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.

Resonant multiple scattering plays an important role in mesoscopic wave phenomena that can also be reached with electromagnetic (EM) fields. In the strong scattering regime, interference of different scattering paths between discrete scatterers can result in, e.g., light localization [1]-an effect analogous to the Anderson localization of electrons in solids. Metamaterials comprise artificially structured media of plasmonic resonators interacting with EM fields. Due to several promising phenomena, such as the possibility for diffraction-free lenses resulting from negative refractive index [2], there has been a rapidly increasing interest in fabrication and theoretical modeling of such systems. The discrete nature of closelyspaced resonators in typical metamaterial arrays raises the possibility to observe also strong collective radiative effects in these systems.
In recent experiments Fedotov et al. observed a dramatic suppression of radiation losses in a 2D planar metamaterial array [3]. The transmission spectra through the metamolecular sheet was found to be strongly dependent upon the number of interacting metamolecules in the system. The transmission resonance quality factor increased as a function of the total number of active resonators, finally saturating at about 700 metamolecules. The metamaterial unit cell in the experiment was formed by an asymmetric split-ring (ASR) resonator, consisting of two circular arcs of slightly unequal lengths. The currents in these ASRs may be excited symmetrically (antisymmetrically), yielding a net oscillating electric (magnetic) dipole; Fig. 1.
In this letter we theoretically analyze the collective metamaterial response, observed experimentally in Ref. [3]. We find that strong interactions between a discrete set of resonators, mediated by the EM field, characterize the response of the ensemble and results in collective resonance linewidths and frequencies. We show how the cooperative response of sufficiently closely-spaced resonators is responsible for the observed dramatic narrowing of the transmission resonance linewidth (increasing quality factor) with the number of resonators [3]. In particular, the system exhibits a collective mode with an almost purely magnetic excitation, uniform phase profile, The amplitude and phase of magnetic (green) and electric (blue) dipoles are represented by the length and direction of the arrows, respectively. and strongly suppressed radiative properties with each ASR possessessing a nearly equal magnetic dipole moment. We show in detail how this mode can be excited by an incident plane wave propagating perpendicular to the array through an electric dipole coupling to an ASR, even when the magnetic dipole moments are oriented parallel to the propagation direction. We calculate the resonance linewidth of the phase-coherent collective magnetic mode that dramatically narrows as a function of the number of ASRs, providing an excellent agreement with experimental observations. The linewidth is also sensitive to the spacing of the unit cell resonators, with the closelyspaced ASRs exhibiting cooperative response, due to enhanced dipole-dipole interactions. Moreover, at the resonance, and with appropriately chosen parameters, nearly all the excitation can be driven into this mode. Due to its suppressed decay rate, the transmission spectrum displays a narrow resonance. Our analysis demonstrates how essential features of the collective effects of the experiment in Ref. [3] can be captured by a simple, computationally efficient model in which we treat each meta-atom as a discrete scatterer, exhibiting a single mode of current oscillation and possessing appropriate electric and magnetic dipole moments. Interactions with the EM field then determine the collective interactions within the ensemble. Moreover, our analysis indicates the necessity of accounting for the strong collective response of metamaterial systems and interference effects in multiple scattering between the resonators in understanding the dynamics and design of novel meta-materials. Strong interactions between resonators can find important applications in metamaterial systems, providing, e.g., precise control and manipulation of EM fields on a sub-wavelength scale [4], in developments of a lasing spaser [5], and disorder-related phenomena [6].
In order to analyze the experimental observations of the transmission spectra, we provide a simple theoretical model that is computationally efficient in large systems, which are required to generate strong collective effects, but that captures the essential features of interactions between the resonators. We consider an ensemble of N metamaterial unit elements, metamolecules, each formed by n discrete meta-atoms, with the position of the metaatom j denoted by r j (j = 1, . . . , n × N ). This ensemble is driven by an external beam with electric field E in (r, t) and magnetic field B in (r, t) with wavelengths much larger than the size of the meta-atoms. Consequently, we treat meta-atoms as radiating dipoles and ignore the higherorder multipole-field interactions. We assume that each meta-atom j supports a single eigenmode of current oscillation with the charge Q j (t) as its dynamic variable. Then the associated electric and magnetic dipole moment for the meta-atoms are d j = Q j h jdj and m j = I j A jmj , respectively, whered j andm j denote unit vectors and I j (t) = dQ j /dt is the current. Here h j and A j are the corresponding proportionality coefficients (with the units of length and area), depending on the specific geometry of the resonators. In the dipole approximation the polarization and magnetization are given in terms of the density of electric and magnetic dipoles P(r) = j P j (r) and M(r) = j M j (r), where the polarization and the magnetization of the resonator j are P j (r, t) = d j δ(r − r j ) and M j (r, t) = m j δ(r−r j ), respectively. Here r j denotes the position of the meta-atom j.
