## 2 Materials and Method  
The experimental setup and the instrumentation used in this experiment are very similar to those used in previous works on laser solitons in our group [8; 12]. A detailed representation of this arrangement is shown in Figure 1. The VCSEL on this experiment is a large aperture device with a diameter of 200 mm. The emission takes place in the 975-980 nm region through the n-doped Bragg reflector and the transparent substrate, the so-called bottom emitter [16]. A Peltier element with a feedback circuit is utilized to stabilize the VCSEL temperature at 20 \({}^{\circ}\)C.  
Figure 1: Experimental setup: A volume Bragg grating (VBG) provides frequency-selective feedback to a vertical-cavity surface-emitting laser (VCSEL). BS: beam sampler, CCD: charge-coupled device camera, PD: photo-detector, LP: linear polarizer, QWP: quarter-wave plate, HWP: half-wave plate. The upper arm is used to measure the spatially resolved Stokes parameter at high magnification (CCD1), the lower monitors power (PD) and near (CCD3) and far field (CCD2) distributions of potentially the whole laser. The output of the VCSEL is collimated by an aspheric lens of \(f_{1}=8\) mm focal length. A second lens with \(f_{2}=50\) mm is used to focus the light onto the frequency-selective element, a volume Bragg grating (VBG). These two lenses are adjusted to form an a focal telescope, i.e., the external cavity is self-imaging after a round trip. The VBG has a narrow-band reflection peak of 95% at \(\lambda_{g}=978.1\) nm, with a reflection bandwidth of 0.1 nm full-width half-maximum (FWHM). For monitoring the output, a wedged glass plate with an uncoated facet at the front and an anti-reflection coated facet at the back serves as an outcoupler or beam sampler (BS). The reflection is relying on Fresnel reflection and therefore is polarization dependent. The reflectivity is on the order of 10% for s-polarized light and 1% for p-polarized light. Note that the polarization asymmetry is much smaller (1:1.1) in transmission. Via a half-wave plate and an optical isolator, polarization resolved light-current (LI) characteristics as well as near and far field intensity distributions can be obtained with a photo-diode and CCD cameras, respectively. The use of the intra-cavity BS also allows measurements without feedback. However, the main results for the polarization distributions are obtained by observation after the VBG, as the intra-cavity polarization state can be accessed directly from there. The light that goes through the VBG is re-imaged onto another CCD-camera (CCD1) by two telescopic systems, providing enough magnification to accurately resolve the different polarization zones. Within the collimated range between the two lenses of the second telescope, a linear polarizer (LP) and a quarter-wave plate (QWP) are used with different combinations in its orientation. To analyze the state of polarization of the VCSEL's emission, all our work relies on the spatially resolved measurement of the well-known Stokes parameters [17]. The data needed to accomplish this task are the different intensity patterns obtained for the different orientations of the polarizer's axis. The polarization orientation is defined with respect to the propagation direction of the beam, i.e., in the plane orthogonal (\(x,y\)) to its wave vector. The measurements necessary to determine the Stokes parameters are:  
* \(I_{x}\): Horizontally polarized component of the intensity. * \(I_{y}\): Vertically polarized component of the intensity
* \(I_{45}\): Intensity component diagonally polarized. Usually, this study is restricted to the linear components of the polarization but it is important to comment that in our case the results obtained confirm the existence of a non-negligible quantity of circular light. To measure this factor, a fourth value of the intensity is taken:  
* \(I_{circ}\): Circular component of the emission. In this case, a QWP is used in addition to the linear polarizer.