Abstract

The Maréchal balancing theory for constant-amplitude circular pupils is extended to radially symmetric nonnegative amplitude distributions. A convenient technique based on the solution of a set of linear equations is proposed. It permits the determination of optimum balanced wave fronts in the weighted least-squares sense by using corresponding moments of the relevant apodizing function. Relations with the fundamental theory of orthogonal polynomials are discussed. Formulas for fast evaluation of peak intensity degradation in the far-field pattern and corresponding Maréchal tolerances are derived. Numerical analysis of optical systems for use with laser beams. results presented may be important in the design and analysis of optical systems for use with laser beams.

© 1982 Optical Society of America

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  1. M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, 1975), Chap. 9.
  2. E. H. Linfoot, Fourier Methods in Optical Image Evaluation (Focal, London, 1966), Chaps. I, V.
  3. A. Maréchal and M. Françon, Diffraction-Structure des Images (Editions de la Revue d'Optique Théorique et Instrumentale, Paris, 1960), Chap. 8.
  4. E. L. O Neill, Introduction to Statistical Optics (Addison-Wesley, Reading, Mass., 1963), Chap. 4.
  5. F. Zernike, "Beugungstheorie des Schneidenverfahrens und seiner verbesserten Form, des Phasenkontrastmethode," Physica (Utrecht) 1, 689–704 (1934).
  6. S. N. Bezdidko, "The use of Zernike polynomials in optics," Sov. J. Opt. Technol. 41, 425–429 (1974).
  7. If a wave aberration is expressed by the Zernike polynomial, Rnm(r)cos mθ, the aberration average over the pupil is zero. By using the Maréchal balancing technique, however, we always are able to satisfy the above condition: for a rotationally symmetric aberration (m = 0), the constant term, which represents a shift in the radius of the reference sphere, should be treated as a variable aberration coefficient. This immediately follows from general considerations given in Sections 3 and 5. See also Ref. 24.
  8. B. Tatian, "Aberration balancing in rotationally symmetric lenses," J. Opt. Soc. Am. 64, 1083–1091 (1974).
  9. 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).
  10. R. Barakat, "Optimum balanced wave-front aberrations for radially symmetric amplitude distributions: generalizations of Zernike polynomials," J. Opt. Soc. Am. 70, 739–742 (1980).
  11. 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).
  12. D. D. Lowenthal, "Maréchal intensity criteria modified for Gaussian beams," Appl. Opt. 13, 2126–2133 (1974).
  13. V. N. Mahajan, "Zernike annular polynomials for imaging systems with annular pupils," J. Opt. Soc. Am. 71, 75–85 (1981).
  14. W. H. Steel, "The problem of optical tolerances for systems with absorption," Appl. Opt. 8, 2297–2299 (1969).
  15. S. Szapiel, "Maréchal intensity criteria modified for circular apertures with nonuniform intensity transmission: Dini series approach," Opt. Lett. 2, 124–126 (1978).
  16. V. N. Mahajan, "Luneburg apodization problem I," Opt. Lett. 5, 267–269 (1980).
  17. See, for example, D. C. Champeney, Fourier Transforms and Their Physical Applications (Academic, New York, 1973), Chaps. 2 and 3.
  18. P. Jacquinot and B. Roizen-Dossier, "Apodisation," in Progress in Optics (North-Holland, Amsterdam, 1964), Vol. III, pp. 31–188.
  19. H. Bateman and A. Erdélyi, Higher Transcendental Functions (McGraw-Hill, New York, 1953), Vol. II, Chap. 10.
  20. T. S. Chihara, "An Introduction to Orthogonal Polynomials," in Mathematics and Its Applications, A Series of Monographs and Texts (Gordon and Breach, New York, 1978), Vol. 13, especially Chap. 1.
  21. Also note that the Gram determinant [Eq. (4.6)] can be treated as a discriminant of symmetrical quadratic form (4.8); since the form is positive-definite, all the determinants Gp are positive.
  22. Note that, having the values of appropriate moments, another recursive method of orthogonalization may be also proposed. Namely, any three connective orthogonal polynomials are connected by a single recurrence relation (see Ref. 20, pp. 18 and 19, for example).
  23. R. Barakat and A. Houston, "Transfer function of an optical system in the off-axis aberrations," J. Opt. Soc. Am. 55, 1142–1148 (1965).
  24. K. Pietraszkiewicz, "Determination of the optimal reference sphere," J. Opt. Soc. Am. 69, 1045–1046 (1979).
  25. L. D. Dickson, "Characteristics of a propagating Gaussian beam," Appl. Opt. 9, 1854–1861 (1970).
  26. D. A. Holmes, J. E. Korka, and P. V. Avizonis, "Parametric study of apertured focused Gaussian beams," Appl. Opt. 11, 565–574 (1972).
  27. When the truncation ratio is r0 1/n, the pupil radius is equal to n times the 1/e2 beam radius at the lens.
  28. D. D. Lowenthal, Ref. 12, p. 2132, Fig. 6.
  29. H. Bateman and A. Erdélyi, Higher Transcendental Functions (McGraw-Hill, New York, 1953), Chap. 9, Eqs. (16) and (17).
  30. Y. L. Luke, Mathematical Functions and Their Approximations (Academic, New York, 1975), Chap. 4, especially Secs. 4.2 and 4.5.
  31. D. D. Lowenthal, Ref. 12, p. 2129.
  32. D. D. Lowenthal, Ref. 12, p. 2132, Eq. (13); note that certain factors on p. 2132 are misprinted.
  33. V. N. Mahajan, Ref. 13, p. 82, especially Eqs. (62), (68), and (75)–(77); "Zernike annular polynomials for imaging systems with annular pupils: errata," J. Opt. Soc. Am. 71, 1408 (1981).

