## 1 Introduction  
Recently, several new wireless applications and emerging technologies such as Internet of Things (IoT), artificial intelligence (AI), and smart systems have been deployed commercially over 5G networks. The current 5G wireless networks are known as heterogeneous networks, which are designed to operate in both microwave (Mw) and millimeter-wave (MMw) frequency bands simultaneously. Mw bands such as sub-6 GHz provide good far-field coverage for long-range applications, while MMw new radio frequency bands, which are defined by the 3 rd Generation Partnership Project (3 GPP), allow for high-gainbeams. As an example, the n257 band (26.5-29.5 GHz) offers high-gain pencil beams that enable high data rates [1; 2]. The success of 5G has motivated researchers to investigate instant wireless services and connectivity, leading to the expected launch of 6G mobile systems around 2030 [1; 2]. Such 6G mobile systems are expected to allow a shift towards upper MMw frequencies above 100 GHz, primarily in the terahertz band, which is divided into two regions: sub-terahertz (sub-THz) (0.1-0.3 THz) and terahertz (THz) region (0.3-10 THz) [3; 4; 5; 6]. This advanced feature may allow 6G technology to support data rates of around 1 Tbps and lower the latency by 1 microsecond compared to 5G [7]. Thus, it is anticipated that Mw, MMw, and terahertz (THz) heterogeneous networks will work together in the future, paving the way for the era of shared-aperture antennas that can operate for Mw, MMw, and THz wave applications simultaneously. Initially, traditional shared-aperture antennas provided a good solution for designing multi-band antennas [8; 9; 10; 11; 12; 13; 14; 15; 16; 17; 18]. However, these designs make use of open-structure planar antennas such as slots [8; 9; 10; 11], patches [12; 18], and monopoles [13; 14; 15; 16]. These structures are not appropriate for high-frequency applications using MMw because of their low total efficiency and inadequate radiation performance [19]. This can be blamed on their ohmic losses, dielectric losses, and open-structure design [19]. The size of the antenna also influences its impedance-matching characteristics, making it difficult to match at smaller sizes. Hence, traditional planar structures are not efficient for operating in MMw bands, especially in sub-terahertz frequencies. Given the marked differences in size and design between sub-6 GHz, MMw, and terahertz antennas, constructing a multi-band shared-aperture antenna with a broad frequency range, especially for sub-6 GHz, MMw, and terahertz applications, is a challenging task. Over recent years, antennas that are designed based on substrate-integrated waveguides (SIW) have gained researchers' interest due to their compact design, low cost, lightweight, multi-band capabilities, and ease of integration into compact devices [19; 20; 21; 22; 23; 24; 25; 26; 27; 28; 29; 30; 31; 32; 33; 34; 35; 36; 37; 38; 39]. In this regard, three well-known design approaches have been used:  
First, the conventional SIW cavity-based antenna design approach [19; 20; 21; 22; 23; 24; 25; 26; 27; 28; 29], which has been shown to be suitable for dual- and multi-band applications [13; 20; 21; 22; 23; 24; 25; 26; 27; 28; 29]. As shown in Figure 1, this approach can be implemented as a full-mode cavity antenna (Figure 1a) or a sub-mode cavity antenna (Figure 1b-d). The full-mode cavity disadvantage is the large footprint [30; 31; 32; 33; 34; 35]. On the other hand, sub-mode SIW-cavity antennas offer a method of miniaturization by cutting the original cavity along virtual magnetic walls (e.g., A-A', B-B', C-C', and D-D'), as shown in Figure 1a. For instance, bisecting the dimensions of an SIW cavity creates three sub-cavities: half-mode SIW (HMSIW) [20], quarter-mode SIW (QMSIW) [21], and eighth-mode SIW (EMSIW) [22], as shown Figure 1b, Figure 1c, and Figure 1d, respectively. This approach results in a compact size while retaining the electric field distribution of the original full-sized SIW cavity.