## 4 The Influence of Error Sources on the Surface Shape Measurement Accuracy
### The Influence of the Feed Offset on the Surface Shape Measurement Error  
In the previous section, the far-field data caused by the feed offset error was obtained. The far-field data is subjected to the inverse Fourier transform, and the phase distribution is calculated. Substitute it into the Equation (3) to solve the surface measurement error distribution caused by the feed offset error, as shown in Figure 10. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Diameter (m)** & **30** & **50** & **70** & **90** & **110** \\ \hline \(s_{gl}\) & \(-1.37\times 10^{-5}\) & \(-3.92\times 10^{-5}\) & \(-8.61\times 10^{-5}\) & \(-1.91\times 10^{-4}\) & \(-3.92\times 10^{-4}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Relationships between scanning error gain loss coefficients and focal diameter ratios. Figure 8: The far-field pattern caused by the scanning error: (**a**) u-plane intercept 2D far-field pattern; (**b**) v-plane intercept 2D far-field pattern. Figure 9: Antenna gain loss due to scanning error. Figure 10a shows the effect of feed axial offset \(0.5\lambda\) on the surface shape measurement accuracy after microwave holographic measurement, and the root mean square (RMS) error is about 4.97 mm. Figure 10b shows the effect of feed lateral offset \(0.5\lambda\) on the surface shape measurement accuracy after microwave holographic measurement, and the RMS error is about 6.98 mm. If there is no feed offset error, the surface shape measurement error distribution in Figure 10 will be an ideal planar distribution, which will not interfere with the actual surface error distribution of the antenna reconstructed by holographic measurement. However, when there is feed offset error, it is equivalent to adding the false error distribution shown in Figure 10 to the reconstructed actual surface error distribution, which interferes with the accuracy of holographic measurement results. By calculating the surface measurement error RMS caused by the feed offset error, the relationships between the feed offset error and the surface measurement error RMS are established, as shown in Figure 11. Figure 11a shows the relationships between the RMS of the surface measurement error and the axial offset error of the feed when \(f/d\) takes different values. Figure 11b shows the relationships between the surface shape measurement error RMS and the lateral offset error of the feed when \(f/d\) takes different values. It can be seen from Figure 11a,b that the smaller the \(f/d\) value, the faster the RMS of the surface shape measurement error rises, and the larger the RMS of the surface shape measurement error caused by the feed offset error. Figure 11: The relationships between the surface shape measurement error of different antenna focal diameter ratios and the feed offset error: (**a**) the influence of the axial offset error of the feed with different focal diameter ratios on the surface shape measurement error of the antenna; (**b**) the influence of the lateral offset error of the feed with different focal diameter ratios on the surface shape measurement error of the antenna. Figure 10: The surface shape measurement error distribution caused by the feed offset: (**a**) the axial offset; (**b**) the lateral offset. The relationships between the feed offset and the surface shape measurement error RMS are shown in Figure 11, and the relationship is:  
\[\varepsilon_{\text{rms}}=p\frac{\delta}{\lambda} \tag{15}\]  
where \(p_{a}\) is the RMS coefficient of measurement error of the feed axial offset surface shape and \(p_{l}\) is the RMS coefficient of measurement error of the feed lateral offset surface shape, \(\delta\) is the feed offset value. Table 3 shows the RMS coefficients \(p_{a}\) and \(p_{l}\) of feed offset and surface shape measurement error under different focal diameter ratio. According to the comparison of the data in the table, it can be concluded that the influence of feed offset on the surface shape measurement error decreases with the increase of antenna focal diameter ratio.