The target scattering characteristics are crucial for radar systems. However, there is very little study conducted for the recently developed orbital angular momentum (OAM) based radar system. To illustrate the role of OAM-based radar cross section (ORCS), conventional radar equation is modified by taking characteristics of the OAM waves into account. Subsequently, the ORCS is defined in analogy to classical radar cross section (RCS). The unique features of the incident OAM-carrying field are analyzed. The scattered field is derived, and the analytical expressions of ORCSs for metal plate and cylinder targets are obtained. Furthermore, the ORCS and RCS are compared to illustrate the influences of OAM mode number, target size and signal frequency on the ORCS. Analytical studies demonstrate that the mirror-reflection phenomenon disappears and peak values of ORCS are in the non-specular direction. Finally, the ORCS features are summarized to show its advantages in radar target detection. This work can provide theoretical guidance to the design of OAM-based radar as well as the target detection and identification applications.

## I. INTRODUCTION

As described in radar theory,^{1,2} the target scattering characteristics play an important role in the development of radar technology, e.g., radar and target design. Radar cross section (RCS)^{2} is a measure of the power scattered in a given direction when a target is illuminated by an incident wave, which can be used to characterize the target characteristics and benefit radar target detection and imaging.^{3,4}

In 2007, B. Thidé^{5} firstly introduced the orbital angular momentum (OAM) into radio wireless communications, which attracted much attention in the research of OAM and its applications. In the microwave and millimeter-wave bands,^{6} when the traditional electromagnetic (EM) wave is modulated with OAM, a helical wavefront shape occurs, referred to as “vortex electromagnetic wave”. Hitherto, the research of OAM and vortex EM waves has mainly focused on the generation of OAM beams, including OAM antenna design,^{7} the phased array approach^{8} and metasurface,^{9–11} OAM-based wireless communications,^{12,13} and OAM-based radar target detection and imaging.^{6,14} With additional rotational degrees of freedom generated by OAM and the helical wavefront distribution, OAM-based radar has several distinguished merits, e.g., rotational frequency shift can be enhanced and detected without radial translation between radar and target,^{15} the azimuthal information of targets can be estimated without any relative moves or beam scanning as traditional methods required,^{6} the OAM-based radar can make a breakthrough on the Raleigh limit and achieve super-resolution for target imaging.^{14,16} Moreover, many approaches have been applied to collimate and converge the OAM beams to enhance the radar operating distance, including biconvex lens,^{17} intensity controlled masks,^{18} and reflectors antennas.^{19}

Considering the significance of RCS in the development of radar technology,^{1} the target scattering characteristics for OAM-based radar should be investigated. Due to the spiral wavefront distribution of vortex EM fields, the target scattering phenomenon illuminated by OAM beams would be much different from classical RCS. For comparison, we define the concept of OAM-based radar cross section (ORCS) for OAM-based radar, in this paper. Currently, several open articles have discussed the unique features of the vortex EM waves,^{20–22} e.g., the phase velocity in free space is greater than *c* and the energy velocity is less than *c* (*c* is the light speed in the vacuum). The properties of the EM scattering of a Bessel vortex beam by the water sphere were analyzed in Ref. 23. However, to the best of the authors’ knowledge, the target scattering properties of OAM beams and the ORCS have not been reported in open literature.

In this paper, we focus on the features of vortex EM waves and the OAM-based radar target scattering characteristics. The radar range equation is modified to show the significance of ORCS in the development of OAM-based radar. The definition of ORCS and the unique features of the incident vortex EM fields are studied. Moreover, the scattered fields are calculated for typical targets illuminated by OAM beams, and the ORCSs are obtained based on the method of physical optics. The rest of this paper is organized as follows. In Section II, the OAM-based radar target detection is briefly introduced and the radar range equation is modified according to the propagation character of OAM beams. In Section III, the OAM-based radar cross section is defined, and the unique features of the incident vortex EM fields are analyzed. In Section IV, the ORCSs for typical targets including metal flat plate and cylinder are calculated. Moreover, a series of simulations are carried out to analyze the factors that would affect the ORCS distributions. The conclusions are drawn in Section V.

## II. OAM-BASED RADAR RANGE EQUATION

In contrast to a traditional radar system, the OAM-based radar transmits vortex EM waves to illuminate the targets,^{6} as illustrated in Figure 1. A circular array is commonly used to generate the vortex electromagnetic waves (radiation field) with different modes. A single receiving antenna located at the position near the circular array is used to receive the scattering field from the target. Then the azimuthal positions of the targets can be identified by correlation processing based on the approximate dual relationship between the OAM mode number and the azimuthal angle.^{6} In the target detection process, the target scattering characteristics is an important procedure, which would affect the design of transmitted signals and the target identification.

As aforementioned, the target detection of OAM-based radar is different from traditional detection process. To illustrate the operating range of the OAM-based radar as a function of the radar characteristics (especially the ORCS), the propagation characteristics of the vortex EM waves should be analyzed and the OAM-based radar equation should be defined. In the cylindrical coordinate (*ρ*, *ϕ*, *z*), for the TE case and wave traveling in the *z* direction, the analytical expressions of the vortex EM field can be given by^{21}

where *E*_{0} is the wave amplitude, *η*_{0} indicates the wave impedance in the vacuum, and *J*_{l}(*k*_{ρ}*ρ*) is the *l*^{th} order Bessel function of the first kind. *η* denotes the wave impedance of the vortex EM wave and *ω* is the circular frequency. The phase constants are in the formula

where *c* is the light speed in the vacuum and the angle *δ* is between 0 and *π*/2.

The helical wavefront and the corresponding Poynting vector of a vortex EM wave are depicted in Figure 2. At any instant, the Poynting vector is perpendicular to the wavefront surface.^{24} Then, the tilt angle *θ*_{tilt} of the Poynting vector with respect to the beam axis can be expressed as

where *λ* denotes the wavelength and the tilt angle satisfies *θ*_{tilt} ∈ (0,*π*/2).

According to the propagation characteristics of OAM waves, traditional radar equation^{1} is modified. Assuming that the transmitted power of radar is *P*_{t} and the gain of antenna is *G*, we can obtain the power density *S*_{1} at the target’s position

where *R* denotes the distance from radar to target.

Then, the power density *S*_{2} of the scattering echo at the radar’s position can be given by

where *σ*_{O} represents the ORCS, which measures the scattering ability of the target.

where *f* is the radar operating frequency, and *c* is the light speed in the vacuum.

It can be seen from Eq. (6) that ORCS can significantly affect the echo power scattered from the targets, i.e., the signal-to-noise (SNR) is dependent on the ORCS. In radar detection theory,^{25} the SNR plays an important role in the detection performance, e.g., for higher SNR, a better detection performance can be achieved in general. Compared with conventional radar systems, a problem arises that the received power is reduced by a factor of (cos *θ*_{tilt})^{2}. But it can be easily resolved by increasing the transmitting power in practical systems. On the other hand, benefits can be brought by the unique helical wavefront of OAM beams. Most importantly, the azimuthal resolution can be enhanced.^{14,16}

## III. OAM-BASED RADAR CROSS SECTION

For comparison, the ORCS *σ*_{O} is defined in analogy to traditional RCS *σ*_{C}:

where $Eoams$ and $Eoami$ denote the scattered field vector at the radar position and the incident field vector at the target position, respectively. *R* is the distance between the radar and the center of target.

As aforementioned, when the EM field carries OAM, the field will have a helical wavefront distribution with an azimuthal phase dependence of exp(*ilϕ*) and the Poynting vector spirals along a curve with reference to the beam axis. These features would lead to different target scattering phenomenon for OAM-based radar. According to the time-harmonic field expressions in classical EM theory^{26} and Eq. (1), in the Cartesian coordinate (xyz), the incident vortex field $Eoami$ can be given by

where *E*_{0} is a unit vector perpendicular to the wave propagation direction *z*, *α*_{1} and *α*_{2} are real coefficients. If *α*_{1} = 0 or *α*_{2} = 0, Eq. (8) denotes a linear polarization wave. When *α*_{1} = *α*_{2} ≠ 0, it is in circular polarization wave; when *α*_{1} ≠ *α*_{2} ≠ 0, it is in ellipse polarization. Symbols “+” and “-” indicate the left-hand and right-hand polarization, respectively. *l* is the topological charge and *ϕ* denotes the azimuthal angle with respect to the beam axis. *k* is the wave number (phase constant), *ω* is the circular frequency, and *|E*_{0}| is the amplitude vector of the electric field.

According to Eq. (8), we can obtain the time-domain expression of the incident field $Eoami$ in linear polarization:

The electric field vector on a rectangular plane perpendicular to the Poynting vector can be described in Figure 3. Compared with traditional EM wave, the vortex EM wave is not only time-like but also space-like, i.e., it requires both temporal and spatial coherence. It is clear that the change period of the field vector is corresponding to the topological charge *l* (OAM mode number).

