J. Eur. Opt. Society-Rapid Publ. 21, 14( 2025) 157
Fig
. 7. Simultaneous, dual monitoring of the cascade laser: 2.3-lm Q-switch laser( blue) and 1.9-lm laser( red). First line: oscillograms. Second line: 2.3 lm laser power versus 1.9 lm laser power.
his phase space trajectory. This allows to get“ independent variables” and since the build-up time of the pulse is linked to the precedent pulse, we keep a correlation between I [ n ] and I [ n + sf e ]. Choosing the dimension of the Y [ n ] vectors is less trivial. However, Kennel et al. [ 24 ] have proposed to evaluate the proportion of false nearest neighbours in the trajectory.
Figure 6 shows the reconstruction for the intermittent and chaotic regimes presented in Figure 3. We observe an embedding dimension of our system between 6 and 10, depending on the dynamics adopted by the laser. From our study, we can deduce([ 23 ], p. 46) an upper bound for the attractor dimension of our system: d A de 5, where
2
d A is the attractor dimension and d e is the embedding dimension.
In our case, we are able to observe simultaneously both laser emissions at 2.3 lm and1.9lm givingusthe opportunity to access another projection of the phase space. Indeed, it is interesting to look at the 1.9 lm laserrelaxation operation after the 2.3 lm Q-switch pulse to observe whether the 3 F 4 population oscillates synchronously with the Q-switch tempo. For example, the first column of Figure 7a represent stable train where both 2.3 lm Q-switched pulses and 1.9 lm relaxation are synchronized. Plotting( Fig. 7c) the phase diagram I 2. 3 lm versus I 1. 9 lm then leads to a well-defined cycle. To evaluate the intermittency regime in comparison, we study the particular interesting case where intermittencies occur after a stable regime( Fig. 7b). In this case it is interesting to observe the synchronization before and during the inherencies. Indeed, whereas in the stable regime the cycling regime( Fig. 7d) indicates a good synchronization, as soon as the intermittencies occur this synchronization is lost and the relaxation oscillations of the 1.9 lm laser become erratic( Fig. 7b) leading to absence of stable cycling( Fig. 7e). This study is interesting since it seems to indicate that a hypothesis of a loss of synchronization between the Q-switch laser at 2.3 lm and the population of the 3 F 4 metastable regime as an aggravating phenomenon to explain the chaotic instabilities seem very probable.
4 Conclusion
In conclusion, this study has provided insights into the dynamic behavior of Q-switched Tm: YLF lasers operating on the 3 H 4? 3 H 5 transition at 2.3 lm, particularly focusing on their chaotic and intermittent regimes. The investigation highlights a unique intermittency route to chaos, which is atypical in passively Q-switched lasers and appears to be inherently tied to the cascade energy transitions of trivalent thulium ions. The analysis of stable, intermittent, and