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Abstract
The vortex electromagnetic wave has shown great prospects of radar applications, due to the orbital angular momentum (OAM) degree of freedom. However, the radiation energy convergence of the OAM beam remains a hard problem to be solved for radar target imaging in realistic scenario. In this paper, an OAM beam generation method is developed exploiting the OAM and waveform degrees of freedom simultaneously, which can collimate the beams with different OAM modes. Furthermore, the echo demodulation and the imaging methods are proposed to reconstruct the target profiles in the range and azimuth domain. Simulation and experimental results both validate that the OAM-based radar imaging can achieve azimuthal super-resolution beyond the diffraction limit of the array aperture. This work can advance the system design of vortex electromagnetic wave radar and its real-world applications.
© 2024 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Inspired by the optical orbital angular momentum (OAM) [1], the electromagnetic (EM) vortex as a type of structured EM field has been widely researched at radio frequencies [2,3], in recent years. Similar to optical vortex [4,5], the EM vortex beam carrying OAM has helical phase structure and singularity with respect to the propagation axis. Hitherto, there are already some articles on the EM vortex generation [6,7], OAM antenna design [8], OAM-based wireless communications [9], and vortex wave target detection [10–12].
The radar transmitting vortex EM wave for target detection is usually called vortex EM wave radar or OAM-based radar. Due to the distinguished phase distribution characteristics of the EM vortex [13], the OAM-based radar can benefit the target recognition in the cross-range domain, and has the potential to achieve super-resolution beyond the diffraction limit of the antenna aperture. However, the phase singularity of EM vortex leads to the doughnut-shaped energy distribution, which limits the radar system operation distance, for vehicular-borne imaging applications in realistic scenario [14,15].
Currently, some articles are focusing on the array design to collimate the beam, such as concentric uniform circular arrays (CUCAs) [16], helical circular array [17], and spherical conformal array [18]. These array design approaches are effective, but causing great complexity for a radar system. In contrast, the waveform is another degree of freedom to design the radiation energy distribution of OAM beams and solve the radar illumination problem, which can easily be implemented in reality and reduce the system cost. In this paper, we exploit the OAM and waveform degrees of freedom simultaneously to generate a new form of vortex wave with converged energy distribution, and develop a super-resolution imaging method. The generation method of OAM beam with converged energy distribution is proposed based on the phase coded waveform, and the influence of the antenna pattern is analyzed. Then, the echo signal scattered by the point target is derived, and the imaging algorithm is proposed. Simulation and proof-of-concept experiments are carried out to demonstrate the effectiveness of the proposed method.
2. Energy-converged vortex wave generation
Typically, the uniform circular array (UCA) is usually applied to generate vortex EM wave, due to its symmetric characteristic. To design the transmitted signal for vortex EM wave radar imaging, the UCA is selected as the array configuration in this paper. Without loss of generality, N antennas are equally located along the circle and the azimuthal angle of each antenna is denoted as ${\varphi _n}$, shown in Fig. 1.
To generate vortex EM wave with converged energy, the gradient phase modulation and the phase coded waveform are designed together for each antenna feeding. For an arbitrary point $P(r,\theta ,\phi )$ in the space, the transmitted signal ${s_n}(k,l)$ of the $n\textrm{th}$ antenna can be written as
Based on Eq. (1), the total radiation signal ${s_t}(k,l)$ of the UCA is
To show the intensity distribution characteristics of the designed field, simulations are conducted and the main parameters are set in Table 1. In Fig. 2, the accumulated radiation signal over one waveform cycle is sampled on a plane parallel to the x-o-y plane, which is 50 m away along the z axis. And the sampled plane size is 80m × 80 m in the x and y direction, respectively. It can be seen from Fig. 2 that the accumulated radiation energy is converged with the design of the phase coded waveform, and the converged area is mainly decided by the antenna pattern, which is much different from the conventional OAM beam without waveform design in Fig. 2(d). Moreover, the conventional intensity distribution usually suffers from the influence of the superposition OAM modes for the area away from the beam axis, whereas the proposed generation method is not affected.
Fig. 2. Intensity distribution with different antenna pattern for OAM mode l = 3. (a) f1, (b) f2, (c) f3, (d) Conventional OAM beam with f3.
Table 1. Main parameters for signal generation
Furthermore, the intensity distribution does not change much for different OAM modes, as shown in Fig. 3(a) and (b), which indicates that the potential imaging area can be always illuminated by the incident wave carrying different OAM modes. Other array configuration parameters, e.g., array radius and antenna number, have little influence on the radiation energy distribution. Thus, for one UCA, the OAM modulation combined with the coded phase waveform can provide a solution to solve the illumination problem in radar applications, especially for different OAM modes. As for the phase distribution, the instantaneous field of the proposed vortex wave changes as a function of the waveform in one cycle. However, the regular phase distribution with respect to conventional OAM beams can be recovered by the demodulation method in the next section.
