Summary
Interferometrists often like to represent wavefronts as Zernike polynomials; optical designers are more accustomed to Seidel aberration polynomials. It is sometimes stated that the first few Zernike coefficients are related to the Seidel aberrations. For example, the Zernike coefficient Z8 is compared to the primary spherical aberration SA3. This is misleading, because it is not generally possible to compute Z8 from SA3, nor vice versa. When high-order aberrations are present, the primary Seidel aberrations are spread among the various Zernike coefficients in a way that makes the Zernike aberrations orthogonal. Only when high-order aberrations are negligible, is it possible to relate the first few Zernike coefficients to the Seidel aberrations.
The Zernike coefficients are ordinarily found using a least-squares fit to a grid of exact ray data, while the Seidel coefficients can be computed from paraxial ray data . Until recently, the computational cost of Zernike coefficients severely limited their use in optical design, but the availability of low-cost high-speed computers has rekindled interest in the use of Zernike coefficients both to specify aspheric surfaces, and to build error functions.
Discussion
The Seidel aberrations were developed in the mid 19th century to account for the monochromatic geometrical aberrations of centered optical systems, i.e. defects from perfect imagery in optical systems that have an optical axis. The types of aberrations can be developed from considerations of symmetry, and are given names such as spherical aberration, coma, astigmatism, field curvature, and distortion[1,2]. In addition to their type, aberrations are usually specified according to their order (third-order, fifth-order, etc.) although sometimes they are called primary, secondary, etc. Higher order aberrations introduce new types. For example, fifth-order aberrations add the types oblique spherical aberration and elliptical coma[3].
The magnitude of the aberrations is expressed in terms of an aberration polynomial. The aberration polynomial coefficients can be found either from a series expansion of the law of refraction, or using an iteration procedure introduced by Buchdahl[4], which considers rotational invariants and builds upon low-order coefficients to develop the next higher-order coefficients. Whether one works with transverse ray displacements, longitudinal ray displacements, or wavefronts is not fundamentally significant. The equations needed to calculate aberration coefficients are complicated, but can be readily evaluated numerically, and most optical design software provides values for the third-order coefficients (OSLO also provides the fifth-order coefficients and the seventh-order spherical term).
The Zernike polynomials were introduced by F. Zernike in the early 20th century and later developed by several workers to describe the diffraction theory of aberrations[5]. Like the Seidel aberration aberration polynomial, Zernike polynomials describe defects from perfect imagery, but the nature of the Zernike expansion is different from the Seidel expansion. Zernike polynomials describe the properties of an aberrated wavefront without regard to the symmetry properties of the system that gave rise to the wavefront. The Zernike polynomials have some interesting and useful properties: they form a complete set, they are readily separated into radial and angular functions, and the individual polynomials are orthogonal over the (entire) unit circle[6].
Notwithstanding their desirable properties, Zernike polynomials must be used with caution in real design situations. The ad cites a common misconception: that the low-order Zernike polynomials have a one-to-one correspondence to the common Seidel aberrations. If the aberration function, defined continuously over the unit circle, having no aberrations higher than third order, completely describes a system, then there is such a correspondence. But if fifth or higher-order aberrations are present, then the equality of the third-order relationships is not maintained.
Other subtleties can mislead users of Zernike polynomials. For example, although the polynomials are orthogonal over the entire unit circle, they are not orthogonal over portions of the unit circle, so that if the polynomials are generated by fitting to a limited set of data points, the orthogonality relationships may not be maintained.[7]
One fairly new application of Zernike polynomials in optical design is for the representation of aspheric surfaces (either refractive or diffractive). Here, the orthogonality of the coefficients is useful for introducing higher-order terms that are independent of the lower-order terms. Another new development is the use of Zernike terms as optimization operands, also now available in OSLO. Here the advantage of Zernike terms is more problematical because of the orthogonality problem described above, but there is at least the potential for describing a complicated image field in terms of orthogonal polynomials, which may be helpful.
1. Bruce Walker, "Optical Engineering Fundamentals", SPIE Press Vol. TT30, 1997.
2. Warren J. Smith, "Modern Optical Engineering (Second Edition)", McGraw-Hill 1990, ISBN 0-07-059174-1.
3. OSLO Version 5 Optics Reference, Sinclair Optics (1997).
4. H.A. Buchdahl, "Optical Aberration Coefficients", Dover Publications (1968).
5. Max Born and Emil Wolf, "Principles of Optics", Pergamon Press, ISBN 0-08-018018 3.
6. C-J. Kim and R.R. Shannon, "Catalog of Zernike Polynomials", in Applied Optics and Optical Engineering, Vol. X, Chapter 4, Academic Press (1987)
7. J.C. Wyant and Katherine Creath, "Basic Wavefront Aberration Theory for Optical Metrology", in Applied Optics and Optical Engineering, Vol. XI, Chapter 1, Academic Press (1992)