Abstract

The formulation for the extinction and scattering cross sections of closely spaced parallel infinite cylinders in a dielectric medium of finite thickness is presented. We consider the general case of dissimilar refractive indices for the half-spaces on both sides of the slab, and the diameter and refractive index of each cylinder can be different. The formulation accounts for the coherent scattering between the cylinders and scattering of the multiply reflected internal waves inside the slab. Discontinuity in the refractive index across the dielectric slab interfaces results in boundary reflections that modify the angular distribution of the scattered intensity in both forward and backward directions. The extinction cross section, which is derived by a formal application of the optical theorem, is shown to consist of both a forward and a backward component. The general solution is applied to obtain the formulas for the cases of cylinders in front of a reflecting plane, cylinders inside a semi-infinite dielectric medium, and cylinders in free space.

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References

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1999 (1)

1998 (1)

1997 (1)

1996 (3)

1994 (2)

T. C. Rao and R. Barakat, "Plane wave scattering by a finite array of conducting cylinders partially buried in a ground plane: TM polarization," Pure Appl. Opt. 3, 1023-1048 (1994).
[CrossRef]

D. Felbacq, G. Tayeb, and D. Maystre, "Scattering by a random set of parallel cylinders," J. Opt. Soc. Am. A 11, 2526-2538 (1994).
[CrossRef]

1993 (1)

M. Born and A. Wolf, Principles of Optics (Pergamon, 1993).

1992 (1)

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

1990 (2)

J. R. Wait, "Note on solution for scattering from parallel wires in an interface," J. Electromagn. Waves Appl. 4, 1151-1155 (1990).
[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]

1987 (1)

1981 (1)

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

1970 (1)

1969 (1)

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

1966 (1)

1952 (1)

V. Twersky, "Multiple scattering of radiation by anarbitrary configuration of parallel cylinders," J. Acoust. Soc. Am. 24, 42-46 (1952).
[CrossRef]

Barabas, M.

Barakat, R.

T. C. Rao and R. Barakat, "Plane wave scattering by a finite array of conducting cylinders partially buried in a ground plane: TM polarization," Pure Appl. Opt. 3, 1023-1048 (1994).
[CrossRef]

Borghi, R.

Born, M.

M. Born and A. Wolf, Principles of Optics (Pergamon, 1993).

Farone, W. A.

Felbacq, D.

Frezza, F.

Gori, F.

Grzesik, J. A.

Kerker, M.

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

Lee, S. C.

Maystre, D.

Ngo, D.

Oloafe, G. O.

Querfeld, C. W.

Rao, T. C.

T. C. Rao and R. Barakat, "Plane wave scattering by a finite array of conducting cylinders partially buried in a ground plane: TM polarization," Pure Appl. Opt. 3, 1023-1048 (1994).
[CrossRef]

Santarsiero, M.

Schettini, G.

Tayeb, G.

Twersky, V.

V. Twersky, "Multiple scattering of radiation by anarbitrary 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, 1981).

Videen, G.

Wait, J. R.

J. R. Wait, "Note on solution for scattering from parallel wires in an interface," J. Electromagn. Waves Appl. 4, 1151-1155 (1990).
[CrossRef]

Wolf, A.

M. Born and A. Wolf, Principles of Optics (Pergamon, 1993).

J. Acoust. Soc. Am. (1)

V. Twersky, "Multiple scattering of radiation by anarbitrary 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. Electromagn. Waves Appl. (1)

J. R. Wait, "Note on solution for scattering from parallel wires in an interface," J. Electromagn. Waves Appl. 4, 1151-1155 (1990).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (8)

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

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

Pure Appl. Opt. (1)

T. C. Rao and R. Barakat, "Plane wave scattering by a finite array of conducting cylinders partially buried in a ground plane: TM polarization," Pure Appl. Opt. 3, 1023-1048 (1994).
[CrossRef]

Other (3)

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

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

M. Born and A. Wolf, Principles of Optics (Pergamon, 1993).

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Figures (7)

Fig. 1
Fig. 1

Schematic diagram showing the pertinent parameters of the problem.

Fig. 2
Fig. 2

Isometric view showing the propagation of the primary source waves.

Fig. 3
Fig. 3

Three silicon-coated silica cylinders inside a dielectric slab.

Fig. 4
Fig. 4

Scattering by a single cylinder in an infinite homogeneous medium.

Fig. 5
Fig. 5

Scattering by the set of three cylinders in an infinite homogeneous medium.

Fig. 6
Fig. 6

Scattering by the set of three cylinders inside a finite dielectric slab with free space on both sides.

