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Journal of the Optical Society of America

Journal of the Optical Society of America

  • Vol. 72, Iss. 7 — Jul. 1, 1982
  • pp: 947–956

Aberration-balancing technique for radially symmetric amplitude distributions: a generalization of the Maréchal approach

Stanisław Szapiel  »View Author Affiliations


JOSA, Vol. 72, Issue 7, pp. 947-956 (1982)
http://dx.doi.org/10.1364/JOSA.72.000947


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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

Citation
Stanisław Szapiel, "Aberration-balancing technique for radially symmetric amplitude distributions: a generalization of the Maréchal approach," J. Opt. Soc. Am. 72, 947-956 (1982)
http://www.opticsinfobase.org/josa/abstract.cfm?URI=josa-72-7-947


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References

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  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).
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  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).

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