Fig. 1. Geometry of sources, scattering plane, observation plane, and propagation system.

Fig. 2. Simulations of intensity pattern prior to the diffuser when the polarizer orientation is 45°. The topological charges of the vortex beams are 1, 2, and 3, respectively.

Fig. 3. Experimental setup for measuring the topological charge of the vortex beams from laser speckle. Laser, He–Ne laser; MO, microscope objective; P, pinhole; $L_1$, $L_2$, and $L_3$, lens; HWP, half-wave plate; BS, beam splitter; M, mirror; SLM, spatial light modulator; GG, ground glass; CCD, charge coupled device.

Fig. 4. Experimental results and numerical simulations of the Fourier transform of the cross-covariance for the vortex beam with topological charge 1. (a)–(d) are the experimental results for rotation angles of polarizer 0°, 45°, 90°, and −45°. (e)–(h) are the simulations corresponding to (a)–(d), respectively.

Fig. 5. Experimental results and numerical simulations of the Fourier transform of the cross-covariance for vortex beam case without a polarizer. (a)–(c) are the experimental results for topological charges 1, 2, and 3. (d)–(f) are the simulations corresponding to (a)–(c), respectively.