Calibrated real-time detection of nonlinearly propagating strain waves

Epitaxially grown metallic oxide transducers support the generation of ultrashort strain pulses in SrTiO${}_3$ (STO) with high amplitudes up to 0.5$\%$. The strain amplitudes are calibrated by real-time measurements of the lattice deformation using ultrafast x-ray diffraction. We determine the speed at which the strain fronts propagate by broadband picosecond ultrasonics and conclude that, above a strain level of approx. 0.2$\%$, the compressive and tensile strain components travel at considerably different sound velocities, indicating nonlinear wave behavior. Simulations based on an anharmonic linear-chain model are in excellent accord with the experimental findings and show how the spectrum of coherent phonon modes changes with time.

Figure 1

(a) Static $\theta$-$2\theta$ scan of the (002) peaks of the LSMO-STO sample showing a weak and broad layer (LSMO) peak and a much brighter and narrower STO substrate peak. (b) Transient UXRD signal after pumping with a laser pulse. The layer peak shifts to smaller angles indicating the expansion of LSMO within 6 ps (logarithmic color code). (c) Two cuts of panel (b) with pump-probe delays $-2$ ps (before pumping) and 12 ps (when the strain pulse has left the layer).

Figure 2

Relative optical reflectivity change of the LSMO-STO sample for a pump fluence of (a) 14 $\mathrm{mJ}/{\mathrm{cm}}^2$ and (b) 47 $\mathrm{mJ}/{\mathrm{cm}}^2$. The low-frequency background was subtracted by high pass filtering. The probe pulse wavelength is given by $\lambda$ (left axis). Both measurements show oscillations which are attributed to Brillouin backscattering of a photon from a phonon with wave vector $k_P$ (right axis). For strong excitation conditions (b) we observe a beating in these oscillations.

Figure 3

Measured sound velocity distribution of the induced strain pulse in STO. The different pump fluences were calibrated with the UXRD measurement to the resulting induced strains of the LSMO layer which is directly linked to the strain amplitude of the bipolar strain pulse in the STO. The narrow distribution for 0.14$\%$ strain implies that the entire strain pulse essentially propagates with a speed around 8 nm/ps. With increasing strain amplitude the sound velocity distribution gets broader and eventually a double-peak distribution is established. At high excitation levels different parts of the strain pulse propagate with different velocities. The stars indicate the sound velocities of the self-steepened sound pulses simulated in Fig. 4a.

Figure 4

(a) Spatial profile of the bipolar strain pulse in the STO for different propagation times in a frame of reference propagating with the speed of sound $v_s$ for high amplitude (blue line, 0.47% strain) and low amplitude (black line, 0.14% strain). For large amplitude, the tensile part of the pulse propagates with subsonic speed and the compressive part propagates with supersonic speed indicated by the stars in Fig. 3. (b) Measured amplitude of oscillations for each wavelength connected to the wave vectors by the Brillouin backscattering condition. The region between the vertical black lines indicates the wave vectors that can be accessed by the optical white light. (c) Phonon amplitude divided by the wavelength $\lambda$ (see text) as a function of wave vector calculated from Fourier transforms of the simulated strain profile, showing good agreement with the measurement in panel (b).