Abstract

Semiclassical scattering phenomena appearing in the far-zone scattered intensity of a point source of electromagnetic radiation inside a spherical particle are examined in the context of both ray theory and wave theory, and the evolution of the phenomena is studied as a function of source position. A number of semiclassical effects that do not occur for plane-wave scattering by the sphere appear prominently for scattering by an interior source. These include a series of scattering resonances and a new family of rainbows in regions of otherwise total internal reflection. Diffractive effects accompanying the semiclassical phenomena are also examined.

© 2001 Optical Society of America

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References

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  1. H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981), pp. 119–126.
  2. M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969), pp. 39–49.
  3. C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), pp. 82–101.
  4. 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., Selected Papers on Geometrical Aspects of Scattering (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1994), pp. 198–204.
  5. B. van der Pol, H. Bremmer, “The diffraction of electromagnetic waves from an electrical point source round a finitely conducting sphere, with applications to radiotelegraphy and the theory of the rainbow,” Philos. Mag. 24, 141–176, 825–864 (1937).
  6. Ref. 1, pp. 210–214.
  7. H. M. Nussenzveig, “High-frequency scattering by a transparent sphere. 1. Direct reflection and transmission,” J. Math. Phys. 10, 82–124 (1969).
    [CrossRef]
  8. H. M. Nussenzveig, “High-frequency scattering by a transparent sphere. 2. Theory of the rainbow and the glory,” J. Math. Phys. 10, 125–176 (1969).
    [CrossRef]
  9. H. M. Nussenzveig, Diffraction Effects in Semiclassical Scattering (Cambridge U. Press, Cambridge, UK, 1992), pp. 45–51.
  10. H. Chew, P. J. McNulty, M. Kerker, “Model for Raman and fluorescent scattering by molecules embedded in small particles,” Phys. Rev. A 13, 396–404 (1976).
    [CrossRef]
  11. H. Chew, “Transition rates of atoms near spherical surfaces,” J. Chem. Phys. 87, 1355–1360 (1987).
    [CrossRef]
  12. S. D. Druger, S. Arnold, L. M. Folan, “Theory of enhanced energy transfer between molecules embedded in spherical dielectric particles,” J. Chem. Phys. 87, 2649–2659 (1987).
    [CrossRef]
  13. S. C. Hill, H. I. Saleheen, K. A. Fuller, “Volume current method for modeling light scattering by inhomogeneously perturbed spheres,” J. Opt. Soc. Am. A 12, 905–915 (1995).
    [CrossRef]
  14. S. C. Hill, H. I. Saleheen, M. D. Barnes, W. B. Whitten, J. M. Ramsey, “Modeling fluorescence collection from single molecules in microspheres: effects of position, orientation, and frequency,” Appl. Opt. 35, 6278–6288 (1996).
    [CrossRef] [PubMed]
  15. S. C. Hill, G. Videen, J. D. Pendleton, “Reciprocity method for obtaining the far fields generated by a source inside or near a microparticle,” J. Opt. Soc. Am. B 14, 2522–2529 (1997).
    [CrossRef]
  16. J. D. Pendleton, S. C. Hill, “Collection of emission from an oscillating dipole inside a sphere: analytical integration over a circular aperture,” Appl. Opt. 36, 8729–8737 (1997).
    [CrossRef]
