PolSAR Image Modulation Using a Flexible Metasurface with Independently Controllable Polarizations

Abstract

Recent advances in time-modulated metasurfaces (TMMs) have introduced approaches for controlling target features in radar imaging. These technologies enable dynamic reconstruction of scattering center locations and intensities by flexibly manipulating radar echoes. However, most existing methods focus on amplitude and phase modulation, lacking joint control over the polarimetric scattering characteristics of targets. As a result, the modulated outputs tend to exhibit limited polarimetric diversity and remain strongly tied to the targets’ physical structures. To address this limitation, this paper proposes a modulation method for polarimetric synthetic aperture radar (PolSAR) images based on a flexible metasurface with independently controllable polarizations (FM-ICP). The method independently controls the echo energy distribution in two polarization channels, enabling target representations in PolSAR images to exhibit polarimetric characteristics beyond their physical geometry—for example, rendering a flat plate as a cylinder, or vice versa. In addition, the method can generate synthetic scattering centers with controllable locations and polarimetric properties, which can be precisely tuned via modulation parameters. This work offers a practical approach for target feature manipulation and shows potential in PolSAR image simulation and feature reconstruction.

1. Introduction

Time-modulated metasurfaces (TMMs) have emerged as a research focus due to their ability to dynamically control electromagnetic (EM) wave characteristics [1,2,3]. By integrating switching elements with subwavelength structures and applying time-coded sequences to control conductive states in real time, TMMs can produce time-varying surface impedance distributions. When interacting with incident waves, they enable precise spatiotemporal control of fundamental EM wave properties, including amplitude, phase, frequency, and polarization state [4,5,6,7]. This dynamic control sets TMMs apart from conventional metamaterials. They demonstrate unique advantages in applications such as frequency conversion and beam steering, where traditional structures typically lack real-time adaptability [8,9,10,11].

With their flexible electromagnetic manipulation capabilities, TMMs have been actively applied in various areas of radar target signature modulation, including radar cross section (RCS) reduction [12,13], high-resolution range profile (HRRP) deception [14,15,16], doppler signature synthesis [17], and two-dimensional image transformation [18,19,20]. Particularly in synthetic aperture radar (SAR) imaging scenarios, TMMs enable dynamic reconstruction of the spatial locations and intensities of target scattering centers within the image. For instance, in [21], precise amplitude and phase-coded modulation was employed to achieve multi-mode control, including simulation of original target images and generation of decoy targets, making the resulting SAR images difficult to distinguish. A periodic phase modulation method was proposed in [22], where TMMs effectively reconstructed imaging features of different targets. However, existing methods mainly focus on amplitude and phase modulation, while neglecting the control of polarization characteristics. When applied to multi-polarization radar systems, existing methods suffer from two main limitations. First, when the metasurface used for modulation can only manipulate electromagnetic waves incident with a single polarization [18,23], the resulting effects—such as synthetic scattering centers—will appear in only one channel of a fully polarimetric imaging radar, while remaining absent in the other three channels. Consequently, these scattering centers can be readily removed through multi-channel image comparison. Second, even if the metasurface can affect electromagnetic waves with dual-polarization incidence [24], it cannot independently control the energy of each polarization channel. As a result, the inter-channel energy relationship remains fixed. Targets modulated in this manner inevitably exhibit the intrinsic polarization characteristics dictated by the physical structure of the metasurface, lacking diversity in polarization attributes. Therefore, simulated targets such as aircraft or vehicles [19,22] generated by this modulation appear as combinations of scattering centers with a single polarization signature, which can be easily distinguished using polarimetric decomposition techniques [25,26], rendering the modulation ineffective.

Recent advancements have significantly expanded the functionality of TMMs, including independently controllable polarizations and flexible structural designs [27,28,29,30,31]. The polarization-independent control is achieved by orthogonally arranged unit cells that allow independent manipulation of two orthogonal polarization channels. The flexible design enables conformal deployment on curved target surfaces [32]. These capabilities theoretically lay a technical foundation for the modulation of radar target polarimetric characteristics. However, current research remains heavily focused on material innovation and applications in communication systems, such as multifunctional antennas [33] or polarization-division multiplexing communication [34], with limited exploration into the active and precise modulation of target scattering characteristics in polarimetric imaging radar using TMMs. Critically, unlike the communication field, where the emphasis lies on information carrying and transmission, the primary objective of target feature control is to reconstruct or simulate the physical scattering properties of targets. To date, few studies have addressed how features such as polarization-independent control and conformal design of TMMs can be adapted and applied to tasks requiring accurate control of target characteristics in polarimetric radar, resulting in a significant gap between the development of advanced metasurface capabilities and the demands of polarimetric radar-based target modulation.

Motivated by the discussions above, this paper proposes an orthogonal dual-channel independent control approach based on a flexible metasurface with independently controllable polarizations (FM-ICP), which enables effective modulation of polarimetric synthetic aperture radar (PolSAR) images. The proposed method relies solely on the amplitude modulation capability of the metasurface to redistribute echo energy between dual-polarization channels, thereby achieving joint control over target polarization characteristics. After modulation, the original targets in PolSAR images can exhibit polarization features that transcend their physical geometric structures, such as making a flat plate appear as a cylinder, or vice versa. Moreover, multiple synthetic scattering centers can be generated, whose spatial positions and polarization characteristics can be precisely controlled through modulation parameters. This study establishes a time-varying modulation model of PolSAR echoes based on metasurfaces and achieves alteration of inter-channel energy relationships, thereby overcoming the limitations of existing methods in polarization control.

The content of this article is structured as follows: Section 2 presents the fundamental principles and methods of FM-ICP-based modulation, including its operating mechanism, equivalent signal model, and the PolSAR modulation framework. Section 3 categorizes the modulation strategies into joint control and independent control, and provides simulation results to evaluate the corresponding PolSAR imaging effects. Section 4 further discusses the variation patterns of target polarimetric features, the impact of clutter on the modulation effect, and the influence of the practical performance of the FM-ICP on the modulation behavior. Finally, Section 5 presents the conclusion.

2. Materials and Methods

The foundation for target feature modulation in PolSAR images in this work is the flexible metasurface with independently controllable polarizations (FM-ICP). This section first introduces the working principle of the FM-ICP. Then, an equivalent modulated signal model based on FM-ICP is established. On this basis, a time-varying modulation model of PolSAR imaging based on FM-ICP is developed, and the modulation effects from echo generation to final imaging results are analyzed.

2.1. Working Principle of the FM-ICP

The FM-ICP consists of two orthogonal tunable impedance layers and one dielectric substrate. The impedance layers integrate metal patch structures and PIN diodes, which are equivalent to LC resonant circuits. By changing the applied voltage, the bias state of the diodes is altered, which in turn changes the equivalent impedance of the FM-ICP. The PIN diodes in the two impedance layers independently control two orthogonal polarization channels. When the diodes are in the OFF state, the FM-ICP reflects the electromagnetic wave of the corresponding polarization, corresponding to the high scattering state. When the diodes are in the ON state, the FM-ICP transmits the wave, corresponding to the low scattering state. Therefore, by controlling the ON/OFF states of the two diodes, the reflection coefficients for different polarization states can be flexibly adjusted [32].

