# Angular Dependence of Photonic Crystal Coupled to Photovoltaic Solar Cell
## 3 Results and Discussion  
### Incident Angle Effect  
In this section, analyses of the 1 DPC incident angle dependence, solar spectrum distribution and polarization of the light are shown, as well as their impact on the optical coupling with the solar cell. The first characterization consisted in determining the influence of the incident angle of the light on the reflectivity exhibited by the 1 DPC. For this test, the dielectric periodic structure was characterized using a UV-VIS spectrometer (Figure 4). It was observed that as the incident angle increased, the Bragg peak position shifted to lower wavelengths and its original gaussian profile was shifted from its original position (Figure 4b), and finally, its shape was distorted (Figure 4c). The limit was identified when the incident angle was higher than 60\({}^{\circ}\): under these conditions, Bragg peak reflection shifted about 100 nm and its profile became significantly smooth, so the energy reflected was totally negligible (Figure 4c, yellow line). The largest absorption was obtained with the longest optical paths used by the light across the active layer, together with the reduction of the coupling from higher reflection at the glass-air interface. As the incidence angle changed from the normal direction, transverse electric and transverse mode were modified, but showing different types of dependence. Thus, photonic bandgap varied as a function of the incident angle of the light. Comparing different incident angles, the Bragg reflection was shifted to high energy photons. One of the most important effects observed was for incident angles higher than 60\({}^{\circ}\), so 1 DPC did not work as an irradiance selective reflector due to the Bragg peak smoothing. \begin{table}
\begin{tabular}{c c} \hline Irradiance (W/m\({}^{2}\)) & 1000 \\ \hline Temperature (\({}^{\circ}\)C) & 25.4 \\ Mismatch corrected & Yes \\ Jsc (mA/cm\({}^{2}\)) & 32.6 \\ Voc (mV) & 591 \\ FF (\%) & 77.2 \\ Efficiency & 14.9 \\ Source & IEC 60904-9 \\ Contact Laboratory & Radboud University Nijmegen \\ \hline \end{tabular}
\end{table} 
Table 2: Silicon solar cell datasheet. Figure 5 shows that transparency, calculated as the integrated transmissivity, was also dependent on the incidence angle. Transparency of the 1 DPC was determined by applying the methodology described in the accepted ISO 9050:2003 [28] to evaluate the light and energy transmittance of solar radiation for glazing:  
\[\tau=\frac{\int_{80}^{780}D_{65}(\lambda)\cdot T(\lambda)d\lambda}{\int_{80}^{78 0}D_{65}(\lambda)d\lambda} \tag{1}\]  
where \(D_{65}(\lambda)\) is the normalized spectrum of the incoming radiation, \(T(\lambda)\) is the transmittance of the sample, and \(\tau\) is the integer transparency of the 1 DPC. Figure 5b shows that transparency integrated over the complete irradiance spectrum depended on the incident angle, according to the Bragg peak shifting. In particular, when the incident angle was higher than 60\({}^{\circ}\), the transmission spectrum did not exhibit a photonic band gap, which was the key parameter that characterized the 1 DPC performance. Thus, for these conditions, the solar cell coupled to 1 DPC had a lower transparency than the reference  
Figure 4: Transmittance of 1 DPC as a function of the incident angle. As incident angle is increased, Bragg peak is shifted to a lower wavelength. Figure 5: (**a**) Bragg peak position shifting as function of the incident angle; and (**b**) transparency data obtained from transmittance curves versus incident irradiation angle. sample without any 1 DPC. After fitting Bragg peak position as a function of the incident angle (Figure 5a), a linear relationship was observed, in well concordance with Equation (2), well-known as Wulff-Bragg's condition. \[\lambda_{0}=2d\sqrt{n_{eff}^{2}-sin^{2}\theta} \tag{2}\]  
where \(n_{\text{eff}}\) represents the effective refractive index of the 1 DPC, \(\theta\) is the incident angle, and \(d\) is the photonic crystal thickness.