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

1. Introduction

Engineering light-matter interactions is a longstanding problem in physics and is of prime importance for numerous technological applications such as the photo and thermophovoltaic energy conversion [1, 2], the optical manipulation of nanoobjects [3] or the quantum information treatment [4]. Light interaction with resonant structures embedded inside a material is a natural way to modify its optical properties. To date, a large number of resonant structures have been developped following such a strategy. Among these, metamaterials based on metallo-dielectric structures have been proposed to operate at frequencies ranging from the microwave domain [5] to the visible [6].

The design of artificially constructed magnetodielectric resonators which strongly interact cooperatively is a very recent and promising way to generate metamaterials that highlight innovative physical [7, 8] and transport [9, 10] properties. However, so far, only heuristic approaches have been followed to identify the convenient meta-structures which display target functionalities. In this paper, we present a general theory to describe the multiple scattering interactions mechanisms in discret networks of resonators embedded in a host material and we propose a general method to identify the appropriate inner structure of networks that highlight a targeted optical property either by considering the interacting objects as simple (electric and magnetic) dipoles or as multipoles of arbitrary order. To illustrate the strong potential of cooperative interactions to tailor the optical properties of materials we design a broadband light absorber made with simple binary lattices of metallic nanoparticles immersed in a transparent host material.

2. Scattering by nanoparticle networks in the dipolar approximation

To start, let us consider a set of objects dispersed inside a host material as depicted in Fig. 1. Suppose this system is higlighted by an external harmonic electromagnetic field of wavelength much larger than the typical size of objects. In this condition, we can associate to each object an electric (E) and magnetic (H) dipole moment pm;A(A = E, H) (the higher orders contributions are discussed in the next paragraph). The local electromagnetic field Aext(rm) at the dipoles location rm results from the superposition of external incident field, the field generated by the others dipoles and the auto-induced field which comes from the interactions with the interfaces. Therefore it takes the form [11, 12, 13]

Amext=AminciωB=E,HΓAB(Δ𝔾mmABpm;B+nm𝔾mnABpn;B),
where (ΓEEΓEHΓHEΓHH)=(iωμ0ω/cω/ciωε0)and 𝔾mnAB is the dyadic Green tensor in the host material which takes into account the presence of interfaces [14] and gives the field A at the position rm given a B-dipole located in rn. Δ𝔾AB defined as Δ𝔾AB𝔾AB𝔾0AB gives the contribution of interfaces only.

 

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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Here 𝔾0AB(rm,rn)=exp(ikrmn)4πrmn×{[(1+ikrmn1k2rmn2)1+33ikrmnk2rmn2k2rmn2r^mnr^mn]ifA=Bikrmn#krmn𝕃ifAB the free space Green tensor in the host material defined with the unit vector mnmn/rmn. rmn denotes here the vector linking the center of dipoles m and n, while rij = |rij|, k is the wavector, 1 the unit dyadic tensor and 𝕃=(0r^mn,zr^mn,yr^mn,z0r^mn,xr^mn,yr^mn,x0). Beside the dipoles location the auto-induced part of field does not exist anymore and it takes the simplified form

Aext(r)=Ainc(r)iωB=E,HΓABj𝔾AB(rrj)pj;B.

It immediately follows that, the dipolar moments associated to each object reads

pm;A=χAαm;AAnext
where χA represents either the vacuum permittivity ε0 or the vacuum permeability μ0 and α⃡i,A is the free polarizability tensor of mth object under the action of field A. By inserting the external contribution (1) of local field into relation (23) we get the following system which relates all dipole moments
pm;A=χAαm;A[AminciωnB=E,H𝔾regAB(rm,rn)pn;B].
Here, where we have introduced the regularized Green tensor
𝔾regAB(r,r)={ΓAB𝔾AB(r,r)ifrrΓABΔ𝔾AB(r,r)ifr=r.

In the particular case of n-ary periodic lattices made with n arbitrary dipoles of free electric and magnetic polarizability α⃡m;A=E,H distributed in a unit cell we have, according to the periodicity, the supplementary relations for the incident fields Ajβinc=Aβ˜expik//.rjβ)and for the dipolar moments pjβ;A = β;A exp(ik//.rjβ). Here rjβ is the position vector of the βth dipole inside the unit cell j of lattice and k// is the parallel component of wavector.

Accordingly, Eq. (4) can be solve with respect to the incident field to give

(p˜Ep˜E)=𝒜1(E˜H˜).
Here we have set p˜A=E,H=(p˜1,A,,p˜n,A)t and A˜=(A˜1,,A˜n)t and we have define the block matrixes
=diag(ε0α1;E,...,ε0αn;E,μ0α1;H,...,μ0αn;H)
and
𝒜=((1+𝕌11EE)𝕌12EE𝕌1nEE𝕌11EH𝕌1nEH𝕌21EE(1+𝕌nnEE)𝕌n1,nEE𝕌n1EH𝕌nnEH𝕌n1EE𝕌n,n1EE(1+𝕍11HH)𝕍12HH𝕍1nHH𝕍11HE𝕍1nHE𝕍21HH𝕍n1,nHH𝕍n1HE𝕍nnHE𝕍n1HH𝕍n,n1HH(1+𝕍nnHH))
with
𝕌lkEA=iε0εαl;Ej𝔾regEA(r0l,rjk)eik//·(rjkr0l),
𝕍ikHA=iμ0ωαl;Hj𝔾regHA(r0l,rjk)eik//.(rjkr0l).
These summations can calculated directly or using the Ewald’s method [15, 16] as in solid-state physics. Relation (6) defines the dressed polarizability tensor [17]
Λ𝒜1=(ΛEEΛEHΛHEΛHH)
of resonators system within the unit cell of lattice. It takes into account both the intrinsic properties of isolated objects and their interactions with the environnement [18] (particles and interfaces). The dipspersion relations of resonant modes inside the system of coupled resonators is then given by the eigenvalues of the dress polarizability tensor.

Beside the spectrum of nanoresonators network we can calculate the amount of energy which is dissipated by the electromagnetic field inside each resonator. According to the Pynting’s theorem [19] the power dissipated at a frequency ω inside the mth resonator is given by the rate of doing work by the electric and mgnetic fields inside the resonator volume Vm

𝒫m(ω)=12A=E,HVmRe[jm;A*(r,ω)A(r,ω)]dr.
Here A denotes either the local electric or magnetic field E and H while jE and jH are the corresponding local current density. In the dipolar approximation jm;A = −pm;Aδ(rri), expression (12) can be recasted into the discrete form
𝒫m(ω)=ω2A=E,H{Im[pm;A*(ω)Amext(ω)]ω3μ02pm;A*Im[𝔾0AA(rm,rm)]pm;A}.
By inverting (1) after having replaced the dipole moments by their expression with respect to Amext, we can express Amext in term of Ainc and explicitely calculate the power dissipated in each object under an external lighting.

