Mechanisms of spatiotemporal mode-locking

Abstract

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...

Single-mode output by controlling the spatiotemporal nonlinearities in mode-locked femtosecond multimode fiber lasers

“…Based on our results, one can explain the measured output beam profile evolution in the laser output with the minimum loss principle. 24 An alternative explanation based on the NPE saturable absorber might also be partly responsible for the experimentally obtained beam cleaning. NPE mode-locking requires nonlinearly induced polarization changes to generate saturable absorber behavior.…”

“…However, the majority of the previous sly reported multimode fiber lasers have also employed the nonlinear polarization evolution modelocking technique, and they did not observe a high-quality beam profile. [9][10][11]24 Furthermore, we would like to highlight that modification/shaping of the beam in free-space propagation has a very small impact to the output field, since the light travels through an approximately 4 m coiled multimode fiber with different mode numbers and splice positions. Another important parameter in the demonstrated cavity is the accumulated modal dispersion.…”

Real-time multispeckle spectral-temporal measurement unveils the complexity of spatiotemporal solitons

“…The presence of multiscale disorders owing to spatiotemporal dispersions and strong nonlinear interactions gives rise to challenges in both theoretical and experimental investigations. Theoretical models have recently been developed for understanding the generation of 3D dissipative solitons 15 , 16 . An experimental framework for real-time observation, however, is yet to be demonstrated, whereas the time-averaged measurements of 3D solitons have recently been demonstrated by sampling the laser beam profiles 17 , 18 and manipulating the pump power 19 .…”

“…The real-time observation on 3D soliton dynamics, however, has been largely unexplored, and applying traditional technologies to 3D soliton dynamics is not straightforward 32 – 36 . Spatiotemporal technologies, such as delay-scanning off-axis digital holography 15 , TERMITES 34 , SEA TADPOLE 37 , and other counterparts 38 , 39 , have recently been demonstrated to study 3D femtosecond pulses with high temporal resolutions—powerful tools for the characterization and optimization of ultrashort pulse lasers. Rather than high repetition rate pulse lasers, they are more suitable for low repetition rate pulse lasers with the identical pulse-to-pulse property.…”

Harnessing a multi-dimensional fibre laser using genetic wavefront shaping

“…The design of control schemes for multi-dimensional oscillators is complicated and challenging since many factors, including chromatic dispersion, modal dispersion, nonlinearity, gain and loss, are nonlinearly coupled and contribute to complexity and uncertainty, thus imposing additional rigorous control requirements. In addition, directly applying traditional one-dimensional (1D) technologies to control multi-dimensional optical systems is not straightforward 13 .…”