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Cooperative electromagnetic interactions between nanoparticles for solar energy harvesting

Mathieu Langlais, Jean-Paul Hugonin, Mondher Besbes, and Philippe Ben-Abdallah

Open Access Open Access

Abstract

The cooperative electromagnetic interactions between discrete resonators have been widely used to modify the optical properties of metamaterials. Here we propose a general approach for engineering these interactions both in the dipolar approximation and for any higher-order description. Finally we apply this strategy to design broadband absorbers in the visible range from simple n-ary arrays of metallic nanoparticles.

© 2014 Optical Society of America

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Figures (6)
Fig. 1
Fig. 1 Multiple light scattering interactions in a set of subwavelength plasmonic structures embeded in a transparent host material of refractive index nh. In the dipolar approximation each object is replaced by both a dipolar electric moment and a magnetic moment. The external field felt by each object decomposes into (1) the incident field, (2) the field radiated by the other objects and (3) the auto-induced field which comes from the interface after being emitted by the object itself. All dipoles radiate (4) in their surrounding.

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Fig. 2
Fig. 2 On the first column, absorption of simple and binary hexagonal lattices made with Ag and Au nanoparticles 30 nm radius immersed at h = 100nm from the surface in a transparent host medium of index nh = 1.5 with respect to the density in particles. On the second column, this absoption is compared with the absorption of single particles without multiple scattering interaction and, on the last column, with the results given by the effective medium theory with the same filling factor.

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Fig. 3
Fig. 3 Evolutionary algorithm to optimize a n-ary lattice. (a) A random population of periodic lattices (a physical view of an unit cell is plotted on the left) is randomly generated. (b) The best individus basd on the fitness function are selected as parents for the crossing over. (c) The next generation is created by linear crossing and completed by new individus (d) to keep the total population constant. (e) Mutations are aaplied on a few number of individus (typically 5%) in the current generation.

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Fig. 4
Fig. 4 Light absorption spectrum at normal incidence of a binary Au-Ag lattice (red dashed curve) optimized by GA by taking into account all multipolar interactions until the second order (quadrupoles) and of a multilayer based on Au-Ag films of thickness defined with the filling factor in nanoparticles (i.e. effective medium theory). Circles curve shows the result obtained by solving the Maxwell’s equations with a finite element method.

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Fig. 5
Fig. 5 Local losses at λ = 550nm in the particles of a gold nanoparticle lattice (a) with the same geometric parameters as in the optimized structure. Losses ℑ(ε)|ESG|2 in the single particle lattice are normalized by the maximum loss. In (b) we show the normalized difference ℑ(ε)|EDG|2 − ℑ(ε)|ESG|2 of losses inside Au particles in presence and without Ag particles (white regions). Analogously, in (c) and (d) the cooperative effect induces by the presence of Au particles on the dissipation in the Ag particles is shown at λ = 650nm.

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Fig. 6
Fig. 6 Impact of disorder on the light absorption spectrum at normal incidence in a binary Au-Ag lattice.The spatial location of particles is randomly perturbated by a displacement of 20nm. The red ciurve corresponds to the spectrum (in polarization TM at nomrla incidence) of the optimized structure and the dotted blue curve is the spectrum of a particular random realization (results in polarization TE, not plotted here are similar). The dashed area shows the maximum and minimm values of absorption spectrum of different random realizations. The histogram shows the discrepancy with the optimal fintess for different realizations of the structure. The red line on the histogram shows the mean error with respect to the number of realizations.The disorder is mimicked by using pseudoperiodic particle array with sufficiently large unit cells.

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Equations (31)

