After an ocular wavefront is obtained from an aberrometer with wavefront sensing and reconstruction, the wavefront often needs to be manipulated. In the previous chapter, we discussed the modal wavefront reconstruction with different basis functions, such as Zernike polynomials, Fourier series, and Taylor monomials. When different sets of basis functions are used in the modal wavefront reconstruction, we must have a way to compare these wavefronts. Therefore, there is a need to convert the coefficients between two different sets of basis functions. In this chapter, we discuss the conversions of coefficients of the following basis functions: Seidel series, Zernike polynomials, Fourier series, and Taylor monomials.
Among these four sets of basis functions, the set of Zernike polynomials and the set of Fourier series are orthonormal. Zernike polynomials are orthonormal over circular pupils, and Fourier series are orthonormal over rectangular pupils. Because they are orthonormal, they are also complete. The set of Taylor monomials is not orthonormal over any pupils, however, they are a complete set of basis functions. Any complete set of basis functions can be used to accurately represent any well-behaved functions, such as an ocular wavefront. The set of Seidel series, which is an extension of the classical aberrations, is not orthonormal over any pupils. Nor is it a complete set of basis functions. The inclusion of the set of Seidel series is mainly for the relation to classical aberrations. Because Zernike polynomials are also related to classical aberrations, we discuss the conversion between Zernike polynomials and Seidel series. The conversion of Seidel series with the other sets of basis functions is not discussed. For the other three complete sets of basis functions, namely Zernike polynomials, Fourier series, and Taylor monomials, the conversions between any two of them are discussed.
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