Abstract

The aberrations of imaging systems with uniformly illuminated annular pupils are discussed in terms of a complete set of polynomials that are orthogonal over an annular region. These polynomials, which we call Zernike annular polynomials, are similar to the Zernike circle polynomials and reduce to them as the annulus approaches the full circle. The Zernike-polynomial expansion of an aberration function is compared with its power-series expansion. The orthogonal aberrations given by Zernike annular polynomials describe how a higher-order classical aberration of a power-series expansion is balanced with one or more lower-order classical aberrations to minimize its variance. It is shown that as the obscuration ratio increases, the standard deviation of an orthogonal as well as a classical primary aberration decreases in the case of spherical aberration and field curvature and increases in the case of coma, astigmatism, and distortion. The only exception is the case of orthogonal coma, for which the standard deviation first increases and then decreases. The orthogonal aberrations for nonuniformly illuminated annular pupils are also considered, and, as an example, Gaussian illumination is discussed. It is shown that the standard deviation of an orthogonal primary aberration for a given amount of the corresponding classical aberration is somewhat smaller for a Gaussian pupil than that for a uniformly illuminated pupil.

© 1981 Optical Society of America

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  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); "Beugungstheorie des Schneidenverfahrens und seiner verbesserten Form, der Phasenkontrastmethode," Physica 1, 689–794 (1934).
  2. A. B. Bhatia and E. Wolf, "On the circle polynomials of Zernike and related orthogonal sets," Proc. Cambridge Philos. Soc. 50, 40–48 (1954).
  3. B. R. A. Nijboer, "The diffraction theory of aberrations," Ph.D. thesis (University of Groningen, Groningen, The Netherlands, 1942). See also a paper by the same title, Physica 23, 605–620 (1947).
  4. For early work on the use of Zernike circle polynomials in optics, see M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, New York, 1975), Chap. 9.
  5. See also E. H. Linfoot, Recent Advances in Optics (Oxford U. Press, Oxford, 1955), Chap. 2.
  6. Special issue on adaptive optics, J. Opt. Soc. Am. 67, 269–409 (1977).
  7. S. N. Bezdid'ko, "The use of Zernike polynomials in optics," Sov. J. Opt. Technol. 41, 425–429 (1974); "Determination of the Zernike polynomial expansion coefficients of the wave aberration," Sov. J. Opt. Technol. 42, 426–427 (1975); "Calculation of the Strehl coefficient and determination of the best-focus plane in the case of polychromatic light," Sov. J. Opt. Technol. 42, 514–516 (1975); "Numerical method of calculating the Strehl coefficient using Zernike polynomials," Sov. J. Opt. Technol. 43, 222–225 (1976).
  8. R. J. Noll, "Zernike polynomials and atmospheric turbulence," J. Opt. Soc. Am. 66, 207–211 (1976).
  9. J. Y. Wang, "Phase-compensated optical beam propagation through atmospheric turbulence," Appl. Opt. 17, 2580–2590 (1978).
  10. E. C. Kintner and R. M. Sillitto, "A new 'analytic' method for computing the optical transfer function," Opt. Acta 23, 607–619 (1976).
  11. W. Lukosz, "Zur Übertragungstheorie der inkohärenten optischen Abbildung vom Standpunkt der geometrischen Optik," Opt. Acta 5, 299–305 (1958); "Der Einfluss der Aberrationen auf die optische Übertragungsfunktion bie kleinen Orts-Frequenzen," Opt. Acta 10, 1–19 (1963).
  12. W. J. Tango, "The circle polynomials of Zernike and their applications in optics," Appl. Phys. 13, 327–332 (1977).
  13. R. M. Sillitto, "Diffraction of uniform and Gaussian beams: an application of Zernike polynomials," Optik 48, 271–277 (1977).
  14. P. W. Hawkes, "The diffraction theory of stigmatic orthomorphic optical or electron optical systems containing toric lenses or quadrupoles," Opt. Acta 11, 237–251 (1964).
  15. B. Tatian, "Aberration balancing in rotationally symmetric lenses," J. Opt. Soc. Am. 64, 1083–1091 (1974).
  16. 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).
  17. D. D. Lowenthal, "Maréchal intensity criteria modified for Gaussian beams," Appl. Opt. 13, 2126–2133 (1974).
  18. G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers (McGraw-Hill, New York, 1968), p. 454.
  19. Expressions for some of the radial annular polynomials are given without derivation in "Three-meter telescope study final report," Perkin-Elmer Corporation Rep. No. ER10713, August 1971, p. 36. There is a typographical error in the expression for R⅓(ρ;ε) and its subsequent discussion. Some numerical factors are missing in equations on p. 42.
  20. R. Barakat and A. Houston, "The aberrations of non-rotationally symmetric systems and their diffraction effects," Opt. Acta 13, 1–30 (1966).
  21. E. H. Linfoot and E. Wolf, "Diffraction images in systems with an annular aperture," Proc. Phys. Soc. (London) B66, 145–149 (1953).
  22. W. T. Welford, "Use of annular apertures to increase focal depth," J. Opt. Soc. Am. 50, 749–753 (1960).
  23. R. Barakat and A. Houston, "Transfer function of an annular aperture in the presence of spherical aberration," J. Opt. Soc. Am. 55, 538–541 (1965).
  24. R. Barakat, "Optimum balanced wave-front aberrations for radially symmetric amplitude distributions: generalizations of Zernike polynomials," J. Opt. Soc. Am. 70, 739–742 (1980).
  25. W. H. Southwell, "Wave-front analyzer using a maximum like-lihood algorithm," J. Opt. Soc. Am. 67, 396–399 (1977).
  26. V. N. Mahajan, J. Govignon, and R. J. Morgan, "Adaptive optics without wavefront sensors," Proc. Soc. Photo-Opt. Instrum. Eng. 228, 63–69 (1980).
  27. G. Szegö, Orthogonal Polynomials (American Mathematical Society, Providence, R.I., 1939), Vol. 23, p. 28.
  28. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970), p. 785. The Christoffel-Darboux formula is also given in Ref. 27, p. 41.

