Abstract

A theory of scattering of polarized light by an arbitrary configuration of closely spaced, parallel nonhomogeneous infinite circular cylinders at oblique incidence is presented in this paper. The polarization of the incident wave can be linear, circular, or elliptic. The diameter of the cylinders, the spacing between the cylinders, and the incident wavelength are comparable with each other. Each cylinder can have an arbitrary number of concentric layers of stratification, and the complex index of refraction in each layer can be different. An exact solution of Maxwell’s equations that accounts for near-field multiple scattering and far-field wave interference is developed. Formulas are developed for the extinction and scattering cross sections and the scattering amplitude. The corresponding expressions for an isolated cylinder are also obtained as a special case. The light-scattering characteristics of a collection of cylinders are examined by numerical data for various incidence angles and polarizations.

© 1996 Optical Society of America

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References

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  1. M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969).
  2. H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981).
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    [CrossRef]
  4. V. Twersky, “Multiple scattering of radiation by an arbitrary configuration of parallel cylinders,” J. Acoust. Soc. Am. 24, 42–46 (1952).
    [CrossRef]
  5. G. O. Olaofe, “Scattering by an arbitrary configuration of parallel circular cylinders,” J. Opt. Soc. Am. 60, 1233–1236 (1970).
    [CrossRef]
  6. M. K. Moaveni, A. A. Rizvi, B. A. Kamran, “Plane-wave scattering by gratings of conducting cylinders in an inhomogeneous and lossy dielectric,” J. Opt. Soc. Am. A 5, 834–842 (1988).
    [CrossRef]
  7. D. Felbacq, G. Tayeb, D. Maystre, “Scattering by a random set of parallel cylinders,” J. Opt. Soc. Am. A 11, 2526–2538 (1994).
    [CrossRef]
  8. S.-C. Lee, “Dependent scattering of an obliquely incident plane wave by a collection of parallel cylinders,” J. Appl. Phys. 68, 4952–4957 (1990).
    [CrossRef]
  9. S.-C. Lee, “Scattering by closely-spaced radially-stratified parallel cylinders,” J. Quant. Spectrosc. Radiat. Transfer 48, 119–130 (1992).
    [CrossRef]
  10. G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge U. Press, Cambridge, 1952).
  11. M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1993).
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    [CrossRef] [PubMed]
  13. W. G. Driscoll, ed., Handbook of Optics (McGraw-Hill, New York, 1987).
  14. I. H. Malitson, “Interspecimen comparison of refractive index of fused silica,” J. Opt. Soc. Am. 55, 1206–1210 (1965).
    [CrossRef]

1994 (1)

1992 (1)

S.-C. Lee, “Scattering by closely-spaced radially-stratified parallel cylinders,” J. Quant. Spectrosc. Radiat. Transfer 48, 119–130 (1992).
[CrossRef]

1990 (1)

S.-C. Lee, “Dependent scattering of an obliquely incident plane wave by a collection of parallel cylinders,” J. Appl. Phys. 68, 4952–4957 (1990).
[CrossRef]

1988 (1)

1987 (1)

1980 (1)

1970 (1)

1965 (1)

I. H. Malitson, “Interspecimen comparison of refractive index of fused silica,” J. Opt. Soc. Am. 55, 1206–1210 (1965).
[CrossRef]

1952 (1)

V. Twersky, “Multiple scattering of radiation by an arbitrary configuration of parallel cylinders,” J. Acoust. Soc. Am. 24, 42–46 (1952).
[CrossRef]

Barabas, M.

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1993).

Cohen, A.

Felbacq, D.

Kamran, B. A.

Kerker, M.

M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969).

Lee, S.-C.

S.-C. Lee, “Scattering by closely-spaced radially-stratified parallel cylinders,” J. Quant. Spectrosc. Radiat. Transfer 48, 119–130 (1992).
[CrossRef]

S.-C. Lee, “Dependent scattering of an obliquely incident plane wave by a collection of parallel cylinders,” J. Appl. Phys. 68, 4952–4957 (1990).
[CrossRef]

Malitson, I. H.

I. H. Malitson, “Interspecimen comparison of refractive index of fused silica,” J. Opt. Soc. Am. 55, 1206–1210 (1965).
[CrossRef]

Maystre, D.

