A new method for comparing Zernike circular polynomials with Zernike annular polynomials in annular pupils

To compare the difference between Zernike annular polynomial and Zernike circular polynomial, an approximate mathematic relationship between Zernike annular polynomial coefficients and SEIDEL coefficients is proposed. A new method is applied in comparing experiment. For real interferometric data of annular wave front, the SEIDEL aberration calculated in different ways is removed from the original wave front. Residual astigmatism aberration in annular polynomial way is close to zero, but defocus and sphere aberration are 0.002709λ, -0.00182λ. The residual SEIDEL using annular polynomial is least. Data used in comparison shows that Zernike annular polynomial is more accurate and meaningful for the decomposition of aberrations and the calculation of Seidel aberrations in annular pupil. The calculation in astigmatism aberration is most accurate and those in defocus and sphere aberration are least accurate, when the first 9 terms are used in approximate solving SEIDEL. More coefficients are needed to obtain the real SEIDEL.