Incident EM field drives the excitation of the current oscillations, generating an oscillating electric and magnetic dipole in each meta-atom. The resulting dipole radiation from the metamaterial array is the sum of the scattered electric and magnetic fields from all the meta-atoms E S (r, t) = j E S,j (r, t) and H S (r, t) = j H S,j (r, t), where E S,j (r, t) and H S,j (r, t) denote the electric and magnetic field emitted by the meta-atom j. The Fourier components of the scattered fields have the familiar expressions of electric and magnetic fields radiated by oscillating electric and magnetic dipoles [7].
We express these in terms of the positive frequency components E + S (r, t) and H + S (r, t) for the frequency Ω (k ≡ Ω/c), where the total field E(r) = E + (r) + E − (r) and (E + ) * = E − , Here G denotes the radiation kernel representing the electric (magnetic) field emitted from an electric (magnetic) dipole [7]. The explicit expression for the corresponding radiated field from a dipole with amplitudev reads wherer ≡ r/r. Similarly, G × (r, k) represents the radiation kernel for the magnetic (electric) field of an electric (magnetic) dipole source [7]. Specifically, the corresponding radiated field from a dipole with amplitudev yields Each meta-atom interacts with the incident fields E in (r, t) and H in (r, t), as well as the fields radiated by all other meta-atoms in the metamaterial sample and with its own self-generated radiation. In each meta-atom the oscillating current generates electric and magnetic dipoles which both radiate magnetic and electric fields, according to Eqs. (1) and (2). These fields couple to dynamical variables of charge oscillations in other metaatoms, producing more dipolar radiation. In order to calculate the EM response of the system, we solve the coupled set of equations involving all the resonators and the fields. A system of n × N single-mode resonators then possesses n × N collective modes of current oscillation. Each collective mode exhibits a distinct collective linewidth (decay rate) and resonance frequency, determined by the imaginary and real parts of the corresponding eigenvalue. The resulting dynamics resemble a cooperative response of atomic gases to resonant light in which case the EM coupling between different atoms is due to electric dipole radiation alone [8,9]. The crucial component of the strong cooperative response of closelyspaced scatterers are recurrent scattering events [8,10] -in which a wave is scattered more than once by the same dipole. Such processes cannot generally be modeled by the continuous medium electrodynamics, necessitating the meta-atoms to be treated as discrete scatterers. An approximate calculation of local field corrections in a magnetodielectric medium of discrete scatterers was performed in Ref. [11] where the translational symmetry of an infinite lattice simplifies the response.
In this work we consider asymmetric metamolecules consisting of two meta-atoms with different resonance frequencies ω j , centered around the frequency ω 0 , with |ω j − ω 0 | ≪ ω 0 . The meta-atoms are illuminated by a near-resonant incident field with a narrow bandwidth and dominant frequency Ω 0 . For simplicity, we assume that the radiative electric and magnetic decay rates of each resonator Γ E and Γ M are independent of the resonator j and sufficiently slow Γ E/M ≪ ω 0 . Then the dominant contribution of the dipole radiation in Eqs. (3) and (4) is generated at the wavenumber k ≃ Ω 0 /c and a single isolated meta-atom behaves as a simple linear LC circuit with ω j = 1/ L j C j , where we have introduced an effective capacitance C j and self-inductance L j associated with the circuit current mode [12]. In the damping rate Γ ≡ Γ E + Γ M + Γ O , we add to radiative dissipation also a nonradiative decay Γ O , resulting, e.g., from ohmic losses. The interaction processes between the different resonators, mediated by dipole radiation and originating from Eqs. (1) and (2), are analogous to frequency dependent mutual inductance and capacitance, but due to the radiative long-range interactions, these can substantially differ from the quasi-static expressions for which G × is also absent.
In the transmission resonance experiment [3] an ASR formed a unit cell resonator (metamolecule). A single ASR consists of two separate coplanar circular arcs (meta-atoms), labeled by j ∈ {l, r} and separated by a ≡ r r − r l ; Fig. 1. The current oscillations in each meta-atom produce electric dipoles with orientationd (d ⊥â) and magnetic dipoles with opposite orientationsm r = −m l (m r ⊥â,d). An asymmetry between the rings, in this case resulting from a difference in arc length, manifests itself as a difference in resonance frequencies with ω r = ω 0 + δω and ω l = ω 0 − δω. In a single, isolated ASR the radiative interactions between the two resonators results in eigenstates analogous to superradiant and subradiant states in a pair of atoms. In order to analyze these eigenstates, we consider the dynamics of symmetric c + and antisymmetric c − modes of current oscillation (Fig. 1) that represent the exact eigenmodes of the ASR in the absence of asymmetry δω = 0. Excitations of these modes possess respective net electric and magnetic dipoles and will thus be referred to electric and magnetic dipole excitations. The split ring asymmetry δω = 0, however, introduces an effective coupling between these modes in a single ASR, so thatċ ± = (−γ ± /2 ∓ i∆)c ± − iδωc ∓ + F ± , where γ ± and ∆ denote the decay rates and a frequency shift, respectively, and F ± represents the driving by the incident field. Due to the asymmetry-dependent coupling between the modes, the incident field with E in d and B in ⊥m only excites the electric mode when δω = 0, but for δω = 0 it can resonantly pump the anti-symmetric magnetic mode.
