# Remote Speckle-Based Measurements of Backward Brillouin Acoustic Vibrations in Optical Fibers
## 5 Discussion  
The results that are presented in Table 2 show that the locations of the measured seem to be not consistent with the changes of the modulation frequencies. According to the relation that is shown in Equation (8), a change of 200 kHz in the modulation frequency should lead to a change of 400 kHz at the measured peak. In our experiment, however, the difference between two consecutive peak values is between 10-68 kHz. However, the accepted results could be well explained by the aliasing phenomena, which cause different signals to become indistinguishable when sampled. Since changes of 400 kHz at the acoustic frequency can take us out of the sampling range (i.e., 450 kHz), the measured frequency \(\Omega_{meas}\) will be given by the modulo operation in the following way:  
\[\Omega_{meas}=\text{mod}(\Omega_{c,1},450\text{kHz})=\text{mod}(\Omega_{B,2 }-2\times\omega_{m},450\text{kHz}) \tag{10}\]  
It can be easily seen that for this relation, a change of 400 kHz in the acoustic frequency \(\Omega_{c,1}\) leads to a change of 50 kHz at the measured frequency. This is quite close to the accepted results presented in Table 1 in which the difference between two consecutive peak values is between 10-68 kHz. To clarify this claim, suppose the first frequency peak in the table was sampled without aliasing. According to Equation (8), we can extract the acoustic frequency \(\Omega_{B,2}\):  
\[\Omega_{B,2}=2\times\omega_{m}+\Omega_{c,1} \tag{11}\]  
Which yields:  
\[\Omega_{B,2}=2\times 5.3001\times 10^{9}+215.8\times 10^{3}=10.6004\text{GHz} \tag{12}\]  
If we put this value in Equation (10) together with the different modulation frequencies, we could find the theoretical values that should be measured in our experiment. Those values compared to experimental values are presented in Table 3.  
The small differences between the theoretical values to the experimental results may be explained by small changes in temperature that occurred during the measurement. Equation (10) shows that the measured frequency depends not only on the modulation frequency but also depend on the acoustic frequency \(\Omega_{B,2}\). According to Equation (5), this frequency depends on the acoustic velocity, which is a temperature-depended parameter. From both Equation (5) and Equation (10) it is possible to show that a small change of 0.01 m/s in the acoustic velocity less to change of about 40 kHz in the measured peak, which may explain the small inconsistent of the accepted results.  
Note that in our experimental setup there are some additional acoustic frequencies, besides the frequency presented in Equation (7) that may also be detected by our system. Those frequencies include the whole possible combinations of frequencies presented in Equation (5) and the second  
\begin{table}
\begin{tabular}{c c} \hline
**Modulation Frequency (GHz)** & **Peak (kHz)** \\ \hline
5.3001 & 215.8 \\
5.3003 & 225.8 \\
5.3005 & 294.4 \\
5.3007 & 303.3 \\ \hline \end{tabular}
\end{table}
Table 2: The peak frequencies received for each modulation frequency.  
\begin{table}
\begin{tabular}{c c c} \hline
**Modulation Frequency (GHz)** & **Measured Peak (kHz)** & **Theoretical Value (kHz)** \\ \hline
5.3001 & 215.8 & 215.8 \\
5.3003 & 225.8 & 265.8 \\
5.3005 & 294.4 & 315.8 \\
5.3007 & 303.3 & 355.8 \\ \hline \end{tabular}
\end{table}
Table 3: The measured peak frequencies compare to the theoretical values.  
cross interaction frequency presented in Equation (9), where some of them may be found within the sampling rates or may be detected by the aliasing phenomena. The existence of these additional acoustic frequencies in our system may also be the reason for the secondary peaks accepted in some measurements (see Figure 4c,d). More experimental work is required to find the exact significance of the vibrations which were measured in this setup.