Abstract

The Zernike aberration theory for constant amplitude circular apertures is extended to annular apertures having a Gaussian-like radial taper. Explicit expressions are obtained for the optimum balanced wave-front aberrations in terms of shifted Jacobi polynomials. Properties of the polynomials (e.g., Rodrigues formula, recurrence relations, derivatives, etc.) are investigated.

© 1980 Optical Society of America

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  1. F. Zernike, "Beugungstheorie Des Schneidenverfahrens und seiner verbesserten Form, der Phasenkontrastmethode," Physica (Utrecht) 1, 689–704 (1934).
  2. F. Zernike and H. C. Brinkman, "Hypersphärische Funktionen und die in spharischen Berischen Orthogonalen Polynome," Ned. Akad. Wet. Indag. Math. 38, 161–169 (1935).
  3. B. R. A. Nijboer, "The diffraction theory of aberrations," Physica (Utrecht) 10, 679–692 (1943); 13, 605–620 (1947).
  4. A. B. Bhatia and E. Wolf, "On the circle polynomials of Zernike and related orthogonal sets," Proc. Cambridge Philos. Soc. 50, 40–48 (1954).
  5. E. H. Linfoot, Recent Advances in Optics (Oxford University, Oxford, 1955), pp. 48–66.
  6. M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1964).
  7. P. W. Hawkes, "The diffraction theory of the aberrations of stigmatic orthomorphic optical or electron optical systems containing toric lenses or quadrupoles," Opt. Acta, 11, 237–251 (1964).
  8. R. Barakat and L. Riseberg, "Diffraction theory of the aberrations of a slit aperture," J. Opt. Soc. Am. 55, 878–881 (1965).
  9. D. R. Myrick, "A generalization of the radial polynomials of F. Zernike," J. Soc. Ind. Appl. Math. 14, 476–492 (1966).
  10. S. N. Bezdid'ko, "The use of Zernike polynomials in optics," Sov. J. Opt. Tech. 41, 425–429 (1974).
  11. E. C. Kintner, "Some comments on the use of Zernike polynomials in optics," Opt. Commun. 18, 235–237 (1976).
  12. E. C. Kintner, "A recurrence relation for calculating the Zernike polynomials," Opt. Acta 23, 499–500 (1976).
  13. E. C. Kintner and R. M. Sillitto, "A new 'analytic' method for computing the optical transfer function," Opt. Acta 23, 607–619 (1976).
  14. R. J. Noll, "Zernike polynomials and atmospheric turbulence," J. Opt. Soc. Am. 66, 207–211 (1976).
  15. E. C. Kintner, "On the mathematical properties of the Zernike polynomials," Opt. Acta 23, 679–680 (1976).
  16. W. J. Tango, "The circle polynomials of Zernike and their application in optics," Appl. Phys. 13, 327–332 (1977).
  17. R. M. Sillitto, "Diffraction of uniform and Gaussian beams: an application of Zernike polynomials," Optik 48, 271–283 (1977).
  18. A. Maréchal, Thesis, University of Paris, 1948 (unpublished).
  19. B. Tatian, "Aberration balancing in rotationally symmetric lenses," J. Opt. Soc. Am. 64, 1083–1091 (1974).
  20. A. Arimoto, "Aberration expansion and evaluation of the quasi-Gaussian beam by a set of orthogonal functions," J. Opt. Soc. Am. 64, 850–856 (1974).
  21. G. Szegö, Orthogonal Polynomials (American Mathematical Society, Providence, 1939).
  22. E. D. Rainville, Special Functions (MacMillan, New York, 1960).
  23. G. Freud, Orthogonale Polynome (Birkhauser-Verlag, Basel, 1969).

1977 (2)

W. J. Tango, "The circle polynomials of Zernike and their application in optics," Appl. Phys. 13, 327–332 (1977).

R. M. Sillitto, "Diffraction of uniform and Gaussian beams: an application of Zernike polynomials," Optik 48, 271–283 (1977).

