Abstract

A classical model for Raman and fluorescent scattering by molecules embedded in particles is extended to the case where the particle consists of concentric spheres made of different dielectric materials.

© 1976 Optical Society of America

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References

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  1. H. Chew, M. Kerker, and P. J. McNulty, Phys. Rev. A (to be published).
  2. J. Gelbwachs and M. Birnbaum, Appl. Opt. 12, 2442 (1973).
    [Crossref]
  3. G. J. Rosasco, E. S. Etz, and W. A. Cassatt, Appl. Spectrosc. 29, 396 (1975).
    [Crossref]
  4. X. Yatangas and B. D. Clarkson, J. Histochem. Cytochem. 22, 651 (1974).
    [Crossref]
  5. This applies also to the expansion coefficients aEd(l,m) and aMd(l,m), which involve spherical Bessel functions evaluated at r′.

1975 (1)

1974 (1)

X. Yatangas and B. D. Clarkson, J. Histochem. Cytochem. 22, 651 (1974).
[Crossref]

1973 (1)

Birnbaum, M.

Cassatt, W. A.

Chew, H.

H. Chew, M. Kerker, and P. J. McNulty, Phys. Rev. A (to be published).

Clarkson, B. D.

X. Yatangas and B. D. Clarkson, J. Histochem. Cytochem. 22, 651 (1974).
[Crossref]

Etz, E. S.

Gelbwachs, J.

Kerker, M.

H. Chew, M. Kerker, and P. J. McNulty, Phys. Rev. A (to be published).

McNulty, P. J.

H. Chew, M. Kerker, and P. J. McNulty, Phys. Rev. A (to be published).

Rosasco, G. J.

Yatangas, X.

X. Yatangas and B. D. Clarkson, J. Histochem. Cytochem. 22, 651 (1974).
[Crossref]

Appl. Opt. (1)

Appl. Spectrosc. (1)

J. Histochem. Cytochem. (1)

X. Yatangas and B. D. Clarkson, J. Histochem. Cytochem. 22, 651 (1974).
[Crossref]

Other (2)

This applies also to the expansion coefficients aEd(l,m) and aMd(l,m), which involve spherical Bessel functions evaluated at r′.

H. Chew, M. Kerker, and P. J. McNulty, Phys. Rev. A (to be published).

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Figures (1)

FIG. 1
FIG. 1

Model for N concentric spheres.

Equations (43)

