## 2 Minkowski Fractal Reflectarray Design and Its Sensitivity Analysis  
The main objective here is to design a high-gain reflectarray consisting of a Minkowski fractal unit cell with beam scanning capabilities at 12 GHz. Later, the work focuses on providing insight into the design's maximum capabilities and its limitations for use in Ku-band satellite internet communications. Hence, initially, a simple fixed-size reflectarray with a 1 st iteration Minkowski fractal unit cell [6] and a grounded substrate is designed here. The total number of unit cell grid points is set to 15 \(\times\) 15, utilising a highly efficient RO3003 substrate with an \(\varepsilon\)r of 3 and a thickness of 0.5 mm to maximise the gain and other performance characteristics of the antenna. The design is considered symmetric around one axis, where one half of the reflectarray is a mirror image of the other half with respect to the X-axis. The arrangement of the individual elements, made with different scaling factors, achieves the required gain and aligns the main beam direction with the required orientation. Typically, the reflectarray is illuminated by a horn antenna as a feed. Here, a horn antenna is also considered as a source within the same design frequency range. The distance between the horn feed and reflectarray antenna is considered 50 mm at the centre of the reflectarray. Generally, the reflectarray performs better when the number of elements is seven or more in both the X-axis and Y-axis. The side lobe level will increase further if the design occurs with a number of elements less than 7. As this work involves a significant amount of simulation to determine the sensitivity of design parameters with respect to overall gain and beam scanning, to ease the process and save time, a minimum seven-element configuration is selected. The geometry of the proposed patch antenna is shown in Figure 1.  
The space-filling properties of the Minkowski fractal help reduce the overall size of the array with a fixed-size patch. Here, the miniaturisation capability of the fractal concept is fully exploited by leaving the unit cell length unchanged. The scaling factor of fractal plays an important role in the phase agility or beam scanning of the array. By carefully adjusting the scale factor of each element in the array, the main beam can be oriented towards a particular direction without changing its overall dimension. Initially, a 15 \(\times\) 15 reflectarray is designed for a centre frequency of 12 GHz. Later, the design is optimised to have a gain above 20 dB with a narrow beamwidth of 15 degrees. The beam direction of the reflectarray is fixed towards 30 degrees with respect to its normal to avoid material loss in the feeder. The feeder is placed at a distance of 50 mm with an edge taper of \(-\)10 dB and 0.7 feed distribution efficiency. With this setup, the design is optimised, and the optimised design parameters are presented in Table 1, along with their classifications. The parameters and their values can be mapped to the design layout in Figure 1 to gain a better understanding of the structure. Figure 1: Reflectarray design. (**a**) Flowchart showing fractal reflectarray design and its optimisation process, (**b**) unit cell top view, (**c**) reflectarray top view, (**d**) side view of reflectarray with feeder, and (**e**) final design set up in CST Microwave Studio. The main feature of the reflectarray is the relationship between the beam phase and the design parameters. Thus, carefully adjusting the design parameters results in the required beam orientation. The phase shift in the reflectarray is the function of the path length difference between the incident and reflected waves. This relationship can be deduced from Equation (1). In a fractal-based reflectarray, for a fixed iteration, the scaling factor determines the resonant length of the fractal element, which in turn directly affects the phase shift. The relationship between the resonating length and fractal scaling factor can be found in Equation (2).