For spherical particles of radius R the polarizability is straightforwardly derived from the Mie scattering theory [20, 21]. If those particles, of refractive index nm, are immersed inside a medium of index nh, we have α⃡A = αA1 with

αE1=k03nh6π(CEi),
αH1=k03nh36π(CHi),
Here k0 is the wavevector inside vacuum and
CE=ρm2ρh2ρm2ρh2(Cosρh+ρhSinρh)(SinρmρmCosρm)+ρmCosρhCosρm+ρhSinρhSinρmρh2ρm2ρm2ρh2(SinρhρhCosρh)(SinρmρmCosρm)ρmSinρhCosρm+ρhCosρhSinρm,
CH=ρh2Cosρh(SinρmρmCosρm)+ρm2Sinρm(Cosρh+ρhSinρh)ρh2Sinρh(SinρmρmCosρm)ρm2Sinρm(SinρhρhCosρh)
with ρh = k0nhR and ρm = k0nmR, nm being the refractive index of resonator. According to Eqs. (13), (16) and (17) it follows that the power dissipated in each particle can be expressed both in term of absorption cross-sections and of incident external field
𝒫m(ω)=ω2{ε0nhω36πc3Im[Emext*(CEαE,m*αE,m)Emext]+μ0nh3ω36πc3Im[Hmext*(CHαH,m*αH,m)Hmext]}

3. Generalization of scattering problem beyond the dipolar approximation

So far, we have only considered interactions between electric (resp. magnetic) dipoles. In this paragraph we describe how to take into account the multipolar interactions. The electromagntic field inside a medium of refractive index nh can be expressed in term of ingoing (−) and outgoing (+) vector spherical wave functions (which form a complete basis)

ψpq±=(Epq±Hpq±)
where we have adopted the usual convention for the multipolar index (m, n) which are replaced by a single index p = n(n + 1) + m and where q set the polarization state (i.e. q = 1 for TE waves and q = 2 for TM waves).

The outgoing wave functions ψpq+ are solutions of Maxwell’s equation (using the eiωt convention)

{×Epq+=iωμHpq++HpqS×Hpq+=iωεEpq++EpqS.
with the source term ψpqS=(EpqSHpqS) (i.e. the multipole pq) defined by
{EpqS=0HpqS=inh1/2r.Dnm.
for the magnetic contributions and by
{EpqS=inh1/2r.DnmHpqS=0.
for the electric ones where Dnm expresses in term of partial derivative of the Dirac distribution δ as
Dnm=i(2k0nh)nn!8π(1)m(2n+1)(nm)!n(n+1)(n+m)!×{z(x+iy)(x+iy)z}(n+m)(x+iy)(n)δ.

Let us consider an isolated particle of arbitrary shape immersed inside a host medium of index nh and highlighetd by an external electromagnetic field. By definition, this field can be decomposed on the complete basis of ψpq± as

Ainc(r)=pqAincpqψpq+(r)+ψpq(r)2.
As for the diffracted field, which is an outgoing field, it reads
Adiff(r)=pqAdiffpqψpq+(r).
To calculate the components Adiffpq of the diffracted field we first calculate, by reciprocity and using the Lorentz relations [19], the action of the sources ψpqS=(EpqSHpqS) (with p′ = (n, −m)) on the incident field. Then using the fact that ψpq± functions form a complete basis they satisfy the following orthogonality relations
{<ψpq±,ψpq±>=4iδpq,pq<ψpq±ψpq>0.
where the brackets < .,. > represents the scalar product in the Lorentz sens defined by
<ψpq1,ψpq2>=(E1×H2E2×H1).ndS.
Here, integration in taken over an oriented surface (with the vector n) surrounding the particle. It follows by applying the Lorentz relation with the field ψpq+ generated by a source ψpqS and the incident field Ainc that
<ψpq+,Ainc>=ψpqS(r).Ainc(r)drIψpqS[Ainc].
Note that Iψ[A] is the action of the distribution on the test function ψ [22]. Then using the orthogonality relations (26) and according to (24)
Aincpq=i2IψpqS[Ainc].
Then, using the matrix T which relates the vectors Ainc of components of incident field to the vector Adiff of diffracted field we have
Adiffpq=i2pqTpq,pqIψpqS[Ainc],
Now, interactions between distinct particles dispersed inside a multilayer can be studied using a generalized form of the translation matrix as introduced by Stout et al. [23] to express the field generated by a source inside a particle in term of components of incident field on another particle.

4. Broadband absorber design

Now the general theoretical framework needed to describe the cooperative electromagnetic interaction inside a network of optical resonators we discuss in this paragraph how to use it to design targeted optical properties. To start with this objective ans show the strong potential of cooperative interactions we first consider the simple geometric configurations as illustrated in Fig. 2, that is single and binary metallic [24] particle arrays dispersed in regular hexagonal lattices of side length d and compare their absorption spectra with that ones of isolated particles and of homogeneous metallic film. All lattices are immersed in a transparent material of refractive index nh = 1.5 and are maintained at a distance h = 100nm from the surface. The results plotted in Fig. 2 clearly show that the resonance peaks in single particle lattices are essentialy centered at the resonance frequency of free particles. On the other hand the absorption spectrum of nanoparticles lattices is much broder and does not simply consist in a superposition of single particle spectra. Moreover, we see that the cooperative interactions allow increasing the absorption even in diluted lattices where the filling factor f is smaller than 3%. Finally, the comparison of the overall absorption of nanoparticle lattices with that of simple metallic films with a thickness defined, using the effective medium theory, from the nanoparticle filling factors points out the prime importance of cooperative effects to magnify the absorption level. In binary lattices, new configurationnal resonances add up to the resonances of single lattices and naturally enlarge the absorption spectrum.

 

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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In light of these results we can introduce a rational design of cooperative electromagnetic interactions to optimze the optical properties of a composite structure made with a distribution of nanoparticles. For this purpose we present the inverse design of a broadband absorber in the visible range [6, 25, 26, 27] made with a n-ary array metallic spherical nanoparticles. A n-ary lattice is defined from a unit cell 𝒞 of a two dimensional paving with a certain thickness (see Fig. 3). In the unit cell of a lattice we consider a set of n vectors ri and n positive reals Ri that represent the location of particles center and the radius of particles, respectively. To avoid the particle interpenetration these vectors must satisfy to the supplemental constraint |rirj |> Ri + Rj.