The response of the FM-ICP to electromagnetic waves with different polarization configurations can be quantitatively characterized. Let the reflection coefficient of the FM-ICP for a given polarization configuration be denoted as $R_{xy}\left(f\right)$, where f is the frequency of the incident wave, x is the receiving polarization, and y is the transmitting polarization. To simplify the model, the average reflection coefficient $R_{xy}$ within the operating bandwidth is used to replace $R_{xy}\left(f\right)$ in the following analysis. It should be noted that the two operating states of the FM-ICP do not alter the polarization state of the incident wave. Therefore, its amplitude modulation capability is only effective in co-polarized channels, where the transmitting and receiving polarizations are identical. This is reflected by a dynamic switching of reflection coefficients in co-polarized channels, while the reflection coefficients of cross-polarized channels remain 0.

When the FM-ICP operates in the high scattering state for a given polarization, the reflection coefficient is normalized to 1. When it operates in the low scattering state, the reflection coefficient is normalized to ρ_y. The calculation of ρ_y is as follows

$${\rho }_y={10}^{\left(R_{h-y}-R_{l-y}\right)/20}$$

(1)

where, R_h−y denotes the reflection coefficient of the FM-ICP under the high scattering state for an incident wave with y-polarization, and R_l−y denotes the reflection coefficient under the low scattering state. The value of ρ_y, representing the normalized reflection coefficient in the low scattering state, varies between 0 and 1 depending on the applied voltage.

Therefore, by independently controlling the ON/OFF states of the diodes in the two variable-impedance layers, the reflection coefficient can be switched between 1 and ρ_y. Moreover, by adjusting the external voltages applied to the two layers independently, the value of ρ_y can be precisely controlled. This characteristic results in a dynamic variation of the reflection coefficient of the FM-ICP, referred to as the functional reflection coefficient of the FM-ICP. Under different combinations of the two PIN diodes’ states, the functional reflection coefficient matrix R of the FM-ICP is shown in

Table 1. Functional reflection coefficient matrix under different OFF/ON combinations of PIN diodes.

Operating State	OFF in y-Polarization Direction	ON in y-Polarization Direction
OFF in x-Polarization Direction	${\mathit{R}}_{HH}=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$	${\mathit{R}}_{HL}=\left[\begin{array}{cc}1& 0\\ 0& {\rho }_y\end{array}\right]$
ON in x-Polarization Direction	${\mathit{R}}_{LH}=\left[\begin{array}{cc}{\rho }_x& 0\\ 0& 1\end{array}\right]$	${\mathit{R}}_{LL}=\left[\begin{array}{cc}{\rho }_x& 0\\ 0& {\rho }_y\end{array}\right]$

.

In addition to the reflection changes caused by the switching of diode states, the structural configuration of the FM-ICP itself also affects the reflection characteristics for different polarizations of incident electromagnetic waves. When the FM-ICP is illuminated by electromagnetic waves, the polarization state of the scattered wave will differ from that of the incident wave, exhibiting a specific transformation relationship. This relationship is closely related to physical properties such as the target’s attitude, size, and structure, and is referred to as the polarization conversion effect. This effect is typically described using the polarimetric scattering matrix S, which can also be termed the physical reflection coefficient matrix. Therefore, the overall scattering characteristic S_T of the FM-ICP can be represented as

$${\mathit{S}}_T=\mathit{S}\mathit{R}$$

(2)

where the functional reflection coefficient matrix R is controlled by the externally applied voltage and can vary rapidly. It serves as the core mechanism through which the FM-ICP achieves scattering characteristic modulation. In contrast, the physical reflection coefficient, i.e., the polarimetric scattering matrix S, is determined by the intrinsic structural characteristics of the FM-ICP and remains relatively stable.

2.2. Modulation Signal Model Based on FM-ICP

According to the working principle of the FM-ICP, the high or low scattering state of electromagnetic waves with different polarizations can be independently controlled by adjusting the operating states of the PIN diodes. When a periodically varying control signal is applied, the functional reflection coefficients of the FM-ICP in each channel also vary periodically, resulting in equivalent modulated signals that are periodic in nature.

Assuming the incident wave contains two orthogonal polarizations, horizontal polarization (H) and vertical polarization (V), Figure 1a shows the external excitation signal applied to the H-polarized incident wave (hereafter referred to as the H-channel control signal), with a switching period of T_sH and a low-level pulse width of T_wH. Figure 1b shows the external excitation signal applied to the V-polarized incident wave (hereafter referred to as the V-channel control signal), with a switching period of T_sV and a low-level pulse width of T_wV. In the figures, the ellipsis indicates the continuation of the periodic waveform, where the subsequent segments are identical to the part already shown. This meaning of the ellipsis applies to all figures in the following discussion and will not be repeated. Since the OFF state of the diode corresponds to the high scattering state and the ON state corresponds to the low scattering state, the equivalent modulation signal models for each channel can be derived, as illustrated in Figure 2. In these models, the amplitude coefficients in the signal model correspond to the functional reflection coefficients of the FM-ICP. The equivalent modulation signal model depends on the polarization of the transmitted electromagnetic wave. Since the waveforms of the HH and VH polarization channels are identical, they are hereinafter referred to as the H-channel modulation signal. Similarly, the waveforms of the HV and VV polarization channels are identical and are referred to as the V-channel modulation signal.

The time-domain waveforms of the H-channel and V-channel modulation signals can be expressed as

$$p_H\left(t\right)=\left(1-x_H\right)rect\left({\displaystyle \frac{t}{T_{wH}}}\right)\otimes {\displaystyle \sum _{-\infty }^{+\infty }\delta \left(t-m{T}_{sH}\right)}+x_H$$

(3)

$$p_V\left(t\right)=\left(1-x_V\right)rect\left({\displaystyle \frac{t}{T_{wV}}}\right)\otimes {\displaystyle \sum _{-\infty }^{+\infty }\delta \left(t-n{T}_{sV}\right)}+x_V$$

(4)

where $rect\left(\cdot \right)$ is a rectangular pulse, $\otimes$ denotes convolution, and $\delta \left(\cdot \right)$ is the impulse function. T_wH and T_wV represent the pulse widths corresponding to the high scattering state in the H and V channels, respectively. T_sH and T_sV denote the modulation periods for the H and V channels. All these parameters correspond to the properties of the applied control signals. x_H and x_V represent the amplitude coefficients in the low scattering state for the H and V channels, respectively, and both vary with the applied voltage. m and n are the indices of the pulse sequences. The effective modulation duty cycle for the H-polarized channel is defined as ${\alpha }_H=T_{wH}/T_{sH}$, and for the V-polarized channel as ${\alpha }_V=T_{wV}/T_{sV}$. Overall, the equivalent modulation signals in both polarization channels are linear superpositions of rectangular pulses.

The time-domain signals described in Equations (3) and (4) are transformed using the Fourier transform, and their corresponding frequency-domain expressions are obtained as

$$P_H\left(f\right)=A_{0-H}\delta \left(f\right)+{\displaystyle \sum _{-\infty ,m\ne 0}^{+\infty }{A}_{m-H}\delta \left(f-m{f}_{sH}\right)}$$

(5)

$$P_V\left(f\right)=A_{0-V}\delta \left(f\right)+{\displaystyle \sum _{-\infty ,n\ne 0}^{+\infty }{A}_{n-V}\delta \left(f-n{f}_{sV}\right)}$$

(6)

The modulation frequencies for the H and V channels are $f_{sH}=1/T_{sH}$ and $f_{sV}=1/T_{sV}$, respectively. The spectral amplitude coefficients of the H-channel modulation signal are $A_{0-H}$ and $A_{m-H}$, while those of the V-channel modulation signal are $A_{0-V}$ and $A_{n-V}$. Their expressions are given as follows:

$$A_{0-H}=\left(1-x_H\right)T_{wH}{f}_{sH}+x_H=\left(1-x_H\right){\alpha }_H+x_H$$

(7)

$$A_{m-H}=\left(1-x_H\right)T_{wH}{f}_{sH}\sin c\left(m{T}_{wH}{f}_{sH}\right)=\left(1-x_H\right){\alpha }_H\sin c\left(m{\alpha }_H\right)$$

(8)

$$A_{0-V}=\left(1-x_V\right)T_{wV}{f}_{sV}+x_V=\left(1-x_V\right){\alpha }_V+x_V$$

(9)

$$A_{n-V}=\left(1-x_V\right)T_{wV}{f}_{sV}\sin c\left(n{T}_{wV}{f}_{sV}\right)=\left(1-x_V\right){\alpha }_V\sin c\left(n{\alpha }_V\right)$$

(10)

In summary, the spectrum of the equivalent modulation signals in each channel based on the FM-ICP consists of harmonic components proportional to the modulation frequency of the corresponding channel. Moreover, the amplitude envelope follows a sinc function distribution. The overall spectral magnitude is influenced by the amplitude coefficients of each channel (x_H and x_V) and their modulation duty cycles (α_H and α_V).

