## 3 Results and Discussions
### Low Frequency Mini-Brillouin-Zone Mode
#### 3.2.3 MBZ-Edge Mode Excitation and Detection  
We will now move on to the discussion of the unexpected occurrence of MBZ-edge modes at 15.1 GHz, 78.3 GHz and 110.1 GHz (see Figure 4b marked by green solid dots). Concerning our experimental observation of MBZ-edge modes, firstly, we discuss a scenario that momentum conservation for edge mode could be fulfilled due to the absorption of the excess momentum by free carriers in the doped SL sample, when \(\mid\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\! \(\Rightarrow\omega(0)/2+\omega(0)/2\). Thirdly, because the edge mode excited via stimulated subharmonic decay process is coherent, it introduces a new periodic phonon grating whose period is twice as large as the lattice period. The doubling of the phonon grating period is attributed to the above described excitation of phonon pairs at opposite MBZ edges, which causes lattice vibrations with \(\pi\) phase shift for the adjacent GaAs/Al\({}_{0.95}\)Ga\({}_{0.05}\) units along \(z\)-axis. As a consequence, a new zone folding is required by the dynamically doubled grating. Thus, with respect to the middle of the mini Brillouin zone, \(\pm k_{max}/2\), the original zone-edge phonons are folded to near-center at \(k\cong 0\), \(\omega\cong\omega(0)/2\) where, being Raman-active, they are observable by the probe light. Fourthly, the probe light scattering by dynamic folded phonons with \(k\cong 0\), \(\omega\cong\omega(0)/2\) is a Raman-Nath-like process which enables the detection of multiple higher-order harmonics of zone-edge modes with comparable amplitudes [23; 24]. As a result of the above hypothesis, the MBZ-edge mode around 15 GHz at \(k=\pi/d_{DBR}\), and its higher-order replicas in the signal up to those at 78.3 GHz and 110.1 GHz with the same-order amplitudes could be possibly detectable due to the subharmonic decay of the \(\omega(0)\approx 32\) GHz mode, subsequent dynamic zone folding, and a Raman-Nath-like probe light scattering process in our experiments (see Figure 4b). So far, we have only discussed the core principle of detectability of MBZ-edge modes; however, the actual processes are much more complicated. As illustrated in Figure 6, basically, three steps are involved to create a quite complex MBZ acoustic spectrum in the DBR. At first, a series of MBZ-center modes \(\omega_{i}(0)\) (\(i=1\), \(2\), \(3\),..., approximately at 32, 64, 96 GHz,... ) are excited due to the electrostriction mechanism, and those modes can become detectable by stimulated Brillouin scattering or Raman scattering (see line 1 in Figure 6). Subsequently, the stimulated subharmonic decay of MBZ-center modes \(\omega_{i}(0)\) (\(i=1\), \(2\), \(3\),... ) excites the edge modes \(\omega_{i}(0)/2\) (\(i=1\), \(2\), \(3\),..., approximately at 16, 32, 48, 64, 80, 96, 112 GHz,... ), and then the edge modes become detectable at the zone center due to dynamical zone folding (see lines 2 and 3 in Figure 6). At last, the detection of dynamically-folded \(\omega_{i}(0)/2\) (\(i=1\), \(2\), \(3\),... ) mode at the zone center is a Raman-Nath-like probe light scattering process, which means several series of Raman-Nath-like phonon signals \(m\omega_{i}(0)/2\) (\(m\) and \(i=1\), \(2\), \(3\),... ) can be present at the same time, for example, \(m\omega_{1}(0)/2\) = 16\(m\) (16, 32, 48, 64, 80, 96, 112 GHz,..., see line 4 in Figure 6), \(m\omega_{2}(0)/2\) = 32\(m\) (32, 64, 96,..., see line 5 in Figure 5), \(m\omega_{2}(0)/2\) = 48\(m\) (48, 96, 144 GHz,... see line 6 in Figure 6), and so on. As can be seen, in particular, the subharmonic decay of the first-order MBZ center mode  
Figure 5: Illustration of zone folding induced by a pair of phonons on the opposite MBZ edges forming phonon grating. A pair of phonons is excited by the subharmonic decay of mode \(\omega(0)\). In the DBR, at the MBZ edge wavevector is expressed as \(k=k_{max}=\pi/d_{DBR}\).