Mechanisms of spatiotemporal mode-locking


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.

Access options

Rent or Buy article

Get time limited or full article access on ReadCube.


All prices are NET prices.

Fig. 1: Conceptual outline of STML, attractor dissection and the spatiotemporal maximum-gain principle.
Fig. 2: Identifying the mechanisms of 3D mode-locked pulses for varying intracavity SF size.
Fig. 3: Experimental regimes of STML and results from a reduced laser model.

Data availability

All data in the manuscript and supplementary are available from the corresponding author on reasonable request.

Code availability

The principal components of the codes used in the manuscript have been made publicly available with extensive documentation at Additional codes mostly build on this core code, and may be requested from the corresponding author.


  1. 1.

    Popmintchev, T. et al. Bright coherent ultrahigh harmonics in the keV X-ray regime from mid-infrared femtosecond lasers. Science 336, 1287–1291 (2012).

    ADS  MathSciNet  Article  Google Scholar 

  2. 2.

    England, R. J. et al. Dielectric laser accelerators. Rev. Mod. Phys. 86, 1337–1389 (2014).

    ADS  Article  Google Scholar 

  3. 3.

    Xu, C. & Wise, F. W. Recent advances in fiber lasers for nonlinear microscopy. Nat. Photon. 7, 875–882 (2013).

    ADS  Article  Google Scholar 

  4. 4.

    Cundiff, S. T. & Ye, J. Colloquium: femtosecond optical frequency combs. Rev. Mod. Phys. 75, 325–342 (2003).

    ADS  Article  Google Scholar 

  5. 5.

    Orringer, D. A. et al. Rapid intraoperative histology of unprocessed surgical specimens via fibre-laser-based stimulated Raman scattering microscopy. Nat. Biomed. Eng. 1, 0027 (2017).

    Article  Google Scholar 

  6. 6.

    Tilma, B. W. et al. Recent advances in ultrafast semiconductor disk lasers. Light Sci. Appl. 4, e310–e310 (2015).

    Article  Google Scholar 

  7. 7.

    Fattahi, H. et al. Third-generation femtosecond technology. Optica 1, 45–63 (2014).

    ADS  Article  Google Scholar 

  8. 8.

    Smith, P. W. Mode-locking of lasers. Proc. IEEE 58, 1342–1357 (1970).

    Article  Google Scholar 

  9. 9.

    Auston, D. Transverse mode locking. IEEE J. Quantum Electron. 4, 420–422 (1968).

    ADS  Article  Google Scholar 

  10. 10.

    Haus, H. A. Mode-locking of lasers. IEEE J. Sel. Top. Quantum Electron. 6, 1173–1185 (2000).

    ADS  Article  Google Scholar 

  11. 11.

    Krausz, F. et al. Femtosecond solid-state lasers. IEEE J. Quantum Electron. 28, 2097–2122 (1992).

    ADS  Article  Google Scholar 

  12. 12.

    Renninger, W., Chong, A. & Wise, F. Dissipative solitons in normal-dispersion fiber lasers. Phys. Rev. A 77, 023814 (2008).

    ADS  Article  Google Scholar 

  13. 13.

    Akhmediev, N., Soto-Crespo, J. M. & Grelu, P. Spatiotemporal optical solitons in nonlinear dissipative media: from stationary light bullets to pulsating complexes. Chaos 17, 037112 (2007).

    ADS  MathSciNet  MATH  Article  Google Scholar 

  14. 14.

    Renninger, W. H. & Wise, F. W. Spatiotemporal soliton laser. Optica 1, 101–104 (2014).

    ADS  Article  Google Scholar 

  15. 15.

    Kutz, J. N., Conti, C. & Trillo, S. Mode-locked X-wave lasers. Opt. Express 15, 16022–16028 (2007).

    ADS  Article  Google Scholar 

  16. 16.

    Büttner, T. F. S. et al. Multicore, tapered optical fiber for nonlinear pulse reshaping and saturable absorption. Opt. Lett. 37, 2469–2471 (2012).

    ADS  Article  Google Scholar 

  17. 17.

    Negus, D. K., Spinelli, L., Goldblatt, N. & Feugnet, G. Sub-100 femtosecond pulse generation by Kerr lens mode-locking in Ti:Al2O3. In Proc. Topical Meeting on Advanced Solid State Lasers (eds Dubé, G. & Chase, L.) SPL7 (OSA, 1991).

  18. 18.

    Herrmann, J. Theory of Kerr-lens mode locking: role of self-focusing and radially varying gain. J. Opt. Soc. Am. B 11, 498–512 (1994).

    ADS  Article  Google Scholar 

  19. 19.

    Kalosha, V. P., Müller, M., Herrmann, J. & Gatz, S. Spatiotemporal model of femtosecond pulse generation in Kerr-lens mode-locked solid-state lasers. J. Opt. Soc. Am. B 15, 535–550 (1998).

