## 2 Transmitarray Antenna Design
### Feed Antenna Design  
Typically, patch antenna and a horn antenna are often used as the feed source of TA and RA [12; 13; 14; 15; 18; 19; 20; 21; 22; 23; 26; 27; 28]. The substrate integrated waveguide (SIW) antenna is often used as the feed of MIMO TA antenna for the beam forming, and the Vivaldi antenna can also be used as the feed for the beam steering [25; 39]. The patch antenna was chosen because of its easy manufacturing and small transmission direction size, which shows the inherent advantage of the low-profile property, but the patch antenna is often limited by the narrow bandwidth, in general, the -10 dB bandwidth of patch antenna is less than 10%. Another shortcoming of the patch antenna is that it is difficult to keep the symmetric radiation pattern in the azimuthal direction because the asymmetrical radiation will bring the distortion to the radiation when the TA or RA is feed by off-set source or in the situation of beam steering [18; 19]. The horn antenna itself has the advantages of higher gain and wider bandwidth, so it can bring a higher gain enhancement and broader bandwidth for the TA antenna. Though the horn antenna often suffers from the bulky volume and larger size in the direction of wave propagation, it is easy to achieve the symmetric radiation pattern in the azimuthal direction. However, the conventional pyramid and conical horn antenna fail to perform the symmetric pattern due to the fundamental mode excitation, like the Potter horn or corrugated horn, higher-order mode is introduced into the cavity [15; 19; 20; 27; 28; 38]. In this paper, we utilize the advantages of Potter horn antenna to optimize a feed source at the frequency range of 26-30 GHz with symmetrical radiation pattern for the SLL reduction in full azimuthal directions. Figure 1 shows the cross-sectional view of the Potter horn antenna operating at 28 GHz. The feed structure is 21.16 mm \(\times\) 21.16 mm \(\times\) 27.96 mm as the effective volume, which consists of a circular waveguide, two conjunctions, and a flared section. The horn is made of copper with a thickness of 1 mm. The gain is 14.6 dBi at 28 GHz, and half power beam width (HPBW) is 36\({}^{\circ}\) at E-plane and H-plane in order to simplify the further optimization about the radiation and the aperture efficiency. In Ref. [37], the horn antenna can be molded as the function of the cos\({}^{\rm q}\) (\(\theta\)) in the scalar expression of far-field radiation pattern for a planar antenna array, and it is reasonable that q = 6.5. The curve of cos\({}^{\rm q}\) (\(\theta\)) and the radiation pattern of the [PERSON] horn antenna in E-plane and H-plane are shown in Figure 2. After optimization, it is obvious that the radiations not only in E-plane but also in H-plane match well with the theoretical curve of cos\({}^{\rm q}\) (\(\theta\)) in the theta range from \(-\)45\({}^{\circ}\) to 45\({}^{\circ}\). The values of the final optimized parameters are: \(L_{\rm w}\) = 3.2 mm, \(L_{1}\) = 4.05 mm, \(L_{2}\) = 16.14 mm, \(L_{\rm c}\) = 8.9 mm, \(R_{\rm w}\) = 3.39 mm, \(R_{1}\) = 5.99 mm, \(R_{2}\) = 8 mm, and \(R_{\rm c}\) = 10.58 mm. All the results are simulated by CST Microwave Studio. \(L_{\rm w}\) is the length of the circular waveguide and the waveguide excites the dominant TE\({}_{11}\) mode and generate a directive beam with asymmetric radiation pattern. \(L_{1}\) and \(L_{2}\) are the length of the two abrupt changes of the waveguide in which the first junction keeps the power in the TE\({}_{11}\) mode, the second junction excites the TM\({}_{11}\) mode, and meanwhile, the flared horn is worked to avoid the propagation of TM\({}_{12}\) mode. The simulation result of return loss (S11) with a \(-\)10 dB bandwidth of the feed source is 14%, which is from 26.39 to 30.37 GHz. The side lobe levels are \(-\)26.2 dB and the Half Power Beam Width (HPBW) are about 36\({}^{\circ}\) in both the E-plane and the H-plane. As shown in Figure 3, the electric fields at the cross sections of the optimized Potter horn are presented.