Moaveni, M. K.

Olaofe, G. O.

Rizvi, A. A.

Tayeb, G.

Twersky, V.

V. Twersky, “Multiple scattering of radiation by an arbitrary configuration of parallel cylinders,” J. Acoust. Soc. Am. 24, 42–46 (1952).
[CrossRef]

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981).

Watson, G. N.

G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge U. Press, Cambridge, 1952).

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1993).

J. Acoust. Soc. Am. (1)

V. Twersky, “Multiple scattering of radiation by an arbitrary configuration of parallel cylinders,” J. Acoust. Soc. Am. 24, 42–46 (1952).
[CrossRef]

J. Appl. Phys. (1)

S.-C. Lee, “Dependent scattering of an obliquely incident plane wave by a collection of parallel cylinders,” J. Appl. Phys. 68, 4952–4957 (1990).
[CrossRef]

J. Opt. Soc. Am. (2)

G. O. Olaofe, “Scattering by an arbitrary configuration of parallel circular cylinders,” J. Opt. Soc. Am. 60, 1233–1236 (1970).
[CrossRef]

I. H. Malitson, “Interspecimen comparison of refractive index of fused silica,” J. Opt. Soc. Am. 55, 1206–1210 (1965).
[CrossRef]

J. Opt. Soc. Am. A (3)

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

S.-C. Lee, “Scattering by closely-spaced radially-stratified parallel cylinders,” J. Quant. Spectrosc. Radiat. Transfer 48, 119–130 (1992).
[CrossRef]

Opt. Lett. (1)

Other (5)

W. G. Driscoll, ed., Handbook of Optics (McGraw-Hill, New York, 1987).

M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969).

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981).

G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge U. Press, Cambridge, 1952).

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1993).

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Figures (8)

Fig. 1
Fig. 1

(a) Schematic diagram depicting a polarized incident wave on an arbitrary configuration of a finite number of cylinders, (b) geometry relating to the application of the addition theorem of Bessel functions.

Fig. 2
Fig. 2

Schematic diagram showing three parallel cylinders at equidistance subjected to a polarized incident wave.

Fig. 3
Fig. 3

Influence of clearance-to-diameter ratio (c/d) on extinction cross section.

Fig. 4
Fig. 4

Influence of c/d on scattering cross section.

Fig. 5
Fig. 5

Influence of polarization on extinction cross section.

Fig. 6
Fig. 6

Influence of polarization on scattering cross section.

Fig. 7
Fig. 7

Angular distribution of the scattered intensity for a circularly polarized incident wave.

Fig. 8
Fig. 8

Angular variation of the phase of the scattered wave for a circularly polarized incident wave.

Tables (1)

Tables Icon

Table 1 Extinction and Scattering Efficiencies of a Set of Lossless Cylinders at ϕ = 30° and θ = 0°

Equations (70)

