Fig. 1. Concept of the GS-based non-probe pre-compensation of distorted optical vortices.

Fig. 2. Computing the pre-compensation phase mask C(x,y) based on GS algorithm. (a) Flow chart. (b) C(x,y) can be obtained through subtracting the iteration output φ(x,y) from the initial helical phase of the transmitted optical vortices S(x,y).

Fig. 3. Experimental setup. Col., collimator; L, convex lens; CCD, infrared CCD camera; ID, iris diaphragm; PM, power meter.

Fig. 4. Observed intensity profiles of optical vortices. (a) No turbulence. (b) With turbulence r0=1  mm. (c) With turbulence r0=3  mm.

Fig. 5. Received power of diverse OAM channels with or without pre-compensation when |+2⟩ is transmitted. (a) Case of strong turbulence with Fried parameter r0=1  mm and d/r0=3.46. (b) Case of weak turbulence with Fried parameter r0=3  mm and d/r0=1.15. The number of iterations of the GS algorithm is 100.

Fig. 6. Mode purity of vortex beams (|+2⟩, |+3⟩) with and without compensation for various turbulence realizations (r0=1  mm for |+2⟩, d/r0=3.46 and |+3⟩, d/r0=4; r0=3  mm for |+2⟩, d/r0=1.15 and |+3⟩, d/r0=1.33). The number of iterations of the GS algorithm under all turbulence realizations is 100.

Fig. 7. Mode purities of a beam with topological charge l=2, as a function of number of iterations, at two turbulence values of different strength (Fried parameter r0=1  mm and r0=3  mm).