11 a$_{l}$ = ϵ$_{0}$j$_{l}$ ( ρ$_{0}$ )[ ρj$_{l}$ ( ρ )] ′ − ϵj$_{l}$ ( ρ )[ ρ$_{0}$j$_{l}$ ( ρ$_{0}$ )] ′ ϵ$_{0}$h$_{l}$ ( ρ$_{0}$ )[ ρj$_{l}$ ( ρ )] ′ ϵj$_{l}$ ( ρ )[ ρ$_{0}$h$_{l}$ ( ρ$_{0}$ )] ′ (55) a$_{l}$ = − is$_{l}$ ˜ α$_{l}$k 2 l $^{+1}$, s$_{l}$ = l + 1 l (2 l + 1)[(2 l − 1)!!] $_{2}$, (56) ˜ α$_{l}$ = α$_{l}$ 1 − is$_{l}$k 2 l $^{+1}$α$_{l}$ , (57) G 0 $_{µν}$( r − r $^{′}$) = ( δ$_{µν}$ − ∇ µ ∇ ′ ν k 2 ) g ( r − r $^{′}$) , (46) g ( r ) = e ikr 4 πr (47) ( △ + k $^{2}$) g ( r ) = − δ ( r ) . (48) G 0 $_{µν}$( r ) = 1 4 πk $^{2}$r3 [ 3 r$_{µ}$r$_{ν}$ r 2 − δ$_{µν}$ ] + ik 6 π δ$_{µν}$. (49) α$_{l}$ = R 2 l +1 l ( ϵ − ϵ$_{0}$ ) lϵ + ( l + 1) ϵ$_{0}$ , (58) N$_{lm}$ ( r ) = 1 k √ l ( l + 1)$^{∇}$ × [ h (1) l ( kr ) L Y$_{lm}$ (ˆ r ) ] , (59) e ik | r − r $^{′}$| 4 π | r − r $^{′}$| = ik ∑ lm j$_{l}$ ( kr $^{′}$) h$_{l}$ ( kr ) Y$_{lm}$ ( r ) Y ∗ $_{lm}$( r $^{′}$) , (50) j$_{l}$ ( kr $^{′}$) = ( kr $^{′}$) l (2 l + 1)!! , h$_{l}$ ( kr ) = ( − i ) l $_{+1}$e ikr kr , (51) ∇ × [ h (1) l ( kr ) L Y$_{lm}$ (ˆ r ) ] = i r k $^{2}$h (1) l ( kr ) Y$_{lm}$ (ˆ r ) + i ∇ [ krh (1) ′ l ( kr ) + h (1) l ( kr )] Y$_{lm}$ (ˆ r ) ] , (60) G 0 $_{µν}$( r , r $^{′}$) = ( δ$_{µν}$ − 1 k 2 ∇ µ ∇ ′ ν ) e ikr r [ 1 4 π − ikr ′ 3 ∑ m Y$_{1}$$_{m}$ (ˆ r ) Y ∗ 1 $_{m}$(ˆ r $^{′}$) ] . (52) h (1) l ( kr ) = ( kr ) l (2 l + 1)!! − i (2 l − 1)!! ( kr ) l +1 , (61) N$_{lm}$ ( kr ) = − 1 k √ s$_{l}$ (2 l + 1)$^{∇}$ [ ϕ$_{l}$ ( kr ) Y$_{lm}$ (ˆ r )] , (62) G 0 $_{µν}$( r , r $^{′}$) = e ikr 4 πr [ δ$_{µν}$ − 4 π 3 ∑ m ˆ r$_{µ}$ Y$_{1}$$_{m}$ (ˆ r ) χ ν ∗ 1 $_{m}$( r $^{′}$) ] , (53) ϕ$_{l}$ ( kr ) = 1 ( kr ) l +1 − is$_{l}$ ( kr ) $^{l}$. (63) G s $_{µν}$( r , r $^{′}$, k ) = ik ∑ lm [ a$_{l}$N µ $_{lm}$( r ) N ν $_{lm}$( r $^{′}$) + b$_{l}$M µ $_{lm}$( r ) M ν $_{lm}$( r $^{′}$) ] , (54) G s $_{µν}$( r , r $^{′}$, k ) ≈ ik ∑ lm [ a$_{l}$N µ $_{lm}$( kr ) N ν $_{lm}$( kr $^{′}$) ] (64) ≈ ∑ lm k 2 $^{l}$˜ α$_{l}$ 2 l + 1 ∇ µ [ ϕ$_{l}$ ( kr ) Y$_{lm}$ (ˆ r ) ] ∇ ′ ν [ ϕ$_{l}$ ( kr $^{′}$) Y ∗ $_{lm}$(ˆ r $^{′}$) ] . contribution in Eq.(54) can be neglected as b$_{l}$ ≪ 1. 20–23 The Mie coefficient a$_{l}$ has a form G$_{µν}$ ( r , r $^{′}$) = G 0 $_{µν}$( r , r $^{′}$) + G s $_{µν}$( r , r $^{′}$), where the free-space Green dyadic is with where ρ$_{i}$ = k$_{i}$R , k$_{i}$ = ω c √ ϵ$_{i}$ , and i = ( ϵ, ϵ$_{0}$ ). For kR ≪ 1, it becomes where 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, is NP multipolar polarizability that accounts for plasmon radiative decay, and is the standard NP polarizability. The function N$_{lm}$ ( r ) is given by 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 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 After differentiation, the far field asymptotics takes the form we obtain where 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, Thus, for kr ≪ 1 and kr ′ ≪ 1, the scattered part of the Green dyadic has the form 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