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Characterization of large array of plasmonic nanoparticles on layered substrate: dipole mode analysis integrated with complex image method

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Abstract

In this paper, an efficient analytical method for characterizing large array of plasmonic nanoparticles located over planarly layered substrate is introduced. The model is called dipole mode complex image (DMCI) method since the main idea lies in modeling a subwavelength spherical nanoparticle at its electric scattering resonance with an induced electric dipole and representing the electromagnetic (EM) fields of this electric dipole over the layered substrate in terms of finite complex images. The major advantages of the proposed method are its accuracy and rapid calculation in characterizing various kinds of large periodic and aperiodic arrays of nanoparticles on layered substrates. The computational time can be reduced significantly in compared to the traditional methods. The accuracy of the theoretical model is validated through comparison with numerical integration of Sommerfeld integrals. Moreover, the analytical results are compared well with those determined by full-wave finite difference time domain (FDTD) method. To demonstrate the capability of our technique, the performances of large arrays of nanoparticles on layered silicon substrates for efficient sunlight energy incoupling are studied.

©2011 Optical Society of America

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Figures (18)

Fig. 1
Fig. 1 Geometry of a concentric core-shell particle.
Fig. 2
Fig. 2 Array of plasmonic nanoparticles on a planarly multilayered medium.
Fig. 3
Fig. 3 Integration contours   C r and   C d (a) in complex   k ρ -plane and (b) in complex   k z -plane.
Fig. 4
Fig. 4 (a) Magnitude and (b) phase of the plarizability factor   ξ of a concentric nano core-shell dielectric-plasmonic particle of Fig. 1 vs. a/b for different wavelengths with SiO2 core and silver coating when b = 30 nm.
Fig. 5
Fig. 5 Magnitudes of   Γ 1 ( e ) ,   Γ 1 ( h ) , and   Γ 2 ( e ) of a concentric nano core-shell dielectric-plasmonic particle of Fig. 1 vs. a/b with SiO2 core and silver coating when b = 30 nm and   λ 0 = 500 nm.
Fig. 6
Fig. 6 A three layer substrate with SiO2 (   ε r = 2.2-j0.01) slab of thickness 0 .16   λ 0 , GaAs-AlGaAs (   ε r = 8-j0.1) slab of thickness 0 .3   λ 0 , and gold (   ε r = −2.28-j3.81) slab of thickness 0 .4   λ 0 (at operating wavelength of   λ 0 = 500 nm).
Fig. 7
Fig. 7 Total spectral reflection coefficients vs. modified contour parameter t for (a)   TE z and (b)   TM z modes from the entire substrate via the exact formulation and CI method.
Fig. 9
Fig. 9 Total spectral reflection coefficients vs. modified contour parameter t for (a)   TE z and (b)   TM z modes in the GaAs-AlGaAs layer via the exact formulation and CI method.
Fig. 8
Fig. 8 Total spectral transmission coefficients vs. modified contour parameter t for (a)   TE z and (b)   TM z modes in the GaAs-AlGaAs layer via the exact formulation and CI method.
Fig. 10
Fig. 10 Magnitude of total spatial reflection coefficients for (a)   TE z and (b)   TM z modes from the entire substrate via the exact integration and CI method.
Fig. 12
Fig. 12 Magnitude of total spatial reflection coefficients for (a)   TE z and (b)   TM z modes for the GaAs-AlGaAs layer via the exact integration and CI method.
Fig. 13
Fig. 13 (a) Configuration of three plasmonic nanoparticles. (b) Normalized   | E x | via the array (deposited on the layered substrate) using exact integration and CI method.
Fig. 11
Fig. 11 Magnitude of total spatial transmission coefficients for (a)   TE z and (b)   TM z modes for the GaAs-AlGaAs layer via the exact integration and CI method.
Fig. 14
Fig. 14 (a) Configuration of three unit electric dipoles (with different polarizations) above the layered substrate shown in Fig. 6, (b) DMCI result, and (c) FDTD result for the normalized   | E z | 2 (dB) at the middle of the GaAs-AlGaAs layer.
Fig. 16
Fig. 16 (a) Configuration of eight plasmonic nanoparticles and a z-directed unit electric dipole at the center above the layered substrate shown in Fig. 6, (b) DMCI result, and (c) FDTD result for the normalized   | E z | 2 (dB) at the middle of the GaAs-AlGaAs layer.
Fig. 15
Fig. 15 (a) Configuration of nine unit electric dipoles above the layered substrate shown in Fig. 6, (b) DMCI result, and (c) FDTD result for the normalized   | E z | 2 (dB) at the middle of the GaAs-AlGaAs layer.
Fig. 17
Fig. 17 (a) A periodic array of 100 plasmonic nanoparticles above the substrate of Fig. 6 and (b)   | E z | 2 (dB) at the middle of the GaAs-AlGaAs layer. It is normalized to the case that no nanoparticle exists.
Fig. 18
Fig. 18 (a) An aperiodic array of 100 plasmonic nanoparticles above the substrate of Fig. 6 and (b)   | E z | 2 (dB) at the middle of the GaAs-AlGaAs layer. It is normalized to the case that no nanoparticle exists.

