Abstract

Rainbows, coronas and glories are caused by the scattering of sunlight from water droplets in the atmosphere. Although these optical phenomena are seen fairly frequently, even scientifically minded people sometimes struggle to provide explanations for their formation. This paper offers explanations of these phenomena based on numerical computations of the scattering of a 5fs pulse of red light by a spherical droplet of water. The results reveal the intricate details of the various scattering mechanisms, some of which are essentially undetectable except in the time domain.

© 2011 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. P. Debye, “Das elektromagnetische Feld um einen Zylinder und die Theorie des Regenbogens,” Phys. Z. 9, 775–778(1908).
  2. 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).
  3. E. A. Hovenac and 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]
  4. 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]
  5. K. S. Schifrin and I. G. Zolotov, “Quasi-stationary scattering of electromagnetic pulses by spherical particles,” Appl. Opt. 33, 7798–7804 (1994).
    [CrossRef]
  6. L. Méès, G. Gouesbet, and G. Gréhan, “Scattering of laser pulses (plane wave and focused Gaussian beam) by spheres,” Appl. Opt. 40, 2546–2550 (2001).
    [CrossRef]
  7. L. Méès, G. Gréhan, 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]
  8. Y. P. Han, L. Méès, K. F. Ren, G. Gréhan, Z. S. Wu, and G. Gouesbet, “Far scattered field from a spheroid under a femtosecond pulsed illumination in a generalized Lorenz-Mie theory framework,” Opt. Commun. 231, 71–77 (2004).
    [CrossRef]
  9. H. Bech and A. Leder, “Particle sizing by ultrashort laser pulses—numerical simulation,” Optik 115, 205–217 (2004).
    [CrossRef]
  10. H. Bech and A. Leder, “Particle sizing by time-resolved Mie calculations—A numerical study,” Optik 117, 40–47 (2006).
    [CrossRef]
  11. C. Calba, C. Rozé, T. Girasole, and L. Méès, “Monte Carlo simulation of the interaction between an ultra-short pulse and a strongly scattering medium: The case of large particles,” Opt. Commun. 265, 373–382 (2006).
    [CrossRef]
  12. S. Bakić, 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. S. Bakić, F. Xu, N. Damaschke, and C. Tropea, “Feasibility of extending rainbow refractometry to small particles using femtosecond laser pulses,” Part. Part. Syst. Charact. 26, 34–40 (2009).
    [CrossRef]
  14. P. Laven, “Separating diffraction from scattering: the million dollar challenge,” J. Nanophoton. 4, 041593 (2010).
    [CrossRef]
  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. 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.
  17. J. B. Keller, “Geometrical theory of diffraction,” J. Opt. Soc. Am. 52, 116–130 (1962).
    [CrossRef] [PubMed]
  18. J. A. Lock and P. Laven, “Mie scattering in the time domain. Part I. The role of surface waves,” J. Opt. Soc. Am. A 28, 1086–1095 (2011).
    [CrossRef]
  19. H. M. Nussenzveig, “High-frequency scattering by a transparent sphere. I. Direct reflection and transmission,” J. Math. Phys. 10, 82–124 (1969).
    [CrossRef]
  20. P. Laven, “How are glories formed?” Appl. Opt. 44, 5675–5683 (2005).
    [CrossRef] [PubMed]
  21. P. Laven, “Effects of refractive index on glories,” Appl. Opt. 47, H133–H142 (2008).
    [CrossRef] [PubMed]

2011 (2)

2010 (1)

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

2009 (1)

S. Bakić, F. Xu, N. Damaschke, and C. Tropea, “Feasibility of extending rainbow refractometry to small particles using femtosecond laser pulses,” Part. Part. Syst. Charact. 26, 34–40 (2009).
[CrossRef]

2008 (2)

2006 (2)

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

C. Calba, C. Rozé, T. Girasole, and L. Méès, “Monte Carlo simulation of the interaction between an ultra-short pulse and a strongly scattering medium: The case of large particles,” Opt. Commun. 265, 373–382 (2006).
[CrossRef]

2005 (1)

2004 (2)

Y. P. Han, L. Méès, K. F. Ren, G. Gréhan, Z. S. Wu, and G. Gouesbet, “Far scattered field from a spheroid under a femtosecond pulsed illumination in a generalized Lorenz-Mie theory framework,” Opt. Commun. 231, 71–77 (2004).
[CrossRef]

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

2001 (2)

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

L. Méès, G. Gréhan, 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]

1994 (2)

1992 (1)

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]

1962 (1)

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).

1908 (1)

P. Debye, “Das elektromagnetische Feld um einen Zylinder und die Theorie des Regenbogens,” Phys. Z. 9, 775–778(1908).

Bakic, S.

S. Bakić, F. Xu, N. Damaschke, and C. Tropea, “Feasibility of extending rainbow refractometry to small particles using femtosecond laser pulses,” Part. Part. Syst. Charact. 26, 34–40 (2009).
[CrossRef]

S. Bakić, 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]

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]

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).

