Supplementary File: Single Mode Output By Controlling The Spatiotemporal Nonlinearities In Mode-Locked Femtosecond Multimode Fiber Lasers

The performance of fiber mode-locked lasers is limited due to the high nonlinearity induced by the spatial confinement of the single-mode fiber core. To massively increase the pulse energy of the femtosecond pulses, amplification is performed outside the oscillator. Recently, spatiotemporal mode-locking has been proposed as a new path to fiber lasers. However, the beam quality was highly multimode, and the calculated threshold pulse energy (>100 nJ) for nonlinear beam self-cleaning was challenging to realize. We present an approach to reach high energy per pulse directly in the mode-locked multimode fiber oscillator with a near single-mode output beam. Our approach relies on spatial beam self-cleaning via the nonlinear Kerr effect, and we demonstrate a multimode fiber oscillator with M-2 < 1.13 beam profile, up to 24 nJ energy, and sub-100 fs compressed duration. Nonlinear beam self-cleaning is verified both numerically and experimentally for the first time in a mode-locked multimode laser cavity. The reported approach is further power scalable with larger core sized fibers up to a certain level of modal dispersion and could benefit applications that require high-power ultrashort lasers with commercially available optical fibers.