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Journal of the Optical Society of America

Journal of the Optical Society of America

  • Vol. 52, Iss. 9 — Sep. 1, 1962
  • pp: 985–990

Computation of the Transfer Function of an Optical System from the Design Data for Rotationally Symmetric Aberrations. Part I. Theory

RICHARD BARAKAT  »View Author Affiliations


JOSA, Vol. 52, Issue 9, pp. 985-990 (1962)
http://dx.doi.org/10.1364/JOSA.52.000985


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Abstract

The transfer function can be determined by convolution of the pupil function over the aperture. The pupil function itself is a function of the design data of the lens system (i.e., refractive indices, radii of curvature, etc.) Of particular importance, both practically and theoretically, is the frequency response on-axis where only rotationally symmetric aberrations are present. The aberration function is obtained from an integration over the ray-trace data and is curve-fitted by Chebyschev interpolation. Unlike the least-squares method, the Chebyschev approach allows a uniform approximation over the interval. This data is substituted into the transfer function which is numerically evaluated by application of very high-order Gauss quadrature theory.

Citation
RICHARD BARAKAT, "Computation of the Transfer Function of an Optical System from the Design Data for Rotationally Symmetric Aberrations. Part I. Theory," J. Opt. Soc. Am. 52, 985-990 (1962)
http://www.opticsinfobase.org/josa/abstract.cfm?URI=josa-52-9-985


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References

  1. P. M. Duffieux, L'integrale de Fourier et ses applications l'optique (Rennes, 1946).
  2. K. Miyamoto, "Wave Optics and Geometrical Optics in Optical Design," Progress in Optics, Volume One (North-Hollan Publishing Company, Amsterdam, 1961). This article contains an excellent discussion of the general problem as well as a large bibliography.
  3. H. H. Hopkins, Proc. Roy. Soc. (London) A231, 91 (1955).
  4. M. De, Proc. Roy. Soc. (London) A233, 91 (1955).
  5. G. B. Parrent and C. J. Drane, Optica Acta (Paris) 3, 195 (1956).
  6. G. Black and E. H. Linfoot, Proc. Roy. Soc. (London) A239, 522 (1957).
  7. M. De and B. K. Kath, Optik 15, 739 (1958).
  8. E. L. O'Neill, "Selected Topics in Optics and Communication Theory," (Itek Corporation, Boston, 1959). Contains an extensive bibliography of papers on the frequency response function up to 1959.
  9. H. H. Hopkins, Proc. Phys. Soc. (London) B70, 1002 (1957).
  10. R. Barakat and M. Morello, J. Opt. Soc. Am. 52, 992 (1962).
  11. W. H. Steel, Rev. optique 32, 4, 143, 269 (1953).
  12. J. Focke, Ber. Verhandl. sächs. Akad. Wiss. Leipzig, Math.-naturw. KI. 101, 3 (1954).
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  17. E. W. Hobson, The Theory of Spherical and Ellipsoidal Harmonics (Cambridge University Press, Cambridge, 1931).
  18. H. Mineur, Techniques de calcul numerique Librarie Polytechnique, Paris, 1952).
  19. The number 20 is somewhat of a compromise chosen because the work was originally programmed for the LGP-30 computer, a relatively slow machine. The program is now being redone on the PDP-1, an extremely fast machine, using 24 points (242=576 points in convolved area) for general use and another using 32 points (322=1024 points) for extermely large aberrations.
  20. A. S. Householder, Principles of Numerical Analysis (McGraw-Hill Book Company, Inc., New York, 1953), p. 244.
  21. C. Lanczos, J. Math. Phys. 17, 123 (1938).
  22. See reference 19.

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