## 3 Results and Discussions
### High Frequency Coherent Acoustic Phonon Combs
#### 3.1.4 Broadband Acoustic Spectrum  
In addition to the temporal evolution of the acoustic signal, its frequency spectrum is also worthy of scrutiny, given that our previous result [25] revealed only a part of the features visible in the spectrum detected in the current experiments. After 25 times magnification in the frequency range from 0 to 232 GHz and 15 times magnification in the frequency range from 497 GHz to 1000 GHz, the acoustic spectrum revealed a wide sub-THz comb-like structure with a regular comb spacing of around 33 GHz, which covered the range from \(\sim\)30 GHz to \(\sim\)800 GHz, as illustrated in Figure 2b. The comb spacing \(\sim\)33 GHz (marked by gray vertical lines) corresponds to the reciprocal of time delay of neighboring wave-packets (30.25 ps), which agrees very well with the experimental result in the vicinity of 365 GHz, while there are small offsets in other spectral ranges. The visible non-constant comb spacing can be attributed to several factors. Firstly, one has to notice that the spectral amplitudes in the range 0-232 GHz and in the range 497-1000 GHz are at least one order of magnitude smaller than those around 365 GHz, which could possibly introduce a few GHz offset from the marked positions. Secondly, the low frequency modes could be nearly degenerate with the mini-Brillouin-zone modes  
Figure 3: The illustration of the generation, propagation, and detection of acoustic stress bursts in the sample. Beige areas represent GaAs, violet areas represent QWs. Green area represents Al\({}_{0.95}\)Ga\({}_{0.05}\)As. S\({}_{ir}\) and S\({}_{ib}\) (i = 1, 2, 3) denote the backward and forward emitted from the i-th triple-QW stack stress bursts presented in red and blue colors, respectively. which will be discussed in Section 3.2. Thirdly, there are uncertainties in the determination of the longitudinal acoustic velocity in the sample and in the quantum well stack-to-stack distance. This will lead to slight deviations from the expected constant comb spacing. The mode amplitudes of the frequency comb are modulated by the envelope derived from the first wave-packet FFT (magenta line in Figure 2b), which produces a main lobe with a bandwidth of \(\Delta B=109.6\) GHz (full width at half maximum) and a few side lobes with at least an order of smaller amplitudes than that of the main lobe. Therefore, the five frequency components in the vicinity of 365 GHz dominate the acoustic temporal features, while the lower frequency components (below 100 GHz) and higher frequency components (around 700 GHz) have a much smaller impact on the overall acoustic oscillations. It is worth noting that some of the predicted comb components are missing, which can be explained by the amplitude modulation of their envelope. Because the observed wave-packet is a result of the convolution between the generated stress burst and the detection sensitivity function of the QW-stack, the FFT envelope of the frequency comb is associated with the generation and detection processes. If the generation and detection functions are assumed to be of the rectangular spatial shape with a 33-nm width equal to the QW-stack thickness, due to the spatial restriction, the corresponding FFT envelope is thus sampling-function-like with a main lobe bandwidth \(\Delta B=124\) GHz, which means the amplitude modulation leads to the suppression of comb components close to the dips of the envelope such as those around 260 GHz and 460 GHz. In total, up to 24 periodic frequency components spaced by 33 GHz are resolved in our experiment. Table 3 lists the main high frequency components from experiment and calculation. In comparison with calculations based on \(m\Delta f\) (\(m\) is an integer), the experimental frequency components show a small shift, i.e., of around 6 GHz, for each component in the vicinity of 365 GHz, which can be attributed to the uncertainties of longitudinal sound velocities in the sample and stack-to-stack distances.