1. Introduction

In the traditional meaning, the information modulation methods mainly focus on the time domain, frequency domain, and polarization domain. With the rapid development of information technology, a new degree of freedom has been exploited, which is the orbital angular momentum (OAM) of the electromagnetic (EM) wave. The wave front of EM wave carrying OAM is in helical shape so that it is called OAM vortex wave. The phase of OAM vortex wave equals to l times of its azimuthal angle, where l represents the OAM order. The OAM vortex waves with different orders are orthogonal to each other and can transfer information independently [1]. Thus, it is expected to enhance the communication capacity using OAM multiplexing [2–5]. Due to its promising application prospect, the generation method of OAM vortex wave has drawn lots of attentions. At present, array antenna [6–8], traveling wave antenna [9,10], and metasurface antenna [7,11–14] etc. are the main approaches for OAM vortex wave generation at microwave bands. It is not a problem any more to generate OAM vortex wave.

Along with the deepening of research, the scattering characteristic of OAM vortex wave has gradually attracted attentions. The special wave front structure makes the interaction between the targets and OAM vortex wave quite different from the plane wave. In [15], the target scattering characteristics for OAM-based radar was discussed. The OAM-based radar cross section (ORCS) was defined by modifying the conventional radar equation. Analytical studies demonstrated that the mirror-reflection phenomenon disappeared and peak values of ORCS were in the non-specular direction for metal plate and cylinder targets. In [16], the backward scattering characteristics of the perfect electrical conductor (PEC) sphere and PEC cone were studied. It was concluded that given a specific propagation direction, compared to plane waves, more information would be offered by the OAM beams for object detection and recognition. In [17], experiments showed that different radar cross sections (RCSs) of a specific complex target can be obtained by different OAM waves and the signal-to-noise ratio gain can be achieved due to the appropriate OAM mode selection. OAM vortex waves were decomposed into plane waves in the spectral domain with different elevation and azimuth angles in [18]. The results revealed that the RCS of the dielectric cylinder increased for the OAM wave incidence compared with plane wave incidence, and OAM mode number had a significant effect on the RCS. Therefore, it was concluded that the OAM can promisingly be used to enhance the radar target detection capability. Besides these conventional targets, metamaterial target is also taken into consideration. In [19], the scattering of OAM vortex wave by a perfect electromagnetic conductor (PEMC) sphere using Mie Theory was conducted. The result showed that the co and cross polarized RCS always showed opposite behavior for PEMC sphere. Accordingly, Laguerre–Gaussian (LG) beam scattering by a PEMC cylinder coated with a metamaterial was analyzed in [20]. This research work contained enough information to calculate the RCS by metamaterials for an incident LG beam. The research in [21] focused on the metasurface stealth target (MST). It was found that the echo of MST had a strong correlation with OAM order l. Moreover, a chessboard metasurface (CBM) was taken as an example to illustrate the backscattering enhancement phenomenon by OAM detection approach compared to the plane wave incidence case. This work uncovered the promising application for MST detection using OAM vortex wave.

Phase modulated metasurface (PMM) such as CBM and phase gradient metasurface (PGM) is broadly used to control the transmission state of EM waves. In [22], a polarization-insensitive PGM was proposed to deflect the different LP incident waves to the same direction. In addition, by introducing phase-corrected gradient (PCG) on the metasurface platform, lossless transformation from the incidence to far-field patterns was obtained in [23]. However, the research on PMM is mainly focused on its interaction with plane wave. In this paper, we have further studied the interaction between PMM and the OAM vortex wave. In comparison to [21], the mathematical model has been modified by taking into consideration of oblique incidence case, which is more common in real scenarios. Then, the scattering characteristics of CBM and PGM are analyzed under incidence of OAM vortex wave with different angles and OAM orders. The interaction between PMM and OAM vortex wave may find promising applications in radar detection field, especially for MST detection problem.

