# Demodulation Technique Based on Laser Interference for Weak Photo-Acoustic Signals on Water Surface
## 2 Materials and Methods
### PGC-Based Improved Arc Tangent Demodulation Algorithm  
When the signal frequency under test is lower than 2 kHz, PGCs are used to detect the underwater acoustic source. The test system built for this purpose is shown in Figure 4. A high-frequency underwater acoustic source was placed on the reference optical path, and the high-frequency water surface waves excited by it were introduced into the system as the carriers. This introduction of carriers can avoid the use of high-price optical phase modulators and are easy to realize; meanwhile, the interference effect is better because the reference light has a similar light intensity with the measured light. Furthermore, there are also environmental disturbances on the water surface of the reference pool, which can be reduced with measurement pool disturbances and the anti-interference ability of the system can be improved.  
Figure 3: The function image of \(A_{s}(R)\).  
A high-frequency signal was added to the reference arm, its amplitude, frequency, and initial phase being represented by \(A_{c}\), \(\omega_{c}\), and \(\theta_{c}\), respectively.  
\[\left\{\begin{array}{l}H=\frac{2\pi}{\lambda}A_{c}\\ \theta(t)=\frac{2\pi}{\lambda}[\sum\limits_{i}A_{i}\cos(\omega_{i}t+\theta_{i} )+A_{s}\cos(\omega_{s}t+\theta_{s})]\end{array}\right. \tag{4}\]  
Assuming that the reference optical path carriers have a modulation depth of \(H\), and that the phase caused by water surface disturbance and WSAWs under measurement is \(\theta(t)\), we achieved the following interference signals of the PGCs:  
\[U(t)=A\cos[H\cos(\omega_{c}t+\theta_{c})+\theta(t)+\Omega] \tag{5}\]  
Based on Equation (5), a demodulation was performed with the configuration given in Figure 5. This process starts with a sine/cosine mixing of the carrier frequency, followed by low-pass filtering that generated three mixing signals:  
\[\left\{\begin{array}{l}U_{1s}=-AJ_{1}(H)\sin(\theta_{c})\sin[\theta(t)]\\ U_{1c}=-AJ_{1}(H)\cos(\theta_{c})\sin[\theta(t)]\\ U_{2c}=-AJ_{2}(H)\cos(2\theta_{c})\cos[\theta(t)]\end{array}\right. \tag{6}\]  
It can be seen from the above equation that \(U_{1s}\) and \(U_{1c}\) signals only differ in amplitude. Their amplitude ratio is equal to the tangent of the initial phase \(\theta_{c}\) of the carriers. Initial phase of the carriers can be derived from these two signals, which can be written as:  
\[\theta_{c}=\arctan(\frac{U_{1s}}{U_{1c}}) \tag{7}\]  
With the PGC demodulation technique, an arc tangent function demodulation method [15] was used to achieve electronic mixing, filtering, and phase division of the signals expressed by Equation (6). On this basis, the water surface fluctuations under measurement can be obtained by means of demodulation and expressed as follows:  
\[\theta(t)=\arctan[\frac{I_{2}(H)\cos(2\theta_{c})}{J_{1}(H)\cos(\theta_{c})} \cdot\frac{U_{1c}}{U_{2c}}] \tag{8}\]  
A combination of the carrier modulation depth \(H\) obtained from the spectrum and the carrier initial phase \(\theta_{c}\) in Equation (6) gives complete information on demodulated water surface  
Figure 4: Test system diagram.  
fluctuations, which, after elimination of low-frequency disturbances through high-pass filtering, provides WSAW information.