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E d ( r ) = l , m [ ( i c n σ 2 ω ) a E d ( l , m ) X [ h l ( 1 ) ( k σ r ) Y l l m ] + a M d ( l , m ) h l ( 1 ) ( k σ r ) Y l l m ] ,
B d ( r ) = l , m [ a E d ( l , m ) h l ( 1 ) ( k σ r ) Y l l m - ( i c / ω ) a M d ( l , m ) X [ h l ( 1 ) ( k σ r ) Y l l m ] ]
α E d ( l , m ) = a E d ( l , m ) ,     with j l and j l + 1 replaced by h l ( 1 ) , h l + 1 ( 1 ) ;
α M d ( l , m ) = a M d ( l , m ) ,     with j l replaced by h l ( 1 ) .
E ρ ( r ) + δ ρ σ E d ( r )
B ρ ( r ) + δ ρ σ B d ( r ) ,
E ρ ( r ) = l , m [ ( i c n ρ 2 ω ) β E ρ ( l , m ) X [ g l ρ ( k ρ r ) Y l l m ] + β m ρ ( l , m ) g ¯ l ρ ( k ρ r ) Y l l m ] ,
B ρ ( r ) = l , m [ β E ρ ( l , m ) g l ( k ρ r ) Y l l m - ( i c ω ) β M ρ ( l , m ) X [ g ¯ l ( k ρ r ) Y l l m ] ] .
g l 1 ( k 1 r ) = j l ( k 1 r ) = g ¯ l 1 ( k 1 r ) ;
g l ( k ρ r ) = j l ( k ρ r ) + γ ρ n l ( k ρ r ) , g ¯ l ρ ( k ρ r ) = j l ( k ρ r ) + γ ¯ ρ n l ( k ρ r )     for 1 < ρ N
g l N + 1 ( k N + 1 r ) = h l ( 1 ) ( k r ) = g ¯ l N + 1 ( k n + 1 r ) .
β E 1 ( l , m ) , β E 2 ( l , m ) , γ 2 β E 2 ( l , m ) β E N ( l , m ) , γ N β E N ( l , m ) , β E N + 1 ( l , m ) β E ( l , m )
β M 1 ( l , m ) , β M 2 ( l , m ) , γ ¯ 2 β M 2 ( l , m ) β M N ( l , m ) , γ ¯ N β M N ( l , m ) , β M N + 1 ( l , m ) β M ( l , m ) .
n ρ + 1 2 { β E ρ ( l , m ) [ k ρ a ρ g l ρ ( k ρ a ρ ) ] + δ ρ σ [ k ρ a ρ j l ( k ρ a ρ ) ] a E d ( l , m ) } = n ρ 2 { β E ρ + 1 ( l , m ) [ k ρ + 1 a ρ g l ρ + 1 ( k ρ + 1 a ρ ) ] + δ ρ + 1 , σ [ k ρ + 1 a ρ j l ( k ρ + 1 a ρ ) ] a M d ( l , m ) } ,
β M ρ ( l , m ) g ¯ l ρ ( k ρ a ρ ) + δ ρ σ a M d ( l , m ) j l ( k ρ a ρ ) = β M ρ + 1 ( l , m ) g ¯ l ρ + 1 ( k ρ + 1 a ρ ) + δ ρ + 1 , σ a M d ( l , m ) j l ( k ρ + 1 a ρ ) ,
μ ρ + 1 [ β E ρ ( l , m ) g l ρ ( k ρ a ρ ) + δ ρ σ a E d ( l , m ) j l ( k ρ a ρ ) ] = μ ρ [ β E ρ + 1 ( l , m ) g l ρ + 1 ( k ρ + 1 a ρ ) + δ ρ + 1 , σ a E d ( l , m ) j l ( k ρ + 1 a ρ ) ] ,
μ ρ + 1 { β M ρ ( l , m ) [ k ρ a ρ g ¯ l ρ ( k ρ a ρ ) ] + δ ρ σ a M d ( l , m ) j l ( k ρ a ρ ) } = μ ρ { β M ρ + 1 ( l , m ) [ k ρ + 1 a ρ g ¯ l ρ ( k ρ + 1 a ρ ) ] + δ ρ + 1 , σ a M d ( l , m ) [ k ρ + 1 a ρ j l ( k ρ + 1 a ρ ) ] } .
E ( r ) = l , m [ ( i c n 2 ω ) β E ( l , m ) X [ h l ( 1 ) ( k r ) Y l l m ] + β M ( l , m ) h l ( 1 ) ( k r ) Y l l m ] ,
B ( r ) = l , m [ β E ( l , m ) h l ( 1 ) ( k r ) Y l l m - ( i c ω ) β M ( l , m ) X [ h l ( 1 ) ( k r ) Y l l m ] ] .