2.3. Time-Varying Modulation Model for PolSAR Imaging

PolSAR systems can generally be categorized into two types based on their polarization measurement method: time-sharing polarization measurement and simultaneous polarization measurement. Considering implementation complexity and cost constraints, the time-sharing polarization measurement remains the mainstream architecture for most PolSAR systems. This architecture relies on a pair of orthogonally polarized antennas (commonly referred to as horizontal (H) and vertical (V) polarization) to alternately transmit orthogonally polarized radar signals over two consecutive Pulse Repetition Intervals (PRIs), while simultaneously receiving the backscattered signals through dual-polarization receiver channels. As a result, the system can reconstruct the full polarimetric scattering matrix of the target across two consecutive PRIs, enabling accurate measurement of its polarimetric scattering characteristics.

Figure 3 illustrates the timing sequence of the time-sharing polarization measurement architecture. When the H-polarized signal is transmitted, the receiving system captures the S_HH and S_VH components of the target’s scattering matrix. When the V-polarized signal is transmitted, the S_HV and S_VV components are obtained.

In each PRI, the commonly used transmitted signal in PolSAR systems is a linear frequency modulated (LFM) signal, which can be expressed as

$$s\left(t_r\right)=\mathrm{rect}\left({\displaystyle \frac{t_r}{T_p}}\right)\exp \left[j2\pi \left(f_c{t}_r+{\displaystyle \frac{1}{2}}{K}_r{t}_r^2\right)\right]$$

(11)

where t_r is the fast-time variable, corresponding to the range dimension of the image. T_p is the pulse width, f_c is the carrier frequency, $K_r=B/T_p$ is the range chirp rate, and B is the signal bandwidth.

In this study, the PolSAR system is assumed to operate in an airborne broadside stripmap imaging mode. The target is located within the imaging scene, and it reflects the radar-transmitted signal, which is then received by the radar receiver. After down-conversion and filtering, the obtained baseband echo signal is given by

$$\begin{array}{ll}{s}_b\left(t_r,t_{ay}\right)=& s_b\left(t_r\right)\cdot s_b\left(t_{ay}\right)\\ \mathrm{ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}}=& S_{xy}\cdot rect\left({\displaystyle \frac{t_r-2R\left(t_{ay}\right)/c}{T_p}}\right)\exp \left[j\pi K_r{\left(t_r-{\displaystyle \frac{2R\left(t_{ay}\right)}{c}}\right)}^2\right]\\ \mathrm{ \hspace{1em}\hspace{1em} }& \cdot rect\left({\displaystyle \frac{t_{ay}-x_0/v}{T_L}}\right)\exp \left(-{\displaystyle \frac{j4\pi R\left(t_{ay}\right)}{\lambda }}\right)\end{array}$$

(12)

where t_ay denotes the slow-time variable corresponding to the azimuth direction, aligned with the y-polarized transmission. The subscript xy represents the polarization configuration, where the radar transmits with y-polarization and receives with x-polarization. S_xy denotes the polarimetric scattering matrix coefficient of the target under the corresponding polarization configuration, which corresponds to the physical reflection coefficient of the FM-ICP. $R\left(t_{ay}\right)=R_0+{\left(v{t}_{ay}\right)}^2/2R_0$ is the instantaneous slant range between the radar and the target. R₀ is the closest slant range between the target and the radar platform’s flight path. v is the platform velocity, c is the speed of light in free space, x₀ is the azimuth position of the target, T_L is the synthetic aperture time, and λ is the wavelength.

After applying the Range-Doppler (RD) imaging algorithm to the echo signal, the resulting PolSAR images for each polarization channel can be obtained as follows:

$$I_{xy}\left(t_r^{\prime },t_{ay}^{\prime }\right)=S_{xy}\cdot G\cdot \sin c\left(K_r{T}_p{t}_r^{\prime }\right)\cdot \sin c\left(K_a{T}_L{t}_{ay}^{\prime }\right)$$

(13)

where $t_r^{\prime }=t_r-2R_0/c$, $t_{aj}^{\prime }=t_{aj}-x_0/v$, G is the two-dimensional matched filtering gain, and $K_a=2v^2{f}_c/c{R}_0$ is the azimuth doppler chirp rate.

When the FM-ICP is conformally attached to the target and periodic time-varying control signals are applied to the two independently controlled polarization channels, the resulting modulation signals for each channel are shown in Figure 4. The orange waveform represents the H-channel modulation signal, and the blue waveform represents the V-channel modulation signal. The modulation frequency, duty cycle, and amplitude coefficient in the low scattering state for each channel can be independently controlled. This modulation approach is equivalent to modulating the PolSAR signal in the range direction, which requires the modulation period of each channel to be shorter than the pulse width.

Under the modulation of the FM-ICP, the baseband echo signal received and processed by the radar system is given by

$$\begin{array}{ll}{s}_{mb}\left(t_r,t_{ay}\right)& =p_y\left(t_r\right)\cdot s_b\left(t_r\right)\cdot s_b\left(t_{ay}\right)\\ & =p_y\left(t_r\right)\cdot S_{xy}\cdot rect\left({\displaystyle \frac{t_r-2R\left(t_{ay}\right)/c}{T_p}}\right)\exp \left[j\pi K_r{\left(t_r-{\displaystyle \frac{2R\left(t_{ay}\right)}{c}}\right)}^2\right]\\ \mathrm{ \hspace{1em}\hspace{1em} }& \cdot rect\left({\displaystyle \frac{t_{ay}-x_0/v}{T_L}}\right)\exp \left(-{\displaystyle \frac{j4\pi R\left(t_{ay}\right)}{\lambda }}\right)\end{array}$$