1981 (1)

1980 (2)

1979 (1)

1978 (1)

1974 (4)

1972 (1)

1970 (1)

1969 (1)

1965 (2)

1934 (1)

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

Arimoto, A.

Avizonis, P. V.

Barakat, R.

Bateman, H.

H. Bateman and A. Erdélyi, Higher Transcendental Functions (McGraw-Hill, New York, 1953), Vol. II, Chap. 10.

H. Bateman and A. Erdélyi, Higher Transcendental Functions (McGraw-Hill, New York, 1953), Chap. 9, Eqs. (16) and (17).

Bezdidko, S. N.

S. N. Bezdidko, "The use of Zernike polynomials in optics," Sov. J. Opt. Technol. 41, 425–429 (1974).

Born, M.

M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, 1975), Chap. 9.

Champeney, D. C.

See, for example, D. C. Champeney, Fourier Transforms and Their Physical Applications (Academic, New York, 1973), Chaps. 2 and 3.

Chihara, T. S.

T. S. Chihara, "An Introduction to Orthogonal Polynomials," in Mathematics and Its Applications, A Series of Monographs and Texts (Gordon and Breach, New York, 1978), Vol. 13, especially Chap. 1.

Dickson, L. D.

Erdélyi, A.

H. Bateman and A. Erdélyi, Higher Transcendental Functions (McGraw-Hill, New York, 1953), Chap. 9, Eqs. (16) and (17).

H. Bateman and A. Erdélyi, Higher Transcendental Functions (McGraw-Hill, New York, 1953), Vol. II, Chap. 10.

Françon, M.

A. Maréchal and M. Françon, Diffraction-Structure des Images (Editions de la Revue d'Optique Théorique et Instrumentale, Paris, 1960), Chap. 8.

Holmes, D. A.

Houston, A.

Jacquinot, P.

P. Jacquinot and B. Roizen-Dossier, "Apodisation," in Progress in Optics (North-Holland, Amsterdam, 1964), Vol. III, pp. 31–188.

Korka, J. E.

Linfoot, E. H.

E. H. Linfoot, Fourier Methods in Optical Image Evaluation (Focal, London, 1966), Chaps. I, V.

Lowenthal, D. D.

D. D. Lowenthal, "Maréchal intensity criteria modified for Gaussian beams," Appl. Opt. 13, 2126–2133 (1974).

D. D. Lowenthal, Ref. 12, p. 2132, Fig. 6.

D. D. Lowenthal, Ref. 12, p. 2129.

D. D. Lowenthal, Ref. 12, p. 2132, Eq. (13); note that certain factors on p. 2132 are misprinted.

Luke, Y. L.

Y. L. Luke, Mathematical Functions and Their Approximations (Academic, New York, 1975), Chap. 4, especially Secs. 4.2 and 4.5.

