## 2 Materials and Methods  
### Principle of High-Speed Holographic for Shape and Displacement Measurements  
#### 2.1.1 Multiple Wavelength Holographic Interferometry (MWHI) Method for Shape Measurement  
The multiple wavelength holographic interferometry (MWHI) shape measurements are based on the principle of variations in the wavelength of illumination with a constant optical path length (OPL) (i.e., a stationary sample) [33; 34; 35; 36; 37]. The relation between the interference phase \(\Delta\gamma\), the constant OPL, and wavelengths \(\lambda_{1}\) and \(\lambda_{2}\) at two hologram exposures is described as follows:  
\[\Delta\gamma=\frac{2\pi}{\lambda_{1}}\;OPL-\frac{2\pi}{\lambda_{2}}\;OPL=\; \frac{2\pi}{\Lambda}\;OPL, \tag{1}\]  
where \(\Lambda\) is the synthetic wavelength:  
\[\Lambda=\frac{\lambda_{1}\lambda_{2}}{\lambda_{2}-\lambda_{1}}. \tag{2}\]  
The synthetic wavelength \(\Lambda\) defines the sensitivity of the measurement. Assuming several synthetic wavelengths, the longest one determines the unambiguous measurement range, while the shortest synthetic wavelength gives the lowest uncertainty.  
As described in [31], the MWHI method has been designed to sweep the synthetic wavelength allowing multiple measurements of shape in a short time. A tunable laser operating in open loop mode is used as a coherent light source, and continual wavelength tuning is achieved by changing the laser cavity length while keeping the number of waves in the cavity constant [38]. In this paper, a temporal unwrapping approach is applied to ensure that the measurement always yields the maximum resolution of a given wavelength tuning without spatial unwrapping errors.  
During the wavelength tuning, N phase-shifted high-speed digital holograms are captured. The Pearson correlation of a 3 by 3 pixel kernel of the first image in the high-speed array and the rest are calculated, which gives information about optical phase change \(\Delta\phi\)[32] as follows:\[\rho(I_{1}(\mathrm{x},y),I_{n}(\mathrm{x},y))=\frac{\left(r+1\right)^{2}+4r\cos( \Delta\phi)}{\left(r+1\right)^{2}+4r}, \tag{3}\]  
where \(\rho(\ )\) is the Pearson correlation operator, \(I_{1}\) is the first hologram, \(I_{n}\) is the \(n\)th holograms, and \(r\) is the object and reference beam ratio \(r=I_{o}/I_{r}\).  
Holograms with phase change \(\Delta\phi\) of \(\pi\)/2, \(\pi\), and \(3\pi\)/2 are identified from the image arrays to calculate the optical phase \(F\) of the measurement at each wavelength using the 4 step phase shifting equation [39]:  
\[F=arctan\frac{I_{3\pi/2}-I_{\pi/2}}{I_{1}-I_{\pi}}, \tag{4}\]  
where \(I_{\pi}\), \(I_{\pi/2}\), and \(I_{3\pi/2}\) are the \(\pi\)/2, \(\pi\), and \(3\pi\)/2 phase shifted holograms.  
The optical phase difference \(\Delta F_{N-1}\) of two consecutive high-speed phase measurements are calculated, and 2D Goldstein's branch cut method [40] is applied to remove any phase jumps. The differential approach provides measurements with lower dynamic ranges, and therefore, the spatial unwrapping is more reliable. Moreover, if the number of phase sampling \(N\) is high enough, the optical phase difference has no \(2\pi\) phase jumps, so the spatial unwrapping can be completely avoided.  
The total phase change \(\Delta\Upsilon\) of the entire wavelength tuning, ensuring the lowest uncertainty, is  
\[\Delta\Upsilon=\sum\
olimits_{2}^{N}\Delta\Phi_{k}. \tag{5}\]  
Assuming perpendicular illumination and observation of the object, the surface \(z\) component of a point (\(x\), \(y\)) on the sample is  
\[z(x,y)=\ \frac{\Lambda}{4\pi}\Delta\Upsilon(\mathrm{x},\mathrm{y}). \tag{6}\]