# Dynamics of a Low-Dimensional Model for Short Pulse Mode Locking

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## Abstract

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## 1. Introduction

## 2. Governing Equations and Reduced Models

#### 2.1. Short Pulse Equation

#### 2.2. Alternative Formulations of Short Pulse Dynamics

#### 2.3. Short Pulse Master Mode Locking Equation

#### 2.4. Reduced Model

## 3. Phase-Plane Analysis and Stability of Mode Locking

**Figure 1.**When all mode locking terms are included, stable pulses are possible. Here, parameter values are chosen to be perturbatively small, ${g}_{0}=0.015,\gamma =0.0035,\beta =0.001.$ ODE shows reasonable agreement with the full PDE, but does not reflect the oscillations inherent in the PDE model at this low dissipation. When the magnitude of dissipative perturbation is increased, the ODE agrees only qualitatively with PDE. Parameter values are ${g}_{0}=0.25,\gamma =0.25,\beta =0.05.$ However, the stability of such solutions is greatly enhanced, as evidenced by the robust mode locking in the PDE model, as well as the large negative eigenvalues associated with the ODE.

#### 3.1. Nullclines and Fixed Points

**Figure 2.**As the bandwidth parameter ${\tau}_{4}$ decreases, fixed points move up and to the right. Values of (from the left) ${\tau}_{4}=0.55,0.45,0.35,0.25.$ For ${\tau}_{4}>0.57$, the k nullcline ceases to exist.

**Figure 3.**Pulses become taller and more narrow for large ${\tau}_{4}$ as ${\tau}_{4}$ decreases. At around ${\tau}_{4}=0.196$, fixed points jump to the upper branch, and pulses immediately become much taller and broader (smaller k). The stability of the fixed point changes at almost that point, as well.

#### 3.2. Stability

**Figure 4.**Phase portrait for the two-dimensional ODE shows convergence to fixed point at about $\eta =0.82,k=0.206$ using small parameter perturbation (

**left**) and $\eta =1.05,k=0.58$ using large parameter perturbation (

**right**). The red curve is the η nullcline, and the green curve is the k nullcline. Blue lines are trajectories, all of which are attracted to the fixed point. The larger dissipative terms in the second panel give rise to a shorter pulse (larger k).

## 4. Bifurcations in the SPE Mode Locking Model

**Figure 5.**As ${\tau}_{4}$ decreases, fixed point moves up. When ${\tau}_{4}=0.1962$, there exists a region of k for which ${\eta}^{\prime}>0$. At about ${\tau}_{4}<0.162$, the fixed point exists only on the upper branch of the η nullcline.

**Figure 6.**The fixed point is a stable attractor on the lower branch of the η nullcline until about ${\tau}_{4}=0.1962$, when the fixed point jumps to the upper branch of the nullcline. At ${\tau}_{4}=0.1960$, the fixed point becomes a stable spiral. At about ${\tau}_{4}=0.1940$, the system undergoes a Hopf bifurcation, and solutions tend to a stable limit cycle, which vanishes for ${\tau}_{4}<0.192$.

**Figure 7.**For ${\tau}_{4}<0.194$, the only fixed point is an unstable spiral. However, for a short band of ${\tau}_{4}$, there appears to be a limit cycle. For ${\tau}_{4}<0.19195$, this limit cycle exhibits its instability only after a long period of time. This corresponds to the breathing characteristic of a pulse undergoing harmonic pulse splitting. For smaller values of ${\tau}_{4}$, the transition to instability is nearly instantaneous.

**Figure 8.**As the bandwidth parameter ${\tau}_{4}$ decreases, amplitudes increase until becoming unstable, as predicted by the ODE model. In fact, solutions to the PDE undergo harmonic pulse splitting, as evidenced here.

