Intermediate asymptotics in nonlinear optical systems

Using a fiber laser system as a specific illustrative example, we introduce the concept of intermediate asymptotic states in finite nonlinear optical systems. We show that intermediate asymptotics of nonlinear equations (e.g., coherent structures with a finite lifetime or distance) can be used in applications similar to those of truly stable asymptotic solutions, such as, e.g., solitons and dissipative nonlinear waves. Applying this general idea to a particular, albeit practically important, physical system, we demonstrate a novel type of nonlinear pulse-shaping regime in a mode-locked fiber laser leading to the generation of linearly chirped pulses with a triangular distribution of the intensity.

Figure 1

Evolution of an initial Gaussian pulse in the fiber. (a) Evolution of the misfit parameter to a triangular temporal shape for varying values of $N$ and the (normalized) chirp parameter $C$. Left: $N=6$ (dotted curve), $N=10$ (solid curve), and $N=14$ (dashed curve); $C=1$. Right: $C=0.75$ (dotted curve), $C=1$ (solid curve), and $C=1.25$ (dashed curve); $N=10$. Gray (red) lines define the life distances of the triangular pulses. (b) Evolution of the intensity profile for $N=10$ and $C=1$. The profiles at distances ${\xi }_0\le \xi \le {\xi }_0+{\xi }_{\mathrm{life}}$ are drawn in red and shown in the inset.

Figure 2

Evolution of the pulse width ${\tau }_0\left(\xi \right)$ [gray (red) line] and chirp parameter $b\left(\xi \right)$ (black line) for $N=10$ and $C=1$: solid curves, simulation results; crosses, theoretical predictions from (2b) and (2c) for $\xi \ge {\xi }_0$. The shaded region defines the life distance of the triangular solution.

Figure 3

Schematic model for the laser. SMF, single-mode fiber; OC, output coupler; SA, saturable absorber; DDL, dispersive delay line.

Figure 4

Misfit parameters of the steady-state pulses to parabolic and triangular temporal shapes at the end of the SMF versus net cavity dispersion and total gain. Misfit values >0.1 are rendered with the same color associated with 0.1. The coupling coefficient $R=0.1$.

Figure 5

(a) Temporal [solid (red) curves] and spectral [dash-dotted (black) curves] profiles of the pulse at the end of the SMF for ${\beta }_{\mathrm{net}}^{\left(2\right)}=0.011{\mathrm{ps}}^2$ and $G=20$ dB (left) and for ${\beta }_{\mathrm{net}}^{\left(2\right)}=0.004{\mathrm{ps}}^2$ and $G=60$ dB (right). (b) Temporal intensity profiles on a log scale [solid (black) curves] and chirp profiles [open (black) circles]. Dashed (red) curves: parabolic (left) and triangular (right) fits. The coupling coefficient $R=0.1$.

Figure 6

Evolution of (a) the rms temporal [(red) circles] and spectral [(black) triangles] widths and (b) the chirp parameter [(blue) diamonds] along the cavity for ${\beta }_{\mathrm{net}}^{\left(2\right)}=0.011{\mathrm{ps}}^2$ and $G=20$ dB (left) and for ${\beta }_{\mathrm{net}}^{\left(2\right)}=0.004{\mathrm{ps}}^2$ and $G=60$ dB (right). The coupling coefficient $R=0.1$.