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E = i m k 0 × × ( e z u ) + × ( e z υ ) ,
H = m × ( e z u ) + i k 0 × × ( e z υ ) ,
( u 0 υ 0 ) = ( α I α II ) exp ( i k · ρ ) ,
( E 0 H 0 ) = [ α I α II α II α I ] ( P N P M ) i l 0 exp ( i k · ρ ) ,
S 0 = c 0 8 π l 0 2 ( | α I | 2 + | α II | 2 ) e i ,
E 0 = [ l 0 | α I | exp ( i ɛ I ) P N + l 0 | α II | exp ( i ɛ II ) P M ] × exp ( i k · ρ ) ,
cos δ 0 = Re ( α I α II * ) / | α I α II | ,
( u j ( R p ) υ j ( R p ) ) = ( α I α II ) ɛ j exp ( i k · R j p i h 0 z ) + ( u j s ( R j p ) υ j s ( R j p ) ) + k j N ( u k s ( R k p ) υ k s ( R k p ) ) ,
ɛ j = exp [ i k 0 R j cos ϕ cos ( θ + γ j ) ] ,
exp ( i k · R j p ) = n = ( i ) n exp [ in ( θ + γ j p ) ] J n ( l 0 R j p ) .
( u j s ( R j p ) υ j s ( R j p ) ) = exp ( i h 0 z ) n = ( i ) n exp ( in γ j p ) × H n ( l 0 R j p ) ( b j n a j n ) .
exp ( in ψ k ) H n ( l 0 R k p ) = s = exp ( is ψ j ) × H n + s ( l 0 R j k ) J n ( l 0 R j p ) .
( u k s ( R k p ) υ k s ( R k p ) ) = exp ( i h 0 z ) n = s = ( i ) s × exp ( in γ j p ) exp [ i ( s n ) γ k j ] H s n ( l 0 R j k ) × ( b k s a k s ) J n ( l 0 R j p ) .
( u j ( R p ) υ j ( R p ) ) = exp ( i h 0 z ) n = ( i ) n exp ( in γ j p ) × { [ ( α I α II ) ɛ j exp ( in θ ) k j N s = ( b k s a k s ) G k s j n ] J n ( l 0 R j p ) ( b j n a j n ) H n ( l 0 R j p ) } ,
G k s j n = ( i ) s n exp [ i ( s n ) γ k j ] H s n ( l 0 R j k ) .
( u j ( m ) ( r ) υ j ( m ) ( r ) ) = exp ( i h 0 z ) n = ( i ) n exp ( in γ j p ) × [ ( B j n ( m ) A j n ( m ) ) J n ( l j m r ) ( b j n ( m ) a j n ( m ) ) H n ( l j m r ) ] ,
b j n ( 1 ) = a j n ( 1 ) = 0 ,
E j n ( m ) = e r [ h 0 m j m k 0 u j n ( m ) r + i m j m r υ j n ( m ) ] + e γ [ in h 0 m j m k 0 r u j n ( m ) υ j n ( m ) r ] + e z i l j m 2 m j m k 0 u j n ( m ) ,
H j n ( m ) = e r [ in m j m r u j n ( m ) + h 0 k 0 υ j n ( m ) r ] + e γ [ m j m u j n ( m ) r + in h 0 k 0 r υ j n ( m ) ] + e z i l j m 2 k 0 υ j n ( m ) .
( E j n ( m ) , H j n ( m ) ) · e t = ( E j n ( m + 1 ) , H j n ( m + 1 ) ) · e t
[ δ j k δ n s + ( 1 δ j k ) G k s j n b j n I 0 ( 1 δ j k ) G k s j n b j n II 0 ( 1 δ j k ) G k s j n a j n I 0 δ j k δ n s + ( 1 δ j k ) G k s j n a j n II 0 ] × ( b j n a j n ) = exp ( in θ ) ( ɛ j ( α I b j n I 0 + α II b j n II 0 ) ɛ j ( α I a j n I 0 + α II a j n II 0 ) ) ,
( b j n a j n ) = ( α I b j n I + α II b j n II α I a j n I + α II a j n II ) ,
H n ( l 0 R ) = ( 2 π l 0 R ) 1 / 2 exp [ i l 0 R + i ( 2 n + 1 ) π / 4 ] ,
( u s ( R ) υ s ( R ) ) = ( 2 π l 0 R ) 1 / 2 exp ( i l 0 R i h 0 z i 3 π / 4 ) × ( T u ( ϕ , γ ) T υ ( ϕ , γ ) ) ,
