Abstract

We derive the formula of the Debye-series decomposition for normally incident plane-wave scattering by an infinite multilayered cylinder. A comparison of the scattering diagrams calculated by the Debye series and Mie theory for a graded-index polymer optical fiber is given and the agreement is found to be satisfied. This approach permits us to simulate the rainbow intensity distribution of any single order and the interference of several orders, which is of good use to the study of the scattering characteristics of an inhomogeneous cylinder and to the measurement of the refractive index profile of an inhomogeneous cylinder.

© 2006 Optical Society of America

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References

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  1. R. Li, X. Han, H. Jiang, and K. F. Ren, "Debye series of light scattering by a multilayered sphere," Appl. Opt. 45, 1260-1270 (2006).
    [CrossRef] [PubMed]
  2. 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]
  3. J. A. Lock, J. M. Jamison, and C.-Y. Lin, "Rainbow scattering by a coated sphere," Appl. Opt. 33, 4677-4690 (1994).
    [CrossRef] [PubMed]
  4. Z. S. Wu and Y. P. Wang, "Electromagnetic scattering for multi-layered sphere: recursive algorithms," Radio Sci. 26, 1393-1401 (1991).
    [CrossRef]
  5. 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 multilayered spheres," Appl. Opt. 36, 5188-5198 (1997).
    [CrossRef] [PubMed]
  6. H. Du, "Mie-scattering calculation," Appl. Opt. 43, 1951-1956 (2004).
    [CrossRef] [PubMed]
  7. M. Barabás, "Scattering of a plane wave by a radially stratified tilted cylinder," J. Opt. Soc. Am. A 4, 2240-2248 (1987).
    [CrossRef]
  8. X. Han and H. Jiang, "Characteristics of scattering diagram of a cylinder in rainbow angles with mixed orders," Opt. Commun. 233, 253-259 (2004).
    [CrossRef]
  9. J. A. Adam, "The mathematical physics of rainbows and glories," Phys. Rep. 356, 229-365 (2002).
    [CrossRef]
  10. 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 (1)

2004 (2)

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

X. Han and H. Jiang, "Characteristics of scattering diagram of a cylinder in rainbow angles with mixed orders," Opt. Commun. 233, 253-259 (2004).
[CrossRef]

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]

1987 (1)

Adam, J. A.

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

Barabás, M.

Du, H.

Gouesbet, G.

Gréhan, G.

Guo, L. X.

Han, X.

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

X. Han and H. Jiang, "Characteristics of scattering diagram of a cylinder in rainbow angles with mixed orders," Opt. Commun. 233, 253-259 (2004).
[CrossRef]

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).

Hovenac, E. A.

Jamison, J. M.

Jiang, H.

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

X. Han and H. Jiang, "Characteristics of scattering diagram of a cylinder in rainbow angles with mixed orders," Opt. Commun. 233, 253-259 (2004).
[CrossRef]

Li, R.

Lin, C.-Y.

Lock, J. A.

Ren, K. F.

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.

Appl. Opt. (4)

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

Opt. Commun. (1)

X. Han and H. Jiang, "Characteristics of scattering diagram of a cylinder in rainbow angles with mixed orders," 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]

Other (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).

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Figures (10)

Fig. 1
Fig. 1

Geometry of an l-layer cylinder.

Fig. 2
Fig. 2

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

Fig. 3
Fig. 3

Scattered intensities obtained by our method corresponding to the case of Fig. 5 in Ref. 7 for a doubly clad optical fiber ( r 1 = 5.6   μm , r 2 = 6.3   μm , r 3 = 7.0   μm , m 1 = 1.62 , m 2 = 1.505 , m 3 = 1.56 ) illuminated by a plane wave of wavelength λ = 633   nm of perpendicular and parallel polarizations.

Fig. 4
Fig. 4

Comparison of the scattered intensities calculated by Mie theory and by the Debye series for a 1000-layered cylinder of radius a = 500   μm with a refractive profile described by Eq. (22) ( m s = 1.47 , m c = 1.49 , g = 2.0 ) illuminated by a plane wave of wavelength λ = 632.8   nm of perpendicular polarization. The Debye intensity has been offset by the addition of 5 for clarity.

Fig. 5
Fig. 5

Scheme of the experiment setup. The width of the CCD is 2 .66   cm , the distance from the center of the fiber to the center of the CCD is 15 .22   cm , and the length of the fiber is 8   cm .

Fig. 6
Fig. 6

Comparison of the first rainbow intensity of the experimental and theoretical results for a homogeneous optical fiber ( d = 125   μm , m = 1.46 ) illuminated by a plane wave of wavelength λ = 632.8   nm .

