Lenses used in everything from cameras and telescopes to optical data storage are sophisticated systems containing multiple lens elements. The design of these optical systems has changed significantly in the past few decades. The overall process—picking a starting layout and improving it using experience and numerical optimization software—remains largely the same. However, greater computing power and sophisticated commercial software can now rapidly generate multiple starting points.^{1}

Lens design is a challenging problem that calls for cost-effective solutions. It requires knowledge of first-order optics, wherein lenses are assumed to form perfect images, for system layout. At the same time, determining the best scheme demands an understanding of third-order aberration theory.^{2} This theory breaks down aberrations—unavoidable results of light refraction at spherical surfaces that lead to blurry or distorted images—into components that optical engineers can then correct or reduce.

Before modern computing, when tracing single light rays required manual effort, aberration theory was the quintessential tool for gaining insight into a lens model. The method required cost and time, so a lens was typically developed and built as soon as a reasonable form was obtained. Beginning mid-century, numerical optimization gradually replaced many roles that the theory traditionally filled.

**Figure 1. **Different designs and third-order spherical aberration by surface for a high-numerical-aperture data-storage objective lens. The highlighted surfaces have noticeably large amounts of spherical aberration and are more sensitive to fabrication tolerances.

However, aberration theory is still a useful tool for one critical design process step: the investigation of various starting layouts to choose viable candidates for further development. This is vital to avoid running multiple design efforts and choosing an inferior starting layout. It is indeed entirely possible to start with forms that cannot be improved due to higher-order aberrations.^{3}

**Figure 2. **A deep-UV lithography lens and surface-by-surface third-order Petzval contributions. Analysis of Petzval by surface reveals that the reflective group is intended for field flattening rather than color aberration control. This design is based on a Nikon patent for a catadioptric optical system.

^{4} Aberration theory can augment brute-force numerical optimization and give the designer a much firmer grasp of a lens system's limitations and potential. Designers can gain solid insight into why a lens scheme works, quickly assess its tolerance sensitivity and manufacturability, rapidly find the most sensitive parameters and components, and guide subsequent improvement. They benefit from learning how aberrations are computed, balanced, and corrected, as well as by understanding how fabrication errors and specific lens characteristics induce aberration.^{5–7}

We explore two examples of how aberration analysis can guide the design process, from anticipating manufacturability to changing fundamental lens architecture.

First, consider a monochromatic, high-numerical-aperture small-field objective for optical data storage. Such lenses are generally dominated by spherical aberration. Thus, analyzing surface contributions to third-order aberration should indicate the design's manufacturing sensitivity. Figure 1 shows three different lens forms, each with three all-spherical lenses and roughly the same performance. The figure also shows the six surface contributions to third-order spherical aberration.

Each scheme has a single negative-powered surface, which contributes positive spherical aberration. This balances the negative spherical contributions from the other five surfaces. The key difference between the systems lies in the location of the negative-powered surface and the magnitude of its aberration. Surfaces with large aberrations are, in general, more sensitive to manufacturing errors because the ray angle of incidence is relatively large. Here, the sum of third-order spherical is nearly the same for all three designs.

However, Solution #1 should be less sensitive to manufacturing errors than the other two designs because the individual surface contributions are smaller. For the same reason, we expect Solution #2 to be slightly more sensitive than #3. Indeed, a Monte Carlo analysis of all three lenses with identical tolerances shows that more than 97% of Solution #1, 84% of Solution #3, and less than 50% of Solution #2 systems will meet the top-level performance goal.

A deep-UV lithography lens is a more complex demonstration of how aberration analysis can drive lens architecture. Many designs encompassing refractive, reflective, and catadioptric (combining both) systems have been proposed since the onset of lithographic lens design in the early 1960s.^{8}

Correction of Petzval, a type of third-order aberration, is a major limitation of purely refractive designs. It is commonly corrected using a negative-powered lens near the focal plane. The primary approach is to place a central negative lens group near an intermediate image. This method results in a ‘bulge-waist’ structure in which the diameter of the positive lenses on either side of the central group increases. With increasing field-size requirements, the positive groups exceed material fabrication capabilities and have forced the development of alternative solutions.

One such approach uses a Schupmann subsystem: a negative lens element with a concave mirror. This corrects Petzval and keeps the remaining element diameters relatively small. Such a lens is shown in Figure 2 along with the surface-by-surface third-order Petzval. Reflective surfaces are often employed for color correction, but in this case the strong opposite contribution of the mirror enables the entire design. It would be extremely unlikely for a designer to arrive at the Schupmann scheme starting from a bulge-waist structure using an optimization algorithm. Here, aberration analysis combined with knowledge of the Petzval limitation inherent in the traditional bulge-waist approach guided the development of a new design form.

Although the development of aberration theory began in the 18^{th} century, papers continue to be published even into this century on the fundamental theory applied to optical system design. Recent examples illustrating the theory's continued relevance in lens design include nodal aberration theory^{9} and higher-order wavefront expansions.^{10} Rather than rendering the theory obsolete, contemporary global optimization software has emphasized the need for its application to successful optical systems.

Craig Olson

L-3 Communications/Sonoma EO

Santa Rosa, CA

Craig Olson is a senior optical engineer at L-3 Communications/Sonoma EO. He works in the areas of optical design, sensor modeling, and electro-optic and IR sensor metrology.

Julie Bentley

Institute of Optics, University of Rochester

Rochester, NY

Julie Bentley is an associate professor. She teaches courses in geometrical optics, optical design, and product design.

Richard N. Youngworth

Light Capture, Inc.

Longmont, CO

Richard Youngworth is director of optical engineering at Light Capture, Inc., an optical and mechanical engineering services and high-volume product-development company.

References:

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*Proc. SPIE* 7849, pp. 78490C, 2010. doi:

10.1117/12.8717202. D. Shafer, Global optimization in optical design, *Comput. Phys*. 8, no. 2, pp. 188-195, 1994.

3. D. Shafer, Induced aberrations complicate lens design, *Laser Focus World 27*, no. 8, pp. 103-108, 1991.

4. Y. Omura, N. Shiraishi, S. Owa, Projection exposure apparatus and method, catadioptric optical system, and manufacturing method of devices, *US patent 0268474*, 2007.

5. W. T. Welford, *Aberrations of Optical Systems*, Adam Hilger, Bristol, UK, 1991.

6. R. Kingslake, R. B. Johnson, *Lens Design Fundamentals*, Academic Press, San Diego, CA, 2009.

7. W. J. Smith, *Modern Optical Engineering*, McGraw-Hill, New York, NY, 2007.

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*Proc. SPIE* 7652, pp. 76522B, 2010. doi:

10.1117/12.8685309. K. P. Thompson, T. Schmid, P. Kao, J. P. Rolland, Recent discoveries from nodal aberration theory, *Int'l Opt. Design Conf*., 2010.

10. J. Sasian, Theory of sixth-order wave aberrations,

*Appl. Opt.* 49, pp. D69-D95, 2010. doi:

10.1364/AO.49.000D69