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

82 CHAPTER 5. MODELLING AND ANALYSIS OF THE OPT. PROP.Re []2.22.12.01.91.81.71.61.5substratehostRe []0-20-40-60-80Au NPAu bulk12108642Im []1.4200400600800 [nm]100012001400a. b.-100200400600800 [nm]1000120001400Figure 5.6: Panel a: real parts of the dielectric constants ε s for the substrate (red line) andε h for the host (black line) as employed in the calculations; the corresponding imaginaryparts are null at all wavelengths. Panel b: real and imaginary parts of the dielectricconstant ε m for the metallic inclusions (continuous lines), compared to the correspondingvalues for bulk gold (dashed lines).α γ =∫ 10P(L γ ) α MLWAγ dL γ (5.1)where P(η) can be any arbitrary distribution and the mean values for L γ are chosenaccording to (1.43).The system is excited by an external electric field E ex (r,t) = E ex e i(ωt−q·r) , withfrequency ω and wave vector q; in the quasi-static limit the electric field inside eachinclusion is uniform, thus q ≫ 1/a i . We suppose the incident field to be parallel to oneof the principal axes ξ of the ellipsoids (ξ can be either x, y or z), E ex = E exˆξ. Omittingthe time factors e iωt , the dipolar moment p i induced inside the i-th inclusion, with centerlocated at r i , is proportional to the local electric field E loc,i acting at r i (see (1.41))p i = ε 0 α⊗E loc,i (5.2)The local field E loc,i is the sum of several contributions, as sketched in fig. 5.7:E loc,i = E h,i +E others,i +E sub,i (5.3)E hp jE others, ip ihostE sub, ip jIp iIsubstrateFigure 5.7: Sketch of the dipolar contributions to the local electric field acting on eachparticle, and representation of the image dipoles induced inside the substrate.