Tables (3)

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Table 1 Complex Terms for the Total Reflection Coefficients of the Entire Substrate

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Table 3 Complex Terms for the Total Reflection Coefficients of the GaAs-AlGaAs Layer

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Table 2 Complex Terms for the Total Transmission Coefficients of the GaAs-AlGaAs Layer

Equations (45)

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E i = p = 0 q = p p [ a p q N p q ( 1 ) + b p q M p q ( 1 ) ] E s = p = 0 q = p p [ Γ p ( e ) a p q N p q ( 3 ) + Γ p ( h ) b p q M p q ( 3 ) ] .
Γ p ( e ) = U p ( e ) U p ( e ) j V p ( e )
U p ( e ) = [ j p ( k c a ) j p ( k s a ) y p ( k s a ) 0 j p d ( k c a ) / ε c j p d ( k s a ) / ε s y p d ( k s a ) / ε s 0 0 j p ( k s b ) y p ( k s b ) j p ( k m b ) 0 j p d ( k s b ) / ε s y p d ( k s b ) / ε s j p d ( k m b ) / ε m ] .
J 0 = ξ E t o t a l ξ = 6 π ω ε m k m 3 Γ 1 ( e ) .
E ( r ) = j ω μ ( I + k 2 ) . J 0 exp ( j k r ) 4 π r H ( r ) = × J 0 exp ( j k r ) 4 π r
J 0 = x J x + y J y + z J z I = x x + y y + z z .
J 0 p ( r p ) = ξ p { E e x t . ( r p ) + q = 1 , q p N E d i p . q ( r p ) + E r e f . e x t . ( r p ) + q = 1 N E r e f . q ( r p ) } .
exp ( j k r ) r = + d k ρ k ρ j 2 k z H 0 ( 2 ) ( k ρ ρ ) exp ( j k z | z z ' | )
H 0 ( 2 ) ( k ρ ρ ) = 1 π + d k x 1 k y exp ( j k x ( x x ' ) j k y | y y ' | ) .
E i x q ( r ) = 1 8 π ω ε i { J x q ( ω 2 μ i ε i + 2 x 2 ) + J y q 2 x y + J z q 2 x z } × + d k ρ k ρ k i z H 0 ( 2 ) ( k ρ ρ q ) T i T M [ exp ( j k i z | z z q | ) R i T M exp ( j k i z ( z z q + 2 d i ) ) ]
E i y q ( r ) = 1 8 π ω ε i { J x q 2 y x + J y q ( ω 2 μ i ε i + 2 y 2 ) + J z q 2 y z } × + d k ρ k ρ k i z H 0 ( 2 ) ( k ρ ρ q ) T i T E [ exp ( j k i z | z z q | ) + R i T E exp ( j k i z ( z z q + 2 d i ) ) ]
E i z q ( r ) = 1 8 π ω ε i { J x q 2 z x + J y q 2 z y + J z q ( ω 2 μ i ε i + 2 z 2 ) } × + d k ρ k ρ k i z H 0 ( 2 ) ( k ρ ρ q ) T i T M [ exp ( j k i z | z z q | ) + R i T M exp ( j k i z ( z z q + 2 d i ) ) ]
H i x q ( r ) = j 8 π { J y q z J z q y } × + d k ρ k ρ k i z H 0 ( 2 ) ( k ρ ρ q ) T i T E [ exp ( j k i z | z z q | ) R i T E exp ( j k i z ( z z q + 2 d i ) ) ]
H i y q ( r ) = j 8 π { J x q z + J z q x } × + d k ρ k ρ k i z H 0 ( 2 ) ( k ρ ρ q ) T i T M [ exp ( j k i z | z z q | ) + R i T M exp ( j k i z ( z z q + 2 d i ) ) ]
H i z q ( r ) = j 8 π { J x q y J y q x } × + d k ρ k ρ k i z H 0 ( 2 ) ( k ρ ρ q ) T i T E [ exp ( j k i z | z z q | ) + R i T E exp ( j k i z ( z z q + 2 d i ) ) ]
T i T M = T 1 T M exp ( j k 1 z d 1 ) exp ( j k i z d i 1 ) × l = 1 i 1 { exp ( j k l z ( d l d l 1 ) ) t l , l + 1 T M 1 + r l , l + 1 T M R l + 1 T M exp ( j 2 k l + 1 z ( d l + 1 d l ) ) }
R i T M = r i , i + 1 T M + R i + 1 T M exp ( j 2 k i + 1 z ( d i + 1 d i ) ) 1 + r i , i + 1 T M R i + 1 T M exp ( j 2 k i + 1 z ( d i + 1 d i ) )
