After over 125 years of design, polishing, and testing lens elements with rotationally symmetric, often spherical surfaces placed in round tubes—including single-lens-reflex cameras and microscope objective lenses—computer control has come along to interject advances that are rapidly changing the field. In the 1990s (the late) Harvey Pollicove founded and led the Rochester Center for Optics Manufacturing to automate fabrication of spherical and rotationally symmetric aspheric surfaces under computer numerical control (CNC). Now, this technology is moving forward to fabricate surfaces without rotational symmetry, or more generally freeform surfaces.

These advances are bringing a new paradigm in our millennium. Previously, most optical imaging systems were constrained to spherical surfaces for economic reasons. In such a system, a typical lens element contributes to up to 15 aberrations, or ways to lose, each with individual characteristics. Using a more complicated (freeform) shape for the surface offers an opportunity to introduce innovative 3D packages while simultaneously correcting more directly the limiting aberrations. CNC machines now produce freeform surfaces that depart from rotational symmetry for use at wavelengths as short as one micron. This is revolutionary.

Today, the optical design, fabrication, and testing communities are scrambling to develop a self-consistent structure for describing freeform surfaces through their respective processes. In 2006, Greg Forbes, a leading industry mathematician, sounded the alarm when he showed definitively that the formulation of rotationally symmetric surfaces introduced by Abbe in his 1902 patent was failing dramatically, as optical designers added more and more terms in a misguided effort to achieve a lower value for the optimization merit function. This shortcoming led to the introduction in the optical design environment of so-called Q-polynomial surfaces,^{1} whose effectiveness in reducing assembly sensitivity we have demonstrated.^{2} This Q-polynomial formulation, developed for rotationally symmetric aspheric surfaces now available for optical design, is also becoming available in optical surface fabrication and testing equipment.

**Figure 1. **Left to right: A design with three freeform surfaces and three full-field displays showing new field dependencies for astigmatism in systems with freeform surfaces and asymmetric configurations. Field binodal (left), field linear, field asymmetric (also seen in unobscured three-mirror anastigmats, like the James Webb Space Telescope) (middle), and the latest discovery, field linear, field conjugate astigmatism of freeform surfaces (right), all predicted from nodal aberration theory.^{5}

An effective approach to the mathematical description of freeform surfaces is to use the Zernike polynomials in one of their many forms.^{3} Adopted in the 1970s by the leading interferometric testing companies, this surface departure descriptor continues to dominate optical testing. It has slowly merged with optical design codes initially for analysis of as-fabricated surfaces within an optical system simulation and more recently for optimizing freeform optical systems. However, to keep optical design aligned with the need to reduce the maximum slope for optical testing, the optical testing community is proposing that the ‘default’ freeform optical surface mathematical shape descriptor should be a function like the Q-polynomial of Forbes. Recently, Forbes proposed a freeform surface description that enables rms slope control that is related to the Zernike polynomials.^{4}

In a dramatic development, a path that links the aberration theory for nonsymmetric optical systems to freeform optical surfaces and, more particularly, surfaces whose shape is described as a Zernike polynomial^{5} has been discovered. In 1977, Shack^{6} introduced a vector formulation of aberration theory that we later expanded to create nodal aberration theory through fourth and then sixth order.^{7, 8} The strength of this approach is that it is directly traceable to the aberrations of Seidel, which are still taught in all introductory optics courses. However, this work, while enabling optical design of systems without symmetry, has been restricted to rotationally symmetric surfaces.

Together with Fuerschbach, we discovered that there is a direct path to create an aberration theory for the design of optical systems with Zernike class freeform optical surfaces in an unconstrained geometry. The link is found by extending the paper by Schmid,^{9} which introduced figure error on the primary mirror of large astronomical telescopes into nodal aberration theory. By combining this work with concepts most often attributed to Burch,^{10} we linked the two seemingly disparate fields of research, with exciting implications for 3D optical design.

The tools that enable the immediate application of this discovery are full-field displays,^{11} funded by the Defense Advanced Research Projects Agency in the late 1990s, that we used to design a wide-field-of-view, unobscured all-reflective optical system with fully freeform surfaces (see Figure 1).^{12}Current applications are augmented reality displays, missile seekers, and astronomical telescopes. Now, armed with an underlying theory for optical design and a method of fabrication, the first generation of these radical optical systems are emerging as pathfinder systems.