Mahajan, V. N.

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

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

V. N. Mahajan, Ref. 13, p. 82, especially Eqs. (62), (68), and (75)–(77); "Zernike annular polynomials for imaging systems with annular pupils: errata," J. Opt. Soc. Am. 71, 1408 (1981).

Maréchal, A.

A. Maréchal and M. Françon, Diffraction-Structure des Images (Editions de la Revue d'Optique Théorique et Instrumentale, Paris, 1960), Chap. 8.

Neill, E. L. O

E. L. O Neill, Introduction to Statistical Optics (Addison-Wesley, Reading, Mass., 1963), Chap. 4.

Pietraszkiewicz, K.

Roizen-Dossier, B.

P. Jacquinot and B. Roizen-Dossier, "Apodisation," in Progress in Optics (North-Holland, Amsterdam, 1964), Vol. III, pp. 31–188.

Steel, W. H.

Szapiel, S.

Tatian, B.

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, 1975), Chap. 9.

Zernike, F.

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

Appl. Opt. (4)

J. Opt. Soc. Am. (7)

Opt. Lett. (2)

Physica (1)

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

Sov. J. Opt. Technol. (1)

S. N. Bezdidko, "The use of Zernike polynomials in optics," Sov. J. Opt. Technol. 41, 425–429 (1974).

Other (18)

If a wave aberration is expressed by the Zernike polynomial, Rnm(r)cos mθ, the aberration average over the pupil is zero. By using the Maréchal balancing technique, however, we always are able to satisfy the above condition: for a rotationally symmetric aberration (m = 0), the constant term, which represents a shift in the radius of the reference sphere, should be treated as a variable aberration coefficient. This immediately follows from general considerations given in Sections 3 and 5. See also Ref. 24.

When the truncation ratio is r0 1/n, the pupil radius is equal to n times the 1/e2 beam radius at the lens.

D. D. Lowenthal, Ref. 12, p. 2132, Fig. 6.

H. Bateman and A. Erdélyi, Higher Transcendental Functions (McGraw-Hill, New York, 1953), Chap. 9, Eqs. (16) and (17).

Y. L. Luke, Mathematical Functions and Their Approximations (Academic, New York, 1975), Chap. 4, especially Secs. 4.2 and 4.5.

D. D. Lowenthal, Ref. 12, p. 2129.

D. D. Lowenthal, Ref. 12, p. 2132, Eq. (13); note that certain factors on p. 2132 are misprinted.

V. N. Mahajan, Ref. 13, p. 82, especially Eqs. (62), (68), and (75)–(77); "Zernike annular polynomials for imaging systems with annular pupils: errata," J. Opt. Soc. Am. 71, 1408 (1981).

See, for example, D. C. Champeney, Fourier Transforms and Their Physical Applications (Academic, New York, 1973), Chaps. 2 and 3.

P. Jacquinot and B. Roizen-Dossier, "Apodisation," in Progress in Optics (North-Holland, Amsterdam, 1964), Vol. III, pp. 31–188.

H. Bateman and A. Erdélyi, Higher Transcendental Functions (McGraw-Hill, New York, 1953), Vol. II, Chap. 10.

T. S. Chihara, "An Introduction to Orthogonal Polynomials," in Mathematics and Its Applications, A Series of Monographs and Texts (Gordon and Breach, New York, 1978), Vol. 13, especially Chap. 1.

Also note that the Gram determinant [Eq. (4.6)] can be treated as a discriminant of symmetrical quadratic form (4.8); since the form is positive-definite, all the determinants Gp are positive.

Note that, having the values of appropriate moments, another recursive method of orthogonalization may be also proposed. Namely, any three connective orthogonal polynomials are connected by a single recurrence relation (see Ref. 20, pp. 18 and 19, for example).

M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, 1975), Chap. 9.

E. H. Linfoot, Fourier Methods in Optical Image Evaluation (Focal, London, 1966), Chaps. I, V.

A. Maréchal and M. Françon, Diffraction-Structure des Images (Editions de la Revue d'Optique Théorique et Instrumentale, Paris, 1960), Chap. 8.

E. L. O Neill, Introduction to Statistical Optics (Addison-Wesley, Reading, Mass., 1963), Chap. 4.

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