Figure 1

(Color online) Model of the fiber laser used in our simulations. Notations are as follows: SMF, single mode fiber; SA, saturable absorber; OC, output coupler; and EDF, erbium-doped fiber.Reuse & Permissions

Figure 2

(Color online) Bifurcation diagram constructed by varying $Q_{\text{sat}}$ while keeping the rest of the parameters fixed. The plot is split into two sections (a) and (b) for convenience. Each (red) dot on the diagram represents the absolute maximum of the field amplitude observed at each round trip. The regions with a single maximum for a given $Q_{\text{sat}}$ in (a) correspond to a stable single-pulse operation. The regions with a wide vertical band represent a chaotic output. These regions appear repeatedly when increasing $Q_{\text{sat}}$. Each of them occupies a finite interval along the $Q_{\text{sat}}$ axis, being a potential area for the appearance of rogue waves. Parameters of the system are $D_2=-0.735$, ${\Gamma }_2=3$, $g_0=4$, ${\beta }_2=0.008$, $L_{\text{SMF}}=1.1$, $L_{\text{EDF}}=0.22$, $T_r=70\%$, $T_0=0.3$, $\Delta T=0.7$, and $I_{\text{sat}}=0.6$.Reuse & Permissions

Figure 3

(Color online) Chaotic evolution of the pulse in the cavity for $Q_{\text{sat}}=3$. The evolution variable, $N$, is the number of round trips.Reuse & Permissions

Figure 4

(Color online) Chaotic evolution of a group of pulses when $Q_{\text{sat}}=60$. Note the different scale in the $t$ axis with respect to Fig. 3.Reuse & Permissions

Figure 5

(Color online) (a) Evolution of the absolute maximum amplitude of the field in $t$ with the number of round trips, $N$, for $Q_{\text{sat}}=60$ (solid blue line). The highest maximum in this run is observed when $N=276$. This value is marked by the dashed (red) vertical line. (b) The contour plot of the same simulation in the ($t,z$) plane, that is, inside the cavity for the two nearest round trips. Each of the two focused spots (red online) here (at $t=0$) is a signature of a rogue wave. The output radiation contains only those formed at the coupler position, that is, at the integer values of $N$.Reuse & Permissions

Figure 6

(Color online) Main segment of the transverse field profile of the laser radiation at the round-trip number $N=276$. The whole bunch of pulses occupies the larger temporal interval [$-50,50$]. The extreme pulse in the middle can be considered as a rogue wave. The inset shows the same pulse in higher resolution.Reuse & Permissions

Figure 7

(Color online) A sample of the output radiation from the model laser. Parameters are the same as in the caption of Fig. 2 with $Q_{\text{sat}}=60$.Reuse & Permissions

Figure 8

(Color online) Part of the curve enclosed in the central (red) box in Fig. 7.Reuse & Permissions

Figure 9

(Color online) Contour plots of evolution of two sample pulses when $Q_{\text{sat}}=3$ (left) and $Q_{\text{sat}}=6$ (right). The vertical axis corresponds to $N$, and the horizontal axis corresponds to $t$.Reuse & Permissions

Figure 10

(Color online) Probability density functions for $Q_{\text{sat}}=3$ (red curve) and $Q_{\text{sat}}=6$ (blue curve).Reuse & Permissions

Figure 11

(Color online) Probability density function for the maxima of intensity (red curve) and for the pulse hight from crest to trough (blue line). The two curves are virtually identical. For comparison, the dashed (green) line is drawn for a pure exponential decay of the probability.Reuse & Permissions

Figure 12

(Color online) Significant wave height shown by the dotted (blue) curve versus the low-intensity threshold, $I_{\text{min}}$, when the maxima below $I_{\text{min}}$ are not accounted for. The probability of having a pulse with a peak intensity above the level of $2.2$ times the significant wave height, $I_{\text{SWH}}$ is shown by the dashed (red) line.Reuse & Permissions