Abstract

The Debye series has been a key tool for the understanding of light scattering features, and it is also a convenient method for understanding and improving the design of optical instruments aimed at optical particle sizing. Gouesbet has derived the Debye series formulation for generalized Lorenz–Mie theory (GLMT). However, the scattering object is a homogeneous sphere, and no numerical result is provided. The Debye series formula for plane-wave scattering by multilayered spheres has been derived before. We have devoted our work to the Debye series of Gaussian beam scattering by multilayered spheres. The integral localized approximation is employed in the calculation of beam-shape coefficients (BSCs) and allows the study of the scattering characteristics of particles illuminated by the strongly focused beams. The formula and code are verified by the comparison with the results produced by GLMT and also by the comparison with the result for the case of plane-wave incidence. The formula is also employed in the simulation of the first rainbow by illuminating the particle with one or several narrow beams.

© 2007 Optical Society of America

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References

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  1. G. Gouesbet, B. Maheu, and 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]
  2. F. Onofri, G. Gréhan, and G. Gouesbet, "Electromagnetic scattering from a multilayered sphere located in an arbitrary beam," Appl. Opt. 34, 7113-7124 (1995).
    [CrossRef] [PubMed]
  3. Z. S. Wu, L. X. Guo, K. F. Ren, G. Gouesbet, and G. Gréhan, "Improved algorithms for electromagnetic scattering of plane waves and shaped beams by multilayered spheres," Appl. Opt. 36, 5188-5198 (1997).
    [CrossRef] [PubMed]
  4. G. Gouesbet, "Generalized Lorenz-Mie theory and applications," Part. Part. Syst. Charact. 11, 22-23 (1994).
    [CrossRef]
  5. R. Li, X. Han, H. Jiang, and K. F. Ren, "Debye series for light scattering by a multilayered sphere," Appl. Opt. 45, 1260-1270 (2006).
    [CrossRef] [PubMed]
  6. R. Li, X. Han, H. Jiang, and K. F. Ren, "Debye series of normally incident plane-wave scattering by an infinite multilayered cylinder," Appl. Opt. 45, 6255-6262 (2006).
    [CrossRef] [PubMed]
  7. 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]
  8. J. A. Lock, J. M. Jamison, and C. Y. Lin, "Rainbow scattering by a coated sphere," Appl. Opt. 33, 4677-4690 (1994).
    [CrossRef] [PubMed]
  9. P. Laven, "Simulation of rainbows, coronas, and glories using Mie theory and the Debye series," J. Quant. Spectrosc. Radiat. Transfer 89, 257-269 (2004).
    [CrossRef]
  10. G. Gouesbet, "Debye series formulation for generalized Lorenz-Mie theory with the Bromwich method," Part. Part. Syst. Charact. 20, 382-386 (2003).
    [CrossRef]
  11. K. F. Ren, G. Gouesbet, and G. Gréhan, "Integral localized approximation in generalized Lorenz-Mie theory," Appl. Opt. 37, 4218-4225 (1998).
    [CrossRef]
  12. J. P. A. J. van Beeck and M. L. Riethmuller, "Rainbow phenomena applied to the measurement of droplet size and velocity and to the detection of nonsphericity," Appl. Opt. 35, 2259-2266 (1996).
    [CrossRef] [PubMed]
  13. L. W. Davis, "Theory of electromagnetic beams," Phys. Rev. A 19, 1177-1179 (1979).
    [CrossRef]
  14. K. F. Ren, G. Gréhan, and G. Gouesbet, "Evaluation of laser sheet beam shape coefficients in generalized Lorenz-Mie theory by using a localized approximation," J. Opt. Soc. Am. A 11, 2072-2079 (1994).
    [CrossRef]
  15. A. Doicu and T. Wriedt, "Computation of the beam-shape coefficients in the generalized Lorenz-Mie theory by using the translational addition theorem for spherical vector wave functions," Appl. Opt. 36, 2971-2978 (1997).
    [CrossRef] [PubMed]
  16. X. Han and H. Jiang, "Characteristics of intensity and angular spectrum of homogeneous files at second rainbow angle," Opt. Commun. 233, 253-259 (2004).
    [CrossRef]
  17. H. Jiang, X. Han, K. F. Ren, F. Y. Pan, and L. Mees, "Reconstuction of intensity of the second and fifth rainbows and its applications to a homogeneous droplet," Acta Opt. Sin. 24, 1561-1566 (2004).
  18. 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, 693-705 (1993).
    [CrossRef]
  19. H. C. van de Hulst, "Light scattering by small particles" (Dover, 1957).
  20. J. A. Lock, "Theory of the observation made of high-order rainbows," Appl. Opt. 26, 5291-5297 (1987).
    [CrossRef] [PubMed]
  21. J. P. A. J. van Beeck, "Rainbow phenomena, development of a laser-based, non-intrusive technique for measuring droplet size, temperature and velocity," Ph.D. dissertation (Technische Universiteit Eindhoven, 1997).
  22. X. Han, K. F. Ren, L. Mees, G. Gouesbet, and G. Gréhan, "Surface waves/geometrical rays interferences: numerical and experimental behavior at rainbow angles," Opt. Commun. 195, 49-54 (2001).
    [CrossRef]
  23. X. Han, "Study of refractometry of rainbow and applications to the measurement of instability and temperature gradient of a liquid jet," Ph.D. dissertation (Rouen University, 2000).

