Abstract

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

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  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).

2008 (1)

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]

2007 (2)

2005 (3)

2004 (2)

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).

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]

2002 (3)

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]

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]

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]

1994 (1)

J. A. Lock and T. A. McCollum, “Further thoughts on Newton’s zero-order rainbow,” Am. J. Phys. 62, 1082–1089 (1994).
[CrossRef]

1990 (1)

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]

1989 (2)

1988 (1)

1985 (2)

C. M. Vest, “Tomography for properties of materials that bend rays: a tutorial,” Appl. Opt. 24, 4089–4094(1985).
[CrossRef] [PubMed]

W. Glantschnig, “How accurately can one reconstruct an index profile from transverse measurement data?” J. Lightwave Technol. 3, 678–683 (1985).
[CrossRef]

1984 (1)

1983 (3)

M. Deutsch and I. Beniaminy, “Inversion of Abel’s integral equation for experimental data,” J. Appl. Phys. 54, 137–143(1983).
[CrossRef]

M. Deutsch, “Abel inversion with a simple analytic representation for experimental data,” Appl. Phys. Lett. 42, 237–239(1983).
[CrossRef]

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]

1982 (1)

M. Deutsch and I. Beniaminy, “Derivative-free inversion of Abel’s integral equation,” Appl. Phys. Lett. 41, 27–28(1982).
[CrossRef]

1979 (4)

1978 (1)

1977 (2)

P. L. Chu, “Nondestructive measurement of index profile of an optical-fibre preform,” Electron. Lett. 13, 736–738 (1977).
[CrossRef]

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]

1976 (2)

P. L. Chu, “Determination of the diameter of unclad optical fibre,” Electron. Lett. 12, 14–16 (1976).
[CrossRef]

P. L. Chu, “Determination of diameters and refractive indices of step-index optical fibres,” Electron. Lett. 12, 155–157(1976).
[CrossRef]

1975 (3)

1974 (2)

1973 (2)

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]

V. R. Eshleman, “The radio occultation method for the study of planetary atmospheres,” Planet. Space Sci. 21, 1521–1531(1973).
[CrossRef]

1971 (1)

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]

1969 (1)

J. L. Lundberg, “Light scattering from large fibers at normal incidence,” J. Colloid Interface Sci. 29, 565–583 (1969).
[CrossRef]

1968 (1)

R. A. Phinney and D. L. Anderson, “On the radio occultation method for studying planetary atmospheres,” J. Geophys. Res. 73, 1819–1827 (1968).
[CrossRef]

1962 (1)

1961 (2)

Abreu, V. J.

Adam, J. A.

Alexopoulos, N. G.

Anderson, D.

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]

Anderson, D. L.

R. A. Phinney and D. L. Anderson, “On the radio occultation method for studying planetary atmospheres,” J. Geophys. Res. 73, 1819–1827 (1968).
[CrossRef]

Barr, W. L.

Beniaminy, I.

M. Deutsch and I. Beniaminy, “Inversion of Abel’s integral equation for experimental data,” J. Appl. Phys. 54, 137–143(1983).
[CrossRef]

M. Deutsch and I. Beniaminy, “Derivative-free inversion of Abel’s integral equation,” Appl. Phys. Lett. 41, 27–28(1982).
[CrossRef]

Berry, M.

M. Berry, Principles of Cosmology and Gravitation(Cambridge University, 1976).

Bertiger, W. I.

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]

Bockasten, K.

Brockman, C. L.

Bullen, K. E.

K. E. Bullen, Introduction to the Theory of Seismology, 3rd ed. (Cambridge University, 1965).

Burgess, M. D. J.

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]

Cattani, F.

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]

Chu, P. L.

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]

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]

P. L. Chu, “Nondestructive measurement of index profile of an optical-fibre preform,” Electron. Lett. 13, 736–738 (1977).
[CrossRef]

P. L. Chu, “Determination of the diameter of unclad optical fibre,” Electron. Lett. 12, 14–16 (1976).
[CrossRef]

P. L. Chu, “Determination of diameters and refractive indices of step-index optical fibres,” Electron. Lett. 12, 155–157(1976).
[CrossRef]

Deutsch, M.