2. Mathematical model

The interaction scenario between OAM vortex wave and PMM is shown in Fig. 1. The metasurface is composed of M × N unit cells denoted by (m, n). A vortex source radiating OAM vortex wave is located above the metasurface. As shown in the figure, θ_inc and φ_inc represent the incident elevation and azimuthal angle of OAM vortex wave, respectively. θ(m, n) is the angle between the axis of the vortex source and unit cell (m, n). The direction of scattered wave is $\mathrm{\hat{u}}$ and it can be calculated by $\hat{u} = {\hat{e}_x}\sin \theta \cos \varphi + {\hat{e}_y}\sin \theta \sin \varphi + {\hat{e}_z}\cos \theta$, where θ and φ are the elevation angle and azimuthal angle in global xyz coordinate system, respectively.

 

Fig. 1. The interaction scenario between OAM vortex wave and PMM.

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According to the array antenna theory, the total scattered wave is the superposition of the waves reflected by all the unit cells of the metasurface. Thus, it is obtained that

Then, we can obtain

When a uniform circular array (UCA) is adopted as the vortex source, the incoming wave at unit cell $(m^{\prime},n^{\prime})$ can be expressed as [21]

In (5), W is the number of the elements in UCA, $\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} (m^{\prime},n^{\prime})$ and ${\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} _w}$ are the position vector of unit cell $(m^{\prime},n^{\prime})$ and wth element of UCA, respectively. ${\varphi _w} = 2\pi w/W$ is the azimuthal angle of wth element with respect to the axis and a is the radius of the UCA. $R(m^{\prime},n^{\prime})$ is the distance between vortex source center and unit cell $(m^{\prime},n^{\prime})$. ${J_l}$ is the lth order Bessel function of the first kind and ${\cos ^{{q_e}}}\theta (m^{\prime},n^{\prime})$ is the element pattern at unit cell $(m^{\prime},n^{\prime})$, where q_e = 1. Using (1)∼(5), the scattering characteristic can be analyzed in detail.

3. Scattering characteristic of CBM

CBM is usually used to reduce the RCS of a target due to the phase cancellation scheme. A CBM divided into four regions is adopted to study the scattering characteristic as shown in Fig. 2. The modulated phase of the adjacent regions is coded by “0” and “1”, of which the phase difference is π. There are 30 × 30 unit cells in total and each area contains 15 × 15 of them. The period of each unit cell is 20mm. A UCA composed of 16 microstrip antenna elements with the radius of 3λ is utilized to generate OAM vortex waves, where λ is the wavelength at the working frequency of 4.8GHz. The distance between the UCA and CBM is 10λ.

 

Fig. 2. The structure of a CBM.

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The scattering patterns of the CBM under different incident angle ${\theta _{inc}}$ (${\varphi _{inc}} = 0$) and OAM orders are shown in Fig. 3. The first row of the figure represents the plane wave case of l = 0. When the incident angle ${\theta _{inc}} = {0^\textrm{o}}$, the scattering pattern has four peaks around the normal direction of the metasurface due to the phase cancellation scheme. As a result, the monostatic RCS can be reduced significantly. With the increase of ${\theta _{inc}}$, the four scattering peaks still exist but they are deflected to the specular direction. The same deflection phenomenon can be observed for other cases as shown in each row of the figure. In the first column, the incident angle keeps unchanged while the OAM order l grows bigger from above to below. There are still four peaks for the scattering patterns of l = ±1, ± 3, and ±4, which reveals that the phase cancellation scheme still works for these OAM modes. Compared to plane wave case, the peaks are extended to four arms and these arms become bigger with the increasing of OAM order. This is reasonable because the divergence level increases with the growing of OAM order. For ${\theta _{inc}} = {15^\textrm{o}}$ and ${\theta _{inc}} = {30^\textrm{o}}$, these arms are deflected to the specular direction and become more and more disordered. However, for l = ±2 case, only one scattering peak is observed instead of four. This peak appears at the normal direction for ${\theta _{inc}} = {0^\textrm{o}}$, indicating that there is strong backward scattering energy. This phenomenon is against the stealth scheme of CBM. It can be inferred that the monostatic RCS will be increased to a great extent. The reason of such phenomenon lies in the phase superposition of CBM and the OAM vortex wave of l = ±2 [21]. For ${\theta _{inc}} = {15^\textrm{o}}$ and ${\theta _{inc}} = {30^\textrm{o}}$, the scattering pattern still has only one peak occurring at the specular direction, manifesting that the scattering energy is also focused instead of dispersed. It indicates that the bistatic RCS of the CBM can be enhanced greatly.

