# Demodulation Technique Based on Laser Interference for Weak Photo-Acoustic Signals on Water Surface
## 2 Materials and Methods  
### General Concept  
An interference system is used for water surface detection, and information on water surface waves is contained in the phase of the detection signals. The disturbances have two components: natural disturbances of the water surface and WSAW. By expressing the water surface disturbances with simple harmonic vibration superimposition, we depict the interference signals in the following equation:  
\[U(t)=A\cos\{\frac{4\pi}{\lambda}[\sum_{i}A_{i}\cos(\omega_{i}t+\theta_{i})+A_{ s}\cos(\omega_{s}t+\theta_{s})]+\Omega\} \tag{1}\]  
where \(A\) is the system gain of the interference signals; \(\lambda\) is the laser wavelength; and \(\Omega\) is the fixed phase caused by the interference optical path. \(A_{i}\), \(\omega_{i}\), and \(\theta_{i}\) represent amplitude, angular frequency, and initial phase of the natural disturbances waves, respectively, and \(i\) represents number of harmonic waves. Similarly, \(A_{s}\), \(\omega_{s}\), and \(\theta_{s}\) are amplitude, angular frequency, and initial phase of the WSAW, respectively. In Appendix A, the above equation receives a trigonometric function expansion and a Bessel function simplification in turn, followed by a simulation at an underwater acoustic source frequency of 1 kHz.  
The resulting signal spectrum of Table 1 is shown in Figure 1. The dense spectral lines in the low-frequency zone are attributed to natural fluctuations of water surface and various low-frequency disturbance waves. Concentrated frequency bands exist around 1 kHz, i.e., the acoustic source frequency, and exhibit a symmetric distribution. Spectral lines also appear around multiples of 1 kHz such as 2 kHz and 3 kHz, although with negligible amplitude. In reality, the wide low-frequency bands imply superimposition of low frequency acoustic source signals and low-frequency natural water surface fluctuation signals, making it impossible to perform spectrum analysis of the underwater low-frequency signals directly. A new phase demodulation method is therefore required to obtain the vibration information.  
Two demodulation algorithms were used in our study for interpretation of signals acquired in the course of underwater acoustic source detection, as shown in Figure 2. The signals provided by the photodetector underwent a DC filtering process, and a spectrum analysis was then performed on the interference signals. Different demodulation schemes were adopted depending on the frequency. For underwater high-frequency acoustic source signals above 2 kHz, frequency bands independent of the low-frequency ones were created in the spectrum, with a symmetrical distribution around the underwater sound-generating source frequency. After determination of the underwater source frequency by locating the distribution center, a characteristic ratio method was used to obtain the WSAW amplitude. For low-frequency signals below 2 kHz, given their aliasing with low-frequency bands, we were not able to derive further information from the spectrum. To address this, an improved arc tangent function algorithm was employed to demodulate the phase generated carriers (PGCs) signals, thus characterizing phase changes caused by WSAWs, and obtaining amplitude and frequency of the WSAWs.