Abstract

We computed the Debye series p=1 and p=2 terms of the Mie scattered intensity as a function of scattering angle and delay time for a linearly polarized plane wave pulse incident on a spherical dielectric particle and physically interpreted the resulting numerical data. Radiation shed by electromagnetic surface waves plays a prominent role in the scattered intensity. We determined the surface wave phase and damping rate and studied the structure of the p=1,2 surface wave glory in the time domain.

© 2011 Optical Society of America

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References

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  1. H. C. van de Hulst, “Rigorous scattering theory for spheres of arbitrary size,” in Light Scattering by Small Particles (Dover, 1957), pp. 114–130.
  2. M. Kerker, “Scattering by a sphere,” in The Scattering of Light and Other Electromagnetic Radiation (Academic, 1969), pp. 27–96.
  3. C. F. Bohren and D. R. Huffman, “Absorption and scattering by a sphere,” in Absorption and Scattering of Light by Small Particles (Wiley-Interscience, 1983), pp. 82–129.
  4. P. Debye “Das Elektromagnetische Feld um einen Zylinder und die Theorie des Regenbogens,” in Geometrical Aspects of Scattering, P.L.Marston, ed., Milestone Series (SPIE, 1994), Vol.  MS89, pp. 198–204.
  5. B. Van der Pol and 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, 825–864 (1937).
  6. E. S. C. Ching, P. T. Leung, and K. Young, “The role of quasinormal modes,” in Optical Processes in MicrocavitiesR.K.Chang and A.J.Campillo, eds. (World Scientific, 1996), pp. 1–75.
    [CrossRef]
  7. R. Greenler, “Rainbows,” in Rainbows, Halos, and Glories (Cambridge University, 1980), pp. 8–10.
  8. L. Mees, G. Gouesbet, and G. Grehan, “Scattering of laser pulses (plane wave and focused Gaussian beam) by spheres,” Appl. Opt. 40, 2546–2550 (2001).
    [CrossRef]
  9. L. Mees, G. Grehan, and G. Gouesbet, “Time-resolved scattering diagrams for a sphere illuminated by plane wave and focused short pulses,” Opt. Commun. 194, 59–65 (2001).
    [CrossRef]
  10. H. Bech and A. Leder, “Particle sizing by ultrashort laser pulses—numerical simulation,” Optik 115, 205–217 (2004).
    [CrossRef]
  11. H. Bech and A. Leder, “Particle sizing by time-resolved Mie calculations—a numerical study,” Optik 117, 40–47 (2006).
    [CrossRef]
  12. S. Bakic, C. Heinisch, N. Damaschke, T. Tschudi, and C. Tropea, “Time integrated detection of femtosecond laser pulses scattered by small droplets,” Appl. Opt. 47, 523–530 (2008).
    [CrossRef] [PubMed]
  13. P. Laven, “Separating diffraction from scattering: the million-dollar challenge,” J. Nanophotonics 4, 041593 (2010).
    [CrossRef]
  14. J. B. Keller, “A geometrical theory of diffraction,” in Calculus of Variations and its Applications, L.M.Graves, ed., Proceedings of Symposia in Applied Mathematics (McGraw-Hill, 1958), Vol.  3, pp. 27–52.
  15. J. A. Lock and P. Laven, “Mie scattering in the time domain. Part 2. The role of diffraction,” J. Opt. Soc. Am. A 28, 1096–1106(2011).
    [CrossRef]
  16. D. Q. Chowdhury, S. C. Hill, and P. W. Barber, “Time dependence of internal intensity of a dielectric sphere on and near resonance,” J. Opt. Soc. Am. A 9, 1364–1373(1992).
    [CrossRef]
  17. E. E. M. Khaled, D. Q. Chowdhury, S. C. Hill, and P. W. Barber, “Internal and scattered time-dependent intensity of a dielectric sphere illuminated with a pulsed Gaussian beam,” J. Opt. Soc. Am. A 11, 2065–2071 (1994).
