## 2 Measurement Procedure
### Data Processing Procedure
#### 2.2.2 Multi-Angle Stitching Process  
Furthermore, the following three steps which need to be completed to recover the surface shape of the large gradient micro-structure accurately. First, the acquired multiple object waves carry different rotation angles, so the coordinate system of these object waves must be unified by employing the diffraction calculation between tilted planes [16; 17]. As is shown in Figure 4, the sample plane X2 which share the origin with the imaging plane X1 was tilted with respect to the imaging plane with the angle of \(\beta\) around the R_Y axis, and CCD can only capture the object wave in the imaging plane. The angular spectrum \(A(fx_{1},fy_{1})\) of wave distribution on the imaging plane is calculated by FFT:  
\[A(fx_{1},fy_{1})=FFT[U(x_{1},y_{1})] \tag{3}\]  
where \(U(x_{1},y_{1})\) the wave is distribution of the imaging plane. The spatial frequencies of sample plane \((fx_{1},fy_{1},fz_{1})\) and imaging plane \((fx_{2},fy_{2},fz_{2})\) satisfy the relation:  
\[(fx_{1},fy_{1},fz_{1})=\left( \begin{array}{ccc}\cos\beta&0&-\sin\beta\\ 0&1&0\\ \sin\beta&0&\cos\beta\end{array} \right)^{-1}(fx_{2},fy_{2},fz_{2}) \tag{4}\]  
However, in the process of transformation, the origin of the imaging Fourier coordinates is shifted to \((-\frac{\sin\beta}{\lambda},0)\), so it is necessary to cancel the offset by shifting angular spectrum reversely. The reconstructed wave field \(U(x_{2},y_{2})\) on the sample plane can be calculated by inverse FFT as follows:  
\[ \begin{array}{ll}U(x_{2},y_{2})&=IFFT[A(fx_{2}-\frac{\sin\beta}{\lambda},fy_ {2})]\\ &=IFFT[A(fx_{2},fy_{2})]\exp[2\pi(-\frac{\sin\beta}{\lambda})x_{2}]\end{array}  \tag{5}\]  
so, the phase maps of multi-angle holograms can be gained by \(\varphi=\arg[U(x_{2},y_{2})]\). The second step was topography reconstruction algorithm with high accuracy. Generally, the unwrapped sample phase is converted to surface feature of sample by using thin element approximation method (TEA), but it cannot accurately recover the topography of large gradient micro-structure. Therefore, the tilted local ray approximation algorithm (TLRA) was developed for the topography reconstruction of the micro-structure of large gradient in inclined plane wave illumination [18]. Figure 4: Schematic of the algorithm for diffraction calculation. The algorithm was derived from the analysis of relation of local ray's optical path difference with the unwrapped sample phase. The principle of the TLRA algorithm for transmission configuration is shown in Figure 5 and the sample of height can be computed using the formula as follows:  
\[h^{TL}(x+x_{s})=\varphi(x)(\frac{nk_{ox}(x)k_{ix}-k_{0}{}^{2}}{k_{ox}(x)}+k_{iz} )^{-1} \tag{6}\]  
\[x_{s}=h^{TL}(x_{2})\frac{k_{ox}}{k_{ox}}=\varphi(x)(nk_{ix}-\frac{k_{0}{}^{2}}{ k_{ox}}+\frac{nk_{ix}k_{oz}}{k_{ox}})^{-1} \tag{7}\]  
where \(\varphi\) is the unwrapped phase of specimen, \(x_{s}=x_{2}-x\) is a transverse shift of imaging ray, \(n\) represent the refractive index of specimen material, \(k_{0}\) is the wave number and \(k_{i}\) and \(k_{o}\) are the wave vectors of illumination and refracted ray, respectively:  
\[k_{i}=[k_{ix},k_{iz}]=[nk_{0}\sin a_{1},nk_{0}\cos a_{1}] \tag{8}\]  
\[k_{o}=[k_{ox},k_{oz}]=[-\
abla\varphi(x)+k_{ix},(k_{o}{}^{2}-(\
abla\varphi(x )-k_{ix})^{2})^{1/2}] \tag{9}\]  
The last step was stitching the algorithm based on overlapped region. We have got multiple sub-surfaces of sample at different angles of tilt through the above diffraction calculation and topography reconstruction algorithm. However, due to the existence of mechanical motion, it was impossible to assure precise position and pose parameters of the tested sample. As shown in Figure 6a, the alignment tolerance \(\Delta\theta\), \(\Delta\beta\) of two rotary axes R_X, R_Y is introduced, respectively, because of the addition of rotation angle of sample. So, the coordinate of the acquired sub-surfaces and the coordinate of the real sub-surfaces have the following relationship:  
\[\left[