Analysis of bistatic multiphoton quantum radar cross section for the cylindrical surface

Zhifu Tian; Tao Hu; Di Wu; Shu Wang

doi:10.1364/OE.520224

1. Introduction

Quantum remote sensing is one of the most advanced fields in applying quantum technology [1]. Quantum remote sensing uses all available quantum properties to improve remote sensing performance. Quantum remote sensing has many good characteristics. Compared with classical remote sensing, single photon remote sensing can provide lower power and high sensitivity for long-distance detection and high resolution in the optical frequency band [2,3]. Quantum illumination can provide high detection probability, anti-interference, and anti-interception ability [4,5]. Quantum imaging can offer the ability to resist imaging in strongly scattering media [6]. Applying these good characteristics of quantum technology can realize a radar with better detection performance and better adapt to the increasingly complex target environment.

The Scattering rate of a target is described chiefly as a small constant in theoretical research of quantum sensing [7–9], which is reasonable because the scattering rate of a target with a fixed posture, material, and size is indeed a constant for the system of a radiation source with a fixed wavelength and photon number. The photon emission rate of the current photon source is not high enough, and the scattering rate of an actual target is extremely low, which leads to a too-long photon accumulation time. At present, the target is modeled chiefly as a beam splitter with low reflectivity [10–12], or no target is placed in experiments [13]. This setting is highly simplified, and the quantum scattering characteristics of the target should be related to the target's attitude, material, and size, the frequency of the incident signal, and the number of photons. The quantum scattering characteristics of the target are concentrated in the QRCS for the target. Understanding the results of the interaction between quantum state signals and atoms on the target surface will help optimize the structure design of the stealth target and the detection strategy.

Like the application of other quantum technologies, QRCS needs to be re-discussed by using quantum theory to make the best use of quantum effects [14,15]. Currently, the calculation models of QRCS are divided into numerical calculation model [14] and Fourier integral model [16]. When M photon is incident on the target surface of N atom, the numerical calculation model of QRCS is

(1)$${\sigma _Q} = 4\pi {A_ \bot }\frac{{{{\left|{\sum\limits_{i = 1}^N {{e^{jk\Delta {R_i}}}} } \right|}^{2M}}}}{{\int_0^{2\pi } {\int_0^{\frac{\pi }{2}} {{{\left|{\sum\limits_{i = 1}^N {{e^{jk\Delta {R_i}}}} } \right|}^{2M}}\sin \theta d\theta d\varphi } } }}, $$

where ${A_ \bot }$ is the total projected area of the target perpendicular to the direction of the incident wave and k is the wavenumber. The azimuth and pitch angles of the radar relative to the rectangular plate are $\varphi$ and $\theta$, respectively. $\Delta {R_i}$ is the interference distance from the ith atom on the target surface to the detector.

Lanzagorta established a single-photon version of the model in Eq. (1) by using quantum electrodynamics (QED), calculated the pattern of QRCS for the rectangular plates, and observed the sidelobe structure of QRCS for the rectangular plates [14]. On this basis, Liu et al. further discussed the influence of the size-wavelength ratio on the QRCS for rectangular plates [17], quantified the influence of QRCS on target detection performance, and preliminarily explored the QRCS pattern for cylindrical surface and dihedral corner reflectors [18]. In our previous work, the influencing factors of QRCS for the double scattering of dihedral corner reflectors were discussed in detail [19,20]. The difficulty in calculating the projection area of a complex three-dimensional convex surface target is one of the critical reasons for preventing the calculation expansion from the QRCS for two-dimensional targets to the QRCS for three-dimensional targets. Fang et al. proposed an improved projection and rotation transformation method to calculate the projection area of arbitrary three-dimensional convex targets [21]. Based on this method, Fang et al. took the lead in calculating the QRCS patterns for many typical three-dimensional targets, such as cubes [22], pyramids [21], cones [23], and cylinders [24]. Xu et al. gave the QRCS pattern for the cone-cylinder composite target [25]. Overall, the model of Eq. (1) has been widely used. The model of Eq. (1) can calculate the QRCS for planar targets with known boundary conditions, and it has universality, which is the basis of its wide application. However, the amount of computation increases significantly with the increase of the atomic sampling interval and the target size in the model [26]. At the same time, the more accurate calculation results of QRCS require a smaller atomic sampling interval, which leads to a large amount of calculation for the accurate QRCS of an electrically large target. The method of accelerating computation based on GPU has been proposed [26], but more is needed to solve this problem fundamentally. In addition, with the structure of Eq. (1), it is not convenient to analyze the influence of transmitted signal parameters and target parameters on QRCS, and the scattering characteristics of the target can only be summarized through multiple calculations. Brandsema et al. deeply felt these problems and creatively proposed the Fourier integral model of QRCS [16]

(2)$${\sigma _Q} = 4\pi {A_ \bot }\frac{{{{\left|{\int\!\!\!\int {{e^{j({{\boldsymbol k^{\prime}} - {\boldsymbol k}} ){{{\boldsymbol x^{\prime}}}_i}}}dS} } \right|}^{2M}}}}{{\int\!\!\!\int {{{\left|{\int\!\!\!\int {{e^{j({{\boldsymbol k^{\prime}} - {\boldsymbol k}} ){{{\boldsymbol x^{\prime}}}_i}}}} dS} \right|}^{2M}}\sin \theta ^{\prime}d\theta ^{\prime}d\varphi ^{\prime}} }}, $$

where ${{\boldsymbol x}_i}^\prime$ is the distance vector from the target center to the ith atom on the target surface, and $k\;\textrm{and}\;k^\prime$ are the incident and scattering wave vectors, respectively.

