## 3 Electrostatic Scattering by a Sphere  
A standard way to analyze subwavelength plasmonic resonances is by implementing an electrostatic scattering picture for the proposed problem. The analysis can be seen as the perturbation that a spherical object causes on a constant electric field. The formulated model can be expressed as a boundary value problem for the following quantities: the external potential, \(\Phi_{\text{inc}}=-E_{0}r\cos\theta\), the perturbed (scattered) potential \(\Phi_{\text{sca}}=\frac{B_{0}}{r^{2}}\cos\theta\) and the internal potential \(\Phi_{\text{in}}=A_{1}r\cos\theta\). The boundary problem requires the continuity of the tangential field and normal flux density on the boundary, i.e., \(\Phi_{\text{out}}(r,\theta)=\Phi_{\text{in}}(r,\theta)\), at \(r=r_{1}\) and \(\varepsilon_{\text{host}}\frac{\partial}{\partial r}\Phi_{\text{out}}(r, \theta)=\varepsilon_{1}\frac{\partial}{\partial r}\Phi_{\text{in}}(r,\theta)\) at \(r=r_{1}\). The resulted scattering amplitudes, \(B_{0}\) and \(A_{0}\), read  
\[B_{0}=\frac{\varepsilon_{1}-\varepsilon_{\text{host}}}{\varepsilon_{1}+2 \varepsilon_{\text{host}}}E_{0}r_{1}^{3} \tag{5}\]  
and  
\[A_{1}=-\frac{3\varepsilon_{\text{host}}}{\varepsilon_{1}+2\varepsilon_{\text {host}}}E_{0} \tag{6}\]  
The proportionality constant of the dipolar scattering amplitude, \(B_{0}\), is commonly recognized as the normalized polarizability of the inclusion [64], viz.,  
\[\alpha=3\frac{\varepsilon_{1}-\varepsilon_{\text{host}}}{\varepsilon_{1}+2 \varepsilon_{\text{host}}} \tag{7}\]  
Its form reveals an interesting property, the field enhancement around the value \(\varepsilon_{1}=-2\varepsilon_{\text{host}}\), known as polarization enhancement of Frohlich frequency [65; 66]. A very informative section can be found in [PERSON] and Huffman (in [19], Section 12.1). Actually, the general rule describing the resonant condition of each \(n\)-th multipole term, i.e., \(\varepsilon_{1}=-2\varepsilon_{\text{host}}\) (dipole), \(\varepsilon_{1}=-\frac{3}{2}\varepsilon_{\text{host}}\) (quadrupole), etc., reads  
\[\varepsilon_{1}=-\frac{n+1}{n}\varepsilon_{\text{host}} \tag{8}\]  
Figure 4 illustrates the scattering efficiency of an extremely small sphere (\(x=0.01\)), where two main plasmonic resonances are visible. The position of these resonances is described by the condition found in Equation (8). Figure 4: The total scattering efficiency of a deeply subwavelength sphere (\(x=0.01\)) in the plasmonic region. One main electric dipole resonance is observed at \(-2\), while a second ultra-sharp resonance at \(-1.5\). The field distribution for the dipolar resonance is characterized by a dipole-like external field, with a constant field at the inside domain of the sphere. This resonance can be seen in Figure 2 (inset picture). Assuming an ideal lossless Drude-like material, i.e., \(\varepsilon(\omega)=\varepsilon_{\infty}-\frac{\omega_{p}^{2}}{\omega^{2}}\), the resonant condition of a solid sphere is met when  
\[\omega_{-}^{2}=\frac{\omega_{p}^{2}}{\varepsilon_{\infty}+2\varepsilon_{\rm host}} \tag{9}\]  
often also called the symmetric (bonding) resonance of a sphere [1]. A realistic Drude-like material includes also an imaginary damping term, i.e., \(\varepsilon_{1}(\omega)=\varepsilon_{\infty}-\frac{\omega_{p}^{2}}{\omega^{2 }+\eta\omega}\)[50; 51]. This model gives the free-electron description inside metals, such as gold or silver [67], and it is extensively used for the analysis of plasmonic resonances. For this type of materials, negative values for the real part of the permittivity occur. Hence, the above plasmonic resonance can be captured by the simple electrostatic model presented above. A very important conclusion is that these resonances can in principle occur even for deeply subwavelength sizes of the scatterer. This is the reason why plasmonic resonances are particularly attractive for the localization of energy even far below the diffraction limit [50].