The orthonormal polynomials for various pupils discussed in the preceding chapters represent balanced aberrations for those pupils, just as the Zernike circle polynomials (discussed in Chapter 4) do for a circular pupil. In this chapter, we consider the use of circle polynomials for the analysis of a noncircular wavefront. Since the circle polynomials form a complete set, any wavefront, regardless of the shape of the pupil (which defines the perimeter of the wavefront), can be expanded in terms of them. Moreover, since each orthonormal polynomial is a linear combination of the circle polynomials [see Eq. (3-18)], the wavefront fitting with the former set of polynomials is as good as that with the latter. However, we illustrate the pitfalls of using circle polynomials for a noncircular pupil by considering an annular and a hexagonal pupil.
It is shown that, unlike the orthonormal coefficients, the circle coefficients generally change as the number of polynomials used in the expansion changes. Although the wavefront fit with a certain number of circle polynomials is the same as that with the corresponding orthonormal polynomials, the piston circle coefficient does not represent the mean value of the aberration function, and the sum of the squares of the other coefficients does not yield its variance. While the interferometer setting errors of tip, tilt, and defocus from a 4-circle-polynomial expansion are the same as those from the orthonormal polynomial expansion, these errors obtained from, say, an 11-circlepolynomial expansion, and removed from the aberration function yield wrong polishing by zeroing out the residual aberration function. If the common practice of defining the center of an interferogram and drawing a circle around it is followed, and determining the circle coefficients in the same manner as for a circular interferogram, then the circle coefficients of a noncircular interferogram do not yield a correct representation of the aberration function. Moreover, in this case, some of the higher-order coefficients of aberrations that are nonexistent in the aberration function are also nonzero. Finally, the circle coefficients, however obtained, do not represent coefficients of the balanced aberrations for a noncircular pupil. Such results are illustrated analytically and numerically by considering annular and hexagonal Seidel aberration functions as examples.
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