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...

We demonstrate a fiber source with the best performance from an ultrafast fiber oscillator to date. The ring-cavity Mamyshev oscillator produces ~50-nJ and ~40-fs pulses. The peak power is an order of magnitude higher than that of previous lasers with similar fiber mode area. This performance is achieved by designing the oscillator to support parabolic pulse formation which enables the management of unprecedented nonlinear phase shifts. Experimental results are limited by available pump power. Numerical simulations reveal key aspects of the pulse evolution, and realistically suggest that (after external compression) peak powers that approach 10 MW are possible from ordinary single-mode fiber. The combination of practical features such as environmental stability, established previously, with the performance described here make the Mamyshev oscillator extremely attractive for applications.

Building on the scientific understanding and technological infrastructure of single-mode fibers, multimode fibers are being explored as a means of adding new degrees of freedom to optical technologies such as telecommunications, fiber lasers, imaging, and measurement. Here, starting from a baseline of single-mode nonlinear fiber optics, we introduce the growing topic of multimode nonlinear fiber optics. We demonstrate a new numerical solution method for the system of equations that describes nonlinear multimode propagation, the generalized multimode nonlinear Schrödinger equation. This numerical solver is freely available, implemented in MATLAB ® and includes a number of multimode fiber analysis tools. It features a significant parallel computing speed-up on modern graphical processing units, translating to orders-of-magnitude speed-up over the split-step Fourier method. We demonstrate its use with several examples in graded-and step-index multimode fibers. Finally, we discuss several key open directions and questions, whose answers could have significant scientific and technological impact.

Whereas traditional cryptography encrypts a secret message into an unintelligible form, steganography conceals that communication is taking place by encoding a secret message into a cover signal. Language is a particularly pragmatic cover signal due to its benign occurrence and independence from any one medium. Traditionally, linguistic steganography systems encode secret messages in existing text via synonym substitution or word order rearrangements. Advances in neural language models enable previously impractical generation-based techniques. We propose a steganography technique based on arithmetic coding with large-scale neural language models. We find that our approach can generate realistic looking cover sentences as evaluated by humans, while at the same time preserving security by matching the cover message distribution with the language model distribution.

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