Abstract

Generalized Lorenz–Mie theory describes electromagnetic scattering of an arbitrary light beam by a spherical particle. The localized approximation is an analytical function that accurately models the beam-shape coefficients that give the decomposition of a focused Gaussian beam into partial waves. A mathematical justification and physical interpretation of the localized approximation is presented for a focused off-axis Gaussian beam that propagates parallel to but not along the z axis.

© 1994 Optical Society of America

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References

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  1. T. Baer, “Continuous-wave laser oscillation in a Nd:YAG sphere,” Opt. Lett. 12, 392–394 (1987).
    [CrossRef] [PubMed]
  2. J.-Z. Zhang, D. H. Leach, R. K. Chang, “Photon lifetime within a droplet: temporal determination of elastic and stimulated Raman scattering,” Opt. Lett. 13, 270–272 (1988).
    [CrossRef] [PubMed]
  3. J. P. Barton, D. R. Alexander, S. A. Schaub, “Internal fields of a spherical particle illuminated by a tightly focused laser beam: focal point positioning effects at resonance,” J. Appl. Phys. 65, 2900–2906 (1989).
    [CrossRef]
  4. E. E. M. Khaled, S. C. Hill, P. W. Barber, D. Q. Chowdhury, “Near-resonance excitation of dielectric spheres with plane waves and off-axis Gaussian beams,” Appl. Opt. 31, 1166–1169 (1992).
    [CrossRef] [PubMed]
  5. H. C. van de Hulst, R. T. Wang, “Glare points,” Appl. Opt. 30, 4755–4763 (1991). See especially the section on reciprocity and an incident narrow beam on p. 4762.
    [CrossRef] [PubMed]
  6. J. A. Lock, “Contribution of high-order rainbows to the scattering of a Gaussian laser beam by a spherical particle,” J. Opt. Soc. Am. A 10, 693–706 (1993).
    [CrossRef]
  7. G. Gouesbet, G. Gréhan, “Sur la généralisation de la théorie de Lorenz–Mie,”J. Opt. (Paris) 13, 97–103 (1982).
    [CrossRef]
  8. G. Gouesbet, G. Gréhan, B. Maheu, “Scattering of a Gaussian beam by a Mie scatter center using a Bromwich formalism,”J. Opt. (Paris) 16, 83–93 (1985).
    [CrossRef]
  9. G. Gouesbet, B. Maheu, G. Gréhan, “Light scattering from a sphere arbitrarily located in a Gaussian beam, using a Bromwich formulation,” J. Opt. Soc. Am. A 5, 1427–1443 (1988).
    [CrossRef]
  10. B. Maheu, G. Gouesbet, G. Gréhan, “A concise presentation of the generalized Lorenz–Mie theory for an arbitrary location of the scatterer in an arbitrary incident profile,”J. Opt. (Paris) 19, 59–67 (1988).
    [CrossRef]
  11. J. P. Barton, D. R. Alexander, S. A. Schaub, “Internal and near-surface electromagnetic fields for a spherical particle irradiated by a focused laser beam,” J. Appl. Phys. 64, 1632–1639 (1988).
    [CrossRef]
  12. G. Gréhan, B. Maheu, G. Gouesbet, “Scattering of laser beams by Mie scatter centers: numerical results using a localized approximation,” Appl. Opt. 25, 3539–3548 (1986).
    [CrossRef] [PubMed]
  13. B. Maheu, G. Gréhan, G. Gouesbet, “Generalized Lorenz–Mie theory: first exact values and comparisons with the localized approximation,” Appl. Opt. 26, 23–25 (1987).
    [CrossRef] [PubMed]
  14. G. Gouesbet, G. Gréhan, B. Maheu, “Computations of the gncoefficients in the generalized Lorenz–Mie theory using three different methods,” Appl. Opt. 27, 4874–4883 (1988).
    [CrossRef] [PubMed]
  15. G. Gouesbet, G. Gréhan, B. Maheu, “Localized interpretation to compute all the coefficients gnmin the generalized Lorenz–Mie theory,” J. Opt. Soc. Am. A 7, 998–1007 (1990).
    [CrossRef]
  16. B. Maheu, G. Gréhan, G. Gouesbet, “Ray localization in Gaussian beams,” Opt. Commun. 70, 259–262 (1989).
    [CrossRef]
  17. J. A. Lock, G. Gouesbet, “Rigorous justification of the localized approximation to the beam-shape coefficients in generalized Lorenz–Mie theory. I. On-axis beams,” J. Opt. Soc. Am. A 11, 2503–2515 (1994).
    [CrossRef]
  18. L. W. Davis, “Theory of electromagnetic beams,” Phys. Rev. A 19, 1177–1179 (1979).
    [CrossRef]
  19. J. R. Reitz, F. J. Milford, R. W. Christy, Foundations of Electromagnetic Theory, 3rd ed. (Addison-Wesley, Reading, Mass., 1979), Sec. 17.5.
  20. G. Gouesbet, B. Maheu, G. Gréhan, “The order of approximation in a theory of the scattering of a Gaussian beam by a Mie scatter center,”J. Opt. (Paris) 16, 239–247 (1985).
    [CrossRef]
  21. J. P. Barton, D. R. Alexander, “Fifth order corrected electromagnetic field components for a fundamental Gaussian beam,” J. Appl. Phys. 66, 2800–2802 (1989).
    [CrossRef]
  22. G. P. Können, J. H. de Boer, “Polarized rainbow,” Appl. Opt. 18, 1961–1965 (1979).
    [CrossRef] [PubMed]
  23. 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]
  24. J. A. Lock, “Improved Gaussian beam scattering algorithm,” submitted to Appl. Opt.
  25. F. Corbin, G. Gréhan, G. Gouesbet, “Top-hat beam technique: improvements and application to bubble measurements,” Part. Part. Syst. Charact. 8, 222–228 (1991).
    [CrossRef]
  26. E. A. Hovenac, J. A. Lock, “Calibration of the forward scattering spectrometer probe: modeling scattering from a multimode laser beam,”J. Atmos. Oceanic Technol. 10, 518–525 (1993).
    [CrossRef]

