Cross-eye jamming method based on 1-bit digital coding metasurface

Summary

Monopulse radar angle measurement technology is crucial for modern missile precision guidance systems due to its high accuracy and real-time capabilities. Cross-eye jamming (CEJ) is recognized as one of the most effective countermeasures against monopulse radar. However, traditional CEJ implementation requires complex amplitude and phase modulation through specialized hardware. The digital coding metasurface (DCM) is an advanced electromagnetic (EM) wave modulation technology. Synchronized modulation of phase and amplitude can be achieved by dynamically adjusting the digital coding state of each individual unit of the DCM. In this paper, a cross-eye jamming method based on DCM is proposed. The EM modulation model of DCM is derived, and it is theoretically demonstrated that cross-eye jamming signals can be generated by two DCMs. The simulation results demonstrate that the method attains more than twice the cross-eye gain when the error of the DCM phase modulation is maintained within 40°.

Graphical abstract

Highlights

• •A cross-eye jamming method based on metasurfaces is proposed and demonstrated
• •The metasurface design offers low cost and efficient control of jamming signal properties
• •This approach provides enhanced flexibility for electromagnetic interference applications

Photonics; Applied sciences; Engineering

Introduction

Monopulse radar can extract target angular information from a single pulse echo, offering high angular accuracy, fast measurement speed, and strong antijamming capabilities. This technology is widely applied in missile guidance, target tracking, and related fields.^1,2,3 Developing more effective jamming techniques against monopulse radar remains a key research focus in the field of radar countermeasures.

Cross-eye jamming (CEJ) is considered to be one of the most effective techniques used for deceiving monopulse radar.^4,5 The concept of cross-eye jamming originated from research on glint jamming. Glint is caused by interference between echoes from multiple points on a radar target, and can produce an apparent target that lies outside the real target.⁶ In 1958, the concept of cross-eye jamming was first proposed based on angular glint jamming studies.⁷ Cross-eye jamming transmits two signals with approximately equal amplitudes and opposite phases to emulate the case of glint. Therefore, cross-eye jamming is sometimes described as “artificial glint.” In 1963, Redmill et al. proposed the first mathematical model of cross-eye jamming.⁸ Subsequently, the development of cross-eye jamming progressed through two phases: two-element retrodirective cross-eye jamming (TRCJ) stage and multi-element retrodirective cross-eye jamming (MRCJ) stage. In 2009, Du Plessis et al. conducted the first study on TRCJ. They provided a rigorous mathematical derivation of retrodirective cross-eye jamming and analyzed the impact of retrodirective antenna structures on monopulse radar.⁹ Additionally, Du Plessis systematically studied and validated the theoretical derivation of the retrodirective cross-eye jamming technique. He analyzed tolerance requirements, the impact of platform skin echoes on jamming effectiveness, and the cumulative distribution function of total cross-eye gain. These analyses provided critical insights for theoretical refinement and practical application.^10,11 However, the effectiveness of TRCJ is limited by strict parameter tolerances and high jammer-to-signal ratio (JSR) requirements.

To address this issue, Harwood et al. proposed the concept of multi-element cross-eye jamming (MCJ).¹² However, no mathematical derivation was provided. Since 2013, Tianpeng Liu et al. have developed MRCJ techniques based on linear and circular retrodirective antenna arrays.^13,14,15 These techniques induce larger angular errors in monopulse radar and offer improved parameter tolerance and JSR performance. Songyang Liu et al. proposed the rotating orthogonal MRCJ technique. This scheme ensures that the jamming system maintains effective jamming capability even when the monopulse radar is rotating or detected by a monopulse radar from different directions.¹⁶ In the study by Chen et al.,¹⁷ a multi-loop retrodirective cross-eye jamming configuration is introduced. This method uses unequal spacing of antenna array elements to induce larger angular errors and enhance jamming performance against monopulse radar. In summary, these methods fundamentally operate by increasing the degrees of freedom in jamming systems to enhance jamming performance.

Digital coding metasurfaces (DCMs) have emerged as an effective approach for radar jamming,^18,19,20,21 due to their powerful capability in manipulating electromagnetic (EM) waves.^22,23,24,25 By modulating the DCMs code in fast-time domain, radar high-resolution range profiles (HRRP) false point target generation can be realized.²⁶ In the study by Yu et al.,²⁷ a dual-metasurface-based imaging radar deception jamming system is proposed to address the angular sensitivity of metasurfaces in radar signal modulation. The system enables the generation of multiple artificial targets while concealing the original target in HRRP under oblique incidences. Furthermore, DCMs demonstrate the capability to generate micro-Doppler effects for target fine-feature jamming. In the study by Lai et al.,²⁸ a method for introducing artificial Doppler and micro-Doppler effects using a phase-coded metasurface is proposed. By temporally modulating the phase of reflected waves, the stationary metasurface can emulate the Doppler shift and micro-Doppler characteristics of moving targets. The powerful EM wave manipulation capabilities of DCM also enable emerging applications in cross-eye jamming systems.