  17. M. Kerker, S. D. Druger, “Raman and fluorescent scattering by molecules embedded in spheres with radii up to several multiples of the wavelength,” Appl. Opt. 18, 1172–1179 (1979).
    [CrossRef] [PubMed]
  18. J. Zhang, D. R. Alexander, “Hybrid inelastic-scattering models for particle thermometry: unpolarized emissions,” Appl. Opt. 31, 7132–7139 (1992).
    [CrossRef] [PubMed]
  19. J. Zhang, D. R. Alexander, “Hybrid inelastic-scattering models for particle thermometry: polarized emissions,” Appl. Opt. 31, 7140–7146 (1992).
    [CrossRef] [PubMed]
  20. J. Zhang, L. A. Melton, “Numerical simulations and restorations of laser droplet-slicing images,” Appl. Opt. 33, 192–200 (1994).
    [CrossRef] [PubMed]
  21. N. Velesco, G. Schweiger, “Geometrical optics calculation of inelastic scattering on large particles,” Appl. Opt. 38, 1046–1052 (1999).
    [CrossRef]
  22. R. Domann, Y. Hardalupas, “Spatial distribution of fluorescence intensity within large droplets and its dependence on dye concentration,” Appl. Opt. 40, 3586–3597 (2001).
    [CrossRef]
  23. P. L. Marston, “Geometrical and catastrophe optics methods in scattering,” Phys. Acoust. 21, 1–234 (1992), p. 187, Fig. 76.
  24. D. S. Langley, M. J. Morrell, “Rainbow-enhanced forward and backward glory scattering,” Appl. Opt. 30, 3459–3467 (1991).
    [CrossRef] [PubMed]
  25. C. F. Bohren, A. B. Fraser, “Newton’s zero-order rainbow: unobservable or nonexistent?” Am. J. Phys. 59, 325–326 (1991).
    [CrossRef]
  26. J. A. Lock, T. A. McCollum, “Further thoughts on Newton’s zero-order rainbow,” Am. J. Phys. 62, 1082–1089 (1994).
    [CrossRef]
  27. J. A. Lock, E. A. Hovenac, “Internal caustic structure of illuminated liquid droplets,” J. Opt. Soc. Am. A 8, 1541–1552 (1991).
    [CrossRef]
  28. S. C. Hill, M. D. Barnes, W. B. Whitten, J. M. Ramsey, “Collection of fluorescence from single molecules in microspheres: effects of illumination geometry,” Appl. Opt. 36, 4425–4437 (1997).
    [CrossRef] [PubMed]
  29. 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]
  30. H. M. Nussenzveig, “Complex angular momentum theory of the rainbow and the glory,” J. Opt. Soc. Am. 69, 1068–1079, 1193–1194 (1979).
    [CrossRef]
  31. 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]
  32. C. M. Mount, D. B. Theissen, P. L. Marston, “Scattering observations for tilted transparent fibers: evolution of the Airy caustics with cylinder tilt and the caustic merging transition,” Appl. Opt. 37, 1534–1539 (1998).
    [CrossRef]
  33. D. Q. Chowdhury, D. H. Leach, R. K. Chang, “Effect ofthe Goos–Hanchen shift on the geometrical-optics model for spherical-cavity mode spacing,” J. Opt. Soc. Am. A 11, 1110–1116 (1994).
    [CrossRef]
  34. N. H. Tran, L. Dutriaux, Ph. Balcou, A. Le Floch, F. Bretenaker, “Angular Goos–Hanchen effect in curved dielectric microstructures,” Opt. Lett. 20, 1233–1235 (1995).
    [CrossRef] [PubMed]
  35. S. C. Hill, P. Nachman, S. Arnold, J. M. Ramsey, M. D. Barnes, “Fluorescence image of a single molecule in a microscope: model,” J. Opt. Soc. Am. B 16, 1868–1873 (1999).
    [CrossRef]
  36. Ref. 35, footnote 30.
  37. S. C. Hill, V. Boutou, J. Yu, S. Ramstein, J.-P. Wolf, Y.-L. Pan, S. Holler, R. K. Chang, “Enhanced backward-directed multiphoton-excited fluorescence from dielectric microcavities,” Phys. Rev. Lett. 85, 54–57 (2000).
    [CrossRef] [PubMed]

2001 (1)

2000 (1)

S. C. Hill, V. Boutou, J. Yu, S. Ramstein, J.-P. Wolf, Y.-L. Pan, S. Holler, R. K. Chang, “Enhanced backward-directed multiphoton-excited fluorescence from dielectric microcavities,” Phys. Rev. Lett. 85, 54–57 (2000).
[CrossRef] [PubMed]

1999 (2)

1998 (1)

1997 (4)

1996 (1)

1995 (2)

1994 (3)

1992 (4)

1991 (3)

1987 (2)

H. Chew, “Transition rates of atoms near spherical surfaces,” J. Chem. Phys. 87, 1355–1360 (1987).
[CrossRef]

S. D. Druger, S. Arnold, L. M. Folan, “Theory of enhanced energy transfer between molecules embedded in spherical dielectric particles,” J. Chem. Phys. 87, 2649–2659 (1987).
[CrossRef]

1979 (2)

1976 (1)

H. Chew, P. J. McNulty, M. Kerker, “Model for Raman and fluorescent scattering by molecules embedded in small particles,” Phys. Rev. A 13, 396–404 (1976).
[CrossRef]

1969 (2)

H. M. Nussenzveig, “High-frequency scattering by a transparent sphere. 1. Direct reflection and transmission,” J. Math. Phys. 10, 82–124 (1969).
[CrossRef]

H. M. Nussenzveig, “High-frequency scattering by a transparent sphere. 2. Theory of the rainbow and the glory,” J. Math. Phys. 10, 125–176 (1969).
[CrossRef]

1937 (1)

B. van der Pol, H. Bremmer, “The diffraction of electromagnetic waves from an electrical point source round a finitely conducting sphere, with applications to radiotelegraphy and the theory of the rainbow,” Philos. Mag. 24, 141–176, 825–864 (1937).

Adler, C. L.

Alexander, D. R.

Arnold, S.

S. C. Hill, P. Nachman, S. Arnold, J. M. Ramsey, M. D. Barnes, “Fluorescence image of a single molecule in a microscope: model,” J. Opt. Soc. Am. B 16, 1868–1873 (1999).
[CrossRef]

S. D. Druger, S. Arnold, L. M. Folan, “Theory of enhanced energy transfer between molecules embedded in spherical dielectric particles,” J. Chem. Phys. 87, 2649–2659 (1987).
[CrossRef]

Balcou, Ph.

Barnes, M. D.

Bohren, C. F.

C. F. Bohren, A. B. Fraser, “Newton’s zero-order rainbow: unobservable or nonexistent?” Am. J. Phys. 59, 325–326 (1991).
[CrossRef]

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), pp. 82–101.

Boutou, V.

S. C. Hill, V. Boutou, J. Yu, S. Ramstein, J.-P. Wolf, Y.-L. Pan, S. Holler, R. K. Chang, “Enhanced backward-directed multiphoton-excited fluorescence from dielectric microcavities,” Phys. Rev. Lett. 85, 54–57 (2000).
[CrossRef] [PubMed]

Bremmer, H.

B. van der Pol, H. Bremmer, “The diffraction of electromagnetic waves from an electrical point source round a finitely conducting sphere, with applications to radiotelegraphy and the theory of the rainbow,” Philos. Mag. 24, 141–176, 825–864 (1937).

Bretenaker, F.

Chang, R. K.

S. C. Hill, V. Boutou, J. Yu, S. Ramstein, J.-P. Wolf, Y.-L. Pan, S. Holler, R. K. Chang, “Enhanced backward-directed multiphoton-excited fluorescence from dielectric microcavities,” Phys. Rev. Lett. 85, 54–57 (2000).
[CrossRef] [PubMed]

D. Q. Chowdhury, D. H. Leach, R. K. Chang, “Effect ofthe Goos–Hanchen shift on the geometrical-optics model for spherical-cavity mode spacing,” J. Opt. Soc. Am. A 11, 1110–1116 (1994).
[CrossRef]

Chew, H.

H. Chew, “Transition rates of atoms near spherical surfaces,” J. Chem. Phys. 87, 1355–1360 (1987).
[CrossRef]

H. Chew, P. J. McNulty, M. Kerker, “Model for Raman and fluorescent scattering by molecules embedded in small particles,” Phys. Rev. A 13, 396–404 (1976).
[CrossRef]

Chowdhury, D. Q.

Debye, P.

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., Selected Papers on Geometrical Aspects of Scattering (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1994), pp. 198–204.

Domann, R.

Druger, S. D.

