Journal of Modern Optics

The Laser Modelocking Master Equation: 50 Years of Impact and Innovation

Journal of Modern Optics Special Issue

The Laser Modelocking Master Equation: 50 Years of Impact and Innovation

Modelocked lasers [1–10], known for generating the shortest and most intense light pulses, have become indispensable across a wide range of applications, from precision surgery and industrial manufacturing to advanced fields like metrology and spectroscopy [11]. As enablers of ultrafast science and technology, modelocked lasers have profoundly impacted modern society, as underscored by four Nobel Prizes awarded for their advancements and applications: femtochemistry (1999), frequency metrology (2005), chirped-pulse amplification (2018), and attosecond science (2024).

In 2025, we commemorate the 50th anniversary of Hermann Haus’ annus mirabilis, when he established the modern theoretical foundation of laser modelocking by addressing the three key processes still in use today: active modelocking [12], and passive modelocking with fast [13], and with slow [14] saturable absorbers. Haus’ Master Equation [1] (HME), which expands and generalizes theoretical work by several distinguished pioneers [15–17], remains one of the most influential frameworks in optics. It provides a comprehensive understanding of the dynamics and stability of one of the most pivotal technologies in history: the laser.

This Special Issue aims to celebrate the legacy of the HME, its continued relevance in the evolution of laser modelocking [18–23], and its future potential, particularly considering the plethora of recent advancements in technologies, principles, materials, and platforms [24–29]. These developments have broadened the range of available timescales in modelocked lasers, necessitating further refinement of the theoretical approach [30–33]. We invite contributions that explore the enduring significance of the HME, including not only theoretical papers but also, and especially, experimental studies that test, expand upon, or challenge these theoretical predictions.

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

[2] Keller, U. Ultrafast lasers; Springer: Cham, Switzerland, 2022.

[3] Hargrove, L.E.; Fork, R.L.; Pollack, M.A. Locking of He-Ne laser modes induced by synchronous intracavity modulation. Appl. Phys. Lett. 1964, 5, 4–6.

[4] Mocker, H.W.; Collins, J.R. Mode competition and self-locking effects in a Q-switched ruby laser. Appl. Phys. Lett. 1965, 7, 270–272.

[5] DeMaria, A. J.; Stetser, D. A.; Heynau, H. Self mode-locking of lasers with saturable absorbers. Appl. Phys. Lett. 1966, 8, 174–176.

[6] Ippen, E.P.; Shank, C.V.; Dienes, A. Passive mode locking of the cw dye laser. Appl. Phys. Lett. 1972, 21, 348–350.

[7] Spence, D.E.; Kean, P. N.; Sibbett, W. 60-fsec pulse generation from a self-mode-locked Ti:Sapphire laser. Opt. Lett. 1991, 16, 42–44.

[8] Keller, U.; Miller, D.A.B.; Boyd, G.D.; Chiu, T.H.; Ferguson, J.F.; Asom, M.T. Solid-state low-loss intracavity saturable absorber for Nd:YLF lasers: an antiresonant semiconductor Fabry–Perot saturable absorber. Opt. Lett. 1992, 17, 505–507.

[9] de Barros, M.R.X.; Becker, P.C. Two-color synchronously mode-locked femtosecond Ti:sapphire laser. Opt. Lett., 1993, 18, 631–633.

[10] Huang, X.; Stintz, A.; Li, H.; Lester, L.F.; Cheng, J.; Malloy, K.J. Passive mode-locking in 1.3 μm two-section InAs quantum dot lasers. Appl. Phys. Lett. 2001, 78, 2825–2827.

[11] Fermann, M.A.; Galvanauskas, A.; Sucha G. (eds.) Ultrafast Lasers: Technology and Applications; Marcel Dekker: New York, USA, 2002.

[12] Haus, H.A. A theory of forced mode locking. IEEE J. Quantum Electron. 1975, 11, 323–330.

[13] Haus, H.A. Theory of mode locking with a fast saturable absorber. J. Appl. Phys. 1975, 46, 3049–3058.

[14] Haus, H.A. Theory of mode locking with a slow saturable absorber. IEEE J. Quantum Electron. 1975, 11, 736–746.

