Characteristics of the extreme events observed in the Kerr-lens mode-locked Ti:sapphire laser

Kerr-lens mode-locked Ti:sapphire lasers are known to display three coexistent modes of operation: continuous wave, transform limited pulses (P1), and positive chirped pulses (P2). Extreme events (sometimes also called optical rogue waves), in the form of pulses of high energy appearing much often than in a Gaussian distribution, are observed in the chaotic regime of the P2 mode, but not of P1. The extreme events in P2 appear unpredictably, but their separation (measured in number of round trips) is a simple combination of the numbers 11 and 12 (which were named “magic numbers”). The existence of extreme events in P2 and not in P1, and also of the magic numbers, have been successfully reproduced by numerical simulations based on a five-variables iterative map, but the intuitive insight on the physical causes has been limited. In this paper, we present evidence that the extreme events in this laser appear if the amount of self-phase modulation on the pulses is above a certain threshold, and also that the P1 mode becomes unstable before crossing that threshold. This explains why the extreme events are observed in P2, and not in P1. Remarkably, even though the values of self-phase modulation on all the pulses (in the chaotic regime) are widely spread, the values inside the set of extreme events are relatively well defined. Finally, the magic numbers are found to be the residuals of the periodical orbits of the “cold” laser cavity when they are perturbed by the opposite effects of a dissipative term, due to the presence of transversal apertures, and an expansive term, due to the self-focusing.

Figure 1

Scheme of the laser. LB: pump focusing lens $(f=10\mathrm{cm})$; ${\mathrm{M}}_{2,3}$: curved mirrors $(R=10\mathrm{cm})$; $\mathrm{M}{\mathrm{P}}_{1,2}$: plane HR mirrors; ${\mathrm{P}}_{1,2}$: pair of prisms. Distances in mm: ${\mathrm{M}}_3–\mathrm{R}=\mathrm{R}–{\mathrm{M}}_2=50,{\mathrm{M}}_2–\mathrm{M}{\mathrm{P}}_2=140,\mathrm{M}{\mathrm{P}}_2–{\mathrm{M}}_1=465,{\mathrm{M}}_3–{\mathrm{P}}_1=297,{\mathrm{P}}_1–\mathrm{M}{\mathrm{P}}_1=198,\mathrm{M}{\mathrm{P}}_1–{\mathrm{P}}_2=415,{\mathrm{P}}_2–{\mathrm{M}}_4=109$. The prisms’ positions are adjusted to get negative total GVD.

Figure 2

Chaotic regime of the P2 mode. The horizontal dashed line indicates the threshold of EEs for the full time series according to the 2×AI criterion. (a) Experimental time series, zoom of ≈2000 pulses of 9978 with a total of 205 EEs, kurtosis = 4.91; (b) further zoom of the same; (c) theoretical time series obtained from the five-variables iterative map, zoom of ≈2000 iterations of ${10}^4$ with a total of 226 EEs, kurtosis = 4.98; (d) further zoom of the same. Note the intermittent excursions to a regime of pulses of higher energy in both series. Be aware that each point in (b) and (d) is not the sample of a digital oscilloscope, but the energy of a single pulse in the mode-locking train. The periodicities behind the magic numbers (Sec. 4) are clearly visible. Average pulse duration: 80 fs.

Figure 3

Histograms of the number of EEs according to the distance (in cavity round trips) to the next EE in the chaotic regime of P2; left: experimental results; right: theoretical ones.

Figure 4

Bifurcation diagrams for the pulse energy of the modes P1 (left) and P2 (right) as the GVD is varied, as obtained from their own five-variables maps. The energy is scaled to $U^{*}$; the fixed point energy value of P1 at $\mathrm{GVD}=–130\mathrm{f}{\mathrm{s}}^2$.