S. D. Druger, S. Arnold, L. M. Folan, “Theory of enhanced energy transfer between molecules embedded in spherical dielectric particles,” J. Chem. Phys. 87, 2649–2659 (1987).
[CrossRef]

M. Kerker, S. D. Druger, “Raman and fluorescent scattering by molecules embedded in spheres with radii up to several multiples of the wavelength,” Appl. Opt. 18, 1172–1179 (1979).
[CrossRef] [PubMed]

Dutriaux, L.

Folan, L. M.

S. D. Druger, S. Arnold, L. M. Folan, “Theory of enhanced energy transfer between molecules embedded in spherical dielectric particles,” J. Chem. Phys. 87, 2649–2659 (1987).
[CrossRef]

Fraser, A. B.

C. F. Bohren, A. B. Fraser, “Newton’s zero-order rainbow: unobservable or nonexistent?” Am. J. Phys. 59, 325–326 (1991).
[CrossRef]

Fuller, K. A.

Hardalupas, Y.

Hill, S. C.

Holler, S.

S. C. Hill, V. Boutou, J. Yu, S. Ramstein, J.-P. Wolf, Y.-L. Pan, S. Holler, R. K. Chang, “Enhanced backward-directed multiphoton-excited fluorescence from dielectric microcavities,” Phys. Rev. Lett. 85, 54–57 (2000).
[CrossRef] [PubMed]

Hovenac, E. A.

Huffman, D. R.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), pp. 82–101.

Kerker, M.

M. Kerker, S. D. Druger, “Raman and fluorescent scattering by molecules embedded in spheres with radii up to several multiples of the wavelength,” Appl. Opt. 18, 1172–1179 (1979).
[CrossRef] [PubMed]

H. Chew, P. J. McNulty, M. Kerker, “Model for Raman and fluorescent scattering by molecules embedded in small particles,” Phys. Rev. A 13, 396–404 (1976).
[CrossRef]

M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969), pp. 39–49.

Langley, D. S.

Le Floch, A.

Leach, D. H.

Lock, J. A.

Marston, P. L.

McCollum, T. A.

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

McNulty, P. J.

H. Chew, P. J. McNulty, M. Kerker, “Model for Raman and fluorescent scattering by molecules embedded in small particles,” Phys. Rev. A 13, 396–404 (1976).
[CrossRef]

Melton, L. A.

Morrell, M. J.

Mount, C. M.

Nachman, P.

Nussenzveig, H. M.

H. M. Nussenzveig, “Complex angular momentum theory of the rainbow and the glory,” J. Opt. Soc. Am. 69, 1068–1079, 1193–1194 (1979).
[CrossRef]

H. M. Nussenzveig, “High-frequency scattering by a transparent sphere. 1. Direct reflection and transmission,” J. Math. Phys. 10, 82–124 (1969).
[CrossRef]

H. M. Nussenzveig, “High-frequency scattering by a transparent sphere. 2. Theory of the rainbow and the glory,” J. Math. Phys. 10, 125–176 (1969).
[CrossRef]

H. M. Nussenzveig, Diffraction Effects in Semiclassical Scattering (Cambridge U. Press, Cambridge, UK, 1992), pp. 45–51.

Pan, Y.-L.

S. C. Hill, V. Boutou, J. Yu, S. Ramstein, J.-P. Wolf, Y.-L. Pan, S. Holler, R. K. Chang, “Enhanced backward-directed multiphoton-excited fluorescence from dielectric microcavities,” Phys. Rev. Lett. 85, 54–57 (2000).
[CrossRef] [PubMed]

Pendleton, J. D.

Ramsey, J. M.

Ramstein, S.

S. C. Hill, V. Boutou, J. Yu, S. Ramstein, J.-P. Wolf, Y.-L. Pan, S. Holler, R. K. Chang, “Enhanced backward-directed multiphoton-excited fluorescence from dielectric microcavities,” Phys. Rev. Lett. 85, 54–57 (2000).
[CrossRef] [PubMed]

Saleheen, H. I.

Schweiger, G.

Theissen, D. B.

Tran, N. H.

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981), pp. 119–126.

van der Pol, B.

