## 2 Electrodynamic Scattering: Mie Coefficients  
As discussed in the introductory section, the scattering problem by a sphere subject to a plane wave has been rigorously solved by following the aforementioned assumptions. The key concepts are the Lorenz-Mie (or simply Mie) coefficients that rigorously quantify the material and size contributions to the overall scattering behavior. These coefficients are described by a set of fractional functions consisting of spherical Bessel, Hankel, and Riccati-Bessel and Riccati-Hankel functions, viz.,  
\[a_{n}=\frac{m^{2}j_{n}(mx)\left[xj_{n}(x)\right]^{\prime}-\mu_{1}j_{n}(x)\left[ mxj_{n}(mx)\right]^{\prime}}{m^{2}j_{n}(mx)\left[xh_{n}^{(1)}(x)\right]^{ \prime}-\mu_{1}h_{n}^{(1)}(x)\left[mxj_{n}(mx)\right]^{\prime}} \tag{1}\]  
\[b_{n}=\frac{\mu_{1}j_{n}(mx)\left[xj_{n}(x)\right]^{\prime}-j_{n}(x)\left[mxj_ {n}(mx)\right]^{\prime}}{\mu_{1}j_{n}(mx)\left[xh_{n}^{(1)}(x)\right]^{\prime }-h_{n}^{(1)}(x)\left[mxj_{n}(mx)\right]^{\prime}} \tag{2}\]  
where \(a_{n}\) and \(b_{n}\) denote the electric and magnetic coefficients, respectively [19]. The size parameter, \(x=ka\), is a function of the sphere's radius \(a\) and host medium wavenumber; \(\varepsilon_{1}\) and \(\mu_{1}\) are the sphere's material parameters. Finally, \(m=\frac{\sqrt{\varepsilon_{1}\mu_{1}}}{\sqrt{k_{\mathrm{host}}}H_{\mathrm{host }}}\) is the contrast parameter. Similar expressions are also available for the case of core-shell structures [54], while iterative relations of the Mie coefficients can be found for the case of \(n\)-layered structures [55]. Likewise, cases of inhomogeneous [56; 57] or anisotropic inclusions [58; 59] are available, being, however, beyond the scope of this review. The Mie coefficients can also describe the energy balance between the different mechanisms on a sphere: scattering and absorption. A common way for expressing these energies is by looking the far-field characteristics, i.e., the scattering and extinction (scattering + absorption) efficiencies. These efficiencies can be expressed as a combination of the Mie coefficients, i.e.,  
\[Q_{\mathrm{sca}}=\frac{2}{x^{2}}\sum_{n=1}^{\infty}\left(2n+1\right)\left(|a_{ n}|^{2}+|b_{n}|^{2}\right) \tag{3}\]  
and  
\[Q_{\mathrm{ext}}=\frac{2}{x^{2}}\sum_{n=1}^{\infty}\left(2n+1\right)\Re\{a_{n} +b_{n}\} \tag{4}\]  
as can be found in text-books [9; 18; 19]. Figure 2 depicts the scattering efficiency spectrum as a function of both \(\varepsilon_{1}\) and \(x\). We can observe several lines of radiation enhancement in both plasmonic (\(\varepsilon_{1}<0\)) and dielectric (\(\varepsilon_{1}>0\)) region, respectively. In the computation of Figure 2 the infinite sum of Equation (3) has been truncated in accordance to the [PERSON]'s criterion, that correlates the sufficient number of modes with the size parameter of the sphere [60]. The plasmonic region resonances correspond to the enhancement of the electric Mie coefficients, \(a_{n}\), while the dielectric region exhibits both magnetic \(b_{n}\) and electric \(a_{n}\) resonances. This fact is depicted in Figure 3, where the scattering efficiency and the first three magnetic and electric Mie coefficients are visible, for a given size parameter (\(x=0.75\)). The inset figures depict the charge and surface current distribution for the \(a_{1},a_{2},a_{3}\) and \(b_{1},b_{2},b_{3}\) coefficients, respectively. It is evident that the magnetic coefficients exhibit pronounced circulating charge distributions, resembling in such manner the magnetic dipole, quadrupole, and octupole configurations. Actually, this is particularly interesting phenomenon that enables the magnetic response of a dielectric sphere the optical and near-infrared regime, where magnetic materials are not traditionally available [61; 62]. Likewise, for the electric coefficients the surface charge accumulates in a dipolar, quadrupolar, and octupolar manner. Note that both plasmonic and dielectric electric resonances exhibit the same surface strength and charge distribution.