Calba, C.

C. Calba, C. Rozé, T. Girasole, and L. Méès, “Monte Carlo simulation of the interaction between an ultra-short pulse and a strongly scattering medium: The case of large particles,” Opt. Commun. 265, 373–382 (2006).
[CrossRef]

Chowdhury, D. Q.

Damaschke, N.

S. Bakić, F. Xu, N. Damaschke, and C. Tropea, “Feasibility of extending rainbow refractometry to small particles using femtosecond laser pulses,” Part. Part. Syst. Charact. 26, 34–40 (2009).
[CrossRef]

S. Bakić, 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]

Debye, P.

P. Debye, “Das elektromagnetische Feld um einen Zylinder und die Theorie des Regenbogens,” Phys. Z. 9, 775–778(1908).

Girasole, T.

C. Calba, C. Rozé, T. Girasole, and L. Méès, “Monte Carlo simulation of the interaction between an ultra-short pulse and a strongly scattering medium: The case of large particles,” Opt. Commun. 265, 373–382 (2006).
[CrossRef]

Gouesbet, G.

Y. P. Han, L. Méès, K. F. Ren, G. Gréhan, Z. S. Wu, and G. Gouesbet, “Far scattered field from a spheroid under a femtosecond pulsed illumination in a generalized Lorenz-Mie theory framework,” Opt. Commun. 231, 71–77 (2004).
[CrossRef]

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

L. Méès, G. Gréhan, 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]

Gréhan, G.

Y. P. Han, L. Méès, K. F. Ren, G. Gréhan, Z. S. Wu, and G. Gouesbet, “Far scattered field from a spheroid under a femtosecond pulsed illumination in a generalized Lorenz-Mie theory framework,” Opt. Commun. 231, 71–77 (2004).
[CrossRef]

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

Han, Y. P.

Y. P. Han, L. Méès, K. F. Ren, G. Gréhan, Z. S. Wu, and G. Gouesbet, “Far scattered field from a spheroid under a femtosecond pulsed illumination in a generalized Lorenz-Mie theory framework,” Opt. Commun. 231, 71–77 (2004).
[CrossRef]

Heinisch, C.

Hill, S. C.

Hovenac, E. A.

Keller, J. B.

J. B. Keller, “Geometrical theory of diffraction,” J. Opt. Soc. Am. 52, 116–130 (1962).
[CrossRef] [PubMed]

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.

Khaled, E. E. M.

Laven, P.

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]

Lock, J. A.

Méès, L.

C. Calba, C. Rozé, T. Girasole, and L. Méès, “Monte Carlo simulation of the interaction between an ultra-short pulse and a strongly scattering medium: The case of large particles,” Opt. Commun. 265, 373–382 (2006).
[CrossRef]

Y. P. Han, L. Méès, K. F. Ren, G. Gréhan, Z. S. Wu, and G. Gouesbet, “Far scattered field from a spheroid under a femtosecond pulsed illumination in a generalized Lorenz-Mie theory framework,” Opt. Commun. 231, 71–77 (2004).
[CrossRef]

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

L. Méès, G. Gréhan, 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]

Nussenzveig, H. M.

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

Ren, K. F.

Y. P. Han, L. Méès, K. F. Ren, G. Gréhan, Z. S. Wu, and G. Gouesbet, “Far scattered field from a spheroid under a femtosecond pulsed illumination in a generalized Lorenz-Mie theory framework,” Opt. Commun. 231, 71–77 (2004).
[CrossRef]

Rozé, C.

C. Calba, C. Rozé, T. Girasole, and L. Méès, “Monte Carlo simulation of the interaction between an ultra-short pulse and a strongly scattering medium: The case of large particles,” Opt. Commun. 265, 373–382 (2006).
[CrossRef]

Schifrin, K. S.

Tropea, C.

S. Bakić, F. Xu, N. Damaschke, and C. Tropea, “Feasibility of extending rainbow refractometry to small particles using femtosecond laser pulses,” Part. Part. Syst. Charact. 26, 34–40 (2009).
[CrossRef]

S. Bakić, 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]

Tschudi, T.

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).

Wu, Z. S.

Y. P. Han, L. Méès, K. F. Ren, G. Gréhan, Z. S. Wu, and G. Gouesbet, “Far scattered field from a spheroid under a femtosecond pulsed illumination in a generalized Lorenz-Mie theory framework,” Opt. Commun. 231, 71–77 (2004).
[CrossRef]

Xu, F.

S. Bakić, F. Xu, N. Damaschke, and C. Tropea, “Feasibility of extending rainbow refractometry to small particles using femtosecond laser pulses,” Part. Part. Syst. Charact. 26, 34–40 (2009).
[CrossRef]

Zolotov, I. G.

