## 3 Results and Discussion
For the rod, the best cavity was the one presented in Figure 6, whereas for the nail, it was the one presented in Figure 2:1 These two cavities had the fact that they were found in a constant cross-section region in common. Then, their performance factors, \(Q\) and \(F\), were calculated in order to compare both cavities. Figure 2:2 Analysis of the whispering gallery modes in region (ii) of the nail. The side length “a” measured in the (**a**) SEM images was 1.04 μm. (**b**) \(\upmu\)-PL spectra for all the polarization conditions, (**c**) interference order \(N\) of the peaks observed in the spectra, and (**d**) estimation of refractive index from the wavelength, which virtually overlapped for both polarizations. The quality factor, \(Q\), and the finesse, \(F\), can be calculated using the following equations:  
Figure 2:3  
Figure 2:4  
where \(\Gamma\) stands for the full-width at the half maximum (FWHM) of the peak at \(\lambda\). The FWHM was calculated by means of a Lorentzian fitting of the peaks at each \(\lambda\) in the TM mode (the set of resonant peaks that are generally more intense [6,22]). This value allowed for estimating \(Q\) and \(F\) (Table 1). \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multicolumn{2}{c}{**Cross–Cavity of Figure 6**} & \multicolumn{2}{c}{**Nail–Cavity of Figure 9**} \\ \hline
**Wavelength (nm)** & \(Q\) **Factor** & \(F\) & **Wavelength (nm)** & \(Q\) **Factor** & \(F\) \\ \hline \multicolumn{5}{c}{648.353} & 98.91 & 4.17 \\  & & 620.998 & 97.23 & 3.88 \\
673.26 & 182.37 & 13.59 & 596.248 & 133.40 & 5.15 \\
623.08 & 230.02 & 15.39 & 573.21 & 177.34 & 6.51 \\
581.4 & 198.95 & 12.07 & 552.167 & 227.87 & 7.96 \\
546.12 & 212.29 & 11.75 & 532.888 & 229.21 & 7.49 \\
515.9 & 224.28 & 11.17 & 515.485 & 283.90 & 8.75 \\
490.21 & 214.94 & 9.73 & 499.93 & 494.94 & 14.45 \\
468.02 & 159.80 & 6.49 & 485.004 & 589.00 & 16.20 \\  & & 471.665 & 624.54 & 16.14 \\  & & 459.472 & 671.14 & 16.44 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Calculated quality (\(Q\)) factor and finesse (\(F\)) for each wavelength in the cross and the nail structures. In order to compare the results from the different types of structures, data were centered at 515 nm, which was a common peak. Figure 10: Analysis of the whispering gallery modes in region (iii). The measured length side “a” measured in the (**a**) SEM images was 1.39 \(\upmu\)m. (**b**) \(\upmu\)-PL spectra for all the polarization conditions and spectra (**c**) shows a detailed second set of resonant peaks from the Fabry-Pérot modes. (**d**) Interference order \(N\) of the peaks observed in the spectra. (**e**) Estimation of the refractive index from the wavelength. Due to the thicker section in this structure, the calculated refractive indexes were less accurate, showing a higher difference for both polarizations. In the case of the cavity of the cross, a clear trend could not be observed in the values of \(Q\) and \(F\) with the wavelength. However, in the case of the cavity of the nail, an increase in the \(Q\) factor and \(F\) was observed as the wavelength increased. This behavior could be associated with the increase in the refractive index with the wavelength. The \(Q\) factors obtained were in agreement with those reported in the literature for ZnO hexagonal cavities [2; 23], although there is still some space for improvement [24]. A third relevant parameter to characterize a resonant cavity is the reflectivity \(R\) of the faces between which the resonant mode is established. In the case of a regular polygonal cavity, as was our case for the WGM, the reflectivity was related to the quality factor via the expression [25]:  
\[Q=\frac{\pi nmDR^{\frac{\pi}{2}}}{2\lambda\Big{(}1-R^{\frac{\pi}{2}}\Big{)}} \sin\frac{2\pi}{m}, \tag{9}\]  
where \(n\) is the refractive index, \(m\) the number of polygon facets (for our case 6), \(D\) the diameter of the circle circumscribing the polygon, and \(R\) the reflectivity.