In order to model the experimentally observed collective response [3] we study an ensemble of identical ASRs (with a = aê x and d = dê y ; Fig. 1) arranged in a 2D square lattice within a circle of radius r c , with lattice spacing u, and lattice vectors (uê x , uê y ). The sample is illuminated by a cw plane wave E + in (r) = 1 2 Eê y e ik·r with k = kê z , coupling to the electric dipole moments of the ASRs. In the experimentally measured transmission resonance through such a sheet [3] the number of active ASRs, N , was controlled by decoupling the ASRs with r r c from the rest of the system with approximately circular shaped metal masks with varying radii r c . The resonance quality factor increased with the total number of active ASRs, saturating at about N = 700.
We find that an incident plane-wave drives all metamolecules uniformly and is phase-matched to collective modes in which the electric and/or magnetic dipoles oscillate in phase. In the absence of a split-ring asymmetry, only modes involving oscillating electric dipoles can be driven. These modes strongly emit perpendicular to the array (into the ±ê z directions) enhancing incident wave reflection. The magnetic dipoles, however, dominantly radiate into EM field modes within the ASR plane. This radiation may become trapped through recurrent scattering processes in the array, representing modes with suppressed emission rates and reflectance, and resulting in a transmission resonance. In order to quantify the effect, we study the radiation properties (for Γ E = Γ M ) of the numerically calculated collective magnetic eigenmode v m of the system [ Fig. 1(d)] which maximizes the overlap 2N (all ASRs oscillating anti-symmetrically). We then show that the introduction of an asymmetry δω in the resonances allows the excitation of v m by the incident field. This mode closely resembles that responsible for the experimentally observed transmission resonance [3,13]. Figure 2a shows dependence of the resonance linewidth γ of the collective mode v m , which is the closest representation of the phase-coherent uniform magnetic excitation, on the number of metamolecules N for different lattice spacings u (λ denotes the resonance wavelength of v m ). In the absence of ohmic losses and for sufficiently small u and δω, γ ∝ 1/N for large N . The split ring asymmetry only weakly affects v m . For δω = 0.01Γ, the curves representing γ are indistinguishable from those for δω = 0. For the relatively large δω = 0.1Γ, however, γ is increased for N 200. The cooperative response and linewidth narrowing sensitively depends on u. For larger u (e.g., u = 3/2λ), γ becomes insensitive to N , indicating the limit of independent scattering of isolated metamolecules and a diminished role of cooperative effects.
As with an isolated ASR, an asymmetry δω = 0 generates an effective two-step coupling between collective electric and magnetic modes: one leg of the transition is provided by the coupling of the incident field to collective electric dipole excitations, while the other is provided by the coupling introduced by the asymmetry. We illustrate this in Fig. 2(b), showing the relative population O m (b f ) of the resonantly driven collective magnetic mode v m , where b f is the state induced by the uniform driving resonant on v m . We find that for γ ≪ Γ and δω γ, one can excite a state in which more than 98% of the energy is in the target mode v m . For δω ≪ γ, any excitation that ends up in v m is radiated away before it can accumulate; the array behaves as a collection of radiating electric dipoles. For larger δω, the population of v m decreases since the increased strength of the coupling between the symmetric and anti-symmetric modes begins to excite also other modes with nearby resonance frequencies. Although the density of modes which may be excited increases linearly with N , the corresponding reduction of γ means that a smaller δω is needed to excite the target mode, and there is a range of asymmetries for which v m is excited. The narrowing in γ combined with the near exclusive excitation of this mode implies that for larger arrays the radiation from the sheet is suppressed and hence the transmission enhanced as in Ref. [3]. In Fig. 2(c) we compare the experimentally observed transmission resonance [3] to our numerics for Γ O ≃ 0.14Γ that results in the expected saturation of quality factor with N . We use the experimental spacing u ≃ 0.28λ, when δω ≃ 0.3Γ is chosen consistently with the measurements in Ref. [13]. The excellent agreement of our simplified model that only includes dipole radiation contributions from each meta-atom can be understood by a notably weaker quadrupole than dipolar radiation field from an ASR [14]. The result also confirms the importance of the uniform magnetic mode v m on the observed transmission resonance. The observed saturation is due to a combination of a fixed δω, which in larger arrays leads to the population of several other modes in addition to v m , and ohmic losses in the resonators which set an ultimate limit to the narrowing of γ. Since in the experiment δω γ, the resonance narrowing could be improved by using a smaller δω, while a sufficient transfer of excitations to v m can still be achieved; Fig. 2(b).
In conclusion, we analyzed the recent observations of transmission spectra in a metamaterial array of ASRs. We showed that the system can exhibit a strong cooperative response in the case of sufficiently closely-spaced resonators. Moreover, we demonstrated how an asymmetry in the split rings leads to excitation of collective magnetic modes by a field which does not couple directly to ASR magnetic moments. The excitation of this uniform phase-coherent mode results in cooperative response exhibiting a dramatic resonance linewidth narrowing, explaining the experimental findings [3].
We gratefully acknowledge N. Papasimakis, V. Fedotov, and N. Zheludev for discussions and providing the measurement data for the quality factor. This work was financially supported by the EPSRC.