Fluorescence Emission Near a Metal Nanosphere

We conclude this first chapter with the most important case in molecular plasmonics an emitting dipole near a metallic nanosphere. For this problem the exact electrod5mamic solution can be obtained, as shown by Ruppin [41], Chew [42] and by Kim et al. [43]. 

The emitter is described as a classical dipole positioned at a distance d from the surface of a sphere with radius R we consider a parallel (i.e. tangential) or perpendicular (i.e. radial) orientation (see Fig. 1.19). The sphere has a complex dielectric constant Sp co) and it is in the vacuum. Generalization with a sphere embedded in an external medium with a real dielectric constant Sout can be directly obtained by considering the sphere with a relative dielectric constant ep(ty) /Sout- 

The normalized (with respect the vacuum radiative rate) total 

The expressions for the radiative rates, Eqs. (1.371,1.373) have been obtained considering the flux of the Poynting vector of the total field (the dipolar source plus the field scattered from the sphere) over a spherical surface with radius r oo. The expressions for the total decay rates have been obtained from Eq. (1.345). 

If the imaginary part of p( w) is not zero, e.g. if the sphere is metallic, the dissipation inside the sphere has to be considered. The non-radiative decay rate (see Eq. (1.343) has been obtained by Ruppin [41], computing the total electric field inside the sphere. These expressions are quite complicated and it is easier to obtain the non-radiative rate as a difference between the total and the radiative rates. It can be shown that [44] 

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