OSA's Digital Library

Applied Optics

Applied Optics


  • Editor: Joseph N. Mait
  • Vol. 50, Iss. 28 — Oct. 1, 2011
  • pp: F50–F59

Zero-order bows in radially inhomogeneous spheres: direct and inverse problems

John A. Adam  »View Author Affiliations

Applied Optics, Vol. 50, Issue 28, pp. F50-F59 (2011)

View Full Text Article

Enhanced HTML    Acrobat PDF (440 KB)

Browse Journals / Lookup Meetings

Browse by Journal and Year


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools



Zero-order ray paths are examined in radially inhomogeneous spheres with differentiable refractive index profiles. It is demonstrated that zero-order and sometimes twin zero-order bows can exist when the gradient of refractive index is sufficiently negative. Abel inversion is used to “recover” the refractive index profiles; it is therefore possible in principle to specify the nature and type of bows and determine the refractive index profile that induces them. This may be of interest in the field of rainbow refractometry and optical fiber studies. This ray-theoretic analysis has direct similarities with the phenomenon of “orbiting” and other phenomena in scattering theory and also in seismological, surface gravity wave, and gravitational “lensing” studies. For completeness these topics are briefly discussed in the appendixes; they may also be of pedagogic interest.

© 2011 Optical Society of America

OCIS Codes
(000.3860) General : Mathematical methods in physics
(000.3870) General : Mathematics
(080.2720) Geometric optics : Mathematical methods (general)
(080.5692) Geometric optics : Ray trajectories in inhomogeneous media
(110.3200) Imaging systems : Inverse scattering
(260.2710) Physical optics : Inhomogeneous optical media

Original Manuscript: April 7, 2011
Revised Manuscript: July 25, 2011
Manuscript Accepted: July 26, 2011
Published: September 6, 2011

John A. Adam, "Zero-order bows in radially inhomogeneous spheres: direct and inverse problems," Appl. Opt. 50, F50-F59 (2011)

