## 6 Directional Scattering Characteristics: Forward and Backscattering Conditions
### Directive Scattering Conditions for Dielectric Resonances  
The asymmetry factor gives a complete picture about the underlying physics for the directional scattering characteristics. Since \(a_{n}\) and \(b_{n}\) are complex-valued in general, it is important to explore for which values of \(a_{n}\) and \(b_{n}\) the described scattering conditions might hold. A more practical form of the above conditions can be extracted if we re-write the forward and backward scattering efficiencies in the following manner [18; 21],  
\[Q_{\rm B}=\frac{1}{x^{2}}\left|\sum_{n}\left(2n+1\right)\left(-1\right)^{n} \left(a_{n}-b_{n}\right)\right|^{2} \tag{31}\]  
and  
\[Q_{\rm F}=\frac{1}{x^{2}}\left|\sum_{n}\left(2n+1\right)\left(a_{n}+b_{n} \right)\right|^{2} \tag{32}\]  
A close observation of the above equations reveals that \(Q_{\rm B}\) and \(Q_{\rm F}\) can be minimized for certain \(a_{n}\) and \(b_{n}\) combinations. For the case of the first dipole terms, i.e., \(a_{1}\) and \(b_{1}\), the following conditions can be satisfied  
\[K_{\rm F}=a_{1}-b_{1}=0\text{, for maximum forward scattering} \tag{33}\]  
and  
\[K_{\rm B}=a_{1}+b_{1}=0\text{, for maximum backward scattering} \tag{34}\]  
Figure 9: The normalized asymmetry factor for the case of a lossless sphere of size \(x=0.5\). The maximum peak (forward scattering) occurs at \(\epsilon_{\rm F}^{\rm Mie}=30.06\) and the minimum valley (maximum backscattering) at \(\epsilon_{\rm B}^{\rm Mie}=49.02\). commonly recognized as the first and second _Kerker_ conditions [103]. However, these are not the only available conditions, e.g., a minimum backscattering can be also obtained when \(3a_{1}-5a_{2}=0\). Here, we will concentrate only on the case where the first electric and magnetic dipole interfere accordingly, producing maximum forward or backward scattering for an dielectric sphere. To do so we need to utilize the aforementioned Pade-Mie heuristic method and apply it to both conditions \(K_{\text{F}}\) and \(K_{\text{B}}\). Indeed, after some calculations (that are here omitted), the Kerker conditions are satisfied for  
\[\varepsilon_{\text{F}}\approx\left(\frac{2.7437}{x}\right)^{2} \tag{35}\]  
and  
\[\varepsilon_{\text{B}}\approx-8.766+\left(\frac{3.7263}{x}\right)^{2} \tag{36}\]  
These expressions were extracted using a \([5/25]\) Pade expansion of Equations (33) and (34), respectively. As an illustrative example, we evaluated the required permittivities for maximum forward and backscattering of an \(x=0.5\) dielectric (lossless) sphere. In this case, Equations (35) and (36) predict resonant maxima at \(\varepsilon_{\text{F}}=30.11\) and \(\varepsilon_{\text{B}}=46.78\), respectively. A comparison with the the actual values, i.e., \(\varepsilon_{\text{F}}^{\text{Mie}}=30.06\) and \(\varepsilon_{\text{B}}^{\text{Mie}}=49.02\) (see Figure 9), gives a relative error of 0.2% and 4.5%, respectively. An easy mnemonic rule can be found for the Equation (35) since Euler number is \(e\approx 2.718281\). Therefore, Equation (35) recasts to the simpler approximative form  
\[\varepsilon_{\text{F}}\approx\left(\frac{e}{x}\right)^{2} \tag{37}\]  
Finally, we present the radiation patterns and surface field distributions for both cases, as illustrated in Figure 10. The surface distribution agrees perfectly with the physical picture; the forward scattering pattern is created by two, spatially aligned dipoles, and the maximum scattering is a result of two perpendicular ones. Figure 10: The radiation and surface field patterns of maximum forward scattering (left set of images) and maximum backscattering (right set of images). **Left**: The central subfigure illustrates the far field radiation pattern (in dB) of the maximum forward scattering for both perpendicular planes, \(E\) and \(H\), spanning between 0 degrees (backward scattering direction) to 180 degrees of angle (forward scattering direction). The inset subfigures depict: **(a)** 3D far-field pattern; **(b)** the surface electric field; and **(c)** the tangential magnetic field.