1994 (1)

1993 (2)

E. A. Hovenac, J. A. Lock, “Calibration of the forward scattering spectrometer probe: modeling scattering from a multimode laser beam,”J. Atmos. Oceanic Technol. 10, 518–525 (1993).
[CrossRef]

J. A. Lock, “Contribution of high-order rainbows to the scattering of a Gaussian laser beam by a spherical particle,” J. Opt. Soc. Am. A 10, 693–706 (1993).
[CrossRef]

1992 (2)

1991 (2)

F. Corbin, G. Gréhan, G. Gouesbet, “Top-hat beam technique: improvements and application to bubble measurements,” Part. Part. Syst. Charact. 8, 222–228 (1991).
[CrossRef]

H. C. van de Hulst, R. T. Wang, “Glare points,” Appl. Opt. 30, 4755–4763 (1991). See especially the section on reciprocity and an incident narrow beam on p. 4762.
[CrossRef] [PubMed]

1990 (1)

1989 (3)

B. Maheu, G. Gréhan, G. Gouesbet, “Ray localization in Gaussian beams,” Opt. Commun. 70, 259–262 (1989).
[CrossRef]

J. P. Barton, D. R. Alexander, S. A. Schaub, “Internal fields of a spherical particle illuminated by a tightly focused laser beam: focal point positioning effects at resonance,” J. Appl. Phys. 65, 2900–2906 (1989).
[CrossRef]

J. P. Barton, D. R. Alexander, “Fifth order corrected electromagnetic field components for a fundamental Gaussian beam,” J. Appl. Phys. 66, 2800–2802 (1989).
[CrossRef]

1988 (5)

J.-Z. Zhang, D. H. Leach, R. K. Chang, “Photon lifetime within a droplet: temporal determination of elastic and stimulated Raman scattering,” Opt. Lett. 13, 270–272 (1988).
[CrossRef] [PubMed]

G. Gouesbet, G. Gréhan, B. Maheu, “Computations of the gncoefficients in the generalized Lorenz–Mie theory using three different methods,” Appl. Opt. 27, 4874–4883 (1988).
[CrossRef] [PubMed]

G. Gouesbet, B. Maheu, G. Gréhan, “Light scattering from a sphere arbitrarily located in a Gaussian beam, using a Bromwich formulation,” J. Opt. Soc. Am. A 5, 1427–1443 (1988).
[CrossRef]

B. Maheu, G. Gouesbet, G. Gréhan, “A concise presentation of the generalized Lorenz–Mie theory for an arbitrary location of the scatterer in an arbitrary incident profile,”J. Opt. (Paris) 19, 59–67 (1988).
[CrossRef]

J. P. Barton, D. R. Alexander, S. A. Schaub, “Internal and near-surface electromagnetic fields for a spherical particle irradiated by a focused laser beam,” J. Appl. Phys. 64, 1632–1639 (1988).
[CrossRef]

1987 (2)

1986 (1)

1985 (2)

G. Gouesbet, G. Gréhan, B. Maheu, “Scattering of a Gaussian beam by a Mie scatter center using a Bromwich formalism,”J. Opt. (Paris) 16, 83–93 (1985).
[CrossRef]

G. Gouesbet, B. Maheu, G. Gréhan, “The order of approximation in a theory of the scattering of a Gaussian beam by a Mie scatter center,”J. Opt. (Paris) 16, 239–247 (1985).
[CrossRef]

1982 (1)

G. Gouesbet, G. Gréhan, “Sur la généralisation de la théorie de Lorenz–Mie,”J. Opt. (Paris) 13, 97–103 (1982).
[CrossRef]

1979 (2)

G. P. Können, J. H. de Boer, “Polarized rainbow,” Appl. Opt. 18, 1961–1965 (1979).
[CrossRef] [PubMed]

L. W. Davis, “Theory of electromagnetic beams,” Phys. Rev. A 19, 1177–1179 (1979).
[CrossRef]

Alexander, D. R.

J. P. Barton, D. R. Alexander, S. A. Schaub, “Internal fields of a spherical particle illuminated by a tightly focused laser beam: focal point positioning effects at resonance,” J. Appl. Phys. 65, 2900–2906 (1989).
[CrossRef]

J. P. Barton, D. R. Alexander, “Fifth order corrected electromagnetic field components for a fundamental Gaussian beam,” J. Appl. Phys. 66, 2800–2802 (1989).
[CrossRef]

J. P. Barton, D. R. Alexander, S. A. Schaub, “Internal and near-surface electromagnetic fields for a spherical particle irradiated by a focused laser beam,” J. Appl. Phys. 64, 1632–1639 (1988).
[CrossRef]

Baer, T.

