Abstract

We calculate the intensity of the light scattered by an infinite, radially stratified cylinder. The incident light is a plane wave having an arbitrary angle of incidence and an arbitrary polarization. The Hertz potentials of the scattered wave are represented as superpositions of conical waves, and the boundary-value method is used to derive an infinite set of systems of linear equations for the expansion coefficients. The intensity and the polarization of the far-field scattered wave is expressed in terms of these expansion coefficients. Numerical results showing the angular distribution of the scattered intensity corresponding to different angles of incidence are also presented for the case of a doubly clad image-transmitting fiber illuminated by a He–Ne laser.

© 1987 Optical Society of America

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References

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  1. H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).
  2. M. Kerker, The Scattering of Light (AcademicNew York, 1969).
  3. M. Kerker, E. Matijevic, “Scattering of electromagnetic waves from concentric infinite cylinders,”J. Opt. Soc. Am. 51, 506–508 (1961).
    [CrossRef]
  4. W. A. Farone Querfeld, “Electromagnetic scattering from radially inhomogeneous infinite cylinders at oblique incidence,”J. Opt. Soc. Am. 56, 476–480 (1966).
    [CrossRef]
  5. A. Cohen, A. Gross, “Radiation pressure and 360° scattering diagrams for infinite cylinders,” Opt. Lett. 8, 253–255 (1983).
    [CrossRef] [PubMed]
  6. A. Cohen, “Scattering of linearly polarized incidence at arbitrary angle to incident plane of infinite tilted cylinders,” Opt. Lett. 5, 150–152 (1980).
    [CrossRef] [PubMed]
  7. A. Cohen, C. Acquista, “Light scattering by tilted cylinders: properties of partial wave coefficients,”J. Opt. Soc. Am. 72, 531–534 (1982).
    [CrossRef]
  8. S. N. Samaddar, “Scattering of plane electromagnetic waves by radially inhomogeneous infinite cylinders,” Nuovo Cimento Ital. Fis. B 66, 33–50 (1970).
    [CrossRef]
  9. L. W. Pearson, “A construction of the fields radiated by z directed point sources of current in the presence of a cylindrically layered obstacle,” Radio Sci. 21, 559–569 (1986).
    [CrossRef]
  10. M. Abramowitz, I. A. Stegun, eds., Handbook of Mathematical Functions (Dover, New York, 1968).
  11. M. Barabás, A. Podmaniczky, “Ray optical analysis of backscattering by an obliquely illuminated doubly clad fiber,” in Symposium Optika’84, G. Lupkovics, A. Podmaniczky, eds., Proc. Soc. Photo-Opt. Instrum. Eng.473, 126–129 (1984).
    [CrossRef]
  12. A. Podmaniczky, M. Barabás, “Microinterferometric method for relative diameter measurement of optical fibres,” in 1984 European Conference on Optics, Optical Systems, and Applications, B. Bolger, H. A. Ferwerda, eds., Proc. Soc. Photo-Opt. Instrum. Eng.492, 334–340 (1984).
    [CrossRef]
  13. G. A. Korn, T. M. Korn, Mathematical Handbook for Scientists and Engineers (McGraw-Hill, New York, 1973).

1986 (1)

L. W. Pearson, “A construction of the fields radiated by z directed point sources of current in the presence of a cylindrically layered obstacle,” Radio Sci. 21, 559–569 (1986).
[CrossRef]

1983 (1)

1982 (1)

1980 (1)

1970 (1)

S. N. Samaddar, “Scattering of plane electromagnetic waves by radially inhomogeneous infinite cylinders,” Nuovo Cimento Ital. Fis. B 66, 33–50 (1970).
[CrossRef]

1966 (1)

1961 (1)

Acquista, C.

Barabás, M.

M. Barabás, A. Podmaniczky, “Ray optical analysis of backscattering by an obliquely illuminated doubly clad fiber,” in Symposium Optika’84, G. Lupkovics, A. Podmaniczky, eds., Proc. Soc. Photo-Opt. Instrum. Eng.473, 126–129 (1984).
[CrossRef]

A. Podmaniczky, M. Barabás, “Microinterferometric method for relative diameter measurement of optical fibres,” in 1984 European Conference on Optics, Optical Systems, and Applications, B. Bolger, H. A. Ferwerda, eds., Proc. Soc. Photo-Opt. Instrum. Eng.492, 334–340 (1984).
[CrossRef]

Cohen, A.

