## 1 Introduction  
ZnO is a well-known material with applications in very different fields from photonics to cosmetics. A wide bandgap, relatively high refractive index, large exciton binding energy, and non-ohmic behavior are behind many of these applications. A very good review about this material and its applications was published by [PERSON] et al. [1]. In particular, the lasing capabilities at room temperature have received a lot of attention in the last few years, with special attention being paid to hexagonal structures, rods, plates, or nails at the micro- and nanoscale [2; 3]. Resonant cavities play an important role in a variety of devices, such as musical instruments or microwave generators; however, in the last few years, the need for miniaturized devices has focused the attention on optical micro- and nanocavities. Dielectric whispering gallery resonators are the focus of this application. Typically, they are microstructures with disk, cylindrical, or spherical shape, in which the light circulates around the lateral walls, causing multiple internal reflections. This effect usually leads to high \(Q\) factors, and consequently, low lasing threshold powers. The wurtzite crystal structure offers unique properties for creating microcavities for this purpose [2; 3; 4; 5; 6; 7]. In elongated structures, such as the hexagonal rods obtained using ZnO, different resonant modes may appear. The easiest is, of course, the Fabry-Perot mode that is established along the structure due to the reflection at both ends of the structure. However, although there are papers that report the use of Fabry-Perot modes for optical applications [8], these modes are not optimal for lasing or resonant cavity applications due to the low reflectivity for normal incident light of the end facets. Much better conditions are obtained when the confinement is produced in the cross-section of the structure. The resonant modes would then be established on the perimeter of the rod and propagate along the structure. As shown in the scheme of Figure 1, in a hexagonal cavity, three types of optical modes can be set, each with a different optical path, hence giving rise to different resonance conditions. In most cases, the observed resonances are compatible with whispering gallery modes (WGM1) and quasi-whispering-gallery modes (WGM2). The corresponding equations for the wavelength position of the resonances, depending on the polarization (TM--transverse magnetic, TE--transverse electric) and the optical mode, may be written as [5, 9, 10]:  
\[\lambda_{TM}=\frac{n\cdot\delta_{op}}{N+\frac{6}{n}\cdot\arctan\left(\frac{1} {n}\sqrt{3n^{2}-4}\right)}, \tag{1}\]  
\[\lambda_{TE}=\frac{n\cdot\delta_{op}}{(N-3)+\frac{6}{n}\arctan\left(n\sqrt{3n ^{2}-4}\right)}, \tag{2}\]  
for WGM1, and:  
\[\lambda_{Quasi-TM}=\frac{n\cdot\delta_{op}}{N+\frac{3}{n}\arctan\left(\frac{1 }{n}\sqrt{\frac{n^{2}-4}{3}}\right)}, \tag{3}\]  
\[\lambda_{Quasi-TE}=\frac{n\cdot\delta_{op}}{\left(N-\frac{3}{2}\right)+\frac{ 3}{n}\arctan\left(n\sqrt{\frac{n^{2}-4}{3}}\right)}, \tag{4}\]  
for WGM2, where \(N\) is the interference order, \(n\) the refractive index and \(\delta_{op}\) is the optical path. Previous work has shown wave-guiding behavior in Er- and Li-codoped ZnO that were assigned to Fabry-Perot and whispering gallery modes [11]. Then, we had a particular interest in investigating the behavior of the Li-doped structures when no rare earth (RE) dopant is present. Furthermore, some works have reported an improvement in microsphere optical resonant cavities with Li doping [12]. In this work, the evolution of the whispering gallery modes found at the cross-section of ZnO:Li elongated structures was studied. It was found that not all the possible resonant cavities in these structures had the same behavior. In this paper, the evolution and quality of the resonant peaks in relation to the size and shape of the resonant cavity and to the reflectance efficiency of the facets are described.