β E ( l , m ) = i 2 a E d ( l , m ) / μ 1 ( k 1 a ) ( k 2 b ) D E ( 2 ) ,
D E ( 2 ) = { 1 j l ( k 1 a ) [ k 2 a j l ( k 2 a ) ] - 2 j l ( k 2 a ) [ k 1 a j l ( k 1 a ) ] } × { h l ( 1 ) ( k b ) [ k 2 b n l ( k 2 b ) ] - 2 n l ( k 2 b ) [ k b h l ( 1 ) ( k b ) ] } + { 1 j l ( k 1 a ) [ k 2 a n l ( k 2 a ) ] - 2 n l ( k 2 a ) [ k 1 a j l ( k 1 a ) ] } × { 2 j l ( k 2 b ) [ k b h l ( 1 ) ( k b ) ] - h l ( 1 ) ( k b ) [ k 2 b j l ( k 2 b ) ] }
β M ( l , m ) = i μ 2 a M d ( l , m ) / ( k 1 a ) ( k 2 b ) D M ( 2 ) ,
D M ( 2 ) = { μ 2 j l ( k 2 a ) [ k 1 a j l ( k 1 a ) ] - μ 1 j l ( k 1 a ) [ k 2 a j l ( k 2 a ) ] } × { μ 2 n l ( k 2 b ) [ k b h l ( 1 ) ( k b ) ] - h l ( 1 ) ( k b ) [ k 2 b n l ( k 2 b ) ] } + { μ 2 n l ( k 2 a ) [ k 1 a j l ( k 1 a ) ] - μ 1 j l ( k 1 a ) [ ( k 2 a ) ] } × { h l ( 1 ) ( k b ) [ k 2 b j l ( k 2 b ) ] - μ 2 j l ( k 2 b ) [ k b h l ( 1 ) ( k b ) ] } .
β E ( l , m ) = N E ( 2 ) a E ( l , m ) / μ 2 k 2 b D E ( 2 ) ,
β M ( l , m ) = N M ( 2 ) a M d ( l , m ) / k 2 b D M ( 2 ) ,
N E ( 2 ) = 2 [ k 1 a j l ( k 1 a ) ] [ ξ e j l ( k 1 a ) - h l ( 1 ) ( k 1 a ) ] - 1 j l ( k 1 a ) { ξ e [ k 2 a j l ( k 2 a ) ] - [ k 2 a h l ( 1 ) ( k 2 a ) ] } , N M ( 2 ) = μ 2 [ k 1 a j l ( k 1 a ) ] [ ξ m j l ( k 2 a ) - h l ( 1 ) ( k 2 a ) ] - μ 1 j l ( k 1 a ) { ξ m [ k 2 a j l ( k 2 a ) ] - [ k 2 a h l ( 1 ) ( k 2 a ) ] } ,
ξ e = α E d ( l , m ) / a E d ( l , m ) , ξ m = α M d ( l , m ) / a M d ( l , m ) = h l ( 1 ) ( k 2 r ) / j l ( k 2 r )
β E ( l , m ) = N E ( 3 ) a E d ( l , m ) / Δ E ( 3 ) , β M ( l , m ) = N M ( 3 ) a M d ( l , m ) / Δ M ( 3 ) ,
Δ E ( 3 ) = | n 2 2 ψ l ( k 1 a 1 ) - n 1 2 ψ l ( k 2 a 1 ) - n 1 2 ν l ( k 2 a 1 ) 0 0 0 0 n 3 2 ψ l ( k 2 a 2 ) n 3 2 ν l ( k 2 a 2 ) - n 2 2 ψ l ( k 3 a 2 ) - n 2 2 ν l ( k 3 a 2 ) 0 0 0 0 ψ l ( k 3 a 3 ) ν l ( k 3 a 3 ) - n 3 2 ζ l ( 1 ) ( k a 3 ) μ 2 j l ( k 1 a 1 ) - μ 1 ψ l ( k 2 a 1 ) - μ 1 ν l ( k 2 a 1 ) 0 0 0 0 μ 3 j l ( k 2 a 2 ) μ 3 n l ( k 2 a 2 ) - μ 2 j l ( k 3 a 2 ) - μ 2 n l ( k 3 a 2 ) 0 0 0 0 j l ( k 3 a 3 ) n l ( k 3 a 3 ) - μ 3 h l ( 1 ) ( k a 3 ) | ,
ψ l ( x ) x j l ( x ) , ν l ( x ) x n l ( x ) , ζ l ( 1 ) ( x ) ψ l ( x ) + i ν l ( x ) , ψ l ( x ) ( d / d x ) ψ l ( x ) , etc .
( - n 2 2 ζ l ( 1 ) ( k 1 a 1 ) 0 0 - μ 2 h l ( 1 ) ( k 1 a 1 ) 0 0 ) ,
( n 1 2 ξ e ψ l ( k 2 a 1 ) - n 2 3 ζ l ( 1 ) ( k 2 a 2 ) 0 μ 1 ξ e j l ( k 2 a 1 ) - μ 3 h l ( 1 ) ( k 2 a 2 ) 0 ) ,
( 0 n 2 2 ξ e ψ l ( k 3 a 2 ) - n 2 ζ l ( 1 ) ( k 3 a 3 ) 0 μ 2 ξ e j l ( k 3 a 2 ) - μ h l ( 1 ) ( k 3 a 3 ) ) .