[15] Haken, H.; Pauthier, M. Nonlinear theory of multimode action in loss modulated lasers. IEEE J. Quantum Electron. 1968, 4, 454–459.

[16] Kuizenga, D.; Siegman, A. FM and AM mode locking of the homogeneous laser–Part I: Theory. IEEE J. Quantum Electron. 1970, 6, 694–708.

[17] New, G.H.C. Pulse evolution in mode-locked quasi-continuous lasers. IEEE J. Quantum Electron. 1974, 10, 115–124.

[18] Martínez, O.E.; Fork, R.L.; Gordon, J.P. Theory of passively mode-locked lasers for the case of a nonlinear complex-propagation coefficient. J. Opt. Soc. Am. B 1985, 2, 753–760.

[19] Haus, H.A.; Fujimoto, J.G.; Ippen, E.P. Analytic theory of additive pulse and Kerr lens mode locking. IEEE J. Quantum Electron. 1992, 28, 2086–2096.

[20] Komarov, A.; Leblond H.; Sanchez, F. Quintic complex Ginzburg-Landau model for ring fiber lasers. Phys. Rev. E 2005, 72, 025604(R).

[21] Chong, A.; Buckley, J.; Renninger, W.; Wise, F. All-normal-dispersion femtosecond fiber laser. Opt. Express 2006, 14, 10095–10100.

[22] Oktem, B.; Ülgüdür, C.; Ilday, F. Soliton–similariton fibre laser. Nat. Photonics 2010, 4, 307–311.

[23] Grelu, P.; Akhmediev, N. Dissipative solitons for mode-locked lasers. Nat. Photonics 2012, 6, 84–92.

[24] Link, S.M.; Klenner, A.; Mangold, M.; Zaugg, C.A.; Golling, M.; Tilma, B.W.; Keller, U. Dual-comb modelocked laser. Opt. Express 2015, 23, 5521–5531.

[25] Wright, L.G.; Sidorenko, P.; Pourbeyram, H.; Ziegler, Z.M.; Isichenko, A.; Malomed, B.A.; Menyuk, C.R.; Christodoulides, D.N.; Wise, F.W. Mechanisms of spatiotemporal mode-locking. Nat. Phys. 2020, 16, 565–570.

[26] Hillbrand, J.; Opačak, N.; Piccardo, M.; Schneider, H.; Strasser, G.; Capasso, F.; Schwarz, B. Mode-locked short pulses from an 8 μm wavelength semiconductor laser. Nat. Commun. 2020, 11, 1.

[27] Täschler, P.; Bertrand, M.; Schneider, B.; Singleton, M.; Jouy, P.; Kapsalidis, F.; Beck, M.; Faist, J. Femtosecond pulses from a mid-infrared quantum cascade laser. Nat. Photonics 2021, 15, 919–924.

[28] Guo, Q.; Gutierrez, B.K.; Sekine, R.; Gray, R.M.; Williams, J.A.; Ledezma, L.; Costa, L.; Roy, A.; Zhou, S.; Liu, M.; Marandi, A. Ultrafast mode-locked laser in nanophotonic lithium niobate. Science 2023, 382, 708–713.

[29] Leefmans, C. R.; Parto, M.; Williams, J.; Li, G.H.Y.; Dutt, A.; Nori, F.; Marandi, A. Topological temporally mode-locked laser. Nat. Phys. 2024, 20, 852–858.

[30] Perego, A.M.; Garbin, B.; Gustave, F.; Barland, S.; Prati. F.; de Valcárcel, G.J. Coherent master equation for laser modelocking. Nat. Commun. 2020, 11, 311.

[31] Hausen, J.; Lüdge, K.; Gurevich, S.V.; Javaloyes, J. How carrier memory enters the Haus master equation of mode-locking. Opt. Lett. 2020, 45, 6210-6213

[32] Nizette M.; Vladimirov, A.G.; Generalized Haus master equation model for mode-locked class-B lasers. Phys. Rev. E 2021, 104, 014215.

[33] Prati, F; Perego, A.M; Redondo, J.; de Valcárcel, G.J. The master equation for passive modelocking. EPJ Web of Conferences 2023, 287, 08013.

Germán J. de Valcárcel¹ and Franco Prati²

¹Universitat de València, Spain

²Università dell’Insubria, Italy