Barber, P. W.

Barton, J. P.

J. P. Barton, D. R. Alexander, S. A. Schaub, “Internal fields of a spherical particle illuminated by a tightly focused laser beam: focal point positioning effects at resonance,” J. Appl. Phys. 65, 2900–2906 (1989).
[CrossRef]

J. P. Barton, D. R. Alexander, “Fifth order corrected electromagnetic field components for a fundamental Gaussian beam,” J. Appl. Phys. 66, 2800–2802 (1989).
[CrossRef]

J. P. Barton, D. R. Alexander, S. A. Schaub, “Internal and near-surface electromagnetic fields for a spherical particle irradiated by a focused laser beam,” J. Appl. Phys. 64, 1632–1639 (1988).
[CrossRef]

Chang, R. K.

Chowdhury, D. Q.

Christy, R. W.

J. R. Reitz, F. J. Milford, R. W. Christy, Foundations of Electromagnetic Theory, 3rd ed. (Addison-Wesley, Reading, Mass., 1979), Sec. 17.5.

Corbin, F.

F. Corbin, G. Gréhan, G. Gouesbet, “Top-hat beam technique: improvements and application to bubble measurements,” Part. Part. Syst. Charact. 8, 222–228 (1991).
[CrossRef]

Davis, L. W.

L. W. Davis, “Theory of electromagnetic beams,” Phys. Rev. A 19, 1177–1179 (1979).
[CrossRef]

de Boer, J. H.

Gouesbet, G.

J. A. Lock, G. Gouesbet, “Rigorous justification of the localized approximation to the beam-shape coefficients in generalized Lorenz–Mie theory. I. On-axis beams,” J. Opt. Soc. Am. A 11, 2503–2515 (1994).
[CrossRef]

F. Corbin, G. Gréhan, G. Gouesbet, “Top-hat beam technique: improvements and application to bubble measurements,” Part. Part. Syst. Charact. 8, 222–228 (1991).
[CrossRef]

G. Gouesbet, G. Gréhan, B. Maheu, “Localized interpretation to compute all the coefficients gnmin the generalized Lorenz–Mie theory,” J. Opt. Soc. Am. A 7, 998–1007 (1990).
[CrossRef]

B. Maheu, G. Gréhan, G. Gouesbet, “Ray localization in Gaussian beams,” Opt. Commun. 70, 259–262 (1989).
[CrossRef]

G. Gouesbet, G. Gréhan, B. Maheu, “Computations of the gncoefficients in the generalized Lorenz–Mie theory using three different methods,” Appl. Opt. 27, 4874–4883 (1988).
[CrossRef] [PubMed]

G. Gouesbet, B. Maheu, G. Gréhan, “Light scattering from a sphere arbitrarily located in a Gaussian beam, using a Bromwich formulation,” J. Opt. Soc. Am. A 5, 1427–1443 (1988).
[CrossRef]

B. Maheu, G. Gouesbet, G. Gréhan, “A concise presentation of the generalized Lorenz–Mie theory for an arbitrary location of the scatterer in an arbitrary incident profile,”J. Opt. (Paris) 19, 59–67 (1988).
[CrossRef]

B. Maheu, G. Gréhan, G. Gouesbet, “Generalized Lorenz–Mie theory: first exact values and comparisons with the localized approximation,” Appl. Opt. 26, 23–25 (1987).
[CrossRef] [PubMed]

G. Gréhan, B. Maheu, G. Gouesbet, “Scattering of laser beams by Mie scatter centers: numerical results using a localized approximation,” Appl. Opt. 25, 3539–3548 (1986).
[CrossRef] [PubMed]

G. Gouesbet, G. Gréhan, B. Maheu, “Scattering of a Gaussian beam by a Mie scatter center using a Bromwich formalism,”J. Opt. (Paris) 16, 83–93 (1985).
[CrossRef]

G. Gouesbet, B. Maheu, G. Gréhan, “The order of approximation in a theory of the scattering of a Gaussian beam by a Mie scatter center,”J. Opt. (Paris) 16, 239–247 (1985).
[CrossRef]

G. Gouesbet, G. Gréhan, “Sur la généralisation de la théorie de Lorenz–Mie,”J. Opt. (Paris) 13, 97–103 (1982).
[CrossRef]

Gréhan, G.

F. Corbin, G. Gréhan, G. Gouesbet, “Top-hat beam technique: improvements and application to bubble measurements,” Part. Part. Syst. Charact. 8, 222–228 (1991).
[CrossRef]

G. Gouesbet, G. Gréhan, B. Maheu, “Localized interpretation to compute all the coefficients gnmin the generalized Lorenz–Mie theory,” J. Opt. Soc. Am. A 7, 998–1007 (1990).
[CrossRef]

B. Maheu, G. Gréhan, G. Gouesbet, “Ray localization in Gaussian beams,” Opt. Commun. 70, 259–262 (1989).
[CrossRef]

G. Gouesbet, G. Gréhan, B. Maheu, “Computations of the gncoefficients in the generalized Lorenz–Mie theory using three different methods,” Appl. Opt. 27, 4874–4883 (1988).
[CrossRef] [PubMed]

B. Maheu, G. Gouesbet, G. Gréhan, “A concise presentation of the generalized Lorenz–Mie theory for an arbitrary location of the scatterer in an arbitrary incident profile,”J. Opt. (Paris) 19, 59–67 (1988).
[CrossRef]