B. van der Pol, H. Bremmer, “The diffraction of electromagnetic waves from an electrical point source round a finitely conducting sphere, with applications to radiotelegraphy and the theory of the rainbow,” Philos. Mag. 24, 141–176, 825–864 (1937).

Velesco, N.

Videen, G.

Whitten, W. B.

Wolf, J.-P.

S. C. Hill, V. Boutou, J. Yu, S. Ramstein, J.-P. Wolf, Y.-L. Pan, S. Holler, R. K. Chang, “Enhanced backward-directed multiphoton-excited fluorescence from dielectric microcavities,” Phys. Rev. Lett. 85, 54–57 (2000).
[CrossRef] [PubMed]

Yu, J.

S. C. Hill, V. Boutou, J. Yu, S. Ramstein, J.-P. Wolf, Y.-L. Pan, S. Holler, R. K. Chang, “Enhanced backward-directed multiphoton-excited fluorescence from dielectric microcavities,” Phys. Rev. Lett. 85, 54–57 (2000).
[CrossRef] [PubMed]

Zhang, J.

Am. J. Phys. (2)

C. F. Bohren, A. B. Fraser, “Newton’s zero-order rainbow: unobservable or nonexistent?” Am. J. Phys. 59, 325–326 (1991).
[CrossRef]

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

Appl. Opt. (11)

D. S. Langley, M. J. Morrell, “Rainbow-enhanced forward and backward glory scattering,” Appl. Opt. 30, 3459–3467 (1991).
[CrossRef] [PubMed]

S. C. Hill, H. I. Saleheen, M. D. Barnes, W. B. Whitten, J. M. Ramsey, “Modeling fluorescence collection from single molecules in microspheres: effects of position, orientation, and frequency,” Appl. Opt. 35, 6278–6288 (1996).
[CrossRef] [PubMed]

S. C. Hill, M. D. Barnes, W. B. Whitten, J. M. Ramsey, “Collection of fluorescence from single molecules in microspheres: effects of illumination geometry,” Appl. Opt. 36, 4425–4437 (1997).
[CrossRef] [PubMed]

C. M. Mount, D. B. Theissen, P. L. Marston, “Scattering observations for tilted transparent fibers: evolution of the Airy caustics with cylinder tilt and the caustic merging transition,” Appl. Opt. 37, 1534–1539 (1998).
[CrossRef]

J. D. Pendleton, S. C. Hill, “Collection of emission from an oscillating dipole inside a sphere: analytical integration over a circular aperture,” Appl. Opt. 36, 8729–8737 (1997).
[CrossRef]

M. Kerker, S. D. Druger, “Raman and fluorescent scattering by molecules embedded in spheres with radii up to several multiples of the wavelength,” Appl. Opt. 18, 1172–1179 (1979).
[CrossRef] [PubMed]

J. Zhang, D. R. Alexander, “Hybrid inelastic-scattering models for particle thermometry: unpolarized emissions,” Appl. Opt. 31, 7132–7139 (1992).
[CrossRef] [PubMed]

J. Zhang, D. R. Alexander, “Hybrid inelastic-scattering models for particle thermometry: polarized emissions,” Appl. Opt. 31, 7140–7146 (1992).
[CrossRef] [PubMed]

J. Zhang, L. A. Melton, “Numerical simulations and restorations of laser droplet-slicing images,” Appl. Opt. 33, 192–200 (1994).
[CrossRef] [PubMed]

N. Velesco, G. Schweiger, “Geometrical optics calculation of inelastic scattering on large particles,” Appl. Opt. 38, 1046–1052 (1999).
[CrossRef]

R. Domann, Y. Hardalupas, “Spatial distribution of fluorescence intensity within large droplets and its dependence on dye concentration,” Appl. Opt. 40, 3586–3597 (2001).
[CrossRef]

J. Chem. Phys. (2)

H. Chew, “Transition rates of atoms near spherical surfaces,” J. Chem. Phys. 87, 1355–1360 (1987).
[CrossRef]

S. D. Druger, S. Arnold, L. M. Folan, “Theory of enhanced energy transfer between molecules embedded in spherical dielectric particles,” J. Chem. Phys. 87, 2649–2659 (1987).
[CrossRef]

