Theoretical formalism for collective electromagnetic response of discrete metamaterial systems

Stewart D. Jenkins and Janne Ruostekoski

Phys. Rev. B 86, 085116 – Published 13 August 2012

Abstract

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 are 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.

Received 1 June 2012

DOI:https://doi.org/10.1103/PhysRevB.86.085116

Authors & Affiliations

Stewart D. Jenkins and Janne Ruostekoski

School of Mathematics and Centre for Photonic Metamaterials, University of Southampton, Southampton SO17 1BJ, United Kingdom

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Issue

Vol. 86, Iss. 8 — 15 August 2012

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Images

Figure 1
A schematic illustration of a split ring resonator. An excitation in the left meta-atom ( $l$ ) produces an oscillating electric dipole (indicated by the blue arrow) in the direction $\hat{d}$ and a magnetic dipole (indicated by the red arrow) in the direction $- \hat{m}$ , while an excitation in the right meta-atom ( $r$ ) produces an electric dipole in the direction $\hat{d}$ and a magnetic dipole in the direction $\hat{m}$ . The meta-atoms, in the point source approximation, are separated by a vector $u$ . When the meta-atoms are excited in phase, the electric dipoles reinforce each other and the magnetic dipoles cancel out.Reuse & Permissions
Figure 2
The distributions of collective radiative emission rates in a $33 \times 33$ square array of SRRs. The distribution is represented as a histogram of the $\log_{10}$ of the collective emission rates. (a) The distribution for an ensemble of SRRs with lattice spacing $a = 0.5 λ$ . (b) The distribution when the lattice spacing $a = 1.4 λ$ . When the lattice spacing is larger, interactions become weaker, and cooperative effects are diminished, leading to a narrower distribution of collective mode linewidths.Reuse & Permissions
Figure 3
An illustration of the most subradiant of the collective modes in a $33 \times 33$ array of SRRs. The height of the surface represents the energy of the SRR symmetric oscillations $| c_{+} |^{2}$ normalized to the peak SRR energy $E_{0} = \max_{ℓ} (| c_{+, ℓ} |^{2} + c_{-, ℓ} |^{2})$ . The colored patches indicate the phase of the electric dipole oscillations for each SRR. The black dots indicate the position of each SRR, while their height indicates the normalized total energy within each unit cell, so that the energy in the magnetic dipole oscillations is given by the difference between the black dots' height and the height of the surface. This subradiant mode is ferroelectric in nature, and has a radiative emission rate $1.2 \times 10^{- 5} Γ$ . The lattice spacing $a = 0.5 λ$ , the meta-atom separation within the SRRs $u = 0.12 λ$ , and $Γ_{E} = Γ_{M}$ .Reuse & Permissions
Figure 4
An illustration of the most superradiant of the collective modes in a $33 \times 33$ array of SRRs. The height of the surface represents the energy of the SRR antisymmetric oscillations $| c_{-} |^{2}$ normalized to the peak SRR energy $E_{0} = \max_{ℓ} (| c_{+, ℓ} {|^{2} + c_{-, ℓ} |}^{2}$ . The black dots indicate the position of each SRR, while their height indicates the normalized total energy within each unit cell. The total excitation, $| c_{+, ℓ} |^{2}$ , of the electric dipole oscillations is given by the difference between the black dots' height and the height of the surface. All parameters of the ensemble are as in Fig. 3. This superradiant mode has a radiative emission rate of about $11 Γ$ and consists largely of magnetic dipole oscillations whose phase variation is matched with EM waves propagating in the $\pm x$ directions.Reuse & Permissions
Figure 5
An illustration of the uniform electric collective mode in a $33 \times 33$ array of SRRs. The height of the surface represents the excitation energy of the SRR symmetric oscillations $| c_{+} |^{2}$ normalized to the peak SRR energy $E_{0} = \max_{ℓ} (| c_{+, ℓ} {|^{2} + c_{-, ℓ} |}^{2}$ . The colored patches indicate the phase of the electric dipole oscillations for each SRR. The black dots indicate the position of each SRR, while their height indicates the normalized total energy within each unit cell. All parameters of the ensemble are as in Fig. 3. This mode consists of the split ring electric dipoles oscillating in phase and has a radiative emission rate of approximately the single meta-atom emission rate $Γ$ .Reuse & Permissions
Figure 6
An illustration of the uniform magnetic collective mode in a $33 \times 33$ array of SRRs. The height of the surface represents the energy of the SRR antisymmetric oscillations $| c_{-} |^{2}$ normalized to the peak SRR energy $E_{0} = \max_{ℓ} (| c_{+, ℓ} {|^{2} + c_{-, ℓ} |}^{2}$ . The black dots indicate the position of each SRR, while their height indicates the normalized total energy within each unit cell. All parameters of the ensemble are as in Fig. 3. This mode is composed of all SRRs oscillating antisymmetrically, producing magnetic dipoles in phase. The radiative emission rate of this mode is $0.02 Γ$ .Reuse & Permissions

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