Inelastic light scattering by trains of ultrashort acoustic solitons in sapphire

We monitor the development of a picosecond strain wave packet into a train of ultrashort acoustic solitons along the $c$ axis of sapphire using Brillouin scattering. One-dimensional propagation yields an intricate oscillation pattern of the scattered intensity against distance that is interpreted in terms of optical interference and Bragg resonances of light reflected from the moving soliton train. By exploring this pattern over a range of scattering angles, we derive quantitative information on the soliton parameters using only a simple analytical framework. Further more, the analogy between the wave packet and a diffraction grating is explored that provides a direct estimate of the amount of solitons in the train from the observed number of oscillation periods.

Figure 1

Visualization of the $\mathrm{KdV}$ scattering problem, showing the time varying potential $-\eta (\xi ,\tau )$ and the incident, reflected and transmitted wavefunctions ${\Psi }_{\mathrm{in}}$, ${\Psi }_R$ and ${\Psi }_T$. Potentials $-\eta (\xi ,0)$ are the normalized Gaussian derivative (line) and hyperbolic secants (thick dash) functions, and the resulting soliton train for a value of $\sigma \approx 13$. Horizontal lines (dash) denote eigenvalues $E_n={\lambda }_n∕{\sigma }^2$ for the Gaussian derivative.

Figure 2

(a) Simulated development of the leading, compressive part of a bipolar strain pulse during the first $200\mu \mathrm{m}$ in a sapphire [0001] crystal, showing the initial stages of self-steepening of the wavepacket and breakup into a soliton train. (b) Development of the acoustic spectrum of the total bipolar wave packet.

Figure 3

Pump-probe reflectivity data of a $100\text{−}\mathrm{nm}$ chromium film on lead molybdate (line) and sapphire (dash) single crystals. Smooth lines denote fits to the data. (a) Spectrum of the first acoustic echo as obtained from fits to the data.

Figure 4

(a) Dependence of Brillouin scattering intensity on the electric-field polarization angle of argon-ion laser, minimum at $\vec{E}\perp c$, maximum at $\vec{E}\Vert c$. (b) Power spectrum of strain waveform gauged against thermal phonon background, $\circ$ experimental data, (line) simulation. Calibrated strain wave form obtained from (b), consistent with reflectometry data of Fig. 3.

Figure 5

Brillouin scattering intensity as a function of propagation distance at 5 different scattering angles, at a pump fluence of $6.5\mathrm{mJ}∕{\mathrm{cm}}^2$. Inset: spatial Fourier transforms of data $\left(k=1∕\lambda \right)$, arrows indicate highest wavevector in the scans.

Figure 6

Highest spatial frequency ${\nu }_x$ as a function of Brillouin frequency ${\nu }_B$ for 4 typical pump fluences $E=2.6(◀)$, $3.2(◇)$, $4.9(▵)$ and $6.5(엯)\mathrm{mJ}∕{\mathrm{cm}}^2$. Lines are fits to data, following Eq. (11). Inset: initial strain amplitude $\mid s_0\mid$ obtained from the slope of fitted data at different pump fluences $E$. Line denotes linear dependence.

Figure 7

(a) Analytical soliton trains after $0.5\mathrm{mm}$, using 1 real form, Eq. (7) (solid) and 2 equidistant approximation, Eq. (8) (dash). (b) Power spectrum of the trains 1. (solid) and 2. (dash) of (a), arrows indicate positions of local minima in the spectrum of 1.