  Transmission matrix (TM) allows light control through complex media such as
multimode fibers (MMFs), gaining great attention in areas like biophotonics
over the past decade. The measurement of a complex-valued TM is highly desired
as it supports full modulation of the light field, yet demanding as the
holographic setup is usually entailed. Efforts have been taken to retrieve a TM
directly from intensity measurements with several representative phase
retrieval algorithms, which still see limitations like slow or suboptimum
recovery, especially under noisy environment. Here, a modified non-convex
optimization approach is proposed. Through numerical evaluations, it shows that
the nonconvex method offers an optimum efficiency of focusing with less running
time or sampling rate. The comparative test under different signal-to-noise
levels further indicates its improved robustness for TM retrieval.
Experimentally, the optimum retrieval of the TM of a MMF is collectively
validated by multiple groups of single-spot and multi-spot focusing
demonstrations. Focus scanning on the working plane of the MMF is also
conducted where our method achieves 93.6% efficiency of the gold standard
holography method when the sampling rate is 8. Based on the recovered TM, image
transmission through the MMF with high fidelity can be realized via another
phase retrieval. Thanks to parallel operation and GPU acceleration, the
nonconvex approach can retrieve an 8685$\times$1024 TM (sampling rate=8) with
42.3 s on a regular computer. In brief, the proposed method provides optimum
efficiency and fast implementation for TM retrieval, which will facilitate wide
applications in deep-tissue optical imaging, manipulation and treatment.