(14)

where $p_y\left(t_r\right)$ represents the modulation signal of the y-polarized channel, and its amplitude corresponds to the functional reflection coefficient of the FM-ICP. By substituting the modulation signals expressed in (3) and (4) into (14) and applying imaging processing, the PolSAR imaging results for each channel can be obtained as

$$\left[\begin{array}{c}{I}_{HH}\left(t_r^{\prime },t_{aH}^{\prime }\right)\\ I_{HV}\left(t_r^{\prime },t_{aV}^{\prime }\right)\\ I_{VH}\left(t_r^{\prime },t_{aH}^{\prime }\right)\\ I_{VV}\left(t_r^{\prime },t_{aV}^{\prime }\right)\end{array}\right]=\left[\begin{array}{c}{\displaystyle \sum _{m=-M}^{+M}{S}_{HH}\cdot A_{m-H}G\left(1-\left|\frac{t_r^{\prime }}{T_p}\right|\right)\cdot \sin c\left[K_r\left(T_p-\left|t_r^{\prime }\right|\right)\left(t_r^{\prime }+\frac{m{f}_{sH}}{K_r}\right)\right]}\cdot \sin c\left(K_a{T}_L{t}_{aH}^{\prime }\right)\\ {\displaystyle \sum _{n=-N}^{+N}{S}_{HV}\cdot A_{n-V}G\left(1-\left|\frac{t_r^{\prime }}{T_p}\right|\right)\cdot \sin c\left[K_r\left(T_p-\left|t_r^{\prime }\right|\right)\left(t_r^{\prime }+\frac{n{f}_{sV}}{K_r}\right)\right]}\cdot \sin c\left(K_a{T}_L{t}_{aV}^{\prime }\right)\\ {\displaystyle \sum _{m=-M}^{+M}{S}_{VH}\cdot A_{m-H}G\left(1-\left|\frac{t_r^{\prime }}{T_p}\right|\right)\cdot \sin c\left[K_r\left(T_p-\left|t_r^{\prime }\right|\right)\left(t_r^{\prime }+\frac{m{f}_{sH}}{K_r}\right)\right]}\cdot \sin c\left(K_a{T}_L{t}_{aH}^{\prime }\right)\\ {\displaystyle \sum _{n=-N}^{+N}{S}_{VV}\cdot A_{n-V}G\left(1-\left|\frac{t_r^{\prime }}{T_p}\right|\right)\cdot \sin c\left[K_r\left(T_p-\left|t_r^{\prime }\right|\right)\left(t_r^{\prime }+\frac{n{f}_{sV}}{K_r}\right)\right]}\cdot \sin c\left(K_a{T}_L{t}_{aV}^{\prime }\right)\end{array}\right]$$

(15)

Equation (15) shows that the modulation effect of the FM-ICP on target features in PolSAR images is reflected in two aspects. First, in addition to the scattering center at the original target position, multiple synthetic scattering centers are generated along the range direction, and their positions can be controlled. Second, the original polarization scattering matrix S of the target is scaled by the amplitude coefficients of each channel, making it possible to alter the polarization characteristics by adjusting the modulation parameters.

After modulation by the FM-ICP, the position of the m-th-order harmonic peak along the range direction in the HH and VH channels is

$$R_{rH}={\displaystyle \frac{m{f}_{sH}c}{2K_r}}$$

(16)

The position of the n-th-order harmonic peak along the range direction in the HV and VV channels is

$$R_{rV}={\displaystyle \frac{n{f}_{sV}c}{2K_r}}$$

(17)

The polarization scattering matrix of the zero-order peak, that is, the target at its original location, becomes

$${\mathit{S}}_0=\left[\begin{array}{cc}{S}_{HH}& S_{HV}\\ S_{VH}& S_{VV}\end{array}\right]\left[\begin{array}{cc}{A}_{0-H}& 0\\ 0& A_{0-V}\end{array}\right]=\left[\begin{array}{cc}{S}_{HH}\cdot \left[\left(1-x_H\right){\alpha }_H+x_H\right]& S_{HV}\cdot \left[\left(1-x_V\right){\alpha }_V+x_V\right]\\ S_{VH}\cdot \left[\left(1-x_H\right){\alpha }_H+x_H\right]& S_{VV}\cdot \left[\left(1-x_V\right){\alpha }_V+x_V\right]\end{array}\right]$$

(18)

For harmonic peaks above the zero order, the form of the polarization scattering matrix is determined by the positions of the harmonic components in each channel. When the modulation frequencies of the two polarization channels are the same, the harmonic peaks appear at the same positions. In this case, the polarization scattering matrix of the m-th-order harmonic peak is

$${\mathit{S}}_m=\left[\begin{array}{cc}{S}_{HH}& S_{HV}\\ S_{VH}& S_{VV}\end{array}\right]\left[\begin{array}{cc}{A}_{m-H}& 0\\ 0& A_{m-V}\end{array}\right]=\left[\begin{array}{cc}{S}_{HH}\cdot \left[\left(1-x_H\right){\alpha }_H\sin c\left(m{\alpha }_H\right)\right]& S_{HV}\cdot \left[\left(1-x_V\right){\alpha }_V\sin c\left(m{\alpha }_V\right)\right]\\ S_{VH}\cdot \left[\left(1-x_H\right){\alpha }_H\sin c\left(m{\alpha }_H\right)\right]& S_{VV}\cdot \left[\left(1-x_V\right){\alpha }_V\sin c\left(m{\alpha }_V\right)\right]\end{array}\right]$$

(19)

Therefore, the range positions of the synthetic scattering centers generated in the image are proportional to the modulation frequencies of the corresponding channels (f_sH and f_sV), which means that they are inversely proportional to the modulation periods (T_sH and T_sV). The polarization characteristics of the scattering centers depend on the original target’s polarization scattering matrix and are also influenced by the amplitude coefficients during the low scattering state (x_H and x_V) and the modulation duty cycles (α_H and α_V) of each channel.

Figure 5 illustrates the complete signal processing flow for PolSAR imaging based on the FM-ICP as introduced in this study. The modulation of the FM-ICP must be applied during the radar echo generation stage. After imaging processing, the resulting PolSAR image contains multiple synthetic scattering centers with controllable positions and polarimetric characteristics. By extracting the polarimetric scattering matrices of these scattering centers, their polarimetric features can be determined, thereby verifying the effectiveness of the proposed method.

3. Results

According to theoretical analysis, the modulation effect of the FM-ICP introduces multiple synthetic scattering centers in the PolSAR image. By adjusting the modulation parameters of each channel, the positions and polarization characteristics of these scattering centers can be altered. Based on whether the modulation parameters of the two polarization channels are identical, the modulation method can be classified into joint control and independent control. In this section, we first introduce the evaluation metrics employed in this study, followed by separate simulations of PolSAR imaging results under the modulation of two different methods. These simulations are used to investigate the qualitative variation patterns of the positional and polarimetric characteristics of the synthetic scattering centers with respect to the modulation parameters. Finally, a quantitative analysis is performed.

3.1. Metrics and Parameter Settings

To evaluate the characteristics of target scattering points in the modulated PolSAR images, specific metrics are defined for both the positional and polarimetric characteristics of the scattering points. For the positional characteristics, the distance between the generated synthetic scattering centers and the original target points is extracted, and the relative position error between them is calculated, which is expressed as

$$RPE={\displaystyle \frac{\left|AD-TD\right|}{TD}}\times 100\%$$

(20)

where TD denotes the theoretical distance of the synthetic scattering centers in the image, and AD denotes the actual displayed distance.

For the polarimetric characteristics, the polarimetric scattering matrix of the synthetic scattering centers is first extracted, which contains the complete polarimetric information of the scattering points. To more intuitively demonstrate the capability of the proposed method in controlling polarimetric characteristics, a classification algorithm based on Cameron decomposition is employed to characterize the polarimetric properties of the targets. This algorithm performs coherent decomposition using the target’s polarization scattering matrix, enabling effective identification of deterministic scatterers such as spheres, flat plates, dihedrals, dipoles, and helices. It is widely used in the interpretation of man-made targets.

Table 2. Polarization scattering matrices of typical deterministic scatterers.