Fig. 7
Fig. 7

Scattering by the set of three cylinders inside a finite dielectric slab with a medium of higher refractive index on one side.

Tables (1)

Tables Icon

Table 1 Extinction and Scattering Cross Sections for the Various Cases

Equations (80)

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E o σ = i o σ ( u o σ P o N σ + v o σ P o M σ ) , H o σ = i m σ o σ ( v o σ P o N σ u o σ P o M σ ) ,
( u o σ v o σ ) = ( α I σ α I I σ ) exp ( i k o σ ρ ) ,
ψ j ( R p ) = τ = ± ψ o τ ( R p ) + ψ j s ( R j p ) + k j N ψ k j s ( R j p ) + τ = ± k = 1 N ψ k j r τ ( R j p ) .
( u j s v j s ) = n = ( i ) n exp ( i n γ j p ) H n ( R j p ) ( b j n a j n ) ,
k t = γ i e x + η e y + h e z ,
k τ = τ β e x + η e y + h e z ,
k t + = γ t e x + η e y + h e z ,
β 2 + η 2 = 1
[ A ] [ Z ] = [ B 1 ] X k + [ B 2 ] Y k + [ B 3 ] X k + + [ B 4 ] Y k + ,
( X k τ Y k τ ) = 1 π s = τ s ( β + τ i η ) s β ( b k s a k s )
[ Z ] T = [ X k r + X k t Y k r + Y k t X k r X k t + Y k r Y k t + ] ,
{ R u τ u τ , R u τ v τ , R v τ u τ , R v τ v τ } = { X k r τ X k τ , Y k r τ X k τ , X k r τ Y k τ , Y k r τ Y k τ } ,
{ T u τ u τ , T u τ v τ , T v τ u τ , T v τ v τ } = { X k t τ X k τ , Y k t τ X k τ , X k t τ Y k τ , Y k t τ Y k τ } .
[ δ j k δ n s + [ ( 1 δ j k ) G k s j n + R u k s u j n ] b j n o , I + R u k s v j n b j n o , I I [ ( 1 δ j k ) G k s j n + R v k s v j n ] b j n o , I I + R v k s u j n b j n o , I [ ( 1 δ j k ) G k s j n + R u k s u j n ] a j n o , I + R u k s v j n a j n o , I I δ j k δ n s + [ ( 1 δ j k ) G k s j n + R v k s v j n ] a j n o , I I + R v k s u j n a j n o , I ] [ b k s a k s ] = τ = ± ϵ j τ τ n exp ( τ i n θ ) { α I τ [ b j n o , I a j n o , I ] + α I I τ [ b j n o , I I a j n o , I I ] } ,
G k s j n = ( i ) s n exp [ i ( s n ) γ k j ] H s n ( R j k ) .
R ψ k s ψ j n = 1 π 1 β f j k + [ R ψ + ψ + ( β + i η ) s n + ( 1 ) s R ψ ψ + ( β + i η ) ( s + n ) ] d η + ( 1 ) n π 1 β f j k [ R ψ + ψ ( β + i η ) s + n + ( 1 ) s R ψ ψ ( β + i η ) ( s n ) ] d η ,
f j k τ = exp [ τ i β ( x k + x j ) + i η ( y k y j ) ]
( u k t τ v k t τ ) = + exp ( i k τ R k i k t τ R p ) ( X k t τ Y k t τ ) d η .
( u k t τ v k t τ ) = exp ( i t τ R p ) 2 i π t τ R p t τ cos γ k = 1 N ( T u k τ ( γ ) T v k τ ( γ ) ) ,
T ψ k + ( γ ) = exp [ i ( β ̂ + x k + η ̂ + y k ) ] β ̂ + τ = ± s = ( τ ) s ( β ̂ + + τ i η ̂ + ) s { T ̂ u τ ψ + b k s + T ̂ v τ ψ + a k s } ,
( γ ̂ t + i η ̂ + ) t = exp ( i γ ) .
T ψ k ( γ ) = exp [ i ( β ̃ x k + η ̃ y k ) ] β ̃ τ = ± s = ( τ ) s ( β ̃ + τ i η ̃ ) s { u τ ψ b k s + T ̃ v τ ψ a k s } ,