    [CrossRef]
  18. K. S. Schifrin and I. G. Zolotov, “Quasi-stationary scattering of electromagnetic pulses by spherical particles,” Appl. Opt. 33, 7798–7804 (1994).
    [CrossRef]
  19. The International Association for the Properties of Water and Steam, “Release on refractive index of ordinary water substance as a function of wavelength, temperature and pressure” (International Association for the Properties of Water and Steam, 1997), www.iapws.org/relguide/rindex.pdf.
  20. R. M. Pope and E. S. Fry, “Absorption spectrum (380–700 nm) of pure water. II. Integrating cavity measurements,” Appl. Opt. 36, 8710–8723 (1997).
    [CrossRef]
  21. W. J. Wiscombe, “Improved Mie scattering algorithms,” Appl. Opt. 19, 1505–1509 (1980).
    [CrossRef] [PubMed]
  22. H. M. Nussenzveig, “High-frequency scattering by a transparent sphere. I. Direct reflection and transmission,” J. Math. Phys. 10, 82–124 (1969).
    [CrossRef]
  23. H. M. Nussenzveig, “High-frequency scattering by an impenetrable sphere,” Ann. Phys. 34, 23–95 (1965).
    [CrossRef]
  24. M.Abramowitz and I.A.Stegun, eds., “Bessel functions of integer order,” in Handbook of Mathematical Functions (National Bureau of Standards, 1964), p. 366, Eq. (9.3.3).
  25. H. C. van de Hulst, “The reflected and refracted light,” in Light Scattering by Small Particles (Dover, 1957), p. 212.
  26. K. W. Ford and J. A. Wheeler, “Semiclassical description of scattering,” Ann. Phys. 7, 259–286 (1959).
    [CrossRef]
  27. M. V. Berry and K. E. Mount, “Semiclassical approximations in wave mechanics,” Rep. Prog. Phys. 35, 315–397(1972).
    [CrossRef]
  28. V. Khare, “Short-wavelength scattering of electromagnetic waves by a homogeneous dielectric sphere,” Ph.D.dissertation (University of Rochester, 1975), Eqs. 8.32–8.35, 8.40–8.42.
  29. M.Abramowitz and I.A.Stegun, eds., “Bessel functions of integer order,” in Handbook of Mathematical Functions (National Bureau of Standards, 1964), p. 366, Eq. (9.3.4).
  30. M.Abramowitz and I.A.Stegun, eds., “Bessel functions of fractional order,” in Handbook of Mathematical Functions (National Bureau of Standards, 1964), p. 478, Table 10.13.
  31. H. M. Nussenzveig, “Uniform approximation in scattering by spheres,” J. Phys. A 21, 81–109 (1988).
    [CrossRef]
  32. H. M. Nussenzveig and W. J. Wiscombe, “Complex angular momentum approximation to hard-core scattering,” Phys. Rev. A 43, 2093–2112 (1991).
    [CrossRef] [PubMed]
  33. I. S. Gradshteyn and I. M. Ryzhik, “Definite integrals of elementary functions,” in Table of Integrals, Series, and Products(Academic, 1965), p. 495, Eq. (3.952.1).
  34. H. C. van de Hulst, “Waves at the surface of a perfect conductor,” in Light Scattering by Small Particles (Dover, 1957), p. 368.
  35. H. C. van de Hulst, “Intensity,” in Light Scattering by Small Particles (Dover, 1957), p. 205.
  36. H. C. van de Hulst, “Theory based on Mie’s formula,” in Light Scattering by Small Particles (Dover, 1957), p. 253.
  37. V. Khare, “Short-wavelength scattering of electromagnetic waves by a homogeneous dielectric sphere,” Ph.D. dissertation (University of Rochester, 1975), Eqs. 8.10, 8.14b.
  38. G. Arfken, “Bessel functions,” in Mathematical Methods for Physicists, 3rd ed. (Academic, 1985), p. 620.
  39. H. C. van de Hulst, “Phase,” in Light Scattering by Small Particles (Dover, 1957), p. 207.