 

Fig. 3. The normalized scattering pattern of CBM under illumination of OAM vortex wave with different orders and incident angles. (a)-(c) l = 0, (d)-(f) l = ±1, (g)-(i) l = ±2, (j)-(l) l = ±3, (m)-(o) l = ±4.

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4. Scattering characteristic of PGM

PGM can realize abnormal reflection by the phase gradient at certain direction, which obeys the generalized Snell’s law. A typical PGM with the phase difference of π/3 is shown in Fig. 4. According to the generalized Snell’s law, the incident angle ${\theta _{inc}}$ and reflection angle ${\theta _{ref}}$ satisfy

 

Fig. 4. The PGM with phase difference of π/3.

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The scattering pattern of this PGM with OAM vortex wave incidence under different orders and incident angles are shown in Fig. 5, including the magnitude and phase distributions. In the simulation, $d\phi = \pi /3$, $dx = 100mm$ and the working frequency is 4.8GHz. When the incident angle ${\theta _{inc}} = {0^\textrm{o}}$, the scattered waves are deflected away from the normal direction due to the phase gradient of the metasurface. For l = 0, the scattered wave has one peak, which are in accord with the plane wave case. For l ≠ 0, the scattered waves still keep the characteristic of doughnut-like shape and helical phase. However, the energy and phase distributions get disturbed with the increase of OAM order l and incident angle ${\theta _{inc}}$. The two-dimensional scattering patterns at $\varphi = 0$ plane are depicted in Fig. 6. Notice that all the patterns are normalized to the l = 0 case. It is found that the scattered beams are deflected by 6° and 22° for ${\theta _{inc}} = {0^\textrm{o}}$ and ${\theta _{inc}} = {15^\textrm{o}}$, respectively. The theoretical deflection angle is 6° and 21° according to (6). The results obtained by mathematical model agree well with the theoretical prediction, which demonstrates that the generalized Snell’s law is also suitable to the OAM vortex wave.

 

Fig. 5. The PGM scattering patterns of normalized magnitude and phase under OAM vortex wave with different orders and incident angles. (a)-(d) l = 0, (e)-(h) l = ±1, (i)-(l) l = ±2, (m)-(p) l = ±3, (q)-(t) l = ±4.

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Fig. 6. The two-dimensional scattering patterns at plane of φ=0 under the incidence angles of (a) ${\theta _{inc}} = {0^\textrm{o}}$ and (b) ${\theta _{inc}} = {15^\textrm{o}}$.

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5. Conclusion

The scattering characteristic of OAM vortex wave at radio bands has gradually attracted attentions. The interaction targets include the metal plate, PEC cone, PEMC and so on. We furtherly studied the interaction between PMM and OAM vortex wave in this paper. The array theory was adopted to establish the mathematical model, which can be utilized to compute the scattering pattern under arbitrary incident angles and OAM orders. CBM and PGM are taken as examples to study the scattering characteristics. For CBM, it is found that the stealth scheme can be broken when OAM order l equals to ±2 under both normal incidence and oblique incidence. For PGM, the computed results demonstrate that we can still use the generalized Snell’s law to predict the reflection direction of OAM vortex wave and the reflected wave still keeps the OAM features. Although the scattering characteristics are only considered for CBM and PGM, the proposed method can also be used for other types of PMM. The findings of this paper have enriched the scattering scenario of OAM vortex wave and may find applications for PMM targets detections.