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G$_{µν}$ ( r , r $^{′}$) = G 0 $_{µν}$( r , r $^{′}$) + G s $_{µν}$( r , r $^{′}$), where the free-space Green dyadic is 

with 

satisfying a scalar equation 

Consider first the free space part. Its near field expression can be obtained in the long wave approximation, i.e. by expanding in kr ≪ 1. In the first order, 

In the far field limit, i.e., kr ≫ 1 and kr ′ ≪ 1, the free- space part can be expanded via Bessel functions, 

which are approximated as 

yielding 

After differentiation, the far field asymptotics takes the form 

where we introduced χ µ $_{lm}$( r ) = ∇ $_{µ}$[ r $^{l}$Y$_{lm}$ (ˆ r ) and ψ µ $_{lm}$( r ) = ∇ $_{µ}$[ r − l − $^{1}$Y$_{lm}$ (ˆ r )]. Now turn to the scattered part of the Green dyadic 

derived from solution of Mie problem for electromagnetic wave scattered on single sphere, 

where the first and second terms are electric and mag- netic contributions and a$_{l}$ and b$_{l}$ are the Mie coefficients. In the long wave approximation, kR ≪ 1, the magnetic 

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contribution in Eq.(54) can be neglected as b$_{l}$ ≪ 1. 20–23 The Mie coefficient a$_{l}$ has a form 

where ρ$_{i}$ = k$_{i}$R , k$_{i}$ = ω c √ ϵ$_{i}$ , and i = ( ϵ, ϵ$_{0}$ ). For kR ≪ 1, it becomes 

where 

is NP multipolar polarizability that accounts for plasmon radiative decay, and 

is the standard NP polarizability. The function N$_{lm}$ ( r ) is given by 

where L = − i ( r × ∇ ) is angular momentum operator. Using the following identity, 

prime standing for derivative, and expanding h (1) l ( kr ) = j$_{l}$ ( kr ) + in$_{l}$ ( kr ) in kr as 

we obtain 

where 

Thus, for kr ≪ 1 and kr ′ ≪ 1, the scattered part of the Green dyadic has the form 

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