content_image/432708.jpg content_image/432706.jpg content_image/432711.jpg content_image/432714.jpg content_image/432709.jpg content_image/432710.jpg content_image/432712.jpg content_image/432713.jpg content_image/432715.jpg content_image/432716.jpg content_image/432707.jpg
and for parallel orientation ( e$_{j}$ · ˆr$_{j}$ = 0), we similarly get 

<img src='content_image/432708.jpg'>

The decay matrix , Γ$_{jk}$ = − 2ImΣ$_{jk}$, can be decomposed into radiative and nonradiative terms, Γ$_{jk}$ = Γ r jk + Γ nr $_{jk}$, as follows 

<img src='content_image/432706.jpg'>

The diagonal elements Γ jj and Γ jj describe, respectively, plasmon-enhanced radiative decay rate of an isolated dipole near a NP, and nonradiative transfer of its en- ergy to electronic excitations in metal. The non-diagonal elements of Γ r jk describe plasmon-enhanced radiative cou- pling that generalizes the Dicke mechanism responsible for cooperative emission by incorporating local field en- hancement into near field radiative coupling. On the other hand, the non-diagonal terms in Γ nr jk describe non- radiative energy transfer between dipoles mediated by NP plasmons. The latter coupling is absent in the pho- tonic Dicke effect but it plays important role in the plas- monic Dicke effect, as we will see below. 

Since the numerical calculations below are carried for normal dipoles orientations, we provide here the corre- sponding expressions for self-energy matrix Σ$_{jk}$. The de- cay matrix has the form 

<img src='content_image/432711.jpg'>

where P$_{l}$ is Legendre polynomial and γ$_{jk}$ is the angle between positions of dipoles j and k measured from NP center (here cos γ$_{jk}$ = e$_{j}$ · e$_{k}$ ). The real part of self-energy matrix has the form 

<img src='content_image/432714.jpg'>

where the second term describes NP-induced interac- tions. The latter in turn consists of two terms, first coming from plasmon-enhanced radiative coupling and second coming from non-radiative coupling. 

Note that both NP-induced terms are weaker than their counterparts in Γ$_{jk}$ while having same symmetry and therefore is not expected to significantly alter the eigenstates. On the other hand, the dipole-dipole inter- action term, Eq. (10), has different symmetry and can become large for two dipoles in a close proximity to each other. The effect of interactions on cooperative emission 

is studied in Sec. IV. 

In this section, we derive general expression for total energy radiated by an ensemble of dipoles near a metal nanostructure. The radiated energy in the unit frequency interval is obtained by integrating the far field ( r → ∞ ) spectral intensity over solid angle 29 

and then averaging the result over initial random phases of individual dipoles, ϕ$_{j}$ . The electric field E ( r , ω ) is given by Eq. (7), where v$_{j}$ is the solution of Eq. (4). Then the energy density takes the form 

<img src='content_image/432709.jpg'>

with G ( r , r$_{j}$ , ω ) = G $^{0}$( r , r$_{j}$ , ω ) + G $^{s}$( r , r$_{j}$ , ω ) replaced by its far field asymptotics (see Appendix), 

<img src='content_image/432710.jpg'>

where first two terms come from the free space part, G $^{0}$, and the last term comes from the scattered part, G $^{s}$, of the Green diadic. The angular integral in Eq. (25) can be performed using the relations Eq. (19). The free space contribution yields 

<img src='content_image/432712.jpg'>

The other integrals in the product Eq. (26) are evaluated using the relations 

<img src='content_image/432713.jpg'>

and 

<img src='content_image/432715.jpg'>

yielding 

<img src='content_image/432716.jpg'>

## III. RADIATED ENERGY 

<img src='content_image/432707.jpg'>