1980 (2)

V. N. Mahajan, J. Govignon, and R. J. Morgan, "Adaptive optics without wavefront sensors," Proc. Soc. Photo-Opt. Instrum. Eng. 228, 63–69 (1980).

R. Barakat, "Optimum balanced wave-front aberrations for radially symmetric amplitude distributions: generalizations of Zernike polynomials," J. Opt. Soc. Am. 70, 739–742 (1980).

1978 (1)

1977 (3)

W. H. Southwell, "Wave-front analyzer using a maximum like-lihood algorithm," J. Opt. Soc. Am. 67, 396–399 (1977).

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

R. M. Sillitto, "Diffraction of uniform and Gaussian beams: an application of Zernike polynomials," Optik 48, 271–277 (1977).

1976 (2)

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).

1974 (3)

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).

B. Tatian, "Aberration balancing in rotationally symmetric lenses," J. Opt. Soc. Am. 64, 1083–1091 (1974).

S. N. Bezdid'ko, "The use of Zernike polynomials in optics," Sov. J. Opt. Technol. 41, 425–429 (1974); "Determination of the Zernike polynomial expansion coefficients of the wave aberration," Sov. J. Opt. Technol. 42, 426–427 (1975); "Calculation of the Strehl coefficient and determination of the best-focus plane in the case of polychromatic light," Sov. J. Opt. Technol. 42, 514–516 (1975); "Numerical method of calculating the Strehl coefficient using Zernike polynomials," Sov. J. Opt. Technol. 43, 222–225 (1976).

1966 (1)

R. Barakat and A. Houston, "The aberrations of non-rotationally symmetric systems and their diffraction effects," Opt. Acta 13, 1–30 (1966).

