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

84 CHAPTER 5. MODELLING AND ANALYSIS OF THE OPT. PROP.where S ξξ ≡ ∑ j sξξ ij e−iq·rij and r ij = r j −r i .We can now introduce the effective polarizability tensor A, which expresses the polarizabilityof the metallic inclusions accounting for their reciprocal dipolar interactions:A γγ =Then, the induced dipoles can be finally written asα γ1−ε 0 α γ Sγγ, γ = x,y,z (5.10)p = ε o ε h A⊗E ex (5.11)We can now formulate an effective medium approximation for the metallic inclusions.Given the isolated nature of the inclusions, we follow an approach similar to the Maxwell-Garnett EMA (§1.2), modified in order to account the ordered arrangement of the inclusionsand the presence of the substrate (cfr. also refs. [209–211]).We note that the induced dipoles (5.11) within the array have the same expression(1.41) for a single isolated particle, except for the polarizability tensor α replaced by A.Then, we can treat the inclusions as isolated, and write the total polarization density asP = np = ε 0 ε h nA⊗E ex , (5.12)where n = 1/d x d y d eff is the number of inclusions per unit volume, and d eff the thicknessof the host. The effective dielectric tensor ε eff for the inclusions can be evaluated fromthe definition of the electric displacement field DD = ε 0 ε eff ⊗E ex = ε 0 ε h E ex +P = ε 0 ε h E ex +ε 0 ε h nA⊗E ex (5.13)where we can see that ε eff is diagonal similar to A. Solving separately for the threeprincipal components we finally obtainε ξξeffε h= 1+nA ξξ , ξ = x,y,z (5.14)Using the dielectric tensor ε eff we can represent the inclusions and the host as an homogeneousand biaxially anisotropic film (fig. 5.5(c)), still preserving all the informationson the NPs shapes and interactions. The optical response of the whole system can thenbe easily calculated using the Fresnel coefficients (1.26) for the thin-film/substrate model.In this respect, however, the refractive indices for the s and p components of the electricfield are required, whereas the dielectric tensor ε eff is expressed in xyz coordinates. Weneed therefore to rewrite the components of ε eff in sp coordinates.In general the reflection of light from a biaxially anisotropic film is a quite complexphenomenon, especially when the principal axes of the dielectric tensor are not parallel tothe interfaces or the plane of incidence [172, 212]. In this case, however, we just need toreplicate the geometries applied in the experiments, i.e. plane of incidence either parallel(y axis) or perpendicular (x axis) the ripples.Therefore, we suppose the plane of incidence parallel to the xz plane, as in fig. 5.8,correspondingtotheperpendicular(⊥)geometry; thecalculationsinparallel(||)geometryare exactly the same, except swapping the axes x and y.We consider an EM plane wave travelling in the xz plane and approaching the surfaceat an angle of incidence θ. When the electric field is s-polarized, it always oscillates along