Theoretical formalism for collective electromagnetic response of discrete metamaterial systems

We develop a general formalism to describe the propagation of a near-resonant electromagnetic field in a medium composed of magnetodielectric resonators. As the size and the spatial separation of nanofabricated resonators in a metamaterial array is frequently less than the wavelength, we describe them as discrete scatterers, supporting a single mode of current oscillation represented by a single dynamic variable. We derive a Lagrangian and Hamiltonian formalism for the coupled electromagnetic fields and oscillating currents in the length gauge, obtained by the Power-Zienau-Woolley transformation. The response of each resonator to electromagnetic field is then described by polarization and magnetization densities that, to the lowest order in a multipole expansion, generate electric and magnetic dipole excitations. We derive a closed set of equations for the coherently scattered field and normal mode amplitudes of current oscillations of each resonator both within the rotating wave approximation, in which case the radiative decay rate is much smaller than the resonance frequency, and without such an assumption. The set of equations includes the radiative couplings between a discrete set of resonators mediated by the electromagnetic field, fully incorporating recurrent scattering processes to all orders. By considering an example of a two-dimensional split ring resonator metamaterial array, we show that the system responds cooperatively to near-resonant field, exhibiting collective eigenmodes, resonance frequencies, and radiative linewidths that result from strong radiative interactions between closely-spaced resonators.Comment: 34 pages, 6 figure

Controlled manipulation of light by cooperative response of atoms in an optical lattice

We show that a cooperative atom response in an optical lattice to resonant incident light can be employed for precise control and manipulation of light on a subwavelength scale. Specific collective excitation modes of the system that result from strong light-mediated dipole-dipole interactions can be addressed by tailoring the spatial phase-profile of the incident light. We demonstrate how the collective response can be used to produce optical excitations at well-isolated sites on the lattice.Comment: 8 pages, 1 figur

Cooperative resonance linewidth narrowing in a planar metamaterial

We theoretically analyze the experimental observations of a spectral line collapse in a metamaterial array of asymmetric split ring resonators [Fedotov et al., Phys. Rev. Lett. 104, 223901 (2010)]. We show that the ensemble of closely-spaced resonators exhibits cooperative response, explaining the observed system-size dependent narrowing of the transmission resonance linewidth. We further show that this cooperative narrowing depends sensitively on the lattice spacing and that significantly stronger narrowing could be achieved in media with suppressed ohmic losses.Comment: 19 pages, 6 figures, to appear in New Journal of Physic

Stochastic methods for light propagation and recurrent scattering in saturated and nonsaturated atomic ensembles

We derive equations for the strongly coupled system of light and dense atomic ensembles. The formalism includes an arbitrary internal-level structure for the atoms and is not restricted to weak excitation of atoms by light. In the low-light-intensity limit for atoms with a single electronic ground state, the full quantum field-theoretical representation of the model can be solved exactly by means of classical stochastic electrodynamics simulations for stationary atoms that represent cold atomic ensembles. Simulations for the optical response of atoms in a quantum degenerate regime require one to synthesize a stochastic ensemble of atomic positions that generates the corresponding quantum statistical position correlations between the atoms. In the case of multiple ground levels or at light intensities where saturation becomes important, the classical simulations require approximations that neglect quantum fluctuations between the levels. We show how the model is extended to incorporate corrections due to quantum fluctuations that result from virtual scattering processes. In the low-light-intensity limit, we illustrate the simulations in a system of atoms in a Mott-insulator state in a two-dimensional optical lattice, where recurrent scattering of light induces strong interatomic correlations. These correlations result in collective many-atom subradiant and superradiant states and a strong dependence of the response on the spatial confinement within the lattice site

Point dipole and quadrupole scattering approximation to collectively responding resonator systems

