# 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

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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