G. Gouesbet, B. Maheu, G. Gréhan, “Light scattering from a sphere arbitrarily located in a Gaussian beam, using a Bromwich formulation,” J. Opt. Soc. Am. A 5, 1427–1443 (1988).
[CrossRef]

B. Maheu, G. Gréhan, G. Gouesbet, “Generalized Lorenz–Mie theory: first exact values and comparisons with the localized approximation,” Appl. Opt. 26, 23–25 (1987).
[CrossRef] [PubMed]

G. Gréhan, B. Maheu, G. Gouesbet, “Scattering of laser beams by Mie scatter centers: numerical results using a localized approximation,” Appl. Opt. 25, 3539–3548 (1986).
[CrossRef] [PubMed]

G. Gouesbet, G. Gréhan, B. Maheu, “Scattering of a Gaussian beam by a Mie scatter center using a Bromwich formalism,”J. Opt. (Paris) 16, 83–93 (1985).
[CrossRef]

G. Gouesbet, B. Maheu, G. Gréhan, “The order of approximation in a theory of the scattering of a Gaussian beam by a Mie scatter center,”J. Opt. (Paris) 16, 239–247 (1985).
[CrossRef]

G. Gouesbet, G. Gréhan, “Sur la généralisation de la théorie de Lorenz–Mie,”J. Opt. (Paris) 13, 97–103 (1982).
[CrossRef]

Hill, S. C.

Hovenac, E. A.

E. A. Hovenac, J. A. Lock, “Calibration of the forward scattering spectrometer probe: modeling scattering from a multimode laser beam,”J. Atmos. Oceanic Technol. 10, 518–525 (1993).
[CrossRef]

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]

Khaled, E. E. M.

Können, G. P.

Leach, D. H.

Lock, J. A.

Maheu, B.

G. Gouesbet, G. Gréhan, B. Maheu, “Localized interpretation to compute all the coefficients gnmin the generalized Lorenz–Mie theory,” J. Opt. Soc. Am. A 7, 998–1007 (1990).
[CrossRef]

B. Maheu, G. Gréhan, G. Gouesbet, “Ray localization in Gaussian beams,” Opt. Commun. 70, 259–262 (1989).
[CrossRef]

G. Gouesbet, G. Gréhan, B. Maheu, “Computations of the gncoefficients in the generalized Lorenz–Mie theory using three different methods,” Appl. Opt. 27, 4874–4883 (1988).
[CrossRef] [PubMed]

G. Gouesbet, B. Maheu, G. Gréhan, “Light scattering from a sphere arbitrarily located in a Gaussian beam, using a Bromwich formulation,” J. Opt. Soc. Am. A 5, 1427–1443 (1988).
[CrossRef]

B. Maheu, G. Gouesbet, G. Gréhan, “A concise presentation of the generalized Lorenz–Mie theory for an arbitrary location of the scatterer in an arbitrary incident profile,”J. Opt. (Paris) 19, 59–67 (1988).
[CrossRef]

B. Maheu, G. Gréhan, G. Gouesbet, “Generalized Lorenz–Mie theory: first exact values and comparisons with the localized approximation,” Appl. Opt. 26, 23–25 (1987).
[CrossRef] [PubMed]

G. Gréhan, B. Maheu, G. Gouesbet, “Scattering of laser beams by Mie scatter centers: numerical results using a localized approximation,” Appl. Opt. 25, 3539–3548 (1986).
[CrossRef] [PubMed]

G. Gouesbet, G. Gréhan, B. Maheu, “Scattering of a Gaussian beam by a Mie scatter center using a Bromwich formalism,”J. Opt. (Paris) 16, 83–93 (1985).
[CrossRef]

G. Gouesbet, B. Maheu, G. Gréhan, “The order of approximation in a theory of the scattering of a Gaussian beam by a Mie scatter center,”J. Opt. (Paris) 16, 239–247 (1985).
[CrossRef]

Milford, F. J.

J. R. Reitz, F. J. Milford, R. W. Christy, Foundations of Electromagnetic Theory, 3rd ed. (Addison-Wesley, Reading, Mass., 1979), Sec. 17.5.

Reitz, J. R.

J. R. Reitz, F. J. Milford, R. W. Christy, Foundations of Electromagnetic Theory, 3rd ed. (Addison-Wesley, Reading, Mass., 1979), Sec. 17.5.

Schaub, S. A.

J. P. Barton, D. R. Alexander, S. A. Schaub, “Internal fields of a spherical particle illuminated by a tightly focused laser beam: focal point positioning effects at resonance,” J. Appl. Phys. 65, 2900–2906 (1989).
[CrossRef]

J. P. Barton, D. R. Alexander, S. A. Schaub, “Internal and near-surface electromagnetic fields for a spherical particle irradiated by a focused laser beam,” J. Appl. Phys. 64, 1632–1639 (1988).
[CrossRef]

van de Hulst, H. C.

H. C. van de Hulst, R. T. Wang, “Glare points,” Appl. Opt. 30, 4755–4763 (1991). See especially the section on reciprocity and an incident narrow beam on p. 4762.
[CrossRef] [PubMed]

Wang, R. T.

H. C. van de Hulst, R. T. Wang, “Glare points,” Appl. Opt. 30, 4755–4763 (1991). See especially the section on reciprocity and an incident narrow beam on p. 4762.
[CrossRef] [PubMed]

Zhang, J.-Z.