We develop a theoretical formalism for collectively responding point scatterers where the radiating electromagnetic fields from each emitter are considered in the electric dipole, magnetic dipole, and electric quadrupole approximation. The contributions of the electric quadrupole moment to electromagnetically-mediated interactions between the scatterers are derived in detail for a system where each scatterer represents a linear RLC circuit resonator, representing common metamaterial resonators in radiofrequency, microwave, and optical regimes. The resulting theory includes a closed set of equations for an ensemble of discrete resonators that are radiatively coupled to each other by propagating electromagnetic fields, incorporating potentially strong interactions and recurrent scattering processes. The effective model is illustrated and tested for examples of pairs of interacting point electric dipoles, where each pair can be qualitatively replaced by a model point emitter with different multipole radiation moments

Stochastic electrodynamics simulations for collective atom response in optical cavities

We study the collective optical response of an atomic ensemble confined within a single-mode optical cavity by stochastic electrodynamics simulations that include the effects of atomic position correlations, internal level structure, and spatial variations in cavity coupling strength and atom density. In the limit of low light intensity the simulations exactly reproduce the full quantum field-theoretical description for cold stationary atoms and at higher light intensities we introduce semiclassical approximations to atomic saturation that we compare with the exact solution in the case of two atoms. We find that collective subradiant modes of the atoms, with very narrow linewidths, can be coupled to the cavity field by spatial variation of the atomic transition frequency and resolved at low intensities, and show that they can be specifically driven by tailored transverse pumping beams. We show that the cavity optical response, in particular both the subradiant mode profile and the resonance shift of the cavity mode, can be used as a diagnostic tool for the position correlations of the atoms and hence the atomic quantum many-body phase. The quantum effects are found to be most prominent close to the narrow subradiant mode resonances at high light intensities. Although an optical cavity can generally strongly enhance quantum fluctuations via light confinement, we show that the semiclassical approximation to the stochastic electrodynamics model provides at least a qualitative agreement with the exact optical response outside the subradiant mode resonances even in the presence of significant saturation of the atoms

Point-dipole approximation for small systems of strongly coupled radiating nanorods

Systems of closely-spaced resonators can be strongly coupled by interactions mediated by scattered electromagnetic fields. In large systems the resulting response has been shown to be more sensitive to these collective interactions than to the detailed structure of individual resonators. Attempts to describe such systems have resulted in point-dipole approximations to resonators that are computationally efficient for large resonator ensembles. Here we provide a detailed study for the validity of point dipole approximations in small systems of strongly coupled plasmonic nanorods, including the cases of both super-radiantand subradiant excitations, where the characteristics of the excitation depends on the spatial separation between the nanorods. We show that over an appreciable range of rod lengths centered on 210 nm, when the relative separation kl in terms of the resonance wave number of light k satisfies kl >pi/2, the point electric dipole model becomes accurate. However, when theresonators are closer, the finite-size and geometry of the resonators modifies the excitation modes, in particular the cooperative mode line shifts of the point dipole approximation begin to rapidly diverge at small separations. We also construct simplified effective models by describing a pair of nanorods as a single effective metamolecule

Cooperative field localization and excitation eigenmodes in disordered metamaterials

We investigate numerically and experimentally the near-field response of disordered arrays comprising asymmetrically split ring resonators that exhibit a strong cooperative response. Our simulations treat the unit cell split-ring resonators as discrete pointlike oscillators with associated electric and magnetic point dipole radiation, while the strong cooperative radiative coupling between the different split rings is fully included at all orders. The methods allow us to calculate local field and Purcell factor enhancement arising from the collective electric and magnetic excitations. We find substantially increased standard deviation of the Purcell enhancement with disorder, making it increasingly likely to find collective excitation eigenmodes with very high Purcell factors that are also stronger for magnetic than electric excitations. We show that disorder can dramatically modify the cooperative response of the metamaterial even in the presence of strong dissipation losses, as is the case for plasmonic systems. Our analysis in terms of collective eigenmodes paves the way for controlled engineering of electromagnetic device functionalities based on strongly interacting metamaterial arrays

Development of a generic activities model of command and control

This paper reports on five different models of command and control. Four different models are reviewed: a process model, a contextual control model, a decision ladder model and a functional model. Further to this, command and control activities are analysed in three distinct domains: armed forces, emergency services and civilian services. From this analysis, taxonomies of command and control activities are developed that give rise to an activities model of command and control. This model will be used to guide further research into technological support of command and control activities