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A m ext = A m inc − i ω ∑ B = E , H Γ A B ( Δ 𝔾 m m A B p m ; B + ∑ n ≠ m 𝔾 m n A B p n ; B ) ,
A ext ( r ) = A inc ( r ) − i ω ∑ B = E , H Γ A B ∑ j 𝔾 A B ( r − r j ) p j ; B .
p m ; A = χ A α ↔ m ; A A n ext
p m ; A = χ A α ↔ m ; A [ A m inc − i ω ∑ n ∑ B = E , H 𝔾 reg A B ( r m , r n ) p n ; B ] .
𝔾 reg A B ( r , r ′ ) = { Γ A B 𝔾 A B ( r , r ′ )   i f   r ≠ r ′ Γ A B Δ 𝔾 A B ( r , r ′ )   i f   r = r ′ .
( p ˜ E p ˜ E ) = 𝒜 − 1 ℋ ( E ˜ H ˜ ) .
ℋ = diag ( ε 0 α ↔ 1 ; E , ... , ε 0 α ↔ n ; E , μ 0 α ↔ 1 ; H , ... , μ 0 α ↔ n ; H )
𝒜 = ( ( 1 + 𝕌 11 E E ) 𝕌 12 E E ⋯ 𝕌 1 n E E 𝕌 11 E H ⋯ 𝕌 1 n E H 𝕌 21 E E ⋱ ⋮ ⋮ ⋮ ⋮ ⋱ ( 1 + 𝕌 n n E E ) 𝕌 n − 1 , n E E 𝕌 n 1 E H ⋯ 𝕌 n n E H 𝕌 n 1 E E ⋯ 𝕌 n , n − 1 E E ( 1 + 𝕍 11 H H ) 𝕍 12 H H ⋯ 𝕍 1 n H H 𝕍 11 H E ⋯ 𝕍 1 n H E 𝕍 21 H H ⋱ ⋱ ⋮ ⋮ ⋮ ⋮ ⋱ ⋱ 𝕍 n − 1 , n H H 𝕍 n 1 H E ⋯ 𝕍 n n H E 𝕍 n 1 H H ⋯ 𝕍 n , n − 1 H H ( 1 + 𝕍 n n H H ) )
𝕌 l k E A = i ε 0 ε α ↔ l ; E ∑ j 𝔾 reg E A ( r 0 l , r j k ) e i k / / · ( r j k − r 0 l ) ,
𝕍 i k H A = i μ 0 ω α ↔ l ; H ∑ j 𝔾 reg H A ( r 0 l , r j k ) e i k / / . ( r j k − r 0 l ) .
Λ ≡ 𝒜 − 1 ℋ = ( Λ E E Λ E H Λ H E Λ H H )
𝒫 m ( ω ) = 1 2 ∑ A = E , H ∫ V m Re [ j m ; A * ( r , ω ) ⋅ A ( r , ω ) ] d r .
𝒫 m ( ω ) = − ω 2 ∑ A = E , H { Im [ p m ; A * ( ω ) ⋅ A m ext ( ω ) ] − ω 3 μ 0 2 p m ; A * Im [ 𝔾 0 A A ( r m , r m ) ] p m ; A } .
α E − 1 = k 0 3 n h 6 π ( C E − i ) ,
α H − 1 = k 0 3 n h 3 6 π ( C H − i ) ,
C E = ρ m 2 − ρ h 2 ρ m 2 ρ h 2 ( Cos ρ h + ρ h Sin ρ h ) ( Sin ρ m − ρ m Cos ρ m ) + ρ m Cos ρ h Cos ρ m + ρ h Sin ρ h Sin ρ m ρ h 2 − ρ m 2 ρ m 2 ρ h 2 ( Sin ρ h − ρ h Cos ρ h ) ( Sin ρ m − ρ m Cos ρ m ) − ρ m Sin ρ h Cos ρ m + ρ h Cos ρ h Sin ρ m ,
C H = − ρ h 2 Cos ρ h ( Sin ρ m − ρ m Cos ρ m ) + ρ m 2 Sin ρ m ( Cos ρ h + ρ h Sin ρ h ) ρ h 2 Sin ρ h ( Sin ρ m − ρ m Cos ρ m ) − ρ m 2 Sin ρ m ( Sin ρ h − ρ h Cos ρ h )
𝒫 m ( ω ) = − ω 2 { ε 0 n h ω 3 6 π c 3 Im [ E m ext * ( C E α ↔ E , m * α ↔ E , m ) E m ext ] + μ 0 n h 3 ω 3 6 π c 3 Im [ H m ext * ( C H α ↔ H , m * α ↔ H , m ) H m ext ] }
ψ p q ± = ( E p q ± H p q ± )
{ ∇ × E p q + = i ω μ H p q + + H p q S ∇ × H p q + = − i ω ε E p q + + E p q S .
{ E p q S = 0 H p q S = i n h 1 / 2 r . D n m .
{ E p q S = − i n h − 1 / 2 r . D n m H p q S = 0 .
D n m = i ( 2 k 0 n h ) n n ! 8 π ( − 1 ) m ( 2 n + 1 ) ( n − m ) ! n ( n + 1 ) ( n + m ) ! × { z ( ∂ ∂ x + i ∂ ∂ y ) − ( x + i y ) ∂ ∂ z } ( n + m ) ( − ∂ ∂ x + i ∂ ∂ y ) ( n ) δ .
A inc ( r ) = ∑ p q A inc p q ψ p q + ( r ) + ψ p q − ( r ) 2 .
A diff ( r ) = ∑ p q A diff p q ψ p q + ( r ) .
{ < ψ p q ± , ψ p ′ q ′ ± > = − 4 i δ p q , p ′ q ′ < ψ p q ± ψ p ′ q ′ ∓ > 0 .
< ψ p q 1 , ψ p q 2 > = ∮ ( E 1 × H 2 − E 2 × H 1 ) . n d S .
< ψ p ′ q + , A inc > = ∫ ψ p ′ q S ( r ) . A inc ( r ) d r ≡ I ψ p ′ q S [ A inc ] .
A inc p q = i 2 I ψ p ′ q S [ A inc ] .
A diff p q = i 2 ∑ p q T p q , p ′ q ′ I ψ p ′ q S [ A inc ] ,
A ( λ ) = ∑ m ∈ Cell 𝒫 m ( λ ) 𝒮 ϕ inc ( λ ) .
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