Fig. 1. Sketch map of optical OAM reconstitution through scattering media. (a) Without wavefront shaping, the OAM beam forms a disordered speckle pattern behind the scattering medium. (b) With an appropriate SLM phase mask applied in advance, optical OAM is reconstituted after scattering. Using an SPP with an opposite topological charge as a detection component, a focal point can be generated on the screen.

Fig. 2. Experimental setup for OAM restoration behind the strongly scattering media. HWP, half-wavelength plate; $P_1$, $P_2$, linear polarizer; BS, beam splitter; SLM, spatial light modulator; M, reflecting mirror; SPP, spiral phase plate, which has a center-symmetrical phase distribution as shown; $L_{1-5}$, lens; f₁₋₅, 30, 200, 200, 200, 100 mm. Inset: a scanning electron microscopy image of the ${\mathrm{TiO}}_2$ powder.

Fig. 3. Experimental demonstration of measuring OAM and calibration. (a) An example for the OAM state with $\ell =6$ generated by SLM. (b)–(f) Far-field diffraction patterns while the OAM beam in (a) passes through an SPP with different topological charges.

Fig. 4. (a) Calibration of detecting system with an SPP of $\ell =-6$. (b) Direct imaging for the OAM states through scattering media before algorithm optimization. (c) During algorithm optimization, the light intensity of the target area is significantly enhanced. The red circles indicate the target area to be optimized marked by the calibration.

Fig. 5. (a) Enhancement factor curves of different topological charges (ℓ = 4–7). In our experiment, a typical value is 150 after 500 generations. (b) Stability measurement of the focusing relative intensity over 10 h.

Fig. 6. Reconstitution of OAMs in different spatial directions. (a) and (b) have a deflection angle of around $\pm 10°$, while (c) and (d) have a pitch angle of around $\pm 6°$. The red cross is the optical axis direction of the original optical path.