## 5 Conclusions  
This work shows the possibilities of an electrically tunable q-plate to generate different kinds of vector beams and study their dynamics. This device is commercially available, therefore useful for researchers with no fabrication facilities. We employed a liquid-crystal q-plate from the company Arcoptix, with retardance addressable by voltage and a broadband operation that covers the VIS and NIR ranges. While in the standard operation it is used to generate pure vector beams at any wavelength by tuning the device to half-wave retardance (HW q-plate), here we employed it at quarter-wave retardance (QW q-plate) to generate different kinds of hybrid vector beams. Standard hybrid vector beams and more complex petal-like hybrid vector beams were generated when the input beam on the device is a Gaussian beam, a vortex or a pure vector beam. We built a simple setup to experimentally analyze their propagation dynamics by comparing the intensity and polarization patterns in the near field and in the far field. Since the propagation of a hybrid vector beam is affected by the Gouy phase of the different LG modes constituting the beam, we developed an analytical approximation that describes the polarization patterns in terms of mode superpositions and their transformation upon propagation based on their different Gouy phase. The experimental results bear good agreement with numerical and analytical results within the frame of this approximation. We illustrated how simple hybrid vector beams (superposition of RCP and LCP components with one single but different LG mode) generate the same polarization pattern in the near field, provided the half-difference \(m=1/2\) is the same. We showed that these hybrid beams are affected by the different Gouy phase of the circular polarization components, causing a significant rotation of the polarization pattern in the far field. As the charge of our q-plate is \(q=1/2\), it creates hybrid vector beams of \(m=1/2\) when detuned to a QW retardance. Thus, when focused, the far field RCP and LCP components show a good overlap and the resulting pattern is fairly well described as a polarization rotation driven by the Gouy phase difference between components. We analyzed a second set of hybrid beams, which are the superposition of two different LG modes on each RCP and LCP component. These more complex beams are generated by illuminating the QW  
Figure 6: Results for a QW q-plate illuminated with a radially polarized beam of order \(\ell=2\). (a) Results in the near field; (b) results in the far field. Simulated and experimental polarization maps, with equivalent color representation as in Figure 3. Experimental images captured for different orientations of the analyzer, indicated as white arrows (NA indicates no analyzer). q-plate with a homogeneous linearly polarized beam or with a higher-order pure vector beam. In this case the output hybrid vector beam presents lobes along the azimuthal coordinate that propagate differently for the RCP and LCP components, but overlap at the far field. In summary, we used a simple setup to generate hybrid vector beams that is fully based on an electrically tunable commercial q-plate. Although it does not provide the vast flexibility offered by SLM-based systems, this q-plate-based setup holds advantages that could be relevant for technological applications of vector beams. Namely, it is more compact, of easier alignment, and of higher efficiency, since SLM losses caused by pixelization are not present. In addition, similarly to SLM schemes, it can generate a wide variety of pure and hybrid vector beams, allowing to switch from one kind of beam to the other by simply addressing a voltage. Furthermore, although these results were obtained for a wavelength of 633 nm, they can be extended to a broad range of wavelengths. As this tunable q-plate is \(q=1/2\), we are limited to generate hybrid vector beams where the different modes differ in azimuthal order (topological charge) by only one.