2011 (1)

2010 (1)

P. Laven, “Separating diffraction from scattering: the million-dollar challenge,” J. Nanophotonics 4, 041593 (2010).
[CrossRef]

2008 (1)

2006 (1)

H. Bech and A. Leder, “Particle sizing by time-resolved Mie calculations—a numerical study,” Optik 117, 40–47 (2006).
[CrossRef]

2004 (1)

H. Bech and A. Leder, “Particle sizing by ultrashort laser pulses—numerical simulation,” Optik 115, 205–217 (2004).
[CrossRef]

2001 (2)

L. Mees, G. Grehan, and G. Gouesbet, “Time-resolved scattering diagrams for a sphere illuminated by plane wave and focused short pulses,” Opt. Commun. 194, 59–65 (2001).
[CrossRef]

L. Mees, G. Gouesbet, and G. Grehan, “Scattering of laser pulses (plane wave and focused Gaussian beam) by spheres,” Appl. Opt. 40, 2546–2550 (2001).
[CrossRef]

1997 (1)

1994 (2)

1992 (1)

1991 (1)

H. M. Nussenzveig and W. J. Wiscombe, “Complex angular momentum approximation to hard-core scattering,” Phys. Rev. A 43, 2093–2112 (1991).
[CrossRef] [PubMed]

1988 (1)

H. M. Nussenzveig, “Uniform approximation in scattering by spheres,” J. Phys. A 21, 81–109 (1988).
[CrossRef]

1980 (1)

1972 (1)

M. V. Berry and K. E. Mount, “Semiclassical approximations in wave mechanics,” Rep. Prog. Phys. 35, 315–397(1972).
[CrossRef]

1969 (1)

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

1965 (1)

H. M. Nussenzveig, “High-frequency scattering by an impenetrable sphere,” Ann. Phys. 34, 23–95 (1965).
[CrossRef]

1959 (1)

K. W. Ford and J. A. Wheeler, “Semiclassical description of scattering,” Ann. Phys. 7, 259–286 (1959).
[CrossRef]

1937 (1)

B. Van der Pol and 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, 825–864 (1937).

Arfken, G.

G. Arfken, “Bessel functions,” in Mathematical Methods for Physicists, 3rd ed. (Academic, 1985), p. 620.

Bakic, S.

Barber, P. W.

Bech, H.

H. Bech and A. Leder, “Particle sizing by time-resolved Mie calculations—a numerical study,” Optik 117, 40–47 (2006).
[CrossRef]

H. Bech and A. Leder, “Particle sizing by ultrashort laser pulses—numerical simulation,” Optik 115, 205–217 (2004).
[CrossRef]

Berry, M. V.

M. V. Berry and K. E. Mount, “Semiclassical approximations in wave mechanics,” Rep. Prog. Phys. 35, 315–397(1972).
[CrossRef]

Bohren, C. F.

C. F. Bohren and D. R. Huffman, “Absorption and scattering by a sphere,” in Absorption and Scattering of Light by Small Particles (Wiley-Interscience, 1983), pp. 82–129.

Bremmer, H.

B. Van der Pol and 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, 825–864 (1937).

Ching, E. S. C.

E. S. C. Ching, P. T. Leung, and K. Young, “The role of quasinormal modes,” in Optical Processes in MicrocavitiesR.K.Chang and A.J.Campillo, eds. (World Scientific, 1996), pp. 1–75.
[CrossRef]

Chowdhury, D. Q.

Damaschke, N.

Debye, P.

P. Debye “Das Elektromagnetische Feld um einen Zylinder und die Theorie des Regenbogens,” in Geometrical Aspects of Scattering, P.L.Marston, ed., Milestone Series (SPIE, 1994), Vol.  MS89, pp. 198–204.

Ford, K. W.

K. W. Ford and J. A. Wheeler, “Semiclassical description of scattering,” Ann. Phys. 7, 259–286 (1959).
[CrossRef]

Fry, E. S.

Gouesbet, G.

L. Mees, G. Grehan, and G. Gouesbet, “Time-resolved scattering diagrams for a sphere illuminated by plane wave and focused short pulses,” Opt. Commun. 194, 59–65 (2001).
[CrossRef]

L. Mees, G. Gouesbet, and G. Grehan, “Scattering of laser pulses (plane wave and focused Gaussian beam) by spheres,” Appl. Opt. 40, 2546–2550 (2001).
[CrossRef]

Gradshteyn, I. S.

I. S. Gradshteyn and I. M. Ryzhik, “Definite integrals of elementary functions,” in Table of Integrals, Series, and Products(Academic, 1965), p. 495, Eq. (3.952.1).

Greenler, R.

R. Greenler, “Rainbows,” in Rainbows, Halos, and Glories (Cambridge University, 1980), pp. 8–10.

Grehan, G.

L. Mees, G. Grehan, and G. Gouesbet, “Time-resolved scattering diagrams for a sphere illuminated by plane wave and focused short pulses,” Opt. Commun. 194, 59–65 (2001).
[CrossRef]

L. Mees, G. Gouesbet, and G. Grehan, “Scattering of laser pulses (plane wave and focused Gaussian beam) by spheres,” Appl. Opt. 40, 2546–2550 (2001).
[CrossRef]

Heinisch, C.

