## 1 Introduction
The q-plate device is a geometric phase element consisting of a half-wave retarder where the c-axis makes \(q\) turns around the azimuthal orientation \(\theta\) around a central point. The \(q\) value is the topological charge of the singularity appearing at the center point and it is fixed on fabrication [23; 24]. Q-plates convert input circularly polarized light into the opposite circular polarization and add a helical phase \(\exp(\pm 2q\theta)\), where the sign is given by the input polarization helicity. Therefore, they have been widely used to impart and manipulate orbital angular momentum (OAM). In addition, because they couple the spin angular momentum (SAM, associated to circular polarization) to the OAM (associated to the wavefront's helicity) q-plates are also employed to generate non-separable superpositions of local polarization and wavefront modes, i.e., vector beams. Namely, it is well known that a q-plate converts input homogeneous linear polarization into a cylindrical vector beam. Typically, q-plates are designed to introduce a retardance of \(\pi\) radians (half-wave retardance) at a given operating wavelength. In this condition they provide maximum conversion efficiency. When made of liquid-crystal materials, the device can incorporate electrodes to tune the retardance [25], thus allowing operation at different wavelengths whilst keeping the maximum conversion efficiency [26]. However, it is also possible to operate at a retardance different than the standard \(\pi\) value. A q-plate with \(q=1/2\) and \(\pi/2\) retardance (QW q-plate) was identified to generate a hybrid radially polarized beam when being illuminated with circular polarization [13; 27]; the generated beam was in fact a full Poincare beam of charges \(\ell_{R}=0\) and \(\ell_{L}=2\). This situation was also achieved with a non-tunable commercial q-plate by using an operating wavelength where it exhibits quarter-wave retardance [28]. Furthermore, the implementation of quantum walks and wavepacket dynamics has been reported using such QW q-plates [29]. In this work we use an electrically tunable liquid-crystal commercial q-plate [30]. The purpose of this paper is to review different kinds of hybrid vector beams that can be generated with a q-plate with detuned retardance and provide a comprehensive explanation of their polarization patterns within a simple theoretical approach. We consider different input beams illuminating the device, including input homogenous polarizations, vortex beams, and vector beams. In all cases the output is regarded as a hybrid vector beam consisting of the collinear superposition of different modes in the two circular polarization components. The resulting hybrid vector beam shows more or less complicated structure depending on the input beam illuminating the q-plate. Pure and hybrid vector beams display a similar polarization pattern in the near field, since it depends mostly on the \(\ell_{R}-\ell_{L}\) difference. However, their propagation dynamics are rather different in the far field. While pure vector beams retain their polarization pattern upon propagation, the Gouy phase difference of the two circular polarization components yields transformations in the polarization pattern of hybrid vector beams in the far field [31; 32; 33]. Therefore, these different kinds of generated hybrid vector beams are analyzed both in the near field and in the far field, in order to evidence their different propagation dynamics. Their theoretical intensity patterns and polarization maps are compared with the experimental results, bearing good agreement. The paper is organized as follows: after this introduction, Section 2 briefly recalls the essential aspects of q-plates and vector beams required in this work and describes our experimental system and the tunable commercial q-plate employed [30]. For the sake of clarity, the experimental results and their discussion are divided in two sections.