n ρ 2 μ ρ ,             μ ρ 1 ,             ξ e ξ m .
β E ( l , m ) = N E a E d ( l , m ) / Δ E
β M ( l , m ) = N M a M d ( l , m ) / Δ M ,
Δ E = | n 2 2 ψ l ( k 1 a 1 ) - n 1 2 ψ l ( k 2 a 1 ) - n 1 2 ν l ( k 2 a 1 ) 0 0 0 0 n 3 2 ψ l ( k 2 a 2 ) n 3 2 ν l ( k 2 a 2 ) - n 2 2 ψ l ( k 3 a 2 ) - n 2 2 ν l ( k 3 a 2 ) 0 0 0 0 n 4 2 ψ l ( k 3 a 3 ) n 4 2 ν l ( k 3 a 3 ) - n 3 2 ψ l ( k 4 a 3 ) - n 3 2 ν l ( k 4 a 3 ) - - - - - - - 0 0 0 0 n 2 ψ l ( k N a N ) n 2 ν l ( k N a N ) - n N 2 ζ l ( 1 ) ( k a N ) μ 2 l l ( k 1 a 1 ) - μ 1 j l ( k 2 a 1 ) - μ 1 n l ( k 2 a 1 ) 0 0 0 0 0 μ 3 j 1 ( k 2 a 2 ) μ 3 n l ( k 2 a 2 ) - μ 2 j l ( k 3 a 2 ) - μ 2 n l ( k 3 a 2 ) 0 0 0 0 0 μ 3 j l ( k 2 a 2 ) μ 3 n l ( k 2 a 2 ) - μ 2 j l ( k 3 a 2 ) - μ 2 n l ( k 3 a 2 ) - - - - - - - 0 0 0 0 μ j l ( k N a N ) μ n l ( k N a N ) - μ N h l ( 1 ) ( k a N ) |
( - n 2 2 ζ l ( 1 ) ( k 2 a 1 ) 0 - μ 2 h l ( 1 ) ( k 1 a 1 ) 0 ) N th row .
( 0 n σ - 1 2 ξ e ψ l ( k σ a σ - 1 ) - n σ + 1 2 ζ l ( 1 ) ( k σ a σ ) 0 μ σ - 1 ξ e j l ( k σ a σ - 1 ) - μ σ + 1 h l ( 1 ) ( k σ a σ ) ) ( σ - 1 ) th row σ th row ( N + σ - 1 ) th row ( N + σ ) th row .
n ρ 2 μ ρ ,             μ ρ 1 ,             ξ e ξ m .
a E ( l , m ) = 2 π i k 1 3 [ l ( l + 1 ) ( 2 l + 1 ) ] - 1 / 2 j l ( k 1 r ) p · [ ( l + 1 ) ] ( 2 l - 1 ) - 1 / 2 j l - 1 ( k 1 r ) - + l ( 2 l + 3 ) - 1 / 2 j l + 1 ( k 1 r ) + ] , a M ( l , m ) = - 2 π i k 2 ( ω / c ) [ l ( l + 1 ) ] - 1 / 2 j l ( k 1 r ) p · M ,
x - = [ ( l + m ) ( l + m - 1 ) ] - 1 / 2 Y l - 1 , m - 1 * ( r ˆ ) - [ ( l - m ) ( l - m - 1 ) ] 1 / 2 Y l - 1 , m + 1 * ( r ˆ ) , y - = - i { [ ( l + m ) ( l + m - 1 ) ] 1 / 2 Y l - 1 , m - 1 * ( r ˆ ) + [ ( l - m ) ( l - m - 1 ) ] 1 / 2 Y l - 1 , m + 1 * ( r ˆ ) } , z - = - 2 [ ( l + m ) ( l - m ) ] 1 / 2 Y l - 1 , m * ( r ˆ ) , x + = [ ( l + m + 1 ) ( l + m + 2 ) ] 1 / 2 Y l + 1 , m + 1 * ( r ˆ ) - [ ( l - m + 1 ) ( l - m + 2 ) ] 1 / 2 Y l + 1 , m - 1 * ( r ˆ ) , y + = i { [ ( l + m + 1 ) ( l + m + 2 ) ] 1 / 2 Y l + 1 , m + 1 * ( r ˆ ) + [ ( l - m + 1 ) ( l - m + 2 ) ] 1 / 2 Y l + 1 , m - 1 * ( r ˆ ) } , z + = - 2 [ ( l + m + 1 ) ( l - m + 1 ) ] 1 / 2 Y l + 1 , m * ( r ˆ ) ;
M x = [ ( l - m ) ( l + m + 1 ) ] 1 / 2 Y l , m + 1 * ( r ˆ ) + [ ( l + m ) ( l - m + 1 ) ] 1 / 2 Y l , m - 1 * ( r ˆ ) , M y = i { [ ( l - m ) ( l + m + 1 ) ] 1 / 2 Y l , m + 1 * ( r ˆ ) - [ ( l + m ) ( l - m + 1 ) ] 1 / 2 Y l , m - 1 * ( r ˆ ) } , M z = 2 m Y l m * ( r ˆ ) .