Fig. 1. Steps of identifying superposition states of two different LG modes by measuring light intensity distribution. Three boxes in the first row represent characteristics of light intensity distribution used, and three boxes on right of the second row represent information obtained at each step

Fig. 2. Experimental device diagram of vortex beam's superposition state transmitted in water channel

Fig. 3. Processing of experimental images. (a) LG superposition state's light intensity distribution diagram whose background light noise is filtered ;(b) binarization result which is used to identify number and region of petals; (c) maximum extraction and Pratt method circle fitting^[27]

Fig. 4. Evaluation function values in different combinations of $(l_{\mathrm{1}},l_{\mathrm{2}})$

Fig. 5. Light intensity diagram of vortex beam's superposition states transmitted through a water channel

Fig. 6. Theoretical diagrams and experimental results of light intensity distribution

Fig. 7. Recognition results of superposition states of 16 vortex beams. (a) Minimum value of evaluation function $\gamma$ of 16 quantum states and ratio $\kappa$ between minimum value of evaluation function and its second minimum value; (b) fidelity F of 16 quantum states; 1 to 16 respectively represent topological charge combinations of (1,-4), (-2,3), (1,-5), (-2,4), (-4,3), (-5,2), (5,-3), (6, -2)、(-6, 3), (5, -4), (-7, 3), (6, -4), (-6, 5), (-7, 4), (-7, 5), and (8, -4)

Fig. 8. Experimental results of different combined topological charges under different salinities. (a), (b), (c), and (d) respectively represent minimum value of evaluation function $\gamma$ of (-3,4),(-3,5),(-4,5),(-4,6) and ratio $\kappa$ between minimum value of evaluation function $\gamma$ and its second minimum value