t i , i + 1 T M = 2 ε i + 1 k i z ε i + 1 k i z + ε i k i + 1 z r i , i + 1 T M = ε i + 1 k i z ε i k i + 1 z ε i + 1 k i z + ε i k i + 1 z .
t i , i + 1 T E = 2 μ i + 1 k i z μ i + 1 k i z + μ i k i + 1 z r i , i + 1 T E = μ i + 1 k i z μ i k i + 1 z μ i + 1 k i z + μ i k i + 1 z .
F ( r ) = + d k ρ f ( k z ) k ρ j 2 k z H 0 ( 2 ) ( k ρ ρ ) exp ( j k z | z z ' | ) ,
f ( k z ) = n a n exp ( k z b n ) ,
F ( r ) = + d k ρ n a n exp ( k z b n ) k ρ j 2 k z H 0 ( 2 ) ( k ρ ρ ) exp ( j k z ( z z ' ) ) = n a n + d k ρ k ρ j 2 k z H 0 ( 2 ) ( k ρ ρ ) exp ( j k z ( z z ' j b n ) ) = n a n exp ( j k R n ) R n
R n = ( x x ' ) 2 + ( y y ' ) 2 + ( z z ' j b n ) 2 .
T i = f ( k 1 z , k 2 z , , k i z ) R i = g ( k i z , k i + 1 z , , k N z ) .
k j z 2 = k j 2 k ρ 2 ,
k j z 2 = k j 2 k i 2 + k i z 2 .
T i T M ( k i z ) = l α i l exp ( k i z β i l ) T i T E ( k i z ) = n α i n ' exp ( k i z β i n ' )
T i T M R i T M ( k i z ) = m γ i m exp ( k i z δ i m ) T i T E R i T E ( k i z ) = o γ i o ' exp ( k i z δ i o ' )
E i x q ( r ) = j 4 π ω ε i { J x q ( ω 2 μ i ε i + 2 x 2 ) + J y q 2 x y + J z q 2 x z } × [ l α i l exp ( j k i P i l q ) P i l q m γ i m exp ( j k i Q i m q ) Q i m q ]
E i y q ( r ) = j 4 π ω ε i { J x q 2 y x + J y q ( ω 2 μ i ε i + 2 y 2 ) + J z q 2 y z } × [ n α i n ' exp ( j k i P i n q ' ) P i n q ' + o γ i o ' exp ( j k i Q i o q ' ) Q i o q ' ]
E i z q ( r ) = j 4 π ω ε i { J x q 2 z x + J y q 2 z y + J z q ( ω 2 μ i ε i + 2 z 2 ) } × [ l α i l exp ( j k i P i l q ) P i l q + m γ i m exp ( j k i Q i m q ) Q i m q ]
H i x q ( r ) = 1 4 π { J y q z + J z q y } [ n α i n ' exp ( j k i P i n q ' ) P i n q ' o γ i o ' exp ( j k i Q i o q ' ) Q i o q ' ]
H i y q ( r ) = 1 4 π { J x q z J z q x } [ l α i l exp ( j k i P i l q ) P i l q + m γ i m exp ( j k i Q i m q ) Q i m q ]
H i z q ( r ) = 1 4 π { J x q y + J y q x } [ n α i n ' exp ( j k i P i n q ' ) P i n q ' + o γ i o ' exp ( j k i Q i o q ' ) Q i o q ' ]
P i l q = ρ q 2 + ( z z q j β i l ) 2 P i n q ' = ρ q 2 + ( z z q j β i n ' ) 2
Q i m q = ρ q 2 + ( z z q + 2 d i + j δ i m ) 2 Q i o q ' = ρ q 2 + ( z z q + 2 d i + j δ i o ' ) 2 .
k i z = k i [ j t + ( 1 t T ) ] 0 t T .
F ( t ) n = 1 N A n exp ( B n t )
F ( t m ) = F ( t 1 ) , F ( t 2 ) , , F ( t 2 N )
t m = T 2 N 1 ( m 1 ) m = 1 , 2 , , 2 N
[ F ( t N ) F ( t N 1 ) F ( t N + 1 ) F ( t N ) F ( t 1 ) F ( t 2 ) F ( t 2 N 1 ) F ( t 2 N 2 ) F ( t N ) ] [ C N C N 1 C 1 ] = [ F ( t N + 1 ) F ( t N + 2 ) F ( t 2 N ) ] ,
ρ N + C N ρ N 1 + C N 1 ρ N 2 + + C 1 = 0.
[ ρ 1 t 1 ρ 2 t 1 ρ 1 t 2 ρ 2 t 2 ρ N t 1 ρ N t 2 ρ 1 t N ρ 2 t N ρ N t N ] [ A 1 A 2 A N ] = [ F ( t 1 ) F ( t 2 ) F ( t N ) ] .
F ( k z ) n = 1 N α n exp ( β n k z )
α n = A n exp ( T B n 1 + j T ) β n = T B n k ( 1 + j T ) .
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