Optical Shop Testing

Chapter 13 - Zernike Polynomials and Wavefront Fitting

13.1.  INTRODUCTION

Optical imaging systems generally have an axis of rotational symmetry, and their
pupil is circular or annular, as in the case of the systems with mirrors. The wave
aberration function of such systems can be expanded in a power series or a complete
set of orthogonal polynomials. Although introduced by Zernike (1934a,b) for testing
the figure of a circular mirror by a knife edge test or his phase contrast method,
Zernike circle polynomials were studied extensively by Nijboer (1943, 1947) and
Nienhuis and Nijboer (1949) to describe the diffraction theory of aberrations. We
show that these polynomials, which are orthogonal over a circular pupil, represent
balanced aberrations that yield minimum variance. For small aberrations, a minimum
of aberration variance yields a maximum of Strehl ratio (Mahajan, 1982, 1983,
1993). Relationships between the coefficients of a power-series expansion and those
of a Zernike-polynomial expansion are given. In particular, the conversion of Seidel
coefficients into Zernike coefficients or vice versa is discussed. Similarly, Zernike
annular polynomials representing balanced aberrations with minimum variance for
systems with annular pupils are discussed (Mahajan, 1981a,b; 1984; 1994). Balanced
aberrations that yield minimum ray aberration variance, often called the root mean
square spot radius, are different and are not discussed here (Braat, 1987). Finally, the
determination of Zernike coefficients from a discrete set of wavefront error data
obtained by ray tracing, as in optical design, or by measurement, as in optical testing,
is discussed, regardless of the shape of the pupil. The procedure described is thus
applicable, for example, to elliptical pupils obtained in off-axis imaging, odd-shaped
vignetted pupils, and hexagonal pupils of a segmented mirror with hexagonal
segments.

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