Farone Querfeld, W. A.

Gross, A.

Kerker, M.

Korn, G. A.

G. A. Korn, T. M. Korn, Mathematical Handbook for Scientists and Engineers (McGraw-Hill, New York, 1973).

Korn, T. M.

G. A. Korn, T. M. Korn, Mathematical Handbook for Scientists and Engineers (McGraw-Hill, New York, 1973).

Matijevic, E.

Pearson, L. W.

L. W. Pearson, “A construction of the fields radiated by z directed point sources of current in the presence of a cylindrically layered obstacle,” Radio Sci. 21, 559–569 (1986).
[CrossRef]

Podmaniczky, A.

M. Barabás, A. Podmaniczky, “Ray optical analysis of backscattering by an obliquely illuminated doubly clad fiber,” in Symposium Optika’84, G. Lupkovics, A. Podmaniczky, eds., Proc. Soc. Photo-Opt. Instrum. Eng.473, 126–129 (1984).
[CrossRef]

A. Podmaniczky, M. Barabás, “Microinterferometric method for relative diameter measurement of optical fibres,” in 1984 European Conference on Optics, Optical Systems, and Applications, B. Bolger, H. A. Ferwerda, eds., Proc. Soc. Photo-Opt. Instrum. Eng.492, 334–340 (1984).
[CrossRef]

Samaddar, S. N.

S. N. Samaddar, “Scattering of plane electromagnetic waves by radially inhomogeneous infinite cylinders,” Nuovo Cimento Ital. Fis. B 66, 33–50 (1970).
[CrossRef]

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).

J. Opt. Soc. Am. (3)

Nuovo Cimento Ital. Fis. B (1)

S. N. Samaddar, “Scattering of plane electromagnetic waves by radially inhomogeneous infinite cylinders,” Nuovo Cimento Ital. Fis. B 66, 33–50 (1970).
[CrossRef]

Opt. Lett. (2)

Radio Sci. (1)

L. W. Pearson, “A construction of the fields radiated by z directed point sources of current in the presence of a cylindrically layered obstacle,” Radio Sci. 21, 559–569 (1986).
[CrossRef]

Other (6)

M. Abramowitz, I. A. Stegun, eds., Handbook of Mathematical Functions (Dover, New York, 1968).

M. Barabás, A. Podmaniczky, “Ray optical analysis of backscattering by an obliquely illuminated doubly clad fiber,” in Symposium Optika’84, G. Lupkovics, A. Podmaniczky, eds., Proc. Soc. Photo-Opt. Instrum. Eng.473, 126–129 (1984).
[CrossRef]

A. Podmaniczky, M. Barabás, “Microinterferometric method for relative diameter measurement of optical fibres,” in 1984 European Conference on Optics, Optical Systems, and Applications, B. Bolger, H. A. Ferwerda, eds., Proc. Soc. Photo-Opt. Instrum. Eng.492, 334–340 (1984).
[CrossRef]

G. A. Korn, T. M. Korn, Mathematical Handbook for Scientists and Engineers (McGraw-Hill, New York, 1973).

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).

M. Kerker, The Scattering of Light (AcademicNew York, 1969).

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Figures (6)

Fig. 1
Fig. 1

Cartesian and cylindrical coordinates. The z axis coincides with the axis of the cylinder, and the propagation vector of the incident beam lies in the (x, z) plane.

Fig. 2
Fig. 2

Equation of the scattering coefficients of order m for a doubly coated dielectric cylinder. [See also Eqs. (22)(27).]

Fig. 3
Fig. 3

Unit vectors used in Eqs. (49)(53).

Fig. 4
Fig. 4

Wave front and Poynting vector in the far field.

Fig. 5
Fig. 5

Angular distribution of intensity of the far-field scattered wave. The cylinder is a doubly clad optical fiber (n1 = 1.62, n2 = 1.505, n3 = 1.56, r1 = 5.6 μm, r2 = 6.3 μm, r3 = 7.0 μm). The incident wave propagating along the x axis (φ = 0) is of wavelength λ0 = 0.633 μm and is plane polarized (a) in the (x, z) plane, (b) in the plane y = x, or (c) in the (x, y) plane.