Typical Scatterers	Scattering Matrix	Typical Scatterers	Scattering Matrix
trihedral reflectors	${\displaystyle \frac{1}{\sqrt{2}}}\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$	cylindrical scatterers	${\displaystyle \frac{1}{\sqrt{5}}}\left[\begin{array}{cc}2& 0\\ 0& 1\end{array}\right]$
dihedral reflectors	${\displaystyle \frac{1}{\sqrt{2}}}\left[\begin{array}{cc}1& 0\\ 0& -1\end{array}\right]$	narrow dihedral reflectors	${\displaystyle \frac{1}{\sqrt{5}}}\left[\begin{array}{cc}2& 0\\ 0& -1\end{array}\right]$
dipole scatterers	$\left[\begin{array}{cc}1& 0\\ 0& 0\end{array}\right]$	left-handed helices	${\displaystyle \frac{1}{2}}\left[\begin{array}{cc}1& i\\ i& -1\end{array}\right]$

presents the standard polarization scattering matrices of several typical deterministic scatterers. The basic principle of the Cameron algorithm is to classify targets by calculating reciprocity, symmetry, and the degree of matching with scattering models. The degree of matching between the scattering matrix of the synthetic scattering centers and that of various ideal scatterers can be quantified by the coherence coefficient, which is defined as

$$CC\left(\mathit{S},{\mathit{S}}_T\right)={\displaystyle \frac{\left|tr\left(\mathit{S}{{\mathit{S}}_T}^H\right)\right|}{{‖\mathit{S}‖}_F\cdot {‖{\mathit{S}}_T‖}_F}}$$

(21)

where S represents the polarimetric scattering matrix of the synthetic scattering centers in the image, ${\mathit{S}}_T$ represents the polarimetric scattering matrix of an ideal scatterer, $tr\left(\cdot \right)$ denotes the trace of a matrix, the superscript ^H denotes the conjugate transpose, and ${‖\cdot ‖}_F$ denotes the Frobenius norm, i.e., the square root of the sum of the squares of all matrix elements.

Table 3. Simulation parameter settings for PolSAR system.

Parameters	Value	Parameters	Value
squint angle	0°	signal bandwidth	150 MHz
nadir angle	45°	signal pulse width	2 us
platform altitude	1 km	oversampling factor	1.4
platform velocity	200 m/s	pulse repetition frequency (PRF)	350 Hz
center frequency	3 GHz	beamwidth	0.03 rad

provides the simulation parameters of the airborne PolSAR system. The imaging scene is set to a display area of 200 m × 200 m, with the target placed at the center position (0, 0). To ensure non-zero signals in all channels, any zero entries in the polarization scattering matrix for a given target category are replaced with 0.01 during condition setup. The original imaging results without modulation are shown in Figure 6. The extracted polarization characteristics of the scattering centers match the predefined category settings.

3.2. Simulation Experiment Results

3.2.1. Joint Control

Joint control refers to the case where the modulation parameters of the two channels are identical, i.e., the modulation periods (T_sH and T_sV), modulation duty cycles (α_H and α_V), and amplitude coefficients in the low scattering state (x_H and x_V) are the same. According to the theoretical analysis, this modulation method enables both channels to generate synthetic scattering centers, with the scattering centers appearing at the same locations. In addition, the modulation does not alter the energy relationship between the channels, so the category of each scattering center remains consistent with that of the original target. The following section discusses the effects of each modulation parameter in detail.

1.

Modulation periods (T_sH and T_sV)

The modulation duty cycles of both channels are fixed at α_H = α_V = 0.4, and the amplitude coefficients in the low scattering state are set to x_H = x_V = 0.2. The modulation periods (T_sH and T_sV) of the two channels are then varied simultaneously. Figure 7 shows the imaging results under simultaneous modulation period variation, where the HH channel result represents the modulation effect of the H channel, and the HV channel result represents that of the V channel. It can be observed that multiple synthetic scattering centers are generated in each channel, and the spacing between the scattering centers is inversely proportional to the modulation period. By extracting the zero-, first-, and second-order scattering centers from each channel, the classification results are summarized in

Table 4. Classification results under joint control of modulation period for each channel.

Parameter Conditions		Scattering Matrix	Cameron
cylindrical scatterers TsH = TsV = 0.4 us	zero-order peak	$\left[\begin{array}{cc}0.9399+0.0163i& 0.0047+0.0001i\\ 0.0047+0.0001i& 0.4701+0.0081i\end{array}\right]$	cylindrical scatterers
	first-order peak	$\left[\begin{array}{cc}0.4692+0.0915i& 0.0023+0.0004i\\ 0.0023+0.0005i& 0.2340+0.0437i\end{array}\right]$	cylindrical scatterers
	second-order peak	$\left[\begin{array}{cc}0.1369-0.0138i& 0.0007-0.0001i\\ 0.0007-0.0001i& 0.0681-0.0074i\end{array}\right]$	cylindrical scatterers
cylindrical scatterers TsH = TsV = 0.2 us	zero-order peak	$\left[\begin{array}{cc}0.9013-0.0248i& 0.0045-0.0001i\\ 0.0045-0.0001i& 0.4505-0.0123i\end{array}\right]$	cylindrical scatterers
	first-order peak	$\left[\begin{array}{cc}0.3919-0.0725i& 0.0020-0.0004i\\ 0.0020-0.0004i& 0.1957-0.0368i\end{array}\right]$	cylindrical scatterers
	second-order peak	$\left[\begin{array}{cc}-0.0839-0.1132i& -0.0004-0.0006i\\ -0.0004-0.0006i& -0.0423-0.0558i\end{array}\right]$	cylindrical scatterers
trihedral scatterers TsH = TsV = 0.4 us	zero-order peak	$\left[\begin{array}{cc}0.7431+0.0129i& 0.0074+0.0001i\\ 0.0074+0.0001i& 0.7434+0.0128i\end{array}\right]$	trihedral scatterers
	first-order peak	$\left[\begin{array}{cc}0.3709+0.0724i& 0.0037+0.0007i\\ 0.0037+0.0007i& 0.3700+0.0691i\end{array}\right]$	trihedral scatterers
	second-order peak	$\left[\begin{array}{cc}0.1082-0.0109i& 0.0011-0.0001i\\ 0.0011-0.0001i& 0.1076-0.0117i\end{array}\right]$	trihedral scatterers

. It is evident that simultaneous variation of the modulation periods does not alter the energy relationship between the channels, and the category of each scattering center remains consistent with that of the original target.

2.

Modulation duty cycles (α_H and α_V)

With the modulation period fixed at T_sH = T_sV = 0.2 us and the amplitude coefficient in the low scattering state set as x_H = x_V = 0.2 for both channels, the modulation duty cycle (α_H and α_V) is varied. The imaging results under different modulation duty cycles are shown in Figure 8. It can be observed that the duty cycle primarily affects the amplitude of the scattering centers. Notably, when the duty cycle is 0.5, even-order scattering centers disappear. This is because the sinc term corresponding to even-order harmonic peaks becomes zero, which is consistent with the theoretical analysis.

Table 5. Classification results under joint control of modulation duty cycle for each channel.