This paper proposes a cross-eye jamming method based on 1-bit digital coding metasurface. The proposed method can generate jamming signals with nearly equal amplitudes and a 180° phase difference using DCM, thereby achieving effective cross-eye jamming against monopulse radar systems. This approach offers several advantages over traditional antennas. Digital coding metasurfaces enable enhanced dynamic beam-steering flexibility and programmable electromagnetic response control, thereby reducing system complexity, hardware requirements, and operational costs. Additionally, the use of digital coding enables the integration of reconfigurable and adaptive functionalities that are difficult to achieve with traditional antenna designs. These features highlight the novelty and practical significance of our research and demonstrate the great potential of digital coding metasurfaces for next-generation radar systems. Simulation results demonstrate that the method achieves more than twice the cross-eye gain when the error in the DCM phase modulation remains within 40°, thereby confirming the method’s effectiveness.

1-bit DCM model

The monopulse radar system emits EM waves and then receives the echo signal from the target to determine its direction. In order to achieve cross-eye jamming of a monopulse radar, two jamming signals with a phase difference of 180° must be emitted to the monopulse radar on both sides of the real target. The metasurface is capable of directly modulating the amplitude, phase, and direction of the radar incident signal. Consequently, cross-eye jamming signal can be generated by directly modulating the radar incident signal with a metasurface.

Figure 1 presents the model of the reflective DCM for EM wave modulation. The DCM comprises multiple controllable unit structures, each capable of providing distinct quantized phase compensation for incident EM waves. After phase compensation, the reflected beam achieves the same wavefront in the specified direction, thereby enabling direction control of the reflected beam.

Figure 1.

Schematic of the EM wave modulation model based on the DCM

As shown in Figure 1, an xoy plane is created with the center of the metasurface as the origin, and the horn antenna emits an incident signal to the metasurface. The direction of the reflected main beam is denoted as $\left(\alpha ,\beta \right)$, where $\alpha$ is the pitch angle (the angle with the z axis) and $\beta$ is the azimuthal angle (the angle of the beam’s projection in the xoy-plane with the x axis). The spatial coordinates of the horn antenna are denoted as $\left(x_s,y_s,z_s\right)$, while the coordinates of the i-th element of the metasurface are $\left(x_i,y_i,z_i\right)$. Each element on the metasurface compensates for the phase of the incident EM wave. After phase compensation, the reflected sub-waves are all parallel to the main beam direction, thereby forming an equiphase surface. For the i-th element on the metasurface, the reflected phase of the EM wave can be expressed as:

$${\varphi }_i^{out}={\varphi }_i^{comp}+{\varphi }_i^{in}$$ (Equation 1)

where ${\varphi }_i^{comp}$ is the compensation phase of the i-th element, and ${\varphi }_i^{in}$ is the incident phase of the EM wave at the i-th element.

Herein, the incident phase ${\varphi }_i^{in}$ of the i-th element can be written as:

$${\varphi }_i^{in}=-k_0{r}_i=-k_0\sqrt{{\left(x_i-x_s\right)}^2+{\left(y_i-y_s\right)}^2+{\left(z_i-z_s\right)}^2}$$ (Equation 2)

where $k_0$ denotes the wavenumber, and $r_i$ denotes the geometrical path length of the EM wave.

According to array synthesis theory, when the main beam is steered to the direction $\left(\alpha ,\beta \right)$, the reflected phase of the i-th reflective element can be calculated as:

$${\varphi }_i^{out}=-k_0\sin \alpha \left(x_i\cos \beta +y_i\sin \beta \right)$$ (Equation 3)

Based on Equations 1, 2, and 3, the compensation phase of the i-th element can be expressed as:

$${\varphi }_i^{comp}={\varphi }_i^{out}-{\varphi }_i^{in}=k_0\sqrt{{\left(x_i-x_s\right)}^2+{\left(y_i-y_s\right)}^2+{\left(z_i-z_s\right)}^2}-k_0\sin \alpha \left(x_i\cos \beta +y_i\sin \beta \right)$$ (Equation 4)