Sort:  Author  |  Year  |  Journal  |  Reset  


  1. J. A. Lock and T. A. McCollum, “Further thoughts on Newton’s zero-order rainbow,” Am. J. Phys. 62, 1082–1089 (1994). [CrossRef]
  2. J. A. Adam and P. Laven, “Rainbows from inhomogeneous transparent spheres: a ray-theoretic approach,” Appl. Opt. 46, 922–929 (2007). [CrossRef] [PubMed]
  3. M. Deutsch and I. Beniaminy, “Derivative-free inversion of Abel’s integral equation,” Appl. Phys. Lett. 41, 27–28(1982). [CrossRef]
  4. M. Deutsch and I. Beniaminy, “Inversion of Abel’s integral equation for experimental data,” J. Appl. Phys. 54, 137–143(1983). [CrossRef]
  5. M. Deutsch, “Abel inversion with a simple analytic representation for experimental data,” Appl. Phys. Lett. 42, 237–239(1983). [CrossRef]
  6. M. Deutsch, A. Notea, and D. Pal, “Reconstruction of discontinuous density profiles of cylindrically symmetric objects from single x-ray projections,” Appl. Opt. 27, 3962–3964(1988). [CrossRef] [PubMed]
  7. M. Deutsch, A. Notea, and D. Pal, “Abel reconstruction of piecewise constant radial density profiles from x-ray radiographs,” Appl. Opt. 28, 3183–3186 (1989). [CrossRef] [PubMed]
  8. M. Deutsch, A. Notea, and D. Pal, “Inversion of Abel’s integral equation and its application to NDT by x-ray radiography,” NDT Int. 23, 32–38 (1990). [CrossRef]
  9. M. A. Sharaf, A. A. Sharaf, and H. Selim, “Analytical solution of Abel’s equation for stellar density in globular clusters,” Rom. Astr. J. 14, 107–114 (2004).
  10. V. R. Eshleman, E. M. Gurrola, and G. F. Lindal, “On the black hole lens and its foci,” Adv. Space Res. 9, 119–122 (1989). [CrossRef]
  11. M. Kerker and E. Matijevic, “Scattering of electromagnetic waves from concentric infinite cylinders,” J. Opt. Soc. Am. 51, 506–508 (1961). [CrossRef]
  12. J. L. Lundberg, “Light scattering from large fibers at normal incidence,” J. Colloid Interface Sci. 29, 565–583 (1969). [CrossRef]
  13. H. M. Presby, “Refractive index and diameter measurements of unclad optical fibers,” J. Opt. Soc. Am. 64, 280–284(1974). [CrossRef]
  14. L. S. Watkins, “Scattering from side-illuminated clad glass fibers for determination of fiber parameters,” J. Opt. Soc. Am. 64, 767–772 (1974). [CrossRef]
  15. D. Marcuse and H. M. Presby, “Light scattering from optical fibers with arbitrary refractive-index distributions,” J. Opt. Soc. Am. 65, 367–375 (1975). [CrossRef]
  16. J. W. Y. Lit, “Radius of uncladded optical fiber from back-scattered radiation pattern,” J. Opt. Soc. Am. 65, 1311–1315(1975). [CrossRef]
  17. D. Marcuse, “Light scattering from unclad fibers: ray theory,” Appl. Opt. 14, 1528–1532 (1975). [CrossRef] [PubMed]
  18. P. L. Chu, “Determination of the diameter of unclad optical fibre,” Electron. Lett. 12, 14–16 (1976). [CrossRef]
  19. P. L. Chu, “Determination of diameters and refractive indices of step-index optical fibres,” Electron. Lett. 12, 155–157(1976). [CrossRef]
  20. P. L. Chu, “Nondestructive measurement of index profile of an optical-fibre preform,” Electron. Lett. 13, 736–738 (1977). [CrossRef]
  21. C. Saekeang and P. L. Chu, “Backscattering of light from optical fibers with arbitrary refractive index distributions: uniform approximation approach,” J. Opt. Soc. Am. 68, 1298–1305 (1978). [CrossRef]
  22. D. Marcuse, “Refractive index determination by the focusing method,” Appl. Opt. 18, 9–13 (1979). [CrossRef] [PubMed]
  23. D. Marcuse and H. M. Presby, “Focusing method for nondestructive measurement of optical fiber index profiles,” Appl. Opt. 18, 14–22 (1979). [CrossRef] [PubMed]
  24. H. M. Presby and D. Marcuse, “Optical fiber preform Diagnostics,” Appl. Opt. 18, 23–30 (1979). [CrossRef] [PubMed]
  25. C. Saekeang and P. L. Chu, “Nondestructive determination of refractive index profile of an optical fiber: backward light scattering method,” Appl. Opt. 18, 1110–1116 (1979). [CrossRef] [PubMed]
  26. R. A. Phinney and D. L. Anderson, “On the radio occultation method for studying planetary atmospheres,” J. Geophys. Res. 73, 1819–1827 (1968). [CrossRef]
  27. G. Fjeldbo, A. J. Kliore, and V. R. Eshleman, “The neutral atmosphere of Venus as studied with the Mariner V radio occultation experiments,” Astron. J. 76, 123–140 (1971). [CrossRef]
