# The Quest for Low Loss High Refractive Index Dielectric Materials for UV Photonic Applications
## 4 Results
### Directionality Properties: The Full-Forward Condition  
The full-forward condition, also known as the first Kerker condition, maximizes the scattered intensity in the forward direction while it minimizes the backward one. This situation has found a great utility in light guiding and light trapping applications [54; 55; 11; 56; 57]. In order to evaluate the full-forward condition, we have considered nanospheres of different radii made of AlP, AlAs, AlN, CeO\({}_{2}\), TiO\({}_{2}\) and Si and we have analyzed the following parameters: photon  
Figure 2: Near-field enhancement averaged over the surface \(\langle\)1 \(E\)1 / _I_E_0 1\(\rangle\) of spheres with radius \(R\) (between 30 and 90 nm). A white dashed line represents the energy where \(\epsilon_{2}\) = 1, and a white solid line indicates the energy where we consider the UV regime starts, 3 eV. energy where the zero-backward condition is produced (\(E_{ZB}\)), the scattering efficiency (\(Q_{sca}\)) and the forward to backward scattering ratios (\(I_{Fwd}\)/\(I_{Bck}\)) at \(E_{ZB}\). In addition, the combined effect of these two last magnitudes, the _zero backward efficiency_, defined as \(Q_{ZB}=Q_{sca}\cdot I_{Fwd}\)/\(I_{Bck}\) will be also analyzed. Figure 3a shows the photon energies where the zero-backward condition holds (\(E_{ZB}\)) for spheres of radius \(R\) between 30 and 90 nm. The zero-forward condition is not satisfied for all the materials in the analyzed size range. For example, for CeO\({}_{2}\) NPs, \(R\) should be larger than 60 nm. On the contrary, nanospheres made of either AIP or diamond, exhibit this condition for all analyzed radii. The rest of materials also shows the zero-backward condition in the UV but not for all radii. Notice that all of them satisfy this condition at higher energies than Si. Those that show the zero-backward condition above 3 eV are AIP (\(30\leq R\leq 50\) nm), AlN (\(40\leq R\leq 80\) nm), diamond (\(40\leq R\leq 70\) nm), CeO\({}_{2}\) (\(60\leq R\leq 80\) nm), and TiO\({}_{2}\) (\(R=50\) nm). High values of \(Q_{sca}\) are desirable in order to exploit the maximum amount of incoming intensity in the analyzed condition, the zero-backward [38]. Figure 3b shows the scattering efficiency \(Q_{sca}\) at the photon energy of the zero-backward condition as a function of the sphere radius, \(R\). Among the studied materials, those that show the highest values of \(Q_{sca}\) at the zero-backward condition are diamond, CeO\({}_{2}\) and AlN. AIP, one of the promising candidates for directionality applications in the UV, presents values of \(Q_{sca}\) around two times smaller than diamond and CeO\({}_{2}\), and very similar to those of TiO\({}_{2}\). It is worth noting that all the analyzed materials show higher values of \(Q_{sca}\) than Si. Figure 3c shows the forward to backward intensity ratio \(I_{Fwd}\)/\(I_{Bck}\) at the photon energy of the zero-backward condition as a function of \(R\). This magnitude will allow us to quantify the asymmetry introduced in the scattered intensity along the propagating direction when imposing the zero-backward conditions. Diamond, CeO\({}_{2}\) and AlN are the materials that present higher ratios \(I_{Fwd}\)/\(I_{Bck}\). Values remain almost constant for all the considered sizes. TiO\({}_{2}\) and AlP present the same values of \(I_{Fwd}\)/\(I_{Bck}\), and very similar to that of AlAs. Once again, Si shows the lowest ratio. The ideal zero backward condition is a combination of two effects: a high scattering efficiency and a high forward to backward ratio. Therefore, forward directionality requires the presence of both.