Similarly, the time-domain expression of the vortex EM wave in right-hand circular polarization can be written as

Based on Eq. (10), the instantaneous electric field vectors for the right-hand circular polarization are shown in Figure 4. It can be seen from Figure 4 that the electric field vector of the circular polarization changes on a circle in reference to the beam axis, which is different from the linear polarization cases. Figure 3 and Figure 4 indicate that the target scattering distributions illuminated by vortex EM waves might be different from the incidence of plane waves. For simplicity, the ORCS in linear polarization is studied in this paper.

## IV. ORCS CHARACTERISTICS FOR TYPICAL TARGETS

In this section, the EM vortex scattering phenomena by two typical targets, namely, metal plate and cylinder, are studied. The scattering field is calculated based on the physical optics approximation^{2} and the factors that would affect the ORCS characteristics are analyzed in detail.

### A. ORCS characteristics of plate

The flat plate is taken as an example to derive the analytical expressions of ORCS, shown in Figure 5. For brevity, the incident wave is assumed to be linearly polarized and the influences of polarization on the target scattering is not studied in this paper. Based on Eq. (8), we rewrite the incident field with linear polarization in the complex form. Without loss of generality, it is reasonable to assume the incident electric field $Eoami$ in the $\theta ^$ direction and magnetic field $Hoami$ in the $\varphi ^$ direction, which takes the form

where $r^$ is the unit vector in the scattering direction and ** r′** represents the position of the integral area.

*ρ*is the distance between the target and the beam axis,

*k*

_{p}denotes the

*ρ*component of the wave number

*k*, and

*ϕ*′ is the azimuthal angle with respect to the beam axis.

*E*

_{0}and

*H*

_{0}are the wave amplitudes of the electric field and magnetic field, respectively.

*E*

_{0}=

*η*

_{0}

*H*

_{0}and

*η*

_{0}indicates the wave impedance in the vacuum.

Based on the Stratton-Chu equations and the physical optics approximation,^{2} the back-scattered electric field vector $Eoams$ for the plate target can be expressed as

where $n^$ is the normal vector perpendicular to the flat plate.

where

and *k*_{ρ} = *βk*, *β* ∈ (0,1), which is decided by the topological charge *l* of the incident OAM beams.

If the absorption effect is ignored, energy conservation during the OAM-based radar-target interaction process requires that all the incident energy has to be reflected. Then, we obtain the following formula

where *A*_{⊥} indicates the orthogonal projection area of the plate with respect to the LOS.

Based on Eqns. (11), (13) and (14), the analytical expression of ORCS *σ*_{O} for the plate target can be represented as

where

*u* = sin*θ* cos*ϕ* and *v* = sin*θ* sin*ϕ*.

A typical expression of RCS *σ*_{C} for the flat plate is in the formula^{2}

where *A* = *a* × *b* denotes the area of the target, *u* = sin*θ* cos*ϕ* and *v* = sin*θ* sin*ϕ*.

Based on Eqns. (15) and (16), simulations are carried out and the comparisons of the ORCS and RCS are shown in Figure 6. In the simulation, the operating frequency is set as *f* = 10 GHz and the size of the plate target is (*a*,*b*) = (0.5 m,0.4 m). It can be seen from Figure 6(a) that the ORCS characteristic is just the same as RCS when the topological charge is equal to zero *l* = 0. The differences between RCS and ORCS for large incident angles are mainly caused by the approximation error during the derivation. For OAM beams *l* ≠ 0, the peak values of ORCS are not in the specular direction (*θ* = 0°) but occur in the angle where the side lobes of traditional RCS are located. Moreover, the main-lobe angle increases with the increasing the topological charge. The OAM carried by the incident vortex EM waves contributes to the differences between ORCS and traditional RCS. Generally, the unusual features of ORCS distributions indicate that the OAM-based radar might offer a new solution to the object identification problem.^{27,28}

Furthermore, the influences of the target size on the ORCS is depicted in Figure 7. It can be seen from Figure 7 that the peak values increase and the width of the peak becomes narrower as the side *a* increases, for targets with the same area. When the target area increases, both the peak and the side lobes increase. The influences of signal frequency on the ORCS are analyzed in Figure 8. In the simulation, the first order OAM beam is transmitted to illuminate the target and the target size is set as (*a*,*b*) = (0.5 m,0.4 m). Results show that the peak values increase with the crease of signal frequency.

### B. ORCS characteristics of cylinder

The ORCS for the metal cylinder target is derived and analyzed. The sketch map of the interaction between the vortex EM wave and the cylinder target is shown in Figure 9. Based on Eqns. (11) and (12), the scattered field of the cylinder can be given by

where

and *k*_{ρ} = *βk*, *β* ∈ (0,1), which is decided by the topological charge *l* of the incident OAM beams.

