Fig. 1. TM retrieval with fast Fourier transform (FFT) and phase correction from intensity measurements without reference. (a) Comparison of data acquisition between the reference-less methods and the reference-based methods. The reference-based methods measure the complex fields with a reference beam, while the reference-less methods take only intensity without any reference, leading to a simpler and more stable experimental setup. (b) Computational efficiency improvement using FFT. In the conventional methods, the incident fields are directly generated with random phases, and the forward model of scattering has to be computed by matrix–vector multiplication. Our method designs the incident fields based on the Fourier transform matrix. Thus the forward model of scattering can be computed by FFT, which significantly improves computational efficiency. It also allows the inverse algorithm of TM retrieval to be implemented with FFT. (c) Correction of the error of phase offset using defocus intensity images. The estimated TM from the intensity images measured at one defocus plane has the error of phase offset. Our algorithm corrects the phase errors using the defocus intensity images.

Fig. 2. Experimental setup. The light-modulation module on the left of the MMF simultaneously generates the incident complex fields for both polarizations, while the calibration module on the right measures the intensity distribution of the transmitted complex fields. The abbreviations are defined as follows: L1 to L5, lens; DMD, digital micromirror device; M1 to M3, mirror; HWP, half-wave plate; QWP, quarter-wave plate; PBS1-2, polarization beam splitter; OBJ1-2, objective lens; LP, linear polarizer; CMOS, complementary metal oxide semiconductor.

Fig. 3. Full procedure of the TM retrieval method. The intensity images are measured at one fixed camera plane.

Fig. 4. Error of recovered TM using the simulated datasets. (a) Normalized amplitude error of the recovered TM for datasets of different M. The errors of $M=3$ to 5 share the same color bar on the top right, while the errors of other data sets share the color bar on the bottom. The plot at the bottom left shows the RMSE of amplitude of the recovered TM. (b) Phase error of the recovered TM for data sets of different $M$. The errors of $M=3$ to 5 share the top right color bar, while the errors of the other data sets share the bottom color bar. The plot shows the RMSE of phase of the recovered TM for different $M$.

Fig. 5. Comparison of the foci generated using the recovered TM and the TM measured by the off-axis holography method. (a) Series of measured speckle intensity images. We give an example of the measured images using the data set of $M=7$. (b) Prepossessing step half-samples the measured $128\times 128$ images into $64\times 64$ images. (c) Binary fiber mask ($M=7$). The white region of the mask indicates the distal end of the MMF fiber. (d) Recovered PR of $M=4$ to 8 and the recovered PR of the holography method. For the cases of $M=7$ and 8, the PR of the foci is near to that of the case of holography. The number inside the image is the average of the top 2000 PR. (e) Histogram of the top 2000 PR. The TM of $M=8$ has 1424 foci that have PRs higher than 0.60, whereas the holography method has 1266 foci above 0.60. (f) Sum projection of selected foci.

Fig. 6. Simulation for the TM retrieval algorithm with phase correction. (a) Defocus intensity images measured for the phase correction. (b) Recovered phase offset by the phase correction algorithm. (c) Amplitude error and phase error of the recovered TM with phase correction. The amplitude error is obtained by subtracting the amplitudes of the corrected TM with the true TM. The phase error is the difference between the phases of the corrected TM and the true TM after removing a constant phase offset. The RMSE of all rows of the TM are organized in $128\times 128$ grids, corresponding to the distal end of the MMF. The numbers inside the images are the RMSE over all rows.

Fig. 7. Correction of the phase offset error in the TM using defocus intensity images. (a) Stack of defocus images. (b) Recovered phase offset by the phase correction algorithm. (c) The intensity images generated using the TM with the error of phase offset and the recovered TM with phase correction. The top row shows the measured intensity images using the TM with the error of phase offset. The foci scattered at large defocus distances. The bottom row shows the measured intensity images using the recovered TM with phase correction. The top row images of $-40$, $-60$, $-80$, and $-100\mu \mathrm{m}$ share the top color bar, whereas the other images share the bottom color bar.

Fig. 8. Sum projection of selected foci measured at different defocus distances.

Table 1. TM retrieval algorithm with FFT achieves 1200× speedup.

Table 1. Optimization of recovering a row of TM tk.