Once the positions of the horn antenna and metasurface are determined, the phase compensation value for each metasurface element can be calculated by substituting the corresponding positional and structural parameters into Equation 4. However, since the DCM uses discrete values for phase compensation, ${\varphi }_i^{comp}$ must undergo further quantization according to the number of coded bits. For a 1-bit DCM, the coded value is limited to two states, “0” and “1”, which correspond to phase compensations of 0° and 180°, respectively. Thus, the theoretical compensated phase value ${\varphi }_i^{comp}$ must first be normalized within the range of (0°, 360°). If the theoretical compensated phase falls within (0°, 180°), the actual compensated phase is set to 0°; if the theoretical compensated phase falls within (180°, 360°), the actual compensated phase is set to 180°. Consequently, the actual compensated phase value for the i-th element of the metasurface can be given by:

$${\varphi }_{i\left(actual\right)}^{comp}=\{\begin{array}{cc}0°& 0°\le {\varphi }_i^{comp}<180°\\ 180°& 180°\le {\varphi }_i^{comp}<360°\end{array}$$ (Equation 5)

The code corresponding to the actual phase compensation value can be expressed as:

$$u_i=\{\begin{array}{cc}0& {\varphi }_{i\left(actual\right)}^{comp}=0^{°}\\ 1& {\varphi }_{i\left(actual\right)}^{comp}={180}^{°}\end{array}$$ (Equation 6)

It should be noted that in this work, we adopt the phase quantization method based on the interval lower bound. Although different quantization strategies, such as the interval lower bound or center value, can result in different quantized phase values for individual metasurface elements, applying a consistent quantization strategy ensures that the overall phase gradient across the metasurface remains unchanged. As a result, the main beam direction is not influenced by the particular quantization method chosen.

Proposed method

Jamming scenario

The jamming geometry of a two-element cross-eye jamming system employing DCM is shown in Figure 2. The jamming system consists of two DCMs. Each metasurface receives the radar signal, modulates its amplitude and phase, and reflects the modified signal back to the monopulse radar to generate jamming signals. In this paper, platform skin return is not considered.

Figure 2.

The geometric model of the cross-eye jamming method based on DCM

As shown in Figure 2, the true target is positioned along the radar boresight (x axis). Two DCMs are symmetrically placed on either side of the true target along the radar boresight. We denote the angle to the first metasurface measured from the radar boresight by ${\theta }_r$, the false target angle measured under cross-eye jamming by ${\theta }_j$, and the spacing of the phase centers of the monopulse antenna elements by d.

Cross-eye jamming requires a 180° phase difference between two jamming signals. By applying the inverse operation to the “0/1” coding states of each DCM element, the beam direction is preserved while the phase is inverted by 180°, thus meeting the phase requirement for cross-eye jamming. Assuming that the encoded value of each element of the first DCM is $u_i^1$ and follows the encoding rule of Equation 6. Then the second DCM follows the opposite coding rule, and each element is encoded with the value:

$$u_i^2=\{\begin{array}{cc}0& u_i^1=1\\ 1& u_i^1=0\end{array}$$ (Equation 7)

In the jamming scenario illustrated in Figure 2, the two DCMs are symmetrically positioned with respect to the radar boresight. Therefore, it is only necessary to compute the coded value of the first DCM. Then, the element codes of the second DCM are set to be the inverse of those of the first DCM, and the second metasurface is rotated by 180°. This allows the two DCMs to emit jamming signals with a 180° phase difference, symmetric to the radar boresight. The placement diagram of the two DCMs is shown in Figure 3.

Figure 3.

Schematic of DCM placement

Mathematical analysis of the monopulse error

The cross-eye jamming model is mathematically analyzed in the following text. The jamming scenario is shown in Figure 2. Assuming that the wavelength of the jamming signal is $\lambda$, the signal reflected from the first DCM to the center of monopulse radar antenna is $\epsilon$. Hence, the signal received by the sum-channel can be expressed as:

$$S=2\epsilon \left(1-a\right)\cos \left(\mathrm{\pi }d\sin \left({\theta }_r\right)/\lambda \right)$$ (Equation 8)

The signal received by the difference-channel can be expressed as:

$$D=2\epsilon \left(1+a\right)j\sin \left(\mathrm{\pi }d\sin \left({\theta }_r\right)/\lambda \right)$$ (Equation 9)

where a is the amplitude ratio.

The monopulse radar extracts angle information by normalizing the difference-channel echo with the sum-channel echo. The ratio of the difference-channel echo to the sum-channel echo is defined as the monopulse ratio or monopulse error. Hence, the monopulse error can be given by:

$$M_J=\text{Im}\left(\frac{D}{S}\right)=\text{Im}\left[\frac{2\left(1+a\right)j\sin \left(\mathrm{\pi }d\sin \left({\theta }_r\right)/\lambda \right)}{2\left(1-a\right)\cos \left(\mathrm{\pi }d\sin \left({\theta }_r\right)/\lambda \right)}\right]=\frac{\left(1+a\right)}{\left(1-a\right)}\tan \left[\frac{\mathrm{\pi }d}{\lambda }\sin {\theta }_r\right]$$ (Equation 10)

where Im(·) denotes taking the imaginary part.

