A. Experimental setup

The experimental setup and operation principle are illustrated as Fig. 1. The light source illuminates the digital micromirror array device (DMD) through an isolator and a collimator. The training and testing patterns are loaded onto the DMD. The diffracted light from the DMD passes through an objective, which images the patterns on the input facet of a 1.5 m-long MMF (Nufern, MM-S400/440-22A). After the input light field is distorted by MMF, a speckle field is formed by superimposition of modes on the output facet. It is imaged by another objective to an industrial camera. The speckles are recovered to the original images by an optimized approximate ITM or trained neural networks.We experimentally compare the performance of five kinds of fiber light sources in the system: a broadband ASE fiber source (BASE), an ASE fiber source with narrow linewidth (NASE), a common fiber laser with FBG-based cavity (CFL), a tunable NLL, and an RFL. The specific characteristics and parameters of these lasers are shown in the Results section. Control experiments are designed by changing the light sources while keeping the back-end part fixed. The information of devices and experimental details are presented in the supplementary material.

B. Principle of image transmission

There exist several methods of single-fiber image transmission currently as mentioned above. Structurally, the conventional dual-arm interferometric measurement method66. Y. Choi, C. Yoon, M. Kim, T. D. Yang, C. Fang-Yen, R. R. Dasari, K. J. Lee, and W. Choi, “Scanner-free and wide-field endoscopic imaging by using a single multimode optical fiber,” Phys. Rev. Lett. 109, 203901 (2012). https://doi.org/10.1103/physrevlett.109.203901 requires a reference optical path beyond the MMF. The reference light is coherently superimposed with the signal light to obtain the phase information. The interferometric results are inevitably related to the polarization state, stability, and coherence length of the light source, making the relationship between the system performance and the source more complicated.

Therefore, we adopt a simple interference-free structure. It can reconstruct images by only acquiring the amplitude distribution of the output fields to optimize the ITM. This single-arm structure is more conducive to analyzing the impact of source characteristics on the system. The reflective structure is closer to practical application scenarios.

Whether solving the TM in the frequency domain or in the spatial domain, it is necessary to obtain the set of different input responses. The TM can be viewed as a Green’s function that satisfies between an origin array and a response array.2626. E. G. van Putten, “Disorder-enhanced imaging with spatially controlled light,” Ph.D. Thesis (University of Twente, 2011), pp. 68–69. Specifically, the optical path between the input images and the captured speckles (i.e., from the DMD surface to the sensing surface of the camera in Fig. 1) is viewed as an overall transmission system. The inverse transmission matrix of this system is optimized iteratively so that the product of the output speckle and the ITM can gradually approximates the target image.We optimize the inverse transmission matrix model2727. P. Caramazza, O. Moran, R. Murray-Smith, and D. Faccio, “Transmission of natural scene images through a multimode fibre,” Nat. Commun. 10, 2029 (2019). https://doi.org/10.1038/s41467-019-10057-8 to achieve better results with shorter run time. Apart from parameter adjustment, we modify the loss function and optimizer. The cosine similarity between the real input image and reconstructed image is used as the loss function L,L=∑i=1n2YRi×Yi∑i=1n2YRi2×∑i=1n2Yi2,(1)where YRi and Yi are the elements in the reconstructed vector and the real vector, respectively.

The difference between the output value and the true value is fed back to the ITM. We compute the derivatives of the loss function L with respect to the elements of the ITM using adaptive moment estimation (Adam) as the optimizer. It is an extension of the classical stochastic gradient descent (SGD) method and can update the network weights more efficiently. The ITM is optimized iteratively with a specific step size and the loss function converges to a minimum value. The step size of the optimization (i.e., learning rate) also varies adaptively using the loss function as a monitor to enable faster convergence. In our experiment, the acquisition time of training set is 1 h 15 min, the training time is 54 min 35 s, and the average single-step training time is about 31 s.

The ITM after each epoch of optimization is judged by the peak signal-to-noise ratio (PSNR). The PSNR after each epoch is compared with the previous one. The model of this checkpoint is saved if the PSNR increases. The PSNR tends to be stable after several iterations. The ITM method based on dataset training used in this paper is intermediate between the traditional solution of the real TM and the deep learning neural networks with absolutely no assumption for the system. Compared to solving the complete TM, it omits the reference optical path and eliminates the need to collect phase information; and compared to traditional deep learning methods, in this particular physical circumstance, it restrict the function describing the system to the form of a TM, making the network concrete. This results in simple, efficient, and high-quality image transmission.

There are still some limitations of this method for image reconstruction, such as the large amount of training data in the early stage, and the difficulty in recovering three-dimensional information of the input light field. We need to retrain after changing the light source. It is important to ensure the stability and alignment of the fiber system. Deformation of optical fiber will interfere with the comparison between sources. We should try to avoid it, although detection with different bend radii can be achieved by joint learning.2828. L. Wang, Y. Yang, Z. Liu, J. Tian, Y. Meng, T. Qi, T. He, D. Li, P. Yan, M. Gong, Q. Liu, and Q. Xiao, “High-speed all-fiber micro-imaging with large depth of field,” Laser Photonics Rev. 16, 2100724 (2022). https://doi.org/10.1002/lpor.202100724 In addition, in order to evaluate the impact of the image demodulation method on the comparison of light sources, we also apply the neural network approach to training the same database as a benchmark. The processing results of the classical U-net type network verify the experimental conclusions similarly. The specific network structure is shown in Fig. S5, and parameter settings are described in the supplementary material.

C. RFL structure

Figure 2 shows the structure of the RFL used in this paper. To achieve a stable temporal output from the RFL oscillator, the RFL needs to operate well above the lasing threshold.2929. O. A. Gorbunov, S. Sugavanam, and D. V. Churkin, “Intensity dynamics and statistical properties of random distributed feedback fiber laser,” Opt. Lett. 40, 1783–1786 (2015). https://doi.org/10.1364/ol.40.001783 However, too high output power is unsuitable for illumination and image transmission system. Therefore, the RFL adopts a half-open-cavity structure with a 30 km passive fiber (CDSEI, SM-G652D) to decrease the lasing threshold.3030. Z. Wang, H. Wu, M. Fan, L. Zhang, Y. Rao, W. Zhang, and X. Jia, “High power random fiber laser with short cavity length: Theoretical and experimental investigations,” IEEE J. Sel. Top. Quantum Electron. 21, 10–15 (2015). https://doi.org/10.1109/jstqe.2014.2344293 We premeasured the threshold power of the random laser before the experiment to ensure that the RFL operates at a steady state. Specific experimental details are provided in the supplementary material. Random distributed Rayleigh scattering of the 30 km passive fiber and the point-action reflection of the high-reflectivity fiber Bragg grating (HR-FBG, with 0.21 nm bandwidth, core/cladding diameters are 10/125 µm) forms the half-open resonator. An LD (DILAS, 975.6 nm, 25 W, core/cladding diameters are 105/125 µm) serves as the pump source and the pump light is injected into the ytterbium-doped fiber (Nufern, LMA-YDF-10/130-VIII) through a (2 + 1) × 1 combiner. The gain of the ytterbium ions excites a 1064 nm random laser. To reduce feedback, an isolator is inserted before the output and the other pigtail of the HR-FBG is angle cleaved. The output power of the RFL is about 0.8 mW, which is the same as other light sources to ensure the same illumination condition. The structures of the other four sources are shown in Figs. S1–S4 of the supplementary material.