J. Eur. Opt. Society-Rapid Publ. 21, 14( 2025) 153
Fig. 1.( a) Experimental setup allowing co-lasing at 1.9 lm and 2.3 lm. The saturable absorber is placed in the 2.3 lm laser cavity exclusively.( b) Normalized impulsion train at 2.3 lm and laser intensity at 1.9 lm for a stable Q-switch.( c) Energy levels of Tm 3 +. GSA: Ground State Absorption, NR: Non Radiative transition.
using the function“ sampen” from the“ nolds” Python library. The parameters are the following: the embedded dimension is m = 4, and the tolerance is 2 %. The experiment then shows an increase, with the selected parameters, of this entropy from 0 when stable up to 1.55 in the chaotic regime. The results showed in Figure 3 will be more discussed in the later part of the article.
3 Discussion and interpretation
Fig. 2. Dependence of the pulse repetition rate on the pump power at 780 nm indicating the points where the oscillograms( a)–( c) from the Figure 3 have been taken. The laser emission at 1.9 lm is not permitted in this case.
with and without cascade laser. In these configurations, Q-switched regime is obtained between 1 and 30 kHz with a constant pulse duration of 4.7 ls( Fig 1c). The repetition rate of the Q-switching increases, as expected, linearly versus the pump power( Fig. 2). The error bars indicate the root mean square( rms) value of the repetition rate measurements. In both cases( with or without cascade laser), chaotic zones can be clearly identified when the dispersion of the repetition rate explodes [ 18 ]. On the opposite, stable zone occurs with relatively low dispersion in the repetition rates. Different typical regimes exploring the route to chaos has been further analysed( shown in Figs. 2 and 3). These different setting points are the stable regime( a), the chaotic regime( c) and the intermittency regime( b) in between.
We will focus now on the transition regime from stable Q-switching to chaotic regime around 20 kHz. To analyse the different regimes, we plot in Figure 3 the oscillograms( first line). Further analysis is possible extracting from these oscillograms the pulse peak powers which allows to have access to key parameters for chaos study, such as sample entropy and Poincaré map( Fig. 3 second line) [ 19, 20 ]. Another parameter useful to quantify the route to chaos is the entropy. The corresponding histograms are given in Figure 3 third line. The entropy of the peaks is evaluated
Let us focus on the intermittency regime. In this region the entropy is around 0.55; an intermediate value due to the observation of laminar phases [ 19, 20 ] closed to a stable regime interrupted by stochastic intermittencies. One can then isolate the laminar phases in the Poincaré diagram which consists in plotting the peak powers versus their antecedent: I n + 1 = f( I n). During the laminar phase, the points are above the bisector line. In the case of type I intermittency scenario, these laminar phase points can be fitted by a polynomic( typically cubic). This fit is plotted in orange in the Poincaré map, column b in Figure 3. Moreover, one can observe, still in the type-I intermittency scenario accordance, that this fitting curve approaches the bisector line around the point( 1, 1). To perform this analysis, we select another dataset in the same configuration with six laminar phases with high regularity in the laminar length( between 7 and 8); this dataset point is plot in Figure 4b. The minimum distance( e) between the fitting curve and the bisector line is then related to the laminar length( L): L = e �k with k = 0.5 within the theory of the type-I intermittency scenario. In our case e = 0.0165 and the mean value of L = 7.75, this leads to k = 0.499 corroborating the type I intermittency scenario.
Using this fit it is then possible to simulate the theoretical evolution of the peak power. As shown in Figure 4a, even if the theoretical curve can be extended beyond the( 1, 1) point, the real peak power is confined inside the Poincare map, which extends over the square with side 1. Then, when I n + 1 = f( I n) > 1, the peak power is folded under the bissector line. The folding process respects the Baker transformation which consists of a two steps transformation: