## 5 Plasmonic and Dielectric Resonances: Summary  
We conclude the analysis of Mie coefficients by introducing a set of general rules regarding the pole conditions of all three type of resonances and summarizing their particular characteristics. After some extensive heuristic calculations using high order Pade expansion, a pattern can be seen for all three resonances. This general pattern can be found in Table 1, where formulas up to the first five (\(n=1,2,3,4,5\)) magnetic \(b_{n}\) coefficients, and the first four electric \(a_{n}\) coefficients for the dielectric case with parametric values can be found by combining the values found in Table 2.  
The above generalized conditions give the value of the complex permittivity value required for the system to resonate. These expressions can be readily used as rule-of-thumb design equations. The pole conditions of the electric resonances, found in Table 1, reveal a similar but not identical imaginary part; the \(a_{1}\) (dipole resonance) exhibits the same amplitude of the \(x^{3}\) term compared to the plasmonic case. This difference states that the behavior of the electric resonances in the plasmonic and dielectric region is close but not completely identical. Similarly, the \(b_{n}\) coefficients posses a different radiative damping term with respect to the electric ones both in amplitude and in the order of \(x\). Note that the \(c_{n}\) coefficients correspond to the first root of the \(n\)-th spherical Bessel function of the first kind (\(j_{n}(c_{n})=0\)). The dots in Table 2 denote that the higher order \(t_{n}\) values can be extracted in a similar way for resonances above \(n=4\).  
A small homogeneous sphere may exhibit a series of resonances that correspond to the maximization of each of the Mie coefficients. Figures 2 and 3 revealed the existence of electric and magnetic multipole resonances as a function of both permittivity and size parameter. One distinctive category of resonances are the plasmonic resonances; these resonances can occur in deeply subwavelength spheres exposing an electric multipole character [52]. Increasing either the size or the material contrast of the scatterer, a second type of resonances occur, i.e., the dielectric resonances, of either magnetic or electric multipole character. Note that the electric resonances are resonances due to the \(a_{n}\) Mie coefficient, siblings to the plasmonic cases, occurring however at a different permittivity range.  
Therefore an obvious question appears: do the electric multipole resonances exhibit any significant differences on their field distributions? One way to approach this question is by exposing the electric field distribution near and inside the sphere. In particular, the amplitude of the scattered field for the first two plasmonic (Figure 7a,b and dielectric Figure 7c,d) electric dipole and quadrupole resonances for a relatively large sphere (\(x=0.75\)) as can be seen in Figure 7. One observes that although the external (scattered) field is identical, the inner field distributions reveals difference in the origin of these resonances. The main differences lie in the fact that the plasmonic resonances exhibit a rather uniform internal field distribution, while their dielectric counterparts exhibit a spatial variation; dielectric resonances are effectively large, especially in the inner domain, due to the high permittivity contrast, while the same does not hold for the plasmonic case.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \(n\) & 1 & 2 & 3 & 4 & 5 \\ \(c_{n}\) & \(\pi\) & 4.4934 & 5.7634 & 6.9879 & 8.1428 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Values of uses parameters for the \(b_{n}\) and \(a_{n}\) pole resonances.