## 3 Results and Discussions
### Low Frequency Mini-Brillouin-Zone Mode
#### 3.2.2 MBZ-Center Mode Excitation
As a result, because in our DBR the wavenumber of the pump light is very close to the MBZ boundary, in other words, half of the pump light wavelength is close to the DBR period, the photo-generated phonon is very close to MBZ center. However, this is not the only channel for the generation of the GHz CLAPs in the DBR. In the DBR, not only the periodically oscillating part, but also the averaged part of the intensity distribution of the pump light interference pattern can generate high frequency CLAPs due to the spatial periodic modulation of the electrostriction parameter in the DBR. This mechanism is absent in spatially-homogeneous, non-structured media. In principle, in this second channel, all \(k\) = 0 modes of the phononic DBR are photo-generated. From the above analysis, it follows that, because in our DBR \(\lambda_{pump}^{DBR}/2\) and \(d_{DBR}\) are very close, the constructive generation of the lowest energy \(k\cong 0\) mode in the MBZ of the DBR via two described channels could be possible, in spite of transparency of the DBR for the pump light. The local electrostriction force \(\overset{\rightarrow}{f}\) can be presented as [43]  
\[\overset{\rightarrow}{f}_{\text{\tiny es}}=\frac{1}{2}\
abla(\rho\frac{d \epsilon}{d\rho}E^{2}), \tag{5}\]  
where \(\rho\) denotes the total mass per unit volume, \(\epsilon\) denotes the permittivity and \(E\) denotes the electric field. The right side of Equation (5) can be expanded into two terms:  
\[\overset{\rightarrow}{f}_{\text{\tiny es}}=\frac{1}{2}\rho\frac{d\epsilon}{d \rho}\
abla E^{2}+\frac{1}{2}E^{2}\
abla(\rho\frac{d\epsilon}{d\rho}). \tag{6}\]  
The first term describes the force \(\overset{\rightarrow}{f}_{\text{\tiny es1}}\) stemming from the spatial modulation of the electric intensity \(E^{2}\); the second term describes the force \(\overset{\rightarrow}{f}_{\text{\tiny es2}}\) caused by the spatial modulation of the electrostriction coefficient \(\rho\frac{d\epsilon}{d\rho}\). The combined effects described by Equation (6) cause the resonant excitation of MBZ mode at \(k\cong 0\) (fundamental \(\sim\)32 GHz mode and its higher-order harmonics) in the DBR, given the condition \(\lambda_{pump}^{DBR}/2\approx d_{DBR}\) in our sample. Thus, the peaks at 32.0 GHz, 63.1 GHz and 95.1 GHz in Figure 4b can be attributed to MBZ-zone center modes, which are resonantly coherently excited by the light field in the DBR via simultaneous spatial modulation of the electric field intensity and of the electrostriction coefficient. One may notice that amplitudes of the MBZ center modes do not show a monotonic decrease with increasing order. On the contrary, the second order mode 63.1 GHz significantly dominates in the spectrum. The amplitude ratios among the three modes are approximately 0.2:1:0.05. In order to explain this unusual behavior, we need to take two factors into account. Firstly, as demonstrated in Section 3.1, the acoustic frequency comb excited in the In\({}_{0.27}\)Ga\({}_{0.73}\)As QWs has multiple components equidistantly spaced by \(\Delta f\approx 33\) GHz over a wide frequency range, which means the first three components are at the locations \(\Delta f\), \(2\Delta f\) and \(3\Delta f\), respectively. Therefore, the first three comb components are approximately coincident with the MBZ center modes of the first three orders in the DBR. In consequence, the experimental frequency at around 63.1 GHz could acquire more contribution from the photo-excited acoustic frequency comb in the QWs than the experimental frequencies at around 32.0 GHz and 95.1 GHz do. This is due to the fact that in the lowest frequency region, the side lobe of the envelope determined by the individual wave-packet FFT shows a peak close to the comb component \(2\Delta f\), while it exhibits dips close to the comb components \(\Delta f\) and \(3\Delta f\) (see Figure 2b).