## 2 Principle of Heterodyne Phase Unwrapping
### Principle of Heterodyne Phase Unwrapping  
Multi-frequency heterodyne phase unwrapping [15] is a phase measurement method based on the principle of interference, which is an improvement of the heterodyne phase unwrapping method. Compared to dual-frequency heterodyne unwrapping methods, it can use optical signals with more frequencies to improve the accuracy and reliability of phase measurements.  
In multi-frequency heterodyne phase unwrapping, three or more optical signals with similar frequencies are used for interference, and the phase information on the measured object surface or medium can be obtained by calculating and processing the interference signals of different frequencies. Specifically, these optical signals with similar frequenciesare mixed to obtain a different frequency signal at several frequencies. Phase solving and synthesis of these differential frequency signals can obtain the absolute phase distribution of the measured object. In this paper, the three-frequency heterodyne phase unwrapping is utilized to solve the absolute phase, as shown in Figure 1:  
Firstly, three sinusoidal gratings with pitches \(P_{1}\), \(P_{2}\), and \(P_{3}\) are projected onto the measured object, respectively. Then, after superposition, \(P_{1}\) and \(P_{2}\) generate fringe pattern 12 with pitch \(P_{12}\), and \(P_{2}\) and \(P_{3}\) generate fringe pattern 23 with pitch \(P_{23}\), where \(P_{1}<P_{2}<P_{3}<P_{12}<P_{23}\). For a certain point on the measured object [15], the following relationships exist:  
\[P_{23}=\frac{P_{2}P_{3}}{P_{3}-P_{2}},\ P_{23}=\frac{P_{2}P_{3}}{P_{3}-P_{2}} \tag{3}\]  
\[P_{1}n_{1}=P_{2}n_{2}=P_{3}n_{3}=P_{12}n_{12}=P_{23}n_{23} \tag{4}\]  
\[\Delta n_{i}=\frac{\varphi_{i}}{2\pi},\ i\ =\ 1,2,12,23,123 \tag{5}\]  
\[\theta_{i}=2\pi n_{i}=2\pi(\Delta n_{i}+N_{i})\ N_{i}\in Z \tag{6}\]  
\[\left\{\begin{array}{l}\Delta n_{12}=\left\{\begin{array}{l}\Delta n_{1} \ -\ \Delta n_{2}\ \ \Delta n_{1}\ \geq\ \Delta n_{2}\\ \Delta n_{1}\ -\ \Delta n_{2}\ +\ 1,\ \Delta n_{1}\ <\ \Delta n_{2}\\ \Delta n_{2}\ -\ \Delta n_{3}\ +\ 1,\ \Delta n_{2}\ <\ \Delta n_{3}\end{array}\right.\end{array}\right. \tag{7}\]  
where, \(\varphi_{i}\) and \(\theta_{i}\) refer to the phase principal value and absolute phase of the corresponding fringe pattern, respectively; \(n_{i}\) is the fringe order of the corresponding fringe pattern; \(\Delta n_{i}\) and \(N\) represent the decimal part and integer part of the fringe order, respectively. Fringe pattern 123 is generated by the superposition of fringe patterns 12 and 23. A suitable choice of \(P_{1}\), \(P_{2}\), and \(P_{3}\) allows the fringe pitch of fringe pattern 123 to cover the full field. At this time [16; 17; 18], \(N\) = 0, so  
\[\Delta n_{123}=\left\{\begin{array}{l}\Delta n_{12}\ -\ \Delta n_{23}\,\ \Delta n_{12}\ \geq\ \Delta n_{23}\\ \Delta n_{12}\ -\ \Delta n_{23}\ +\ 1,\ \Delta n_{12}\ <\ \Delta n_{23}\end{array}\right. \tag{8}\]  
\[n_{12}=\frac{P_{23}(N_{123}\ +\ \Delta n_{123})}{P_{23}-P_{12}}, \tag{9}\]  
\[N_{12}=\text{floor}(n_{12}) \tag{10}\]  
\[\theta_{12}=2\pi\times N_{12}+\varphi_{12}, \tag{11}\]  
Figure 1: Principle of three-frequency heterodyne phase unwrapping.