Hill, S. C.

Huffman, D. R.

C. F. Bohren and D. R. Huffman, “Absorption and scattering by a sphere,” in Absorption and Scattering of Light by Small Particles (Wiley-Interscience, 1983), pp. 82–129.

Keller, J. B.

J. B. Keller, “A geometrical theory of diffraction,” in Calculus of Variations and its Applications, L.M.Graves, ed., Proceedings of Symposia in Applied Mathematics (McGraw-Hill, 1958), Vol.  3, pp. 27–52.

Kerker, M.

M. Kerker, “Scattering by a sphere,” in The Scattering of Light and Other Electromagnetic Radiation (Academic, 1969), pp. 27–96.

Khaled, E. E. M.

Khare, V.

V. Khare, “Short-wavelength scattering of electromagnetic waves by a homogeneous dielectric sphere,” Ph.D. dissertation (University of Rochester, 1975), Eqs. 8.10, 8.14b.

V. Khare, “Short-wavelength scattering of electromagnetic waves by a homogeneous dielectric sphere,” Ph.D.dissertation (University of Rochester, 1975), Eqs. 8.32–8.35, 8.40–8.42.

Laven, P.

J. A. Lock and P. Laven, “Mie scattering in the time domain. Part 2. The role of diffraction,” J. Opt. Soc. Am. A 28, 1096–1106(2011).
[CrossRef]

P. Laven, “Separating diffraction from scattering: the million-dollar challenge,” J. Nanophotonics 4, 041593 (2010).
[CrossRef]

Leder, A.

H. Bech and A. Leder, “Particle sizing by time-resolved Mie calculations—a numerical study,” Optik 117, 40–47 (2006).
[CrossRef]

H. Bech and A. Leder, “Particle sizing by ultrashort laser pulses—numerical simulation,” Optik 115, 205–217 (2004).
[CrossRef]

Leung, P. T.

E. S. C. Ching, P. T. Leung, and K. Young, “The role of quasinormal modes,” in Optical Processes in MicrocavitiesR.K.Chang and A.J.Campillo, eds. (World Scientific, 1996), pp. 1–75.
[CrossRef]

Lock, J. A.

Mees, L.

L. Mees, G. Gouesbet, and G. Grehan, “Scattering of laser pulses (plane wave and focused Gaussian beam) by spheres,” Appl. Opt. 40, 2546–2550 (2001).
[CrossRef]

L. Mees, G. Grehan, and G. Gouesbet, “Time-resolved scattering diagrams for a sphere illuminated by plane wave and focused short pulses,” Opt. Commun. 194, 59–65 (2001).
[CrossRef]

Mount, K. E.

M. V. Berry and K. E. Mount, “Semiclassical approximations in wave mechanics,” Rep. Prog. Phys. 35, 315–397(1972).
[CrossRef]

Nussenzveig, H. M.

H. M. Nussenzveig and W. J. Wiscombe, “Complex angular momentum approximation to hard-core scattering,” Phys. Rev. A 43, 2093–2112 (1991).
[CrossRef] [PubMed]

H. M. Nussenzveig, “Uniform approximation in scattering by spheres,” J. Phys. A 21, 81–109 (1988).
[CrossRef]

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

H. M. Nussenzveig, “High-frequency scattering by an impenetrable sphere,” Ann. Phys. 34, 23–95 (1965).
[CrossRef]

Pope, R. M.

Ryzhik, I. M.

I. S. Gradshteyn and I. M. Ryzhik, “Definite integrals of elementary functions,” in Table of Integrals, Series, and Products(Academic, 1965), p. 495, Eq. (3.952.1).

Schifrin, K. S.

Tropea, C.

Tschudi, T.

van de Hulst, H. C.

H. C. van de Hulst, “The reflected and refracted light,” in Light Scattering by Small Particles (Dover, 1957), p. 212.

H. C. van de Hulst, “Theory based on Mie’s formula,” in Light Scattering by Small Particles (Dover, 1957), p. 253.

H. C. van de Hulst, “Waves at the surface of a perfect conductor,” in Light Scattering by Small Particles (Dover, 1957), p. 368.

H. C. van de Hulst, “Rigorous scattering theory for spheres of arbitrary size,” in Light Scattering by Small Particles (Dover, 1957), pp. 114–130.

H. C. van de Hulst, “Intensity,” in Light Scattering by Small Particles (Dover, 1957), p. 205.

H. C. van de Hulst, “Phase,” in Light Scattering by Small Particles (Dover, 1957), p. 207.

Van der Pol, B.

B. Van der Pol and 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, 825–864 (1937).

Wheeler, J. A.

K. W. Ford and J. A. Wheeler, “Semiclassical description of scattering,” Ann. Phys. 7, 259–286 (1959).
[CrossRef]

Wiscombe, W. J.

H. M. Nussenzveig and W. J. Wiscombe, “Complex angular momentum approximation to hard-core scattering,” Phys. Rev. A 43, 2093–2112 (1991).
[CrossRef] [PubMed]

W. J. Wiscombe, “Improved Mie scattering algorithms,” Appl. Opt. 19, 1505–1509 (1980).
[CrossRef] [PubMed]

Young, K.

E. S. C. Ching, P. T. Leung, and K. Young, “The role of quasinormal modes,” in Optical Processes in MicrocavitiesR.K.Chang and A.J.Campillo, eds. (World Scientific, 1996), pp. 1–75.
[CrossRef]

Zolotov, I. G.

Ann. Phys. (2)

H. M. Nussenzveig, “High-frequency scattering by an impenetrable sphere,” Ann. Phys. 34, 23–95 (1965).
[CrossRef]

K. W. Ford and J. A. Wheeler, “Semiclassical description of scattering,” Ann. Phys. 7, 259–286 (1959).
[CrossRef]

Appl. Opt. (5)

J. Math. Phys. (1)

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

J. Nanophotonics (1)

P. Laven, “Separating diffraction from scattering: the million-dollar challenge,” J. Nanophotonics 4, 041593 (2010).
[CrossRef]

J. Opt. Soc. Am. A (3)

J. Phys. A (1)

H. M. Nussenzveig, “Uniform approximation in scattering by spheres,” J. Phys. A 21, 81–109 (1988).
[CrossRef]

Opt. Commun. (1)

L. Mees, G. Grehan, and G. Gouesbet, “Time-resolved scattering diagrams for a sphere illuminated by plane wave and focused short pulses,” Opt. Commun. 194, 59–65 (2001).
[CrossRef]

Optik (2)

H. Bech and A. Leder, “Particle sizing by ultrashort laser pulses—numerical simulation,” Optik 115, 205–217 (2004).
[CrossRef]

H. Bech and A. Leder, “Particle sizing by time-resolved Mie calculations—a numerical study,” Optik 117, 40–47 (2006).
[CrossRef]

Philos. Mag. (1)

B. Van der Pol and 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, 825–864 (1937).

Phys. Rev. A (1)

H. M. Nussenzveig and W. J. Wiscombe, “Complex angular momentum approximation to hard-core scattering,” Phys. Rev. A 43, 2093–2112 (1991).
[CrossRef] [PubMed]

Rep. Prog. Phys. (1)

M. V. Berry and K. E. Mount, “Semiclassical approximations in wave mechanics,” Rep. Prog. Phys. 35, 315–397(1972).
[CrossRef]

Other (20)

V. Khare, “Short-wavelength scattering of electromagnetic waves by a homogeneous dielectric sphere,” Ph.D.dissertation (University of Rochester, 1975), Eqs. 8.32–8.35, 8.40–8.42.

M.Abramowitz and I.A.Stegun, eds., “Bessel functions of integer order,” in Handbook of Mathematical Functions (National Bureau of Standards, 1964), p. 366, Eq. (9.3.4).

M.Abramowitz and I.A.Stegun, eds., “Bessel functions of fractional order,” in Handbook of Mathematical Functions (National Bureau of Standards, 1964), p. 478, Table 10.13.

J. B. Keller, “A geometrical theory of diffraction,” in Calculus of Variations and its Applications, L.M.Graves, ed., Proceedings of Symposia in Applied Mathematics (McGraw-Hill, 1958), Vol.  3, pp. 27–52.

M.Abramowitz and I.A.Stegun, eds., “Bessel functions of integer order,” in Handbook of Mathematical Functions (National Bureau of Standards, 1964), p. 366, Eq. (9.3.3).

H. C. van de Hulst, “The reflected and refracted light,” in Light Scattering by Small Particles (Dover, 1957), p. 212.

E. S. C. Ching, P. T. Leung, and K. Young, “The role of quasinormal modes,” in Optical Processes in MicrocavitiesR.K.Chang and A.J.Campillo, eds. (World Scientific, 1996), pp. 1–75.
[CrossRef]

R. Greenler, “Rainbows,” in Rainbows, Halos, and Glories (Cambridge University, 1980), pp. 8–10.

H. C. van de Hulst, “Rigorous scattering theory for spheres of arbitrary size,” in Light Scattering by Small Particles (Dover, 1957), pp. 114–130.

M. Kerker, “Scattering by a sphere,” in The Scattering of Light and Other Electromagnetic Radiation (Academic, 1969), pp. 27–96.

C. F. Bohren and D. R. Huffman, “Absorption and scattering by a sphere,” in Absorption and Scattering of Light by Small Particles (Wiley-Interscience, 1983), pp. 82–129.

P. Debye “Das Elektromagnetische Feld um einen Zylinder und die Theorie des Regenbogens,” in Geometrical Aspects of Scattering, P.L.Marston, ed., Milestone Series (SPIE, 1994), Vol.  MS89, pp. 198–204.

I. S. Gradshteyn and I. M. Ryzhik, “Definite integrals of elementary functions,” in Table of Integrals, Series, and Products(Academic, 1965), p. 495, Eq. (3.952.1).

H. C. van de Hulst, “Waves at the surface of a perfect conductor,” in Light Scattering by Small Particles (Dover, 1957), p. 368.

H. C. van de Hulst, “Intensity,” in Light Scattering by Small Particles (Dover, 1957), p. 205.

H. C. van de Hulst, “Theory based on Mie’s formula,” in Light Scattering by Small Particles (Dover, 1957), p. 253.

V. Khare, “Short-wavelength scattering of electromagnetic waves by a homogeneous dielectric sphere,” Ph.D. dissertation (University of Rochester, 1975), Eqs. 8.10, 8.14b.

G. Arfken, “Bessel functions,” in Mathematical Methods for Physicists, 3rd ed. (Academic, 1985), p. 620.

H. C. van de Hulst, “Phase,” in Light Scattering by Small Particles (Dover, 1957), p. 207.

The International Association for the Properties of Water and Steam, “Release on refractive index of ordinary water substance as a function of wavelength, temperature and pressure” (International Association for the Properties of Water and Steam, 1997), www.iapws.org/relguide/rindex.pdf.

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Figures (9)

Fig. 1
Fig. 1

Scattered intensity in the time domain calculated using Mie theory as a function of the delay time t and scattering angle θ for a raised cosine pulse with k 0 = 9.67 × 10 6 m 1 and τ = 5 fs incident on a spherical particle of radius a = 10 μm and refractive index 1.3326 + i ( 1.67 × 10 8 ) .

Fig. 2
Fig. 2

p = 1 component of the scattered intensity in the time domain as a function of the delay time t and scattering angle θ for the pulse and particle parameters of Fig. 1. The false color scale for intensity in Fig. 1 has been reset in Fig. 2 to emphasize the low-intensity components. The ray theory scattering angle as a function of delay time of Eqs. (9, 10, 11) has been overlaid on the figure. The grid ticks on the ray theory line correspond to the ray impact parameter b in intervals of Δ b = 0.1 .

Fig. 3
Fig. 3

Pictorial representation of the p = 1 surface waves A and B with θ < 180 ° , and C and D with θ > 180 ° . A and D are the short path length surface waves, and B and C are the long path length surface waves.

Fig. 4
Fig. 4

Similar to Fig. 2, but also showing scattering for θ > 180 ° . The surface waves A + C and D + B of Fig. 3 intersect at θ = 180 ° to form the p = 1 surface wave glory.

Fig. 5
Fig. 5

p = 2 component of the scattered intensity in the time domain as a function of the delay time t and scattering angle θ for the pulse and particle parameters of Fig. 1. The ray theory scattering angle as a function of delay time of Eqs. (12, 13) has been overlaid on the figure.

Fig. 6
Fig. 6

The quantity v / c , where v is the effective surface wave velocity on the sphere surface, as a function of the scattering angle θ obtained [blue (bottom) curves] from the angular advancement of the phase of the p = 1 component of the scattered field when a monochromatic plane wave with k = 9.67 × 10 6 m 1 is incident on a spherical particle with the parameters of Fig. 1, and from the temporal maximum of the p = 1 scattered pulse [red (top) curves]. The pulse and particle parameters are the same as in Fig. 1 except that the temporal pixel width is Δ t 0.004 fs .