1976 (5)

E. C. Kintner, "Some comments on the use of Zernike polynomials in optics," Opt. Commun. 18, 235–237 (1976).

E. C. Kintner, "A recurrence relation for calculating the Zernike polynomials," Opt. Acta 23, 499–500 (1976).

E. C. Kintner and R. M. Sillitto, "A new 'analytic' method for computing the optical transfer function," Opt. Acta 23, 607–619 (1976).

R. J. Noll, "Zernike polynomials and atmospheric turbulence," J. Opt. Soc. Am. 66, 207–211 (1976).

E. C. Kintner, "On the mathematical properties of the Zernike polynomials," Opt. Acta 23, 679–680 (1976).

1974 (3)

1966 (1)

D. R. Myrick, "A generalization of the radial polynomials of F. Zernike," J. Soc. Ind. Appl. Math. 14, 476–492 (1966).

1965 (1)

1964 (1)

P. W. Hawkes, "The diffraction theory of the aberrations of stigmatic orthomorphic optical or electron optical systems containing toric lenses or quadrupoles," Opt. Acta, 11, 237–251 (1964).

1954 (1)

A. B. Bhatia and E. Wolf, "On the circle polynomials of Zernike and related orthogonal sets," Proc. Cambridge Philos. Soc. 50, 40–48 (1954).

1943 (1)

B. R. A. Nijboer, "The diffraction theory of aberrations," Physica (Utrecht) 10, 679–692 (1943); 13, 605–620 (1947).

1935 (1)

F. Zernike and H. C. Brinkman, "Hypersphärische Funktionen und die in spharischen Berischen Orthogonalen Polynome," Ned. Akad. Wet. Indag. Math. 38, 161–169 (1935).

1934 (1)

F. Zernike, "Beugungstheorie Des Schneidenverfahrens und seiner verbesserten Form, der Phasenkontrastmethode," Physica (Utrecht) 1, 689–704 (1934).

Arimoto, A.

Barakat, R.

Bezdid’ko, S. N.

S. N. Bezdid'ko, "The use of Zernike polynomials in optics," Sov. J. Opt. Tech. 41, 425–429 (1974).

Bhatia, A. B.

A. B. Bhatia and E. Wolf, "On the circle polynomials of Zernike and related orthogonal sets," Proc. Cambridge Philos. Soc. 50, 40–48 (1954).

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1964).

Brinkman, H. C.

F. Zernike and H. C. Brinkman, "Hypersphärische Funktionen und die in spharischen Berischen Orthogonalen Polynome," Ned. Akad. Wet. Indag. Math. 38, 161–169 (1935).

Freud, G.

G. Freud, Orthogonale Polynome (Birkhauser-Verlag, Basel, 1969).

Hawkes, P. W.

P. W. Hawkes, "The diffraction theory of the aberrations of stigmatic orthomorphic optical or electron optical systems containing toric lenses or quadrupoles," Opt. Acta, 11, 237–251 (1964).

Kintner, E. C.

E. C. Kintner, "Some comments on the use of Zernike polynomials in optics," Opt. Commun. 18, 235–237 (1976).

E. C. Kintner, "On the mathematical properties of the Zernike polynomials," Opt. Acta 23, 679–680 (1976).

E. C. Kintner and R. M. Sillitto, "A new 'analytic' method for computing the optical transfer function," Opt. Acta 23, 607–619 (1976).

E. C. Kintner, "A recurrence relation for calculating the Zernike polynomials," Opt. Acta 23, 499–500 (1976).

Linfoot, E. H.

E. H. Linfoot, Recent Advances in Optics (Oxford University, Oxford, 1955), pp. 48–66.

Maréchal, A.

A. Maréchal, Thesis, University of Paris, 1948 (unpublished).

Myrick, D. R.

D. R. Myrick, "A generalization of the radial polynomials of F. Zernike," J. Soc. Ind. Appl. Math. 14, 476–492 (1966).

Nijboer, B. R. A.

B. R. A. Nijboer, "The diffraction theory of aberrations," Physica (Utrecht) 10, 679–692 (1943); 13, 605–620 (1947).

Noll, R. J.

Rainville, E. D.

E. D. Rainville, Special Functions (MacMillan, New York, 1960).

Riseberg, L.

Sillitto, R. M.