1965 (1)

1964 (1)

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

1960 (1)

1958 (1)

W. Lukosz, "Zur Übertragungstheorie der inkohärenten optischen Abbildung vom Standpunkt der geometrischen Optik," Opt. Acta 5, 299–305 (1958); "Der Einfluss der Aberrationen auf die optische Übertragungsfunktion bie kleinen Orts-Frequenzen," Opt. Acta 10, 1–19 (1963).

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).

1953 (1)

E. H. Linfoot and E. Wolf, "Diffraction images in systems with an annular aperture," Proc. Phys. Soc. (London) B66, 145–149 (1953).

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); "Beugungstheorie des Schneidenverfahrens und seiner verbesserten Form, der Phasenkontrastmethode," Physica 1, 689–794 (1934).

Abramowitz, M.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970), p. 785. The Christoffel-Darboux formula is also given in Ref. 27, p. 41.

Arimoto, A.

Barakat, R.

Bezdid’ko, S. N.

S. N. Bezdid'ko, "The use of Zernike polynomials in optics," Sov. J. Opt. Technol. 41, 425–429 (1974); "Determination of the Zernike polynomial expansion coefficients of the wave aberration," Sov. J. Opt. Technol. 42, 426–427 (1975); "Calculation of the Strehl coefficient and determination of the best-focus plane in the case of polychromatic light," Sov. J. Opt. Technol. 42, 514–516 (1975); "Numerical method of calculating the Strehl coefficient using Zernike polynomials," Sov. J. Opt. Technol. 43, 222–225 (1976).

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.

For early work on the use of Zernike circle polynomials in optics, see M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, New York, 1975), Chap. 9.

Govignon, J.

V. N. Mahajan, J. Govignon, and R. J. Morgan, "Adaptive optics without wavefront sensors," Proc. Soc. Photo-Opt. Instrum. Eng. 228, 63–69 (1980).

Hawkes, P. W.

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

Houston, A.

R. Barakat and A. Houston, "The aberrations of non-rotationally symmetric systems and their diffraction effects," Opt. Acta 13, 1–30 (1966).

R. Barakat and A. Houston, "Transfer function of an annular aperture in the presence of spherical aberration," J. Opt. Soc. Am. 55, 538–541 (1965).

Kintner, E. C.

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

Korn, G. A.

G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers (McGraw-Hill, New York, 1968), p. 454.

Korn, T. M.

G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers (McGraw-Hill, New York, 1968), p. 454.

Linfoot, E. H.

E. H. Linfoot and E. Wolf, "Diffraction images in systems with an annular aperture," Proc. Phys. Soc. (London) B66, 145–149 (1953).

See also E. H. Linfoot, Recent Advances in Optics (Oxford U. Press, Oxford, 1955), Chap. 2.

Lowenthal, D. D.

Lukosz, W.

W. Lukosz, "Zur Übertragungstheorie der inkohärenten optischen Abbildung vom Standpunkt der geometrischen Optik," Opt. Acta 5, 299–305 (1958); "Der Einfluss der Aberrationen auf die optische Übertragungsfunktion bie kleinen Orts-Frequenzen," Opt. Acta 10, 1–19 (1963).

Mahajan, V. N.

V. N. Mahajan, J. Govignon, and R. J. Morgan, "Adaptive optics without wavefront sensors," Proc. Soc. Photo-Opt. Instrum. Eng. 228, 63–69 (1980).

Morgan, R. J.

V. N. Mahajan, J. Govignon, and R. J. Morgan, "Adaptive optics without wavefront sensors," Proc. Soc. Photo-Opt. Instrum. Eng. 228, 63–69 (1980).

Nijboer, B. R. A.

B. R. A. Nijboer, "The diffraction theory of aberrations," Ph.D. thesis (University of Groningen, Groningen, The Netherlands, 1942). See also a paper by the same title, Physica 23, 605–620 (1947).

Noll, R. J.

Sillitto, R. M.

R. M. Sillitto, "Diffraction of uniform and Gaussian beams: an application of Zernike polynomials," Optik 48, 271–277 (1977).

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

Southwell, W. H.

