# Generation of Beam Tilt through Three-Dimensional Printed Surface
## 2 Unit Cell Analysis and Surface Generation
The tilting direction (\(\theta\)) can be calculated by Equation (1) [35],  
\[\theta=\arcsin\frac{\lambda\gamma}{d2\pi} \tag{1}\]  
where \(\lambda\) is the wavelength of the operating frequency, \(\gamma\) is the phase difference assumed between the adjacent radiating unit cells, and \(d\) is the separation distance between those radiating elements. This Equation (1) can be rewritten to derive the phase difference values, as shown in Equation (2). \[\gamma=\frac{\omega d\sin\theta}{c} \tag{2}\]  
where \(c\) is the velocity of the electromagnetic wave, and \(\omega\) is angular frequency. This shows that there is a time delay, \(\tau\), between the adjacent radiating elements, which is represented as in Equation (3). \[\tau=\frac{d\gamma}{d\omega}=\frac{d\sin\theta}{c} \tag{3}\]  
Equation (3) shows that the beam direction angle can be achieved through tuning the value of \(\tau\). In this case, the phase gradient is \(40^{\circ}\) as considered in the generation of CRS, which is eventually used to design BDS, as discussed in a later section. The study of tilting behavior begins with the step to make more of a planar wavefront above the feed horn source. The phase distribution obtained above the horn fed at \(\lambda/2\) distance is mentioned in [31], whereby noting these relative phase values, we have started to maintain the five-layer cube distribution for eight different rounds from the center towards the aperture end, moving from a distance of 3.75 mm from the center of aperture to 56.25 mm towards the end of aperture. Round 1, Round 2, Round 3, Round 4, Round 5, Round 6, Round 7 and Round 8 signify the respective distances of 3.75, 11.25, 18.75, 26.25, 33.75, 41.25, 48.75, 56.25 mm along the aperture distribution. Rounds are the defined aperture positions from the center of the structure. These respective rounds are shown in Figure 4, where the center position is of Round 1 with a radial distance of 3.75 mm from the center of the structure. These rounds are maintained for radial distances of 11.25, 18.75, 26.25, 33.75, 41.25, 48.75 and 56.25 mm as noted from the center of the structure and are indicated by respective rounds (i.e., Round 2, Round 3, Round 4, Round 5, Round 6 and Round 7). Thus, this will arrange the respective cubes in defined apertures of 16\(\lambda\)\(\times\) 16\(\lambda\). This will make the overall phase distribution more uniform, which will be used to generate the BDS, as detailed by the synthesis algorithm and flow chart. The synthesis algorithm is detailed below:  
Step 1: Note the phase values at \(\lambda/2\) distance above the aperture of Horn fed for respective distances of 3.75, 11.25, 18.75, 26.25, 33.75, 41.25, 48.75, 56.25 mm along the aperture positions. Step 2: Normalize the noted phase values and choose the values greater than the maximum phase value. The chosen maximum values run through a complete 360\({}^{\circ}\) phase cycle. Step 3: Generate the five-layer structure of the proposed cubes with eight different rounds that will be able to decrease the overall phase variation along the defined aperture of 16\(\lambda\)\(\times\) 16\(\lambda\). Thus, appropriate cubes sizes are selected for defined aperture positions by giving the command from Matlab, which is interlinked with Computer Simulation Technology (CST) software. Step 4: Conical Rotating Surface (CRS) is generated for an increasing phase gradient of 40\({}^{\circ}\) from the left towards the right direction. These phase values are added to the phase of the eight chosen different rounds from Step 3, which will design the BDS. These surfaces are placed along the E-plane of the feed horn source. Step 5: In order to study the beam deflection, BDS is placed above the feed horn source and the angle of rotation is noted for \(0^{\circ}\), \(90^{\circ}\), \(180^{\circ}\) and \(270^{\circ}\) in the counterclockwise direction. Step 6: CRS is rotated \(180^{\circ}\) in the anticlockwise direction and placed 1 mm above the BDS, which is kept just above the horn aperture.