The monopulse indicated angle ${\theta }_i$ and the monopulse error $M_J$ have the relationship as follows:

$$M_J=\tan \left[\frac{\mathrm{\pi }d}{\lambda }\sin {\theta }_i\right]$$ (Equation 11)

Thus, the monopulse indicated angle can be converted from the monopulse error using the following relationship:

$${\theta }_i=\arcsin \left(\frac{\arctan \left(M_J\right)\lambda }{\mathrm{\pi }d}\right)$$ (Equation 12)

Within the 3 dB main lobe of the sum beam in a monopulse radar, the relationship between $M_J$ and ${\theta }_i$ in Equation 12 is approximately linear. Therefore, the linear fit analysis is usually done using the following equation:

$${\theta }_i=\xi M_J$$ (Equation 13)

where $\xi$ is a constant that represents the coefficient of the linear regression.

Defining $\left(1+a\right)/\left(1-a\right)$ in Equation 10 as the cross-eye gain $G$. From (Equation 11), (Equation 12), (Equation 13), the jamming angle ${\theta }_j$ measured under cross-eye jamming can be expressed as:

$${\theta }_j=\xi \frac{\left(1+a\right)}{\left(1-a\right)}\tan \left[\frac{\mathrm{\pi }d}{\lambda }\sin {\theta }_r\right]=\frac{\left(1+a\right)}{\left(1-a\right)}\xi \tan \left[\frac{\mathrm{\pi }d}{\lambda }\sin {\theta }_r\right]=G{\theta }_r$$ (Equation 14)

As shown in Equation 14, when the angle between the metasurface and the radar boresight remains constant, the jamming angle ${\theta }_j$ increases as cross-eye gain G increases. When a is in the range (0,1), G increases with $a$ and approaches infinity as $a$ approaches 1. When $a$ is held constant, ${\theta }_j$ increases with ${\theta }_r$. Therefore, by controlling the amplitude ratio to approach 1 and increasing the angle between the metasurface and the radar boresight, a larger jamming angle can be obtained.

Analysis of monopulse error in practical applications

For a DCM using PIN diodes, the unit state switches between 0 and 1 by controlling the diode’s on-off state. In practice, due to the influence of component parameters, EM wave frequency, and other factors, an error exists between the reflection phase difference of the units in the two states and 180°. Furthermore, the reflection magnitudes in the two states are not exactly equal. Figure 4 presents a schematic diagram of the 1-bit unit cell. The structure consists of two metallic patches connected by a PIN diode on the top reflective layer. This configuration is supported by two stacked dielectric substrates with different permittivities.

Figure 4.

Schematic diagram of the 1-bit unit cell

Figure 5 shows the reflecting phase and amplitude curves between the two states of the 1-bit metasurface unit across the frequency range from 0 to 15 GHz²⁹ Figure 6 illustrates the phase and magnitude differences between the two states. As shown in Figure 6A, the phase difference remains within the range of 155°–205° over the frequency interval from 6.88 GHz to 13.62 GHz, which is sufficiently close to the ideal 180° required for effective 1-bit coding. This ensures that the proposed method can achieve robust performance within this band. Furthermore, this interval completely covers the X-band (8–12 GHz). Figure 6B shows the amplitude difference curve in the X-band. As seen from the curve, the amplitude difference error remains within 1 dB, indicating that both the phase and amplitude differences between the two states are minimal in the X-band. Therefore, the proposed method is especially suitable for X-band applications. In this section, the effect of phase difference and amplitude difference within a certain tolerance range on the interference performance is analyzed.

Figure 5.

Reflecting phase and magnitude curves of the 1-bit unit cell

Figure 6.

Phase difference and magnitude difference curves

(A) Phase difference.

(B) Magnitude difference.

The jamming scenario and parameter settings are the same as those in Figure 2. The signal of the sum-channel can be rewritten as:

$$S_{\left(e\right)}=2\epsilon \left(1+a{e}^{j\phi }\right)\cos \left(\mathrm{\pi }d\sin \left({\theta }_r\right)/\lambda \right)$$ (Equation 15)

The difference-channel signal can be rewritten as:

$$D_{\left(e\right)}=2\epsilon \left(1-a{e}^{j\phi }\right)j\sin \left(\mathrm{\pi }d\sin \left({\theta }_r\right)/\lambda \right)$$ (Equation 16)

where a is the amplitude ratio and $\phi$ is the phase difference of the two DCMs reflection signals.

