Fig. 1. Theoretical comparisons of TM retrieval performances of prVBEM, GGS 2-1, and our RAF 2-1. (a) Schematic diagram of TM retrieval for an MMF. (b) Focusing efficiency achieved by different algorithms under a range of running times when N=576 and γ=4. (c) The iterations taken by different algorithms versus running times for the case of (b). (d) Focusing efficiency achieved by different algorithms under a range of γ when using N=576 and a running time of 20 s. Note the error bars denote the standard deviations of 30 repeated tests.

Fig. 2. Simulated (a) focusing efficiency and (b) focusing uniformity achieved by prVBEM, GGS 2-1, RAF, and RAF 2-1 under different SNR levels when using N=1024, γ=5, and a running time of 50 s. Note the error bars denote the standard deviations of 30 repeated tests.

Fig. 3. Experimental configuration for retrieving the TM of an MMF with speckle-intensity measurements. CW: continuous wave; C, collimator; DMD, digital mirror device; L, lens; P, polarizer; MMF, multimode fiber; Obj, objective lens; λ/4, quarter-wave plate.

Fig. 4. Comparison of light-focusing results through MMF with the TMs retrieved by different algorithms. (a) The histograms of normalized PBR of 20×20 foci and (b) the results of multispot focusing (pentagram) in the output field of MMF, both obtained by prVBEM, GGS 2-1, RAF, and RAF 2-1 with N=1024 and γ=5. Note the crosses in (a) represent the mean values, and the white dashed circles in (b) show the fiber region. The scale bar in (b)–(e) is 20μm.

Fig. 5. Comparison of the PBR maps of focusing on the working plane of the MMF using the TMs (a) measured by off-axis holography and (b) retrieved by RAF 2-1 under a range of γ with N=1024. The scale bar in (a) and (b) is 20μm.

Fig. 6. Comparison of image transmission results through MMF using the retrieved TMs by RAF 2-1 under a range of γ with N=1024. (a) Normalized reconstructed images for an object of the Chinese character “光” with the values of PCC to the object labeled. (b) The progression curves of PCC during the iterative reconstruction.

Fig. 7. Focusing efficiency achieved by GGS 2-1 and RAF 2-1 under a series of iteration ratios during their two-step gradient iterations, with 30 repeated tests.

Fig. 8. Normalized curves of error as a function of running time for RAF and RAF 2-1 when N=576 and γ=4. Note the error bars denote the standard deviations of 30 repeated tests.

Fig. 9. (a) Normalized curves of error as a function of running time for RAF with a fixed step size (μ) or an adaptive one. (b) Normalized curves of error as a function of running time for RAF with SD, NCG, or L-BFGS. Note the error bars denote the standard deviations of 30 repeated tests.