( T u ( ϕ , γ ) T υ ( ϕ , γ ) ) = ( α I T 11 ( ϕ , γ ) + α II T 21 ( ϕ , γ ) α I T 12 ( ϕ , γ ) + α II T 22 ( ϕ , γ ) ) ,
[ T 11 ( ϕ , γ ) T 12 ( ϕ , γ ) T 21 ( ϕ , γ ) T 22 ( ϕ , γ ) ] = j = 1 N n = [ b j n I a j n I b j n II a j n II ] exp [ in γ + i l 0 R j cos ( γ γ j ) ] .
( E s H s ) = i l 0 ( 2 π l 0 R ) 1 / 2 exp ( i l 0 R i h 0 z i 3 π / 4 ) × [ T u ( ϕ , γ ) T υ ( ϕ , γ ) T υ ( ϕ , γ ) T u ( ϕ , γ ) ] ( e c e γ ) ,
I s ( ϕ , γ ) = c 0 8 π S 0 · e i Re ( E s × H s * ) = ( 2 π l 0 R ) 1 | α I | 2 + | α II | 2 [ | T u ( ϕ , γ ) | 2 + | T υ ( ϕ , γ ) | 2 ] e s ,
C s ( ϕ ) = 0 2 π ( I s · e r ) R d γ = 4 k 0 1 | α I | 2 + | α II | 2 j = 1 N k = 1 N n = s = exp [ i ( n s ) ( γ k j π / 2 ) ] J s n ( l 0 R j k ) ( b j n b k s * + a j n a k s * ) .
C s ( ϕ ) = 4 k 0 1 | α I | 2 + | α II | 2 j = 1 N k = 1 N n = s = exp [ i ( n s ) ( γ k j π / 2 ) ] J s n ( l 0 R j k ) { | α I | 2 ( b j n I b k s I * + a j n I a k s I * ) + | α II | 2 ( b j n II b k s II * + a j n II a k s II * ) + 2 Re [ α I α II * ( b j n I b k s II * + a j n I a k s II * ) ] } .
C e ( ϕ ) = c 0 8 π S 0 · e i Re { 0 2 π [ ( E 0 × H s * + E s × H 0 * ) · e r ] R d γ } .
C e ( ϕ ) = 4 k 0 1 | α I | 2 + | α II | 2 Re [ α I * T u ( ϕ , γ = θ ) + α II * T υ ( ϕ , γ = θ ) ] ,
C e ( ϕ ) = 4 k 0 1 | α I | 2 + | α II | 2 Re [ | α I | 2 T 11 ( ϕ , γ = θ ) + | α II | 2 T 22 ( ϕ , γ = θ ) + α I α II * T 12 ( ϕ , γ = θ ) + α I * α II T 21 ( ϕ , γ = θ ) ] .
( C e TM ( ϕ ) C e TE ( ϕ ) ) = 4 k 0 Re { j = 1 N n = ( b j n I a j n II ) × exp [ i l 0 R j cos ( θ + γ j ) in θ ] } ,
( C s TM ( ϕ ) C s TE ( ϕ ) ) = 4 k 0 Re j = 1 N k = 1 N n = s = exp [ i ( n s ) ( γ k j π / 2 ) ] J s n ( l 0 R j k ) ( b j n I b k s I * + a j n I a k s I * b j n II b k s II * + a j n II a k s II * ) ,
( I s Q s U s V s ) = ( l 0 2 ( | T u | 2 + | T υ | 2 ) l 0 2 ( | T u | 2 | T υ | 2 ) 2 l 0 2 Re ( T u T υ * ) 2 l 0 2 Im ( T u T υ * ) ) ,
cos δ s = Re ( T u T υ * ) / | T u T υ | ,
I s ( ϕ = 0 , γ ) = ( 2 π k 0 R ) 1 | α I | 2 + | α II | 2 [ | α I | 2 | T 11 ( γ ) | 2 + | α II | 2 | T 22 ( γ ) | 2 ] e r ,
C s = 4 k 0 1 | α I | 2 + | α II | 2 j = 1 N k = 1 N n = s = exp [ i ( n s ) × ( γ k j π / 2 ) ] J s n ( l 0 R j k ) ( | α I | 2 b j n I b k s I * + | α II | 2 a j n II a k s II * ) ,
C e = 4 k 0 1 | α I | 2 + | α II | 2 Re { j = 1 N n = ( | α I | 2 b j n I + | α II | 2 a j n II ) exp [ i l 0 R j cos ( θ + γ j ) in θ ] } .
cos δ s = Re ( α I α II * T 11 T 22 * ) / | α I α II T 11 T 22 | .
