Spectral shape of stimulated Brillouin scattering in crystals

We derived a formula to describe the stimulated Brillouin spectral shape in crystals for various temperatures ranging from room temperature to liquid-helium temperature. We modeled a sample as a one-dimensional system with a finite thickness in which the optically induced phonon propagates, partly interacting with the pump and probe laser beams. When the sample length is shorter than the propagation distance (i.e., the mean free path) of phonons, the spectral shape becomes multipeaked due to the multiple phonon reflections in the sample. Such a situation can be realized in a thin film or a bulk sample at low temperatures. We experimentally measured the Brillouin gain spectra with a multipeak structure in ${\mathrm{TeO}}_2$ and ${\mathrm{PbMoO}}_4$ crystals at low temperatures. We found that these spectra were reproduced by our formula for both the coaxial and off-axis phonon propagations with respect to the laser beams. It was revealed that our formula is very useful in estimating the phonon attenuation coefficient from the observed spectra, which gradually change from Lorentzian shape to a multipeak spectrum with decreasing temperature.

Figure 1

Brillouin gain in a one-dimensional medium of length $L$. The electric field, angular frequency, and wave vector of the incident probe and pump beams are denoted by $E_i, {\omega }_i$, and ${\mathit{k}}_i$ ($i=1,2$), respectively. The external force oscillating with $\mathrm{\Omega }={\omega }_1-{\omega }_2, \mathit{q}={\mathit{k}}_1-{\mathit{k}}_2$ drives the sound wave.

Figure 2

Mirror image of $G^{\prime }(z,t,t_0)$ due to reflection. The $C\left(z\right)$-shaped waves propagate along the $z$ axis with decreasing amplitude and a separation between each wave of $2L$.

Figure 3

Spectral shapes of the response function depending on the acoustic attenuation from Eq. (25). $\mathrm{\Gamma }/2=0.001$, 0.01, and 0.1 are shown by the red, green, and blue curves, respectively. The center frequency region for the case $\mathrm{\Gamma }/2=0.001$ (red curve) and the Lorentzian (gray curve) with the same attenuation is magnified in the inset.

Figure 4

Left column: Schematic configurations of the incident laser beams and the propagating sound wave. Right column: The electric field intensity distributions experienced by the propagating sound wave. The sound wave generated by (a) plane collimated and (d) focusing beams propagates coaxially. Two types of off-axis propagation are shown in (b) and (c). (b) The sound wave walks off and deviates from the generating area in a low symmetric direction of an anisotropic media. The retroreflection of the sound wave occurs at the end of the media. (c) Even when the laser beams are misaligned from normal incidence if the crystal surface is oriented in the highly symmetric direction, the generated sound wave can propagate and multireflect.

Figure 5

Brillouin gain spectra of ${\mathrm{TeO}}_2$ in the [110] direction. Red points show the experimentally measured spectra at (a) 45.9, (b) 31.0, and (c) 6.6 K. The blue curves are the results of parameter fitting via Eq. (25).

Figure 6

(a) Brillouin gain spectrum for ${\mathrm{PbMoO}}_4$ in the [100] direction. The red curve is the experimentally obtained spectrum. The blue curve is derived from Eq. (23) by substituting Eq. (27). The center frequency region of the spectra is magnified in (b) to further clarify the fine structure present in the spectrum.