Appl. Opt. (6)

J. Appl. Phys. (3)

J. P. Barton, D. R. Alexander, “Fifth order corrected electromagnetic field components for a fundamental Gaussian beam,” J. Appl. Phys. 66, 2800–2802 (1989).
[CrossRef]

J. P. Barton, D. R. Alexander, S. A. Schaub, “Internal and near-surface electromagnetic fields for a spherical particle irradiated by a focused laser beam,” J. Appl. Phys. 64, 1632–1639 (1988).
[CrossRef]

J. P. Barton, D. R. Alexander, S. A. Schaub, “Internal fields of a spherical particle illuminated by a tightly focused laser beam: focal point positioning effects at resonance,” J. Appl. Phys. 65, 2900–2906 (1989).
[CrossRef]

J. Atmos. Oceanic Technol. (1)

E. A. Hovenac, J. A. Lock, “Calibration of the forward scattering spectrometer probe: modeling scattering from a multimode laser beam,”J. Atmos. Oceanic Technol. 10, 518–525 (1993).
[CrossRef]

J. Opt. (Paris) (4)

B. Maheu, G. Gouesbet, G. Gréhan, “A concise presentation of the generalized Lorenz–Mie theory for an arbitrary location of the scatterer in an arbitrary incident profile,”J. Opt. (Paris) 19, 59–67 (1988).
[CrossRef]

G. Gouesbet, G. Gréhan, “Sur la généralisation de la théorie de Lorenz–Mie,”J. Opt. (Paris) 13, 97–103 (1982).
[CrossRef]

G. Gouesbet, G. Gréhan, B. Maheu, “Scattering of a Gaussian beam by a Mie scatter center using a Bromwich formalism,”J. Opt. (Paris) 16, 83–93 (1985).
[CrossRef]

G. Gouesbet, B. Maheu, G. Gréhan, “The order of approximation in a theory of the scattering of a Gaussian beam by a Mie scatter center,”J. Opt. (Paris) 16, 239–247 (1985).
[CrossRef]

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

Opt. Commun. (1)

B. Maheu, G. Gréhan, G. Gouesbet, “Ray localization in Gaussian beams,” Opt. Commun. 70, 259–262 (1989).
[CrossRef]

Opt. Lett. (2)

Part. Part. Syst. Charact. (1)

F. Corbin, G. Gréhan, G. Gouesbet, “Top-hat beam technique: improvements and application to bubble measurements,” Part. Part. Syst. Charact. 8, 222–228 (1991).
[CrossRef]

Phys. Rev. A (1)

L. W. Davis, “Theory of electromagnetic beams,” Phys. Rev. A 19, 1177–1179 (1979).
[CrossRef]

Other (2)

J. R. Reitz, F. J. Milford, R. W. Christy, Foundations of Electromagnetic Theory, 3rd ed. (Addison-Wesley, Reading, Mass., 1979), Sec. 17.5.

J. A. Lock, “Improved Gaussian beam scattering algorithm,” submitted to Appl. Opt.

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Figures (4)

Fig. 1
Fig. 1

Electric-field profile Ex(x, 0, 0) of the localized beam of Eqs. (18), (19), and (41) for λ = 0.6328 μm, w0 = 10 μm, and three values of xf (dashed curves). The solid curves are Gaussian profiles.

Fig. 2
Fig. 2

Electric-field profile Ex(x, 0, 0) of the localized beam of Eqs. (18), (19), and (41) for λ = 0.6328 μm, w0 = 1 μm, and three values of xf (dashed curves). The solid curves are Gaussian profiles.

Fig. 3
Fig. 3

Gaussian beam incident off axis upon a spherical water droplet and the ray trajectories for specular reflection (p = 0), transmission (p = 1), one internal reflection (p = 2), and two internal reflections (p = 3).

Fig. 4
Fig. 4

(a) Scattered intensity |S2(θ, ϕ)|2 for ϕ = 0°, −180° ≤ θ ≤ 180°, a = 31.58 μm, N = 1.333, λ = 0.6328 μm, w0 = 10 μm, xf = 30 μm, and yf = zf = 0; (b) the p = 0, p = 1, p = 2, p = 3 Debye series contributions to |S2|2; (c) the p = 3 Debye series contribution, showing the spurious second-order rainbow at θ = −129°.

Equations (50)