TM incidence : ( I s , Q s , U s , V s ) = ( k 0 2 | T 11 ( γ ) | 2 , k 0 2 | T 11 ( γ ) | 2 , 0 , 0 ) ,
TE incidence : ( I s , Q s , U s , V s ) = ( k 0 2 | T 22 ( γ ) | 2 , k 0 2 | T 22 ( γ ) | 2 , 0 , 0 ) ,
( b n 0 a n 0 ) = ( α I b n I 0 + α II b n II 0 α I a n I 0 + α II a n II 0 ) .
b n I 0 = b n I 0 , a n II 0 = a n II 0 , a n I 0 = b n II 0 , a 0 I 0 = 0 ,
b n II 0 = b n II 0 , a n I 0 = a n I 0
b n II 0 = a n I 0 = 0
( T u 0 ( ϕ , γ ) T υ 0 ( ϕ , γ ) ) = n = ( α I b n I 0 + α II b n II 0 α I a n I 0 + α II a n II 0 ) exp ( in γ ) ,
I s 0 ( ϕ , γ ) = ( 2 π l 0 R ) 1 | α I | 2 + | α II | 2 × { | α I [ b 0 I 0 + 2 n = 1 b n I 0 cos ( n γ ) ] + 2 i α II n = 1 b n II 0 sin ( n γ ) | 2 + | α II [ a 0 II 0 + 2 n = 1 a n II 0 cos ( n γ ) ] + 2 i α I n = 1 a n I 0 sin ( n γ ) | 2 } e s .
C s 0 ( ϕ ) = 4 k 0 1 | α I | 2 + | α II | 2 × { | α I | 2 [ | b 0 I | 2 + 2 n = 1 ( | b n I | 2 + | a n I | 2 ) ] + | α II | 2 [ | a 0 II | 2 + 2 n = 1 ( | a n II | 2 + | a n I | 2 ) ] } ,
C e 0 ( ϕ ) = 4 k 0 1 | α I | 2 + | α II | 2 [ | α I | 2 Re ( b 0 I 0 + 2 n = 1 b n I 0 ) + | α II | 2 Re ( a 0 II 0 + 2 n = 1 a n II 0 ) ]
C e 0 ( TM ) = 10.2888
C e 0 ( TE ) = 4.1667
C s 0 ( TM ) = 10.2888
C s 0 ( TM ) = 4.1667
M j n X j n = α j n P j n + β j n Q j n ,
( α j n β j n ) = ( α I α II ) ɛ j exp ( in θ ) k j N s = ( b k s a k s ) G k s j n
X j n = ( a j n b j n a j n ( L ) b j n ( L ) A j n ( L ) B j n ( L ) . . . a j n ( 2 ) b j n ( 2 ) A j n ( 2 ) B j n ( 2 ) A j n ( 1 ) B j n ( 1 ) ) T ,
P j n = ( in h 0 k 0 r j L J n ( l 0 r j L ) i l 0 2 k 0 J n ( l 0 r j L ) l 0 J n ( l 0 r j L ) 0 0 0 . . . ) T ,
Q j n = ( l 0 J n ( l 0 r j L ) 0 in h 0 k 0 r j L J n ( l 0 r j L ) l 0 2 J n ( l 0 r j L ) 0 0 . . . ) T ,
M j n ( 1 ) = ( M 00 M 01 0 0 0 . . . ) ,
M j n ( L ) = ( 0 0 0 . . . M L 1 , L 1 M L 1 , L ) ,
M j n ( p ) = ( 0 0 . . . M p p M p , p + 1 0 0 0 . . . ) ,
M j n = ( M 00 M 01 ) ,
M j n = [ M 00 M 01 0 0 M 11 M 12 ] ,
M j n = [ M 00 M 01 0 0 0 M 11 M 12 0 0 0 M 22 M 23 ] .
M j n X j n σ 0 = α I P j n + α II Q j n ,
X j n σ 0 = ( a j n σ 0 b j n σ 0 a j n σ ( L ) 0 b j n σ ( L ) 0 A j n σ ( L ) 0 B j n σ ( L ) 0 . . . a j n σ ( 2 ) 0 b j n σ ( 2 ) 0 A j n σ ( 2 ) 0 B j n σ ( 2 ) 0 A j n σ ( 1 ) 0 B j n σ ( 1 ) 0 ) T ,
σ = { blank arbitrary α I and α II I ( TM ) α I = 1 , α I I = 0 II ( TE ) α I = 0 , α I I = 1 .
X j n = ɛ j exp ( i n θ ) ( α I X j n I 0 + α II X j n II 0 ) X j n I 0 k j N s = b k s G k s j n X j n II 0 k j N s = a k s G k s j n ,

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