## 1 Introduction  
Modern wireless communication systems have adopted frequency duplex (FD) and polarization duplex (PD) as solutions for simultaneous transmission and reception to meet the growing demand for higher transmission data rates [1, 2, 3, 4]. These techniques are used in numerous applications, including satellite communication [5, 6], mobile networks [7, 8], vehicular communication [9], and radio-frequency identification (RFID) systems [10, 11], resulting in an increase in channel capacity and a decrease in data collision rates. In recent years, as a result of advances in electromagnetic meta-surface research and development, frequency selective surfaces (FSS) and polarization selective surfaces (PSS) have emerged as promising duplex technologies in wireless systems, covering areas such as antenna design [12, 13], analogue circuit absorption [14, 15], polarization diversity [16], spatial filtering [17], and beam-switching [18, 19]. Dual-band FSSs can be designed by cascading multi-layer structures [20, 21, 22], employing multi-resonant FSS units [23], designing fractal elements [24, 25, 26], and adopting the perturbation of a single-layer FSS [27]. In Ref. [22], arrays of double square loops and wire grids are cascaded to create an X- and Ka-band transmission-type dual-band FSS. An ultra-thin single-layer multi-resonant dual-band FSS is proposed in Ref. [27], which consists of a square patch with eight L-shaped arms. The L-shaped arm's vertical and horizontal portions produce a stopband, respectively. In addition, the emergence of three-dimensional FSS offers a new design method for dual-band FSSs [28, 29]. On the other hand, multi-layer film stacks, grating design, and cascading planar resonators are the primary design techniques in the electromagnetic field for PSSs [30; 31; 32; 33; 34]. Ref. [31] describes a simple U-shaped metallic structure that generates a first-order passband for transverse electric (TE) and transverse magnetic (TM) incident waves, respectively. Due to the use of a single resonating mode and a single substrate layer, however, the performance of selectivity and polarization isolation is limited. In Ref. [32], the polarization grid in a folded reflectarray antenna is replaced with a FSS that has two elliptical patches separated by ground, and an elliptical slot. This FSS is designed to have different polarization selectivity in each of the two frequency bands that the antenna is meant to operate in. This modification allows the reflectarray to support dual-polarization and operate in both frequency bands. In Ref. [33], a novel PSS composed of three metallic layers is proposed. Based on the resonating mechanism of the Jerusalem cross, this design offers two second-order passbands under the TE wave at 6.5\(\sim\)8.07 GHz and 25.42\(\sim\)28.46 GHz, respectively. In the frequency range of 9.94 GHz to 12.88 GHz, the best reflectivity for TM wave is approximately \(-\)40 dB. Similarly, in Ref. [34], a PSS composed of three metallic layers, each with a square loop and circular patch with a twisted slot, generates two orthogonal linearly polarized waves in the lower (7.63\(\sim\)8.97 GHz) and upper (13.26\(\sim\)13.87 GHz) frequency bands. Still, there are some design disadvantages and application limitations in the works above:  
* Current designs feature a frequency ratio that is notably high, making the design of the feeding networks and signal processing modules in frequency division duplex (FDD) systems more difficult and costly, particularly in satellite communication and base station applications. In addition, despite the widespread use of circuit analysis in the literature, the relationship between circuit components, physical structure, achievable performance, and the general design process applicable to different frequency bands has not been thoroughly explored;
* There are few studies on the in-band flatness and out-of-band rejection in the current research on polarization and frequency selective surfaces of duplex systems.