2006 (2)

2004 (3)

X. Han and H. Jiang, "Characteristics of intensity and angular spectrum of homogeneous files at second rainbow angle," Opt. Commun. 233, 253-259 (2004).
[CrossRef]

H. Jiang, X. Han, K. F. Ren, F. Y. Pan, and L. Mees, "Reconstuction of intensity of the second and fifth rainbows and its applications to a homogeneous droplet," Acta Opt. Sin. 24, 1561-1566 (2004).

P. Laven, "Simulation of rainbows, coronas, and glories using Mie theory and the Debye series," J. Quant. Spectrosc. Radiat. Transfer 89, 257-269 (2004).
[CrossRef]

2003 (1)

G. Gouesbet, "Debye series formulation for generalized Lorenz-Mie theory with the Bromwich method," Part. Part. Syst. Charact. 20, 382-386 (2003).
[CrossRef]

2001 (1)

X. Han, K. F. Ren, L. Mees, G. Gouesbet, and G. Gréhan, "Surface waves/geometrical rays interferences: numerical and experimental behavior at rainbow angles," Opt. Commun. 195, 49-54 (2001).
[CrossRef]

2000 (1)

X. Han, "Study of refractometry of rainbow and applications to the measurement of instability and temperature gradient of a liquid jet," Ph.D. dissertation (Rouen University, 2000).

1998 (1)

1997 (3)

1996 (1)

1995 (1)

1994 (3)

1993 (1)

1992 (1)

1988 (1)

1987 (1)

1979 (1)

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

Acta Opt. Sin. (1)

H. Jiang, X. Han, K. F. Ren, F. Y. Pan, and L. Mees, "Reconstuction of intensity of the second and fifth rainbows and its applications to a homogeneous droplet," Acta Opt. Sin. 24, 1561-1566 (2004).

Appl. Opt. (9)

J. A. Lock, "Theory of the observation made of high-order rainbows," Appl. Opt. 26, 5291-5297 (1987).
[CrossRef] [PubMed]

A. Doicu and T. Wriedt, "Computation of the beam-shape coefficients in the generalized Lorenz-Mie theory by using the translational addition theorem for spherical vector wave functions," Appl. Opt. 36, 2971-2978 (1997).
[CrossRef] [PubMed]

F. Onofri, G. Gréhan, and G. Gouesbet, "Electromagnetic scattering from a multilayered sphere located in an arbitrary beam," Appl. Opt. 34, 7113-7124 (1995).
[CrossRef] [PubMed]

Z. S. Wu, L. X. Guo, K. F. Ren, G. Gouesbet, and G. Gréhan, "Improved algorithms for electromagnetic scattering of plane waves and shaped beams by multilayered spheres," Appl. Opt. 36, 5188-5198 (1997).
[CrossRef] [PubMed]

R. Li, X. Han, H. Jiang, and K. F. Ren, "Debye series for light scattering by a multilayered sphere," Appl. Opt. 45, 1260-1270 (2006).
[CrossRef] [PubMed]