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k = 2 π n λ = ω n c
ψ inc TE ( R , θ , ϕ ) = - E 0 n = 1 m = - n n ( - i ) n 2 n + 1 n ( n + 1 ) × ( g n m ) TE j n ( R ) P n m ( cos θ ) exp ( i m ϕ ) , ψ inc TM ( R , θ , ϕ ) = - E 0 n = 1 m = - n n ( - i ) n 2 n + 1 n ( n + 1 ) × ( g n m ) TM j n ( R ) P n m ( cos θ ) exp ( i m ϕ ) ,
R k r .
( g n m ) TE = - 1 4 π ( i n - 1 ) R j n ( R ) ( n - m ) ! ( n + m ) ! 0 π sin θ d θ × 0 2 π d ϕ P n m ( cos θ ) exp ( - i m ϕ ) c B inc rad ( R , θ , ϕ ) n E 0 , ( g n m ) TM = - 1 4 π ( i n - 1 ) R j n ( R ) ( n - m ) ! ( n + m ) ! 0 π sin θ d θ × 0 2 π d ϕ P n m ( cos θ ) exp ( - i m ϕ ) E inc rad ( R , θ , ϕ ) E 0 .
lim r I ( r , θ , ϕ ) = n E 0 2 2 μ 0 c 1 R 2 [ S 1 ( θ , ϕ ) 2 + S 2 ( θ , ϕ ) 2 ] ,
S 1 ( θ , ϕ ) = n = 1 m = - n n 2 n + 1 n ( n + 1 ) [ ( g n m ) TM a n m π n m ( θ ) + i ( g n m ) TE b n τ n m ( θ ) ] exp ( i m ϕ ) , S 2 ( θ , ϕ ) = n = 1 m = - n n 2 n + 1 n ( n + 1 ) [ i ( g n m ) TE b n m π n m ( θ ) + ( g n m ) TM a n τ n m ( θ ) ] exp ( i m ϕ ) ,
π n m ( θ ) = 1 sin θ P n m ( cos θ ) , τ n m ( θ ) = d d θ P n m ( cos θ )
s = 1 k w 0 ,
E Davis ( r ) = E 0 D exp [ - i k ( z - z f ) ] × exp { - D [ ( x - x f ) 2 + ( y - y f ) 2 ] / w 0 2 } × [ u ^ x + 2 i s D ( x - x f ) w 0 u ^ z ] , B Davis ( r ) = B 0 D exp [ - i k ( z - z f ) ] × exp { - D [ ( x - x f ) 2 + ( y - y f ) 2 ] / w 0 2 } × [ u ^ y + 2 i s D ( y - y f ) w 0 u ^ z ] ,
B 0 = n E 0 c ,
D = [ 1 - 2 i ( z - z f ) s w 0 ] - 1 .
E rad Davis = E x Davis sin θ cos ϕ + E y Davis × sin θ sin ϕ + E z Davis cos θ = E 0 D exp ( i k z f ) exp ( - i R cos θ ) × exp ( - s 2 R 2 D sin 2 θ ) exp [ - D ( x f 2 + y f 2 ) / w 0 2 ] × exp [ 2 s R D sin θ ( x f w 0 cos ϕ + y f w 0 sin ϕ ) ] × [ sin θ cos ϕ ( 1 + 2 i s 2 R D cos θ ) - 2 i s D x f w 0 cos θ ] ,
B rad Davis = B x Davis sin θ cos ϕ + B y Davis × sin θ sin ϕ + B z Davis cos θ = B 0 D exp ( i k z f ) exp ( - i R cos θ ) × exp ( - s 2 R 2 D sin 2 θ ) exp [ - D ( x f 2 + y f 2 ) / w 0 2 ] × exp [ 2 s R D sin θ ( x f w 0 cos ϕ + y f w 0 sin ϕ ) ] × [ sin θ sin ϕ ( 1 + 2 i s 2 R D cos θ ) - 2 i s D y f w 0 cos θ ] .
E rad Davis = E 0 2 F Davis ( R , θ ) j = 0 p = 0 j Ψ j , p Davis ( R , θ ) × { exp [ i ( j - 2 p + 1 ) ϕ ] + exp [ i ( j - 2 p - 1 ) ϕ ] } + E 0 x f w 0 G Davis ( R , θ ) j = 0 p = 0 j Ψ j , p Davis ( R , θ ) × exp [ i ( j - 2 p ) ϕ ] ,
B rad Davis = B 0 2 i F Davis ( R , θ ) j = 0 p = 0 j Ψ j , p Davis ( R , θ ) × { exp [ i ( j - 2 p + 1 ) ϕ ] - exp [ i ( j - 2 p - 1 ) ϕ ] } + B 0 y f w 0 G Davis ( R , θ ) j = 0 p = 0 j Ψ j , p Davis ( R , θ ) × exp [ i ( j - 2 p ) ϕ ] ,
F Davis ( R , θ ) = D 2 sin θ exp ( - i R cos θ ) exp ( i k z f ) × exp ( - s 2 R 2 D sin 2 θ ) × exp [ - D ( x f 2 + y f 2 ) / w 0 2 ] ( 1 + 2 i s z f w 0 ) , G Davis ( R , θ ) = - 2 i s D 2 cos θ exp ( - i R cos θ ) × exp ( i k z f ) exp ( - s 2 R 2 D sin 2 θ ) × exp [ - D ( x f 2 + y f 2 ) / w 0 2 ] ,
Ψ j , p Davis ( R , θ ) = ( s R D sin θ ) j × ( x f - i y f w 0 ) j - p ( j - p ) ! ( x f + i y f w 0 ) p p ! .