J. Math. Phys. (2)

H. M. Nussenzveig, “High-frequency scattering by a transparent sphere. 1. Direct reflection and transmission,” J. Math. Phys. 10, 82–124 (1969).
[CrossRef]

H. M. Nussenzveig, “High-frequency scattering by a transparent sphere. 2. Theory of the rainbow and the glory,” J. Math. Phys. 10, 125–176 (1969).
[CrossRef]

J. Opt. Soc. Am. (1)

H. M. Nussenzveig, “Complex angular momentum theory of the rainbow and the glory,” J. Opt. Soc. Am. 69, 1068–1079, 1193–1194 (1979).
[CrossRef]

J. Opt. Soc. Am. A (5)

J. Opt. Soc. Am. B (2)

Opt. Lett. (1)

Philos. Mag. (1)

B. van der Pol, H. Bremmer, “The diffraction of electromagnetic waves from an electrical point source round a finitely conducting sphere, with applications to radiotelegraphy and the theory of the rainbow,” Philos. Mag. 24, 141–176, 825–864 (1937).

Phys. Acoust. (1)

P. L. Marston, “Geometrical and catastrophe optics methods in scattering,” Phys. Acoust. 21, 1–234 (1992), p. 187, Fig. 76.

Phys. Rev. A (1)

H. Chew, P. J. McNulty, M. Kerker, “Model for Raman and fluorescent scattering by molecules embedded in small particles,” Phys. Rev. A 13, 396–404 (1976).
[CrossRef]

Phys. Rev. Lett. (1)

S. C. Hill, V. Boutou, J. Yu, S. Ramstein, J.-P. Wolf, Y.-L. Pan, S. Holler, R. K. Chang, “Enhanced backward-directed multiphoton-excited fluorescence from dielectric microcavities,” Phys. Rev. Lett. 85, 54–57 (2000).
[CrossRef] [PubMed]

Other (7)

Ref. 35, footnote 30.

Ref. 1, pp. 210–214.

H. M. Nussenzveig, Diffraction Effects in Semiclassical Scattering (Cambridge U. Press, Cambridge, UK, 1992), pp. 45–51.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981), pp. 119–126.

M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969), pp. 39–49.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), pp. 82–101.

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., Selected Papers on Geometrical Aspects of Scattering (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1994), pp. 198–204.

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Figures (13)

Fig. 1
Fig. 1

p=1 ray trajectory. The source is located at z0 on the z axis of the sphere. The ray makes an angle α with the z axis, and the angles of incidence and transmission at the sphere surface are θi and θt.

Fig. 2
Fig. 2

p=1 family of scattered rays. p=1 rays do not exit the sphere between ΘTIR- and ΘTIR+.

Fig. 3
Fig. 3

(a) Scattering angle θ as a function of α of the p=1 ray family for N=4/3 and q=0.748 and (b) p=1 ray scattering intensity exhibiting the scattering resonance at θ132°.

Fig. 4
Fig. 4

Deflection angle of the p=1 ray family as a function of α for N=4/3 and q=0.90. The total internal reflection region extends from ΘTIR- to ΘTIR+, and the total internal reflection rainbow is indicated by TIRR.

Fig. 5
Fig. 5

Deflection angle Θ of the p=2 ray family as a function of α for N=4/3 and q=0.95. The rainbow is indicated by R, and the total internal reflection rainbow is indicated by TIRR. A ray theory glory occurs at Θ=180° for q0.91.

Fig. 6
Fig. 6

Deflection angle Θ of the p=3 ray family as a function of α for N=4/3 and q=0.85. The rainbow is indicated by R, and the total internal reflection rainbow is indicated by TIRR. Both α1 paraxial rays and the α>90° rays contributing to the total internal reflection rainbow extend into the total internal reflection region.