Fig. 6
Fig. 6

Angular intensity distribution of the light scattered by the same fiber as in Fig. 5 for an incident wave whose propagation direction is tilted by φ = 45° with respect to the (x, y) plane and which is (a) TM polarized or (c) TE polarized or (b) vibrating parallel to the vector −ex sin φ +ey +ez cos φ.

Equations (69)

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E = i n k 0 rot rot u + rot v ,
H = - n rot u + i k 0 rot rot v ,
[ Δ + ( n k 0 ) 2 ] ( u v ) = 0 ,
u = ( 0 , 0 , u ) ,             v = ( 0 , 0 , v ) .
( u v ) = ( u inc v inc ) + ( u scat v scat ) .
( u inc v inc ) = ( α β ) exp [ - i k 0 ( x cos φ + z sin φ ) ]
( u inc v inc ) = ( α β ) m = - F m , h inc ( ϑ , z ) J m ( l inc r ) ,
F m , h ( ϑ , z ) = ( - i ) m exp [ - i ( h z + m ϑ ) ] ,
h inc = k 0 sin φ ,
l inc = ( k 0 2 - h inc 2 ) 1 / 2 ,
S = c 8 π Re ( E × H * )
I inc = S = c 8 π k 0 2 ( cos 2 φ ) ( α 2 + β 2 ) .
( u m ( j ) v m ( j ) ) = [ ( a m ( j ) A m ( j ) ) J m ( l j r ) + ( b m ( j ) B m ( j ) ) Y m ( l j r ) ] F m , h j ( ϑ , z ) ,
l j = ( n j 2 k 0 - h j 2 ) 1 / 2 ;
( u ( j ) v ( j ) ) = m = - ( u m ( j ) v m ( j ) ) .
h j - h inc = k 0 sin φ             ( j = 1 , 2 , N + 1 ) .
b m ( 1 ) = B m ( 1 ) = 0.
H m ( l r ) = J m ( l r ) - i Y m ( l r ) ,
b m ( N + 1 ) = - i a m ( N + 1 ) , B m ( N + 1 ) = - i A m ( N + 1 ) .
a m ( 1 ) , A m ( 1 ) , a m ( j ) , A m ( j ) , b m ( j ) , B m ( j )             ( j = 2 , 3 N ) , a m ( N + 1 ) , A m ( N + 1 ) .
E r m ( j ) = h n j k 0 r u m - i m r v m , E ϑ m ( j ) = - i m h n j k 0 r u m - r v m , E z m ( j ) = i l j 2 n j k 0 u m , H r m ( j ) = i m n j r u m + h k 0 r v m , H ϑ m ( j ) = n j r u m - i m h k 0 r v m , H z m ( j ) = i l j 2 k 0 v m .
( u m v m ) = ( u m ( N + 1 ) v m ( N + 1 ) ) + ( α β ) J m ( l inc r ) F m , h ( ϑ , z ) .
E ϑ m ( j ) ( r j , ϑ , z ) = E ϑ m ( j + 1 ) ( r j , ϑ , z ) ,
E z m ( j ) ( r j , ϑ , z ) = E z m ( j + 1 ) ( r j , ϑ , z ) ,
H ϑ m ( j ) ( r j , ϑ , z ) = H ϑ m ( j + 1 ) ( r j , ϑ , z ) ,
H z m ( j ) ( r j , ϑ , z ) = H z m ( j + 1 ) ( r j , ϑ , z ) ,
H ϑ ( r 1 ) , E z ( r 1 ) , H ϑ ( r 2 ) , E z ( r 2 ) , , H ϑ ( r N ) , E z ( r N ) , E ϑ ( r 1 ) , H z ( r 1 ) , E ϑ ( r 2 ) , H z ( r 2 ) , , E ϑ ( r N ) , H z ( r N ) ,
ν p = h / ( n p k 0 ) ,
μ p = l p / ( n p k 0 )             ( p = 1 , 2 , , N + 1 ) ,
ρ q = k 0 r q             ( q = 1 , 2 , , N ) ,
J p q = J m ( n p μ p ρ q ) ,
Y p q = Y m ( n p μ p ρ q )             ( p = 1 , 2 , , N + 1 ; q = 1 , 2 , , N ; p - q = 0 or 1 ) ,
