3 II. PLASMONIC COUPLING OF RADIATING DIPOLES ¨ d$_{j}$ + ω 2 $_{0}$d$_{j}$ = q m E ( r$_{j}$ , t ) · e$_{j}$ (4) j ( t ) = q ∑ j ˙ d$_{j}$ ( t ) e$_{j}$ δ ( r − r$_{j}$ ) , (3) ϵ ( r , ω ) ω 2 c 2 E ( r , ω ) − ∇ × ∇ × E ( r , ω ) = − 4 πiω c 2 j ( r , ω ) , (2) agreement of measured distance dependences with single- dipole models. 20–23 In the case of molecular layer, the am- biguities caused by uncertain molecules number and their separation from the metal surface prevent, in general, de- termination of system characteristics from fluorescence variations. However, in cooperative regime, the ambigu- ity related to molecules number is removed, and fluores- cence intensity is essentially determined by Eq. (1) with distance-averaged single-molecule quantum efficiency. plasmon pole, the relative strength of dipole-dipole in- teractions is effectively reduced as compared to purely photonic case. Second, there are additional corrections to emitters’ frequencies, one originating from plasmon- enhanced radiative coupling and another from nonradia- tive coupling. While neither of those corrections diverges as dipoles approach each other, the latter becomes very large as dipoles approach the metal surface due to contri- bution of high- l plasmons. Therefore, the actual system eigenstates are determined by both the interactions and the energy exchange, and their frequencies and decays rates must be found simultaneously. It is precisely the goal of this paper to calculate the full spectrum of inter- acting emitters near a metal NP. The paper is organized as follows. In Sec. II, the derivation of plasmonic coupling for an ensemble of dipoles distributed near metal NP is given. In Sec. III, the general expression for radiated energy is obtained. Our numerical results and discussion are presented in Sec. IV. Conclusions and some technical details are pro- vided, respectively, in Sec. V and in the Appendix. Specifically, we consider a common situation when emitters, e.g., fluorescent molecules or quantum dots, are attached to NP surface via flexible linkers. Typically, flu- orophores bound to linker molecules have certain orienta- tion of their dipole moments with respect to NP surface and, due to repulsive interactions, their angular positions are ordered rather then random. 16 Therefore, we assume here that angular positions of emitters coincide with the sites of spherical lattice, such as fullerenes. Specifically, we perform our numerical simulations for C20, C60, and C80 configurations for respective number N of dipoles; we also study the effect of deviations from ideal lattice. We find that not too close to metal surface, the system eigenstates fall into three groups, each dominated by a particular coupling mechanism: three superradiant states dominated by plasmon-enhanced radiative coupling, one state dominated by direct dipole-dipole interactions, and the rest dominated by non-radiative coupling via NP. Importantly, superradiant states are not significantly af- fected by dipole-dipole interactions whose main effect is a large frequency shift of subradiant state with the smallest decay rate. We consider a system of N emiters, such as fluorescing molecules or quantum dots, with dipole moments d$_{j}$ = d$_{j}$ e$_{j}$ , where d$_{j}$ and e$_{j}$ are magnitude and orientation, re- spectively, located at positions r$_{j}$ near a metal NP with radius R . Throughout the paper, we assume that charac- teristic size of the system (NP+dipoles) is much smaller than the radiation wavelength, | r$_{j}$ − r$_{k}$ | ≡ | r$_{jk}$ | ≪ λ$_{0}$ . We also assume that emission events by individual molecules are uncorrelated, i.e., after excitation each molecule re- laxes through its own internal nonradiative transitions before emitting a photon. Then the ensemble emission can be described within classical approach by consid- ering dipoles as identical Lorentz oscillators with ran- dom initial phases driven by common electric field, i.e., one created by all dipoles in the presence of metal NP. The frequency-dependent electric field, E ( r , ω ), satisfies Maxwell’s equation where dielectric permittivity ϵ ( r , ω ) is that of the metal inside NP, ϵ ( ω ) for r < R and that of outside dielectric ϵ$_{0}$ for r > R . Here j ( r , ω ) = − i ∞ ∫ 0 e $^{iωt}$j ( t ) dt , is the Laplace transform of dipole current We also address the effect of individual emitters’ in- ternal nonradiative processes on the ensemble quantum efficiency. Internal relaxation is known to inhibit the photonic Dicke effect if the corresponding decay rate, Γ nr 0 , is sufficiently high, i.e., relaxation time is shorter than the lifetime of superradiant state. 3 However, near metal nanostructure, the plasmon-enhanced radiative de- cay rate, Γ $^{r}$, is significantly larger than Γnr 0 , so that plasmon-mediated superradiance is less sensitive to quan- tum yield of individual emitters. Specifically, we perform numerical calculations for high-yield and low-yield emit- ters to show that Eq. (1) holds for both types. More precisely, in cooperative regime, it holds almost exactly for high-yield emitters while for low-yield emitters Q$_{ens}$ is somewhat larger than 3 Q due to the effective N -fold suppression of Γ nr 0 by superradiant states’ decay rate. where dipole displacements d$_{j}$ ( t ) are driven by the com- mon electric field at dipoles’ positions with the initial conditions (at t = 0): d$_{j}$ = d$_{0}$ e$_{j}$ sin ϕ$_{j}$ , ˙ d$_{j}$ = ω$_{0}$d$_{0}$ e$_{j}$ cos ϕ$_{j}$ , and E = 0 (dot stands for time- derivative). Hereafter, ω$_{0}$ , q , m , and ϕ$_{j}$ are oscillators’ An obvious application of plasmon-mediated coopera- tive emission is related to fluorescence of a large but un- certain number of molecules at some average distances from metal nanostructure. For single-molecule case, flu- orescence intensity variation with distance was proposed to serve as nanoscopic ruler, 27 owing to the excellent