Fig. 7
Fig. 7

Deflection angle Θ of the p=4 ray family as a function of α for N=4/3 and q=0.77, producing a rainbow-enhanced glory at ΘR=360°. The rainbow is indicated by R, and the total internal reflection rainbow is indicated by TIRR. For q<0.77, the rainbow angle is ΘR>360°. For 0.77<q<0.80, ΘR<360° and the rainbow is accompanied by two glory rays. For q>0.80, ΘTIR-<360° and one of the glory rays ceases to exist in ray theory.

Fig. 8
Fig. 8

Ray scattering intensity (dashed curves) for N=4/3 and p=1, and the wave-scattered intensity (solid curves) as a function of θ for ϕ=π/2 for an electric dipole source pointing in the x direction with λ=0.6328 µm, a=30.0 µm, N=4/3, and p=1 for (a) q=0.50, (b) q=0.748, and (c) q=0.99. The ray and wave intensities are normalized to the same value at θ=0°.

Fig. 9
Fig. 9

Interference of p=1 rays (A) with the radiation shed by p=1 electromagnetic surface waves (B,C) in the vicinity of ΘTIR0 for qqc.

Fig. 10
Fig. 10

Wave scattering intensity as a function of θ for ϕ=π/2 for an electric dipole source pointing in the x direction with λ=0.6328 µm, a=30.0 µm, N=4/3, q=0.99, and (a) p=1, (b) p=2, (c) p=3, and (d) p=4.

Fig. 11
Fig. 11

(a) Wave scattering intensity of Figs. 10(a), 10(b), 10(c), and 10(d) for 1p4 along with the p=5 intensity, showing the confluence of the p+1 rainbow with the p total internal reflection rainbow as q1. (b) Total wave scattering intensity as a function of θ for ϕ=π/2 for an electric dipole source pointing in the x direction with λ=0.6328 µm, a=30.0 µm, N=4/3, and q=0.99.

Fig. 12
Fig. 12

Wave scattering intensity averaged over all electric dipole orientations as a function of θ for λ=0.6328µm, a=30.0 µm, N=4/3, and (a) q=0.10, (b) q=0.75, and (c) q=0.99.

Fig. 13
Fig. 13

Wave scattering intensity averaged over all electric dipole orientations as a function of θ for the conditions of the TE-polarized MDR in partial wave 309, λ=0.6328 µm, a=29.940913 µm, N=1.333333, and (a) q=0.70 and (b) q=0.99.

Equations (53)