## 5. Conclusion

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

- Backus, S.; Durfee, C.G.; Murname, M.M.; Kapteyn, H.C. High power ultrafast lasers. Rev. Sci. Instrum.
**1998**, 69, 1207–1233. [Google Scholar] [CrossRef] - Spielmann, C.; Curley, P.F.; Brabec, T.; Krausz, F. Ultrabroadband femtosecond lasers. IEEE J. Quantum Electron.
**1994**, 1100, 1100–1114. [Google Scholar] [CrossRef] - Sutter, D.H.; Jung, I.D.; Kärtner, F.X.; Matuschek, N.; Morier-Genoud, F.; Scheuer, V.; Tilsch, M.; Tschudi, T.; Keller, U. Self-starting 6.5-fs pulses from a Ti:Sapphire laser using a semiconductor saturable absorber and double-chirped mirrors. IEEE J. Sel. Top. Quantum Electron.
**1998**, 4, 169–178. [Google Scholar] [CrossRef] - Ell, R.; Angelow, G.; Seitz, W.; Lederer, M.J.; Huber, H.; Kopf, D.; Birge, J.R.; Kärtner, F.X. Quasi-synchronous pumping of modelocked few-cycle Titanium Sapphire lasers. Opt. Express
**2005**, 13, 9292–9298. [Google Scholar] [CrossRef] [PubMed] - Kärtner, F.X.; Morgner, U.; Ell, R.; Schibli, T.; Fujimoto, J.G.; Ippen, E.P.; Scheuer, V.; Angelow, G.; Tschudi, T. Ultrabroadband Double-Chirped Mirror Pairs for Generation of Octave Spectra. J. Opt. Soc. Am. B
**2001**, 19, 382–385. [Google Scholar] [CrossRef] - Keller, U. Ultrafast solid-state lasers. Progr. Opt.
**2004**, 46, 1–115. [Google Scholar] - Hentschel, M.; Kienberger, R.; Spielmann, Ch.; Reider, G.A.; Milosevic, N.; Brabec, T.; Corkum, P.; Heinzmann, U.; Drescher, M.; Krausz, F. Attosecond metrology. Nature
**2001**, 414, 509–513. [Google Scholar] [CrossRef] [PubMed] - Schafer, K.; Gaarde, M.; Heinrich, A.; Biegert, J.; Keller, U. Strong Field Quantum Path Control Using Attosecond Pulse Trains. Phys. Rev. Lett.
**2004**, 92, 023003. [Google Scholar] [CrossRef] [PubMed] - Silberberg, Y. Physics at the attosecond frontier. Nature
**2001**, 414, 494–495. [Google Scholar] [CrossRef] [PubMed] - Scrinzi, A.; Ivanov, M.Y.; Kienberger, R.; Villeneuve, D.M. Attosecond physics. J. Phys. B At. Mol. Opt. Phys.
**2006**, 39, R1–R37. [Google Scholar] [CrossRef] - Cundiff, S.T. Attosecond Physics: Better by half. Nature Phys.
**2007**, 3, 16–18. [Google Scholar] [CrossRef] - Cundiff, S.T. Femtosecond comb technology. J. Korean Phys. Soc.
**2006**, 48, 1181–1187. [Google Scholar] - Cundiff, S.T.; Ye, S.J.; Hall, J. Rulers of light. Sci Am.
**2008**, 298, 74–81. [Google Scholar] [CrossRef] [PubMed] - Haus, H.A. Mode-Locking of Lasers. IEEE J. Sel. Top. Quant. Elec.
**2000**, 6, 1173–1185. [Google Scholar] [CrossRef] - Kutz, J.N. Mode-locked soliton lasers. SIAM Rev.
**2006**, 48, 629–678. [Google Scholar] [CrossRef] - Haus, H.A.; Fujimoto, J.G.; Ippen, E.P. Structures for additive pulse mode locking. J. Opt. Soc. Am. B
**1991**, 8, 2068–2076. [Google Scholar] [CrossRef] - Schafer, T.; Wayne, C.E. Propagation of ultra-short optical pulses in cubic nonlinear media. Phys. D
**2004**, 196, 90–105. [Google Scholar] [CrossRef] - Chung, Y.; Jones, C.K.R.; Schafer, T. Ultra-short pulses in linear and nonlinear media. Nonlinearity
**2005**, 18, 1351–1374. [Google Scholar] [CrossRef] - Amiranashvili, Sh.; Vladimirov, A.G.; Bandelow, U. Solitary-wave solutions for few-cycle optical pulses. Phys. Rev. A
**2008**, 77, 063821. [Google Scholar] [CrossRef] - Pietrzyk, M.; Kanattsikov, I.; Bandelow, U. On the propagation of vector ultra-short pulses. JNMP
**2008**, 15, 162–170. [Google Scholar] [CrossRef] - Farnum, E.; Kutz, J.N. Master mode-locking theory for few-femtosecond pulses. Opt. Lett.
**2010**, 35, 3033–3035. [Google Scholar] [CrossRef] [PubMed] - Farnum, E.; Kutz, J.N. Mode Locking in the Few-Femtosecond Regime Using Waveguide Arrays and the Coupled Short-Pulse Equations. IEEE J. Sel. Top. Quantum Electron.
**2012**, 18, 113–118. [Google Scholar] [CrossRef] - Elgin, J. Perturbations of optical solitons. Phys. Rev. A
**1993**, 47, 4331–4341. [Google Scholar] [CrossRef] [PubMed] - Kaup, D.J. Perturbation theory for solitons in optical fibers. Phys. Rev. A
**1990**, 42, 5689–5694. [Google Scholar] [CrossRef] [PubMed] - Gordon, J.P.; Haus, H.A. Random walk of coherently amplified solitons in optical fiber transmission. Opt. Lett.
