Reconfigurable intelligent surfaces (RISs), as a special form of two-dimensional metamaterials^[1], enable effective manipulation of electromagnetic (EM) waves in terms of frequency, phase, amplitude, polarization, and wave vector^[2–7]. Moreover, RISs can modulate the EM waves in a programmable manner in real time by cooperating with the field programmable gate array (FPGA). They are characterized by low cost, low power consumption, low complexity, and easy deployment, which make them more useful and valuable in realistic scenarios^[8–10]. For example, RIS has emerged as a transformative technology in modern wireless communication systems, significantly enhancing signal propagation and network efficiency^[11,12]. Among various types of RISs, dual-polarized RISs^[13–15] stand out due to their ability to manipulate the polarization states of EM waves, thereby providing more robust and versatile control over the signal environment. Dual-polarized RISs are particularly of priority in mitigating polarization mismatch and enhancing the channel capacity in complex urban and indoor scenarios^[16–18].

To further accelerate the development of RISs, there is a significant focus on strategies to improve RIS design efficiency, including machine learning-assisted (ML-assisted) methods^[19–21] and equivalent circuit modeling methods^[22–25]. In ML-assisted RIS methods, utilizing ML models for parameter analysis can replace high computational EM simulation models and thus shorten the design cycle of RISs. However, the ML-assisted method overly relies on modeling EM samples. For some complex RISs, such as dual-polarized RISs, the demand for EM samples has to be large to obtain good accuracy. In addition, the equivalent circuit modeling methods can also efficiently calculate the EM responses of RIS elements without full-wave simulation. However, it only can be applied to elements with regular-shaped structures. Due to the complexity of the RIS structure, it is not easy to establish high-precision equivalent circuit models quickly.

Recently, a macro-modeling method of RISs based on the dual-port network theory has been proposed^[26], which can efficiently predict the reflection coefficients by changing the active device parameters of RISs. Furthermore, a two-stage optimization framework^[27] for the rapid design of RISs was also established based on the multi-port network model to quickly design RIS elements by calculating the reflection EM responses of the elements with various passive structures and tunable devices. However, it is only limited to the conventional single-polarized RIS elements.

In this work, the equivalent model for the dual-polarized RIS element is established using the multiport network theory, which enables the quick and accurate prediction of the reflection EM responses in $x$- and $y$-polarization directions under various passive structures and tunable device parameters, with only one full-wave EM simulation. Additionally, utilizing the equivalent model, an optimized structure parameter for the element is determined using the genetic algorithm (GA), enabling rapid dual-polarized RIS design without EM simulations. Finally, a $10\times 10$ 1-bit dual-polarized RIS is designed, fabricated, and measured to validate the proposed method, with consistent simulated and experimental EM responses.

As is well known, a dual-polarized RIS element illuminated by TE and TM mode polarized EM waves has reflection EM responses analogous to transmission line reflection coefficients. As shown in Fig. 1, the space wave impedance $Z_0$ can be equivalent to the transmission line characteristic impedance. ${\mathit{Z}}_{\mathit{S}}(\omega )$ is the transmission line input impedance^[26]. The reflection EM responses of the element can be calculated using the reflection coefficients $\mathit{\Gamma }(\omega )$ according to the multiport network theory. $Z_0$, as the space wave impedance, is 377 Ω in the free space.

As shown in Fig. 1(a), a quasi-symmetric structural dual-polarized RIS element with a multiport is selected as an initial element. The element with ($2N+2$) ports is divided into $2N$-internal and two external ports. Two external ports connect the incident EM waves, considering TE and TM polarization types and $2N$-internal ports are the discrete load parameters used to change the passive topology of the element, thereby altering its performance.

