# The Quest for Low Loss High Refractive Index Dielectric Materials for UV Photonic Applications
## 2 Methods  
### Electromagnetic Simulations  
The electromagnetic interaction between isolated nanospheres and light has been modeled following the Lorenz-Mie formalism for scattering and absorption of light by small particles [36; 37]. The scattering efficiency by a spherical particle is given by  
\[Q_{sca}=\frac{2}{x^{2}}{\sum_{n=1}^{\infty}}(2n+1)(|a_{n}|^{2}+|b_{n}|^{2}), \tag{1}\]  
where \(a_{n}\) and \(b_{n}\) are the scattering coefficients of the Lorenz-Mie theory. These depend on both particle size and optical properties of the NP relative to its surrounding medium. The size parameter \(x\) is given by \(2\pi m_{med}R/\lambda\), where \(\lambda\) is the wavelength of incident light in the surrounding medium of refractive index \(m_{med}\), and \(R\) is the radius of the sphere. From a physical point of view, coefficients \(a_{n}\) and \(b_{n}\) represent the weighting factors of the different multipolar contributions of order \(n\), electric or magnetic, respectively. In particular, \(a_{1}\) and \(b_{1}\) correspond to the electric and magnetic dipolar modes, respectively. When \(a_{1}=b_{1}\) the zero backward condition is fulfilled and, as a result [9], there is null scattering in the backward direction, assuming that the rest of the coefficients are negligible (small size limit).  
In the far-field region, the scattered electric field \(E^{sca}\) is related to the incident one \(E^{inc}\) through the scattering amplitude matrix \(S(\theta,\phi)\)[36]. The scattered \(p\)-/\(s\)-components are given by  
\[\left(\begin{array}{c}E^{sca}_{p}\\ E^{sca}_{s}\end{array}\right)\propto\left(\begin{array}{cc}S_{2}(\theta, \phi)&S_{3}(\theta,\phi)\\ S_{4}(\theta,\phi)&S_{1}(\theta,\phi)\end{array}\right)\left(\begin{array}{c }E^{inc}_{p}\\ E^{inc}_{s}\end{array}\right), \tag{2}\]  
where \(\theta\) and \(\phi\) are the scattering and azimuthal angles, respectively. The \(S_{i}\) (\(\theta\),\(\phi\)) elements are related to the coefficients \(a_{n}\) and \(b_{n}\), and specific angular functions [36].  
In order to quantify the efficiency of the zero backward scattering condition, the forward to backward scattering intensity ratio (\(I_{Fud}\)/\(I_{Bck}\)) is introduced. This magnitude is defined as the ratio between the integrated scattered intensity in the forward and backward hemispheres. A key point to consider is that although a particle can have a high value of \(I_{Fud}\)/\(I_{Bck}\), its scattering efficiency may be negligible, making its scattering directionality properties useless. Therefore, it is important to find NPs made of materials that have high values of both \(I_{Fud}\)/\(I_{Bck}\) and \(Q_{sca}\)[38]. Keeping this in mind, we have evaluated the product between these two magnitudes: \(Q_{ZB}=I_{Fud}\)/\(I_{Bck}\)-\(Q_{sca}\). We will refer to this magnitude as _zero backward efficiency_ (\(Q_{ZB}\)).  
As aforementioned, the electromagnetic response of the nanospheres excited with light has been modeled using the Lorenz-Mie formalism. On the other hand, the interaction of light with dimers has been solved with finite element (FEM) calculations through commercial software COMSOL Multiphysics 5.2 (COMSOL Inc., Burlington, MA, USA) [39].