Mode-locking is a process in which different modes of an optical resonator establish, through nonlinear interactions, stable synchronization. This self-organization underlies light sources that enable many modern scientific applications, such as ultrafast and high-field optics and frequency combs. Despite this, mode-locking has almost exclusively referred to self-organization of light in a single dimension -time. Here we present a theoretical approach, attractor dissection, for understanding three-dimensional (3D) spatiotemporal mode-locking (STML). The key idea is to find, for each distinct type of 3D pulse, a specific, minimal reduced model, and thus to identify the important intracavity effects responsible for its formation and stability. An intuition for the results follows from the "minimum loss principle," the idea that a laser strives to find the configuration of intracavity light that minimizes loss (maximizes gain extraction). Through this approach, we identify and explain several distinct forms of STML. These novel phases of coherent laser light have no analogues in 1D and are supported by experimental measurements of the three-dimensional field, revealing STML states comprising more than 10 7 cavity modes. Our results should facilitate the discovery and understanding of new higher-dimensional forms of coherent light which, in turn, may enable new applications.The supplementary material in this document is organized as follows.Section 1 provides a description of spatiotemporal mode-locking (STML) in the frequency domain, in terms of the resonant frequencies of the cavity's modes.Section 2 details our primary numerical models, based on propagation of the laser field through the gain medium using generalized nonlinear Schrödinger equations (NLSEs), plus additional effects. These models include most relevant effects, but our use of them mainly focuses on the computationally-efficient case where only a small number of transverse modes are considered. These models are generalizations of the most widely used models for describing ultrafast lasers and nonlinear pulse propagation in the modern literature.Section 3 details the treatment of individual effects within the cavity in the simplified description of STML outlined in the paper. Specifically, we develop the nonlinear projection operations for each relevant effect and show the calculation of the attractors for that effect through Eqn. 2 in the main article.Section 4 describes reduced models which incorporate a subset of the effects considered in the primary numerical models, by combining components described in Section 3. Extending the approach from Section 3, these models are quite approximate as whole-laser simulations but are computationally compact enough to be applied to conditions relevant to our experiments in a 90transverse mode fiber, and simple enough to be interpreted easily. Accordingly, they help bridge the gap between experiments and the nonlinear wave physics of STML studied primarily in the few-mode case.Section 5 summarizes relevant findin...

Stimulated Raman scattering cascade spanning the wavelength range of 523 to 1750 nm using a graded-index multimode optical fiber We report on the generation of a Raman cascade spanning the wavelength range of 523 to 1750 nm wavelength range, in a standard telecommunication graded-index multimode optical fiber. Despite the highly multimode nature of the pump, the Raman peaks are generated in specific modes of the fiber, confirming substantial beam cleanup during the stimulated Raman scattering process.

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A short piece of commercial-grade SMF-28 optical fiber is pumped with a 680 ps high-peak power green laser. Red Stokes and blue anti-Stokes beams are generated spontaneously from vacuum noise in different modes in the fiber via intermodal four-wave mixing. Detailed experimental and theoretical analyses are performed and are in reasonable agreement. The large spectral shifts from the pump protect the Stokes and anti-Stokes from contamination by spontaneous Raman scattering noise. This work highlights the predictive power and limitations of a theoretical model to explain the experimental results for a process that relies on the amplification of quantum vacuum energy over more than 11 orders of magnitude.

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