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. Nanophoton. (1)

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

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (4)

Opt. Commun. (3)

L. Méès, G. Gréhan, 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]

Y. P. Han, L. Méès, K. F. Ren, G. Gréhan, Z. S. Wu, and G. Gouesbet, “Far scattered field from a spheroid under a femtosecond pulsed illumination in a generalized Lorenz-Mie theory framework,” Opt. Commun. 231, 71–77 (2004).
[CrossRef]

C. Calba, C. Rozé, T. Girasole, and L. Méès, “Monte Carlo simulation of the interaction between an ultra-short pulse and a strongly scattering medium: The case of large particles,” Opt. Commun. 265, 373–382 (2006).
[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]

Part. Part. Syst. Charact. (1)

S. Bakić, F. Xu, N. Damaschke, and C. Tropea, “Feasibility of extending rainbow refractometry to small particles using femtosecond laser pulses,” Part. Part. Syst. Charact. 26, 34–40 (2009).
[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. Z. (1)

P. Debye, “Das elektromagnetische Feld um einen Zylinder und die Theorie des Regenbogens,” Phys. Z. 9, 775–778(1908).

Other (1)

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.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (14)

Fig. 1
Fig. 1

Results of Mie theory calculations for scattering of red light ( 650 nm ) by a spherical droplet of water of radius r = 10 μm . The refractive index of the sphere n 1 = 1.33257 + i 1.67 E 08 , while the refractive index of the medium n 0 = 1 .

Fig. 2
Fig. 2

As Fig. 1, except for the use of Debye series calculations. The symbol ⊥ denotes perpendicular polarization, while the symbol / / denotes parallel polarization.

Fig. 3
Fig. 3

Geometric rays for p 5 that contribute to scattering at θ = 150 ° assuming that the refractive index of sphere n 1 = 1.33257 and the refractive index of medium n 0 = 1 .

Fig. 4
Fig. 4

Impulse response of a water droplet of radius r = 10 μm for a 5 fs pulse of red light (nominal wavelength λ = 650 nm ) at scattering angle θ = 150 ° . Graph (a) takes account of dispersion (i.e., due to the varying refractive index n 1 of the sphere across the bandwidth of the pulse). Graph (b) assumes that the refractive index of the sphere n 1 = 1.33257 + i 1.67 E 08 and that this does not change with wavelength. Graph (c) uses Debye series calculations to identify scattering caused by specific values of p. The letters A to H correspond to the time delays τ specified in Table 1.

Fig. 5
Fig. 5

Mie theory calculations of the impulse response of a water droplet of radius r = 10 μm for a 5 fs pulse of red light (nominal wavelength λ = 650 nm ) as a function of scattering angle θ. The refractive index of the sphere n 1 = 1.33257 + i 1.67 E 08 and the refractive index of the medium n 0 = 1 are assumed to be constant across the bandwidth of the pulse. The intensity of the scattered pulses is coded according to the false- color scale shown above the diagram.

Fig. 6
Fig. 6

p = 0 impulse response as a function of scattering angle θ calculated using (a) Debye series calculations, (b) geometrical optics calculations for reflection from exterior of the droplet, and (c) the diffraction term. Note that (a) shows the result of combining (b) and (c).

Fig. 7
Fig. 7

Intensity of p = 1 scattering as a function of scattering angle θ, calculated using Debye series and geometrical optics. The symbol ⊥ denotes perpendicular polarization, while the symbol / / denotes parallel polarization.

Fig. 8
Fig. 8

p = 1 impulse response as a function of scattering angle θ calculated using (a) Debye series and (b) geometric optics.

Fig. 9
Fig. 9

Two p = 1 propagation paths involving surface waves resulting in θ = 150 ° .

Fig. 10
Fig. 10

p = 2 impulse response as a function of scattering angle θ calculated using the Debye series.

Fig. 11
Fig. 11

p = 2 propagation path involving surface waves resulting in θ = 150 ° .

Fig. 12
Fig. 12

Three p = 2 propagation paths resulting in θ = 175 ° . Ray A is a geometrical ray, whereas rays B and C involve surface waves.

Fig. 13
Fig. 13

Intensity of p = 2 scattering for propagation paths of type A, B and C as defined in Fig. 12 derived from the time domain results summarized in Fig. 10.

Fig. 14
Fig. 14

p = 3 impulse response as a function of scattering angle θ calculated using the Debye series.

Tables (1)

Tables Icon

Table 1 Propagation Parameters for Geometric Rays Resulting in θ = 150 ° , Assuming a Sphere of Radius r = 10 μm , Refractive Index of Sphere n 1 = 1.33257 and Refractive Index of Medium n 0 = 1

Equations (1)

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

τ = 2 r / c [ n 0 [ 1 ( 1 b 2 ) ] + n 1 p cos [ arcsin ( b n 0 / n 1 ) ] ] ,

Metrics