Hence, the monopulse error can be given by:

$$M_{J\left(e\right)}=\text{Im}\left(\frac{D_{\left(e\right)}}{S_{\left(e\right)}}\right)=\text{Im}\left[\frac{2\epsilon \left(1-a{e}^{j\phi }\right)j\sin \left(\mathrm{\pi }d\sin \left({\theta }_r\right)/\lambda \right)}{2\epsilon \left(1+a{e}^{j\phi }\right)\cos \left(\mathrm{\pi }d\sin \left({\theta }_r\right)/\lambda \right)}\right]=\frac{\left(1-a^2\right)}{\left[1+a^2+2a\cos \left(\phi \right)\right]}\tan \left[\frac{\mathrm{\pi }d}{\lambda }\sin {\theta }_r\right]$$ (Equation 17)

Similar to the derivation process of Equation 14, the jamming angle can be expressed as:

$${\theta }_{j\left(e\right)}=\xi \frac{\left(1-a^2\right)}{1+a^2+2a\cos \left(\phi \right)}\tan \left[\frac{\mathrm{\pi }d}{\lambda }\sin {\theta }_r\right]=G_c{\theta }_r$$ (Equation 18)

where the cross-eye gain is denoted as $G_c$:

$$G_c=\frac{\left(1-a^2\right)}{1+a^2+2a\cos \left(\phi \right)}$$ (Equation 19)

(Equation 11), (Equation 12), (Equation 13) and (Equation 11), (Equation 12), (Equation 13) reveal that as the amplitude ratio approaches 1 and the phase difference approaches 180°, the cross-eye jamming gain increases. A deviation in the amplitude ratio from 1 or in the phase difference from 180° may result in a decrease in cross-eye gain, consequently influencing the jamming performance. Therefore, the amplitude ratio should be controlled to approach 1, and the phase difference should approach 180° to maximize the angular interference effect. Additionally, a larger ${\theta }_r$ can also cause a larger angular measurement error.

Results and discussions

DCM beam modulation

In this subsection, the EM beam modulation capability of DCM is verified. Assuming that the horn antenna transmits a signal frequency of 10 GHz, the horn height is 175 mm, the projection is located at the center of the DCM, and the DCM is coded with 20 × 20 square elements, each with a side length of 12 mm. The angle of the reflected beam from the first DCM $\left(\alpha ,\beta \right)$ is set to (2°, 0°). The simulations were conducted using MATLAB 2019a, with the optimization toolbox for algorithm implementation. The hardware used for the experiments consisted of a desktop computer with an AMD Ryzen 9 7945HX processor and an NVIDIA GTX 4060 GPU. The compensated phases of each element in the first metasurface are shown in Figure 7A. In order to realize the 180° phase difference of the two metasurfaces transmit signals, the coding rule of the second metasurface is opposite to that of the first. The unit coded as 0 in the first metasurface is coded as 1 in the corresponding position element in the second one, and the unit coded as 1 is similarly changed to 0. Meanwhile, in order to make the direction of the two beams symmetric, the second metasurface should be flipped by 180°. The compensated phases of the elements in the second metasurface are finally obtained as shown in Figure 7B. The three-dimensional radiation patterns of the signals emitted by the two metasurfaces are shown in Figure 8A and 8B, respectively. The normalized pitch-plane radiation patterns of the signals emitted by the two metasurfaces are shown in Figures 9A and 9B, respectively.

Figure 7.

The phase compensation of the DCMs

(A) The first metasurface.

(B) The second metasurface.

Figure 8.

Three-dimensional radiation pattern

(A) The first metasurface.

(B) The second metasurface.

Figure 9.

Normalized pattern of the signals emitted by the two DCMs

(A) The first metasurface.

(B) The second metasurface.

As illustrated in Figure 9, the two DCMs steer the emitted signals to symmetrical directions following the described methodology. Although there are minor deviations between the actual steering directions and the predetermined values, the symmetry of the two directions ensures that no additional phase difference is introduced, remaining within acceptable tolerance.

Sum- and difference-channel signals

This subsection validates the sum- and difference-channel signal formulas for the ideal 180° phase difference case. Radar frequency $f=9\text{GHz}$ and monopulse antenna spacing distance $d=2.54\lambda$ are taken for the simulation.

