## 3 Results and Discussion
In order to analyze the resonant peaks in this region, the spectra recorded at the middle point (\(-40\)\(\upmu\)m) were selected  
Figure 3: **(a)** SEM image of the selected rod, where the Y-axis scale used to perform the maps in (**b,c**) is shown. The directions of the transverse electric (TE) and transverse magnetic (TM) polarizations with respect to the c-axis of the structure are also indicated. (**b,c**) The maps of the evolution of the \(\upmu\)-PL emission along the Y-direction marked in (**a**): (**b**) total photoluminescence (PL) signal, (**c**) TE polarization, and (**d**) TM polarization. and are shown in Figure 4a. The length side \"a\" of the cavity at this point, measured using an SEM image, was 1.34 \(\upmu\)m. It can be seen in the spectra of Figure 4a that both the TE and TM polarizations presented optical resonant modes and the positions of the TM peaks were slightly shifted toward shorter wavelengths. Let us try to understand what type of optical resonant modes were established in this cavity. As it was described in the introduction, three types of resonant modes can appear: Fabry-Perot modes, whispering gallery modes (WGM1), and quasi-whispering gallery modes (WGM2). The two types of whispering gallery modes are more favorable, as the confinement of the light is based on the total internal reflection (TIR). For ZnO, with an initial estimation of the refractive index around 2 in the visible range, the critical angle for the TIR is:  
\[\theta_{c}=arcsin\left(\frac{n_{air}}{n_{ZnO}}\right)=30^{\circ}. \tag{5}\]  
The incident angle for the quasi-WG modes was 30\({}^{\circ}\) (WGM2 in Figure 1), which is quite close to the condition for TIR. Then, the most favorable conditions for the optical confinement were found for the WG modes (WGM1 in Figure 1, angle of incidence of 60\({}^{\circ}\)). Furthermore, the WG optical path (\(L_{0}=6.96\)\(\upmu\)m) fits well with the separation of the resonant peaks observed in Figure 4a. Less intense  
Figure 4: Analysis of the whispering gallery modes observed in region (i) of the structure in Figure 3a. The length side “a” of the cavity was 1.34 \(\upmu\)m at the selected point. (**a**) \(\upmu\)-PL spectra for all the polarization conditions. (**b**) Interference order \(N\) of the peaks observed in the spectra of part (**a**). (**c**) Refractive index estimated from the wavelength position of the resonant peaks. (**d**) Variation of the order \(N\) wavelength position as a function of the position along the structure, i.e., as a function of the size of the cavity). modes that are marked with an asterisk were associated with the quasi-WG modes (incidence of 30\({}^{\circ}\), less confinement due to TIR, and broader peaks). Then, Equations (1) and (2) were used to analyze the peaks of TM and TE polarizations, respectively. However, the refractive index that appears in these equations is also a function of the wavelength. In our case, for ZnO, we used the formula reported by Bond for bulk material [20]:  
\[n^{2}=2.81418+\frac{0.87968\lambda^{2}}{\lambda^{2}-0.3042^{2}}-0.00711\lambda^ {2}, \tag{6}\]  
in order to have an initial value of the refractive index as a function of the wavelength. Then, performing the calculations as described, for example, in [5], the interference order \(N\) (Figure 4b) and an estimation of the refractive index \(n\) (Figure 4c) can be obtained for the wavelength position of the resonant modes and the calculated optical path. As has been mentioned, the change in the size of the optical cavity produced a change in the position of the optical resonances. This is shown in part d of Figure 4, where the spectra obtained at points located at different positions of the rod showed a slight shift toward longer wavelengths, indicating that there was a small increase of the cross-section size as the distance from the start of the structure (i.e., the ball) became longer.