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

16 CHAPTER 1. THEORYε 1 (λ) = N 2 (λ) = A+ ∑ jB j λ 2λ 2 −λ 2 , ε 2 (λ) = 0 (1.19)0where A and B j are numerical parameters.The Cauchy model is a further approximation of the Sellmeier model, obtained fromthe series expansion of (1.18):Kramers-Kronig relationshipsN(λ) = A+ B λ 2 + C +··· , K(λ) = 0 (1.20)λ4 We conclude this section with an important consideration on the relation between thereal and the imaginary parts of the complex dielectric function ε. Discussing the Lorentzmodel, we have seen qualitatively that ε 1 and ε 2 are not independent parameters but arerelated to each other. This is indeed a true property of ε, which follows from the principleof causality. In fact, from the definition (1.4) we can write the polarization of the mediumas P = ε 0 (ε−1)E, which explicitly shows that ε(ω)−1 is the response function for theapplication of electric fields. Therefore, we can apply the laws of causality and derive thegeneral Kramers-Kronig (KK) relationships between the real and the imaginary parts ofε(ω) (but also of the complex refractive index ñ(ω)):ε 1 (ω) = 1+ 1 π P ∫ ∞ε 2 (ω) = − 1 π P ∫ ∞−∞−∞ε 2 (ω ′ )ω ′ −ω dω′ε 1 (ω ′ )−1ω ′ −ω dω′(1.21a)(1.21b)where P indicates the principal value of the integral.The KK relations can be very useful because they allow, for example, to calculate thedispersion of the dielectric constant and of the refractive index by measuring the frequencydependence of only the optical absorption. They also provide a tool for checking thephysical consistency of the dielectric constant approximations. For example, the Lorentzand Drude expressions for ε(ω) satisfy the KK relations, contrarily to the Sellmeier andCauchy parametrizations, where ε 2 (ω) = 0 and K(λ) = 0 are not physically reasonable.1.1.2 Light refractionWhen EM waves travel in homogeneous media they propagate according to (1.8), maintainingconstant direction, frequency and wavelength. Instead, when light crosses differentmaterials, it experiences a discontinuity of the refractive index, which strongly affects itspropagation. As a consequence of this two main effects are observed (refraction of light):the transmitted beam propagates along a different direction and with a different wavelength,and a reflected beam is generated. This is illustrated in the following example.Let us consider a flat interface between two media a and b with complex refractiveindices ñ a and ñ b ; an EM wave is approaching from a at an angle θ i with respect to thesurface normal, while the reflected and the transmitted waves leave at angles θ r and θ t(fig. 1.5). (In the following the subscripts i, r and t will be used for the incident, thereflected and the transmitted beams, respectively). All the three beams are contained inthe same plane, called the plane of incidence.