# Single-Shot Full Characterization of the Spatial Wavefunction of Light Fields via Stokes Tomography
## 2 Principle and Methods  
To prove the principle with specific examples, we chose paraxial structured Gaussian beams as unknown signals to be characterized. The transverse SCA of a beam upon propagation (referred to as the complete 3D spatial wavefunction) can be fully described (or predicted) by a solution of the paraxial wave equation, an analog of the Schrodinger equation for free particles with time \(t\) replaced by the propagation coordinate \(z\), expressed as follows:  
\[\psi_{\text{sig.}}(\mathbf{r},z)=u(\mathbf{r},z)\text{exp}[iv(\mathbf{r},z)], \tag{1}\]  
where \(u(\mathbf{r},z)\) and \(v(\mathbf{r},z)\) are the spatial amplitude and wavefront, respectively, and \(\mathbf{r}\) denotes the transverse coordinates. In particular, Equation (1) belongs to an eigensolution of the paraxial wave equation (commonly referred to as a paraxial eigenmode) when the beam profile on \(z\) is self-similar; otherwise, it can be represented as a superposition of eigensolutions with different modal orders [14]. Notably, the full 3D spatial wavefunction can be retrieved from a transverse SCA in an arbitrary propagation plane using the diffraction (or path) integral [3; 15]. Namely, to fully characterize a light beam, it is required to record both \(u(\mathbf{r})\) and \(v(\mathbf{r})\) exactly at the same measuring plane \(z_{m}\). For this task, as shown in Figure 1a, we introduce an ancillary beam, denoted as \(\psi_{\text{anc. }}(\mathbf{r},z)\), and combine it with the signal beam, both carrying orthogonal circular state of polarization (SoP) \(\hat{e}_{\pm}\), to generate a vector spatial mode given by [16; 17; 18]. \[\Psi(\mathbf{r},z)=\sqrt{a}\psi_{\text{anc.}}(\mathbf{r},z)\hat{e}_{+}+e^{i\theta}\sqrt {1-a}\psi_{\text{sig.}}(\mathbf{r},z)\hat{e}_{-}. \tag{2}\]  
Here, \(a\in[0,\,1]\) is a weighting coefficient that controls the degree of spin-orbit non-separability (also known as concurrence) between \(\psi_{\text{sig. }}\) and \(\psi_{\text{anc. }}\) overlapping positions, which is the maximum for the position \(\mathbf{r}_{0}\) as \(\sqrt{a}g(\mathbf{r}_{0})=\sqrt{1-au}(\mathbf{r}_{0})\); \(\theta\) is the intramodal phase between the two polarization components. For convenience, we take a well-known ancillary mode, the fundamental Gaussian mode, whose minimum beam width, acquired for \(z=0\), coincides with the measuring plane. That is, \(\psi_{\text{anc. }}(\mathbf{r},z_{m})=g(\mathbf{r})\text{exp}[ikz]\), where \(g(\mathbf{r})\) denotes a 2D Gaussian amplitude envelope whose cross section can cover the signal mode well. In this way, the vector mode at the measuring plane can be further represented as follows:  
\[\Psi(\mathbf{r},z_{m})=\sqrt{a}g(\mathbf{r})\hat{e}_{+}+e^{i[v(\mathbf{r},z_{m})]}\sqrt{1 -au}(\mathbf{r},z_{m})\hat{e}_{-}. \tag{3}\]  
Here, we assumed the global \(\hat{e}_{\pm}\) intramodal phase \(\theta=0\) for simplicity, which can be realized in experiments via SoP control. In mode (3), the ancillary light works as a reference beam that copropagates with the signal along the same beam axis. Therefore, the vector mode can be regarded as a phase-stable polarizing interferometer. We now perform a spatially resolved Stokes measurement on the vector mode given by Equation (3), namely, \(\langle\Psi(\mathbf{r},z_{m})|\phi_{\text{1-3}}(\mathbf{r})|\Psi(\mathbf{r},z_{m})\rangle\), where \(\phi_{\text{1-3}}\) are Pauli matrices for SoP tomography [19; 20]. The measured expectation values are a group of mutually unbiased observables named spatial Stokes distributions \(S_{\text{1-3}}(\mathbf{r})\) (see Appendix C for details), from which it is possible to reconstruct the spatially variant SoP of the vector mode. This measurement provides an in situ visualization for vector modes generated and manipulated in experiments, so it has been widely used in the structured light community [21; 22; 23; 24].