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]

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]

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]

M. Deutsch and I. Beniaminy, “Inversion of Abel’s integral equation for experimental data,” J. Appl. Phys. 54, 137–143(1983).
[CrossRef]

M. Deutsch, “Abel inversion with a simple analytic representation for experimental data,” Appl. Phys. Lett. 42, 237–239(1983).
[CrossRef]

M. Deutsch and I. Beniaminy, “Derivative-free inversion of Abel’s integral equation,” Appl. Phys. Lett. 41, 27–28(1982).
[CrossRef]

Drosdoff, D.

D. Drosdoff and A. Widom, “Snell’s law from an elementary particle viewpoint,” Am. J. Phys. 73, 973–975 (2005).
[CrossRef]

Eshleman, V. R.

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]

V. R. Eshleman, “The radio occultation method for the study of planetary atmospheres,” Planet. Space Sci. 21, 1521–1531(1973).
[CrossRef]

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]

Fjeldbo, G.

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]

Glantschnig, W.

W. Glantschnig, “How accurately can one reconstruct an index profile from transverse measurement data?” J. Lightwave Technol. 3, 678–683 (1985).
[CrossRef]

Grandy, W. T.

W. T. Grandy, Jr., Scattering of Waves from Large Spheres (Cambridge University, 2000).
[CrossRef]

Guo, P.

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]

Gurrola, E. M.

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]

Haase, J.

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]

Haase, J. S.

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]

Hajj, G. A.

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]

Han, X.

Hays, P. B.

Healy, S. B.

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]

Hong, Z-J.

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]

Huang, C.

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]

Hunter, A. M.

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]

Jiang, H.

Kerker, M.

Kliore, A. J.

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]

Kursinski, E. R.

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]

Laven, P.

Leroy, S. S.

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]

Lesne, O.

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]

Li, R.

Li, X.

Lindal, G. F.

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]

Lisak, M.

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]

Lit, J. W. Y.

Liu, M.

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]

Lock, J. A.

J. A. Lock and T. A. McCollum, “Further thoughts on Newton’s zero-order rainbow,” Am. J. Phys. 62, 1082–1089 (1994).
[CrossRef]

Lundberg, J. L.

J. L. Lundberg, “Light scattering from large fibers at normal incidence,” J. Colloid Interface Sci. 29, 565–583 (1969).
[CrossRef]

Lundgren, L.

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]

Luther-Davies, B.

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]

Marcuse, D.

Marklund, M.

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]

Matijevic, E.

McCollum, T. A.

J. A. Lock and T. A. McCollum, “Further thoughts on Newton’s zero-order rainbow,” Am. J. Phys. 62, 1082–1089 (1994).
[CrossRef]

Mei, C. C.

C. C. Mei, The Applied Dynamics of Ocean Surface Waves (World Scientific, 1989).

Newton, R. G.

R. G. Newton, Scattering Theory of Waves and Particles (Springer-Verlag, 1982).

Notea, A.

Nussenzveig, H. M.

H. M. Nussenzveig, Diffraction Effects in Semiclassical Scattering (Cambridge University, 1992).
[CrossRef]

Officer, C. B.

C. B. Officer, Introduction to Theoretical Geophysics (Springer-Verlag, 1974).

Pal, D.

Phinney, R. A.

R. A. Phinney and D. L. Anderson, “On the radio occultation method for studying planetary atmospheres,” J. Geophys. Res. 73, 1819–1827 (1968).
[CrossRef]

Presby, H. M.

Riethmuller, M.

Riethmuller, M. L.

Romans, L. J.

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]

Saekeang, C.

Schreiber, P. W.

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]

Selim, H.

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).

Sharaf, A. A.

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).

Sharaf, M. A.

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).

Smith, D. R.

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]

Solomon, S. C.

Stacey, F. D.

F. D. Stacey, Physics of the Earth, 2nd ed. (Wiley, 1977).

Syndergaard, S.

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]

Tallents, C. J.

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]

van Beeck, J. P.

Vest, C. M.

Vetrano, M. R.

Watkins, L. S.

Widom, A.

D. Drosdoff and A. Widom, “Snell’s law from an elementary particle viewpoint,” Am. J. Phys. 73, 973–975 (2005).
[CrossRef]

Xie, F.