 

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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To design the n-ary lattice in order to maximize its overalll absorption we have to explore the large and complex space of all possible configurations. To do that we employ a genetic algorithm (GA) [28] which is a stochastic global optimisation method that is based on natural selection rules in a similar way to the Darwin’s theory of evolution. Evolutionary optimization has been yet successfully applied in numerous fields of optics [29, 30].

Basically, a GA uses an initial population (Fig. 3) of typically few hundreds of structures also called individus which are randomly generated in size and position. For each individu we calculate the fitness parameter which is here the mean absorption (for a given polarization) A¯=(λmaxλmin)1minλmaxA(λ)dλ over the spectral range [λmin; λmax] where we want to increase the absorption. The monochromatic absorption A(λ) at a given wavelength is simply given by the the sum of power dissipated inside the particles of the unit cell normalized by the incident flux ϕinc(λ) on its surface 𝒮 that is

A(λ)=mCell𝒫m(λ)𝒮ϕinc(λ).
The GA consists in maximizing the fitness function of structures (i.e. Āmax). To do so, we select 90% of the highest fitness as future parents for the next generation of selecting process. Those parents are linear crossed and the new ’child’ generation is completed by new individual structures (randomly generated) to keep the same total number of lattices for any generation. To avoid the convergence toward local extrema, every m (typically 10) generations, we introduce also some mutations that is to say random perturbations with a probability of about 5% on the value of parameters we are optimizing. The results presented in Fig. 4 for superposed binary Au-Ag lattices (with the radius rAu = 77nm and rAg = 39nm, the separation distances from the surface hAu = 120nm and hAg = 242nm, the lattice constants dAu = dAg = 200nm and the off-centring ex = 56nm and ey = 10nm) exhibit a broad absorption band in the visible range. By taking into account the multipolar interactions until the second order (i.e. quadrupolar interactions) we see that the level of aborption becomes close to one. The comparaison of these results with full electromagnetic simulations based on the finite element method shows that the higher order multipole moments do not contribute significantly to the overall absorption. The absorption enhancement can be understood by examinating the electromagnetic cooperative effects inside the structure. These effects are highlighted in Fig. 5 at two different wavelengths by plotting the local losses inside the gold (resp. silver) nanoparticles within the optimized binary lattice in presence or without silver (resp. gold) nanoparticles. At λ = 550nm, that is to say, at the resonance of gold particles (which corresponds to the region where we observe in Fig. 4 an important bump in the absorption spectrum when it is calculated in the dipolar approximation) we see that the presence of silver nanoparticles enhance by 20% the losses inside the gold particles. Similarly, at λ = 650nm, the gold particles enhance by a factor of 60% the dissipation inside the silver particles. However, because of the weakness of intrinsic losses inside isolated Ag particles this cooperative effect is not sufficiently important to increase the overall absorption of the structure. At low wavelength we have checked (not plotted in Fig. 5) that the high absorption levels as shown in Fig. 4 results from cooperative effects between the gold particles themselves. The silver particles do not play any role in the exaltation mechanism.

 

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 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 (ε)|ESG|2 in the single particle lattice are normalized by the maximum loss. In (b) we show the normalized difference (ε)|EDG|2(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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Interestingly, the numerical simulations have shown also that the cooperative effects are not very sensitive to the presence of disorder. In Fig. 6 we show, by disturbing the optimal structure with a -random perturbation of particles locations by a maximum displacement of 20nm, that the discrepancy between the optimal structure and the perturbed ones, given by the mean square error ζ=[λminλmax(A(λ)Aopt(λ))2dλ]1/2, remains small. For some realization a broadening of spectrum can be observed around 650 nm. This effect can attributed to the presence of new modes supported which give rise to new channels for dissipating light energy within the structure. However, the detailed study of random structures goes far beyond the scope of the present work and it will be carried out in a future work.

 

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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5. Conclusion

In conclusion, we have proposed a general method for engineering the cooperative electromagnetic interactions in resonators networks both in the dipolar approximation and for arbitrary multipolar orders. Our results have demonstrated the strong potential of these interactions to tailor the optical properties spectrum. We believe that this approach opens the way to a rational design of metamaterials and it could find broad applications in various fields of applied physics, as for instance, in the domain of photovoltaic energy conversion for the conception of more efficient solar cells, in optical information treatement for the design of quantum information systems and, according to the reciprocity principle [31], in light extraction technologies to improve the performances of light emitting diodes.

Acknowledgments

J.-P. H. acknowledges discussions with J.J. Greffet. P. B.-A. gratefully acknowledges the support of Total news energies.

References and links

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6. K. Aydin, V. E. Ferry, R. M. Briggs, and H. A. Atwater, “Broadband polarization-independent resonant light absorption using ultrathin plasmonic super absorbers,” Nat. Commun. 2, 517 (2011). [CrossRef]   [PubMed]  

