Efficient reference-less transmission matrix retrieval for a multimode fiber using fast Fourier transform

Transmission matrix (TM) linearly maps the incident and transmitted complex fields, and has been used widely due to its ability to characterize scattering media. It is computationally demanding to reconstruct the TM from intensity images measured by a reference-less experimental setup. Removing reference beam for interference gains the advantage of simple experimental setup. However, the long computational time still limits its practical application. We propose an efficient reference-less TM retrieval method for multimode fiber (MMF). Our method adopts a data acquisition scheme which employs Fourier transform matrix in the design of the incident fields. We develop a nonlinear optimization algorithm to solve the TM retrieval problem in a parallel manner. The data acquisition scheme allows the algorithm to be implemented with fast Fourier transform (FFT), and hence achieves great efficiency improvement. Further, our method acquires intensity images at a defocus plane and correct the error of relative phase offset of TM recovered from the intensity images measured at one fixed plane. We validate the proposed TM retrieval method with both simulations and experiments. By using FFT, our TM retrieval algorithm achieves 1200x speed-up in computational time, and recovers $2286 \times 8192$ TM of a 0.22 NA and $50 \ \mu m$ diameter MMF with 124.9 seconds by a computer of 32 CPU cores. With the advantages of efficiency and the correction of phase offset, our method paves the way for the application of reference-less TM retrieval in real practice.