# Semi-Coprime Array with Staggered Beam-Steering of Sub-Arrays
## 1 Introduction  
The design of an array of antennas or sensors has been the area of research for a long time with the objective of achieving a half-power beam-width (HPBW) as small as possible, peak-to-side lobe ratio (PSLR) as high as possible, and using as few antenna elements as possible. To achieve this objective, researchers have analysed different geometries, inter-element distances, aperture sizes and excitation amplitudes or weights. Linear geometry has been tested for uniform [1, 2, 3, 4] and non-uniform distances [5, 6, 7, 8, 9, 10, 11, 12, 13, 14]. Uniform linear arrays (ULAs) have been investigated for different inter-element spacing ranging from a fraction to few multiples of a wavelength (\(\lambda\)). The most common ULA, called a standard ULA (SULA) hereafter, has a standard spacing of \(\lambda/2\). Larger spacing causes larger aperture offering smaller HPBW, i.e., better resolution but spacing more than \(\lambda/2\) produces grating lobes which are not desirable. On the other hand, increasing the aperture size while maintaining the standard spacing requires more elements which means higher system costs. To increase the aperture size with fewer elements (\(L\)), while avoiding grating lobes, non-uniform linear arrays (NULAs) were proposed. One of the techniques to create an NULA is sparsification or thinning of the SULA in which many of the elements are removed. Other methods fordesigning NULAs include finding inter-element spacing using either a non-linear function such as a logarithm [5], parabolic relation [13], etc., or a heuristic optimization technique such as genetic algorithm (GA) [8], firefly algorithm (FA) [6], particle swarm optimization (PSO) [15], invasive weed optimization (IWO) [9], chicken swarm optimization (CSO) [16], grasshopper optimization (GO) [17], etc. To achieve either a small HPBW or high PSRL, a number of non-uniform weighting techniques have been applied to both ULAs and NULAs. These analytical techniques include Dolph-Chebyshev [4], binomial, Blackman [1; 18], Hanning [5], etc., as well as heuristic optimization algorithms such as PSO [10; 11], IWO [3], strawberry algorithm [12]. The Dolph-Chebyshev technique pioneered by [PERSON] [19] in the first half of the 20 th century is still employed in advanced antenna systems. For instance, in adaptive beam-forming, a method of calculating initial weights was proposed using Chebyshev distribution which suppressed the side lobes in the 0-30 dB range [20]. modified Chebyshev window was also developed for a non-linear frequency-modulated waveform [21]. This modified Chebyshev window was able to suppress the peak side lobe level ratio by approximately \(-41\) dB and integrated side lobe level ratio by approximately \(-31\) dB. The Dolph-Chebyshev slow-wave transmission line feed networks have also been implemented, achieving side lobe levels of \(-30\) dB [22]. In other research, iterative Fourier transform and Chebyshev-based random thinning were presented to lower the side lobe. However, this is preferred for small arrays only where it can suppress the side lobe level by \(-36.9\) dB at \(29^{\circ}\)[23]. Apart from weighting techniques, there are various alternative approaches reported in the literature that leverage advanced antenna designs and meta-surface technologies to enhance the PSRL. For instance, circularly polarized antenna arrays in [24] refer to an antenna configuration that utilizes not only circularly polarized elements but also decoupled quad vortex beams to steer the main beam and control the side lobe levels. Chirality-assisted high-efficiency meta-surfaces utilize the concept of manipulating the chirality in electromagnetic waves to design efficient and beam-steerable antennas [25]. These meta-surfaces employ sub-wavelength structures created to control the radiation properties, including the suppression of side lobes. Dual-mode transmissive meta-surfaces operate in two distinct modes, usually with orthogonal polarizations or different frequencies [26].