Self-similar propagation of Hermite-Gauss water-wave pulses

We demonstrate both theoretically and experimentally propagation dynamics of surface gravity water-wave pulses, having Hermite-Gauss envelopes. We show that these waves propagate self-similarly along an 18-m wave tank, preserving their general Hermite-Gauss envelopes in both the linear and the nonlinear regimes. The measured surface elevation wave groups enable observing the envelope phase evolution of both nonchirped and linearly frequency chirped Hermite-Gauss pulses, hence allowing us to measure Gouy phase shifts of high-order Hermite-Gauss pulses for the first time. Finally, when increasing pulse amplitude, nonlinearity becomes essential and the second harmonic of Hermite-Gauss waves was observed. We further show that these generated second harmonic bound waves still exhibit self-similar Hermite-Gauss shapes along the tank.

Figure 1

The envelope evolutions of the nonchirped and linearly chirped Hermite-Gauss wave pulses, with parameters of $a_0$=6 mm, $t_0=2.5$ s, and $C=0$ (a, b), $C= -2.19$ (c, d) for different order of $m$ (see the top). In the experiments, the pulse envelopes were obtained using Hilbert transform of the measured elevations, in a frame of reference moving at speed $c_g$. The color bar units of the envelope are millimeters. (a, c) The experimental measurements and (b, d) the theoretical results based on Eq. (3).

Figure 2

The temporal profiles of the measured elevation wave groups at $x=1$ m (before the focus), 6 m (nearly the focus), and 11 m (after the focus), with different orders: (a) $m=1$, (b) $m=2$, and (c) $m=3$. The parameters are set as $a_0=6$ mm, $t_0=2.5$ s, and $C=-2.19$. Red curves are the measured elevations while the blue curves denote the constructed envelopes by Hilbert transform.

Figure 3

(a, b) The measured square roots of second-order moments of HG wave pulses, as a function of locations. The parameters used here are $a_0=6$ mm, $t_0=2.5$ s, and $C=0$ (a), $C=-2.19$ (b). (c, d) The calculated $M$ factor as a function of locations, with $C=0$ (c), $C=-2.19$ (d). In all these figures, the lines denote the theoretical results; while the scattered dots represent the experimental outcomes.

Figure 4

The carrier-envelope phase of the nonchirped HG pulses demodulated from the measured elevations, with $a_0=6$ mm, $t_0=2.5$ s, and $C=0$. The obtained envelope phase was illustrated as a cosine function of time. (a) Two-dimensional phase map calculated from Eq. (3) for different order of $m$. (b, c) The envelope phase profiles at $x=1$ m (b) and $x=11$ m (c). The blue curves in (b) and (c) correspond to the theory, while the red scattered dots denote the experiments.

Figure 5

The same as described in the caption of Fig. 4, but in the case of chirped pulses, i.e., $C=-2.19$.

Figure 6

The obtained Gouy phase ${\psi }_G$ (a, b) and quadratic coefficient $\gamma$ (c, d) by fitting the envelope phase with the method of least square. The parameters are set as $a_0=6$ mm, $t_0=2.5$ s, and $C$ = 0 (a, c); $C=-2.19$ (b, d). The scattered data points correspond to the experimental outcomes, while the solid lines represent the theoretical results.

Figure 7

Experimental results for: (a) nonlinear propagation dynamics of HG pulses and (b) the generated second harmonic bound waves with $a_0=21$ mm ($\varepsilon =0.17$), $t_0=2.5$ s, and $C=0$. Both (a) and (b) represent the envelope in a moving system. The color bar units are millimeters.

Figure 8

The spectra of the nonchirped HG wave pulses calculated at the location of $x=7.39$ m, with $t_0=2.5$ s and $C=0$. (a) The measured linear spectra with $a_0$=6 mm ($\varepsilon =0.05$); (b) the measured nonlinear spectra with $a_0$ = 21 mm ($\varepsilon =0.17$). Blue curves correspond to simulations based on Eq. (10); while the red curves denote the experiments.

Figure 9

The spectra of the chirped HG pulses with order (a)–(c) $m=0$ and (d)–(f) $m=1$, calculated at three different locations: (a, d) $x=1.0$ m (before the focus), (b, e) $x=6.5$ m (near the focus) and (c, f) $x=12.0$ m (after the focus). The parameters are set as $a_0=8$ mm, $t_0=3.3$ s, and $C=-4.11$. Blue curves correspond to simulations based on Eq. (10); while the red curves denote the experiments.