First, the sum- and difference-channel signal results expressed in (Equation 11), (Equation 12), (Equation 13) and (Equation 11), (Equation 12), (Equation 13) are verified. The monopulse radar sum- and difference-channel signals, obtained for values of a as −0.7 dB, −0.5 dB, and −0.3 dB, are compared with the results without cross-eye jamming. The absence of cross-eye jamming means that the monopulse radar measures only a single target with an angle of ${\theta }_r$. In the absence of jamming, the sum- and difference-channel signals are given as follows:

$$S=2\epsilon \cos \left(\mathrm{\pi }d\sin \left({\theta }_r\right)/\lambda \right)$$ (Equation 20)

$$D=2\epsilon j\sin \left(\mathrm{\pi }d\sin \left({\theta }_r\right)/\lambda \right)$$ (Equation 21)

According to (Equation 11), (Equation 12), (Equation 13), (Equation 11), (Equation 12), (Equation 13), (Equation 11), (Equation 12), (Equation 13), and (Equation 11), (Equation 12), (Equation 13), when a is less than 0 dB, the amplitude of the sum-channel signal decreases compared to the absence of jamming, while the difference-channel signal increases. The closer the amplitude approachs 0 dB, the greater the decrease and increase. Using the maximum amplitudes of the sum- and difference-channel signals of the monopulse radar in the absence of cross-eye jamming as a reference, the maximum normalized amplitude values of the sum- and difference-channel signals for each jamming amplitude ratio are obtained based on (Equation 11), (Equation 12), (Equation 13), (Equation 11), (Equation 12), (Equation 13), (Equation 11), (Equation 12), (Equation 13), and (Equation 11), (Equation 12), (Equation 13), as shown in Table 1. The results of the simulation experiment are shown in Figures 10 and 11.

Table 1.

Maximum normalized amplitude of sum- and difference-channel signals

	Maximum amplitude of sum-channel signal (dB)	Maximum amplitude of difference-channel signal (dB)
Without CEJ	0	0
a = −0.3 dB	−67.628	13.520
a = −0.5 dB	−57.648	13.295
a = −0.7 dB	−51.279	13.077

Figure 10.

Sum-channel signal

Figure 11.

Difference-channel signal

The simulation results of the maximum amplitude of sum- and difference-channel signals show good agreement with the theoretical calculations presented in Table 1. As shown in Figure 8, the absence of cross-eye jamming yields a significantly higher sum-channel signal amplitude relative to the interfered case. For amplitude ratios of less than 0 dB, as the ratio approaches 0 dB, the sum-channel amplitude decreases more significantly. At $a=-0.3\text{dB}$, the normalized maximum amplitude drops to −67.6279 dB, matching the corresponding value in Table 1. Results for other amplitude ratios also show strong agreement with Table 1, confirming the validity of the derived sum channel signal equation. As shown in Figure 9, cross-eye jamming has been found to elevate the difference-channel signal amplitude relative to the interference-free case. Similarly, as the amplitude ratio approaches 0 dB, the increase in difference-channel amplitude is more pronounced. At $a=-0.3\text{dB}$, the normalized maximum amplitude rises to 13.52 dB, matching the Table 1 value. Results for other amplitude ratios also align with the calculations in Table 1, validating the difference channel equation. In this study, only amplitude ratios less than 0 dB were analyzed. For amplitude ratios greater than 0 dB, the results remain similar, as the two signals simply exchange their relative amplitudes.

Analysis of jamming performance

The effect of jamming can be intuitively represented by the monopulse indicated angle. Figure 12 shows the monopulse indicated angle obtained from the Equation 12 under various amplitude ratios. The black line represents the indicated angle in the absence of jamming. As shown in Figure 12, amplitude ratios approaching 0 dB lead to increased angular errors. Notably, the measured maximum angle approaches 11.3503° due to the trigonometric constraints inherent in Equation 12. Figure 13 shows the monopulse indicated angle obtained from the Equation 13 under various amplitude ratios. As shown in Figure 13, as the amplitude ratio a approaches 0 dB, the gain G increases, which resulting in a greater deviation of the measured monopulse indicated angle from the true curve. A larger ${\theta }_r$ also results in a larger goniometric error, which is consistent with the result in mathematical analysis of the monopulse error.

Figure 12.

Indicated angle calculated using trigonometric formula

Figure 13.

Indicated angle obtained through linear approximation

Finally, the analysis of monopulse error in practical applications is validated through simulation experiments. According to Equation 19, the cross-eye gain is influenced by both the phase difference and the amplitude ratio. To enhance realism, the metasurface element compensation phase is set to (160°, 200°), and the amplitude ratio is varied between −6 dB and 6 dB. The relationship between system parameters and $G_c$ can be analyzed using contour plots, as shown in Figure 14. When the amplitude ratio and phase difference values lie on a specific contour, the cross-eye jamming results in the corresponding gain. When the values fall within the contour, the cross-eye jamming produces a gain greater than the corresponding value.

Figure 14.

Cross-eye gain

As can be seen in Figure 14, the cross-eye gain contours are symmetric about the amplitude ratio $a=0\text{dB}$. Within the range of phase difference at (160°, 200°) and amplitude ratio at (-6 dB, 6 dB), the cross-eye gain is still able to reach more than twice, which is capable of jamming the monopulse radar. Moreover, as the amplitude ratio approaches 0 dB and the phase difference approaches 180°, the cross-eye gain increases significantly. Furthermore, it is evident that an increase in cross-eye gain leads to higher demands on each parameter. This means that the parameter tolerances become stricter, and the accuracy requirements for the metasurface device increase.

Conclusions

In conclusion, this paper presents a cross-eye jamming method based on DCM. The proposed method replaces the antenna structure with a DCM in TRCJ, which enables effective monopulse radar jamming with the advantages of cost-effectiveness and flexible control. This study derives the EM wave modulation model of DCM and demonstrates that jamming of a monopulse radar can be achieved by using DCMs to generate cross-eye jamming signals. Simulation results show that when the phase difference of the metasurface element is controlled within the range of (160°, 200°), this method can achieve more than twice the cross-eye gain, thereby validating the effectiveness and feasibility of the proposed method. Future work will focus on improving jamming performance and minimizing parameter tolerances.

Limitations of the study

Although this study presents progress in cross-eye jamming based on metasurfaces, several limitations should be acknowledged. First, the metasurface design demonstrates optimal reflection characteristics only within specific frequency bands, which may limit the generalizability of the results. Second, more complex jamming scenarios were not thoroughly investigated, and these factors may influence the overall performance. In addition, all experiments were conducted under simulation conditions, and further validation in more diverse or real-world environments is required. Future research addressing these aspects will help to strengthen and extend the conclusions of this study.

Resource availability

Lead contact

Further information and requests for resources and reagents should be directed to and will be fulfilled by the lead contact, Xiaolong Su (suxiaolong16@nudt.edu.cn).

Materials availability

This study did not generate new reagents.

Data and code availability

• •Data reported in this paper will be shared by the lead contact upon request.
• •This paper does not report original codes.
• •Any additional information required to reanalyze the data reported in this paper is available from the lead contact upon request.

Acknowledgments

This work was supported in part by the National Natural Science Foundation of China under grant 62401585, in part by the Research Program of National University of Defense Technology under grant ZK23-18, and in part by the National Natural Science Foundation of China under grant 62201588.

Author contributions

Conceptualization, Z.W. and X.S.; methodology, Z.W. and X.S.; software, Z.W.; validation, Z.W., X.S., H.S., and Z.L.; formal analysis, Z.W., X.S., and Z.L.; investigation, Z.W. and X.S.; resources, X.S., P.H., and T.L.; data curation, Z.W. and X.S.; writing-original draft preparation, Z.W.; writing-review and editing, X.S. and P.H.; visualization, Z.W. and H.S.; supervision, X.S., P.H., and T.L.; project administration, Z.W. and X.S.; funding acquisition, X.S. and T.L. All authors have read and agreed to the published version of the manuscript.

Declaration of interests

The authors declare no conflicts of interest.

STAR★Methods

Key resources table

REAGENT or RESOURCE	SOURCE	IDENTIFIER
Software and algorithms
MATLAB 2019a	MathWorks	https://www.mathworks.com/matlabcentral/answers/601606-download-matlab-2019a; RRID: SCR_001622
Origin 2022	OriginLab	https://www.originlab.com/2022; RRID: SCR_014212

Method details

Derivation of sum- and difference-channel signals under ideal CEJ

The jamming signals received by the top and bottom antennas from the first metasurface can be respectively given by

$$P_t^1=\epsilon e^{-j\mathrm{\pi }d\sin \left({\theta }_r\right)/\lambda }$$ (Equation 22)

and

$$P_b^1=\epsilon e^{j\mathrm{\pi }d\sin \left({\theta }_r\right)/\lambda }$$ (Equation 23)

The second DCM inverts the phase and modulates the amplitude of the jamming signal by a factor of $a$. The signals received by the two monopulse radar antennas from the second metasurface can be respectively given by

$$P_t^2=-a\epsilon e^{j\mathrm{\pi }d\sin \left({\theta }_r\right)/\lambda }$$ (Equation 24)

and

$$P_b^2=-a\epsilon e^{-j\mathrm{\pi }d\sin \left({\theta }_r\right)/\lambda }$$ (Equation 25)