Fig. 7
Fig. 7

Debye intensities of single order for a 1000-layered cylinder of radius 500   μm and refractive profile parameters m s = 1.47 , m c = 1.49 , and g = 2 [Eq. (22)]. Solid and dashed curves are, respectively, for multilayered and homogeneous fibers.

Fig. 8
Fig. 8

Linear intensity presentation of parts a and c in Fig. 4.

Fig. 9
Fig. 9

Rainbow intensities simulated by the Debye-series decomposition for a 1000-layered cylinder of radius 500   μm with refractive index parameters m s = 1.47 , m c = 1.49 , and g = 2 [Eq. (22)] illuminated by a plane wave of wavelength λ = 632.8   nm .

Fig. 10
Fig. 10

Comparison of scattered intensities of a 1000-layered sphere and cylinder of radius 100   μm with refractive index parameters m s = 1.3216 , m c = 1.3216 , and g = 6 [Eq. (22)] illuminated by a plane wave of wavelength λ = 632.8   nm .

Equations (25)

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I 1     l ( θ ) = | S 1     l ( θ ) | 2 , I 2     l ( θ ) = | S 2     l ( θ ) | 2 ,
S 1     l ( θ ) = a 0     l + 2 n = 1 a n     l cos ( n θ ) ,
S 2     l ( θ ) = b 0     l + 2 n = 1 b n     l cos ( n θ ) .
a n     l = 1 2 [ 1 Q 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 Q n       j 1 R n       j , j + 1 , 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 ,
Ψ = { cos n ϕ sin n ϕ } ξ n     ( 2 ) ( m j + 1 k ρ ) e i ω t
Ψ j = T n       j + 1 , j { cos n ϕ sin n ϕ } ξ n     ( 2 ) ( m j k ρ ) e i ω t ,
Ψ j + 1 = [ ξ n     ( 2 ) ( m j + 1 k ρ ) + R n       j + 1 , j , j + 1 ξ n     ( 1 ) ( m j + 1 k ρ ) ] × { cos n ϕ sin n ϕ } e i ω t .
R n       j + 1 , j , j + 1 = α ξ n     ( 2 ) ( m j + 1 k r j ) ξ n     ( 2 ) ( m j k r j ) β ξ n     ( 2 ) ( m j k r j ) ξ n     ( 2 ) ( m j + 1 k r j ) β ξ n     ( 2 ) ( m j k r j ) ξ n     ( 1 ) ( m j + 1 k r j ) α ξ n     ( 1 ) ( m j + 1 k r j ) ξ n     ( 2 ) ( m j k r j ) ,
T n       j + 1 , j = α β ξ n     ( 2 ) ( m j + 1 k r j ) ξ n     ( 1 ) ( m j + 1 k r j ) ξ n ( 1 ) ( m j + 1 k r j ) ξ n     ( 2 ) ( m j + 1 k r j ) β ξ n     ( 2 ) ( m j k r j ) ξ n     ( 1 ) ( m j + 1 k r j ) α ξ n     ( 1 ) ( m j + 1 k r j ) ξ n     ( 2 ) ( m j k r j ) .
R n      j , j + 1 , j = α ξ n     ( 1 ) ( m j + 1 k r j ) ξ n     ( 1 ) ( m j k r j ) β ξ n     ( 1 ) ( m j k r j ) ξ n     ( 1 ) ( m j + 1 k r j ) β ξ n     ( 2 ) ( m j k r j ) ξ n     ( 1 ) ( m j + 1 k r j ) α ξ n     ( 1 ) ( m j + 1 k r j ) ξ n     ( 2 ) ( m j k r j ) ,
T n         j , j + 1 = ξ n     ( 1 ) ( m j k r j ) ξ n     ( 2 ) ( m j k r j ) ξ n     ( 1 ) ( m j k r j ) ξ n     ( 2 ) ( m j k r j ) β ξ n     ( 2 ) ( m j k r j ) ξ n     ( 1 ) ( m j + 1 k r j ) α ξ n     ( 1 ) ( m j + 1 k r j ) ξ n     ( 2 ) ( m j k r j ) ,
α = { 1 TE   wave m j m j + 1 TM   wave , β = { m j m j + 1 TE   wave 1 TM   wave .
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 ) .
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 1 ] / z .
B n ( z ) = ξ n     ( 1 ) ( z ) ξ n     ( 2 ) ( z ) = ξ n 1               ( 1 ) ( z ) ξ n 1               ( 2 ) ( z ) [ [ n 1 ] / z + A n 1                 4 ( z ) ] [ [ n 1 ] / z + A n 1                 3 ( z ) ] .
T n       j = T n       j + 1 , j T n         j , j + 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 = α β 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 ) ,
m ( r ) = { m c [ 1 2 Δ ( r / a ) g ] 1 / 2 0 r a m s r = a ,
Δ = m c 2 m s 2 2 m c 2 .

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