The equivalent multiport model of the element is presented in Fig. 1(b), the remaining $2N$ discrete ports (internal ports) consist of tunable devices and simple circuits (“open circuit” or “short circuit”). The short circuit is replaced by a $1\times {10}^{-6}\mathrm{\Omega }$ resistor, and the open circuit is replaced by a $1\times {10}^6\mathrm{\Omega }$ resistor to calculate the reflection EM responses of the dual-polarized RIS element accurately. For a passive structure, the metallic line represents the short circuit, and the metallic gap is used to aminate the open circuit condition. ${\mathit{Z}}_{\mathit{S}}(\omega )$ represents the surface impedance of the dual-polarized RIS element in the $x$- and $y$-polarization directions, respectively, as defined by $${\mathit{Z}}_{\mathit{S}}(\omega )={[\begin{array}{cc}{Z}_S^{\mathrm{TM}}(\omega )& Z_S^{\mathrm{TE}}(\omega )\end{array}]}^T,$$(1)where $Z_S^{\mathrm{TM}}(\omega )$ and $Z_S^{\mathrm{TE}}(\omega )$ are the input impedances of the ports under the TE and TM mode polarized excitation waves, respectively.

To reduce the dimensionality of the internal port matrix and simplify the reflection EM response calculations, the simple circuits are encoded by 0 and 1. The vector $\mathit{c}$, with a dimension of $(2N-2)\times 1$, consists of these 0 and 1 codes. For example, when the discrete load ports are all open circuits, $\mathit{c}$ = ‘1, …, 1’; when the discrete port loads are all short circuits, $\mathit{c}$ = ‘0, …, 0’. There is a one-to-one correspondence between $\mathit{c}$ and ${\mathit{Z}}_{\mathit{LP}}(\omega )$. The first two internal port loads serve as the Schottky diode impedances, which are denoted by $Z_x(\omega )$ and $Z_y(\omega )$, respectively. To optimize the passive structure, the remaining ($2N-2$) internal ports are chosen as variables, which are denoted by ${\mathit{Z}}_{\mathit{LP}}(\omega )={[Z_1,\dots ,Z_{2N-2}]}^T$, ${\mathit{Z}}_{\mathit{L}}(\omega )=\mathrm{diag}{[Z_x,Z_y,Z_1,\dots ,Z_{2N-2}]}^T$, which represent the $2N\times 2N$ impedance matrix of all discrete load ports.

In the following, let ${\mathit{Z}}_{\mathit{P}}(\omega )$ represent the impedance matrix of the passive structure, which is obtained through only one full-wave simulation. It is worth noting that, in the equivalent model, we group simple circuits into discrete load port parameters. The passive structure of the element can be considered to be fixed, and ${\mathit{Z}}_{\mathit{P}}(\omega )$ only varies with the operation frequency $\omega$. ${\mathit{Z}}_{\mathit{P}}(\omega )$ is expressed as $${\mathit{Z}}_{\mathit{P}}(\omega )=\left[\begin{array}{ccc}{\mathit{Z}}_{\mathit{II}}& {\mathit{Z}}_{\mathbf{2}\mathit{N}}^{\mathit{1}}& {\mathit{Z}}_{\mathit{2}\mathit{N}}^{\mathit{2}}\\ {\mathit{Z}}_{\mathit{N}\mathit{2}}^1& {\mathit{Z}}_{\mathit{X}\mathit{X}}& {\mathit{Z}}_{\mathit{X}\mathit{Y}}\\ {\mathit{Z}}_{\mathit{N}\mathit{2}}^{\mathit{2}}& {\mathit{Z}}_{\mathit{Y}\mathit{X}}& {\mathit{Z}}_{\mathit{Y}\mathit{Y}}\end{array}\right],$$(2)where ${\mathit{Z}}_{\mathit{II}}$ represents the $2\times 2$ impedance matrix of the excitation ports, and ${\mathit{Z}}_{\mathbf{2}\mathit{N}}$ and ${\mathit{Z}}_{\mathit{N}\mathbf{2}}$ are the $2\times N$ and $N\times 2$ impedance matrices, respectively, representing the reciprocal impedance between the exciting ports and load ports, respectively. The $N\times N$ impedance matrices consist of four submatrices ${\mathit{Z}}_{\mathit{X}\mathit{X}}$, ${\mathit{Z}}_{\mathit{X}\mathit{Y}}$, ${\mathit{Z}}_{\mathit{Y}\mathit{X}}$, and ${\mathit{Z}}_{\mathit{Y}\mathit{Y}}$, which represent the physical impedance relation of the $2N$ discrete ports.