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]

Yan, H-J.

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]

Adv. Space Res. (1)

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]

Am. J. Phys. (3)

J. A. Lock and T. A. McCollum, “Further thoughts on Newton’s zero-order rainbow,” Am. J. Phys. 62, 1082–1089 (1994).
[CrossRef]

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]

D. Drosdoff and A. Widom, “Snell’s law from an elementary particle viewpoint,” Am. J. Phys. 73, 973–975 (2005).
[CrossRef]

Ann. Geophys. (1)

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]

Appl. Opt. (13)

S. C. Solomon, P. B. Hays, and V. J. Abreu, “Tomographic inversion of satellite photometry,” Appl. Opt. 23, 3409–3414(1984).
[CrossRef] [PubMed]

D. Marcuse, “Refractive index determination by the focusing method,” Appl. Opt. 18, 9–13 (1979).
[CrossRef] [PubMed]

D. Marcuse and H. M. Presby, “Focusing method for nondestructive measurement of optical fiber index profiles,” Appl. Opt. 18, 14–22 (1979).
[CrossRef] [PubMed]

H. M. Presby and D. Marcuse, “Optical fiber preform Diagnostics,” Appl. Opt. 18, 23–30 (1979).
[CrossRef] [PubMed]

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]

J. A. Adam and P. Laven, “Rainbows from inhomogeneous transparent spheres: a ray-theoretic approach,” Appl. Opt. 46, 922–929 (2007).
[CrossRef] [PubMed]

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]

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]

D. Marcuse, “Light scattering from unclad fibers: ray theory,” Appl. Opt. 14, 1528–1532 (1975).
[CrossRef] [PubMed]

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]

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]

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]

C. M. Vest, “Tomography for properties of materials that bend rays: a tutorial,” Appl. Opt. 24, 4089–4094(1985).
[CrossRef] [PubMed]

Appl. Phys. Lett. (2)

M. Deutsch, “Abel inversion with a simple analytic representation for experimental data,” Appl. Phys. Lett. 42, 237–239(1983).
[CrossRef]

M. Deutsch and I. Beniaminy, “Derivative-free inversion of Abel’s integral equation,” Appl. Phys. Lett. 41, 27–28(1982).
[CrossRef]

Astron. J. (1)

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]

Chinese Astron. Astrophys. (1)

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]

Electron. Lett. (3)

P. L. Chu, “Determination of the diameter of unclad optical fibre,” Electron. Lett. 12, 14–16 (1976).
[CrossRef]

P. L. Chu, “Determination of diameters and refractive indices of step-index optical fibres,” Electron. Lett. 12, 155–157(1976).
[CrossRef]

P. L. Chu, “Nondestructive measurement of index profile of an optical-fibre preform,” Electron. Lett. 13, 736–738 (1977).
[CrossRef]

IEEE Trans. Geosci. Remote Sens. (1)

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]

J. Appl. Phys. (1)

M. Deutsch and I. Beniaminy, “Inversion of Abel’s integral equation for experimental data,” J. Appl. Phys. 54, 137–143(1983).
[CrossRef]

J. Atmos. Sol. Terr. Phys. (1)

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]

J. Colloid Interface Sci. (1)

J. L. Lundberg, “Light scattering from large fibers at normal incidence,” J. Colloid Interface Sci. 29, 565–583 (1969).
[CrossRef]

J. Geophys. Res. (1)

R. A. Phinney and D. L. Anderson, “On the radio occultation method for studying planetary atmospheres,” J. Geophys. Res. 73, 1819–1827 (1968).
[CrossRef]

J. Lightwave Technol. (1)

W. Glantschnig, “How accurately can one reconstruct an index profile from transverse measurement data?” J. Lightwave Technol. 3, 678–683 (1985).
[CrossRef]

J. Opt. Soc. Am. (8)

NDT Int. (1)

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]

Opt. Commun. (1)

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]

Opt. Lett. (1)

Planet. Space Sci. (1)

V. R. Eshleman, “The radio occultation method for the study of planetary atmospheres,” Planet. Space Sci. 21, 1521–1531(1973).
[CrossRef]

Plasma Phys. (1)

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]

Rom. Astr. J. (1)

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).

