## 1 Introduction  
Dramatic advances in the latest active electronically scanned array (AESA) technologies have resulted in the development of high-performance multi-function radars (MFRs) for military ship radar systems [1; 2; 3; 4; 5]. Modern advanced MFRs can simultaneously handle surveillance, tracking, electronic warfare, and missile guidance in addition to a single classical mission [6; 7; 8; 9]. Although various frequency bands are applicable to military ship radar systems, the S-band and X-band are particularly important and are widely adopted [10]. In such system configurations, the S-band radar is often employed for the surveillance of targets, whereas the X-band radar is used for the accurate tracking of targets [11; 12]. In previous studies, various element types for the S-band and X-band radar systems have been reported, such as patch antennas [13; 14; 15; 16; 17; 18], dipole antennas [13; 14], and Vivaldi antennas [19]. In particular, patch-type antennas with a low profile and reduced radar cross section (RCS) characteristics are considered to be more suitable for military ship MFRs. However, since the S-band and X-band antennas are mounted in different spaces on the ship mast, the aperture area of the system is increased, resulting in a high RCS. In addition, since the antenna elements are not generally in the form of expandable modules, whenever the array antenna is applied to a differently sized platform, the full array antennas for a new platform must be re-developed [20]. Even in the case of failure of individual antenna elements, the entire system must be repaired or replaced if not modularized. In order to lower the RCS, it is necessary to reduce the array aperture size by stacking the antennas in the same aperture space, and to increase the efficiency of maintenance, modularization is required. Therefore, one candidate solution to this challenge is to efficiently stack the modularized S-band and X-band antennas on a shared aperture. However, as the space efficiency increases, the elements on the lower layer are inevitably blocked by the elements on the upper layer due to the decreased distance between elements. This physical blockage leads to a deterioration of the electromagnetic radiation characteristics of the antenna [21]. For the multi-layer shared-aperture array, perforated elements were applied to the upper elements to avoid the blockage of the upper-layer elements [22; 23; 24]. Nonetheless, it is difficult to apply this solution for space-efficient shared-aperture arrays, since there are many restrictions on the antenna shape and configuration of the upper and lower layers, which lead to the degradation of the array characteristics. In order to compensate for the performance degradation of the elements, complicated feed structures, such as a feeding network with a stub-loaded resonator (SLR), or a differential feeding method, have also been tried [22; 23]. Even so, these complicated feed structures make the modularization of array elements difficult. In this paper, we propose the electromagnetic (EM) transparent E-shaped antenna, which consists of one S-band element stacked on top of 3 \(\times\) 3 X-band elements in a shared aperture. These ten antenna elements form a basic unit-cell, which can be expanded to a larger array in a modular way. The proposed modular configuration has the advantage of easy maintenance at a low cost, since damaged unit-cells can easily be replaced with an identical module product. The simple E-shaped structure, which has a low profile with a good bandwidth, is employed for the antenna elements. However, due to the limited aperture size of the stacked configuration, the elements on the lower layer can be physically blocked by the upper-layer elements. To reduce this blockage effect, the S-band element is mounted to rotate 90 degrees with respect to X-band elements so that the polarizations between the S- and X-band elements are perpendicular to each other.