( γ ̃ i + i η ̃ ) i = exp [ i ( π γ ) ] .
E t o t a l ( 0 ) = E o i + E o r + E t , H t o t a l ( 0 ) = H o i + H o r + H t for x < 0 ,
E t o t a l ( 2 ) = E o t + E t + , H t o t a l ( 2 ) = H o t + H t + for x > L ,
E t τ = i σ ( u t τ P N t τ + v t τ P M t τ ) , H t τ = i σ m σ ( v t τ P N t τ u t τ P M t τ ) ,
S i = c 0 8 π n o i 2 ( α I i 2 + α I I i 2 ) .
W a = c 0 8 π Re π 2 3 π 2 E t o t a l ( 0 ) × H t o t a l ( 0 ) * e R R p d γ + c 0 8 π Re π 2 + π 2 E t o t a l ( 2 ) × H t o t a l ( 2 ) * e R R p d γ ,
W a = W o + W s + W i r + W ,
W o = c 0 8 π Re π 2 3 π 2 ( E o i + E o r ) × ( H o i + H o r ) * e R R p d γ + c 0 8 π Re π 2 π 2 E o t × H o t * e R R p d γ ,
W i r = c 0 8 π Re π 2 3 π 2 ( E o i × H t * + E t + H o i * ) e R R p d γ ,
W s = c 0 8 π Re π 2 3 π 2 E t × H t * e R R p d γ + c 0 8 π Re π 2 π 2 E t + × H t + * e R R p d γ ,
W = c 0 8 π Re π 2 3 π 2 ( E o r × H t * + E t × H o r * ) e R R p d γ + c 0 8 π Re π 2 π 2 ( E t + × H o t * + E o t × H t + * ) e R R p d γ ,
W = W a + W s = W e ,
W s = c 0 t 2 8 π 2 π k o ( t ) 2 π 2 π 2 ( k = 1 N T u k + 2 + k = 1 N T v k + 2 ) cos 2 γ d γ + c 0 i 2 8 π 2 π k o ( i ) 2 π 2 3 π 2 ( k = 1 N T u k 2 + k = 1 N T v k 2 ) cos 2 γ d γ .
W e = c 0 t 2 8 π 4 k o ( cos θ t ) t Re ( α I t k = 1 N T u k + * + α I I t k = 1 N T v k + * ) γ = θ t + c 0 i 2 8 π 4 k o ( cos θ i ) i Re ( α I r k = 1 N T u k * + α I I r k = 1 N T v k * ) γ = π + θ i .
C s = 2 π k i ( t i ) 2 α I i 2 + α I I i 2 ( t ) 2 π 2 π 2 ( k = 1 N T u k + 2 + k = 1 N T v k + 2 ) cos 2 γ d γ + 2 π k i ( i ) 2 α I i 2 + α I I i 2 π 2 3 π 2 ( k = 1 N T u k 2 + k = 1 N T v k 2 ) cos 2 γ d γ ,
C e = 4 k i t ( t i ) 2 cos θ t α I i 2 + α I I i 2 Re ( α I t k = 1 N T u k + * + α I I t k = 1 N T v k + * ) γ = θ t + 4 k i i cos θ i α I i 2 + α I I i 2 Re ( α I r k = 1 N T u k * + α I I r k = 1 N T v k * ) γ = π + θ i ,
( γ ̂ t + i η ̂ + ) t = exp ( i θ t ) .
( γ ̃ i + i η ̃ ) i = exp ( i θ i ) .
R ψ ψ + = R ψ ψ = R ψ + ψ + = 0 .
[ A 22 ] [ Z L ] = [ B 3 L ] X k + + [ B 4 L ] Y k + ,
R ψ k s ψ j n = ( 1 ) n π 1 β f j k R ψ + ψ ( β + i η ) s + n d η ,
T ψ ψ ( ψ ψ ) = T ψ ψ + = 0 , T ψ ψ = 1 .
[ A ] [ Z ] = [ B 3 ] X k + + [ B 4 ] Y k + ,
T ψ k + ( γ ) = exp [ i ( β ̂ + x k + η ̂ + y k ) ] β ̂ + s = ( β ̂ + + i η ̂ + ) s × { T ̂ u τ ψ + b k s + T ̂ v τ ψ + a k s } .
( T u k ( γ ) T v k ( γ ) ) = exp [ i i R k cos ( γ γ k ) ] cos γ s = { exp ( i s γ ) ( b k s a k s ) + exp [ i s ( π γ ) ] ( T ̃ u + u b k s + T ̃ v + u a k s T ̃ u + v b k s + T ̃ v + v a k s ) } .
