Attenuation process of the longitudinal phonon mode in a ${\mathrm{TeO}}_2$ crystal in the 20-GHz range

We experimentally investigated the hypersonic attenuation process of a longitudinal mode (L-mode) sound wave in ${\mathrm{TeO}}_2$ from room temperature to a lower temperature using Brillouin scattering and impulsive stimulated thermal scattering (ISTS) measurements. For precise measurement of the Brillouin linewidth at low temperatures, whereby the mean free path of the phonon becomes longer than the sample length, it is indispensable that the phonon should propagate along the phonon-resonance direction. To figure out the suitable direction, we defined two indices characterizing a degree of phonon divergence and a purity of propagation direction. The best direction that we found from these indices is [110] direction in ${\mathrm{TeO}}_2$, and it was used to discuss the temperature and frequency dependences of Brillouin spectra. We extracted the temperature dependence of the attenuation rate of $T^4$ from the modulated Brillouin spectra due to the phonon resonance below Debye temperature. The frequency dependence ${\omega }^1$ of the hypersonic attenuation was also estimated from the polarization dependence of the Brillouin linewidth. Theoretically, it predicted that the L-mode phonon attenuation at low temperatures in ${\mathrm{TeO}}_2$ is a result of Herring's process, which shows the attenuation behavior of ${\omega }^2{T}^3$. The ${\omega }^1{T}^4$ dependence is not allowed in Herring's process but is allowed by the $L+L\to L$ process, which has been considered to be forbidden so far. We evaluated the thermal phonon lifetime using ISTS and established that it was finite even at 20 K, thereby allowing the $L+L\to L$ process. Therefore, we conclude that the $L+L\to L$ process dominates the attenuation of an L-mode phonon in ${\mathrm{TeO}}_2$ in the low-temperature region.

Figure 1

Distribution of $\mathbf{div}{\mathbf{v}}_{\mathrm{g}}^{\prime \prime }$ for the L-mode phonon in ${\mathrm{TeO}}_2$ crystal.

Figure 2

The index of pure propagation in an L-mode phonon of ${\mathrm{TeO}}_2$ crystal.

Figure 3

Herring's diagram for the $L_1+L_2\to L_3$ process in ${\mathrm{TeO}}_2$ along the [110] direction. Upper and lower panels depict the cross sections of the slowness surfaces within the $xy$- and $z$-[110] planes, respectively.

Figure 4

Experimental setup for a stimulated Brillouin scattering spectrometer. Monolithic type Nd:YAG lasers are used for the pump and probe lasers. The frequency difference between the pump and probe waves is real-timely monitored by using a frequency counter through the beat frequency between the waves. The counterpropagating laser beams are focused into a sample crystal in a continuous flow type cryostat. FP and ABPR denote a scanning Fabry-Perot interferometer and an auto-balanced photoreceiver, respectively. See Ref. [12] for details.

Figure 5

Temperature dependence of Brillouin spectrum of an L-mode phonon in ${\mathrm{TeO}}_2$, [110] direction measured above (a) and below (b) 50 K. The spectra are offset vertically for clarity.

Figure 6

Temperature dependence of the attenuation rate of an L-mode phonon in ${\mathrm{TeO}}_2$, [110] direction. The corresponding phonon frequency was around 20 GHz as shown in Fig. 5.

Figure 7

Comparison of Brillouin spectra obtained in ${\mathrm{TeO}}_2$, [110] direction, L-mode phonon at 373 K. Green and red curves show the spectra in $n_{\mathrm{e}}$ and $n_{\mathrm{o}}$, respectively.

Figure 8

Frequency dependence of the Brillouin linewidth in the [110] L-mode phonon of ${\mathrm{TeO}}_2$ as a function of temperature.

Figure 9

A schematic of the impulsively stimulated thermal scattering (ISTS) method for measuring thermal phonon lifetimes. The probe laser was scattered by the thermal grating generated by the pump pulses. The scattered probe intensity varying in time was measured with an oscilloscope.

Figure 10

Temperature dependence of the thermal phonon lifetime estimated by the ISTS measurement.

Figure 11

Comparison of attenuation rates between our results and the Herring process in ${\mathrm{TeO}}_2$. Attenuation rates measured in [110] and [001] directions are depicted with red and green point, respectively. Since the phonon resonance effect was not observed in the Brillouin spectra measured in the [001] direction, the determination accuracy of attenuation rate was lower than that in the [110] direction at low temperatures. The blue colored line represents $A{\omega }^2{T}^3$ with $A=7.69\times {10}^{-22}\left[{(\mathrm{rad}./\mathrm{s})}^{-1}{\mathrm{K}}^{-3}\right]$, which is estimated from Eq. (11) for the [110] direction. The pink colored line represents $B{\omega }^1{T}^4$. $B=4\times {10}^{-12}\left({\mathrm{K}}^{-4}\right)$ is observed in Fig. 6. For both lines, $\omega /2\pi =20.3193\mathrm{GHz}$ is used.

Figure 12

Temperature dependence of the estimated value of $\omega \tau$ for $\omega /2\pi =20\mathrm{GHz}.$

Figure 13

Phonon dispersion relation in the [110] direction of ${\mathrm{TeO}}_2$. The data were traced in the low-energy region given by Fig. 3 in Ref. [45]. The vertical axis was reduced from energy to temperature using $1\mathrm{K}\simeq 0.7{\mathrm{cm}}^{-1}$.

Figure 14

The area satisfying ${\omega }_1+{\omega }_2={\omega }_3+\mathrm{\Delta }\omega$ with varying $\mathrm{\Delta }\omega$ around the [110] direction for the $L+L\to L$ process in ${\mathrm{TeO}}_2$. (Upper) mapping in the $\theta ,\varphi$ plane. (Lower) three-dimensional mapping of the slowness surface of the L-mode phonon.

Figure 15

Polar coordinates in a crystalline direction.

Figure 16

Schematics of the proposed index for divergence. (a),(b) Projection of the normalized group velocity onto a sphere of $r=1$. (c) The divergence of the normalized group velocity in a $(\theta ,\varphi )$ plane.

Figure 17

The area satisfying ${\omega }_1+{\omega }_2={\omega }_3+\mathrm{\Delta }\omega$ with varying $\mathrm{\Delta }\omega$ around the [110] direction for the $L+FT\to FT$ process in ${\mathrm{TeO}}_2$.