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While a wavefront represents a surface of constant phase and an aberration function represents its deviation from a spherical surface, called the reference sphere, we will use the two terms, wavefront and aberration function, synonymously. In the previous chapters, we have emphasized that for wavefront analysis, i.e., to determine the content of an aberration function, a set of polynomials that are orthonormal over the pupil of an imaging system and represent balanced classical aberrations for the system must be used. The utility of using the orthonormal polynomials, as opposed to the orthogonal, is that each expansion coefficient is not only independent of the number of polynomials used in the expansion, but also represents the standard deviation or the sigma value of the corresponding polynomial aberration term (with the exception of piston). The variance of the aberration function is then simply equal to the sum of the squares of the aberration coefficients. In this chapter, we consider how best to determine the orthonormal expansion or the aberration coefficients from the wavefront data measured at an array of points, as, for example, in a phase-measuring interferometer. The problem of determining the expansion coefficients when the measured data are the wavefront slopes, as, for example, in a Shack–Hartmann sensor is also discussed. Although we have considered optical imaging systems with several different pupil shapes, our focus in this chapter is on a system with a circular pupil. The analysis given here for such a pupil can be extended to systems with other pupil shapes. In practice, what is needed in both optical design and fabrication is the wavefront. The wavefront aberrations determine the image quality in optical design. In fabrication and testing of an optical surface, the wavefront errors determine surface errors, and thus the polishing requirements to obtain the desired surface. Similarly, in adaptive optics, the signal for the actuators of a deformable mirror to negate the aberrations, such as those introduced by atmospheric turbulence, comes from the wavefront data. Hence, there is a need to determine the Zernike coefficients from the wavefront data measured by a wavefront sensor, or from the slope data provided by a slope sensor. In this chapter, we present the two main mathematical approaches to determine the expansion coefficients: an integration method for orthogonal solutions and the classic least squares approach.
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