## 2 Materials and Methods
### Experimental System  
In this work we use a tunable liquid-crystal commercial q-plate from Arcoptix [30] (named variable spiral plate). To our knowledge, this is the only commercial q-plate device with tunable retardance. The possibility of tuning the retardance via an applied voltage allows us to operate the device at different wavelengths and to tune the polarization conversion efficiency, as described by Equation (1). Here we use it to generate hybrid vector beams and experimentally analyze their propagation.  
The q-value of this device is \(q=1/2\) and its nominal operational wavelength range is 400-1700 nm. The device has a circular clear aperture with a diameter of 1.2 cm (Figure 1a) and we measured a 93% transmittance (ratio of the output intensity to the input intensity). Its retardance is controlled by an AC bias between 0 and 8 V, and it was characterized in a broad spectral range [38]. Mode-division-multiplexing of vector beams at 1550 nm using such commercial q-plate was recently reported [39]. Here, we operate the device at a wavelength of 633 nm and we measured the retardance as a function of the applied voltage following a standard procedure [25; 26]. Figure 1b shows the retardance function obtained at this operating wavelength, which exhibits a total variation of more than 2\(\pi\). We identify the following voltages of interest: (1) at V = 2.45 V the retardance is \(\pi\), and thus the q-plate provides maximum conversion efficiency in generating pure vector beams; (2) at V = 3.92 V and V = 1.95 V the retardance is \(\pi\)/2 and 3\(\pi\)/2, respectively, thus acting as a quarter-wave q-plate (QW q-plate). In the present work the q-plate is always set at 3\(\pi\)/2 retardance.  
Figure 2 shows a scheme of the experimental setup. A He-Ne laser of wavelength 633 nm is spatially filtered and collimated. A linear polarizer and a quarter-wave plate (QWP) placed before the spatial filter makes the light beam circularly polarized, so the next polarizer can be rotated without changing the input intensity. Behind the collimating lens, a polarization state generator (PSG) consisting of another rotatable polarizer-QWP pair generates the state of polarization entering the tunable q-plate (this state will be henceforth referred to as input state). In some cases, the PSG will also incorporate a static q-plate in order to have a vortex beam or a vector beam incident on the tunable q-plate. A polarization state analyzer (PSA) placed behind the q-plate projects the output beam onto a given polarization state. The PSA is composed of a QWP and a linear polarizer. In order to compare the near-field (NF) and the far-field (FF) patterns, we use two different configurations.  
For NF observations (Figure 2a), the beam is projected onto a diffuser screen. Then, a photographic objective is used for imaging this screen onto the camera detector. In the second case (Figure 2b) a converging lens is added behind the PSA, and the detector is placed at the focal plane, where the FF patterns are observed.  
Figure 1: (**a**) Picture of the tunable Arcoptix q-plate. (**b**) Measured optical retardance of the Arcoptix q-plate as a function of the applied voltage for the 633 nm wavelength.