R. M. Sillitto, "Diffraction of uniform and Gaussian beams: an application of Zernike polynomials," Optik 48, 271–283 (1977).

E. C. Kintner and R. M. Sillitto, "A new 'analytic' method for computing the optical transfer function," Opt. Acta 23, 607–619 (1976).

Szegö, G.

G. Szegö, Orthogonal Polynomials (American Mathematical Society, Providence, 1939).

Tango, W. J.

W. J. Tango, "The circle polynomials of Zernike and their application in optics," Appl. Phys. 13, 327–332 (1977).

Tatian, B.

Wolf, E.

A. B. Bhatia and E. Wolf, "On the circle polynomials of Zernike and related orthogonal sets," Proc. Cambridge Philos. Soc. 50, 40–48 (1954).

M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1964).

Zernike, F.

F. Zernike and H. C. Brinkman, "Hypersphärische Funktionen und die in spharischen Berischen Orthogonalen Polynome," Ned. Akad. Wet. Indag. Math. 38, 161–169 (1935).

F. Zernike, "Beugungstheorie Des Schneidenverfahrens und seiner verbesserten Form, der Phasenkontrastmethode," Physica (Utrecht) 1, 689–704 (1934).

Appl. Phys. (1)

W. J. Tango, "The circle polynomials of Zernike and their application in optics," Appl. Phys. 13, 327–332 (1977).

J. Opt. Soc. Am. (4)

J. Soc. Ind. Appl. Math. (1)

D. R. Myrick, "A generalization of the radial polynomials of F. Zernike," J. Soc. Ind. Appl. Math. 14, 476–492 (1966).

Ned. Akad. Wet. Indag. Math. (1)

F. Zernike and H. C. Brinkman, "Hypersphärische Funktionen und die in spharischen Berischen Orthogonalen Polynome," Ned. Akad. Wet. Indag. Math. 38, 161–169 (1935).

Opt. Acta (4)

P. W. Hawkes, "The diffraction theory of the aberrations of stigmatic orthomorphic optical or electron optical systems containing toric lenses or quadrupoles," Opt. Acta, 11, 237–251 (1964).

E. C. Kintner, "A recurrence relation for calculating the Zernike polynomials," Opt. Acta 23, 499–500 (1976).

E. C. Kintner and R. M. Sillitto, "A new 'analytic' method for computing the optical transfer function," Opt. Acta 23, 607–619 (1976).

E. C. Kintner, "On the mathematical properties of the Zernike polynomials," Opt. Acta 23, 679–680 (1976).

Opt. Commun. (1)

E. C. Kintner, "Some comments on the use of Zernike polynomials in optics," Opt. Commun. 18, 235–237 (1976).

Optik (1)

R. M. Sillitto, "Diffraction of uniform and Gaussian beams: an application of Zernike polynomials," Optik 48, 271–283 (1977).

Physica (2)

F. Zernike, "Beugungstheorie Des Schneidenverfahrens und seiner verbesserten Form, der Phasenkontrastmethode," Physica (Utrecht) 1, 689–704 (1934).

B. R. A. Nijboer, "The diffraction theory of aberrations," Physica (Utrecht) 10, 679–692 (1943); 13, 605–620 (1947).

Proc. Cambridge Philos. Soc. (1)

A. B. Bhatia and E. Wolf, "On the circle polynomials of Zernike and related orthogonal sets," Proc. Cambridge Philos. Soc. 50, 40–48 (1954).

Sov. J. Opt. Tech. (1)

S. N. Bezdid'ko, "The use of Zernike polynomials in optics," Sov. J. Opt. Tech. 41, 425–429 (1974).

Other (6)

E. H. Linfoot, Recent Advances in Optics (Oxford University, Oxford, 1955), pp. 48–66.

M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1964).

A. Maréchal, Thesis, University of Paris, 1948 (unpublished).

G. Szegö, Orthogonal Polynomials (American Mathematical Society, Providence, 1939).

E. D. Rainville, Special Functions (MacMillan, New York, 1960).

G. Freud, Orthogonale Polynome (Birkhauser-Verlag, Basel, 1969).

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