**1986**, 11, 665–667. [Google Scholar] [CrossRef] [PubMed] - Gordon, J.P. Theory of the soliton self-frequency shift. Opt. Lett.
**1986**, 11, 662–664. [Google Scholar] [CrossRef] [PubMed] - Kapitula, T.; Kutz, J.N.; Sandstede, B. Stability of Pulses in the Master-Modelocking Equation. J. Opt. Soc. Am. B
**2002**, 19, 740–746. [Google Scholar] [CrossRef] - Gordon, J.P. Dispersive perturbations of solitons of the nonlinear Schrödinger equation. J. Opt. Soc. Am. B
**1992**, 9, 91–97. [Google Scholar] [CrossRef] - Mamyshev, P.V.; Mollenauer, L.F. Soliton collisions in wavelength-division-multiplexed dispersion-managed systems. Opt. Lett.
**1999**, 24, 448–450. [Google Scholar] [CrossRef] [PubMed] - Mamyshev, P.V.; Mollenauer, L.F. Pseudo-phase-matched four-wave mixing in soliton wavelength-division multiplexing transmission. Opt. Lett.
**1996**, 21, 396–398. [Google Scholar] [CrossRef] [PubMed] - Farnum, E.; Kutz, J.N. Short-pulse perturbation theory. J. Opt. Soc. Am. B
**2013**, 30, 2191–2198. [Google Scholar] [CrossRef] - Sakovich, A.; Sakovich, S. The short pulse equation is integrable. J. Phys. Soc. Jpn.
**2005**, 74, 239–241. [Google Scholar] [CrossRef] - Sakovich, A.; Sakovich, S. Solitary wave solutions of the short pulse equation. J. Phys. A
**2006**, 39, 361–367. [Google Scholar] [CrossRef] - Brabec, T.; Krausz, F. Nonlinear optical pulse propagation in the single-cycle regime. Phys. Rev. Lett.
**1997**, 78, 3282. [Google Scholar] [CrossRef] - Porras, M.A. Propagation of single-cycle pulsed light beams in dispersive media. Phys. Rev. A
**1999**, 60, 5069. [Google Scholar] [CrossRef] - Kolesik, M.; Moloney, J.V.; Mlejnek, M. Unidirectional Optical Pulse Propagation Equation. Phys. Rev. Lett.
**2002**, 89, 283902. [Google Scholar] [CrossRef] - Kolesik, M.; Moloney, J.V. Nonlinear optical pulse propagation simulation: From Maxwell's to unidirectional equations. Phys. Rev. E
**2004**, 70, 036604. [Google Scholar] [CrossRef] - Leblond, H.; Sanchez, F. Models for optical solitons in the two-cycle regime. Phys. Rev. A
**2003**, 67, 013804. [Google Scholar] [CrossRef][Green Version] - Rosanov, N.N.; Kozlov, V.V.; Wabnitz, S. Maxwell-Drude-Bloch dissipative few-cycle optical solitons. Phys. Rev. A
**2010**, 81, 043815. [Google Scholar] [CrossRef] - Bondeson, A.; Lisak, M.; Anderson, D. Soliton Perturbations: A Variational Principle for the Soliton Parameters. Phys. Scr.
**1979**, 20, 479. [Google Scholar] [CrossRef] - Bale, B.; Kutz, J.N. Variational method for mode-locked lasers. J. Opt. Soc. Am. B
**2008**, 25, 1193–1202. [Google Scholar] [CrossRef] - Brunelli, J.C. The bi-Hamiltonian structure of the short pulse equation. Phys. Lett. A
**2006**, 353, 475–478. [Google Scholar] [CrossRef] - Feng, L.; Ding, E.; Kutz, J.N.; Wai, P.K.A. Dual transmission filters for enhanced energy in mode-locked fiber lasers. Opt. Express
**2011**, 19, 23408–23419. [Google Scholar] [CrossRef] [PubMed] - Fu, X.; Kutz, J.N. High-energy mode-locked fiber lasers using multiple transmission filters and a genetic algorithm. Opt. Express
**2013**, 21, 6526–6537. [Google Scholar] [CrossRef] [PubMed] - Namiki, S.; Ippen, E.P.; Haus, H.A.; Yu, C.X. Energy rate equations for mode-locked lasers. J. Opt. Soc. Am. B
**1997**, 14, 2099–2111. [Google Scholar] [CrossRef]

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**MDPI and ACS Style**

Farnum, E.D.; Kutz, J.N.
Dynamics of a Low-Dimensional Model for Short Pulse Mode Locking. *Photonics* **2015**, *2*, 865-882.
https://doi.org/10.3390/photonics2030865

**AMA Style**

Farnum ED, Kutz JN.
Dynamics of a Low-Dimensional Model for Short Pulse Mode Locking. *Photonics*. 2015; 2(3):865-882.
https://doi.org/10.3390/photonics2030865

**Chicago/Turabian Style**

Farnum, Edward D., and J. Nathan Kutz.
2015. "Dynamics of a Low-Dimensional Model for Short Pulse Mode Locking" *Photonics* 2, no. 3: 865-882.
https://doi.org/10.3390/photonics2030865