer’s insight into the proper initial configuration and its parameterization is crucial. The future of illumination design will likely be dominated by a combination of direct methods such as the SMS or PDE to provide a starting point for a design, with optimization methods used for the fine tuning.
Pablo Benítez and Juan C. Miñano are with the Universidad Politécnica de Madrid, Cedint, ETSI Telecomunicación, C. Universitaria, in Madrid, Spain, and Light Prescriptions Innovators LLC, in Altadena, Calif.
References and Resources
>> R.K. Luneburg. Mathematical Theory of Optics, University of California Press, Los Angeles (1964).
>> L.A. Piegland and W. Tiller. The NURBS Book, SpringerVerlag, New York, N.Y. (1995).
>> H. Ries and J. Muschaweck. “Tailored freeform optical surfaces,” J. Opt. Soc. Am. A 19, 5905 (2002).
>> B. Cassarly. “Nonimaging Optics,” Handbook of Optics III, OSA, 2003.
>> F. Bociort. “Optical System Optimization,” Encyclopedia of Optical Engineering, Marcel and Decker (2003).
>> V.I. Oliker, “Mathematical aspects of design of beam shaping surfaces in geometrical optics”, in Trends in Nonlinear Analysis (Kirkilionis, M., Kromker, S., Rannacher, R., and Tomi, F., eds.) SpringerVerlag, N.Y. (2003).
>> P. Benítez et al. “SMS Optical Design Method in 3D Geometry” Opt. Eng. 43, 14891502 (2004).
>> R. Winston et al. Nonimaging Optics, Academic Press, Elsevier (2005).
>> P. Benitez. “2006 IODC Illumination Design Problem,” Proc. SPIE Vol. 6342, International Optical Design Conference 2006, G. Groot Gregory, Joseph M. Howard, R. John Koshel, eds.
