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

12 CHAPTER 1. THEORYfrom which, separating the real and imaginary part of the complex wave number, weobtain(E(r,t) = E exp − ωK )c ˆk·r exp[i(ωt−k·r)](= E exp − α 2 ˆk·r)exp[i(ωt−k·r)](1.9)The absorption coefficientα = 2Kωc= 4πKλ(1.10)is defined as the fraction of power absorbed per unit length, as expressed by the Beer lawI(z +d) = I(z) e −αd , where I(z) and I(z +d) are the intensities (optical power per unitarea) at positions z and z + d. Then, we can see that the refractive index N and theextinction coefficient K are responsible, respectively, for the propagation of light and forthe exponential decrease of the EM fields amplitudes.For isotropic non-magnetic media, the complex refractive index and the dielectricconstant are related by the simple expressionor equivalently for the real and imaginary partsε = ñ 2 (1.11)ε 1 = N 2 −K 2ε 2 = 2NK(1.12a)(1.12b)andN =√ √ε21 +ε 2 2 +ε 12(1.12c)K =√ √ε21 +ε 2 2 −ε 12(1.12d)1.1.1 Dipole oscillator modelAs seen before, the dielectric constant can be put in relation with the microscopic characteristicsof the dense medium. In this section, we will therefore shortly discuss the mainfeatures of the microscopic mechanisms that govern the light-matter interactions. In general,the functional form of ε is quite complex, as several kinds of polarizations can beinduced. When an analytical representation is required, the usual approach is thereforeto decompose and analyse the individual contributions, and then merge the results. Inthis respect, several models have been proposed, suitable to describe the specific propertiesof the samples. The dipole oscillator or Lorentz model follows from the classicaltheory of absorption and despite its simplicity it offers a good picture of the polarizationmechanisms.