Parameter Conditions		Scattering Matrix	Cameron
cylindrical scatterers αH = αV = 0.3	zero-order peak	$\left[\begin{array}{cc}0.7656-0.0287i& 0.0038-0.0001i\\ 0.0038-0.0001i& 0.3826-0.0143i\end{array}\right]$	cylindrical scatterers
	first-order peak	$\left[\begin{array}{cc}0.3247-0.0600i& 0.0016-0.0003i\\ 0.0016-0.0003i& 0.1622-0.0305i\end{array}\right]$	cylindrical scatterers
	second-order peak	$\left[\begin{array}{cc}-0.1259-0.1644i& -0.0006-0.0008i\\ -0.0006-0.0008i& -0.0635-0.0811i\end{array}\right]$	cylindrical scatterers
cylindrical scatterers αH = αV = 0.5	zero-order peak	$\left[\begin{array}{cc}1.0717-0.0183i& 0.0054-0.0001i\\ 0.0054-0.0001i& 0.5357-0.0091i\end{array}\right]$	cylindrical scatterers
	first-order peak	$\left[\begin{array}{cc}0.4211-0.1135i& 0.0021-0.0006i\\ 0.0021-0.0006i& 0.2103-0.0573i\end{array}\right]$	cylindrical scatterers
	second-order peak	disappear	disappear
trihedral scatterers αH = αV = 0.3	zero-order peak	$\left[\begin{array}{cc}0.6053-0.0227i& 0.0060-0.0002i\\ 0.0061-0.0002i& 0.6050-0.0226i\end{array}\right]$	trihedral scatterers
	first-order peak	$\left[\begin{array}{cc}0.2567-0.0475i& 0.0026-0.0005i\\ 0.0026-0.0005i& 0.2564-0.0482i\end{array}\right]$	trihedral scatterers
	second-order peak	$\left[\begin{array}{cc}-0.0995-0.1299i& -0.0010-0.0013i\\ -0.0010-0.0013i& -0.1004-0.1283i\end{array}\right]$	trihedral scatterers

presents the classification results of each scattering center, showing that varying the duty cycle simultaneously does not alter the energy relationship between channels, and each scattering center retains the same category as the original target.

3.

Amplitude coefficients in the low scattering state (x_H and x_V)

With the modulation period fixed at T_sH = T_sV = 0.2 us and the modulation duty cycle set as α_H = α_V = 0.4 for both channels, the amplitude coefficient in the low scattering state (x_H and x_V) is varied. Figure 9 shows the imaging results under different amplitude coefficients in the low scattering state. The amplitude coefficient mainly affects the amplitude of the scattering centers. As the amplitude coefficient increases, the amplitude of higher-order peaks decreases accordingly.

Table 6. Classification results under joint control of amplitude coefficient in the low scattering state for each channel.

Parameter Conditions		Scattering Matrix	Cameron
cylindrical scatterers xH = xV = 0.3	zero-order peak	$\left[\begin{array}{cc}1.0122-0.0217i& 0.0051-0.0001i\\ 0.0051-0.0001i& 0.5060-0.0108i\end{array}\right]$	cylindrical scatterers
	first-order peak	$\left[\begin{array}{cc}0.3421-0.0633i& 0.0017-0.0003i\\ 0.0017-0.0003i& 0.1709-0.0322i\end{array}\right]$	cylindrical scatterers
	second-order peak	$\left[\begin{array}{cc}-0.1006-0.0700i& -0.0005-0.0003i\\ -0.0005-0.0003i& -0.0503-0.0347i\end{array}\right]$	cylindrical scatterers
cylindrical scatterers xH = xV = 0.6	zero-order peak	$\left[\begin{array}{cc}1.3451-0.0124i& 0.0067-0.0001i\\ 0.0067-0.0001i& 0.6725-0.0062i\end{array}\right]$	cylindrical scatterers
	first-order peak	$\left[\begin{array}{cc}0.1928-0.0358i& 0.0010-0.0002i\\ 0.0010-0.0002i& 0.0963-0.0184i\end{array}\right]$	cylindrical scatterers
	second-order peak	$\left[\begin{array}{cc}-0.0549-0.0399i& -0.0003-0.0002i\\ -0.0003-0.0002i& -0.0274-0.0197i\end{array}\right]$	cylindrical scatterers
trihedral scatterers xH = xV = 0.3	zero-order peak	$\left[\begin{array}{cc}0.8002-0.0171i& 0.0080-0.0002i\\ 0.0080-0.0002i& 0.8001-0.0171i\end{array}\right]$	trihedral scatterers
	first-order peak	$\left[\begin{array}{cc}0.2704-0.0501i& 0.0027-0.0005i\\ 0.0027-0.0005i& 0.2702-0.0509i\end{array}\right]$	trihedral scatterers
	second-order peak	$\left[\begin{array}{cc}-0.0795-0.0553i& -0.0008-0.0005i\\ -0.0008-0.0006i& -0.0796-0.0549i\end{array}\right]$	trihedral scatterers

presents the classification results of each scattering center, indicating that varying the amplitude coefficient in the low scattering state simultaneously does not change the energy relationship between channels, and each scattering center remains in the same category as the original target.

3.2.2. Independent Control

Independent control refers to the case where the modulation parameters of the two channels differ, meaning one or more of the following parameters are not the same: modulation period (T_sH and T_sV), modulation duty cycle (α_H and α_V), and amplitude coefficient in the low scattering state (x_H and x_V). According to theoretical analysis, when the modulation periods differ, the distribution positions of the synthetic scattering centers generated by each channel are different. Additionally, the modulation alters the energy relationship between channels, resulting in changes in the category of each scattering center relative to the original target. The following discussion focuses on the effect of each modulation parameter.

1.

Modulation periods (T_sH and T_sV)

With the modulation duty cycle fixed at α_H = α_V = 0.4 and the amplitude coefficient in the low scattering state fixed at x_H = x_V = 0.2, the modulation period (T_sH and T_sV) is independently controlled for the two channels. Figure 10 shows the imaging results under different modulation periods for each channel. The distribution positions of the synthetic scattering centers differ between channels. In Figure 10b, the zero-order, first-order, and second-order scattering centers for each channel are marked by red, green, and yellow rectangles, respectively.

Table 7. Classification results under independent control of modulation period for each channel.

Parameter Conditions		Scattering Matrix	Cameron
cylindrical scatterers TsH = 0.4 us, TsV = 0.2 us, H-channel	zero-order peak	$\left[\begin{array}{cc}0.9399+0.0163i& 0.0045-0.0001i\\ 0.0047+0.0001i& 0.4505-0.0123i\end{array}\right]$	cylindrical scatterers
	first-order peak	$\left[\begin{array}{cc}0.4692+0.0915i& 0.0003-0.0001i\\ 0.0023+0.0005i& 0.0321-0.0071i\end{array}\right]$	dipole scatterers
	second-order peak	$\left[\begin{array}{cc}0.1369-0.0138i& 0.0019-0.0002i\\ 0.0007-0.0001i& 0.1892-0.0229i\end{array}\right]$	symmetric scatterers
cylindrical scatterers TsH = 0.4 us, TsV = 0.2 us, V-channel	zero-order peak	$\left[\begin{array}{cc}0.9399+0.0163i& 0.0045-0.0001i\\ 0.0047+0.0001i& 0.4505-0.0123i\end{array}\right]$	cylindrical scatterers
	first-order peak	$\left[\begin{array}{cc}0.1062-0.0203i& 0.0020-0.0004i\\ 0.0005-0.0001i& 0.1957-0.0368i\end{array}\right]$	cylindrical scatterers
	second-order peak	$\left[\begin{array}{cc}0.0823+0.0617i& -0.0006-0.0004i\\ 0.0004+0.0003i& -0.0580-0.0397i\end{array}\right]$	narrow dihedral scatterers

presents the classification results of each scattering center. Independently controlling the modulation periods leads to different scattering center distributions. Scattering centers that appear only in the H or V channel exhibit characteristics that differ from those of the original target.

2.