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References

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  1. H. A. Atwater and A. Polman, “Plasmonics for improved photovoltaic devices,” Nat. Mater. 9, 205–2013 (2010).
    [CrossRef] [PubMed]
  2. 10. P. Bermel, M. Ghebrebrhan, W. Chan, Y. X. Yeng, M. Araghchini, R. Hamam, C. H. Marton, K. F. Jensen, M. Soljacic, J. D. Joannopoulos, S. G. Johnson, and I. Celanovic, “Design and global optimization of high-efficiency thermophotovoltaic systems,” Opt. Express 18, A314–A334 (2010).
    [CrossRef] [PubMed]
  3. A. Ashkin, J. M. Dziedzic, J. E. Bjorkholm, and S. Chu, “Observation of a single-beam gradient force optical trap for dielectric particles,” Opt. Lett. 11, 288–290 (1986).
    [CrossRef] [PubMed]
  4. A. N. Poddubny, P. A. Belov, and Y. S. Kivshar, “Spontaneous radiation of a finite-size dipole in hyperbolic media,” Phys. Rev. A 84, 023807 (2011).
    [CrossRef]
  5. N. I. Landy, S. Sajuyigbe, J. J. Mock, D. R. Smith, and W. J. Padilla, “Perfect metamaterial absorber,” Phys. Rev. Lett. 100, 207402 (2008).
    [CrossRef] [PubMed]
  6. K. Aydin, V. E. Ferry, R. M. Briggs, and H. A. Atwater, “Broadband polarization-independent resonant light absorption using ultrathin plasmonic super absorbers,” Nat. Commun. 2, 517 (2011).
    [CrossRef] [PubMed]
  7. S. D. Jenkins and J. Ruostekoski, “Metamaterial transparency induced by cooperative electromagnetic interactions,” Phys. Rev. Lett. 111, 147401 (2013).
    [CrossRef] [PubMed]
  8. V. A. Fedotov, N. Papasimakis, E. Plum, A. Bitzer, M. Walther, P. Kuo, D. P. Tsai, and N. I. Zheludev, “Spectral collapse in ensembles of metamolecules,” Phys. Rev. Lett. 104, 223901 (2010).
    [CrossRef] [PubMed]
  9. P. Ben-Abdallah, R. Messina, S.-A. Biehs, M. Tschikin, K. Joulain, and C. Henkel, “Heat superdiffusion in plasmonic nanostructure networks,” Phys. Rev. Lett. 111, 174301 (2013).
    [CrossRef] [PubMed]
  10. R. Messina, M. Tschikin, S.-A. Biehs, and P. Ben-Abdallah, “Fluctuation-electrodynamic theory and dynamics of heat transfer in systems of multiple dipoles,” Phys. Rev. B 88, 104307 (2013).
    [CrossRef]
  11. E. M. Purcell and C. R. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys. J. 186, 705 (1973).
    [CrossRef]
  12. B. T. Draine and P. J. Flateau, “Discrete-dipole approximation for periodic targets: theory and tests,” J. Opt. Soc. Am. A. 25, 2693 (2008).
    [CrossRef]
  13. A. B. Evlyukhin, C. Reinhardt, A. Seidel, B. S. Luk’yanchuk, and B. N. Chichkov, “Optical response features of Si-nanoparticle arrays,” Phys. Rev. B 82, 045404 (2010).
    [CrossRef]
  14. M. S. Tomas, “Green function for multilayers: light scattering in planar cavities,” Phys. Rev. A 51, 2545 (1995).
    [CrossRef] [PubMed]
  15. P. P. Ewald, “Die berechnung optischer und elektrostatischer gitterpotentiale,” Annalen der Physik,  369, 253–287 (1921).
    [CrossRef]
  16. F. Capolino, D. Wilton, and W. Johnson, “Efficient computation of the 2-D Green’s function for 1-D periodic structures using the Ewald method,” IEEE Trans. on Antennas and Propaga. 53, 9 (2005).
    [CrossRef]
  17. E. Castanie, R. Vincent, R. Pierrat, and R. Carminati, “Absorption by an optical dipole antenna in a structured environment,” Int. J. Opt. 2012, 452047 (2012).
    [CrossRef]
  18. P. Ben-Abdallah, S.-A. Biehs, and K. Joulain, “Many-body radiative heat transfer theory,” Phys. Rev. Lett. 107, 114301 (2011).
    [CrossRef] [PubMed]
  19. J. D. Jackson, Classical Electrodynamics, 3rd ed. (John Wiley, 1999).
  20. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley Science, New York, 1998).
    [CrossRef]
  21. A. B. Evlyukhin, C. Reinhardt, U. Zywietz, and B. N. Chichkov, “Collective resonances in metal nanoparticle arrays with dipole-quadrupole interactions,” Phys. Rev. B 85, 245411, (2012).
    [CrossRef]
  22. L. Schwartz, Théorie des distributions, Hermann (1951).
  23. B. Stout, J.-C. Auger, and J. Lafait, “A transfer matrix approach to local field calculations in multiple scattering problems,” J. Mod. Opt. 49, 2129–2152 (2002).
    [CrossRef]
  24. E. D. Palik, Handbook of Optical Constants of Solids (Academic Press, New York, 1998).
  25. E. E. Narimanov and A. V. Kildishev, “Optical black hole: broadband omnidirectional ligh absorber,” Appl. Phys. Lett. 95, 041106 (2009).
    [CrossRef]
  26. A. Aubry, D. Y. Lei, A. I. Fernández-Domínguez, Y. Sonnefraud, S. A. Maier, and J. B. Pendry, “Plasmonic light-harvesting devices over the whole visible spectrum,” Nano. Lett. 10, 2574–2579 (2010).
    [CrossRef] [PubMed]
  27. N. P. Sergeant, O. Pincon, M. Agrawal, and P. Peumans, “Design of wide-angle solar selective absorbers using aperiodic metal-dielectric stacks,” Opt. Express 17, 22800–22812 (2009).
    [CrossRef]
  28. J. H. Holland, Adaptation in Natural and Artificial Systems (MIT Press/Bradford Books Edition, Cambridge, MA, 1992).
  29. T. Feichtner, O. Selig, M. Kiunke, and B. Hecht, “Evolutionary optimization of optical antennas,” Phys. Rev. Lett. 109, 127701 (2012).
    [CrossRef] [PubMed]
  30. J. Drevillon and P. Ben-Abdallah, “Ab initio design of coherent thermal sources,” J. Appl. Phys. 102, 114305 (2007).
    [CrossRef]
  31. L. Landau, E. Lifchitz, and L. Pitaevskii, Electromagnetics of Continuous Media(Pergamon, Oxford, 1984).

2013 (3)

S. D. Jenkins and J. Ruostekoski, “Metamaterial transparency induced by cooperative electromagnetic interactions,” Phys. Rev. Lett. 111, 147401 (2013).
[CrossRef] [PubMed]

P. Ben-Abdallah, R. Messina, S.-A. Biehs, M. Tschikin, K. Joulain, and C. Henkel, “Heat superdiffusion in plasmonic nanostructure networks,” Phys. Rev. Lett. 111, 174301 (2013).
[CrossRef] [PubMed]

R. Messina, M. Tschikin, S.-A. Biehs, and P. Ben-Abdallah, “Fluctuation-electrodynamic theory and dynamics of heat transfer in systems of multiple dipoles,” Phys. Rev. B 88, 104307 (2013).
[CrossRef]

2012 (3)

E. Castanie, R. Vincent, R. Pierrat, and R. Carminati, “Absorption by an optical dipole antenna in a structured environment,” Int. J. Opt. 2012, 452047 (2012).
[CrossRef]

A. B. Evlyukhin, C. Reinhardt, U. Zywietz, and B. N. Chichkov, “Collective resonances in metal nanoparticle arrays with dipole-quadrupole interactions,” Phys. Rev. B 85, 245411, (2012).
[CrossRef]

T. Feichtner, O. Selig, M. Kiunke, and B. Hecht, “Evolutionary optimization of optical antennas,” Phys. Rev. Lett. 109, 127701 (2012).
[CrossRef] [PubMed]

2011 (3)

P. Ben-Abdallah, S.-A. Biehs, and K. Joulain, “Many-body radiative heat transfer theory,” Phys. Rev. Lett. 107, 114301 (2011).
[CrossRef] [PubMed]

K. Aydin, V. E. Ferry, R. M. Briggs, and H. A. Atwater, “Broadband polarization-independent resonant light absorption using ultrathin plasmonic super absorbers,” Nat. Commun. 2, 517 (2011).
[CrossRef] [PubMed]

A. N. Poddubny, P. A. Belov, and Y. S. Kivshar, “Spontaneous radiation of a finite-size dipole in hyperbolic media,” Phys. Rev. A 84, 023807 (2011).
[CrossRef]

2010 (5)

V. A. Fedotov, N. Papasimakis, E. Plum, A. Bitzer, M. Walther, P. Kuo, D. P. Tsai, and N. I. Zheludev, “Spectral collapse in ensembles of metamolecules,” Phys. Rev. Lett. 104, 223901 (2010).
[CrossRef] [PubMed]

A. B. Evlyukhin, C. Reinhardt, A. Seidel, B. S. Luk’yanchuk, and B. N. Chichkov, “Optical response features of Si-nanoparticle arrays,” Phys. Rev. B 82, 045404 (2010).
[CrossRef]

A. Aubry, D. Y. Lei, A. I. Fernández-Domínguez, Y. Sonnefraud, S. A. Maier, and J. B. Pendry, “Plasmonic light-harvesting devices over the whole visible spectrum,” Nano. Lett. 10, 2574–2579 (2010).
[CrossRef] [PubMed]

H. A. Atwater and A. Polman, “Plasmonics for improved photovoltaic devices,” Nat. Mater. 9, 205–2013 (2010).
[CrossRef] [PubMed]

10. P. Bermel, M. Ghebrebrhan, W. Chan, Y. X. Yeng, M. Araghchini, R. Hamam, C. H. Marton, K. F. Jensen, M. Soljacic, J. D. Joannopoulos, S. G. Johnson, and I. Celanovic, “Design and global optimization of high-efficiency thermophotovoltaic systems,” Opt. Express 18, A314–A334 (2010).
[CrossRef] [PubMed]

2009 (2)

N. P. Sergeant, O. Pincon, M. Agrawal, and P. Peumans, “Design of wide-angle solar selective absorbers using aperiodic metal-dielectric stacks,” Opt. Express 17, 22800–22812 (2009).
[CrossRef]

E. E. Narimanov and A. V. Kildishev, “Optical black hole: broadband omnidirectional ligh absorber,” Appl. Phys. Lett. 95, 041106 (2009).
[CrossRef]

2008 (2)

N. I. Landy, S. Sajuyigbe, J. J. Mock, D. R. Smith, and W. J. Padilla, “Perfect metamaterial absorber,” Phys. Rev. Lett. 100, 207402 (2008).
[CrossRef] [PubMed]

B. T. Draine and P. J. Flateau, “Discrete-dipole approximation for periodic targets: theory and tests,” J. Opt. Soc. Am. A. 25, 2693 (2008).
[CrossRef]

2007 (1)

J. Drevillon and P. Ben-Abdallah, “Ab initio design of coherent thermal sources,” J. Appl. Phys. 102, 114305 (2007).
[CrossRef]

2005 (1)

F. Capolino, D. Wilton, and W. Johnson, “Efficient computation of the 2-D Green’s function for 1-D periodic structures using the Ewald method,” IEEE Trans. on Antennas and Propaga. 53, 9 (2005).
[CrossRef]

2002 (1)

B. Stout, J.-C. Auger, and J. Lafait, “A transfer matrix approach to local field calculations in multiple scattering problems,” J. Mod. Opt. 49, 2129–2152 (2002).
[CrossRef]

1995 (1)

M. S. Tomas, “Green function for multilayers: light scattering in planar cavities,” Phys. Rev. A 51, 2545 (1995).
[CrossRef] [PubMed]

1986 (1)

1973 (1)

E. M. Purcell and C. R. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys. J. 186, 705 (1973).
[CrossRef]

1921 (1)

P. P. Ewald, “Die berechnung optischer und elektrostatischer gitterpotentiale,” Annalen der Physik,  369, 253–287 (1921).
[CrossRef]

Agrawal, M.

Araghchini, M.

Ashkin, A.

Atwater, H. A.

K. Aydin, V. E. Ferry, R. M. Briggs, and H. A. Atwater, “Broadband polarization-independent resonant light absorption using ultrathin plasmonic super absorbers,” Nat. Commun. 2, 517 (2011).
[CrossRef] [PubMed]

H. A. Atwater and A. Polman, “Plasmonics for improved photovoltaic devices,” Nat. Mater. 9, 205–2013 (2010).
[CrossRef] [PubMed]

Aubry, A.

A. Aubry, D. Y. Lei, A. I. Fernández-Domínguez, Y. Sonnefraud, S. A. Maier, and J. B. Pendry, “Plasmonic light-harvesting devices over the whole visible spectrum,” Nano. Lett. 10, 2574–2579 (2010).
[CrossRef] [PubMed]

Auger, J.-C.

B. Stout, J.-C. Auger, and J. Lafait, “A transfer matrix approach to local field calculations in multiple scattering problems,” J. Mod. Opt. 49, 2129–2152 (2002).
[CrossRef]

Aydin, K.

K. Aydin, V. E. Ferry, R. M. Briggs, and H. A. Atwater, “Broadband polarization-independent resonant light absorption using ultrathin plasmonic super absorbers,” Nat. Commun. 2, 517 (2011).
[CrossRef] [PubMed]

Belov, P. A.

A. N. Poddubny, P. A. Belov, and Y. S. Kivshar, “Spontaneous radiation of a finite-size dipole in hyperbolic media,” Phys. Rev. A 84, 023807 (2011).
[CrossRef]

Ben-Abdallah, P.

R. Messina, M. Tschikin, S.-A. Biehs, and P. Ben-Abdallah, “Fluctuation-electrodynamic theory and dynamics of heat transfer in systems of multiple dipoles,” Phys. Rev. B 88, 104307 (2013).
[CrossRef]

P. Ben-Abdallah, R. Messina, S.-A. Biehs, M. Tschikin, K. Joulain, and C. Henkel, “Heat superdiffusion in plasmonic nanostructure networks,” Phys. Rev. Lett. 111, 174301 (2013).
[CrossRef] [PubMed]

P. Ben-Abdallah, S.-A. Biehs, and K. Joulain, “Many-body radiative heat transfer theory,” Phys. Rev. Lett. 107, 114301 (2011).
[CrossRef] [PubMed]

J. Drevillon and P. Ben-Abdallah, “Ab initio design of coherent thermal sources,” J. Appl. Phys. 102, 114305 (2007).
[CrossRef]

Bermel, P.

Biehs, S.-A.

R. Messina, M. Tschikin, S.-A. Biehs, and P. Ben-Abdallah, “Fluctuation-electrodynamic theory and dynamics of heat transfer in systems of multiple dipoles,” Phys. Rev. B 88, 104307 (2013).
[CrossRef]

P. Ben-Abdallah, R. Messina, S.-A. Biehs, M. Tschikin, K. Joulain, and C. Henkel, “Heat superdiffusion in plasmonic nanostructure networks,” Phys. Rev. Lett. 111, 174301 (2013).
[CrossRef] [PubMed]

P. Ben-Abdallah, S.-A. Biehs, and K. Joulain, “Many-body radiative heat transfer theory,” Phys. Rev. Lett. 107, 114301 (2011).
[CrossRef] [PubMed]

Bitzer, A.

V. A. Fedotov, N. Papasimakis, E. Plum, A. Bitzer, M. Walther, P. Kuo, D. P. Tsai, and N. I. Zheludev, “Spectral collapse in ensembles of metamolecules,” Phys. Rev. Lett. 104, 223901 (2010).
[CrossRef] [PubMed]

Bjorkholm, J. E.

Bohren, C. F.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley Science, New York, 1998).
[CrossRef]

Briggs, R. M.

K. Aydin, V. E. Ferry, R. M. Briggs, and H. A. Atwater, “Broadband polarization-independent resonant light absorption using ultrathin plasmonic super absorbers,” Nat. Commun. 2, 517 (2011).
[CrossRef] [PubMed]

Capolino, F.

F. Capolino, D. Wilton, and W. Johnson, “Efficient computation of the 2-D Green’s function for 1-D periodic structures using the Ewald method,” IEEE Trans. on Antennas and Propaga. 53, 9 (2005).
[CrossRef]

Carminati, R.

E. Castanie, R. Vincent, R. Pierrat, and R. Carminati, “Absorption by an optical dipole antenna in a structured environment,” Int. J. Opt. 2012, 452047 (2012).
[CrossRef]

Castanie, E.

E. Castanie, R. Vincent, R. Pierrat, and R. Carminati, “Absorption by an optical dipole antenna in a structured environment,” Int. J. Opt. 2012, 452047 (2012).
[CrossRef]

Celanovic, I.

Chan, W.

Chichkov, B. N.

A. B. Evlyukhin, C. Reinhardt, U. Zywietz, and B. N. Chichkov, “Collective resonances in metal nanoparticle arrays with dipole-quadrupole interactions,” Phys. Rev. B 85, 245411, (2012).
[CrossRef]

A. B. Evlyukhin, C. Reinhardt, A. Seidel, B. S. Luk’yanchuk, and B. N. Chichkov, “Optical response features of Si-nanoparticle arrays,” Phys. Rev. B 82, 045404 (2010).
[CrossRef]

Chu, S.

Draine, B. T.

B. T. Draine and P. J. Flateau, “Discrete-dipole approximation for periodic targets: theory and tests,” J. Opt. Soc. Am. A. 25, 2693 (2008).
[CrossRef]

Drevillon, J.

J. Drevillon and P. Ben-Abdallah, “Ab initio design of coherent thermal sources,” J. Appl. Phys. 102, 114305 (2007).
[CrossRef]

Dziedzic, J. M.

Evlyukhin, A. B.

A. B. Evlyukhin, C. Reinhardt, U. Zywietz, and B. N. Chichkov, “Collective resonances in metal nanoparticle arrays with dipole-quadrupole interactions,” Phys. Rev. B 85, 245411, (2012).
[CrossRef]

A. B. Evlyukhin, C. Reinhardt, A. Seidel, B. S. Luk’yanchuk, and B. N. Chichkov, “Optical response features of Si-nanoparticle arrays,” Phys. Rev. B 82, 045404 (2010).
[CrossRef]

Ewald, P. P.

P. P. Ewald, “Die berechnung optischer und elektrostatischer gitterpotentiale,” Annalen der Physik,  369, 253–287 (1921).
[CrossRef]

Fedotov, V. A.

V. A. Fedotov, N. Papasimakis, E. Plum, A. Bitzer, M. Walther, P. Kuo, D. P. Tsai, and N. I. Zheludev, “Spectral collapse in ensembles of metamolecules,” Phys. Rev. Lett. 104, 223901 (2010).
[CrossRef] [PubMed]

Feichtner, T.

T. Feichtner, O. Selig, M. Kiunke, and B. Hecht, “Evolutionary optimization of optical antennas,” Phys. Rev. Lett. 109, 127701 (2012).
[CrossRef] [PubMed]

Fernández-Domínguez, A. I.

A. Aubry, D. Y. Lei, A. I. Fernández-Domínguez, Y. Sonnefraud, S. A. Maier, and J. B. Pendry, “Plasmonic light-harvesting devices over the whole visible spectrum,” Nano. Lett. 10, 2574–2579 (2010).
[CrossRef] [PubMed]

Ferry, V. E.

K. Aydin, V. E. Ferry, R. M. Briggs, and H. A. Atwater, “Broadband polarization-independent resonant light absorption using ultrathin plasmonic super absorbers,” Nat. Commun. 2, 517 (2011).
[CrossRef] [PubMed]

Flateau, P. J.

B. T. Draine and P. J. Flateau, “Discrete-dipole approximation for periodic targets: theory and tests,” J. Opt. Soc. Am. A. 25, 2693 (2008).
[CrossRef]

Ghebrebrhan, M.

Hamam, R.

Hecht, B.

T. Feichtner, O. Selig, M. Kiunke, and B. Hecht, “Evolutionary optimization of optical antennas,” Phys. Rev. Lett. 109, 127701 (2012).
[CrossRef] [PubMed]

Henkel, C.

P. Ben-Abdallah, R. Messina, S.-A. Biehs, M. Tschikin, K. Joulain, and C. Henkel, “Heat superdiffusion in plasmonic nanostructure networks,” Phys. Rev. Lett. 111, 174301 (2013).
[CrossRef] [PubMed]

Holland, J. H.

J. H. Holland, Adaptation in Natural and Artificial Systems (MIT Press/Bradford Books Edition, Cambridge, MA, 1992).

Huffman, D. R.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley Science, New York, 1998).
[CrossRef]

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics, 3rd ed. (John Wiley, 1999).

Jenkins, S. D.

S. D. Jenkins and J. Ruostekoski, “Metamaterial transparency induced by cooperative electromagnetic interactions,” Phys. Rev. Lett. 111, 147401 (2013).
[CrossRef] [PubMed]

Jensen, K. F.

Joannopoulos, J. D.

Johnson, S. G.

Johnson, W.

F. Capolino, D. Wilton, and W. Johnson, “Efficient computation of the 2-D Green’s function for 1-D periodic structures using the Ewald method,” IEEE Trans. on Antennas and Propaga. 53, 9 (2005).
[CrossRef]

Joulain, K.

