Abstract

Imaging systems with circular and annular pupils aberrated by primary aberrations are considered. Both classical and balanced (Zernike) aberrations are discussed. Closed-form solutions are derived for the Strehl ratio, except in the case of coma, for which the integral form is used. Numerical results are obtained and compared with Maré-chal’s formula for small aberrations. It is shown that, as long as the Strehl ratio is greater than 0.6, the Maréchal formula gives its value with an error of less than 10%. A discussion of the Rayleigh quarter-wave rule is given, and it is shown that it provides only a qualitative measure of aberration tolerance. Nonoptimally balanced aberrations are also considered, and it is shown that, unless the Strehl ratio is quite high, an optimally balanced aberration does not necessarily give a maximum Strehl ratio.

© 1982 Optical Society of America

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  1. V. N. Mahajan, "Zernike annular polynomials for imaging systems with annular pupils," J. Opt. Soc. Am. 71, 75–85 (1981); 71, 1408 (1981).
  2. Some of this work was presented at the 1981 Annual Meeting of the Optical Society of America, Kissimee, Florida, October 1981. See V. N. Mahajan, "Strehl 'ratio for primary aberrations: some apalytical results," J. Opt. Soc. Am. 71, 1561(A) (1981).
  3. M. Born and E. Wolf, Principles of Optics, 5th qd. (Pergamon, New York, 1975), p. 482.,
  4. A. Maréchal, "Etude de's effets combinés de ja diffraction et des aberrations géométriques sur l'image d'un point lumineux," Rev. d'Opt. 26, 257–277 (1947). See also Ref. 3, p. 469.
  5. B. R. A. Nijboer, "The diffraction theory of aberrations," Ph.D. Thesis (University of Groningen, Groningep, The Netherlands, 1942), p. 17.
  6. See Ref. 3, p. 464.
  7. G. C. Steward, The Symmetrical Optical System (Cambridge U. Press, 1928), Chaps. VI and VII.
  8. The corresponding result for circular pupils has been given by B. R. A. Nijboer in "The diffraction theory of optical aberrations," Physica 14,590–608 (1949).
  9. V. N. Mahajan, "Luneburg apodization problem I," Opt. Lett. 5, 267–269 (1980).
  10. D. A. Holmes, J. E. Korka, and P. V. Avizonis, "Parametric study of apertured focused Gaussian beams," Appl. Opt. 11, 565–574 (1972).
  11. A. Arimoto, "Intensity distribution of aberration-free diffraction patterns due to circular apertures in large F-number optical systems," Opt. Acta 23, 245–250 (1976).
  12. M. A. Gusinow, M. E. Riley, and M. A. Palmer, "Focusing in a large F-number optical system," Opt. Quantum Electron. 9, 465–471 (1977).
  13. Y. Li and E. Wolf, "Focal shifts in diffracted converging spherical waves," Opt. Commun. 39, 211–215 (1981).
  14. Lord Rayleigh, Philos. Mag. 8, 403 (1879). Reprinted in his Scientific Papers (Cambridge U. Press, 1899), Vol. 1, pp. 432–435.
  15. Ref. 3, p. 468.
  16. W. T. Welford, Aberrations of the Symmetrical Optical System (Academic, New York, 1974), p. 207.
  17. R. Barakat, "Rayleigh wavefront criterion," J. Opt. Soc. Am. 55, 572–573 (1965).
  18. R. Barakat, "Diffraction theory of the aberrations of a slit aperture," J. Opt. Soc. Am. 55, 878–881 (1965).
  19. See Ref. 5, p. 48.
  20. 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). Strehl-ratio numbers in Table 3 of this paper are also in error. They need to be divided by (1 - ∊2)2.
  21. R. Barakat, "Diffraction effects of coma," J. Opt. Soc. Am. 54, 1084–1088 (1964), Fig. 12.
  22. W. B. King, "Dependence of the Strehl ratio on the magnitude of the variance of the wave aberration," J. Opt. Soc. Am. 58, 655–661 (1968).
  23. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970), p. 483, Eq. 11.3.9.

1981

V. N. Mahajan, "Zernike annular polynomials for imaging systems with annular pupils," J. Opt. Soc. Am. 71, 75–85 (1981); 71, 1408 (1981).

Some of this work was presented at the 1981 Annual Meeting of the Optical Society of America, Kissimee, Florida, October 1981. See V. N. Mahajan, "Strehl 'ratio for primary aberrations: some apalytical results," J. Opt. Soc. Am. 71, 1561(A) (1981).

Y. Li and E. Wolf, "Focal shifts in diffracted converging spherical waves," Opt. Commun. 39, 211–215 (1981).

1980

1977

M. A. Gusinow, M. E. Riley, and M. A. Palmer, "Focusing in a large F-number optical system," Opt. Quantum Electron. 9, 465–471 (1977).

1976

A. Arimoto, "Intensity distribution of aberration-free diffraction patterns due to circular apertures in large F-number optical systems," Opt. Acta 23, 245–250 (1976).

1972

1968

1965

1964

1949

The corresponding result for circular pupils has been given by B. R. A. Nijboer in "The diffraction theory of optical aberrations," Physica 14,590–608 (1949).

