Abstract

Zernike circle polynomials are in widespread use for wavefront analysis because of their orthogonality over a circular pupil and their representation of balanced classical aberrations. In recent papers, we derived closed-form polynomials that are orthonormal over a hexagonal pupil, such as the hexagonal segments of a large mirror. We extend our work to elliptical, rectangular, and square pupils. Using the circle polynomials as the basis functions for their orthogonalization over such pupils, we derive closed-form polynomials that are orthonormal over them. These polynomials are unique in that they are not only orthogonal across such pupils, but also represent balanced classical aberrations, just as the Zernike circle polynomials are unique in these respects for circular pupils. The polynomials are given in terms of the circle polynomials as well as in polar and Cartesian coordinates. Relationships between the orthonormal coefficients and the corresponding Zernike coefficients for a given pupil are also obtained. The orthonormal polynomials for a one-dimensional slit pupil are obtained as a limiting case of a rectangular pupil.

© 2007 Optical Society of America

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2007 (2)

K. N. LaFortune, R. L. Hurd, S. N. Fochs, M. D. Rotter, P. H. Pax, R. L. Combs, S. S. Olivier, J. M. Brase, and R. M. Yamamoto, "Technical challenges for the future of high energy lasers," Proc. SPIE 6454, 1-11 (2007).

G.-m. Dai and V. N. Mahajan, "Nonrecursive orthonormal polynomials with matrix formulation," Opt. Lett. 32, 74-76 (2007).

2006 (1)

2004 (1)

M. Bray, "Orthogonal polynomials: a set for square areas," Proc. SPIE 5252, 314-320 (2004).

2003 (1)

V. N. Mahajan, "Zernike polynomials and aberration balancing," Proc. SPIE 5173, 1-17 (2003).

1996 (1)

1995 (1)

1994 (1)

1992 (1)

1986 (1)

1984 (1)

V. N. Mahajan, "Zernike annular polynomials for imaging systems with annular pupils," J. Opt. Soc. Am. 1, 685 (1984).

1982 (1)

1981 (2)

1976 (1)

1969 (1)

H. Sumita, "Orthogonal expansion of the aberration difference function and its application to image evaluation," Jpn. J. Appl. Phys. 8, 1027-1036 (1969).

1968 (1)

1965 (1)

1934 (1)

F. Zernike, "Diffraction theory of knife-edge test and its improved form, the phase contrast method," Mon. Not. R. Astron. Soc. 94, 377-384 (1934).

Barakat, R.

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Oxford, 1999).

Brase, J. M.

K. N. LaFortune, R. L. Hurd, S. N. Fochs, M. D. Rotter, P. H. Pax, R. L. Combs, S. S. Olivier, J. M. Brase, and R. M. Yamamoto, "Technical challenges for the future of high energy lasers," Proc. SPIE 6454, 1-11 (2007).

Bray, M.

M. Bray, "Orthogonal polynomials: a set for square areas," Proc. SPIE 5252, 314-320 (2004).

Combs, R. L.

K. N. LaFortune, R. L. Hurd, S. N. Fochs, M. D. Rotter, P. H. Pax, R. L. Combs, S. S. Olivier, J. M. Brase, and R. M. Yamamoto, "Technical challenges for the future of high energy lasers," Proc. SPIE 6454, 1-11 (2007).

Dai, G.-m.

Fochs, S. N.

K. N. LaFortune, R. L. Hurd, S. N. Fochs, M. D. Rotter, P. H. Pax, R. L. Combs, S. S. Olivier, J. M. Brase, and R. M. Yamamoto, "Technical challenges for the future of high energy lasers," Proc. SPIE 6454, 1-11 (2007).

Harbers, G.

Hurd, R. L.

K. N. LaFortune, R. L. Hurd, S. N. Fochs, M. D. Rotter, P. H. Pax, R. L. Combs, S. S. Olivier, J. M. Brase, and R. M. Yamamoto, "Technical challenges for the future of high energy lasers," Proc. SPIE 6454, 1-11 (2007).