According to the operating principle of the phase-comparison monopulse radar, the signal received by the sum-channel can be expressed as

$$S=P_t^1+P_b^1+P_t^2+P_b^2=\epsilon \left(1-a\right)\left(e^{j\mathrm{\pi }d\sin \left({\theta }_r\right)/\lambda }+e^{-j\mathrm{\pi }d\sin \left({\theta }_r\right)/\lambda }\right)=2\epsilon \left(1-a\right)\cos \left(\mathrm{\pi }d\sin \left({\theta }_r\right)/\lambda \right)$$ (Equation 26)

The signal received by the difference-channel can be expressed as

$$D=P_b^1-P_t^1+P_b^2-P_t^2=\epsilon \left(1+a\right)\left(e^{j\mathrm{\pi }d\sin \left({\theta }_r\right)/\lambda }-e^{-j\mathrm{\pi }d\sin \left({\theta }_r\right)/\lambda }\right)=2\epsilon \left(1+a\right)j\sin \left(\mathrm{\pi }d\sin \left({\theta }_r\right)/\lambda \right)$$ (Equation 27)

Derivation of sum- and difference-channel signals under general CEJ

The jamming signals received by the top and bottom antennas from the first metasurface can be respectively given by

$$P_{t\left(e\right)}^1=\epsilon e^{-j\mathrm{\pi }d\sin \left({\theta }_r\right)/\lambda }$$ (Equation 28)

and

$$P_{b\left(e\right)}^1=\epsilon e^{j\mathrm{\pi }d\sin \left({\theta }_r\right)/\lambda }$$ (Equation 29)

The jamming signals received by the top and bottom antennas from the second metasurface can be respectively given by

$$P_{t\left(e\right)}^2=a{e}^{j\phi }\epsilon e^{j\mathrm{\pi }d\sin \left({\theta }_r\right)/\lambda }$$ (Equation 30)

and

$$P_{b\left(e\right)}^2=a{e}^{j\phi }\epsilon e^{-j\mathrm{\pi }d\sin \left({\theta }_r\right)/\lambda }$$ (Equation 31)

where a is the amplitude ratio and $\phi$ is the phase difference of the two DCMs reflection signals.

Accordingly, the signal of the sum-channel can be rewritten as

$$S_{\left(e\right)}=P_{t\left(e\right)}^1+P_{b\left(e\right)}^1+P_{t\left(e\right)}^2+P_{b\left(e\right)}^2=\epsilon \left(1+a{e}^{j\phi }\right)\left(e^{j\mathrm{\pi }d\sin \left({\theta }_r\right)/\lambda }+e^{-j\mathrm{\pi }d\sin \left({\theta }_r\right)/\lambda }\right)=2\epsilon \left(1+a{e}^{j\phi }\right)\cos \left(\mathrm{\pi }d\sin \left({\theta }_r\right)/\lambda \right)$$ (Equation 32)

The difference-channel signal can be rewritten as

$$D_{\left(e\right)}=P_{b\left(e\right)}^1-P_{t\left(e\right)}^1+P_{b\left(e\right)}^2-P_{t\left(e\right)}^2=\epsilon \left(1-a{e}^{j\phi }\right)\left(e^{j\mathrm{\pi }d\sin \left({\theta }_r\right)/\lambda }-e^{-j\mathrm{\pi }d\sin \left({\theta }_r\right)/\lambda }\right)=2\epsilon \left(1-a{e}^{j\phi }\right)j\sin \left(\mathrm{\pi }d\sin \left({\theta }_r\right)/\lambda \right)$$ (Equation 33)

Parameter settings of 1-bit unit cell

The 1-bit wideband reflecting element, illustrated schematically in Figure 4, consists of a layered structure combining metallic and dielectric materials. The upper reflective layer is constituted by two copper patches and a PIN diode. This layer is supported by two dielectric substrates, which are arranged periodically with a period of p. The top substrate is composed of F4B, which has a relative permittivity ${\epsilon }_r$ of 2.65 and a loss tangent $\tan \delta$ of 0.001, while the bottom substrate uses FR-4, which has an ${\epsilon }_r$ of 4.30 and a $\tan \delta$ of 0.025.

The metallic patch exhibits a symmetrical bow-tie shape, comprising an isosceles trapezoid and a rectangular section. The left side of the patch is connected to the ground plane, and the right side is linked to a signal transmission line that is electrically isolated from the central ground plane. A PIN diode (SMP1340-040LF from Skyworks) is placed between the two patch sections. The application of forward or reverse bias currents to the diode results in the rapid switching of the electromagnetic response of the reflecting element, thereby enabling dynamic modulation of its reflective properties.

Quantification and statistical analysis

The simulation data is produced by MATLAB software. Figures shown in the main text were produced by ORIGIN and Microsoft Visio from the raw data.

Published: August 6, 2025