    ADS  Article  Google Scholar 

  20. 20.

    Jirauschek, C., Kärtner, F. X. & Morgner, U. Spatiotemporal Gaussian pulse dynamics in Kerr-lens mode-locked lasers. J. Opt. Soc. Am. B 20, 1356–1368 (2003).

    ADS  Article  Google Scholar 

  21. 21.

    Dunlop, A. M., Firth, W. J. & Wright, E. M. Master equation for spatio-temporal beam propagation and Kerr lens mode-locking. Opt. Commun. 138, 211–226 (1997).

    ADS  Article  Google Scholar 

  22. 22.

    Leonetti, M., Conti, C. & Lopez, C. The mode-locking transition of random lasers. Nat. Photon. 5, 615–617 (2011).

    ADS  Article  Google Scholar 

  23. 23.

    Antenucci, F., Crisanti, A., Ibáñez-Berganza, M., Marruzzo, A. & Leuzzi, L. Statistical mechanics models for multimode lasers and random lasers. Philos. Mag. 96, 704–731 (2016).

    ADS  Article  Google Scholar 

  24. 24.

    Nixon, M. et al. Real-time wavefront shaping through scattering media by all-optical feedback. Nat. Photon. 7, 919–924 (2013).

    ADS  Article  Google Scholar 

  25. 25.

    Wright, L. G., Christodoulides, D. N. & Wise, F. W. Spatiotemporal mode-locking in multimode fiber lasers. Science 358, 94–97 (2017).

    ADS  Article  Google Scholar 

  26. 26.

    Fox, A. G. & Tingye, Li Modes in a maser interferometer with curved and tilted mirrors. Proc. IEEE 51, 80–89 (1963).

    Article  Google Scholar 

  27. 27.

    Kogelnik, H. & Li, T. Laser beams and resonators. Appl. Opt. 5, 1550–1567 (1966).

    ADS  Article  Google Scholar 

  28. 28.

    Siegman, A. E. Lasers (University Science Books, 1986).

  29. 29.

    Haken, H Light: Laser Light Dynamics (North-Holland, 1985).

  30. 30.

    Qin, H., Xiao, X., Wang, P. & Yang, C. Observation of soliton molecules in a spatiotemporal mode-locked multimode fiber laser. Opt. Lett. 43, 1982–1985 (2018).

    ADS  Article  Google Scholar 

  31. 31.

    Matos, L. et al. Direct frequency comb generation from an octave-spanning, prismless Ti:sapphire laser. Opt. Lett. 29, 1683–1685 (2004).

    ADS  Article  Google Scholar 

  32. 32.

    Renninger, W. H., Chong, A. & Wise, F. W. Giant-chirp oscillators for short-pulse fiber amplifiers. Opt. Lett. 33, 3025–3027 (2008).

    ADS  Article  Google Scholar 

  33. 33.

    Xiong, W. et al. Principal modes in multimode fibers: exploring the crossover from weak to strong mode coupling. Opt. Express 25, 2709–2724 (2017).

    ADS  Article  Google Scholar 

  34. 34.

    Carpenter, J., Eggleton, B. J. & Schröder, J. Observation of Eisenbud–Wigner–Smith states as principal modes in multimode fibre. Nat. Photon. 9, 751–757 (2015).

    ADS  Article  Google Scholar 

  35. 35.

    Renninger, W. H. & Wise, F. W. Optical solitons in graded-index multimode fibres. Nat. Commun. 4, 1719 (2013).

    ADS  Article  Google Scholar 

  36. 36.

    Wright, L. G., Renninger, W. H., Christodoulides, D. N. & Wise, F. W. Spatiotemporal dynamics of multimode optical solitons. Opt. Express 23, 3492–3506 (2015).

    ADS  Article  Google Scholar 

  37. 37.

    Guenard, R. et al. Kerr self-cleaning of pulsed beam in an ytterbium doped multimode fiber. Opt. Express 25, 4783–4792 (2017).

    ADS  Article  Google Scholar 

  38. 38.

    Guenard, R. et al. Nonlinear beam self-cleaning in a coupled cavity composite laser based on multimode fiber. Opt. Express 25, 22219–22227 (2017).

    ADS  Article  Google Scholar 

  39. 39.

    Florentin, R. et al. Shaping the light amplified in a multimode fiber. Light Sci. Appl. 6, e16208 (2016).

    Article  Google Scholar 

  40. 40.

    Tzang, O., Caravaca-Aguirre, A. M. & Piestun, R. Wave-front shaping in nonlinear multimode fibers. Nat. Photon. 12, 368–374 (2018).

    ADS  Article  Google Scholar 

  41. 41.

    Iegorov, R., Teamir, T., Makey, G. & Ilday, F. Ö. Direct control of mode-locking states of a fiber laser. Optica 3, 1312–1315 (2016).

