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
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  or the quantum information treatment . 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  to the visible .
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]14] and gives the field A at the position rm given a B-dipole located in rn. Δ𝔾AB defined as gives the contribution of interfaces only.
Here the free space Green tensor in the host material defined with the unit vector r̂mn ≡ r̂mn/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 . Beside the dipoles location the auto-induced part of field does not exist anymore and it takes the simplified form
It immediately follows that, the dipolar moments associated to each object reads
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 and for the dipolar moments pjβ;A = p̃β;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 give15, 16] as in solid-state physics. Relation (6) defines the dressed polarizability tensor  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  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
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 withEqs. (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
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)
The outgoing wave functions are solutions of Maxwell’s equation (using the e−iωt convention)
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 as19], the action of the sources (with p′ = (n, −m)) on the incident field. Then using the fact that functions form a complete basis they satisfy the following orthogonality relations 22]. Then using the orthogonality relations (26) and according to (24) 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  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.
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 |ri − rj |> Ri + Rj.
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)  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) 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 isFig. 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.
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 , 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.
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 , in light extraction technologies to improve the performances of light emitting diodes.
J.-P. H. acknowledges discussions with J.J. Greffet. P. B.-A. gratefully acknowledges the support of Total news energies.
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