J. Eur. Opt. Society-Rapid Publ. 21, 14( 2025) 155
Fig. 4. Poincaré maps representing the laminar phase evolution I n + 1 = f( I n):( a) theoretical evolution of a single laminar phase with L = 7;( b) theoretical points of three successive L = 7 laminar phases considering relaminarization using the baker’ s transformation. The relaminarization process consists of cutting the blue curve f( x), taking the upper part, stretching it and folding it back: explaining then the differences between( a) and( b). Experimental points for six consecutive cycles of L = 7 – 8 laminar phases are plotted for comparison.
Fig
. 5. Poincaré maps representing the laminar phase evolution I n + 1 = f( I n):( a) comparison between two different data acquisitions: six consecutive laminar phases counting around seven peaks each: experimental points in purple with the fit in orange and 16 laminar phases: experimental points in red with the fit in blue;( b) a comparison between the intermittency( red points) and chaotic regime( black points); fits are represented, fixing( black) or not( orange) the intersection value with the bisector line.
approaches the bisector. This corresponds to the region where the laminar phase has its minimum variation and where the standard deviation of the measurements is in the same order of magnitude of this variation. It seems important in our method to have a consequent number of laminar phases to reduce the incertitude on the e factor. Indeed, since e is comparable to typical standard deviations of the measured peak intensities, the accuracy of e is interesting to consider. With a standard deviation of 1.5 % and in the case of six laminar phase measurements, the precision is about 0.0061 and it drops to 0.0037 in the case of 16 laminar phases. According to these considerations, we compare the results of the two datasets we studied:( i) the case of 6 very regular laminar phases and( ii) the case of 16 laminar phases with stronger dispersion in their length. Representing these two sets of data, we observed similar evolution, Figure 5a with comparable fitting curves. One last question interesting to consider appears when observing the Poincaré map representation of the chaotic regime still very similar to the intermittency ones, see Figure 5b, where the points for the chaotic regime are in black. Indeed, we observe a kind of extension of the cubic shape tendency to lower values with a crossing with the bisector. Since the intermittency regime does not reach these relatively low values of relative peak intensity, the fitting curve crosses the bisector around 0.5 whereas that for the chaotic regime has this crossing around 0.2. To go further and to evaluate a potential error in the fitting process, we fitted the intermittency regime but forcing the intersection value with the bisector line at the point( 0.2, 0.2). The fit in this last case is represented in black in Figure 5b( and in Fig. 3) and gives e = 0.0168andk = 0.501 which is very similar to the previous case and shows that points lying far from the accumulation are underestimated in the fitting process but does not impact strongly the e value.
The Poincaré map offers an interesting and simple tool to discriminate the laser dynamics. By examining the structure of the trajectory in a 2D plane, we were able to quantify the stability of the pulse train and characterize the nature of the intermittencies. However, the laser under consideration has many dynamic variables. This diversity of physical quantities cannot be fully measured experimentally, which prevents us from studying the entire trajectory of the system in its phase space. In addition, our simulations