Morphology and plasmonic properties of self-organized arrays of ...

5.2. 2-DIMENSIONAL ARRAYS OF GOLD NANOPARTICLES 83• E h,i = ε h E ex e −iq h·r i: the exciting field propagating inside the host, in the absenceof any inclusion; it differs from E ex,i due to the polarization of the host;• E others,i : the sum of the dipolar fields generated at the position r i by all the otherinclusions;• E sub,i : the sum of the dipolar radiation generated by the inclusions and reflectedfrom the substrate.In first approximation E sub,i can be calculated within the image charge model, sothat the contribution of each dipole p j is equivalent to the field generated by a dipole p I jsituated at specular position with respect to the interface and given by [129, 148]⎡ ⎤p I j = ε −1 0 0s −ε h⎣ 0 −1 0⎦⊗p j = M⊗p j , (5.4)ε s +ε h0 0 1The dipolar fields generated by p j and p I j at r i are expressed according to (1.48), whichwe rewrite using the dipolar interaction tensors t ij and t I ij:E j (r i ) = t ij ⊗p jE I j(r i ) = t I ij ⊗p I jThen, equation (5.2) becomes⎛p i = ε 0 α⊗⎝E h e −iq h·r i+ ∑ t ij ⊗p j + ∑j≠i j⎞t I ij ⊗p I ⎠j (5.5)Introducingwe can rewrite (5.5) in the more compact form⎛s ij = (1−δ ij )t ij +t I ijM (5.6)p i = ε 0 α⊗⎝E h e −iq h·r i+ ∑ js ij ⊗p j⎞⎠ (5.7)The calculation of the sum in (5.7) is greatly simplified for inclusions disposed on arectangular lattice with principal axes aligned the lattice vectors. In fact, the dipoles p iare parallel to the exciting electric field, and the only non-zero component is thus the onelying along ξ, which reads⎛ ⎞p ξ i = ε 0α ξ ⎝E h e −iq h·r i+ ∑ s ξξij pξ ⎠j(5.8)jNeglecting boundary effects, all the induced dipoles have the same magnitude p, whilethe relative phases vary due to the spatial oscillations of the electric field; if we substitutep j ≡ p e −iq h·r jin (5.8) and solve for p, we obtainα ξp ξ = ε 01−ε 0 α ξ S ξξE h = ε 0 ε h1−ε 0 α ξ S ξξE ex (5.9)α ξ