    ADS  Article  Google Scholar 

  42. 42.

    Woodward, R. I. & Kelleher, E. J. R. Towards ‘smart lasers’: self-optimisation of an ultrafast pulse source using a genetic algorithm. Sci. Rep. 6, 37616 (2016).

    ADS  Article  Google Scholar 

  43. 43.

    Angelani, L., Conti, C., Ruocco, G. & Zamponi, F. Glassy behavior of light. Phys. Rev. Lett. 96, 065702 (2006).

    ADS  Article  Google Scholar 

  44. 44.

    Wang, Z., Marandi, A., Wen, K., Byer, R. L. & Yamamoto, Y. Coherent Ising machine based on degenerate optical parametric oscillators. Phys. Rev. A 88, 063853 (2013).

    ADS  Article  Google Scholar 

  45. 45.

    Weill, R., Fischer, B. & Gat, O. Light-mode condensation in actively-mode-locked lasers. Phys. Rev. Lett. 104, 173901 (2010).

    ADS  Article  Google Scholar 

  46. 46.

    Gustave, F. et al. Observation of mode-locked spatial laser solitons. Phys. Rev. Lett. 118, 044102 (2017).

    ADS  Article  Google Scholar 

  47. 47.

    Lucas, E. et al. Spatial multiplexing of soliton microcombs. Nat. Photon. 12, 699–705 (2018).

    ADS  Article  Google Scholar 

  48. 48.

    Jang, J. K. et al. Synchronization of coupled optical microresonators. Nat. Photon. 12, 688–693 (2018).

    ADS  Article  Google Scholar 

  49. 49.

    Guang, Z., Rhodes, M., Davis, M. & Trebino, R. Complete characterization of a spatiotemporally complex pulse by an improved single-frame pulse-measurement technique. J. Opt. Soc. Am. B 31, 2736–2743 (2014).

    ADS  Article  Google Scholar 

  50. 50.

    Guang, Z., Rhodes, M. & Trebino, R. Measuring spatiotemporal ultrafast field structures of pulses from multimode optical fibers. Appl. Opt. 56, 3319–3324 (2017).

    ADS  Article  Google Scholar 

  51. 51.

    Pariente, G., Gallet, V., Borot, A., Gobert, O. & Quéré, F. Space–time characterization of ultra-intense femtosecond laser beams. Nat. Photon. 10, 547–553 (2016).

    ADS  Article  Google Scholar 

  52. 52.

    Shapira, O., Abouraddy, A. F., Joannopoulos, J. D. & Fink, Y. Complete modal decomposition for optical waveguides. Phys. Rev. Lett. 94, 143902 (2005).

    ADS  Article  Google Scholar 

  53. 53.

    Lü, H., Zhou, P., Wang, X. & Jiang, Z. Fast and accurate modal decomposition of multimode fiber based on stochastic parallel gradient descent algorithm. Appl. Opt. 52, 2905–2908 (2013).

    ADS  Article  Google Scholar 

  54. 54.

    Paurisse, M., Lévèque, L., Hanna, M., Druon, F. & Georges, P. Complete measurement of fiber modal content by wavefront analysis. Opt. Express 20, 4074–4084 (2012).

    ADS  Article  Google Scholar 

  55. 55.

    Verrier, N. & Atlan, M. Off-axis digital hologram reconstruction: some practical considerations. Appl. Opt. 50, H136–H146 (2011).

    Article  Google Scholar 

  56. 56.

    Kim, M. K. Principles and techniques of digital holographic microscopy. J. Photon. Energy 1, 018005 (2010).

    Article  Google Scholar 

  57. 57.

    Ding, Y. et al. Spatiotemporal mode-locking in lasers with large modal dispersion. Preprint at (2019).

Download references


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.

Author information




L.G.W. performed experiments and simulations and developed the theoretical models. P.S. constructed, with help from A.I. and L.G.W., the 3D pulse measurement and mode decomposition device and software. H.P. performed additional experiments. Z.M.Z. contributed to the numerical codes. B.A.M., C.R.M. and D.N.C. contributed to the formulation and analysis of the theoretical models. L.G.W. and F.W.W. wrote the paper, which was edited by all the authors. F.W.W. supervised the project.

Corresponding author

Correspondence to Logan G. Wright.

Ethics declarations

Competing interests

L.G.W. and F.W.W. hold a US patent, number US20190207361A1 for STML.

Additional information

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

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.

Supplementary information

Supplementary Information

Supplementary Figs. 1–25, discussion and references.

Supplementary Video 1

Shows a tour of the experimental set-up, with examples of typical measurements. Also shows an animated example of multipulse interactions in the SAGE regime.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Wright, L.G., Sidorenko, P., Pourbeyram, H. et al. Mechanisms of spatiotemporal mode-locking. Nat. Phys. 16, 565–570 (2020).

Download citation

Further reading


Quick links

Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing