## 4 Curved Radome Physical Optic Analysis  
The analysis of curved antenna radome can hardly be performed according to general computational electromagnetic full wave methods, because they are inefficient in solving electrically large problems. The high frequency methods can achieve fair precision in practical electrically large engineering problems. In the high frequency method, the physical optic method improves the computational accuracy compared to the geometric ray method. Thus, the usage of the physical optic method in this paper achieves a good balance between computational complexity and results accuracy. The main steps of analyzing the characteristics of the radome by the physical optic method are as follows:x  
1. Use the physical optic method to calculate the incident angle on the radome interior surface for the radome transmission coefficients computation;
2. Obtain the transmitted field on the radome exterior surface by multiplying the incident field with radome transmission coefficients;
3. Compute the antenna far field radiation by integrating the transmitted field on the radome exterior surface. A radome surface is designed with a 1 m height and a bottom circle radius of 0.25 m, as shown in Figure 5. The radome exterior shape is described by the revolution of the curve \(C\) rotated about the z axis, which satisfies the polynomial function relation:  
\[C=\ A_{0}+A_{1}x+A_{2}x^{2}+A_{3}x^{3}+A_{4}x^{4}+A_{5}x^{5}, \tag{21}\]  
where the polynomial coefficients are the fitting results of the radome from reference [26] and are shown in Table 1. The PM FSS partially cover the radome, as in Figure 5a the blue doubled imaginary line is shown. This partial coverage is for the measurement and fabrication convenience because only the upper space radiation patterns need to be considered. Ultimately, 19,488 periodic FSS units are needed for radome manufacture. Figure 4: The transmission coefficients and transmission coefficients phase comparisons of simulation and vector fitting with different incident TE wave: (**a**) Vertical incident normalized transmission coefficients; (**b**) vertical incident transmission coefficients phase; (**c**) 50\({}^{\circ}\) incident normalized transmission coefficients and (**d**) 50\({}^{\circ}\) incident transmission coefficients phase. In the case with no radome, the radiation patterns of the horn antenna in the \(x\) polarization direction or \(y\) polarization direction with uniform current distribution can be calculated by the following formula:  
\[E(\theta,\ \varphi)\ =\ \sum_{m,n}A_{mn}(\theta,\ \varphi)\exp[-j(ksin(\theta) \cos(\varphi)x_{mn}+ksin(\theta)\cos(\varphi)y_{mn})]. \tag{22}\]  
With the radome influencing, the radiation patterns can be expressed as:  
\[E_{r}\ =\ \sum_{m,n}A_{mn}(\theta,\ \varphi)\{T(\theta,\ \varphi)\exp[j\delta( \theta,\ \varphi)]\}\cdot\exp[-j(ksin(\theta)\cos(\varphi)x_{mn}+ksin(\theta)\cos( \varphi)y_{mn})], \tag{23}\]  
where \(T(\theta,\ \varphi)\) is the absolute radome transmission coefficients and \(\delta(\theta,\ \varphi)\) is the radome transmission coefficients phase. The radome transmission coefficients is closely linked to the incident angle. Figure 6 shows the 10 GHz planar PM FSS transmission coefficient curves of the TE and TM modes. Based on the theory described in Section 3, the FSS unit transmission coefficient will change accordingly with the change of incident angle. The data are collected at incident angles of 0, 15, 25, 30, 50 and 75\({}^{\circ}\). As shown in the figure, the transmission coefficient range of the proposed FSS unit is close to 1 in both TE and TM modes but the transmission coefficient phase varies within the range of \(-6^{\circ}\) to \(-37^{\circ}\). As discussed at the end of Section 3, the TE/TM insensitivity peculiarity of the PM FSS structure is verified again at 10 GHz. The 0 to 75\({}^{\circ}\) incident angle variation can affect the normalized transmission coefficients slightly. The larger incident angle may cause a longer propagation length, which will increase the phase difference.