# Single-Shot Full Characterization of the Spatial Wavefunction of Light Fields via Stokes Tomography
## 3 Experimental Results
This characterization method also relates to the generalized beam quality measurement for high-order Gaussian beams, i.e., the \(M^{2}\) value, and the ideal value is \(M^{2}=N+1\)[14]. Figure 2b provides a visual description of the method, with the LG\({}_{2,1}\) mode as an example, in which the two 3D beam profiles used for the comparison are obtained via slice-by-slice observation (using the digital propagation technique) and theoretical reconstruction (using the diffraction integral). The similarity between both makes it difficult to determine the difference between the two 3D beam profiles with the naked eye. Figure 2c further provides an intuitive comparison for all five LG modes, where all the observed beam sizes(dots) fit closely on their corresponding predictions (curves) calculated through numerical diffraction, confirming the observation accuracy in Figure 2a. Thus, not surprisingly, the beam quality factors (\(M^{2}\) values) calculated with the two approaches (dots and curves) as input are almost the same. Notably, beam characterization based on the \(M^{2}\) measurement has many limitations for high-order modes, especially for superposition modes. We further proved the principle by characterizing the beam structure evolution of two superpositions of LG modes, given by \(c_{0,0}\text{LG}_{0,0}+c_{4,0}\text{LG}_{4,0}\) and \(c_{1,0}\text{LG}_{1,0}+c_{1,2}\text{LG}_{1,2}\), and, here, we set all complex coefficients \(c_{\ell,p}=\sqrt{1/2}\) for simplicity. Owing to their different Gouy phases \((N+1)\text{arctan}(z/z_{R})\), the beam structures would no longer be propagation-invariant. Relevant studies have indicated that the changes in beam structure manifest in azimuthal rotation and radial oscillation, and both exhibit pattern revival in the far field [30; 31]. This inference was confirmed by the results in Figure 3a,b, which show the beam structure evolution of the two superposed LG modes. That is, \(\psi_{\text{obs}}(\mathbf{r},z)\) is recorded at the \(z=0\), \(z=z_{R}\), and \(z=z_{\infty}\) (Fourier) planes, and \(\psi_{\text{num}}(\mathbf{r},z)\) is calculated with \(z=z_{R}\) and \(z=z_{\infty}\) by using the observation at \(z=0\) as input. The numerical reconstruction looks the same as the previously measured one. To make a quantitative examination, we calculated the fidelity of reconstruction, defined by \(F=\langle\psi_{\text{obs}}(\mathbf{r},z)|\psi_{\text{num}}(\mathbf{r},z)\rangle\), and the high fidelity shown in Figure 3c validates the accuracy of the characterization. Figure 2: Experimental results for paraxial eigenmodes. **(a)** Observed HLG modes with orders \(N\) from 1 to 5, where the colored arrow displays the path of the astigmatic transformation from LG to HG modes on the modal sphere; **(b)** visual comparison for numerically reconstructed (diffraction integral) and experimentally observed (via digital propagation) 3D profiles of the measured LG\({}_{2,1}\) mode; and **(c)** comparison of observed and predicted changes in beam size of measured LG modes as well as associated \(M^{2}\) values. Finally, beyond morphological observations, we can further derive the modal constitution of light fields (i.e., quantum state or density matrix) in terms of spatial degrees of freedom. This task is usually accomplished by using quantum state tomography [19, 20], which requires multiple projections on all mutually unbiased bases, leading to a complicated and time-consuming operation, especially for high-dimensional states. Moreover, recent works showed that astigmatic operation and machine-learning-based pattern recognition can also be used to determine simple OAM superimposition states [32, 33]. In contrast, the SCA characterization provides complete information on light fields and, thus, enables the direct and easy determination of the complex-valued modal spectrum by computing projections on a complete orthogonal basis, such as the LG basis set; see Appendix B.