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Enhanced scattering and absorption due to the presence of a particle close to an interface

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Abstract

We study the influence of the presence of an interface on the scattering by a Rayleigh scatterer. The influence of an interface on the spontaneous emission has been known for many years. Here, we study the influence on the extinction cross-section and absorption cross-section. We provide a detailed analysis of interference and near-field effects. We show that the presence of a Rayleigh scatterer may enhance the specular reflection or specular transmission under certain conditions. Finally, we analyze the enhancement of absorption in the bulk in the presence of a small scatterer.

© 2012 Optical Society of America

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Figures (6)

Fig. 1
Fig. 1 Schematic set-up of scattering from a small particle – a dielectric sphere with refractive index ns = 2 and radius a = 50 nm modeled by a dipole moment P0.
Fig. 2
Fig. 2 Absorption efficiency depends on the particle height from the surface for transparent (line), lossy (dash), and metallic-like (dash-dot) substrates. Illumination in normal direction.
Fig. 3
Fig. 3 Absorption efficiency as a function of the parallel wave vector, k||, in medium 2 due to (a) particle on the interface (z0 = a), and (b) particle at z0 = λ/2 for transparent (line), lossy (dash), and metallic-like (dash-dot) substrates. The dotted vertical lines present the wave vectors in medium 1 and 2.
Fig. 4
Fig. 4 Absorption efficiency in the (k, ω) plane when the substrate is silver and the particle is placed on the surface. Illumination in normal incident direction corresponds to wavelength between 300 nm and 1200 nm. The white line denotes the light-line in medium 1.
Fig. 5
Fig. 5 Extinction efficiency of reflected (dash), transmitted (dash-dot) and total power (line, scale on the right) vs. the particle height from the surface for (a) transparent, (b) lossy, and (c) metallic-like substrates. Normal incident illumination.
Fig. 6
Fig. 6 Extinction efficiency of reflected (dash), transmitted (dash-dot) and total power (line, scale on the right) as a function of the angle of incidence for (a) transparent, (b) lossy, and (c) metallic-like substrates. Illuminating the particle on the interface (z0 = a), by an unpolarized light.

Equations (50)