H p q = H m ( n p μ p ρ q ) ,
M x = α ξ + β η ,
x = [ n 1 a ( 1 ) , - n 2 a ( 2 ) , - n 2 b ( 2 ) , , ( - 1 ) N a ( N + 1 ) , n 1 A ( 1 ) , - n 2 A ( 2 ) , - n 2 B ( 2 ) , , ( - 1 ) N A ( N + 1 ) ] .
M = [ M 11 M 12 M 21 M 22 ] ,
ξ = ( ξ 1 ξ 2 ) ,             η = ( η 1 η 2 ) .
p = [ C / 2 ] + 1             ( C = 1 , 2 , , 2 N ) ,
q = [ ( L + 1 ) / 2 ]             ( L = 1 , 2 , , 2 N ) ,
m L C = μ p 2 Z p q             ( L = 1 , 3 , , 2 N - 1 ) ,
m L C = n p μ p Z p q             ( L = 2 , 4 , , 2 N ) ,
Z = J             if             C = 1 , 2 , 4 , , 2 N - 2.
Z = Y             if             C = 3 , 5 , , 2 N - 1 ,
Z = H             if             C = 2 N .
m L C ( lower right ) = { m L C ( upper left ) / n p if L is odd m L C ( upper left ) n p if L is even .
m L C ( upper right ) = { - i m ν p μ p 2 ρ q m L + 1 , C ( upper left ) if L is odd 0 if L is even .
m L C ( lower left ) = - m L C ( upper right ) / n p ( L , C = 1 , 2 , 2 N ) .
ξ L ( upper ) = { 0 if 1 L 2 N - 1 μ N + 1 J N + 1 , N if L = 2 N - 1 μ N + 1 2 J N + 1 , N if L = 2 N .
ξ 2 N - 1 ( lower ) = i m ν N + 1 ρ N J N + 1 , N .
η 1 = - ξ 2 ,             η 2 = ξ 1 .
M 11 x 1 = ξ 1 , M 22 x 2 = η 2 ,
a m ( N + 1 ) = α a m ( TM ) + β a m ( TE ) , A m ( N + 1 ) = α A m ( TM ) + β A m ( TE ) ,
H m ( l r ) = ( 2 π l r ) 1 / 2 exp [ - i ( l r - 2 m + 1 4 π ) ]
[ S MM S EM S ME S EE ] = m = - [ a m ( TM ) a m ( TE ) A m ( TM ) A m ( TE ) ] exp ( - i m ϑ ) ,
( u v ) = ( 2 π l r ) 1 / 2 exp [ - i ( l r + h z - π 4 ) ] [ S MM S EM S ME S EE ] ( α β ) .
E = - i k 0 cos φ ( u f - v e ) , H = - i k 0 cos φ ( v f + u e ) ,
e x = - sin ϑ ,             e y = cos ϑ ,             e z = 0 ,
f x = sin φ cos ϑ ,             f y = sin φ sin ϑ ,             f z = - cos φ .
[ E ( TM ) E ( TE ) ] = - i ( 2 k 0 cos φ π r ) 1 / 2 exp [ - i ( r k 0 cos φ + z k 0 sin φ - π 4 ) ] [ S MM S EM S ME S EE ] ( α β ) ,
S = c k 0 cos φ 4 π 2 r ( α S MM + β S EM 2 + α S ME + β S EE 2 ) s ,
s = f × e .
r k 0 cos φ + z k 0 sin φ ,
[ S MM S EM S ME S EE ]
i ( r , ϑ ) = S I inc = 2 π r k 0 cos φ ( α S MM + β S EM 2 + α S ME + β S EE 2 ) ,
a M ( TM ) , a M ( TE ) a 1 ( TM ) , a 1 ( TE )             ( M > m max ) .
m max k 0 r N cos φ ,
f ( r ) = - r 0 r ( u x / x , u y / y , 0 ) d s , g ( r ) = - r 0 r ( v x / x , v y / y , 0 ) d s
J m ( x ) ~ 1 ( 2 π m ) 1 / 2 ( e x 2 m ) m , Y m ( x ) ~ - ( 2 π m ) 1 / 2 ( e x 2 m ) - m .
m k 0 r N cos φ ,

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