Modulation duty cycles (α_H and α_V)

With the modulation period fixed at T_sH = T_sV = 0.2 us and the amplitude coefficient in the low scattering state fixed at x_H = x_V = 0.2, the modulation duty cycles (α_H and α_V) are independently controlled for the two channels. Figure 11 shows the imaging results under different modulation duty cycles for the two channels, and

Table 8. Classification results under independent control of modulation duty cycle for each channel.

Parameter Conditions		Scattering Matrix	Cameron
cylindrical scatterers αH = 0.3, αV = 0.7	zero-order peak	$\left[\begin{array}{cc}0.7656-0.0287i& 0.0069-0.0001i\\ 0.0038-0.0001i& 0.6894-0.0057i\end{array}\right]$	trihedral scatterers
	first-order peak	$\left[\begin{array}{cc} 0.3247-0.0600i& 0.0017-0.0003i\\ 0.0016-0.0003i& 0.1711-0.0325i\end{array}\right]$	cylindrical scatterers
	second-order peak	$\left[\begin{array}{cc}-0.1259-0.1644i& 0.0008+0.0008i\\ -0.0006-0.0008i& 0.0812+0.0761i\end{array}\right]$	narrow dihedral scatterers
trihedral scatterers αH = 0.7, αV = 0.3	zero-order peak	$\left[\begin{array}{cc}1.0902-0.0091i& 0.0060-0.0002i\\ 0.0109-0.0001i& 0.6050-0.0226i\end{array}\right]$	cylindrical scatterers
	first-order peak	$\left[\begin{array}{cc}0.2709-0.0504i& 0.0026-0.0005i\\ 0.0027-0.0005i& 0.2564-0.0482i\end{array}\right]$	trihedral scatterers
	second-order peak	$\left[\begin{array}{cc}0.1301+0.1217i& -0.0013-0.0010i\\ 0.0013+0.0012i& -0.1279-0.0975i\end{array}\right]$	dihedral scatterers

presents the classification results of each scattering center. Independently controlling the modulation duty cycles alters the energy relationship between channels, causing the scattering centers to exhibit characteristics that differ from those of the original structure.

3.

Amplitude coefficients in the low scattering state (x_H and x_V)

With the modulation period fixed at T_sH = T_sV = 0.2 us and the modulation duty cycle fixed at α_H = α_V = 0.4, the amplitude coefficients in the low scattering state (x_H and x_V) are independently controlled for the two channels. Figure 12 shows the imaging results under different amplitude coefficients in the low scattering state for the two channels, and

Table 9. Classification results under independent control of amplitude coefficient in the low scattering state for each channel.

Parameter Conditions		Scattering Matrix	Cameron
cylindrical scatterers xH = 0.2, xV = 0.8	zero-order peak	$\left[\begin{array}{cc}0.9013-0.0248i& 0.0078-0.0000i\\ 0.0045-0.0001i& 0.7834-0.0031i\end{array}\right]$	trihedral scatterers
	first-order peak	$\left[\begin{array}{cc}0.3919-0.0725i& 0.0005-0.0001i\\ 0.0020-0.0004i& 0.0465-0.0092i\end{array}\right]$	dipole scatterers
	second-order peak	$\left[\begin{array}{cc}-0.0839-0.1132i& 0.0000-0.0001i\\ -0.0004-0.0006i& 0.0009-0.0130i\end{array}\right]$	dipole scatterers
trihedral scatterers xH = 0.8, xV = 0.2	zero-order peak	$\left[\begin{array}{cc}1.2388-0.0049i& 0.0071-0.0002i\\ 0.0124-0.0000i& 0.7123-0.0195i\end{array}\right]$	cylindrical scatterers
	first-order peak	$\left[\begin{array}{cc}0.0737-0.0138i& 0.0031-0.0006i\\ 0.0007-0.0001i& 0.3095-0.0582i\end{array}\right]$	symmetric scatterers
	second-order peak	$\left[\begin{array}{cc}-0.0193-0.0157i& -0.0009-0.0006i\\ -0.0002-0.0002i& -0.0916-0.0628i\end{array}\right]$	symmetric scatterers

presents the classification results of each scattering center. It can be observed that independently controlling the amplitude coefficients in the low scattering state alters the energy relationship between the channels, resulting in scattering centers exhibiting characteristics that differ from those of the original structure.

In summary, the PolSAR images modulated by the FM-ICP can generate synthetic scattering centers with tunable positions and polarimetric characteristics. Specifically, the modulation periods of each channel (T_sH and T_sV) determine the positions of the scattering centers, with the spacing between scattering centers inversely proportional to the respective modulation periods. The modulation duty cycles (α_H and α_V) and amplitude coefficients in the low scattering state (x_H and x_V) determine the intensity of the scattering centers in each channel, thereby influencing their polarimetric characteristics. For the joint control method, each channel generates a series of scattering centers whose polarimetric characteristics match those of the original target. In contrast, for the independent control method, the energy relationship between the channels is altered, and the resulting scattering centers exhibit polarimetric characteristics that go beyond the original geometric structure—for instance, a cylinder may be transformed into a trihedral or a dipole.

3.3. Quantitative Analysis of Features

The experimental results in Section 3.2 verified that FM-ICP modulation can control both the positional and polarimetric characteristics of synthetic scattering centers in PolSAR images. This section provides a quantitative assessment of the degree of such control.

For the positional characteristics,

Table 10. Distance errors of synthetic scattering centers generated by FM-ICP modulation.

Parameter Conditions		Theoretical Distance (m)	Actual Distance (m)	Relative Error
Ts = 0.4 us	first-order peak	5.00	5.60	0.12
	second-order peak	10.00	12.67	0.27
Ts = 0.2 us	first-order peak	10.00	11.20	0.12
	second-order peak	20.00	24.33	0.22

presents the relative position errors between the distances of synthetic scattering centers generated in PolSAR images under two typical parameter settings and their theoretical distances, calculated based on Equation (20). It can be observed that all relative errors are below 0.3. It is worth noting that the resolution of the PolSAR system used in this simulation is 1.5 m; for PolSAR systems with relatively lower resolution, the corresponding relative errors would further decrease. Therefore, FM-ICP modulation demonstrates the capability of controlling the positional characteristics of synthetic scattering centers.

For the polarimetric characteristics, the coherence coefficients between the polarimetric scattering matrices of the synthetic scattering centers in Section 3.2 and those of the corresponding ideal scatterers were calculated based on Equation (21). To avoid redundancy,

Table 11. Coherence coefficients between various categories of synthetic scattering centers generated by FM-ICP modulation and ideal scatterers.

	Dipole	Trihedral	Cylindrical	Dihedral	Narrow Dihedral
Coherence coefficient	0.9941	0.9975	0.9982	0.9973	0.9906

lists the minimum coherence coefficient for each category of synthetic scattering center, i.e., the relatively poorer results. All coherence coefficients are above 0.9, indicating that FM-ICP modulation can control the polarimetric characteristics of synthetic scattering centers.

In summary, FM-ICP can generate multiple synthetic scattering centers in PolSAR images, whose positional characteristics—specifically, the distances from the original targets—can be adjusted by controlling the modulation period of the system. The polarimetric categories exhibited by these scattering centers can be modified by adjusting the modulation duty cycle and amplitude coefficient of the system, as illustrated in Figure 13.

4. Discussion

4.1. Analysis of the Transformation Pattern of Polarimetric Features

According to the simulation results, by altering the modulation duty cycles (α_H and α_V) and the amplitude coefficients in the low scattering state (x_H and x_V) of each channel, the polarimetric characteristics of the scattering centers may change relative to the original geometric structure of the target. In the theoretical analysis section, Equations (18) and (19) provide the forms of the polarimetric scattering matrices of the synthetic scattering centers. Based on both the theoretical analysis and simulation results, this section discusses the variation patterns of the polarimetric characteristics.