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k | | = ( k x , k y )
k z j = k j 2 k | | 2
u ^ j + = ( k | | cos ϕ , k | | sin ϕ , k z j ) / k j
u ^ j = ( k | | cos ϕ , k | | sin ϕ , k z j ) / k j .
s ^ = κ ^ × z ^
p ^ j ± = s ^ × u ^ j ± .
e ^ i = { ( 0 , 1 , 0 ) s polarization ( k z 1 , 0 , k | | ) / k 1 p polarization
E d ( r 0 ) = [ ( 1 + r 12 s e 2 i k z 1 z 0 ) s ^ s ^ + ( p ^ 1 p ^ 1 + r 12 p e 2 i k z 1 z 0 p ^ 1 + p ^ 1 ) ] e ^ i
r i j s , p = Q i Q j Q i + Q j , t i j s = 2 Q i Q i + Q j , t i j p = n i n j 2 Q i Q i + Q j ,
p 0 = ε 1 α E d .
α eff = α 0 1 ζ 0 α 0
ζ 0 = i k 1 3 6 π .
α eff = α 0 [ 1 ζ α 0 ] 1 ,
ζ = ζ 0 I + ε 1 G r ( r 0 , r 0 )
G r ( r , r 0 ) = + d 2 k | | ( 2 π ) 2 G r ( k | | ; z z 0 ) e i k | | R
P ( r ) = p 0 δ ( r r 0 ) ,
E ( r ) = d 2 k | | ( 2 π ) 2 F ( k | | ; z ) e i k | | R .
F ( k | | ; z ) = G ( k | | ; z z ) P ( k | | ; z ) d z ,
F ( k | | ; z ) = G ( k | | ; z z 0 ) p 0 .
G = G 0 + G r ,
G 0 ( k | | ; z z 0 ) = i 2 ( ω c ) 2 e i k z 1 ( z z 0 ) k z 1 ( s ^ s ^ + p ^ 1 + p ^ 1 + ) G r ( k | | ; z z 0 ) = i 2 ( ω c ) 2 e i k z 1 ( z + z 0 ) k z 1 ( r 12 s s ^ s ^ + r 12 p p ^ 1 + p ^ 1 ) .
F s 1 ( k | | ; z > z 0 ) = i 2 ( ω c ) 2 e i k z 1 ( z z 0 ) k z 1 × [ ( 1 + r 12 s e 2 i k z 1 z 0 ) s ^ s ^ + ( p ^ 1 + p ^ 1 + + r 12 p e 2 i k z 1 z 0 p ^ 1 + p ^ 1 ) ] p 0 .
F s 2 ( k | | ; z < 0 ) = i 2 ( ω c ) 2 e i k z 1 z 0 k z 1 ( t 12 s s ^ s ^ + t 12 p p ^ 2 p ^ 1 ) e i k z 2 z p 0 .
W ext ( r ) + W ext ( t ) = W sca ( r ) + W sca ( t ) ,
Q ext = W ext W i ( π a 2 ) ; Q sca = W sca W i ( π a 2 ) ,
Q ext ( r ) = 2 π a 2 Re { E r F s 1 * } ; Q ext ( t ) = 2 π a 2 k z 2 k z 1 Re { E t F s 2 * } .
Q abs , 2 = Q sca ( t ) .
E ( r ) = E exc ( r ) + V G ( r , r ) ( ε s ε h ) E ( r ) d 3 r ,
E ( r ) = E exc ( r ) + V [ G 0 ( r , r ) + G r ( r , r ) ] ε h ( m 2 1 ) E ( r ) d 3 r
G 0 ( r , r 0 ) = PV [ k h 2 I + ] exp ( i k h ρ ) 4 π ε h ρ I 3 ε h δ ( r r 0 ) ,
G 0 ( r , r 0 ) I 3 ε h δ ( r r 0 ) + i k h 3 6 π ε h I .
E ( r 0 ) = E exc ( r 0 ) + [ i k 1 3 6 π ε h I + G r ( r 0 , r 0 ) ] V ε h ( m 2 1 ) E ( r 0 ) I ( m 2 1 ) 3 E ( r 0 )
E ( r 0 ) = 3 E exc ( r 0 ) m 2 + 2 { I 4 π a 3 m 2 1 m 2 + 2 [ i k h 3 6 π I ε h G r ( r 0 , r 0 ) ] } 1
p = V ( ε s ε h ) E ( r 0 ) = ε h α eff E exc ( r 0 )
α eff = α 0 { I [ i k h 3 6 π I + ε h G r ( r 0 , r 0 ) ] α 0 } 1 .
E 1 = E i + E r
E 2 = E t .
S 1 = 1 2 Re { E 1 × H 1 * } = 1 2 Re { ( E i + E r ) × ( H i * + H r * ) } S 2 = 1 2 Re { E 2 × H 2 * } = 1 2 Re { E t × H t * } .
A S d A = A 1 S 1 d A + A 2 S 2 d A = 0
1 2 Re { E i × H i * } + 1 2 Re { E r × H r * } + 1 2 Re { E t × H t * } = 0 .
E 1 = E i + E r + E s 1 E 2 = E t + E s 2
S 1 = 1 2 Re { ( E i + E r + E s 1 ) × ( H i * + H r * + H s 1 * ) } S 2 = 1 2 Re { ( E t + E s 2 ) × ( H t * + H s 2 * ) } .
A 1 1 2 Re { ( E i × H s 1 * + E s 1 × H i * ) + ( E r × H s 1 * + E s 1 × H r * ) + ( E s 1 × H s 1 * ) } d A + A 2 1 2 { ( E i × H s 2 * + E s 2 × H t * ) + ( E s 2 × H s 2 * ) } d A = 0 .
W ext ( i ) = A 1 1 2 Re { E i × H s 1 * + E s 1 × H i * } d A W ext ( r ) = A 1 1 2 Re { E r × H s 1 * + E s 1 × H r * } d A W sca ( r ) = A 1 1 2 Re { E s 1 × H s 1 * } d A W ext ( t ) = A 2 1 2 Re { E t × H s 2 * + E s 2 × H t * } d A W sca ( t ) = A 2 1 2 Re { E s 2 × H s 2 * } d A
W ext ( i ) + W ext ( r ) + W ext ( t ) = W sca ( r ) + W sca ( t ) .
E i = e ^ i e i k i r H i * = k i * ω μ × E i * E s = d 2 k | | ( 2 π ) 2 F s e i k r H s * = k s * ω μ × E s * = d 2 k | | ( 2 π ) 2 k s * ω μ × F s * e i k r .
W ext = A 1 2 Re { ( E i × H s * + E s × H i * ) } d A .
e i ( Δ k x x + Δ k y y ) d x d y = ( 2 π ) 2 δ ( Δ k x ) δ ( Δ k y ) ,
W ext ( i ) = 0 W ext ( r ) = 1 2 ω μ Re { E r × [ k r * × F s 1 * ( u ^ r ) ] + F s 1 ( u ^ r ) × [ k r * × E r * ] } ( + z ^ ) W ext ( t ) = 1 2 ω μ Re { E t × [ k t * × F s 2 * ( u ^ t ) ] + F s 2 ( u ^ t ) × [ k t * × E t * ] } ( z ^ )
W sca ( r ) = 1 2 ω μ Re d 2 k | | ( 2 π ) 2 { F s 1 ( k | | ) × [ k s * × F s 1 * ( k | | ) ] } ( + z ^ ) W sca ( t ) = 1 2 ω μ Re d 2 k | | ( 2 π ) 2 { F s 2 ( k | | ) × [ k s * × F s 2 * ( k | | ) ] } ( z ^ ) .
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