Stegun, I. A.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970), p. 785. The Christoffel-Darboux formula is also given in Ref. 27, p. 41.

Szegö, G.

G. Szegö, Orthogonal Polynomials (American Mathematical Society, Providence, R.I., 1939), Vol. 23, p. 28.

Tango, W. J.

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

Tatian, B.

Wang, J. Y.

Welford, W. T.

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).

E. H. Linfoot and E. Wolf, "Diffraction images in systems with an annular aperture," Proc. Phys. Soc. (London) B66, 145–149 (1953).

For early work on the use of Zernike circle polynomials in optics, see M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, New York, 1975), Chap. 9.

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); "Beugungstheorie des Schneidenverfahrens und seiner verbesserten Form, der Phasenkontrastmethode," Physica 1, 689–794 (1934).

Appl. Opt. (2)

Appl. Phys. (1)

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

J. Opt. Soc. Am. (7)

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); "Beugungstheorie des Schneidenverfahrens und seiner verbesserten Form, der Phasenkontrastmethode," Physica 1, 689–794 (1934).

Opt. Acta (4)

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

W. Lukosz, "Zur Übertragungstheorie der inkohärenten optischen Abbildung vom Standpunkt der geometrischen Optik," Opt. Acta 5, 299–305 (1958); "Der Einfluss der Aberrationen auf die optische Übertragungsfunktion bie kleinen Orts-Frequenzen," Opt. Acta 10, 1–19 (1963).

R. Barakat and A. Houston, "The aberrations of non-rotationally symmetric systems and their diffraction effects," Opt. Acta 13, 1–30 (1966).

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

Optik (1)

R. M. Sillitto, "Diffraction of uniform and Gaussian beams: an application of Zernike polynomials," Optik 48, 271–277 (1977).

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).

Proc. Phys. Soc. (1)

E. H. Linfoot and E. Wolf, "Diffraction images in systems with an annular aperture," Proc. Phys. Soc. (London) B66, 145–149 (1953).

Proc. Soc. Photo-Opt. Instrum. Eng. (1)

V. N. Mahajan, J. Govignon, and R. J. Morgan, "Adaptive optics without wavefront sensors," Proc. Soc. Photo-Opt. Instrum. Eng. 228, 63–69 (1980).

Sov. J. Opt. Technol. (1)

S. N. Bezdid'ko, "The use of Zernike polynomials in optics," Sov. J. Opt. Technol. 41, 425–429 (1974); "Determination of the Zernike polynomial expansion coefficients of the wave aberration," Sov. J. Opt. Technol. 42, 426–427 (1975); "Calculation of the Strehl coefficient and determination of the best-focus plane in the case of polychromatic light," Sov. J. Opt. Technol. 42, 514–516 (1975); "Numerical method of calculating the Strehl coefficient using Zernike polynomials," Sov. J. Opt. Technol. 43, 222–225 (1976).

Other (8)

B. R. A. Nijboer, "The diffraction theory of aberrations," Ph.D. thesis (University of Groningen, Groningen, The Netherlands, 1942). See also a paper by the same title, Physica 23, 605–620 (1947).

For early work on the use of Zernike circle polynomials in optics, see M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, New York, 1975), Chap. 9.

See also E. H. Linfoot, Recent Advances in Optics (Oxford U. Press, Oxford, 1955), Chap. 2.

Special issue on adaptive optics, J. Opt. Soc. Am. 67, 269–409 (1977).

G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers (McGraw-Hill, New York, 1968), p. 454.

Expressions for some of the radial annular polynomials are given without derivation in "Three-meter telescope study final report," Perkin-Elmer Corporation Rep. No. ER10713, August 1971, p. 36. There is a typographical error in the expression for R⅓(ρ;ε) and its subsequent discussion. Some numerical factors are missing in equations on p. 42.

G. Szegö, Orthogonal Polynomials (American Mathematical Society, Providence, R.I., 1939), Vol. 23, p. 28.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970), p. 785. The Christoffel-Darboux formula is also given in Ref. 27, p. 41.

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