Fig. 3. Intensity and phase distribution with different OAM modes for the antenna pattern f3. (a) Intensity distribution for l = 1, (b) Intensity distribution for l = 2, (c) Instantaneous phase distribution for l = 1, (d) Instantaneous phase distribution for l = 2.
3. Imaging method and experimental results
When the designed vortex EM wave illuminates the target area, one isotropous antenna placed at the coordinate origin is used to receive the target echo, as shown in Fig. 1. Based on Eq. (2), the echo signal ${s_r}(k,l)$ scattered by the target located at the position ${P_\textrm{T}}({r_\textrm{T}},{\theta _\textrm{T}},{\phi _\textrm{T}})$ can be given by
To demodulate the echo signal, the reference matrix $\vec{C}$ is designed according to the waveform matrix
After multiplying the echo signal ${s_r}(k,l)$ by the reference matrix $\vec{C}$, it leads to [6]
Based on Eq. (7), the two dual relationships, i.e., the signal frequency and the target range, the OAM mode and the target azimuth, can be exploited to reconstruct the target’s range and azimuth profiles, respectively. Thus, the proposed imaging algorithm is provided in Table 2.
Table 2. The proposed imaging algorithm
In Fig. 4, simulations are carried out to show the effectiveness of the proposed method. According to the array configuration in Table 1, the OAM modes $l \in [ - 7,7]$ can be applied for imaging, and the signal bandwidth is set as 300 MHz, from 9.5 GHz to 9.8 GHz. In the simulation, two point targets are placed in the imaging area with the coordinates ${P_1}(110\textrm{m},3^\circ ,30^\circ )$ and ${P_2}(110\textrm{m},3^\circ ,50^\circ )$, respectively. Results in Fig. 4(a) show that the two targets can be well reconstructed by OAM-based radar imaging, whereas the conventional real aperture imaging cannot distinguish the two targets, shown in Fig. 4(b) and Fig. 4(c).
Fig. 4. Two-dimensional imaging results. (a) OAM-based radar imaging, (b) Conventional real aperture imaging, (c) Comparison of the azimuthal profiles.
In theory, the cross-range angle resolution ${\delta _a}$ for real aperture imaging can be estimated as
where $\lambda $ and D are the signal wavelength and the array diameter, respectively.In contrast, according to Fig. 4(c), the cross-range angle resolution ${\delta ^{\prime}_a}$ for OAM-based radar imaging is
where $\Delta \phi $ denotes the azimuthal angle along the circle with respect to the beam axis, and $\theta $ is the target’s elevational position.Based on Eqns. (8) and (9), it is shown that about 5.6 times resolution improved results can be achieved by OAM-based radar. The range resolution is decided by the signal bandwidth, which is the same as traditional radar systems. The cross-range angle resolution is mainly decided by the number of OAM modes and the target’s elevation position [19]. To further show the advantages of the proposed OAM radar imaging with phase coded waveform, the imaging results considering the influence of noise are presented in Fig. 5. In the simulation, the signal-to-noise ratio (SNR) is set as 5 dB. Compared to the OAM radar imaging without waveform design, the target profiles can still be well reconstructed by the proposed method, which is more robust against the noise influence. Considering the modulation of the transmitted signal aforementioned, for the demodulation process, it is equal to the echo signal accumulation, and thus the echo SNR can be enhanced.
Finally, the proof-of-concept experiments are also performed, shown in Fig. 6. The operation frequency of the imaging system is from 9.5 GHz to 9.8 GHz. In the imaging scenario, two corner reflectors are placed in the target area to be detected. The distance between radar and the target is about 6.5 m. In Fig. 7, the imaging results of the conventional real aperture and the proposed method are compared. It can be seen from Fig. 7 that the cross-range angle resolution for OAM-based imaging is superior to conventional real aperture imaging, and about 5.1 times resolution improved results can be achieved, as shown in Fig. 7(c). This experimental result demonstrates that vortex wave modulation can make a breakthrough on the antenna aperture resolution limit.
Fig. 7. The imaging results of proof-of-concept experiment. (a) OAM-based radar imaging, (b) Conventional real aperture imaging, (c) Comparison of the azimuthal profiles.
4. Conclusion
In summary, the vortex EM wave radar imaging method exploiting the OAM and waveform degrees of freedom has been studied. The radiation signal generation was studied by designing the waveform, which can converge the radiation energy for different OAM modes. The target echo was demodulated and the imaging model was built. Experimental imaging results validated the effectiveness of the proposed method, which can achieve 5.1 times improved angle resolution compared to conventional real aperture imaging. In future, this work can be exploited in the applications of forward-looking imaging, e.g., unmanned vehicles and helicopter’s landing assistance, which can enhance the cross-range angle resolution of traditional sensing systems [20,21].
Funding
National Key Research and Development Program of China (2022YFB3902400); National Natural Science Foundation of China (62322122, 62171446, 61921001).
Disclosures
The authors declare that there are no conflicts of interest related to this article.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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