P. Ben-Abdallah, R. Messina, S.-A. Biehs, M. Tschikin, K. Joulain, and C. Henkel, “Heat superdiffusion in plasmonic nanostructure networks,” Phys. Rev. Lett. 111, 174301 (2013).
[CrossRef] [PubMed]

P. Ben-Abdallah, S.-A. Biehs, and K. Joulain, “Many-body radiative heat transfer theory,” Phys. Rev. Lett. 107, 114301 (2011).
[CrossRef] [PubMed]

Kildishev, A. V.

E. E. Narimanov and A. V. Kildishev, “Optical black hole: broadband omnidirectional ligh absorber,” Appl. Phys. Lett. 95, 041106 (2009).
[CrossRef]

Kiunke, M.

T. Feichtner, O. Selig, M. Kiunke, and B. Hecht, “Evolutionary optimization of optical antennas,” Phys. Rev. Lett. 109, 127701 (2012).
[CrossRef] [PubMed]

Kivshar, Y. S.

A. N. Poddubny, P. A. Belov, and Y. S. Kivshar, “Spontaneous radiation of a finite-size dipole in hyperbolic media,” Phys. Rev. A 84, 023807 (2011).
[CrossRef]

Kuo, P.

V. A. Fedotov, N. Papasimakis, E. Plum, A. Bitzer, M. Walther, P. Kuo, D. P. Tsai, and N. I. Zheludev, “Spectral collapse in ensembles of metamolecules,” Phys. Rev. Lett. 104, 223901 (2010).
[CrossRef] [PubMed]

Lafait, J.

B. Stout, J.-C. Auger, and J. Lafait, “A transfer matrix approach to local field calculations in multiple scattering problems,” J. Mod. Opt. 49, 2129–2152 (2002).
[CrossRef]

Landau, L.

L. Landau, E. Lifchitz, and L. Pitaevskii, Electromagnetics of Continuous Media(Pergamon, Oxford, 1984).

Landy, N. I.

N. I. Landy, S. Sajuyigbe, J. J. Mock, D. R. Smith, and W. J. Padilla, “Perfect metamaterial absorber,” Phys. Rev. Lett. 100, 207402 (2008).
[CrossRef] [PubMed]

Lei, D. Y.

A. Aubry, D. Y. Lei, A. I. Fernández-Domínguez, Y. Sonnefraud, S. A. Maier, and J. B. Pendry, “Plasmonic light-harvesting devices over the whole visible spectrum,” Nano. Lett. 10, 2574–2579 (2010).
[CrossRef] [PubMed]

Lifchitz, E.

L. Landau, E. Lifchitz, and L. Pitaevskii, Electromagnetics of Continuous Media(Pergamon, Oxford, 1984).

Luk’yanchuk, B. S.

A. B. Evlyukhin, C. Reinhardt, A. Seidel, B. S. Luk’yanchuk, and B. N. Chichkov, “Optical response features of Si-nanoparticle arrays,” Phys. Rev. B 82, 045404 (2010).
[CrossRef]

Maier, S. A.

A. Aubry, D. Y. Lei, A. I. Fernández-Domínguez, Y. Sonnefraud, S. A. Maier, and J. B. Pendry, “Plasmonic light-harvesting devices over the whole visible spectrum,” Nano. Lett. 10, 2574–2579 (2010).
[CrossRef] [PubMed]

Marton, C. H.

Messina, R.

R. Messina, M. Tschikin, S.-A. Biehs, and P. Ben-Abdallah, “Fluctuation-electrodynamic theory and dynamics of heat transfer in systems of multiple dipoles,” Phys. Rev. B 88, 104307 (2013).
[CrossRef]

P. Ben-Abdallah, R. Messina, S.-A. Biehs, M. Tschikin, K. Joulain, and C. Henkel, “Heat superdiffusion in plasmonic nanostructure networks,” Phys. Rev. Lett. 111, 174301 (2013).
[CrossRef] [PubMed]

Mock, J. J.

N. I. Landy, S. Sajuyigbe, J. J. Mock, D. R. Smith, and W. J. Padilla, “Perfect metamaterial absorber,” Phys. Rev. Lett. 100, 207402 (2008).
[CrossRef] [PubMed]

Narimanov, E. E.

E. E. Narimanov and A. V. Kildishev, “Optical black hole: broadband omnidirectional ligh absorber,” Appl. Phys. Lett. 95, 041106 (2009).
[CrossRef]

Padilla, W. J.

N. I. Landy, S. Sajuyigbe, J. J. Mock, D. R. Smith, and W. J. Padilla, “Perfect metamaterial absorber,” Phys. Rev. Lett. 100, 207402 (2008).
[CrossRef] [PubMed]

Palik, E. D.

E. D. Palik, Handbook of Optical Constants of Solids (Academic Press, New York, 1998).

Papasimakis, N.

V. A. Fedotov, N. Papasimakis, E. Plum, A. Bitzer, M. Walther, P. Kuo, D. P. Tsai, and N. I. Zheludev, “Spectral collapse in ensembles of metamolecules,” Phys. Rev. Lett. 104, 223901 (2010).
[CrossRef] [PubMed]

Pendry, J. B.

A. Aubry, D. Y. Lei, A. I. Fernández-Domínguez, Y. Sonnefraud, S. A. Maier, and J. B. Pendry, “Plasmonic light-harvesting devices over the whole visible spectrum,” Nano. Lett. 10, 2574–2579 (2010).
[CrossRef] [PubMed]

Pennypacker, C. R.

E. M. Purcell and C. R. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys. J. 186, 705 (1973).
[CrossRef]

Peumans, P.

Pierrat, R.

E. Castanie, R. Vincent, R. Pierrat, and R. Carminati, “Absorption by an optical dipole antenna in a structured environment,” Int. J. Opt. 2012, 452047 (2012).
[CrossRef]

Pincon, O.

Pitaevskii, L.

L. Landau, E. Lifchitz, and L. Pitaevskii, Electromagnetics of Continuous Media(Pergamon, Oxford, 1984).

Plum, E.

V. A. Fedotov, N. Papasimakis, E. Plum, A. Bitzer, M. Walther, P. Kuo, D. P. Tsai, and N. I. Zheludev, “Spectral collapse in ensembles of metamolecules,” Phys. Rev. Lett. 104, 223901 (2010).
[CrossRef] [PubMed]

Poddubny, A. N.

A. N. Poddubny, P. A. Belov, and Y. S. Kivshar, “Spontaneous radiation of a finite-size dipole in hyperbolic media,” Phys. Rev. A 84, 023807 (2011).
[CrossRef]

Polman, A.