C s = 2 π k i ( t i ) 4 1 α I i 2 + α I I i 2 π 2 π 2 ( k = 1 N T u k + 2 + k = 1 N T v k + 2 ) cos 2 γ d γ + 2 π k i 1 α I i 2 + α I I i 2 j = 1 N k = 1 N n = s = { π exp [ i ( n s ) ( γ k j π 2 ) ] J s n ( i R j k ) + m = m s n 2 ( 1 ) m + n s m + n s sin [ ( m + n s ) π 2 ] exp [ i m ( γ k j π 2 ) ] J m ( i R j k ) } ( b j n b k s * + a j n a k s * ) + 2 π k i 1 α I i 2 + α I I i 2 π 2 3 π 2 ( k = 1 N T u k , 2 2 + k = 1 N T v k , 2 2 ) cos 2 γ d γ ,
( T u k T v k ) = exp [ i i R k cos ( γ k θ i ) ] cos θ i { s = exp [ i s ( π + θ i ) ] ( b k s a k s ) + exp ( i s θ i ) ( T ̃ u + u b k s + T ̃ v + u a k s T ̃ u + v b k s + T ̃ v + v a k s ) γ = π + θ i } ,
C e = 4 k i ( t i ) 3 cos θ t α I i 2 + α I I i 2 Re ( α I t k = 1 N T u k + * + α I I t k = 1 N T v k + * ) γ = θ t + 4 k i cos θ i α I i 2 + α I I i 2 Re ( α I r k = 1 N T u k * + α I I r k = 1 N T v k * ) γ = π + θ i .
R ψ + ψ + = R ψ + ψ = R ψ ψ = 0 .
[ A 11 ] [ Z U ] = [ B 1 U ] X k + [ B 2 U ] Y k ,
R ψ k s ψ j n = ( 1 ) s π 1 β f j k + R ψ ψ + ( β i η ) s + n d η ,
T ψ + ψ + ( ψ ψ ) = T ψ + ψ = 0 , T ψ + ψ + = 1 .
[ A ] [ Z ] = [ B 1 ] X k + [ B 2 ] Y k .
( T u k + ( γ ) T v k + ( γ ) ) = exp [ i t R k cos ( γ γ k ) ] cos γ s = { exp ( i s γ ) ( b k s a k s ) + exp [ i s ( π γ ) ] ( T ̂ u u + b k s + T ̂ v u + a k s T ̂ u v + b k s + T ̂ v v + a k s ) } ,
T ψ k ( γ ) = exp [ i ( β ̃ x k + η ̃ y k ) ] β ̃ s = ( 1 ) s ( β ̃ i η ̃ ) s { T ̃ u ψ b k s + T ̃ v ψ a k s } ,
C s = 2 π k i ( t i ) 2 α I i 2 + α I I i 2 j = 1 N k = 1 N n = s = { π exp [ i ( n s ) ( γ k j π 2 ) ] J s n ( t R j k ) + m = m s n 2 m + n s sin [ ( m + n s ) π 2 ] exp [ i m ( γ k j π 2 ) ] J m ( t R j k ) } ( b j n b k s * + a j n a k s * ) + 2 π k i ( t i ) 2 α I i 2 + α I I i 2 π 2 π 2 ( k = 1 N T u k , 2 + 2 + k = 1 N T v k , 2 + 2 ) cos 2 γ d γ + 2 π k i ( i t ) 2 α I i 2 + α I I i 2 π 2 3 π 2 ( k = 1 N T u k 2 + k = 1 N T v k 2 ) cos 2 γ d γ ,
( T u k + T v k + ) = exp [ i t R k cos ( γ k + θ t ) ] cos θ t s = { exp ( i s θ t ) ( b k s a k s ) + exp [ i s ( π + θ t ) ] ( T ̂ u u + b k s + T ̂ v u + a k s T ̂ u v + b k s + T ̂ v v + a k s ) γ = θ t } .
C e = 4 k i ( t i ) 2 cos θ t α I i 2 + α I I i 2 Re ( α I t k = 1 N T u k + * + α I I t k = 1 N T v k + * ) γ = θ t + 4 k i ( i t ) cos θ i α I i 2 + α I I i 2 Re ( α I r k = 1 N T u k * + α I I r k = 1 N T v k * ) γ = π + θ i .
R ψ τ ψ τ = T ψ τ ψ τ ( ψ ψ , τ τ ) = 0 , T ψ τ ψ τ = 1 ,
[ δ j k δ n s + ( 1 δ j k ) G k s j n b j n o , I ( 1 δ j k ) G k s j n b j n o , I I ( 1 δ j k ) G k s j n a j n o , I δ j k δ n s + ( 1 δ j k ) G k s j n a j n o , I I ] [ b k s a k s ] = ϵ j exp ( i n θ ) { α I [ b j n o , I a j n o , I ] + α I I [ b j n o , I I a j n o , I I ] } ,
