## 2 Principle
 \tag{1}\]  
The received optical power \(P(t)\) of the QD photosensitive surface can be expressed as Equation (2), where \(P_{S}\), \(P_{LO}\) are the DC components, QD denotes the detector photosensitive surface area, and \(\rho\), \(\varepsilon\) denote the transmittance and reflectance of the optical beam splitter (\(0<\rho\), \(\varepsilon<1\)), respectively. The beat frequency signal is the AC component, \(\omega_{het}=\omega_{S}-\omega_{LO}\) denotes the angular frequency difference, \(\Delta\psi(\mathbf{r})=\psi_{S}(\mathbf{r})-\psi_{LO}(\mathbf{r})\) denotes the wavefront phase difference, and \(\Delta\varphi=\varphi_{S}-\varphi_{LO}\) denotes the initial optical phase difference. \[ \begin{array}{l}P(t)=\int\limits_{QD}|\rho E_{S}(\mathbf{r},t)+iE_{LO}(\mathbf{r},t )|^{2}dA\\ =\underbrace{\rho^{2}P_{S}+\varepsilon^{2}P_{LO}}_{DC}+ \underbrace{2\rho\varepsilon\int\limits_{QD}|E_{S}(\mathbf{r})E_{LO}(\mathbf{r})|\cos [(\omega_{het}t+\Delta\psi(\mathbf{r})-\Delta\varphi)]dA}_{AC}\\ P_{S}=\int\limits_{QD}|E_{S}(\mathbf{r})|^{2}dA\quad P_{LO}=\int\limits_{QD}|E_{ LO}(\mathbf{r})|^{2}dA\end{array}  \tag{2}\]  
After the triangular transformation of the AC signal, the received optical power \(P(t)\) can be expressed as Equation (3), where \(\gamma\) denotes the heterodyne efficiency, which is used as a measure of coherent receiver performance. It is expressed in the QD photosensitive surface as the proportion of the overlapping area of the light spot presented by the signal  
Figure 1: Block diagram of the heterodyne coherent angle measurement principle. (**a**) Overall process block diagram. (**b**) Spatial coherent mixing. (**c**) DWS angle phase conversion. (**d**) Phase tracking and detection. and the local oscillation laser beams. \(\Delta\phi\) denotes the phase difference information of the two laser beams. For a 50:50 optical beamsplitter, \(\rho=\epsilon=1/\sqrt{2}\). \[ \begin{split} P(t)&=\rho^{2}P_{S}+\epsilon^{2}P_{LO}+2 \rho\epsilon\sqrt{P_{S}P_{LO}\gamma}\cos(\omega_{het}t+\psi)\\ \gamma&=\frac{\left[\int\limits_{QD}\left|E_{S}( \textbf{r})E_{LO}^{*}(\textbf{r})\big{|}dA\right]^{2}}{\int\limits_{QD}|E_{ S}(\textbf{r})|^{2}dA\int\limits_{QD}|E_{LO}(\textbf{r})|^{2}dA}=\frac{Q^{2}+I^{2}}{P_{S}P _{LO}}\end{split}  \tag{3}\]  
where the \(I\) and \(Q\) variables in Equation (3) are associated with the amount of wavefront phase difference \(\Delta\phi(\textbf{r})\) caused by the angular deviation between the laser beams. This can reflect the changes in amplitude and phase of the beat frequency signal. The expressions of the \(I\) and \(Q\) variables are shown in Equation (4), where dA denotes the integral operation carried out on the QD surface. \[ \begin{split} Q&=\int\limits_{QD}|E_{S}(\textbf{r} )E_{LO}(\textbf{r})|\sin(\Delta\psi(\textbf{r}))dA\\ I&=\int\limits_{QD}|E_{S}(\textbf{r})E_{LO}(\textbf{r })|\cos(\Delta\psi(\textbf{r}))dA\end{split}  \tag{4}\]  
After substituting the above parameters, the received optical power \(P(t)\) and photocurrent \(I(t)\) in each quadrant of the QD can be expressed as Equation (5), where \(\zeta\) denotes the photodiode efficiency, \(q_{e}\) denotes the electron charge (1.602 \(\times\) 10\({}^{-19}\) C), \(\lambda\) denotes the laser wavelength, \(h\) denotes the Planck constant (6.626 \(\times\) 10\({}^{-34}\) J*s), \(c\) denotes the speed of the beam, \(\eta\) denotes the photodiode responsivity, and \(N\) denotes the number of detector quadrants. \[ \begin{split} P(t)&=\frac{P_{S}}{2N}+\frac{P_{t} \omega}{2N}+\frac{1}{N}\sqrt{P_{S}P_{LO}\gamma}\cos(\omega_{het}t+\psi)\\ I(t)&=\zeta\frac{q_{e}A}{hc}P(t)=\eta\Big{[}\frac{P_{S} }{2N}+\frac{P_{t}Q}{2N}+\frac{1}{N}\sqrt{P_{S}P_{LO}\gamma}\cos(\omega_{het}t +\psi)\Big{]}\end{split}  \tag{5}\]  
In actual scenarios, the incident angle deviation of the signal laser beam will cause differences in the phase information of the signals received by different quadrants of the detector.