Other (9)

K. E. Bullen, Introduction to the Theory of Seismology, 3rd ed. (Cambridge University, 1965).

F. D. Stacey, Physics of the Earth, 2nd ed. (Wiley, 1977).

C. B. Officer, Introduction to Theoretical Geophysics (Springer-Verlag, 1974).

R. G. Newton, Scattering Theory of Waves and Particles (Springer-Verlag, 1982).

H. M. Nussenzveig, Diffraction Effects in Semiclassical Scattering (Cambridge University, 1992).
[CrossRef]

W. T. Grandy, Jr., Scattering of Waves from Large Spheres (Cambridge University, 2000).
[CrossRef]

M. Berry, Principles of Cosmology and Gravitation(Cambridge University, 1976).

C. C. Mei, The Applied Dynamics of Ocean Surface Waves (World Scientific, 1989).

J. A. Adam, Mathematics in Nature: Modeling Patterns in the Natural World (Princeton University, 2006).

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Figures (12)

Fig. 1
Fig. 1

Ray path for direct transmission through a radially inhomogeneous sphere for n ( r ) < 0 .

Fig. 2
Fig. 2

Deviation functions for both a homogeneous ( D h ) and inhomogeneous spheres ( D 0 and D 1 ) for the profile (inset)  n 1 ( r ) .

Fig. 3
Fig. 3

Deviation functions ( D 0 and D 1 ) for an inhomogeneous sphere for the profile (inset)  n 2 ( r ) .

Fig. 4
Fig. 4

Deviation function for a primary bow ( D 1 ) from an inhomogeneous sphere with the profile (inset)  n 3 ( r ) .

Fig. 5
Fig. 5

Ray paths for a refractive index profile n ( r ) = n ( 1 ) r m , m = 0.75 here (see [41]).

Fig. 6
Fig. 6

Typical ray paths with corresponding points r c i of closest approach to the center O (only details for ray 3 are shown for clarity).

Fig. 7
Fig. 7

Refractive index profile n ( r ) = ( 5 r ) / 3 and its Abel inversion.

Fig. 8
Fig. 8

Refractive index profile n ( r ) = 1.3 0.2 cos { [ 1.9 ( r 0.85 ) ] 2 } and its Abel inversion.

Fig. 9
Fig. 9

Within-sphere angular deviation Δ ( p ) for the profile Eq. (24) (solid line) and a constant profile (dashed line).

Fig. 10
Fig. 10

η ( r ) = r n ( r ) for the monotonic case. The point of closest approach is r = r c .

Fig. 11
Fig. 11

η ( r ) = r n ( r ) for the nonmonotonic case. The point of closest approach for i > i 2 is r = r c , and a zone of width Δ r exists into which no ray penetrates.

Fig. 12
Fig. 12

Phenomenon of orbiting illustrated schematically associated with a zero of η ( r ) .

Equations (44)

