Full-field single-shot measurement of beam self-cleaning dynamics in graded-index multimode fibers

Kerr beam self-cleaning in graded-index multimode fibers (GRIN MMFs) is a nonlinear effect that drives the spontaneous transformation of complex multimode beams into smooth bell- shaped spatially-coherent output intensity profiles. Its physical origin links to important areas of physics including self-organization, hydrodynamics and nonlinear wave condensation, and from a practical perspective, Kerr beam self-cleaning holds significant promise for spatio-temporal reshaping and high-power beam delivery. However, there is significant debate about the detailed physics of the process, and the fact that previous experimental studies have been based only on spatial intensity measurements has meant that factors such as spectral dynamics, their coupling to modal evolution, and the coherence of beam self-cleaning remain insufficiently understood. In this work, we address these issues directly through a comprehensive experimental study of Kerr self- cleaning in a GRIN MMF where we explicitly reconstruct the complete complex spatial field in real-time using single-shot off-axis digital holography. This interferometric phase-sensitive approach enables us to explicitly resolve the modal structure of the field as a function of spectral content and injected power. This allows us to show that even small-scale spectral broadening arising from self-phase modulation plays a key role in the energy transfer between modes, resulting in nonlinear spatial filtering and spectrally localized self-cleaning. We also show that the dynamics are fully deterministic and perfectly coherent, following a well-defined nonlinear mode evolution rather than being governed by stochastic effects. Our experimental results and physical interpretation are supported by comprehensive numerical simulations. These findings provide conclusive evidence that self-cleaning arises from deterministic and spectrally selective coherent nonlinear mode interactions rather than condensation-like thermodynamic energy redistribution. Beyond addressing the physical understanding of Kerr beam self-cleaning, our approach also opens a more general path to analyze dynamics in multimode systems spectrally, temporarily and spatially. Our insights also have a significant impact for engineered multimode fiber systems where spectral and modal shaping can be co-optimized for advanced applications in nonlinear optics, coherent beam combining, and spatiotemporal control of structured light.