According the definition of ORCS and the energy conservation law, the ORCS for the cylinder is written as

where *A*_{⊥} indicates the orthogonal projection area of the cylinder in reference to the LOS, and

In general, the RCS *σ*_{C} of the cylinder target can be written as^{2}

Based on Eqns. (18) and (19), simulations are conducted to obtain the ORCS and RCS for the cylinder target, shown in Figure 10. In the simulation, the signal frequency is *f* = 10 GHz and the target size is set as (*a*_{r},*h*) = (0.15 m,0.3 m). For OAM beams *l* = 0, the ORCS is almost the same as RCS for the cylinder target. The differences at the direction away from *θ* = 90° are mainly caused by the calculation error and the model approximation error. For the incident vortex EM waves *l* ≠ 0, the peak values of ORCS is no longer in the angle *θ* = 90° and the distance between two peaks increase as the topological charge *l* increases. The influences of the cylinder size on the ORCS characteristics are shown in Figure 11. It can be seen from Figure 11 that the main lobes increase when the height or the radius increases. Furthermore, the influences of signal frequency are illustrated in Figure 12. In the simulations, the target size is set as (*a*_{r},*h*) = (0.15 m,0.3 m). It is clear from Figure 12 that the peak values increase as the signal frequency increases.

### C. ORCS features and its merits in radar detection

According to the definition of ORCS and the analytical results about plate and cylinder targets, the basic properties of ORCS are summarized and listed below.

Independencies: Similar to RCS, ORCS is approximately independent on the distance from radar to the target and the transmitted power of the radar system.

Dependencies: ORCS depends on the properties of the target, e.g., geometry and size, the signal frequency and the OAM mode number.

Asymptotic behavior: When the OAM mode number is equal to zero

*l*= 0, the ORCS gives way to conventional RCS.

Based on the simulation results in Figure 6 and Figure 10, quantitative results are provided to illustrate the enhancement of received power in the non-specular direction, shown in Table I. As shown in Eq. (6), the received power is reduced by a factor of (cos*θ*_{tilt})^{2} due to the propagation of OAM waves, whereas the received power can be enhanced by ORCS in the non-specular direction. The values of reduced power in Table I are obtained from the setup of *θ*_{tilt} in the simulations, i.e., (cos*θ*_{tilt})^{2} = 0.9, 0.7, 0.6 for the listed three cases. It can be seen from Table I that the net power change is always positive in the non-specular direction. Results imply that the OAM-based radar can be applied to improve the target detection performance in the non-specular direction.

. | Flat plate . | Cylinder . | |||
---|---|---|---|---|---|

OAM mode . | Peak value . | Enhanced . | Peak value . | Enhanced . | Reduced . |

number . | direction . | power/dB . | direction . | power/dB . | power/dB . |

l=1 | 3° | 16.2 | 92° | 14.5 | 0.5 |

l=3 | 8.5° | 16.7 | 94.5° | 9.7 | 1.5 |

l=4 | 11.5° | 17.3 | 97.5° | 13.6 | 2.2 |

. | Flat plate . | Cylinder . | |||
---|---|---|---|---|---|

OAM mode . | Peak value . | Enhanced . | Peak value . | Enhanced . | Reduced . |

number . | direction . | power/dB . | direction . | power/dB . | power/dB . |

l=1 | 3° | 16.2 | 92° | 14.5 | 0.5 |

l=3 | 8.5° | 16.7 | 94.5° | 9.7 | 1.5 |

l=4 | 11.5° | 17.3 | 97.5° | 13.6 | 2.2 |

## V. CONCLUSION

The target scattering characteristics for OAM-based radar have been studied for the first time. The conventional radar equation was modified by involving the characteristics of OAM waves. The ORCS was defined, and the analyses of the incident vortex EM fields indicated that the fields had both temporal and spatial coherence. Based on the method of physical optics, the analytical expressions of ORCSs for the metal plate and cylinder were derived. A series of simulations were conducted to compare ORCS with conventional RCS. It has been found that the peak values of ORCS were not in the specular direction anymore and the positions were affected by the OAM mode number. The values of ORCS increased, when the target size or the signal frequency increased. Theoretical results implied that the OAM-based radar can be used to improve the target detection performance in the non-specular direction.

Future works include studying ORCS characteristics for complex targets, and the experimental measurements of the calibrators.

## ACKNOWLEDGMENTS

This work was supported by the National Natural Science Foundation of China under Grant No. 61571011.