Fig. 1. Theoretical prediction of the phase distributions of the input and output OAM beams. (a) ${\ell }_{\text{in}}=3/2$ and $n=2$; (b) ${\ell }_{\text{in}}=4/3$ and $n=3$; (c) ${\ell }_{\text{in}}=8/5$ and $n=5$. The second and the third columns illustrate the phase distributions of the output beam after the transformation without and with $\mathcal{P}(\rho ,\phi )$, respectively.

Fig. 2. Numerical results for FOAM sorting. The first two columns are the intensity and phase distribution of the output beam. The last five columns are the results of topological matching for ${\ell }_{\mathrm{all}}=-2n,-n,0,n,2n$. (a) ${\ell }_{\text{in}}=3/2$ and $n=2$. (b) ${\ell }_{\text{in}}=5/2$ and $n=2$. (c) ${\ell }_{\mathrm{in}}=4/3$ and $n=3$. (d) ${\ell }_{\mathrm{in}}=7/3$ and $n=3$.

Fig. 3. Experimental setup. (a) and (b) Phase maps with different parameters that are used in our experiment. The transformation phase and the correction phase are shown from left to right, respectively. (a) $t=1/2$ and $n=2$. (b) $t=1/3$ and $n=3$. (c) Schematic of the experimental setup. BS, beam splitter (see the text for more details).

Fig. 4. Experimental results of effective sorting of two sets of FOAM modes with different parameters. (a) $t=1/2$ and $n=2$, (b) $t=1/3$ and $n=3$. The first two columns are the intensity distribution before and after the transformation, respectively.

Fig. 5. Mode purity analysis. The numerical and experimental OAM mode purity of the sorted FOAM modes with (a) $t=1/2$ and $n=2$; (b) $t=1/3$ and $n=3$.