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r n ( r ) sin ϕ = constant ,
sin ϕ = r ( θ ) r 2 ( θ ) + ( d r / d θ ) 2 .
Θ ( i ) = sin i r c ( i ) 1 d r r r 2 n 2 ( r ) sin 2 i .
η ( r c ( i ) ) r c ( i ) n ( r c ( i ) ) = sin i .
δ n ( r ) d s = 0 , i.e.   , δ n ( r ) [ 1 + r 2 ( d θ d r ) 2 ] 1 / 2 d r = 0 ,
d d r { n ( r ) r 2 ( d θ / d r ) [ 1 + r 2 ( d θ / d r ) 2 ] 1 / 2 } = 0 ,
i + 2 Θ + ( i | D 0 ( i ) | ) = π | D 0 ( i ) | = 2 i π + 2 Θ .
| D 1 ( i ) | = 2 i π + 4 Θ .
D 0 ( i ) = 2 i 2 r ˜ ( i ) ,
D 1 ( i ) = 2 i + π 4 r ˜ ( i ) ,
n 1 ( r ) = 1.3 0.2 cos { [ 1.9 ( r 0.85 ) ] 2 } .
n 2 ( r ) = 1.1 { 1 [ 0.55 ( 1 r ) ] 2 } 1
n 3 ( r ) = 1.33 + 0.23 cos { [ 1.8 ( r 0.1 ) ] 2 } ,
Δ ( p ) = 2 r c ( i ) 1 p r d r ( η 2 p 2 ) 1 / 2 = 2 p η 0 p r 1 ( η 2 p 2 ) 1 / 2 d r d η d η ,
η i η 0 Δ ( p ) ( p 2 η i 2 ) 1 / 2 d p = η i η 0 d p p η 0 2 p r { 1 ( p 2 η i 2 ) 1 / 2 ( η 2 p 2 ) 1 / 2 } d r d η d η .
η i η 0 d η η i η 2 p r ( p 2 η i 2 ) 1 / 2 ( η 2 p 2 ) 1 / 2 d r d η d p .
η i η p r ( p 2 η i 2 ) 1 / 2 ( η 2 p 2 ) 1 / 2 d p = [ arcsin ( p 2 η i 2 η 2 η i 2 ) ] p = η i p = η = π 2 ,
I ( η i ) η i η 0 Δ ( p ) ( p 2 η i 2 ) 1 / 2 d p = η i η 0 π r d r d η d η = π [ ln r ] r c ( i ) 1 = π ln r i ( i ) .
I ( η i ) = [ Δ ( p ) arccosh ( p η i ) ] η i η 0 η i η 0 arccosh ( p η i ) d Δ d p d p .
I ( η i ) = η i η 0 arccosh ( p η i ) d Δ d p d p = Δ ( η 0 ) Δ ( η i ) arccosh ( p η i ) d Δ = π ln r i ( i ) .
r i ( η i ( i ) ) = exp [ I ( η i ) π ] .
n ( r ( η i ) ) = η i r i ( η i ) = η i exp [ I ( η i ) / π ] .
n ( r ) = r exp [ I ( r ) / π ] .
I ( η i ) η i η 0 Δ ( p ) ( p 2 η i 2 ) 1 / 2 d p .
n ( r ) n 1 ( r ) = 1.3 0.2 cos { [ 1.9 ( r 0.85 ) ] 2 } .
Δ ( p ) = D 0 ( arcsin p ) 2 arcsin p + π .
n ( r ) = 4 r 3 .
Δ 0 l ( p ) = 2 p r c ( p ) 1 d r r ( r 2 n 2 ( r ) p 2 ) 1 / 2 .
r i ( p ) = 2 [ 1 ( 1 3 p 4 ) 1 / 2 ] .
r 2 n 2 ( r ) K 2 = r c 2 n 2 ( r c ) K 2 + d d r [ r 2 n 2 ( r ) ] r c ( r r c ) + 1 2 d 2 d r 2 [ r 2 n 2 ( r ) ] r c ( r r c ) 2 + O ( ( r r c ) 3 ) .
I ( r r c ) 1 / 2 d r ( r r c ) 1 / 2 0 ,
I | r r c | 1 d r ln | r r c | ,
n ( r c ) = n ( r c ) r c < 0 ,
θ = π 2 b a d r r 2 [ 1 b 2 / r 2 V ( r ) / E ] 1 / 2 ,
n ( r ) = [ 1 V ( r ) E ] 1 / 2 ,
n ( r ) = V ( r ) 2 [ 1 V ( r ) / E ] ,
n ( r ¯ ) = 1 2 V ( r ¯ ) c 2 + ,
θ = π 2 Θ π 2 b a d r r 2 [ 1 + ( 2 R s / r ) ( b 2 / r 2 ) ] ,
( i )     cot ( θ 2 ) = b R s ,
d σ d Ω = | b d b sin θ d θ | = R s 2 4 sin 4 ( θ / 2 ) ,
θ = 2 R s b = 4 GM c 2 b . ,
T = 2 r i r 0 η 2 r ( η 2 p 2 ) 1 / 2 d r = 2 r i r 0 { p 2 r 1 ( η 2 p 2 ) 1 / 2 + r 1 ( η 2 p 2 ) 1 / 2 } d r ,
= p Δ + 2 r i r 0 r 1 ( η 2 p 2 ) 1 / 2 d r .
ω = ( g k tanh k h ) 1 / 2 ,

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