R. Li, X. Han, H. Jiang, and K. F. Ren, "Debye series of normally incident plane-wave scattering by an infinite multilayered cylinder," Appl. Opt. 45, 6255-6262 (2006).
[CrossRef] [PubMed]

J. A. Lock, J. M. Jamison, and C. Y. Lin, "Rainbow scattering by a coated sphere," Appl. Opt. 33, 4677-4690 (1994).
[CrossRef] [PubMed]

K. F. Ren, G. Gouesbet, and G. Gréhan, "Integral localized approximation in generalized Lorenz-Mie theory," Appl. Opt. 37, 4218-4225 (1998).
[CrossRef]

J. P. A. J. van Beeck and M. L. Riethmuller, "Rainbow phenomena applied to the measurement of droplet size and velocity and to the detection of nonsphericity," Appl. Opt. 35, 2259-2266 (1996).
[CrossRef] [PubMed]

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

J. Quant. Spectrosc. Radiat. Transfer (1)

P. Laven, "Simulation of rainbows, coronas, and glories using Mie theory and the Debye series," J. Quant. Spectrosc. Radiat. Transfer 89, 257-269 (2004).
[CrossRef]

Opt. Commun. (2)

X. Han and H. Jiang, "Characteristics of intensity and angular spectrum of homogeneous files at second rainbow angle," Opt. Commun. 233, 253-259 (2004).
[CrossRef]

X. Han, K. F. Ren, L. Mees, G. Gouesbet, and G. Gréhan, "Surface waves/geometrical rays interferences: numerical and experimental behavior at rainbow angles," Opt. Commun. 195, 49-54 (2001).
[CrossRef]

Part. Part. Syst. Charact. (2)

G. Gouesbet, "Debye series formulation for generalized Lorenz-Mie theory with the Bromwich method," Part. Part. Syst. Charact. 20, 382-386 (2003).
[CrossRef]

G. Gouesbet, "Generalized Lorenz-Mie theory and applications," Part. Part. Syst. Charact. 11, 22-23 (1994).
[CrossRef]

Phys. Rev. A (1)

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

Other (3)

H. C. van de Hulst, "Light scattering by small particles" (Dover, 1957).

J. P. A. J. van Beeck, "Rainbow phenomena, development of a laser-based, non-intrusive technique for measuring droplet size, temperature and velocity," Ph.D. dissertation (Technische Universiteit Eindhoven, 1997).

X. Han, "Study of refractometry of rainbow and applications to the measurement of instability and temperature gradient of a liquid jet," Ph.D. dissertation (Rouen University, 2000).

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Figures (19)

Fig. 1
Fig. 1

Geometry of the multilayered sphere.

Fig. 2
Fig. 2

Debye model of light scattering by an l-layered sphere.

Fig. 3
Fig. 3

Profile of refractive index with m s = 1.3216 , m c = 1.3316 , and b = 6 .

Fig. 4
Fig. 4

Comparison of the scattered intensities calculated by GLMT and by Debye series for a 100 layered sphere of radius a = 10 μ m with a refractive profile depicted in Fig. 3 illuminated at different positions by a Gaussian beam of wavelength λ = 632.8   nm and beam-waist radius ω 0 = 15 μ m of perpendicular polarization.

Fig. 5
Fig. 5

Comparison of the scattered intensities simulated by Debye series for a 1000 layered sphere of radius a = 100 μ m with a refractive profile depicted in Fig. 3 illuminated by a plane wave and a Gaussian beam of wavelength λ = 632.8   nm [ ω 0 = 150 μ m , x 0 = y 0 = z 0 = 0 μ m] .

Fig. 6
Fig. 6

Debye intensities of single order for a 1000 layered sphere of radius 100 μ m and refractive profile depicted in Fig. 3 illuminated at different positions by Gaussian beam of wavelength λ = 632.8   nm   [ ω 0 = 50 μ m ] .

Fig. 7
Fig. 7

First intensities for a 1000 layered sphere of radius 100 μ m and refractive profile depicted in Fig. 3. illuminated by plane waves of wavelength λ = 632.8   nm .

Fig. 8
Fig. 8

Angular spectrum of the first rainbow when the particle is illuminated by plane waves.

