Rogue waves and extreme events and extreme events in optics
Rogue waves are extremely large waves that exceed expectations based on long-term observations and Gaussian statistics. Ocean rogue waves exceed the significant wave height by a factor two. Similar phenomena have been observed in a multiplicity of optical systems, including nonlinear optical fibers and filaments. In some of these systems, super rogue waves can be observed that exceed a similarly defined significant magnitude by a factor ten.
While it is interesting to note that the rogue wave phenomenon is far more widespread than originally assumed, the question arises on how one can actually benefit from this analogy. Is it really true that ocean rogue waves come out of nowhere and disappear without a trace? Or can one actually identify situations where such large waves appear more often, maybe, even issue a warning?
To answer this question, we recorded the temporal behavior of multifilament beam profiles with high-speed cameras. These recordings indicate a highly dynamic scenario, with occasional short-lived and localized bright flashes, which constitute the rogue events in our system. Employing methods from nonlinear time series analysis, we could confirm that the system bears a certain amount of determinism, which, in turn, makes it predictable. Using the same analysis method for ocean rogue wave data, we find that the analogy between the system holds beyond the linear statistics, i.e., the ocean is also a deterministic system and therefore, at least in principle, predictable.
However, the effective forecast times for a rogue event appear to be fairly short. In the multifilament system, the predictability goes to zero after several 10 milliseconds. Transferring this result to the ocean, one would expect a few 10 seconds of predictability at best. Therefore attempts of predicting individual rogue waves appear to be rather futile. However, nonlinear time series analysis suggests an alternative way of forecasting via estimation of the phase space dimension. In fact, there seems to be a threshold criterion for the possibility of ocean rogue wave formation. Here optics may guide a way to finally understand the true origin of ocean rogue waves.
Non-instantaneous nonlinearities in optics
It is textbook knowledge that the polarization inside a nonlinear material can be presented as a Taylor series. Off resonance in a dielectric material, the coefficients in this series are typically considered as instantaneous. Many characterization techniques for femtosecond pulses explicitly rely on the instantaneity of the nonlinear optical response.
Our study relies on third-harmonic generation in TiO₂ thin films, excited by 800 nm few-femtosecond pulses. Using interferometric frequency-resolved optical gating (IFROG), we retrieved the underlying pulse shape measured with both, the thin films and fused silica as a reference sample. In these experiments, we found observed a substantially longer pulse shape for the TiO₂ thin films, which we explained by a non-instantaneous nonlinearity. In TiO₂, the third-harmonic nonlinearity is resonantly enhanced due to the proximity to the band gap. We conducted numerical simulations of the time-dependent Schrödinger equation, which also confirmed this finding. Mathematically, the underlying response function can be extracted by a deconvolution analysis. This analysis indicates a polarization decay time of about 6.5 fs in TiO₂, which is one of the fastest effects ever measured in femtosecond spectroscopy.
Carrier envelope phase measurement and stabilization
With ever decreasing pulse durations of femtosecond laser pulses, a regime is entered where the relative phase of the electric carrier field relative to the intensity envelope starts to play an important role for the nonlinear interaction of the few-cycle pulses. The most prominent example for this phase dependence is attosecond pulse generation.
Several methods for stabilizing the carrier envelope phase exist, yet they all require modulation of a laser parameter, e.g., via pump power modulation. For practical reasons, such back action on the laser is strongly disfavored. Moreover, stabilization requires maintenance of a phase-locked loop, which is susceptible to dropouts.
At MBI, we therefore developed an alternative stabilization approach, which relies on extra-cavity correction of the carrier-envelope phase by an acousto-optic frequency shifter. This method overcomes the disadvantages of the feedback stabilization scheme and enables a much wider stabilization bandwidth. With the new feed-forward scheme, we were able to push the residual jitter between carrier and envelope to below ten attoseconds, which is the best timing synchronization ever demonstrated.
As the carrier-envelope phase is highly susceptible to environmental influences, e.g., intracavity changes of pressure or temperature, our stabilization scheme can also be exploited for other applications. One such application is the measurement of the Kerr coefficient of air, for which we integrated a resonant high-voltage LC circuit into the cavity of our laser. A second interesting application is the study of phase noise, which may provide insight into amplitude-phase coupling mechanisms of mode-locked lasers.