( g n m ) TE loc = k n m 2 i F Davis ( n + 1 / 2 , π / 2 ) × j = 0 p = 0 Ψ j , p Davis ( n + 1 / 2 , π / 2 ) × ( δ j - 2 p + 1 , m - δ j - 2 p - 1 , m ) , ( g n m ) TM loc = k n m 2 F Davis ( n + 1 / 2 , π / 2 ) × j = 0 p = 0 j Ψ j , p Davis ( n + 1 / 2 , π / 2 ) × ( δ j - 2 p + 1 , m + δ j - 2 p - 1 , m ) ,
k n m = { i n ( n + 1 ) ( n + 1 / 2 ) if m = 0 ( - i n + 1 / 2 ) m - 1 if m 0 .
E rad Davis ( R , θ , ϕ ) = m = - ( E rad Davis ) m exp ( i m ϕ ) ,
( E rad Davis ) m = E 0 2 F Davis ( R , θ ) [ Ψ m - 1 , 0 Davis ( R , θ ) + Ψ m + 1 , 1 Davis ( R , θ ) + + Ψ m + 1 , 0 Davis ( R , θ ) + Ψ m + 3 , 1 Davis ( R , θ ) + ] + E 0 x f w 0 G Davis ( R , θ ) × ( Ψ m , 0 Davis ( R , θ ) + Ψ m + 2 , 1 Davis ( R , θ ) + )
( E rad Davis ) 0 = E 0 2 F Davis ( R , θ ) × [ Ψ 1 , 1 Davis ( R , θ ) + Ψ 1 , 0 Davis ( R , θ ) + Ψ 3 , 2 Davis ( R , θ ) + Ψ 3 , 1 Davis ( R , θ ) + ] + E 0 x f w 0 G Davis ( R , θ ) × [ Ψ 0 , 0 Davis ( R , θ ) + Ψ 2 , 1 Davis ( R , θ ) + ]
( E rad ) m B 1 = E 0 exp ( i k z f ) exp ( - i R cos θ ) × sin m θ 2 ( s R ) m - 1 exp [ - ( x f 2 + y f 2 ) / w 0 2 ] × ( x f - i y f w 0 ) m - 1 ( m - 1 ) ! { 1 - 2 i s z f w 0 ( m - x f 2 + y f 2 w 0 2 ) + s 2 [ - R 2 sin 2 θ - 2 i R cos θ ( x f 2 + y f 2 w 0 2 ) + 2 i ( m + 1 ) R cos θ + R 2 sin 2 θ m ( x f 2 + y f 2 w 0 2 ) + R 2 sin 2 θ m ( m + 1 ) ( x f - i y f w 0 ) 2 - 4 i x f w 0 R cos θ m × ( x f - i y f w 0 ) - 2 m ( m + 1 ) z f 2 w 0 2 + 4 ( m + 1 ) z f 2 w 0 2 × ( x f 2 + y f 2 w 0 2 ) - 2 z f 2 w 0 2 ( x f 2 + y f 2 w 0 2 ) 2 ] + O ( s 3 ) } .
0 π sin m + 1 θ d θ T ( θ ) exp ( - i R cos θ ) P n m ( cos θ )
T ( θ ) = 1 , cos θ , cos 2 θ , sin 2 θ , .
0 π sin m + 1 θ d θ exp ( - i R cos θ ) P n m ( cos θ ) = 2 ( - i ) n + m ( n + m ) ! ( n - m ) ! G n m ( R ) ,
G n m ( R ) j n ( R ) / R m .
R 2 G n m + 2 R ( m + 1 ) G n m + R 2 G n m = L G n m ,
L = ( n - m ) ( n + m + 1 ) = ( n + 1 / 2 ) 2 - ( m + 1 / 2 ) 2
( g n m ) TM B 1 = ½ ( - i ) m - 1 s m - 1 exp [ - ( x f 2 + y f 2 ) / w 0 2 ] exp ( i k z f ) ( x f - i y f w 0 ) m - 1 ( m - 1 ) ! ( 1 - 2 i s z f w 0 ( m - x f 2 + y f 2 w 0 2 ) + s 2 { - L + L m [ x f 2 + y f 2 w 0 2 + 1 ( m + 1 ) ( x f - i y f w 0 ) 2 ] - 2 m ( m + 1 ) z f 2 w 0 2 + 4 ( m + 1 ) z f 2 w 0 2 ( x f 2 + y f 2 w 0 2 ) - 2 z f 2 w 0 2 ( x f 2 + y f 2 w 0 2 ) 2 } + 2 s 3 { i ( m + 1 ) L z f w 0 - 2 i ( m + 1 m ) L z f w 0 ( x f 2 + y f 2 w 0 2 ) + i m L z f w 0 ( x f 2 + y f 2 w 0 2 ) 2 + L m ( m + 1 ) z f w 0 [ i ( x f 2 + y f 2 w 0 2 ) + 2 y f w 0 ( x f - i y f w 0 ) ] ( x f 2 + y f 2 w 0 2 - m - 2 ) + 2 i 3 z f 3 w 0 3 [ m ( m + 1 ) ( m + 2 ) - 3 ( m + 1 ) ( m + 2 ) ( x f 2 + y f 2 w 0 2 ) + 3 ( m + 2 ) ( x f 2 + y f 2 w 0 2 ) 2 - ( x f 2 + y f 2 w 0 2 ) 3 ] } + O ( s 4 R G n m G n m ) ) ,
( g n 0 ) TM B 1 = i s x f w 0 exp ( i k z f ) exp [ - ( x f 2 + y f 2 ) / w 0 2 ] × { n ( n + 1 ) + 2 i s z f w 0 ( x f 2 + y f 2 w 0 2 - 2 ) + s 2 [ 1 / 2 ( x f 2 + y f 2 w 0 2 - 2 ) ( n ( n + 1 ) - 6 ) n ( n + 1 ) + 12 z f 2 w 0 2 ( x f 2 + y f 2 w 0 2 - 1 ) - 2 z f 2 w 0 2 ( x f 2 + y f 2 w 0 2 ) 2 + 4 ( x f 2 + y f 2 w 0 2 - 2 ) R G n 0 G n 0 ] + O ( s 3 ) } ,
