Optical Characterization of a Spheroidal Nanoparticle on a Substrate
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this representation is that the strength and localization of the resonances, when given in terms of the spectral variable, are independent of the dielectric properties of the particle, but depend only on its shape and the dielectric properties of the substrate. With this procedure one is also able to include a larger number of multipoles, allowing the treatment of substrates with a larger contrast in the dielectric constant and particles closer to the substrate. In this work we consider more asymmetric oblate particle than in previous treatments [ 12] and we also study the changes in differentialreflectance spectra for particles made of different materials. FORMALISM We consider an oblate particle located on a substrate. The particle is generated by the rotation around one of the axes of an ellipse with lengths 2a and 2b, with a > b. The symmetry axis of the particle is perpendicular to the substrate, and its center is located at a distance d from the substrate which has a dielectric constant es. The particle has a dielectric function ep and is embedded in an ambient of dielectric constant Ca- We consider that the three media: particle, substrate and ambient are nonmagnetic. The system is excited by light with frequency to and a wavelength k, such that k>> a, b and d. Under this condition, a quasistatic approximation is valid to calculate the electromagnetic fields. In the linear approximation, the dipolar moment of the particle, in the presence of a substrate, linearly depends with the components of the applied external field, throughout the so called effective polarizability tensor oxeff.. Due to the symmetry of the system, Xeff. has only two independent components corresponding to the polarizabilities in the direction normal and perpendicular to the substrate. When the particle is far from the substrate the effective polarizability becomes the polarizability of the isolated particle. But, when the particle is close to the substrate the multipolar interactions induced by the substrate modifies the optical response of the system. The analysis of aq11 for the system described above was done as follows. First, the electric potential induced in the system at any point in space was calculated to all multipolar orders. To find the solution for the induced potential a spectral representation (SR) of the Bergman-Fuchs-Milton type [13] was developed [12]. By identifying the dipole momentp induced in the particle, the components of weffwere obtained. The behavior of the spectral function for different shapes and locations of the particles is analyzed. For a detailed description of the method see Ref. [12]. Within SR, we can write the components of a./in the following form:
7(1)
mv
4 ru(c(O)- nmS where v is the volume of the particle, m denotes the diagonal components of cqef 1 , and u(0o)=[1-aP m m 2 Im ())/a]-l is the spectral variable; Gs2 =(U Is) are the so-called spectral functions where U is an orthogonal matrix that satisfies the relation, yUmHtUm= nm"6.,,.
(2)
I1"
The matrix H H1' depends only on the geometri
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