Fig. 7
Fig. 7

Maximum of the p = 2 intensity of the scattered pulse multiplied by | sin ( θ ) | as a function of scattering angle θ for the pulse and particle parameters of Fig. 1.

Fig. 8
Fig. 8

Maximum of the p = 1 intensity of the scattered pulse multiplied by | sin ( θ ) | for the surface wave A + C of Fig. 3 as a function of the scattering angle θ for the pulse and particle parameters of Fig. 1. The maximum of surface wave A was determined for θ < 180 ° , and the maximum of C was determined for θ > 180 ° .

Fig. 9
Fig. 9

Fine-resolution view of Fig. 4 in the vicinity of the p = 1 surface wave glory region.

Equations (45)

Equations on this page are rendered with MathJax. Learn more.

E pulse ( z , t ) = ( d k / 2 π ) A ( k ) exp [ i k ( z c t ) ] .
E scatt ( t , θ ) = ( d k / 2 π ) A ( k ) E Mie ( k , θ ) exp ( i c k t ) .
E pulse ( z , t ) = E 0 cos 2 [ π ( z c t ) t / 2 c τ ] exp [ i k 0 ( z c t ) ] u x
E pulse ( z , t ) = E 0 exp [ ( z c t ) 2 / σ 2 ] exp [ i k 0 ( z c t ) ] u x ,
A ( k ) = σ E 0 ( π ) 1 / 2 exp [ σ 2 ( k k 0 ) 2 / 4 ] ,
E ray ( k , θ ) = k a B ( θ ) exp [ i k a Φ ( θ ) ] ,
I ray ( t , θ ) = ( k 0 a ) 2 B 2 ( θ ) exp { 2 [ c t a Φ ( θ ) ] 2 / σ 2 ] } ,
t = a Φ ( θ ) / c ,
θ = 2 θ i 2 θ t ,
t = 2 a [ 1 cos ( θ i ) ] / c + 2 a N cos ( θ t ) / c ,
sin ( θ i ) = N sin ( θ t ) ,
θ = π + 2 θ i 4 θ t ,
t = 2 a [ 1 cos ( θ i ) ] / c + 4 a N cos ( θ t ) / c ,
E surface waves ( θ ) = ( k a ) 5 / 6 [ T 2 ( θ ) B 2 ( k , θ ) cos ( φ ) u θ + T 1 ( θ ) B 1 ( k , θ ) sin ( φ ) u φ ] exp [ i k a Φ ( k , θ ) ] .
T j ( θ ) = exp ( i π / 12 ) 2 1 / 6 K j ξ / { [ π ( N 2 1 ) sin ( θ ) ] 1 / 2 [ Ai ( X ) ] 2 } ,
B j ( k , θ ) = exp { ξ [ 3 1 / 2 X ( k a ) 1 / 3 / 2 4 / 3 K j / ( N 2 1 ) 1 / 2 ] } ,
Φ ( k , θ ) = 2 + 2 ( N 2 1 ) 1 / 2 + ξ { 1 + X / [ 2 4 / 3 ( k a ) 2 / 3 ] } .
ξ = θ θ c
v / c = ( Δ Φ / Δ θ ) 1 < 1 .
( Δ Φ / Δ θ ) 1 = { 1 + X / [ 2 4 / 3 ( k a ) 2 / 3 ] } 1 = 0.958 .
E j scatt ( t , θ ) = [ σ T j ( θ ) / 2 π 1 / 2 ] d k exp { [ σ 2 ( k k 0 ) 2 / 4 ] } ( k a ) 5 / 6 B j ( k , θ ) exp [ i k a Φ ( k , θ ) i c k t ] .
B j ( k , θ ) = B j 0 ( k 0 , θ ) + O ( k k 0 ) ,
Φ ( k , θ ) = Φ 0 ( k 0 , θ ) + ( k k 0 ) Φ 1 ( k 0 , θ ) + O [ ( k k 0 ) 2 ] .