H. A. Atwater and A. Polman, “Plasmonics for improved photovoltaic devices,” Nat. Mater. 9, 205–2013 (2010).
[CrossRef] [PubMed]

Purcell, E. M.

E. M. Purcell and C. R. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys. J. 186, 705 (1973).
[CrossRef]

Reinhardt, C.

A. B. Evlyukhin, C. Reinhardt, U. Zywietz, and B. N. Chichkov, “Collective resonances in metal nanoparticle arrays with dipole-quadrupole interactions,” Phys. Rev. B 85, 245411, (2012).
[CrossRef]

A. B. Evlyukhin, C. Reinhardt, A. Seidel, B. S. Luk’yanchuk, and B. N. Chichkov, “Optical response features of Si-nanoparticle arrays,” Phys. Rev. B 82, 045404 (2010).
[CrossRef]

Ruostekoski, J.

S. D. Jenkins and J. Ruostekoski, “Metamaterial transparency induced by cooperative electromagnetic interactions,” Phys. Rev. Lett. 111, 147401 (2013).
[CrossRef] [PubMed]

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N. I. Landy, S. Sajuyigbe, J. J. Mock, D. R. Smith, and W. J. Padilla, “Perfect metamaterial absorber,” Phys. Rev. Lett. 100, 207402 (2008).
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[CrossRef]

Selig, O.

T. Feichtner, O. Selig, M. Kiunke, and B. Hecht, “Evolutionary optimization of optical antennas,” Phys. Rev. Lett. 109, 127701 (2012).
[CrossRef] [PubMed]

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Smith, D. R.

N. I. Landy, S. Sajuyigbe, J. J. Mock, D. R. Smith, and W. J. Padilla, “Perfect metamaterial absorber,” Phys. Rev. Lett. 100, 207402 (2008).
[CrossRef] [PubMed]

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A. Aubry, D. Y. Lei, A. I. Fernández-Domínguez, Y. Sonnefraud, S. A. Maier, and J. B. Pendry, “Plasmonic light-harvesting devices over the whole visible spectrum,” Nano. Lett. 10, 2574–2579 (2010).
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V. A. Fedotov, N. Papasimakis, E. Plum, A. Bitzer, M. Walther, P. Kuo, D. P. Tsai, and N. I. Zheludev, “Spectral collapse in ensembles of metamolecules,” Phys. Rev. Lett. 104, 223901 (2010).
[CrossRef] [PubMed]

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P. Ben-Abdallah, R. Messina, S.-A. Biehs, M. Tschikin, K. Joulain, and C. Henkel, “Heat superdiffusion in plasmonic nanostructure networks,” Phys. Rev. Lett. 111, 174301 (2013).
[CrossRef] [PubMed]

R. Messina, M. Tschikin, S.-A. Biehs, and P. Ben-Abdallah, “Fluctuation-electrodynamic theory and dynamics of heat transfer in systems of multiple dipoles,” Phys. Rev. B 88, 104307 (2013).
[CrossRef]

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[CrossRef]

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V. A. Fedotov, N. Papasimakis, E. Plum, A. Bitzer, M. Walther, P. Kuo, D. P. Tsai, and N. I. Zheludev, “Spectral collapse in ensembles of metamolecules,” Phys. Rev. Lett. 104, 223901 (2010).
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[CrossRef]

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A. Aubry, D. Y. Lei, A. I. Fernández-Domínguez, Y. Sonnefraud, S. A. Maier, and J. B. Pendry, “Plasmonic light-harvesting devices over the whole visible spectrum,” Nano. Lett. 10, 2574–2579 (2010).
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[CrossRef]

M. S. Tomas, “Green function for multilayers: light scattering in planar cavities,” Phys. Rev. A 51, 2545 (1995).
[CrossRef] [PubMed]

Phys. Rev. B (3)

R. Messina, M. Tschikin, S.-A. Biehs, and P. Ben-Abdallah, “Fluctuation-electrodynamic theory and dynamics of heat transfer in systems of multiple dipoles,” Phys. Rev. B 88, 104307 (2013).
[CrossRef]

A. B. Evlyukhin, C. Reinhardt, U. Zywietz, and B. N. Chichkov, “Collective resonances in metal nanoparticle arrays with dipole-quadrupole interactions,” Phys. Rev. B 85, 245411, (2012).
[CrossRef]

A. B. Evlyukhin, C. Reinhardt, A. Seidel, B. S. Luk’yanchuk, and B. N. Chichkov, “Optical response features of Si-nanoparticle arrays,” Phys. Rev. B 82, 045404 (2010).
[CrossRef]

Phys. Rev. Lett. (6)

N. I. Landy, S. Sajuyigbe, J. J. Mock, D. R. Smith, and W. J. Padilla, “Perfect metamaterial absorber,” Phys. Rev. Lett. 100, 207402 (2008).
[CrossRef] [PubMed]

S. D. Jenkins and J. Ruostekoski, “Metamaterial transparency induced by cooperative electromagnetic interactions,” Phys. Rev. Lett. 111, 147401 (2013).
[CrossRef] [PubMed]

V. A. Fedotov, N. Papasimakis, E. Plum, A. Bitzer, M. Walther, P. Kuo, D. P. Tsai, and N. I. Zheludev, “Spectral collapse in ensembles of metamolecules,” Phys. Rev. Lett. 104, 223901 (2010).
[CrossRef] [PubMed]

P. Ben-Abdallah, R. Messina, S.-A. Biehs, M. Tschikin, K. Joulain, and C. Henkel, “Heat superdiffusion in plasmonic nanostructure networks,” Phys. Rev. Lett. 111, 174301 (2013).
[CrossRef] [PubMed]

P. Ben-Abdallah, S.-A. Biehs, and K. Joulain, “Many-body radiative heat transfer theory,” Phys. Rev. Lett. 107, 114301 (2011).
[CrossRef] [PubMed]

T. Feichtner, O. Selig, M. Kiunke, and B. Hecht, “Evolutionary optimization of optical antennas,” Phys. Rev. Lett. 109, 127701 (2012).
[CrossRef] [PubMed]

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J. H. Holland, Adaptation in Natural and Artificial Systems (MIT Press/Bradford Books Edition, Cambridge, MA, 1992).

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J. D. Jackson, Classical Electrodynamics, 3rd ed. (John Wiley, 1999).

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[CrossRef]

L. Schwartz, Théorie des distributions, Hermann (1951).

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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.

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.

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.

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.

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 (ε)|ESG|2 in the single particle lattice are normalized by the maximum loss. In (b) we show the normalized difference (ε)|EDG|2(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.

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.

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 ( E ˜ 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 = ( 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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