  28. V. R. Eshleman, “The radio occultation method for the study of planetary atmospheres,” Planet. Space Sci. 21, 1521–1531(1973). [CrossRef]
  29. S. B. Healy, J. Haase, and O. Lesne, “Abel transform inversion of radio occultation measurements made with a receiver inside the Earth’s atmosphere,” Ann. Geophys. 20, 1253–1256(2002). [CrossRef]
  30. G. A. Hajj, E. R. Kursinski, L. J. Romans, W. I. Bertiger, and S. S. Leroy, “A technical description of atmospheric sounding by GPS occultation,” J. Atmos. Sol. Terr. Phys. 64, 451–469(2002). [CrossRef]
  31. P. Guo, H-J. Yan, Z-J. Hong, M. Liu and C. Huang, “On the singular points of the Abelian integral transformation in the GPS/LEO occultation technique,” Chinese Astron. Astrophys. 28, 441–448 (2004). [CrossRef]
  32. F. Xie, J. S. Haase, and S. Syndergaard, “Profiling the atmosphere using the airborne GPS radio occultation technique: a sensitivity study,” IEEE Trans. Geosci. Remote Sens. 46, 3424–3435 (2008). [CrossRef]
  33. S. C. Solomon, P. B. Hays, and V. J. Abreu, “Tomographic inversion of satellite photometry,” Appl. Opt. 23, 3409–3414(1984). [CrossRef] [PubMed]
  34. K. Bockasten, “Transformation of observed radiances into radial distribution of the emission of a plasma,” J. Opt. Soc. Am. 51, 943–947 (1961). [CrossRef]
  35. W. L. Barr, “Method for computing the radial distribution of emitters in a cylindrical source,” J. Opt. Soc. Am. 52, 885–888 (1962). [CrossRef]
  36. P. W. Schreiber, A. M. Hunter, and D. R. Smith, Jr., “The determination of plasma electron density from refraction measurements,” Plasma Phys. 15, 635–646 (1973). [CrossRef]
  37. C. J. Tallents, M. D. J. Burgess, and B. Luther-Davies, “The determination of electron density profiles from refraction measurements obtained using holographic interferometry,” Opt. Commun. 44, 384–387 (1983). [CrossRef]
  38. K. E. Bullen, Introduction to the Theory of Seismology, 3rd ed. (Cambridge University, 1965).
  39. F. D. Stacey, Physics of the Earth, 2nd ed. (Wiley, 1977).
  40. C. B. Officer, Introduction to Theoretical Geophysics (Springer-Verlag, 1974).
  41. C. L. Brockman and N. G. Alexopoulos, “Geometrical optics of inhomogeneous particles: glory ray and the rainbow revisited,” Appl. Opt. 16, 166–174 (1977). [CrossRef] [PubMed]
  42. M. Marklund, D. Anderson, F. Cattani, M. Lisak, and L. Lundgren, “Fermat’s principle and the variational analysis of an optical model for light propagation exhibiting a critical radius,” Am. J. Phys. 70, 680–683 (2002). [CrossRef]
  43. M. R. Vetrano, J. P. van Beeck, and M. Riethmuller, “Generalization of the rainbow Airy theory to nonuniform spheres,” Opt. Lett. 30, 658–660 (2005). [CrossRef] [PubMed]
  44. C. M. Vest, “Tomography for properties of materials that bend rays: a tutorial,” Appl. Opt. 24, 4089–4094(1985). [CrossRef] [PubMed]
  45. W. Glantschnig, “How accurately can one reconstruct an index profile from transverse measurement data?” J. Lightwave Technol. 3, 678–683 (1985). [CrossRef]
  46. M. R. Vetrano, J. P. van Beeck, and M. L. Riethmuller, “Assessment of refractive index gradients by standard rainbow thermometry,” Appl. Opt. 44, 7275–7281 (2005). [CrossRef] [PubMed]
  47. X. Li, X. Han, R. Li, and H. Jiang, “Geometrical-optics approximation of forward scattering by gradient-index spheres,” Appl. Opt. 46, 5241–5247 (2007). [CrossRef] [PubMed]
  48. R. G. Newton, Scattering Theory of Waves and Particles (Springer-Verlag, 1982).
  49. H. M. Nussenzveig, Diffraction Effects in Semiclassical Scattering (Cambridge University, 1992). [CrossRef]
  50. W. T. Grandy, Jr., Scattering of Waves from Large Spheres (Cambridge University, 2000). [CrossRef]
  51. D. Drosdoff and A. Widom, “Snell’s law from an elementary particle viewpoint,” Am. J. Phys. 73, 973–975 (2005). [CrossRef]
  52. M. Berry, Principles of Cosmology and Gravitation(Cambridge University, 1976).
  53. C. C. Mei, The Applied Dynamics of Ocean Surface Waves (World Scientific, 1989).
  54. J. A. Adam, Mathematics in Nature: Modeling Patterns in the Natural World (Princeton University, 2006).

Cited By

Alert me when this paper is cited

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited