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Applied Optics

Applied Optics

APPLICATIONS-CENTERED RESEARCH IN OPTICS

  • Vol. 39, Iss. 27 — Sep. 20, 2000
  • pp: 5040–5051

Supernumerary spacing of rainbows produced by an elliptical-cross-section cylinder. I. Theory

James A. Lock  »View Author Affiliations


Applied Optics, Vol. 39, Issue 27, pp. 5040-5051 (2000)
http://dx.doi.org/10.1364/AO.39.005040


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Abstract

A sequence of rainbows is produced in light scattering by a particle of high symmetry in the short-wavelength limit, and a supernumerary interference pattern occurs to one side of each rainbow. Using both a ray-tracing procedure and the Debye-series decomposition of first-order perturbation wave theory, I examine the spacing of the supernumerary maxima and minima as a function of the cylinder rotation angle when an elliptical-cross-section cylinder is normally illuminated by a plane wave. I find that the supernumerary spacing depends sensitively on the cylinder-cross-section shape, and the spacing varies sinusoidally as a function of the cylinder rotation angle for small cylinder ellipticity. I also find that relatively large uncertainties in the supernumerary spacing affect the rainbow angle only minimally.

© 2000 Optical Society of America

OCIS Codes
(080.1510) Geometric optics : Propagation methods
(290.0290) Scattering : Scattering
(290.4020) Scattering : Mie theory

History
Original Manuscript: March 2, 2000
Revised Manuscript: June 13, 2000
Published: September 20, 2000

Citation
James A. Lock, "Supernumerary spacing of rainbows produced by an elliptical-cross-section cylinder. I. Theory," Appl. Opt. 39, 5040-5051 (2000)
http://www.opticsinfobase.org/ao/abstract.cfm?URI=ao-39-27-5040


