# A 3-D Metal-Printed Dual-Polarized Ridged Waveguide Slot Array Antenna for X-Band Applications
## 3 Antenna Design and Analysis Procedure
### Characterization of the Radiation Unit Cells  
The dual-polarized array configuration and the arrangement of the two types of longitudinal and V-shaped slots were established for the VP and HP linear arrays, respectively. To achieve uniformity and optimization for both polarizers, a common design process was applied to each linear array. The periodic unit cell structure simulation method was used to analyze the characteristics of the radiation slot pairs, through Ansys high-frequency structure simulator (HFSS) software. This method assumes that the slot to be modeled is in an infinite array medium with identical elements and introduces an infinite array approach to characterize the slots. For designs of planar arrays, the array can be considered infinite in both directions. Therefore, the simulation model for unit cells with the longitudinal slots for VP and the V-shaped slots for HP are shown in Figure 4a,b, respectively. With the expectation of the exact same field distribution in the hypothetical environment of infinite arrays, periodic boundary conditions are applied to the free space boundaries facing each other. Specifically,  
Figure 3: EM field surface distribution on a radiating waveguide of the two HP slots. (**a**) Inclined slot. (**b**) V-shaped slot. if two facing boundaries are periodic, then one is chosen as the \"Master Boundary\", and the other is selected as the \"Slave Boundary\". While the top boundary of the unit cell is designated as the radiation boundary, it is set up for multimode excitation through a Floquet port for periodic boundary conditions. This is based on the modal decomposition that the fields inside each unit cell of a large planar array possess. Simultaneously, the slots in the unit cell must be excited with equal amplitude and desired phase difference. Therefore, two TE10 dominant mode excitation wave ports are placed at each end of the RWG to obtain the required characteristic scattering parameters of the slot. In the design of a resonant slot array antenna, in order to achieve radiation in the boresight, the slots etched in the broad wall must be spaced at a distance equal to half a guide wavelength. At the same time, adjacent slots have differences in symmetry along the transverse and longitudinal axes of the waveguide. Specifically, the longitudinal slots for the VP array differ in the offset direction, which is opposite to that of the adjacent slots. The V-shaped slots for the HP array differ in the mirroring direction, which is inverse to that of the adjacent slots. This makes it necessary to use two slots inside a unit cell during the design process. This approach provides sufficient accuracy along with the influence between adjacent coupling slots, matching the slot parameter requirements when applied to large arrays. The boundaries of the air box surrounding the waveguide are defined as radiation boundaries except for the bottom boundary, which is defined as PEC and has a chosen dimension of \(1\lambda_{0}(x)\times 1\lambda_{g}(y)\times 1\lambda_{0}(z)\). For the slots on the RWG, an equivalent model is constructed with the slots as shunt admittances linked by sections of ideal transmission lines as illustrated in Figure 4c. This model implies that the scattering from the slot is symmetrical with the backscattering (\(B\)) equal to the forward scattering (\(A\)). Figure 4: Unit cell characterization of the two-port two-slot module. (**a**) VP slotted RWG. (**b**) HP slotted RWG. (**c**) Equivalent circuit of the resonant slots. A scattering analysis mechanism is required to control the radiated amplitude and phase of the slots, including the slot offset (\(slot\_x\)), slot length (\(slot\_l\)), slot depth (\(slot\_d\)), and inclined angle (\(slot\_\theta\)). The slot scattering matrix obtained after unit cell simulation is analyzed to derive the equivalent circuit parameter \(Y=G+jB\).