E j scatt ( t , θ ) = [ ( k 0 a ) 5 / 6 σ T j ( θ ) B j 0 ( k 0 , θ ) / 2 π 1 / 2 ] d k exp [ σ 2 ( k k 0 ) 2 / 4 ] × exp { i k a [ Φ 0 ( k 0 , θ ) + ( k k 0 ) Φ 1 ( k 0 , θ ) i c k t ] } ,
I j scatt ( t , θ ) = ( k 0 a ) 5 / 3 [ σ T j ( θ ) B j 0 ( k 0 , θ ) / 2 ] 2 [ ( σ 2 / 4 ) 2 + ( Φ 1 a ) 2 ] 1 / 2 × exp { 2 σ 2 ( c t a Φ 0 k 0 a Φ 1 ) 2 / [ ( σ 2 / 4 ) + ( Φ 1 a ) 2 ] } .
c t = a Φ 0 + k 0 a Φ 1 ,
B j 0 ( k 0 , θ ) = exp { ξ [ 3 1 / 2 X ( k 0 a ) 1 / 3 / 2 4 / 3 K j / ( N 2 1 ) 1 / 2 ] } ,
Φ 0 ( k 0 , θ ) = 2 + 2 ( N 2 1 ) 1 / 2 + ξ + ξ X / [ 2 4 / 3 ( k 0 a ) 2 / 3 ] ,
Φ 1 ( k 0 , θ ) = ( 2 / 3 ) ξ X a / [ 2 4 / 3 ( k 0 a ) 5 / 3 ] ,
v / c = { 1 + ( 1 / 3 ) X / [ 2 4 / 3 ( k 0 a ) 2 / 3 ] } 1 = 0.9855 ,
r = a { 1 + X / [ 2 4 / 3 ( k a ) 2 / 3 ] } = a [ 1 + 0.928 / ( k a ) 2 / 3 ] .
k r = ( k a + 1 / 2 ) + z ( k a + 1 / 2 ) 1 / 3 k a + z ( k a ) 1 / 3
r = a [ 1 + 0.810 / ( k a ) 2 / 3 ] ,
E scatt ( π δ ) = S a ( π δ ) u x S s ( π δ ) [ cos ( 2 φ ) u x + sin ( 2 φ ) u y ] ,
S a ( π δ ) = ( k a ) 4 / 3 ( N 2 1 ) J 0 ( k a δ ) T ( π δ ) B ( k , π δ ) exp [ i k a Φ ( k , π δ ) ] ,
S s ( π δ ) = ( k a ) 4 / 3 ( N 2 + 1 ) J 2 ( k a δ ) T ( π δ ) B ( k , π δ ) exp [ i k a Φ ( k , π δ ) ] ,
T ( π δ ) = exp ( i π / 6 ) ( π θ c ) / { 2 1 / 3 ( N 2 1 ) 1 / 2 [ Ai ( X ) ] 2 } ,
B ( k , π δ ) = exp { ( π θ c ) [ 3 1 / 2 X ( k a ) 1 / 3 / 2 4 / 3 ( N 2 + 1 ) / 2 ( N 2 1 ) 1 / 2 ] } ,
Φ ( k , π δ ) = 2 + 2 ( N 2 1 ) 1 / 2 + ( π θ c ) { 1 + X / [ 2 4 / 3 ( k a ) 2 / 3 ] } .
E scatt ( π δ ) = T ( π δ θ c ) B ( k , π δ θ c ) exp [ i k a Φ ( k , π δ θ c ) ] i T ( π + δ θ c ) B ( k , π + δ θ c ) exp [ i k a Φ ( k , π + δ θ c ) ] .
I s c a t t ( t , θ ) = W S 2 ( t , θ ) + 2 W S ( t , θ ) W L ( t , θ ) sin { k 0 a [ Φ ( k 0 , π + δ θ c ) Φ ( k 0 , π δ θ c ) ] } + W L 2 ( t , θ ) ,
W S ( t , θ ) = ( k 0 a ) 4 / 3 T ( π δ θ c ) B 0 ( k 0 , π δ θ c ) exp { [ c t a Φ 0 ( k 0 , π δ θ c ) ] 2 / σ 2 } ,
W L ( t , θ ) = ( k 0 a ) 4 / 3 T ( π + δ θ c ) B 0 ( k 0 , π + δ θ c ) exp { [ c t a Φ 0 ( k 0 , π + δ θ c ) ] 2 / σ 2 } .
I scatt ( t , θ ) = W S 2 ( t , θ ) + 2 W S ( t , θ ) W L ( t , θ ) sin ( 2 k 0 a δ ) + W L 2 ( t , θ ) ,
δ = ( M 1 / 4 ) π / k 0 a = 1.40 ° , 3.26 ° , 5.12 ° , 6.98 ° , 8.85 ° , 10.71 ° , and 12.57 ° ,

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