# Single-Shot Full Characterization of the Spatial Wavefunction of Light Fields via Stokes Tomography
## 3 Experimental Results  
Figure 1b shows the schematic setup for the proof-of-principle demonstration. A tunable continuous laser operating at a wavelength of 780 nm was used as the light source. We began with a TEM\({}_{00}\) mode collimating from a single-mode fiber for easy control in further SCA shaping. In signal preparation, we used a spatial light modulator (SLM), in which we loaded a specially designed hologram for performing complex-amplitude modulation and digital propagation [25], to shape the signal light (represented by a red beam in the schematic setup) into a desired spatial mode with a controllable diffraction distance. The prepared signal was first relayed by a 4\(f\) imaging system (L1 and L2) and combined afterwards with the ancillary beam via a polarizing beam splitter (PBS) to obtain the vector mode discussed above. The ancillary beam came from the same laser source and was also shaped into a TEM\({}_{00}\) mode by collimating with a single-mode fiber. To avoid the challenge that comes from SoP intramodal-phase jitter, the spatially variant SoP of the vector mode was measured in a single-shot manner. Specifically, the vector mode was divided into four equal parts, which were then projected into horizontal, vertical, diagonal, and circular SoPs. After that, the four SoP projections, which can be converted to \(S_{0-3}(\mathbf{r})\), were recorded simultaneously within the same frame of the camera (not drawn in the figure). Here, a nonpolarizing beam splitter (NPBS) with four output ports was realized by a liquid-crystal geometric phase, and the polarizing filter (PF) comprised waveplates and a polarizer (see Appendix C). Moreover, lens L4 performed a Fourier transformation for the signal beam in combination with the SLM to realize digital propagation; the ancillary beam was expanded two times by lenses L3 and L4 to ensure that it could fully cover the signal beam. In the experiments, we first took the well-known structured Gaussian modes--the Hermite-Laguerre-Gaussian (HLG) family--as signal beams to be characterized [26]. Figure 2a shows the observed SCA patterns of the prepared signal beams at the waist plane (\(z=0\)). The patterns in the top and bottom rows are the Laguerre-Gaussian (denoted as LG\({}_{\ell,p}\)) and Hermite-Gaussian (denoted as HG\({}_{m,n}\)) modes, respectively, and their intermediates are shown in the middle row. The three HLG modes in each column correspond to a continuous astigmatic transformation from the LG polar to the HG equator on the same spatial-mode sphere with the same order \(N\), where \(N=2p+|\ell|=m+n\) and \(\ell=m-n\)[27; 28; 29]. Note that, except for the special case of \(N>1\), the intermediate HLG mode can no longer be represented as a linear superposition of the two orthogonal LG (or HG) modes on the sphere but as a superposition of all the possible LG (or HG) modes with an order of \(N\) (i.e., including modes not on this sphere) [14]. All the measurements agree well with the theoretical SCAs of the desired modes; see extended data in Appendix B. These SCAs provide complete knowledge about the spatial structure of measured beams, with which, in principle, we can exactly predict the diffraction behavior of each beam in free space or an optical system. That is, we can theoretically reconstruct the full spatial wavefunction of a beam, i.e., \(\psi_{\text{sig. }}(\mathbf{r},z)\), by using the diffraction integral with the SCA observed at the measuring plane, i.e., \(\psi_{\text{sig. }}(\mathbf{r},z_{m})\), as the source (or pupil function); see Appendix A for details. Since the modes in Figure 2a all belong to eigensolutions of the paraxial wave equation, their beam profiles should be propagation-invariant apart from an overall change in size. To validate the observations in Figure 2a, we compare the changes in beam diameter upon diffraction obtained from the theoretical reconstruction and the slice-by-slice observation.