# Single-Shot Full Characterization of the Spatial Wavefunction of Light Fields via Stokes Tomography
## Appendix A Hermite-Laguerre-Gaussian Modes  
The transverse structure of a paraxial light field and its corresponding evolution upon propagation can be fully described by a solution of the paraxial wave equation (PWE), which can be regarded as an analog of the Schrodinger equation for free particles. Except for an ideal plane wave, any eigensolution of the wave equation corresponds to a physically realizable \"eigen\" spatial mode, whose transverse structures are propagation-invariant apart from an overall change in size. For instance, the well-known Laguerre-Gauss (LG) and the Hermite-Gauss (HG) modes are eigensolutions of the PWE in cylindrical and Cartesian coordinates, respectively, and their complex amplitudes are as follows:  
\[\text{LG}_{\ell,p}(r,\varphi,z)=\sqrt{\frac{2p!}{\pi(p+|\ell|)!}}\frac{1}{w(z) }\Bigg{(}\frac{\sqrt{2}r}{w(z)}\Bigg{)}^{|\ell|}\exp\bigg{(}\frac{-r^{2}}{w_{ 2}^{2}}\bigg{)}L_{p}^{|\ell|}\bigg{(}\frac{2r^{2}}{w_{2}^{2}}\bigg{)}\,e^{i[kz+ \frac{k^{2}}{2 Rz}+\ell\varphi-(N+1)\phi]}, \tag{1}\]\[\text{HG}_{m,n}(x,y,z)=\sqrt{\frac{2^{1-n-m}}{\pi n!m!}}\frac{1}{w(z)}\text{exp} \left(\frac{-r^{2}}{w_{z}^{2}}\right)H_{m}\left(\frac{\sqrt{2}x}{w_{z}}\right)H_ {m}\left(\frac{\sqrt{2}y}{w_{z}}\right)e^{i[kz+\frac{k^{2}}{2w_{z}^{2}}-(N+1) \phi]}, \tag{10}\]  
where \(z_{R}=kw_{0}^{2}/2\), \(w_{z}=w_{0}\sqrt{1+(z/z_{R})^{2}}\), and \(R_{z}=z^{2}+z_{R}^{2}/z\) are the Rayleigh length, beam waist, and radius of curvature, respectively. \(L_{P}^{|\rho|}(\cdot)\) and \(H_{m/n}(\cdot)\) denote the Laguerre and Hermite polynomials, respectively. The term \(\phi=\arctan(z/z_{R})\) is the propagation-dependent Gouy phase, and their modal orders are defined as \(N=2p+|\ell|=m+n\). The above Hermite- and Laguerre-Gaussian mode families have their own special features in transverse structures but have the same fundamental TEM\({}_{00}\) mode, and each can independently constitute a unique and unbounded 2D Hilbert space. Therefore, in principle, we can represent any Gaussian mode by using coherent superpositions (i.e., complex series) of LG, HG, or their intermediate Hermite-Laguerre-Gaussian (HLG) modes. i.e., the generalized Hermite-Laguerre-Gaussian (HLG) mode. All the possible N-order HLG modes following the relation \(\ell=m-n\) can form the surface of a unit sphere, called a modal sphere, and unitary transformations for states on the sphere can be realized by using astigmatic operations.  
Astigmatic transformation (AT) refers to a unitary operation based on astigmatic propagation that makes a beam have different Gouy phases with respect to a given orthogonal axis. Specifically, the Gouy phase of HG\({}_{m,n}\) can be represented as follows [26; 27]:  
\[\begin{split}(N+1)\phi&=(m+1/2)\phi_{x}+(n+1/2) \phi_{y},\\ \phi_{x,y}&=\arctan(\frac{\Delta z_{x,y}}{z_{Rx,y}}) \end{split} \tag{11}\]  
For instance, the modal conversion HG\({}_{1,0}^{45^{\circ}}\rightarrow\text{LG}_{1,0}\) requires a relative phase retardation \(\delta=\phi_{y}-\phi_{x}=\pi/2\). So, we have the following:  
\[\text{HG}_{1,0}^{45^{\circ}}=\sqrt{1/2}(\text{HG}_{1,0}+\text{HG}_{0,1}) \rightarrow\sqrt{1/2}(\text{HG}_{1,0}+i\text{HG}_{0,1})=\text{LG}_{1,0}, \tag{12}\]  
In order to check the accuracy of spatial-mode generation and observation, Figure 11 shows the theoretical reference of the observed HLG modes in Figure 2a.  
In addition, the observed SCA profile at the measuring plane, denoted as \(\psi(\mathbf{r},z_{m})\), provides complete information about the light field. Thus, we can further retrieve the complete beam structure in the 3D space, denoted as \(\psi(\mathbf{r},z)\), by using the Collins (or Kirchhoff) diffraction integral with \(\psi(\mathbf{r},z_{m})\) as the source, which in cylindrical coordinates can be expressed as follows:\[\psi(r,\varphi,z)=\frac{i}{\lambda z}\textit{exp}(-ikz)\int r_{0}dr_{0}\int d\varphi _{0}\psi(r_{0},\varphi_{0},z_{m})\text{exp}\bigg{\{}-\frac{ik}{2z}\times\Big{[}r_ {0}^{2}-2 rr_{0}\text{cos}(\varphi-\varphi_{0})+r^{2}\Big{]}\bigg{\}}. \tag{10}\]