Fig. 9
Fig. 9

Geometry of ray paths of the first rainbow.

Fig. 10
Fig. 10

Influence of the width of beam A on the first rainbow.

Fig. 11
Fig. 11

Angular spectrum of the first rainbow when the particle is illuminated by beam A.

Fig. 12
Fig. 12

First rainbow intensity when the particle is illuminated by beams A and B.

Fig. 13
Fig. 13

Angular spectrum of the first rainbow when the particle is illuminated by beams A and B.

Fig. 14
Fig. 14

First rainbow intensity when the particle is illuminated by beams A and C.

Fig. 15
Fig. 15

Angular spectrum of the first rainbow when the particle is illuminated by beams A and C.

Fig. 16
Fig. 16

First rainbow intensity when the particle is illuminated by beams A, B, and C.

Fig. 17
Fig. 17

Angular spectrum of the first rainbow when the particle is illuminated by beams A, B, and C.

Fig. 18
Fig. 18

Comparison of first rainbow intensity when the particle is illuminated by beams A, B, and C with that when the particle is illuminated by beams A and B.

Fig. 19
Fig. 19

Comparison of angular spectra of first rainbow intensity when the particle is illuminated by beams A, B, and C with that when the particle is illuminated by beams A and B.

Equations (19)

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s = ω 0 / l = 1 / ( k ω 0 ) ,
E r , T E s = H r , T M s = 0 ,
E r , T M s = 2 U T M s r 2 + k 2 U T M s ,
H r , T E s = 2 U T E s r 2 + k 2 U T E s ,
U T M s = E 0 k n = 1 m = n n C n p w A n m ξ n ( k r ) P n | m | × ( cos   θ ) exp ( i m φ ) ,
U T E s = H 0 k n = 1 m = n n C n p w B n m ξ n ( k r ) P n | m | × ( cos   θ ) exp ( i m φ ) ,
C n p w = 1 k i n 1 ( 1 ) n 2 n + 1 n ( n + 1 ) ,
A n m = a n g n , T M m B n m = b n g n , T E m ,
( E r / E 0 H r / H 0 ) = i Q e i Q γ 2 + i k z 0 e 2 i Q ρ n ( ξ 0   cos   ϕ + η 0   sin   ϕ ) ( E 0   cos   ϕ H 0   sin   ϕ ) .
( g n , T M m i g n , T E m ) = i Q Z n m 4 π e i Q γ 2 + i k z 0 0 2 π e 2 i Q ρ n ( ξ 0 cos   ϕ + η 0   sin   ϕ ) × ( e i ( m 1 ) ± e i ( m + 1 ) ) d ϕ = i Q Z n m 2 e i Q γ 2 + i k z 0 [ e i ( m 1 ) ϕ 0 J m 1 ( 2 Q ρ n ρ 0 ) ± e i ( m + 1 ) ϕ 0 J m + 1 ( 2 Q ρ n ρ 0 ) ] .
( g n , T M m i g n , T E m ) = i Q Z n m e i Q γ 2 + i k z 0 1 2 [ e i ( m 1 ) ϕ 0 J m 1 ( 2 Q ρ n ρ 0 ) ± e i ( m + 1 ) ϕ 0 J m + 1 ( 2 Q ρ n ρ 0 ) ] ,
Q = 1 i 2 z 0 / l ,
ρ n = ( n + 1 / 2 ) s , γ = ρ n 2 + ρ 0 2 , ξ 0 = x 0 ω 0 ,
η 0 = y 0 ω 0 ,
ρ 0 = ( ρ n   cos   ϕ ξ 0 ) 2 + ( ρ n   sin   ϕ η 0 ) 2 .
a n l b n l } = 1 2 ( 1 Q n l ) .
Q n j = R n j + 1 , j , j + 1 + T n j + 1 , j Q n j 1 T n j , j + 1 1 R n j , j + 1 , j Q n j 1 ,
Q n j = R n j + 1 , j , j + 1 + T n j + 1 , j Q n j 1 T n j , j + 1 p = 1 ( R n j , j + 1 , j Q n j 1 ) p 1 .
m ( ρ ) = m c + ( m s m c ) ( e b ρ 1 ) / ( e b 1 ) ,

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