King, W. B.

Korn, G. A.

G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers (McGraw-Hill, 1968).

Korn, T. M.

G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers (McGraw-Hill, 1968).

Kunst, P. J.

LaFortune, K. N.

K. N. LaFortune, R. L. Hurd, S. N. Fochs, M. D. Rotter, P. H. Pax, R. L. Combs, S. S. Olivier, J. M. Brase, and R. M. Yamamoto, "Technical challenges for the future of high energy lasers," Proc. SPIE 6454, 1-11 (2007).

Leibbrandt, G. W. R.

Mahajan, V. N.

Noll, R. J.

Olivier, S. S.

K. N. LaFortune, R. L. Hurd, S. N. Fochs, M. D. Rotter, P. H. Pax, R. L. Combs, S. S. Olivier, J. M. Brase, and R. M. Yamamoto, "Technical challenges for the future of high energy lasers," Proc. SPIE 6454, 1-11 (2007).

Pax, P. H.

K. N. LaFortune, R. L. Hurd, S. N. Fochs, M. D. Rotter, P. H. Pax, R. L. Combs, S. S. Olivier, J. M. Brase, and R. M. Yamamoto, "Technical challenges for the future of high energy lasers," Proc. SPIE 6454, 1-11 (2007).

Rayces, J. L.

Riseberg, L.

Rotter, M. D.

K. N. LaFortune, R. L. Hurd, S. N. Fochs, M. D. Rotter, P. H. Pax, R. L. Combs, S. S. Olivier, J. M. Brase, and R. M. Yamamoto, "Technical challenges for the future of high energy lasers," Proc. SPIE 6454, 1-11 (2007).

Sumita, H.

H. Sumita, "Orthogonal expansion of the aberration difference function and its application to image evaluation," Jpn. J. Appl. Phys. 8, 1027-1036 (1969).

Szapiel, S.

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Oxford, 1999).

Yamamoto, R. M.

K. N. LaFortune, R. L. Hurd, S. N. Fochs, M. D. Rotter, P. H. Pax, R. L. Combs, S. S. Olivier, J. M. Brase, and R. M. Yamamoto, "Technical challenges for the future of high energy lasers," Proc. SPIE 6454, 1-11 (2007).

Zernike, F.

F. Zernike, "Diffraction theory of knife-edge test and its improved form, the phase contrast method," Mon. Not. R. Astron. Soc. 94, 377-384 (1934).

Appl. Opt. (5)

J. Opt. Soc. Am. (6)

J. Opt. Soc. Am. A (1)

Jpn. J. Appl. Phys. (1)

H. Sumita, "Orthogonal expansion of the aberration difference function and its application to image evaluation," Jpn. J. Appl. Phys. 8, 1027-1036 (1969).

Mon. Not. R. Astron. Soc. (1)

F. Zernike, "Diffraction theory of knife-edge test and its improved form, the phase contrast method," Mon. Not. R. Astron. Soc. 94, 377-384 (1934).

Opt. Lett. (2)

Proc. SPIE (3)

K. N. LaFortune, R. L. Hurd, S. N. Fochs, M. D. Rotter, P. H. Pax, R. L. Combs, S. S. Olivier, J. M. Brase, and R. M. Yamamoto, "Technical challenges for the future of high energy lasers," Proc. SPIE 6454, 1-11 (2007).

M. Bray, "Orthogonal polynomials: a set for square areas," Proc. SPIE 5252, 314-320 (2004).

V. N. Mahajan, "Zernike polynomials and aberration balancing," Proc. SPIE 5173, 1-17 (2003).

Other (5)

V. N. Mahajan, "Zernike polynomials and wavefront fitting," in Optical Shop Testing, 3rd ed., D.Malacara, ed. (Wiley, 2007) pp. 498-546.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Oxford, 1999).

V. N. Mahajan, Optical Imaging and Aberrations, Part II: Wave Diffraction Optics (SPIE, 2004).

G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers (McGraw-Hill, 1968).

http://scikits.com/KFacts.html.

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