## 6 Directional Scattering Characteristics: Forward and Backscattering Conditions
### The Asymmetry Factor  
An available figure of merit that quantifies the directivity of scattering process is the asymmetry factor. The asymmetry factor is the ratio of the weighted scattering cross section over the total scattering cross section [19] and can be calculated in terms of the electric and magnetic coefficients, \(a_{n}\) and \(b_{n}\), as  
\[g=Q_{\text{sca}}<\cos\theta>=\frac{4}{x^{2}}\left(\sum_{n}\frac{n(n+2)}{n+1} \Re\left(a_{n}a_{n+1}^{*}+b_{n}b_{n+1}^{*}\right)+\sum_{n}\frac{2n+1}{n(n+1)} \Re\left(a_{n}b_{n}^{*}\right)\right) \tag{28}\]  
where the Bohren-Huffman notation is used for the averaged cosine [19]. Figure 7: The amplitude of the scattered electric field distribution for a sphere of \(x=0.75\) at: (**a**) \(\epsilon_{1}\approx-3.5\); (**b**) \(\epsilon_{1}\approx-1.754\); (**c**) \(\epsilon_{1}\approx 32.9\); and (**d**) \(\epsilon_{1}\approx 57.93\). These resonances correspond to the first two plasmonic and dielectric resonances, i.e., (**a,c**) dipole and (**b,d**) quadrupole, respectively. Physically, this figure quantifies the direction of the scatterer power. For example, for values close to \(g/Q_{\rm sca}=1\), all the energy is forward directed/scatterer, while \(g/Q_{\rm sca}=-1\) is the backscattering case. The \(g=0\) case denotes a balance between the forward and backward direction, hence a dipole-like radiation. Actually, for the last case, the scatterer possesses a \"symmetric\" radiation pattern, i.e., the dipole pattern [19]. In Figure 8, we can see the normalized asymmetry factor \(g/Q_{\rm sca}\) is as a function of sphere's permittivity (assuming no losses) and the size parameter. As we can see, a very vibrant pattern reveals the existence of peaks and valleys of forward and backscattering in the whole spectrum. Let us now analyze the behavior of the asymmetry factor as can be seen in Equation (28). The multipole coefficients can be written in phasor notation, i.e., \(a_{n}=|a_{n}|\,e^{j\varphi_{n}}\) and \(b_{n}=|b_{n}|\,e^{j\varphi_{n}}\), respectively. First, we assume the case of a very small plasmonic scatterer, where the magnetic multipole terms are \(b_{n}\approx 0\). It is obvious that the only the electric multipole terms survive and the asymmetry factor becomes  
\[g=\frac{4}{x^{2}}\left(\sum_{n}\frac{n(n+2)}{n+1}\,|a_{n}|\,|a_{n+1}|\cos( \varphi_{n}-\varphi_{n+1})\right) \tag{29}\]  
having the following special cases : (a) maximum forward scattering if \(\varphi_{n}-\varphi_{n+1}=2k\pi,\;k=0,1,...\), (b) maximum backward scattering if \(\varphi_{n}-\varphi_{n+1}=(2k+1)\pi,\;k=0,1,...\), and (c) dipole-like scattering if \(\varphi_{n}-\varphi_{n+1}=l\frac{\pi}{2},\;l=1,2,...\). The same results hold also for the case of a magnetic sphere where \(a_{n}\approx 0\). Alternatively, a forward or backscattering pattern can happen by the constructive (destructive) interference of two consecutive (or any linear combination) of multipole resonances. However, the second sum of Equation (28) reveals that the asymmetry factor can be also controlled by the interaction of between both electric and magnetic multipole terms. For example, assume the case where only an electric and a magnetic dipole are enabled. In this case, the asymmetry factor reads  
\[g=\frac{6}{x^{2}}\,|a_{1}|\,|b_{1}|\cos(\varphi_{1}-\psi_{1}) \tag{30}\]  
Figure 8: The normalized asymmetry factor, \(\frac{g}{Q_{\rm sca}}\), as a function of sphere’s permittivity and the size parameter. Note the existence of peaks and valleys, which is a direct indication of the interference of resonant multipoles in the studied spectrum. leading to a rather important result, first presented in [103]. We summarize the following three cases where: (a) maximum forward scattering if \(\varphi_{1}-\psi_{1}=2k\pi,\;k=0,1,...\), (b) maximum backward scattering \(\varphi_{1}-\psi_{1}=(2k+1)\pi,\;k=0,1,...\), and (c) dipole-like scattering if \(\varphi_{1}-\psi_{1}=l\frac{\pi}{2},\;l=1,2,...\).