( g n m ) TM mloc = Q n m 2 F Davis ( L 1 / 2 , π 2 ) × j = 0 p = 0 j Ψ j , p Davis ( L 1 / 2 , π 2 ) × ( δ j - 2 p + 1 , m + δ j - 2 p - 1 , m ) ,
Q n m = ( - i / L 1 / 2 ) m - 1 .
[ g n - m ( x f , y f , z f ) ] TM = [ g n m ( x f , - y f , z f ) ] TM .
( g n m ) TE mloc = Q n m 2 i F Davis ( L 1 / 2 , π 2 ) × j = 0 p = 0 j Ψ j , p Davis ( L 1 / 2 , π 2 ) × ( δ j - 2 p + 1 , m - δ j - 2 p - 1 , m )
[ g n m ( x f , y f , z f ) ] TE = ( - i ) m [ g n m ( y f , - x f , z f ) ] TM
[ g n - m ( x f , y f , z f ) ] TE = - [ g n m ( x f , - y f , z f ) ] TE
E r ( R , θ , ϕ ) = - i E 0 n = 1 ( - i ) n ( 2 n + 1 ) j n ( R ) R m = - n n ( g n m ) TM sin θ π n m ( θ ) exp ( i m ϕ ) , E θ ( R , θ , ϕ ) = - E 0 n = 1 ( - i ) n 2 n + 1 n ( n + 1 ) { j n ( R ) m = - n n ( g n m ) TE i m π n m ( θ ) exp ( i m ϕ ) + i [ j n - 1 ( R ) - n R j n ( R ) ] × m = - n n ( g n m ) TM τ n m ( θ ) exp ( i m ϕ ) } , E ϕ ( R , θ , ϕ ) = - E 0 n = 1 ( - i ) n 2 n + 1 n ( n + 1 ) { - j n ( R ) m = - n n ( g n m ) TE τ n m ( θ ) exp ( i m ϕ ) + i [ j n - 1 ( R ) - n R j n ( R ) ] × m = - n n ( g n m ) TM i m π n m ( θ ) exp ( i m ϕ ) } ,
B r ( R , θ , ϕ ) = - i B 0 n = 1 ( - i ) n ( 2 n + 1 ) j n ( R ) R ( g n m ) TE sin θ π n m ( θ ) exp ( i m ϕ ) , B θ ( R , θ , ϕ ) = - B 0 n = 1 ( - i ) n 2 n + 1 n ( n + 1 ) { - j n ( R ) m = - n n ( g n m ) TM i m π n m ( θ ) exp ( i m ϕ ) + i [ j n - 1 ( R ) - n R j n ( R ) ] × m = - n n ( g n m ) TE τ n m ( θ ) exp ( i m ϕ ) } , B ϕ ( R , θ , ϕ ) = - B 0 n = 1 ( - i ) n 2 n + 1 n ( n + 1 ) { j n ( R ) m = - n n ( g n m ) TM τ n m ( θ ) exp ( i m ϕ ) + i [ j n - 1 ( R ) - n R j n ( R ) ] × m = - n n ( g n m ) TE i m τ n m ( θ ) exp ( i m ϕ ) } .
E x ( x , y , z ) = E r sin θ cos ϕ + E θ cos θ cos ϕ - E ϕ sin ϕ , B y ( x , y , z ) = B r sin θ sin ϕ + B θ cos θ sin ϕ + B ϕ cos ϕ .
E x ( x , 0 , 0 ) = { E r ( r = x , θ = π / 2 , ϕ = 0 ) if x > 0 - E r ( r = x , θ = π / 2 , ϕ = π ) if x < 0 ,
B y ( x , 0 , 0 ) = { B ϕ ( r = x , θ = π / 2 , ϕ = 0 ) if x > 0 - B ϕ ( r = x , θ = π / 2 , ϕ = π ) if x < 0 .
E x ( x , 0 , 0 ) E 0 { exp [ - ( x - x f ) 2 / w 0 2 ] + 1 exp [ - ( x + x f ) 2 / w 0 2 ] + δ 1 H ( x ) } , B y ( x , 0 , 0 ) B 0 { exp [ - ( x - x f ) 2 / w 0 2 ] + 2 exp [ - ( x + x f ) 2 / w 0 2 ] + δ 2 H ( x ) } ,
I n m = 0 π sin m + 1 θ d θ P n m ( cos θ ) exp ( - i R cos θ ) ,
I n 0 = 0 π sin θ d θ P n 0 ( cos θ ) exp ( - i R cos θ ) = 2 ( - i ) n j n ( R ) .
sin θ P n m ( cos θ ) = ( n + m ) ( n + m - 1 ) 2 n + 1 P n - 1 m - 1 ( cos θ ) - ( n - m + 1 ) ( n - m + 2 ) 2 n + 1 P n + 1 m - 1 ( cos θ ) ,
I n m = ( n + m ) ( n + m - 1 ) 2 n + 1 I n - 1 m - 1 - ( n - m + 1 ) ( n - m + 2 ) 2 n + 1 I n + 1 m - 1 .
I n m - 1 = 2 ( - i ) n + m - 1 ( n + m - 1 ) ! ( n - m + 1 ) ! j n ( R ) R m - 1 ,
j n - 1 ( R ) + j n + 1 ( R ) = 2 n + 1 R j n ( R ) ,
I n m = 2 ( - i ) n + m ( n + m ) ! ( n - m ) ! j n ( R ) R m .

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