Therefore, the equivalent impedance ${\mathit{Z}}_{\mathit{S}}$ of the element at the operation frequency $\omega$ can be expressed by Eq. (3). Correspondingly, the reflection EM responses in the $x$- and $y$-polarization directions can be expressed at the operation frequency of $\omega$ by Eq. (5), where $\mathit{\Gamma }(\mathit{c})$ is the vector, which consists of the reflection EM responses under the $x$- and $y$-polarization directions at the operation frequency of $\omega$.

Based on the multiport model, the reflection EM responses of the elements with different internal ports can be predicted accurately and quickly. To rapidly design a dual-polarized RIS element, we use the GA to optimize its discrete load port encoding $\mathit{c}$ to obtain the element that satisfies the target EM responses. $U(\mathit{c})$ is defined as the objective function. Then, the target phase difference can be achieved with the selected discrete load ports. Moreover, at the operation frequency $\omega$, the optimization problem is expressed by Eq. (5), where $v_i$ is the load code of the $i$th port for the symmetry of the $x$- and $y$-polarization. $P_x$ and $P_y$ correspond to the phases of the reflection EM responses for the element in the $x$- and $y$-polarization directions. $\mathit{c}*$ is the optimized encoding parameter. $A_x$ and $A_y$ are the amplitude constraints of reflection EM responses under $x$- and $y$-polarization directions, respectively.

The metallic topology is determined by the restriction of the discrete port load encoding $\mathit{c}$, which effectively reduces the design complexity. Furthermore, the GA is adopted to optimize $\mathit{c}$ and thus optimize the element structure. In this work, by combining the equivalent model with the GA, the design method requires only one full-wave EM simulation, enabling the rapid design of a dual-polarized RIS element.

The flowchart of the proposed method is shown in Fig. 2. A detailed overview of the design method consists of two primary steps: the equivalent physical model and the structure optimization of the element, $${\mathit{Z}}_{\mathit{S}}={\left[\begin{array}{cc}{Z}_S^{\mathrm{TM}}& Z_S^{\mathrm{TE}}\end{array}\right]}^T=\mathrm{diag}\{{\mathit{Z}}_{\mathit{II}}-[Z_{2N}^1,Z_{2N}^2]·({\mathit{Z}}_{\mathit{L}}(\mathit{c})+\left[\begin{array}{cc}{\mathit{Z}}_{\mathit{X}\mathit{X}}& {\mathit{Z}}_{\mathit{X}\mathit{Y}}\\ {\mathit{Z}}_{\mathit{Y}\mathit{X}}& {\mathit{Z}}_{\mathit{Y}\mathit{Y}}\end{array}\right])·\left[\begin{array}{c}{Z}_{N2}^1\\ Z_{N2}^2\end{array}\right]\},$$(3)$$\mathit{\Gamma }(\mathit{c})={[{\mathit{\Gamma }}_x(\mathit{c}),{\mathit{\Gamma }}_y(\mathit{c})]}^T=\frac{{\mathit{Z}}_{\mathit{S}}-{[Z_0,Z_0]}^T}{{\mathit{Z}}_{\mathit{S}}+{[Z_0,Z_0]}^T},$$(4)$$\begin{gathered} \mathit{c}*=\underset{\mathit{c}=[v_1,v_2,\cdots v_{2N-1}]}{\arg \min }\left\{\begin{array}{c}U({\mathit{c}}_l)=|P_x({[Z_{x\mathrm{on}},Z_{y\mathrm{on}},\mathit{c}]}^T)-P_x({[Z_{x\text{off}},Z_{y\mathrm{on}},\mathit{c}]}^T-180)|\\ +|P_y({[Z_{x\mathrm{on}},Z_{y\mathrm{on}},\mathit{c}]}^T)-P_y({[Z_{x\mathrm{on}},Z_{y\mathrm{off}},\mathit{c}]}^T-180)|\end{array}\right\} \\ \text{Subject to}\{\begin{array}{c}|{\mathit{\Gamma }}_x({\mathit{c}}_l)|>A_1\\ |{\mathit{\Gamma }}_y({\mathit{c}}_l)|>A_2\end{array}. \end{gathered}$$(5)

