Abstract

We have derived the formula for the Debye-series decomposition for light scattering by a multilayered sphere. This formulism permits the mechanism of light scattering to be studied. An efficient algorithm is introduced that permits stable calculation for a large sphere with many layers. The formation of triple first-order rainbows by a three-layered sphere and single-order rainbows and the interference of different-order rainbows by a sphere with a gradient refractive index, are then studied by use of the Debye model and Mie calculation. The possibility of taking only one single mode or several modes for each layer is shown to be useful in the study of the scattering characteristics of a multilayered sphere and in the measurement of the sizes and refractive indices of particles.

© 2006 Optical Society of America

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References

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  1. 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]
  2. J. A. Lock, J. M. Jamison, and C.-Y. Lin, "Rainbow scattering by a coated sphere," Appl. Opt. 33, 4677-4690 (1994).
    [CrossRef] [PubMed]
  3. 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]
  4. J. A. Lock, "Cooperative effects among partial waves in Mie scattering," J. Opt. Soc. Am. A 5, 2032-2044 (1988).
    [CrossRef]
  5. Z. S. Wu and Y. P. Wang, "Electromagnetic scattering for multi-layered sphere: recursive algorithms," Radio Sci. 26, 1393-1401 (1991).
    [CrossRef]
  6. Z. S. Wu, L. X. Guo, K. F. Ren, G. Gouesbet, and G. Gréhan, "Improved algorithm for electromagnetic scattering of plane waves and shaped beams by multi-layered spheres," Appl. Opt. 36, 5188-5198 (1997).
    [CrossRef] [PubMed]
  7. H. Du, "Mie-scattering calculation," Appl. Opt. 43, 1951-1956 (2004).
    [CrossRef] [PubMed]
  8. J. A. Adam, "The mathematical physics of rainbows and glories," Phys. Rep. 356, 229-365 (2002).
    [CrossRef]
  9. X. Han and H. Jiang, "Characteristics of intensity and angular spectrum of homogeneous fibers at second rainbow angle," Opt. Commun. 233, 253-259 (2004).
    [CrossRef]
  10. H. Jiang, X. Han, K.-F. Ren, F.-Y. Pan, and L. Mees, "Reconstruction of intensity of the second and fifth rainbows and its applications to a homogeneous droplet," Acta Opt. Sin. 24, 1561-1566 (2004).

2004 (4)

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]

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

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

H. Du, "Mie-scattering calculation," Appl. Opt. 43, 1951-1956 (2004).
[CrossRef] [PubMed]

2002 (1)

J. A. Adam, "The mathematical physics of rainbows and glories," Phys. Rep. 356, 229-365 (2002).
[CrossRef]

1997 (1)

1994 (1)

1992 (1)

1991 (1)

Z. S. Wu and Y. P. Wang, "Electromagnetic scattering for multi-layered sphere: recursive algorithms," Radio Sci. 26, 1393-1401 (1991).
[CrossRef]

1988 (1)

Adam, J. A.

J. A. Adam, "The mathematical physics of rainbows and glories," Phys. Rep. 356, 229-365 (2002).
[CrossRef]

Du, H.

Gouesbet, G.

Gréhan, G.

Guo, L. X.

Han, X.

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

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

Hovenac, E. A.

Jamison, J. M.

Jiang, H.

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

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

Laven, P.

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]

Lin, C.-Y.

Lock, J. A.

Mees, L.

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

Pan, F.-Y.

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

Ren, K. F.

Ren, K.-F.

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

Wang, Y. P.

Z. S. Wu and Y. P. Wang, "Electromagnetic scattering for multi-layered sphere: recursive algorithms," Radio Sci. 26, 1393-1401 (1991).
[CrossRef]

Wu, Z. S.

Acta Opt. Sin. (1)

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

Appl. Opt. (3)

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

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. (1)

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

Phys. Rep. (1)

J. A. Adam, "The mathematical physics of rainbows and glories," Phys. Rep. 356, 229-365 (2002).
[CrossRef]

Radio Sci. (1)

Z. S. Wu and Y. P. Wang, "Electromagnetic scattering for multi-layered sphere: recursive algorithms," Radio Sci. 26, 1393-1401 (1991).
[CrossRef]

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Figures (18)

Fig. 1
Fig. 1

Debye model of scattering by a homogeneous sphere.

Fig. 2
Fig. 2

Debye model of scattering by a coated sphere.

Fig. 3
Fig. 3

Debye model of scattering by a three-layered sphere.

Fig. 4
Fig. 4

Debye model of scattering by an l-layered sphere.