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References

  1. M. V. Berry, C. Upstill, “Catastrophe optics: morphologies of caustics and their diffraction patterns,” in Progress in OpticsE. Wolf, ed. (Elsevier, Amsterdam, 1980), Vol. 18, pp. 257–346. [CrossRef]
  2. W. Möbius, “Zur Theorie des Regenbogens und ihrer experimentallen Prufung,” Abh. Kgl. Saechs. Ges. Wiss. Math.-Phys. Kl. 30, 105–254 (1907–1909).
  3. W. Möbius, “Zur Theorie des Regenbogens und ihrer experimentallen Prufung,” Ann. Phys. 33, 1493–1558 (1910). [CrossRef]
  4. G. P. Können, “Appearance of supernumeraries of the secondary rainbow in rain showers,” J. Opt. Soc. Am. A 4, 810–816 (1987). [CrossRef]
  5. H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981), Sect. 13.23, pp. 243–246.
  6. J. D. Walker, “Multiple rainbows from single drops of water and other liquids,” Am. J. Phys. 44, 421–433 (1976). [CrossRef]
  7. R. T. Wang, H. C. van de Hulst, “Rainbows: Mie computations and the Airy approximation,” Appl. Opt. 30, 106–117 (1991). [CrossRef] [PubMed]
  8. C. L. Adler, J. A. Lock, B. R. Stone, “Rainbow scattering by a cylinder with a nearly elliptical cross section,” Appl. Opt. 37, 1540–1550 (1998). [CrossRef]
  9. G. B. Airy, “On the intensity of light in the neighbourhood of a caustic,” Trans. Cambridge Philos. Soc. 6, 397–403 (1838); reprinted in Geometrical Aspects of Scattering, P. L. Marston, ed., Vol. MS 89 of the SPIE Milestone Series (SPIE, Bellingham Wash., 1994), pp. 298–309.
  10. M. Abramowitz, I. A. Stegun, eds., Handbook of Mathematical Functions (National Bureau of Standards, Washington, D.C., 1964), Sect. 10.4, pp. 446–452.
  11. G. P. Können, J. H. de Boer, “Polarized rainbow,” Appl. Opt. 18, 1961–1965 (1979). [CrossRef] [PubMed]
  12. Lord Rayleigh, “On the electromagnetic theory of light,” Philos. Mag. 12, 81–101 (1881); reprinted in Scientific Papers by Lord Rayleigh (Dover, New York, 1964), Vol. 1, pp. 518–536.
  13. P. Debye, “Das elektromagnetische Feld um einen Zylinder und die Theorie des Regenbogens,” Phys. Z. 9, 775–778 (1908); reprinted and translated into English in P. L. Marston, ed., Geometrical Aspects of Scattering, Vol. MS 89 of the SPIE Milestone Series (SPIE, Bellingham Wash., 1994), pp. 198–204.
  14. D. Marcuse, H. M. Presby, “Light scattering from optical fibers with arbitrary refractive index distributions,” J. Opt. Soc. Am. 65, 367–375 (1975). [CrossRef]
  15. Y. Takano, M. Tanaka, “Phase matrix and cross sections for single scattering by circular cylinders: a comparison of ray optics and wave theory,” Appl. Opt. 19, 2781–2793 (1980). [CrossRef] [PubMed]
  16. J. A. Lock, C. L. Adler, “Debye-series analysis of the first-order rainbow produced in scattering of a diagonally incident plane wave by a circular cylinder,” J. Opt. Soc. Am. A 14, 1316–1328 (1997). [CrossRef]
  17. V. Khare, H. M. Nussenzveig, “Theory of the rainbow,” Phys. Rev. Lett. 33, 976–980 (1974). [CrossRef]
  18. H. M. Nussenzveig, “High-frequency scattering by a transparent sphere. I. Direct reflection and transmission,” J. Math. Phys. 10, 82–124 (1969). [CrossRef]
  19. H. M. Nussenzveig, “High-frequency scattering by a transparent sphere. II. Theory of the rainbow and the glory,” J. Math. Phys. 10, 125–176 (1969). [CrossRef]
  20. E. A. Hovenac, J. A. Lock, “Assessing the contributions of surface waves and complex rays to far-field Mie scattering by use of the Debye series,” J. Opt. Soc. Am. A 9, 781–795 (1992). [CrossRef]
  21. A. Nelson, L. Eyges, “Electromagnetic scattering from dielectric rods of arbitrary cross section,” J. Opt. Soc. Am. 66, 254–259 (1976). [CrossRef]
  22. P. W. Barber, S. C. Hill, Light Scattering by Particles: Computational Methods (World Scientific, Singapore, 1990), Chap. 3, pp. 79–185.
  23. M. I. Mishchenko, L. D. Travis, “Capabilities and limitations of a current FORTRAN implementation of the T-matrix method for randomly oriented, rotationally symmetric scatterers,” J. Quant. Spectrosc. Radiat. Transfer 60, 309–324 (1998). [CrossRef]
  24. C. Yeh, “Perturbation approach to the diffraction of electromagnetic waves by arbitrarily shaped dielectric obstacles,” Phys. Rev. A 135, 1193–1201 (1964). [CrossRef]
  25. V. A. Erma, “An exact solution for the scattering of electromagnetic waves from conductors of arbitrary shape. I. Case of cylindrical symmetry,” Phys. Rev. 173, 1243–1257 (1968). [CrossRef]
  26. V. A. Erma, “Exact solution for the scattering of electromagnetic waves from conductors of arbitrary shape. II. General case,” Phys. Rev. 176, 1544–1553 (1968). [CrossRef]
  27. V. A. Erma, “Exact solution for scattering of electromagnetic waves from bodies of arbitrary shape. III. Obstacles with arbitrary electromagnetic properties,” Phys. Rev. 179, 1238–1246 (1969). [CrossRef]
  28. J. A. Lock, “Excitation of morphology-dependent resonances and van de Hulst’s localization principle,” Opt. Lett. 24, 427–429 (1999). [CrossRef]
  29. C. Yeh, “Perturbation method in the diffraction of electromagnetic waves by arbitrarily shaped penetrable obstacles,” J. Math Phys. 6, 2008–2013 (1965). [CrossRef]
  30. R. Schiffer, “Light scattered by perfectly conducting statistically irregular particles,” J. Opt. Soc. Am. A 6, 385–402 (1989). [CrossRef]
  31. G. Arfken, Mathematical Methods for Physicists, 3rd ed. (Academic, New York, 1985), p. 605.
  32. P. W. Barber, S. C. Hill, “Effects of particle nonsphericity on light-scattering,” in G. Gouesbet, G. Gréhan, eds., Optical Particle Sizing: Theory and Practice (Plenum, New York, 1988), pp. 43–53, Figs. 4 and 5. [CrossRef]
  33. J. P. Barton, “Electromagnetic-field calculations for irregularly shaped, layered cylindrical particles with focused illumination,” Appl. Opt. 36, 1312–1319 (1997). [CrossRef] [PubMed]
  34. H.-B. Lin, J. D. Eversole, A. J. Campillo, J. P. Barton, “Excitation localization principle for spherical microcavities,” Opt. Lett. 23, 1921–1923 (1998). [CrossRef]
  35. J. P. Barton, “Effects of surface perturbations on the quality and the focused-beam excitation of microsphere resonance,” J. Opt. Soc. Am. A 16, 1974–1980 (1999). [CrossRef]
  36. Ref. 11, Table 10.13, p. 478.

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