Direct phase-locking algorithm for tiled-aperture coherent beam combing systems

We present a novel method for coherent beam combining (CBC) by exploiting the analytic modal decomposition (MD) method based on the Moore-Penrose inverse matrix. Emphatically, this method produces a highest intensity in the central lobe on the target plane in a non-iterative or direct manner. We verify the feasibility of the proposed method both numerically and experimentally: The related extensive numerical investigations are carried out for 3-, 7-, 19-, and 37-channel CBC configurations, including the power-in-the-bucket (PIB) efficiency, channel phase error, and phase acquisition speed with three different target image resolutions (64×64, 128×128, and 256×256 pixels) under both noise-free and noise-present conditions. The proposed method demonstrates near-perfect performance under noise-free conditions, achieving a PIB efficiency of over 0.999 particularly for 19-channel CBC with an image resolution of 64×64 although the performance degrades somewhat under noisy conditions. We also conduct a proof-of-concept experiment based on a 3-channel CBC system in a triangular tiled aperture format implemented with the proposed method. An average PIB efficiency of 0.912 is achieved with the estimated phase acquisition time of 38.7 ms without dynamic tip-tilt and alignment control. To the best of our knowledge, this proof-of-concept experimental result is the first demonstration of a CBC system realized by the analytic MD method. In addition, we provide a theoretical analysis of how fringe pattern correlation affects the noise susceptibility of the proposed method, and explain that introducing slight irregularity in beam arrangement can mitigate this effect. We expect the proposed method to pave the way to the advancement of the CBC technology as a novel phase control method.