Fig. 5
Fig. 5

Debye-series component intensities obtained by our method and corresponding to the case in Fig. 2 of Ref. 1 for a homogeneous sphere of size parameter x = 100 with refractive index m = 1.333 illuminated by a plane wave of wavelength λ = 632.8 nm of perpendicular polarization.

Fig. 6
Fig. 6

Debye-series component intensities obtained by our method corresponding to the case in Fig. 9 of Ref. 2 for a coated sphere of size parameter x 23 = 900. The refractive indices of the core, the coating, and the surround are, respectively, 1.333, 1.2, and 1.0. Size parameters of the core: (a) x 12 = 600, (b) x 12 = 700, (c) x 12 = 750.

Fig. 7
Fig. 7

Debye-series component intensities obtained by our method corresponding to the case in Fig. 12 of Ref. 2 for a coated sphere of size parameter x 23 = 900. The refractive indices of the core, the coating, and the surround are, respectively, 1.33, 1.5, and 1.0. Size parameters of the core: (a) x 12 = 850, (b) x 12 = 712.5, (c) x 12 = 630.

Fig. 8
Fig. 8

Comparison of the scattered intensities calculated by Mie theory and by Debye series for a 100 layered sphere of radius a = 10 μm with a refractive profile described by Eq. (27) [(a) ms = 1.3216, mc = 1.3316, b= 6; (b) ms = 1.3216 + 0.005i, mc = 1.3316 + 0.005i, b = 6] illuminated by a plane wave of wavelength λ = 632.8 nm of perpendicular polarization.

Fig. 9
Fig. 9

Comparison of the scattering intensities of Debye series and Mie calculation for the formation of twin first-order rainbows by a coated sphere (m 1 = 1.333, m 2 = 1.2, x 12 = 5000, x 23 = 5175) illuminated by a plane wave of wavelength λ = 632.8 nm, corresponding to the case of Fig. 3 of Ref. 2. Component mode p in the Debye series is 2 for the two layers.

Fig. 10
Fig. 10

Debye model for the formation of triple first-order rainbows of a three-layered sphere.

Fig. 11
Fig. 11

Intensities of a single mode in Debye series (p = 2) for a three-layered sphere. The wavelength of the incident plane wave is λ = 632.8 nm. The refractive indices, from the core to the coating, are 1.35, 1.333, and 1.2. The corresponding size parameters are x 12 = 4650, x 23 = 4825, and x 34 = 5000.

Fig. 12
Fig. 12

Comparison of the intensity distribution of a three-layered sphere about the triple first-order rainbows obtained by Debye series and Mie calculation. The wavelength of the incident plane wave is λ = 632.8 nm. The refractive indices, from the core to the coating, are 1.35, 1.333, and 1.2. The corresponding size parameters are x 12 = 4650, x 23 = 4825, and x 34 = 5000. Component mode p is 2 for all three layers.

Fig. 13
Fig. 13

Debye intensities of single order for a 1000 layered sphere of radius 100 μm and refractive profile parameters ms = 1.3216, mc = 1.3316, and b = 6 [Eq. (27)].

Fig. 14
Fig. 14

Rainbow intensities simulated by Debye-series decomposition for a 1000 layered sphere of radius 100 μm with refractive-index parameters ms = 1.3216, mc = 1.3316, and b = 6 [Eq. (27)] illuminated by a plane wave of wavelength λ = 632.8 nm.

Fig. 15
Fig. 15

Distribution of Mie-scattered intensities (same parameters as for Fig. 9).

Fig. 16
Fig. 16

Comparison of rainbow intensities predicted by Debye expansion (Fig. 14) and that extracted by an IFFT from the Mie-scattered intensity in Fig. 15.

Fig. 17
Fig. 17

Comparison of interference of the second- and the fifth-order rainbows obtained by Debye series and Mie calculation for a homogeneous sphere of 200 μm radius and refractive index 1.3216.

Fig. 18
Fig. 18

Comparison of interference of the second- and the fifth-order rainbows obtained by Debye series and Mie calculation for a 1000 layered sphere of radius 200 μm and refractive profile parameters ms = 1.3216, mc = 1.3316, and b = 6 [Eq. (27)].