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sin(θi)=q sin(α),
sin(θt)=N sin(θi),
qz0/a
T(θi)=4N cos(θi)cos(θt)/[N cos(θi)+cos(θt)]2,
R(θi)=[N cos(θi)-cos(θt)]2/[N cos(θi)+cos(θt)]2.
Θ=(p-1)π+α-(2p-1)θi+θt.
θ=Θ-2Mπif2MπΘ(2M+1)π,(2M+2)π-Θif(2M+1)π<Θ<(2M+2)π,
I(θ)=P sin(α)[R(θi)]p-1T(θi)/[4πr2 sin(θ)|dΘ/dα|],
dΘ/dα=1-(2p-1)q cos(α)/cos(θi)+Nq cos(α)/cos(θt).
qc1/N.
ΘTIR0pπ-(2p-1)sin-1(1/N).
αTIR-ααTIR+,
αTIR-=sin-1(1/Nq),αTIR+=π-sin-1(1/Nq),
ΘTIR-ΘΘTIR+,
ΘTIR±=ΘTIR0±[π/2-sin-1(1/Nq)].
ΘTIR±=ΘTIR0±(2N)1/2+O().
z=(-1)pa/(2p-1-N).
qAF1/(2p-1-N),
Θ=(p-1)π-α(/qAF)+(α3/6){[1-qAF3×(2p-1-N3)]+(/qAF)[1-3qAF3×(2p-1-N3)]}+O(α5).
αR(2/qAF)1/2[1-qAF3(2p-1-N3)]-1/2,
ΘR=(p-1)π-(2/3qAF)αR,
αG=31/2αR
ΘR=(p-1)π-2(p-1)α+sin-1[N sin(α)],
cos(α)=[2(p-1)/N](N2-1)1/2[4(p-1)2-1]-1/2.
(dΘ/dα)min=3[N/(N2-1)]1/3[(2p-1)/2]2/3
ΘRES=ΘTIR0-(2p-1)N/[4(N2-1)1/2]+O(4/3).
θt=π/2-[2N/(2p-1)]1/3(N2-1)1/6,
ΘTIRR=ΘTIR0+(3/2)(N)2/3(2p-1)1/3×(N2-1)-1/6+O(),
ΘTIRR=pπ-2pβ+sin-1[N sin(β)],
α=π-β,
cos(β)=(2p/N)(N2-1)1/2(4p2-1)-1/2.
Escatt(r, θ, ϕ)=i[exp(ikr)/(kr)]×[T1(θ)cos(ϕ)uθ-T2(θ)sin(ϕ)uϕ]
T1(θ)=n=1(-i)n{cn[jn+1(Nkz0)/(n+1)-jn-1(Nkz0)/n]τn,1(θ)+idn×[(2n+1)/n(n+1)]jn(Nkz0)πn,1(θ)},
T2(θ)=n=1(-i)n{cn[jn+1(Nkz0)/(n+1)-jn-1(Nkz0)/n]πn,1(θ)+idn×[(2n+1)/n(n+1)]jn(Nkz0)τn,1(θ)}.
k=2π/λ,
cn=[i/N(ka)2][hn(1)(ka)jn(Nka)-Nhn(1)(ka)jn(Nka)-(N2-1)×hn(1)(ka)jn(Nka)/(Nka)]-1,
dn=[i/N(ka)2][Nhn(1)(ka)×jn(Nka)-hn(1)(ka)jn(Nka)]-1,
πn,m(θ)=Pn,m(θ)/sin(θ),τn,m(θ)=(d/dθ)Pn,m(θ),
Escatt(r, θ, ϕ)=i[exp(ikr)/(kr)]×[T1(θ)sin(ϕ)uθ+T2(θ)cos(ϕ)uϕ],
Escatt(r, θ, ϕ)=-i[exp(ikr)/(kr)]T3(θ)uθ,
T3(θ)=n=1(-i)n(2n+1)cn[jn(Nkz0)/(Nkz0)]τn,0(θ).
Iscatt(r, θ, ϕ)=(1/2μ0c)|Escatt(r, θ, ϕ)|2,
Iscattave(r, θ)=(1/2μ0c)[|T1(θ)|2+|T2(θ)|2+|T3(θ)|2]/(3k2r2).
cn=-TnTM/(1-RnTM)=-p=1(RnTM)p-1TnTM,
dn=-TnTE/(1-RnTE)=-p=1(RnTE)p-1TnTE.
TnTM=[-2i/N(ka)2][hn(1)(ka)hn(2)(Nka)-Nhn(1)(ka)hn(2)(Nka)-(N2-1)×hn(1)(ka)hn(2)(Nka)/(Nka)]-1,
TnTE=[-2i/N(ka)2][Nhn(1)(ka)hn(2)(Nka)-hn(1)(ka)hn(2)(Nka)]-1,
RnTM=-[hn(1)(ka)hn(1)(Nka)-Nhn(1)(ka)hn(1)(Nka)-(N2-1)hn(1)(ka)hn(1)(Nka)/(Nka)]/[hn(1)(ka)hn(2)(Nka)-Nhn(1)(ka)hn(2)(Nka)-(N2-1)hn(1)(ka)hn(2)(Nka)/(Nka)],
RnTE=-[Nhn(1)(ka)hn(1)(Nka)-hn(1)(ka)hn(1)(Nka)]/[Nhn(1)(ka)hn(2)(Nka)-hn(1)(ka)hn(2)(Nka)],
xka,
nmax=1+x+20x1/3.
Itotalave(r, θ, ϕ)dλfd3rs|Einterior(λs; rs, θs, ϕs)|2×S(λf)Iscattave(λf; r, θ),
cos(θ)=cos(θ)cos(θs)+sin(θ)sin(θs)cos(ϕ-ϕs).

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