## 2 Error Sources Analysis of Antenna Holographic Measurement
### Scanning Error  
Scanning error occurs in the far-field measurement, resulting in each point on the u-v grid actually recording the far-field data on the neighboring position of the far-field coordinate corresponding to that point, as shown in Figure 5. The scanning error caused the data obtained from the measurement to form a non-regular grid distribution, and due to the fact that the data solving used the FFT algorithm to transform the regular grid data, the non-regular grid data was mistakenly processed as regular grid data, which in turn caused the measurement error.  
The antenna samples the far-field in the presence of the scanning error, and the actual sampling position will deviate from the ideal sampling position. So, in the calculation of the far-field pattern by radiative integration, we use the amplitude and phase of the far-field pattern at the point \((u_{i}+\Delta u,v_{i}+\Delta v)\) instead of the amplitude and phase at the point \((u_{i},v_{i})\) to simulate the effect of the scanning error on the microwave holography measurement process. So, scanning the error far-field pattern, we can use the radiation integral:  
\[T(u+\Delta u,v+\Delta v)=e^{-j2 kf}\iint\limits_{s}A(x,y)e^{ik\delta(x,y)}e^{ik( (u+\Delta u)x+(v+\Delta v)y)}dxdy \tag{6}\]  
where \(\Delta u\) and \(\Delta v\) are the deviation values of the coordinates of the far-field sampling points.  
The scanning deviations \(\Delta u\) and \(\Delta v\) can be expressed as:  
\[\begin{array}{l}\Delta u=\sin(\Delta EL)\cos(\Delta AZ)\\ \Delta v=\sin(\Delta EL)\sin(\Delta AZ)\end{array} \tag{7}\]  
Figure 4: The geometry of the feed offset error: (**a**) the axial offset; (**b**) the lateral offset.  
Figure 5: Antenna scanning error: (**a**) antenna far-field data processing flow; (**b**) u-v grid far-field data; black points are ideal far-field data; yellow points are far-field data with scanning error.  
where \(\Delta EL\) and \(\Delta AZ\) are the scanning elevation angle deviation and azimuth angle deviation during antenna scanning.  
The term \(e^{jk((u+\Delta u)x+(v+\Delta v)y)}\) in Equation (6) is decomposed:  
\[T(u+\Delta u,v+\Delta v)=e^{-j2 kf}\iint\limits_{s}A(x,y)e^{jk(x,y)}e^{jk(\Delta ux +\Delta vy)}e^{jk(ux+\Delta vy)}dxdy \tag{8}\]  
It can be seen from Equation (9) that the phase term \(e^{jk(\Delta ux+\Delta vy)}\) directly reflects the phase change of the additional aperture field caused by the scanning error. Let \(\Delta\varphi(x,y)=k(\Delta ux+\Delta vy)\), and the equation can be simplified as:  
\[T(u+\Delta u,v+\Delta v)=e^{-j2 kf}\iint\limits_{s}A(x,y)e^{j(\delta(x,y)+ \Delta\varphi(x,y))}e^{jk(ux+\Delta vy)}dxdy \tag{9}\]  
where \((\delta(x,y)+\Delta\varphi(x,y))\) is the phase error distribution of the aperture field after adding the scanning error.  
The phase error caused by the scanning error is linearly superimposed on the phase of the aperture field. The distribution of the aperture field can be expressed as:  
\[A(x,y)e^{j(\delta(x,y)+\Delta\varphi(x,y))}=e^{j2 kf}FFT^{-1}(T(u+\Delta u,v+ \Delta v)) \tag{10}\]  
After the phase of the aperture field is calculated by Equation (10) and brought into Equation (3), the surface measurement error caused by the scanning error can be reconstructed.