Abstract

Several low-order Zernike modes are photographed for visualization. These polynomials are extended to include both circular and annular pupils through a Gram-Schmidt orthogonalization procedure. Contrary to the traditional understanding, the classical least-squares method of determining the Zernike coefficients from a sampled wave front with measurement noise has been found numerically stable. Furthermore, numerical analysis indicates that the so-called Gram-Schmidt method and the least-squares method give practically identical results. An alternate method using the orthogonal property of the polynomials to determine their coefficients is also discussed.

© 1980 Optical Society of America

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