Figure 5

Calculated values of β (a measure of the SPM) and the energy; each point represents one pulse of a chaotic time series of ${10}^5$ after removal of a transient of $3\times {10}^4$. Note that almost all the pulses of the P2 mode (in black) have a larger value of β than even the largest ones of the P1 mode (in gray). The total number of EEs (the points to the right of the vertical dotted line) is 331, they seem to be fewer because of the scale of the figure. $\mathrm{GVD}=–42\mathrm{f}{\mathrm{s}}^2$.

Figure 6

Energy histograms (total ${10}^4$ pulses) in the chaotic regime of the P1 mode for increasing small signal gain Γ, $\mathrm{GVD}=–42\mathrm{f}{\mathrm{s}}^2$ (a) at the normal operating value of the small signal gain, $\Gamma ={\Gamma }^{*}$; (b) $\Gamma /{\Gamma }^{*}=1.1$; (c) $\Gamma /{\Gamma }^{*}=1.2$, (d) $\Gamma /{\Gamma }^{*}=1.4$; a total of 16 EEs are observed, kurtosis = 4.55. For (a–c), the value 2×AI is higher than 100 nJ and out of the figure, for (d) 2×AI = 63 nJ.

Figure 7

β and the energy as in Fig. 5 but for the P1 mode only, and $\Gamma /{\Gamma }^{*}=1.4$. Note the changes in shape and position of the cloud, and the appearance of EEs. The total number of EEs (the points to the right of the vertical dotted line) is 16; they seem to be fewer because of the scale of the figure.

Figure 8

Evolution of the pulse starting in the P1 mode (on the left) towards the P2 mode (on the right). All the parameters correspond to the operation point of the actual laser, except $\Gamma /{\Gamma }^{*}=1.4$. Note that a high-energy excursion in P1 (an EE?) is followed by a transition to P2, where the system remains and displays EEs. The transition to P2 occurs after 218 round trips (≈2.5 μs in real time); total length of the run: ${10}^4$ pulses. The variables are scaled to the fixed point of P1.

Figure 9

Variation of β (curve, vertical axis on the left) and of the modulus of the largest eigenvalue (broken line, axis on the right) as a function of the scaled small signal gain $\Gamma /{\Gamma }^{*}$, for the modes P2 (left) and P1 (right); $\mathrm{GVD}=–42\mathrm{f}{\mathrm{s}}^2$. For P2, the SPM crosses ${\beta }_{\mathrm{EE}}\approx {10}^{-6}\mathrm{f}{\mathrm{s}}^{-2}$ at $\Gamma /{\Gamma }^{*}<1$, while the largest eigenvalue crosses 1 (and the mode becomes unstable) at $\Gamma /{\Gamma }^{*}\approx 2.5$. Thus, at $\Gamma /{\Gamma }^{*}=1$ the P2 is stable and displays EEs, as observed. Instead, for P1 the eigenvalue is >1 at $\Gamma /{\Gamma }^{*}\approx 1.1$ (and the mode becomes unstable) and $\mathrm{SPM}>{\beta }_{\mathrm{EE}}$ at $\Gamma /{\Gamma }^{*}\approx 1.4$. Thus, EEs are not observed in P1 because this mode becomes unstable before reaching the instability threshold.

Figure 10

(a) Values of $\{a,A,K_S\}$ that produce stable periodical orbits (period < 100) in the map [Eq. (7)]. Each dot represents an orbit, the fringed pattern is due to the accumulation of the orbits near the values of $A$ that are zeros of $P_3^n$ of low order. (b) The same as (a) but orbits of period 7 are plotted only; note that they are close to the zeros of $P_3^7$: {±0.223, ±0.624, ±0.901} (from Ref. [27]).

Figure 11

Histogram of the number of EEs according to the distance (in cavity round trips) to the next EE, for the cavity modified such that $A$ is a zero of $P_3^7$ and $P_3^5$; the other laser's parameters are the same as in Fig. 3. The magic numbers are now simple combinations of the numbers 5 and 7, as expected. The total number of pulses in this run is 15 000 and there are 239 EEs.