In this model, the multi-port network theory is utilized to rapidly predict the reflection EM responses in the $x$- and $y$-polarization directions. By introducing the multi-port, it can comprehensively represent the dual-polarized RIS element behavior under various element structures. The modeling for the two-mode incident wave excitation ports enables the rapid prediction of reflection EM responses in the $x$- and $y$-polarization directions, respectively.

To achieve the identical reflection EM responses in the $x$- and $y$-polarization directions, the $+45°$ symmetry metallic topology element is employed as an initial dual-polarized RIS element with ($2N+2$) ports.

The specific steps to determine the initial element structure are as follows:

1. 1)To prevent mutual coupling between the elements in the array, the periodic length $P$ of the dual-polarized RIS elements can be determined by $$P\le \frac{c}{2·f_0},$$(6)where $c$ is the speed of the light and $f_0$ is the maximum operating frequency.
2. 2)The discrete load ports are also $+45°$ symmetrical due to the dual-polarized characteristics of the elements. The number of discrete load ports is related to the rectangular metallic sheet, and for an initial multi-port structure containing two metallic rectangles with $m$ rows and $n$ columns, the variable $N$ can be determined by $$N=2·m·n-m-n+1.$$(7)

As shown in Fig. 3(a), for the multi-port structure consisting of two rows and three columns of rectangular metallic, the value of $N$ is 8.

In the second step, structure optimization, the GA is used to optimize the passive structure of the element. In detail, the optimization process focuses on enhancing the performance of the element by carefully tuning its load encoding $\mathit{c}$. The benefits of this method are to easily find optimized solutions in complex design spaces and improve the overall synthetic design of the dual-polarized RIS element.

A schematic of the dual-polarized RIS designed by the method is shown in Figs. 3(a) and 3(b), and the optimized encodingsequence $\mathit{c}*$ is “0^#0^#0^#0^#1^#0^#0^#0^#0^#0^#0^#0^#1^#0^#0^#0^#.” As depicted in Fig. 3(b), the small orange rectangle represents the short-circuiting metallic structure, and the rectangular gap in the middle of the patch represents the open-circuiting structure. The top metallic topology layer is loaded with two Schottky diodes (Infineon BAT15-02LRH), the substrates are F4B (${\epsilon }_r=4.3$, tan $\delta =0.005$), the metallic ground plane is made of 0.035-mm-thick copper, the fourth layer bonds the metallic backplane and substrate, and the bottom layer is the fan patch feed layer providing bias voltage for tunable devices and signal isolation within a fixed frequency band. The dual-polarized RIS employs a biasing layer that feeds each element individually, enabling arbitrary encoding in the $x$- and $y$-polarization directions. As shown in Fig. 3, the dimensional parameters of the element are $P=24.70\mathrm{mm}$, $L=3.26\mathrm{mm}$, $L_1=9.1\mathrm{mm}$, $L_2=3.88\mathrm{mm}$, $L_3=5.74\mathrm{mm}$, $W_1=5.42\mathrm{mm}$, $W_2=5.67\mathrm{mm}$, $R=6.60\mathrm{mm}$, $h_1=4.80\mathrm{mm}$, and $h_2=0.30\mathrm{mm}$. The equivalent RLC parameters of the tunable devices are $R_{\text{on}}=9.8\mathrm{\Omega }$, $L_{\text{on}}=0.6\mathrm{nH}$, $C_{\text{on}}=10.1\mathrm{pF}$, $R_{\mathrm{off}}=9.9\mathrm{\Omega }$, $L_{\mathrm{off}}=5.1\mathrm{nH}$, and $C_{\mathrm{off}}=0.23\mathrm{pF}$. The equivalent impedance of the simple circuit is $R_e$.

