# Single-Shot Full Characterization of the Spatial Wavefunction of Light Fields via Stokes Tomography
## 4 Conclusions  
We have experimentally demonstrated a \"complex-amplitude profiler\" that can completely determine the SCA of a light field with a single-shot measurement. This novel technique exploits spatially resolved polarimetry with ancillary light to determine both  
Figure 3: Experimental results for noneigen paraxial modes. (**a,b**) Observed superposed LG modes at \(z=0\), \(z=z_{R}\), and \(z=z_{\infty}\) (Fourier) planes, and corresponding numerical reconstructions based on observations at \(z=0\) plane; (**c**) SCA reconstruction fidelity relative to the observation; and (**d,e**) modal constitutions and of two superposed LG modes and associated unitary evolution upon propagation.  
the amplitude and wavefront of unknown beams. Specifically, the unknown signal light field to be measured was first combined with a preknown ancillary light via orthogonal polarization states, forming a vector mode equivalent to a phase-stable polarizing interferometer; then, by performing a spatial Stokes measurement on the vector mode, the SCA of the signal can be determined with high precision and spatial resolution. On this basis, one can further theoretically reconstruct the full 3D spatial wavefunction and derive the modal constitution in the spatial degrees of freedom exactly. In particular, the device is capable of performing a single-shot measurement, providing it with real-time characterization and insensitivity to vibrations. In addition, this technique only requires some low-cost and integratable polarization sorting elements that can be easily incorporated into the existing beam characterization system. Notice, however, that the ancillary light and the measured beam should come from the same laser source (for coherent states) or photon (for single-photon states). A feasible way to experiment is to use a single-mode fiber to sample the partial energy (or probability amplitude) of the signal light (or photons).  
Since SCA characterization allows access to complete information about light fields, the principle demonstrated here has a broad spectrum of potential applications in various areas of optics. For instance, this principle can help researchers to unambiguously probe the topology hidden in a structured light field or the high-dimensional quantum state encoded with spatial degrees of freedom [34; 35; 36; 37; 38]. In addition to fundamental studies, this principle can also be coupled with developing optical techniques based on structured light, such as in situ detection on spatially multiplexed channels or structured sensing beams [35; 41]. Beyond paraxial beams, this technique can also be used to completely characterize the complex spectrum of an imaging signal, enabling further implementation of arbitrary spatial analog computing on the signal [42].  
Conceptualization, Z.Z.; Software, H.W.; Validation, C.L., J.J. and B.Z.; Investigation, B.Y. and Z.Z.; Writing--original draft, Z.Z.; Writing--review and editing, C.R.-G. and B.S.; Funding acquisition, Z.Z. All authors have read and agreed to the published version of the manuscript.  
This work was supported by the National Natural Science Foundation of China (Grant Nos. 11934013, 62075050, 61975047, and 62275070), the High-Level Talents Project of Heilongjiang Province (Grant No. 2020 GSP12), and the Heilongjiang Provincial Education Department (Grant No. UNPYSCT-2020185).  
Not applicable.  
The raw data supporting the conclusions of this article will be made available by the authors on request.  
The authors declare no conflict of interest.