1947

A. Maréchal, "Etude de's effets combinés de ja diffraction et des aberrations géométriques sur l'image d'un point lumineux," Rev. d'Opt. 26, 257–277 (1947). See also Ref. 3, p. 469.

Abramowitz, M.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970), p. 483, Eq. 11.3.9.

Arimoto, A.

A. Arimoto, "Intensity distribution of aberration-free diffraction patterns due to circular apertures in large F-number optical systems," Opt. Acta 23, 245–250 (1976).

Avizonis, P. V.

Barakat, R.

Born, M.

M. Born and E. Wolf, Principles of Optics, 5th qd. (Pergamon, New York, 1975), p. 482.,

Gusinow, M. A.

M. A. Gusinow, M. E. Riley, and M. A. Palmer, "Focusing in a large F-number optical system," Opt. Quantum Electron. 9, 465–471 (1977).

Holmes, D. A.

Houston, A.

King, W. B.

Korka, J. E.

Li, Y.

Y. Li and E. Wolf, "Focal shifts in diffracted converging spherical waves," Opt. Commun. 39, 211–215 (1981).

Mahajan, V. N.

V. N. Mahajan, "Zernike annular polynomials for imaging systems with annular pupils," J. Opt. Soc. Am. 71, 75–85 (1981); 71, 1408 (1981).

Some of this work was presented at the 1981 Annual Meeting of the Optical Society of America, Kissimee, Florida, October 1981. See V. N. Mahajan, "Strehl 'ratio for primary aberrations: some apalytical results," J. Opt. Soc. Am. 71, 1561(A) (1981).

V. N. Mahajan, "Luneburg apodization problem I," Opt. Lett. 5, 267–269 (1980).

Maréchal, A.

A. Maréchal, "Etude de's effets combinés de ja diffraction et des aberrations géométriques sur l'image d'un point lumineux," Rev. d'Opt. 26, 257–277 (1947). See also Ref. 3, p. 469.

Nijboer, B. R. A.

The corresponding result for circular pupils has been given by B. R. A. Nijboer in "The diffraction theory of optical aberrations," Physica 14,590–608 (1949).

B. R. A. Nijboer, "The diffraction theory of aberrations," Ph.D. Thesis (University of Groningen, Groningep, The Netherlands, 1942), p. 17.

Palmer, M. A.

M. A. Gusinow, M. E. Riley, and M. A. Palmer, "Focusing in a large F-number optical system," Opt. Quantum Electron. 9, 465–471 (1977).

Rayleigh, Lord

Lord Rayleigh, Philos. Mag. 8, 403 (1879). Reprinted in his Scientific Papers (Cambridge U. Press, 1899), Vol. 1, pp. 432–435.

Riley, M. E.

M. A. Gusinow, M. E. Riley, and M. A. Palmer, "Focusing in a large F-number optical system," Opt. Quantum Electron. 9, 465–471 (1977).

Stegun, I. A.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970), p. 483, Eq. 11.3.9.

Steward, G. C.

G. C. Steward, The Symmetrical Optical System (Cambridge U. Press, 1928), Chaps. VI and VII.

Welford, W. T.

W. T. Welford, Aberrations of the Symmetrical Optical System (Academic, New York, 1974), p. 207.

Wolf, E.

Y. Li and E. Wolf, "Focal shifts in diffracted converging spherical waves," Opt. Commun. 39, 211–215 (1981).

M. Born and E. Wolf, Principles of Optics, 5th qd. (Pergamon, New York, 1975), p. 482.,

Appl. Opt.

J. Opt. Soc. Am.

Opt. Acta

A. Arimoto, "Intensity distribution of aberration-free diffraction patterns due to circular apertures in large F-number optical systems," Opt. Acta 23, 245–250 (1976).

Opt. Commun.

Y. Li and E. Wolf, "Focal shifts in diffracted converging spherical waves," Opt. Commun. 39, 211–215 (1981).

Opt. Lett.

Opt. Quantum Electron.

M. A. Gusinow, M. E. Riley, and M. A. Palmer, "Focusing in a large F-number optical system," Opt. Quantum Electron. 9, 465–471 (1977).

Physica

The corresponding result for circular pupils has been given by B. R. A. Nijboer in "The diffraction theory of optical aberrations," Physica 14,590–608 (1949).

Rev. d’Opt.

A. Maréchal, "Etude de's effets combinés de ja diffraction et des aberrations géométriques sur l'image d'un point lumineux," Rev. d'Opt. 26, 257–277 (1947). See also Ref. 3, p. 469.

Other

B. R. A. Nijboer, "The diffraction theory of aberrations," Ph.D. Thesis (University of Groningen, Groningep, The Netherlands, 1942), p. 17.

See Ref. 3, p. 464.

G. C. Steward, The Symmetrical Optical System (Cambridge U. Press, 1928), Chaps. VI and VII.

M. Born and E. Wolf, Principles of Optics, 5th qd. (Pergamon, New York, 1975), p. 482.,

See Ref. 5, p. 48.

Lord Rayleigh, Philos. Mag. 8, 403 (1879). Reprinted in his Scientific Papers (Cambridge U. Press, 1899), Vol. 1, pp. 432–435.

Ref. 3, p. 468.

W. T. Welford, Aberrations of the Symmetrical Optical System (Academic, New York, 1974), p. 207.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970), p. 483, Eq. 11.3.9.

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