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 107 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.
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All data in the manuscript and supplementary are available from the corresponding author on reasonable request.
The principal components of the codes used in the manuscript have been made publicly available with extensive documentation at https://github.com/WiseLabAEP/GMMNLSE-Solver-FINAL Additional codes mostly build on this core code, and may be requested from the corresponding author.
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Portions of this work were supported by the Office of Naval Research (N00014-13-1-0649 and N00014-16-1-3027) and the National Science Foundation (ECCS-1609129, ECCS-1912742). L.G.W. acknowledges helpful discussions with A. Cerjan and T. Onodera.
L.G.W. and F.W.W. hold a US patent, number US20190207361A1 for STML.
Peer review information Nature Physics thanks Philippe Grelu, Fatih Ilday and Heping Zheng for their contribution to the peer review of this work.
Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Extended Data Fig. 1 Examples of experimental STML states ranging from the very narrow filter regime (a-c) through narrow-intermediate filter sizes (d-h) and large/no spatial filter (i-j).
The panels (left to right) show the modal energy distribution, the whole-field spectrum, and the measured and reconstructed beam profiles. The reconstructed beam profiles obtained from the 3D field modal decomposition are shown to help qualify the mode decomposition. Overall, we see similar trends as in the few-mode simulations. In the narrow-filter regime, STML states concentrate energy into radially-symmetric modes and a small number of low-order modes. The spectra in this regime also resemble the narrow, asymmetric spectra observed in few-mode simulations. For intermediate filter sizes, we see STML states with intermediate features, still tending to concentrate energy into radially-symmetric modes, but with a broader distribution of modes and with broader and less asymmetric, more rectangular spectra. Finally, for the largest spatial filter (here, meaning no spatial filtering besides that occurring in the optical isolator and fibre coupling) we see the experimental manifestation of the SAGE regime: broad symmetric spectra with a broad distribution of modes that is most heavily weighted to low-order modes. The absence of radial symmetric features in the beam profiles also evidences the transition from radially-symmetric low-order modes to the degenerate mode families of less-symmetric higher-order modes that occurs in the SAGE regime. Considering that 90 transverse modes are present in the oscillator, and that disordered linear mode coupling is important especially for low-symmetry modes, the agreement with the few-mode trends is good. The experimental results also agree well with the predictions of the reduced models in Supplementary Material, Section 4.
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Wright, L.G., Sidorenko, P., Pourbeyram, H. et al. Mechanisms of spatiotemporal mode-locking. Nat. Phys. 16, 565–570 (2020). https://doi.org/10.1038/s41567-020-0784-1
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