( T u k + ( γ ) T v k + ( γ ) ) = exp [ i i R k cos ( γ γ k ) ] cos γ s = exp ( i s γ ) ( b k s a k s ) .
C s = 4 k i 1 α I i 2 + α I I i 2 j = 1 N k = 1 N n = s = exp [ i ( n s ) ( γ k j π 2 ) ] × J s n ( k i R j k ) ( b j n b k s * + a j n a k s * ) .
( T u k + ( γ ) T v k + ( γ ) ) γ = θ i = exp [ i i R k cos ( θ i + γ k ) ] cos θ i s = exp ( i s θ i ) ( b k s a k s ) .
C e = 4 k i cos θ i α I i 2 + α I I i 2 Re ( α I i k = 1 N T u k + * + α I I i k = 1 N T v k + * ) γ = θ t ,
R u τ v τ = R v τ u τ = T u τ v τ = T v τ u τ = 0 .
[ 1 k i k E ( x k ) 0 n 1 β n 0 γ i k n 1 β E ( x k ) 0 E ( x k ) 0 E ( L ) k t k E ̂ ( L ) n 1 β E ( x k ) 0 n 1 β E ( L ) n 2 γ t k E ̂ ( L ) ]
[ X k r + X k t X k r X k t + ] = [ 1 n 1 β 0 0 ] X k + [ 0 0 1 n 1 β ] X k + ,
[ β γ i k β E ( x k ) 0 n 1 n 0 k i k n 1 E ( x k ) 0 β E ( x k ) 0 β E ( L ) γ t k E ̂ ( L ) n 1 E ( x k ) 0 n 1 E ( L ) n 2 k t k E ̂ ( L ) ] [ Y k r + Y k t Y k r Y k t + ] = [ β n 1 0 0 ] Y k + [ 0 0 β n 1 ] Y k + .
[ δ j k δ n s + ( 1 δ j k ) G k s j n b j n o , I + b j n o , I R u k s u j n ] [ b k s ] = τ = ± ϵ j τ τ n exp ( τ i n θ ) α I τ b j n o , I ,
[ δ j k δ n s + ( 1 δ j k ) G k s j n a j n o , I I + a j n o , I I R v k s v j n ] [ a k s ] = τ = ± ϵ j τ τ n exp ( τ i n θ ) α I I τ a j n o , I I .
( T u k + ( γ ) T v k + ( γ ) ) = exp [ i k ( β ̂ + x k + η ̂ + y k ) ] β ̂ + τ = ± s = ( τ ) s ( β ̂ + + τ i η ̂ + ) s ( T ̂ u τ u + b k s T ̂ v τ v + a k s ) ,
( T u k ( γ ) T v k ( γ ) ) = exp [ i k ( β ̃ x k + η ̃ y k ) ] β ̃ τ = ± s = ( τ ) s ( β ̃ + τ i η ̃ ) s ( T ̃ u τ u b k s T ̃ v τ v a k s ) ,
[ A ] = [ h k η h k i η β γ i l h k η E ( x k ) 0 β E ( x k ) 0 k i 2 k i 0 0 k E ( x k ) 0 0 0 n 1 β n 0 γ i n 1 h k η n 0 h k i η n 1 β E ( x k ) 0 n 1 h k η E ( x k ) 0 0 0 n 1 k n 0 i 2 k i 0 0 n 1 k E ( x k ) 0 h k η E ( x k ) 0 β E ( x k ) 0 h k η E ( L ) h k t η E ̂ ( L ) β E ( L ) γ t E ̂ ( L ) k E ( x k ) 0 0 0 k E ( L ) t 2 k t E ̂ ( L ) 0 0 n 1 β E ( x k ) 0 n 1 h k η E ( x k ) 0 n 1 β E ( L ) n 2 γ t E ̂ ( L ) n 1 h k η E ( L ) n 2 h k t η E ̂ ( L ) 0 0 n 1 k E ( x k ) 0 0 0 n 1 k E ( L ) n 2 t 2 k t E ̂ ( L ) ] ,
E ( ± x k ) = exp ( ± i 2 β x k ) , E ( L ) = exp ( i 2 β L ) , E ̂ ( L ) = exp [ i ( β γ t ) L ] ,
[ B 1 ] T = [ h η k k n 1 β 0 0 0 0 0 ] ,
[ B 2 ] T = [ β 0 n 1 h η k n 1 k 0 0 0 0 ] ,
[ B 3 ] T = [ 0 0 0 0 h η k k n 1 β 0 ] ,
[ B 4 ] T = [ 0 0 0 0 β 0 n 1 h η k n 1 k ] .

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