The Gerchberg–Saxton (GS) algorithm, which retrieves phase information from the measured intensities on two related planes (the source plane and the target plane), has been widely adopted in a variety of applications when holographic methods are challenging to be implemented. In this work, we showed that the GS algorithm can be generalized to retrieve the unknown propagating function that connects these two planes. As a proof-of-concept, we employed the generalized GS (GGS) algorithm to retrieve the optical transmission matrix (TM) of a complex medium through the measured intensity distributions on the target plane. Numerical studies indicate that the GGS algorithm can efficiently retrieve the optical TM while maintaining accuracy. With the same training data set, the computational time cost by the GGS algorithm is orders of magnitude less than that consumed by other non-holographic methods reported in the literature. Besides numerical investigations, we also experimentally demonstrated retrieving the optical TMs of a stack of ground glasses and a 1-m-long multimode fiber using the GGS algorithm. The accuracy of the retrieved TM was evaluated by synthesizing high-quality single foci and multiple foci on the target plane through these complex media. These results indicate that the GGS algorithm can handle a large TM with high efficiency, showing great promise in a variety of applications in optics.
Characterizing the transmission matrix (TM) of a multimode fiber (MMF) benefits many fiber-based applications and allows in-depth studies on the physical properties. For example, by modulating the incident field, the knowledge of the TM allows one to synthesize any optical field at the distill end of the MMF. However, the extraction of optical fields usually requires holographic measurements with interferometry, which complicates the system design and introduces additional noise. In this work, we developed an efficient method to retrieve the TM of the MMF in a referenceless optical system. With pure intensity measurements, this method uses the extended Kalman filter (EKF) to recursively search for the optimum solution. To facilitate the computational process, a modified speckle-correlation scatter matrix (MSSM) is constructed as a low-fidelity initial estimation. This method, termed EKF-MSSM, only requires 4N intensity measurements to precisely solve for N unknown complex variables in the TM. Experimentally, we successfully retrieved the TM of the MMF with high precision, which allows optical focusing with the enhancement (>70%) close to the theoretical value. We anticipate that this method will serve as a useful tool for studying physical properties of the MMFs and potentially open new possibilities in a variety of applications in fiber optics.
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