Mechanisms of spatiotemporal mode-locking

Mode-locking of lasers can be understood as self-organization, and the three-dimensional case of spatiotemporal mode-locking can described using attractor dissection theory, which helps develop an intuition for this complex case. Mode-locking is a process in which different modes of an optical resonator establish stable synchronization through non-linear interactions. 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 the self-organization of light in a single dimension-time. Here we present a theoretical approach-attractor dissection-to understand three-dimensional spatiotemporal mode-locking. The key idea is to find a specific, minimal reduced model for each distinct type of three-dimensional pulse, and thus 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 spatiotemporal mode-locking. These phases of coherent laser light have no analogues in one dimension and are supported by measurements of the three-dimensional field, which reveals spatiotemporal mode-locked states that comprise 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.