Moving forward, we will next investigate how different freeform surface descriptions may yield advantages not only in various design challenges but also in supporting the manufacture of these surfaces. Recent work using Zernike polynomials extended to thousands of terms has demonstrated that this approach can be applied to describe surface errors from low to mid-spatial frequencies. This extension of surface descriptors to thousands of parameters can be aided by a simultaneous extension to the ray grids used to sample the surfaces in some cases, as we have recently shown.^{13} Ultimately, spanning the application of freeform surfaces to future systems from the IR to the extreme UV region of the spectrum opens great challenges all the way from design to manufacturing.

As exciting as this is, it is likely to be merely an intermediate step to the ultimate integration of computers with the end-to-end process of creating optical systems. Our recent exploratory work has pointed to the effectiveness of multicentric basis functions, which are particularly well adapted to computer control. This work builds on our investigation, with Cakmakci, of radial basis functions for freeform surface description.^{14–16} For those in optical system development, from optical design, to fabrication, to assembly and testing, it may be time to put on an eyeglass format display to view the entire universe that is classical optical systems.

Jannick Rolland

University of Rochester

Rochester, NY

http://www.odalab-spectrum.org

Jannick Rolland is the Brian J. Thompson Professor of Optical Engineering at The Institute of Optics at the University of Rochester and director of the R. E. Hopkins Center for Optical Design and Engineering. She holds appointments in the Department of Biomedical Engineering and in the Center for Visual Science. She is a Fellow of SPIE.

Kevin Thompson

Synopsys Inc.

Pittsford, NY

Kevin Thompson is the group director for research and development/optics at Synopsys Inc. He is a Fellow of SPIE.

References:

1. G. W. Forbes, Shape specification for axially symmetric optical surfaces, *Opt. Express* 15(8), p. 5218-5226, 2007.

2. B. Ma, L. Li, K. P. Thompson, J. P. Rolland, Applying slope constrained Q-type aspheres to develop higher performance lenses, *Opt. Express* 19(22), p. 21174-21179, 2011.

4. G. W. Forbes, Characterizing the shape of freeform optics, *Opt. Express* 20(3), p. 2483-2499, 2012.

5. K. Fuerschbach, J. P. Rolland, K. P. Thompson, Extending Nodal Aberration Theory to include mount-induced aberrations with extension to freeform surface, *Opt. Express*. (Submitted.)

6. R. V. Shack, K. P. Thompson, Influence of alignment errors of a telescope system on its aberration field,

*Proc. SPIE* 251, p. 146-153, 1980.

doi:10.1117/12.959464
7. K. Thompson, Description of the third-order optical aberrations of near-circular pupil optical systems without symmetry, *J. Opt. Soc. Am. A* 22, p. 1389-1401, 2005.

8. K. P. Thompson, Multinodal fifth-order optical aberrations of optical systems without rotational symmetry: the astigmatic aberrations, *J. Opt. Soc. Am. A* 28, p. 821-836, 2011.

9. T. Schmid, J. P. Rolland, A. Rakich, K. P. Thompson, Separation of the effects of astigmatic figure error from misalignments using Nodal Aberration Theory (NAT), *Opt. Express* 18(5), p. 17433-17447, 2010.

10. C. R. Burch, On the optical see-saw diagram, *Mon. Not. R. Astron. Soc.* 103, p. 159-165, 1942.

11. Synopsys, *Full Field Display: CODE V Reference Manual*, 2011.

12. K. Fuerschbach, J. P. Rolland, K. P. Thompson, A new family of optical systems employing φ-polynomial surfaces, *Opt. Express* 19(22), p. 21919-21928, 2011.

13. I. Kaya, K. P. Thompson, J. P. Rolland, Edge clustered fitting grids for φ-polynomial characterization of freeform optical surfaces, *Opt. Express* 19(27), p. 26962-26974, 2011.

14. O. Cakmakci, B. Moore, H. Foroosh, J. P. Rolland, Optimal local shape description for rotationally non-symmetric optical surface design and analysis, *Opt. Express* 16(3), p. 1583-1589, 2008.

15. O. Cakmakci, S. Vo, K. P. Thompson, J. P. Rolland, Application of radial basis functions to shape description in a dual-element off-axis eyewear display: field-of-view limit, *SID 16*, p. 1089-1098, 2008.

16. O. Cakmakci, K. P. Thompson, P. Vallee, J. Cote, J. P. Rolland, Design of a free-form single-element head worn display,

*Proc. SPIE* 7618, p. 761803, 2010.

doi:10.1117/12.840716