Figure 14 presents the theoretical classification results of the zero-order, first-order, and second-order scattering centers under several parameter conditions. For clarity and comparative analysis, two typical target categories—a cylindrical scatterer and a trihedral scatterer—are selected as the original reference conditions. The polarimetric scattering matrix of the cylindrical scatterer is ${\displaystyle \frac{1}{\sqrt{5}}}\left[\begin{array}{cc}2& 0\\ 0& 1\end{array}\right]$, while that of the trihedral scatterer is ${\displaystyle \frac{1}{\sqrt{2}}}\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$. The feature of the former is that the energy in the H channel is twice that in the V channel, whereas the latter is characterized by equal energy in the H and V channels.

Figure 14a shows the classification results of scattering centers at different harmonic orders as a function of the V-channel modulation duty cycle α_V, under the condition that the original target is a cylindrical scatterer, the amplitude coefficients in the low scattering state are x_H = x_V = 0.2, and the H-channel duty cycle is fixed at α_H = 0.3. It can be observed that when α_V reaches approximately twice α_H or higher, the zero-order scattering center changes from a cylindrical to a trihedral type. Correspondingly, Figure 14b presents the results when the original target is a trihedral scatterer, x_H = x_V = 0.2, and α_H = 0.7. It shows that when α_V is approximately half of α_H or lower, the zero-order scattering center changes from a trihedral to a cylindrical type. Therefore, the modulation duty cycle of each channel is proportional to its energy level and plays a decisive role in the classification result of the zero-order scattering center. In addition, the first-order scattering centers remain consistent with the originally set target category, while the second-order scattering centers are more sensitive to changes in the duty cycle and can traverse categories such as dipole and narrow dihedral scatterers. These results correspond to the simulation outcomes in Table 8, confirming the consistency between theoretical predictions and experimental observations.

Figure 14c presents the classification results of scattering centers at various harmonic orders as a function of x_V, with the original target being a cylindrical scatterer, α_H = α_V = 0.4, and x_H = 0.2. When x_V reaches 0.7 or above, the zero-order scattering center changes from a cylindrical to a trihedral type. Similarly, Figure 14d presents the results for a trihedral scatterer with α_H = α_V = 0.4 and x_H = 0.8. It shows that when x_V decreases to 0.3 or below, the zero-order scattering center changes from a trihedral to a cylindrical type. Hence, the amplitude coefficient in the low scattering state of each channel is proportional to its energy level and has a decisive influence on the classification of the zero-order scattering center. Moreover, the classification results for the first- and second-order scattering centers remain the same. These results are consistent with those shown in Table 9 from the simulation section, further validating the theoretical analysis.

To conclude, the FM-ICP enables control of the polarization characteristics of scattering centers by independently adjusting the modulation parameters of each channel. The core principle lies in altering the energy relationship between channels through changes in the modulation duty cycle and the amplitude coefficient in the low scattering state. The energy of each channel is proportional to both its duty cycle and amplitude coefficient. By configuring the energy distribution in proportional relationships, the target can exhibit characteristics that go beyond its original geometric structure. The polarization characteristics of both the first- and second-order scattering centers can be controlled, manifesting as either the same or different categories compared to the original target.

4.2. Analysis of the Clutter Interference Effect

In the experimental section, the modulation effect of the FM-ICP was analyzed under ideal conditions without background clutter and noise. To further validate the effectiveness of the proposed method, this section simulates a scenario including realistic noise and clutter. The ground clutter is modeled as complex Gaussian noise with Rayleigh-distributed amplitude, while thermal noise is independent of clutter. The signal-to-noise ratio is set to 10 dB. Figure 15 shows the imaging results under different signal-to-clutter ratios (SCRs).

Overall, due to the relatively low energy in cross-polarized channels, the scattering centers in these channels are submerged in clutter when clutter is added with reference to the co-polarized channel energy, while the co-polarized channels are less affected. After extracting the corresponding scattering centers from the co-polarized channels and performing classification, it is observed that when the SCR is above −10 dB, the classification results of all scattering orders are consistent with those under ideal conditions. However, when the SCR drops below −10 dB, second-order and higher-order scattering centers are obscured by clutter and their features can no longer be controlled.

In summary, clutter and noise do not impair the FM-ICP’s ability to control the energy distribution between channels, and thus do not affect its feature modulation capability. Under a certain SCR, the FM-ICP can still achieve flexible control over the zero-order, first-order, and second-order peaks. When the SCR significantly affects the detection of scattering centers, the synthetic scattering centers generated by the FM-ICP are suppressed by clutter.

4.3. Analysis of the Impact of FM-ICP Performance

The proposed method is implemented based on the FM-ICP. Its actual operating bandwidth and the switching speed of its diodes both affect the modulation performance. When the operating bandwidth of the FM-ICP is narrower than that of the imaging radar system, its electromagnetic wave absorption functionality is only effective for part of the frequency spectrum. This situation is equivalent to an energy variation in absorption across the system bandwidth, which can be modeled as a change in the amplitude coefficient in the low scattering state of the equivalent FM-ICP modulation waveform. The effect of this parameter on modulation performance has already been analyzed in the experimental section and will not be repeated here.

The diode switching speed primarily affects the modulation period of the equivalent FM-ICP waveform. For practical PIN diodes, the switching time can reach the nanosecond level, which is sufficient to meet the requirements of most imaging radar systems. On the other hand, when the modulation period is relatively long and cannot be matched to the intra-pulse time of PolSAR radar, it can be adjusted to match the inter-pulse time, i.e., to the order of the PRI. Under this condition, the synthetic scattering centers generated by the FM-ICP will appear distributed along the azimuth direction. Figure 16 presents the PolSAR imaging result under this scenario. Since the modulation period does not affect the decomposition of polarimetric characteristics, the scattering features of different orders are determined solely by the modulation duty cycle and the amplitude coefficient in the low scattering state. Under the corresponding parameters, the results remain consistent with those in the experimental section, confirming the continued effectiveness of the proposed method for polarimetric feature modulation.

5. Conclusions

In summary, this study proposes a method for modulating PolSAR images using a flexible metasurface with independently controllable polarizations (FM-ICP). This method enables independent control of the echo energy distribution in dual-polarization channels, thereby achieving joint modulation of the polarimetric scattering characteristics of targets and effectively overcoming the limitations of existing modulation approaches in the polarization domain. After modulation, the resulting PolSAR image not only retains the scattering centers of the original target but also generates synthetic scattering centers with precisely controllable positions and polarimetric characteristics. By adjusting the modulation parameters, the scattering types of these centers can be flexibly tuned to match or differ from those of the original target.

This study provides a solid theoretical foundation and a practical methodological framework for active control of target features in PolSAR images, from the novel perspective of altering inter-channel energy relationships. It holds broad application potential in various areas of PolSAR image processing. In feature reconstruction, it can be used to accurately reconstruct the scattering characteristics of specific regions of interest (ROIs), thereby improving the completeness and usability of image information. In target simulation, it offers an effective tool for generating realistic synthetic PolSAR scenes with specified scattering characteristics, which is valuable for algorithm validation and target recognition system training. In addition, this method introduces a new approach for actively controlling the polarimetric response of radar targets, contributing to a deeper understanding of different polarimetric scattering mechanisms and their interrelationships. Future research will focus on applying this method to more complex real-world target scenarios, exploring its scalability and adaptability in fully polarimetric systems, and systematically evaluating its impact on various target recognition algorithms during the modulation process.