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Applied Optics

Applied Optics


  • Vol. 40, Iss. 25 — Sep. 1, 2001
  • pp: 4463–4472

Two innovations in diffraction calculations for cylindrically symmetrical systems

Eric L. Shirley and Matthew L. Terraciano  »View Author Affiliations

Applied Optics, Vol. 40, Issue 25, pp. 4463-4472 (2001)

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Two mathematical innovations are presented that relate to calculating propagation of radiation through cylindrically symmetrical systems using Kirchhoff diffraction theory. The first innovation leads to an efficient means of computing Lommel functions of two arguments (u and ν), typically denoted by U n (u, ν) and V n (u, ν). This can accelerate computations involving Fresnel diffraction by circular apertures or lenses. The second innovation facilitates calculations of Kirchhoff diffraction integrals without recourse to the Fresnel approximation, yet with greatly improved efficiency like that characteristic of the latter approximation.

© 2001 Optical Society of America

OCIS Codes
(050.1960) Diffraction and gratings : Diffraction theory
(120.5630) Instrumentation, measurement, and metrology : Radiometry

Original Manuscript: July 28, 2000
Revised Manuscript: March 27, 2001
Published: September 1, 2001

Eric L. Shirley and Matthew L. Terraciano, "Two innovations in diffraction calculations for cylindrically symmetrical systems," Appl. Opt. 40, 4463-4472 (2001)

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  1. E. Lommel, “Die Beugungserscheinungen einer kreisrunden Oeffnung und eines kresirunden Schirmschens theoretisch und experimentell Bearbeitet,” Abh. Bayer. Akad. 15, 233–328 (1885).
  2. M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, New York, 1975).
  3. See, for example, W. T. Welford, Aberrations of the Symmetrical Optical System (Academic, New York, 1974), p. 96.
  4. J. Focke, “Total illumination in an aberration-free diffraction image,” Opt. Acta 3, 161–163 (1956). [CrossRef]
  5. W. R. Blevin, “Diffraction losses in radiometry and photometry,” Metrologia 6, 39–44 (1970). [CrossRef]
  6. W. H. Steel, M. De, J. A. Bell, “Diffraction corrections in radiometry,” J. Opt. Soc. Am. 62, 1099–1103 (1972). [CrossRef]
  7. L. P. Boivin, “Diffraction corrections in radiometry: comparison of two different methods of calculation,” Appl. Opt. 14, 2002–2009 (1975). [CrossRef] [PubMed]
  8. L. P. Boivin, “Diffraction corrections in the radiometry of extended sources,” Appl. Opt. 15, 1204–1209 (1976). [CrossRef] [PubMed]
  9. E. L. Shirley, “Revised formulas for diffraction effects with point and extended sources,” Appl. Opt. 37, 6581–6590 (1998). [CrossRef]
  10. K. Schwarzschild, “Die Beugungsfigure im Fernrohr weit ausserhalb des Focus,” Sitzungsber. München Akad. Wiss., Math.-Phys. Kl. 28, 271–294 (1898).
  11. K. D. Mielenz, “Algorithms for Fresnel diffraction at rectangular and circular apertures,” J. Res. Natl. Inst. Stand. Technol. 103, 497–509 (1998). [CrossRef]
  12. In both innovations, exact or full calculations use exact formulas involving sums of Bessel functions [Eqs. (1), (2) and (31)]. However, the Bessel functions are numerically evaluated by sound and reasonably optimized methods.
  13. J. J. Stamnes, Waves in Focal Regions (Hilger, Boston, Mass., 1986).
  14. See, for example, F. W. J. Olver, Asymptotics and Special Functions (Peters, Wellesley, Mass., 1997), p. 238.
  15. See, for example, Ref. 14, p. 285.
  16. In practice the Faddeeva function is most helpful and has been calculated with software downloaded from http://gams.nist.gov ; G. P. M. Poppe, C. M. J. Wijers, “Algorithm 680: evaluation of the complex error function,” ACM (Assoc. Comput. Mach.) Trans. Math. Software 16, 47 (1990). [CrossRef]
  17. A. H. Stroud, D. Secrest, Gaussian Quadrature Formulas (Prentice-Hall, Englewood Cliffs, N.J., 1966).
  18. In this approximation the normal derivatives of the incident radiation field and its Green’s function are assumed to be ±ik times the respective quantities on the surface of an aperture, whichever sign is appropriate. Analogous approximations would be made in the case of diffraction by a lens.
  19. The combination of substitutions being made has the advantage that it leads to the correct answer in the limit of an infinite aperture.
  20. The various boundary conditions are discussed, for example, in J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975).
  21. L. P. Boivin, “Reduction of diffraction errors in radiometry by means of toothed apertures,” Appl. Opt. 17, 3323–3328 (1978). [CrossRef] [PubMed]

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