Convergence and differentiation of Zernike expansion: application for an analysis of odd-order surfaces

Abstract. Odd-order surfaces have begun to be used in optics. In order to investigate the aberration characteristics of such surfaces, Zernike expansion is widely used since it directly and explicitly corresponds to wavefront aberrations. Since the Zernike expansion of an odd-order surface contains an infinite number of terms, the convergence of the expanded sum and the possibility of termwise derivatives are not explicitly guaranteed mathematically. We give a complete proof for these problems. For an application of this result, we analyze the aberration characteristics of odd-order surfaces and present their effectiveness in optical design.

[1]  Akira Yabe.  Sensitivity control to surface irregularity , 2006, International Optical Design Conference.

[2]  B. Dörband,et al.  Handbook of optical systems , 2012 .

[3]  A. Wünsche.  Generalized Zernike or disc polynomials , 2005 .

[4]  Iain A. Neil.  Optical design dependence on technology development , 2009, Optical Engineering + Applications.

[5]  Emil Wolf.  The diffraction theory of aberrations , 1951 .

[6]  畠山弘至,et al.  Projection optical system and projection display apparatus using the same , 2003 .

[7]  von F. Zernike.  Beugungstheorie des schneidenver-fahrens und seiner verbesserten form, der phasenkontrastmethode , 1934 .

[8]  B. Blinov,et al.  Trapped ion imaging with a high numerical aperture spherical mirror , 2009, 0901.4742.

[9]  Theoretical investigation of the meaning of odd-order aspherical surface and numerical confirmation of effectiveness in rotational-symmetric but off-axis optics , 2010 .

[10]  G. Arfken,et al.  Mathematical methods for physicists 6th ed. , 1996 .

[11]  Emil Wolf,et al.  Principles of Optics: Contents , 1999 .

[12]  A. Janssen,et al.  Zernike expansion of derivatives and Laplacians of the Zernike circle polynomials. , 2014, Journal of the Optical Society of America. A, Optics, image science, and vision.

[13]  G. W. Forbes,et al.  Asphere, O asphere, how shall we describe thee? , 2008, Optical Systems Design.

[14]  G. Arfken.  Mathematical Methods for Physicists , 1967 .

[15]  A. Bhatia,et al.  On the circle polynomials of Zernike and related orthogonal sets , 1954, Mathematical Proceedings of the Cambridge Philosophical Society.