Equations (31)

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I 1 ( θ ) = | S 1 1 ( θ ) | 2 ,         I 2 ( θ ) = | S 2 1 ( θ ) | 2 ,
S 1 1 ( θ ) = n = 1 2 n + 1 n ( n + 1 ) [ a n 1 π n ( θ ) + b n 1 τ n ( θ ) ] ,
S 2 1 ( θ ) = n = 1 2 n + 1 n ( n + 1 ) [ a n 1 τ n ( θ ) + b n 1 π n ( θ ) ] .
π n ( θ ) = 1 sin θ P n 1 ( θ ) ,         τ n ( θ ) = d d θ P n 1 ( θ ) ,
a n 1 b n 1 } = 1 2 ( 1 R n 212 T n 21 T n 12 1 R n 121 ) = 1 2 [ 1 R n 212 p = 1 T n 21 ( R n 121 ) p 1 T n 12 ] ,
T n 21 = m 1 m 2 2 i D n 1 , R n 212 = α ξ n ( 2 ) ( m 2 k a ) ξ n ( 2 ) ( m 1 k a ) β ξ n ( 2 ) ( m 2 k a ) ξ n ( 2 ) ( m 1 k a ) D n 1 ,
T n 12 = 2 i D n 1 , R n 121 = α ξ n ( 1 ) ( m 2 k a ) ξ n ( 1 ) ( m 1 k a ) β ξ n ( 1 ) ( m 2 k a ) ξ n ( 1 ) ( m 1 k a ) D n 1 ,
α = { 1    TE  wave m 1 m 2    TM  wave ,       β = { m 1 m 2    TE  wave 1    TM  wave ,
D n 1 = α ξ n ( 1 ) ( m 2 k a ) ξ n ( 2 ) ( m 1 k a ) + β ξ n ( 1 ) ( m 2 k a ) ξ n ( 2 ) ( m 1 k a ) .
ξ n ( 1 ) ( m k r ) = m k r h n ( 1 ) ( m k r ) ,
ξ n ( 2 ) ( m k r ) = m k r h n ( 2 ) ( m k r ) .
Q n 1 = R n 212 + T n 21 T n 12 1 R n 121 = R n 212 + p = 1 T n 21 ( R n 121 ) p 1 T n 12 ,
a n 2 b n 2 } = 1 2 ( 1 Q n 2 ) ,
Q n 2 = R n 323 + T n 32 Q n 1 T n 23 1 R n 232 Q n 1 = R n 323 + p = 1 T n 32 Q n 1 T n 23 ( R n 232 Q n 1 ) p 1 .
a n 3 b n 3 } = 1 2 ( 1 Q n 3 ) ,
Q n 3 = R n 434 + T n 43 Q n 2 T n 34 1 R n 343 Q n 2 = R n 434 + p = 1 T n 43 Q n 2 T n 34 ( R n 343 Q n 2 ) p 1 ,
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 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 ,
A n 3 ( m k r ) = ξ n ( 1 ) ( m k r ) ξ n ( 1 ) ( m k r ) ,
A n 4 ( m k r ) = ξ n ( 2 ) ( m k r ) ξ n ( 2 ) ( m k r ) .
A n = 1 / [ n / z A n 1 ] n / z ,
B n ( z ) = ξ n 1 ( z ) ξ n 2 ( z ) = ξ n 1 1 ( z ) ξ n 1 2 ( z ) [ n / z A n 1 3 ( z ) ] [ n / z A n 1 4 ( z ) ] .
T n l = T n l + 1 , l T n l , l + 1 .
Q n j = R n j + 1 , j , j + 1 + T n     j Q n j 1 p = 1 ( R n j , j + 1 , j Q n j 1 ) p 1 ,
R n j + 1 , j , j + 1 = 1 B n ( m j + 1 k r j ) × α A n 4 ( m j + 1 k r j ) β A n 4 ( m j k r j ) α A n 3 ( m j + 1 k r j ) + β A n 4 ( m j k r j ) ,
R n j , j + 1 , j = B n ( m j k r j ) × α A n 3 ( m j + 1 k r j ) β A n 3 ( m j k r j ) α A n 3 ( m j + 1 k r j ) + β A n 4 ( m j k r j ) ,
T n       j = m j m j + 1 B n ( m j k r j ) B n ( m j + 1 k r j ) × A n 4 ( m j k r j ) A n 3 ( m j k r j ) α A n 3 ( m j + 1 k r j ) + β A n 4 ( m j k r j ) × A n 4 ( m j + 1 k r j ) A n 3 ( m j + 1 k r j ) α A n 3 ( m j + 1 k r j ) + β A n 4 ( m j k r j ) ,
a n b n } = T n 32 R n 212 R n 232 R n 212 T n 23
a n b n } = T n 32 T n 21 R n 11 T n 12 T n 23
m ( ρ ) = m c + ( m s m c ) ( e b ρ 1 ) / ( e b 1 ) ,

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