The RIS element is simulated using a frequency domain solver, with the simulation frequency range set to 4–6 GHz. The type of electric field polarization is set to linear polarization, and along the $x$- and $y$-directions, “unit cell” boundary conditions are set to mimic an infinitely large array. In the $z$-direction, we have the incident wave port for excitation. The incident wave is set to TE and TM polarized waves, respectively.

As depicted in Fig. 3(c), the proposed dual-polarized RIS with $10\times 10$ elements is fabricated using standard printed circuit board (PCB) technology. The overall size of the RIS measures is $247\mathrm{mm}\times 257\mathrm{mm}$. Measurements of the dual-polarized RIS are conducted for both polarization directions in a microwave anechoic chamber. A vector network analyzer (VNA, Agilent N5245A), a direct current (DC) voltage source, and a horn antenna with linear polarization are used for the experimental test. A metallic plate of the same dimensions as the dual-polarized RIS is measured under the same experimental setup to normalize the reflection EM responses.

The reflection EM responses of the element under both operation states (ON/OFF) of the diodes are shown in Fig. 4. The reflection EM responses in $x$-polarization are illustrated in the upper part, and the reflection EM responses in $y$-polarization are illustrated in the lower part. As illustrated in Figs. 4(a) and 4(c), the red lines represent the full-wave simulation and theoretical calculation results, while the blue lines represent the experimental results. As illustrated in Figs. 4(b) and 4(d), the red lines represent the full-wave simulation and theoretical calculation results, the light blue lines represent the phase differences from the full-wave simulation and theoretical calculation results, while the dark blue lines represent the phase differences from the experiment.

These results show that the reflection losses of the dual-polarized RIS are about less than 5 dB in both states within the frequency range of 4.5–5.5 GHz and satisfy the requirement of $180°\pm 30°$ phase difference, verifying its capability for 1-bit dual-polarized phase modulation. The consistency among the simulation, experimental, and theoretical results confirms the validity of the design method. The discrepancies are primarily from the low magnitude of the reflection EM responses in the experimental results. This error originates mainly from the edge truncation effect, where the finite size of the RIS introduces edge effects that play a significant role. These edge effects lead to additional reflections and diffractions, which increase the overall reflection loss. There are inevitable variations and imperfections in the fabrication process, including deviations in the dimensions of the elements, inaccuracies in the dielectric substrate thickness, and misalignments during assembly. Even slight deviations can significantly affect the electromagnetic properties, leading to increased reflection loss.

It is further verified that the designed dual-polarized RIS has the capability of dual-beam regulation, which can be realized by changing the coding sequence of the RIS. The reflected main beam pointing angle ${\theta }_r$ is derived from Snell’s law^[28]. The coding sequence “0110011001” is selected to verify its beam regulation capability. At 5 GHz, the reflected main beams are calculated to be $\pm 40°$ for this coding sequence. The test is conducted in a microwave anechoic chamber. The theoretical, simulation, and test results of the far-field scattering direction pattern of the dual-polarized RIS are shown in Fig. 5, and the three sets of results show general agreement. Subtle differences are primarily due to the experimental environment and processing precision.

This work presents an efficient design method for the dual-polarized RIS. In this proposed method, a low-cost multi-port network model is established for rapidly calculating the reflection coefficients of a dual-polarized RIS element. Based on the developed dual-polarized multiport network model, the GA selects internal port loads quickly and accurately to optimize the element structure. To verify the performance of the proposed method, a 1-bit dual-polarized RIS element is designed, fabricated, and measured. The EM simulation and measurement results closely match the design specifications, revealing a frequency band of 4.5–5.5 GHz with reflection losses well below 5 dB. As depicted in Table 1, the proposed design method takes only 1.6 h, while the traditional EM simulation method takes 130 h. The proposed method is 81.3 times faster than the traditional EM simulation. The proposed method provides an efficient pathway for designing dual-polarized RIS elements, which paves the way for future advancements in adaptive beamforming and integration with next-generation wireless communication systems.