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

14 CHAPTER 1. THEORY≈ ω 0 like ε L 2. Indeed, if ε L 2 were much smaller than ε L 1, it would follow N ≈ √ ε L 1 andK ≈ ε L 2/2 √ ε L 1 . This correspondence, however, is only valid for gaseous phases, wherethe density of atoms is very low, while for solids it is only approximate because the absorptionsare very strong; nevertheless, the absorption peak is generally observed at afrequency very close to ω 0 .Equation (1.13) can be generalized to include the contributions of several concomitantresonances occurring in the same medium at different frequencies, writing:ε(ω) = 1+ ∑ jA jω 2 0j −ω2 +iΓ j ω(1.14)where j is the index numbering the (supposed discrete) oscillators of the medium. Theoverall frequency dependence of the dielectric constants according to eq. (1.14) is schematicallyshown in fig. 1.4. 1 200Drude1Cauchyω 0,1ω 0,2ω 0,3FrequencyFigure 1.4: Schematic diagram of the frequency dependence of the dielectric function ofan hypothetical solid with three resonant frequencies (ω 0,i ). The Drude contribution issketched in dashed lines at the lowest frequencies; an example of region of validity of theCauchy parametrization is highlighted at the center of the figure.This classical picture of the interaction between light and matter offers a simple physicalinterpretation of the dielectric constant. Looking at fig. 1.4, we can identify partswhere ε 1 is slowly varying and ε 2 is almost null, and regions where ε 1 rapidly changesand ε 2 has a maximum. Moreover, every time a resonance is crossed, moving from low tohigh frequencies, the average value of ε 1 decreases, approaching the value of 1 beyond thelast oscillator. This can be understood in terms of polarizability of the material: consideringa single mechanism of polarization, the atoms can follow the external field, i.e. themedium can be polarized, only up to the resonance frequency, where the polarization ismaximum; at higher frequencies the field varies too fast and the average polarization reducesto zero. At very high frequencies, no polarization is more possible, and the materialbecomes completely transparent.