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Chapter 3:
Orthonormal Polynomials and Gram–Schmidt Orthonormalization
In optical design, we trace rays from a point object through a system to determine the aberrations of the wavefront at its exit pupil. In optical testing, we determine the aberrations of a system or an element interferometrically. In both cases, we obtain aberration numbers at an array of points. We can calculate the PSF or other associated image quality measures from these numbers. We can also calculate the aberration variance, which, in turn, gives some idea of the image quality. However, such measures do not shed light on the content of the aberration function. To understand the nature of this function, we want to know the amount of certain familiar aberrations discussed in Chapter 2 that are present, so that perhaps something can be done about them in improving the design or the system under test. A straightforward approach to determine the content of an aberration function is to decompose it into a set of orthogonal polynomials that represent balanced classical aberrations and include wavefront defocus and tilt. The Zernike circle polynomials are in widespread use for this purpose for systems with circular pupils. These polynomials are unique in the sense that they are not only orthogonal across a unit circle, but they also represent balanced aberrations yielding minimum variance, as we shall see in Chapter 4. In this chapter, we discuss the basic properties of the orthogonal polynomials. We also describe the Gram–Schmidt orthogonalization process for obtaining orthogonal polynomials over one domain from those that are orthogonal over another domain, e.g., obtaining polynomials that are orthogonal over an annular pupil from the circle polynomials. We emphasize the use of orthonormal polynomials so that their coefficients represent the standard deviations of the corresponding polynomial aberration terms.
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