The closed-form solution of QRCS can be obtained by the surface integral of the target in Eq. (2), and the advantage of the closed-form solution is that the functional relationship between the incident signal parameters and the target parameters and QRCS can be directly obtained. However, the disadvantage is that the closed-form solution of QRCS for the target with complex boundary conditions is challenging to solve or does not exist.

Bistatic radar can improve the detection ability of stealth targets and has good anti-electronic interference characteristics. Fang et al. extended the application of the QRCS model to the bistatic situation. Based on the model of Eq. (2), the closed-form solution of the bistatic QRCS for rectangular plates is obtained [27]. Fang et al. gave the comparison pattern between the bistatic QRCS and the monostatic QRCS for rectangular plates and analyzed the difference between bistatic QRCS and monostatic QRCS for the rectangular plates. In addition, Fang et al. also established the closed-form solutions of bistatic QRCS for circular and triangular plates and concluded that the envelope changes of bistatic QRCS for circular and triangular plates are different from those for rectangular plates with incident frequency change [28]. However, the QRCS performance of three-dimensional targets under the bistatic system is unclear. Because of the complexity of the boundary conditions of three-dimensional targets, the analytical analysis of the bistatic QRCS for three-dimensional targets has yet to be seen.

Quantum remote sensing at low power determines the scene where few photons interact with the target. The concept of photon number, which is not found in traditional RCS, must be considered separately in QRCS. Lanzagorta discussed the multiphoton case when establishing the numerical calculation model of QRCS. When studying the rectangular plate, Lanzagorta concluded that increasing the number of photons will enhance the main lobe of QRCS and narrow the main lobe width of QRCS [14]. In the follow-up study, Brandsema et al. put forward the simplified versions of Eq. (1) and Eq. (2) of multiphoton QRCS through a lot of calculation and analysis [29] and proved that under the far-field condition of remote sensing, single photon QRCS is equivalent to classical radar cross-section [30]. In this paper, the similarities and differences between single photon QRCS and CRCS are not discussed. The discussion under the condition of a single photon can be regarded as either single photon QRCS or CRCS because there is almost no difference in magnitude and pattern between single photon QRCS and CRCS. Furthermore, the other one of our previous works obtained the partly closed-form solution of multiphoton QRCS for rectangular plates [31], quantified the physical conditions of main lobe enhancement and main lobe width narrowing, and found that besides the main lobe, the sidelobe of multiphoton QRCS may also be enhanced. Multiphoton QRCS of targets with different edge conditions may also have different characteristics from those found in rectangular plate research. The multiphoton QRCS model requires that the maximum delay between the photons has to be smaller than the lifetime of the excited states of the atoms [30]. The magnitude of the lifetime of the excited states of the atoms is about ${10^{ - 8}}$ s, which means that the model can effectively deal with meter-sized targets. Note that this model does not emphasize the distinction between the plane and convex surface targets. Our recent work gives the unique double-mainlobe patterns of multiphoton QRCS for cones [32]. A powerful tool to discover the characteristics of multiphoton QRCS for targets with different boundary conditions is the analytical solution of multiphoton QRCS. Because the influence of multiphoton conditions on QRCS is power-like, it is not easy to obtain the analytical solution of multiphoton QRCS, especially under the complex boundary conditions of three-dimensional targets.

The above researches show that both bistatic QRCS and multiphoton QRCS have their characteristics. The effects of target boundary conditions, monostatic and bistatic settings, and photon number settings on QRCS may not be isolated, but they may affect each other. At the same time, monostatic QRCS is a special case of bistatic QRCS when the incident direction and scattering direction of the signal field are collinear, and QRCS with different photon numbers are special cases of multiphoton QRCS under the condition of specific photon numbers. Therefore, studying complete bistatic multiphoton QRCS is a natural idea. Fang et al. made a joint analysis of multiphoton QRCS and bistatic QRCS for rectangular plates and found that it is possible to detect stealth targets using the main lobe of bistatic quantum radars operating at each mirror symmetry direction in the small photon number register [27]. However, the closed-form solution of the bistatic multiphoton QRCS for rectangular plates has yet to be obtained. Hu et al. tried to derive the closed-form solution of the bistatic multiphoton QRCS for rectangular plates but did not get the final closed-form solution [33]. By changing the values of different parameters for many experiments, the joint setting of multiphoton and bistatic is analyzed in detail. Moreover, the cylinder is one of the commonly used calibration bodies in measuring the cross-section. The cylindrical surface is the main structure in aircraft structures, especially missiles. Therefore, it is of great practical significance to discuss the QRCS for the cylinder. At the same time, the cylindrical surface is a typical three-dimensional target, and the existence of curvature makes it different from the plane target. The study of QRCS for the cylinders has a higher value under the current research situation.

In this paper, the joint analysis of bistatic and multiphoton QRCS is carried out for cylindrical surfaces. The unanalyzable part of the incident field intensity is estimated through theoretical derivation and many joint analysis experiments, and a novel closed-form solution of bistatic multiphoton QRCS is established. By using the closed-form expression, the difference of QRCS between the cylindrical surface and the rectangular plate with the same size and the characteristics of bistatic QRCS are analyzed, and the influences of cylindrical radius, cylindrical length, and incident photon number on the bistatic multiphoton QRCS are discussed. To the best of our knowledge, the first closed-form expressions of bistatic multiphoton QRCS for three-dimensional targets are established in this paper. The concrete method of joint analysis and establishing closed-form solutions can be applied to modeling bistatic multiphoton QRCS of targets with different boundary conditions, thus contributing to many fields of quantum remote sensing.

The rest of this paper is structured as follows. The modeling process of the closed-form model of the bistatic multiphoton QRCS for cylindrical surfaces is given in Section 2. The accuracy of the proposed model is verified by comparing the proposed model with the numerical calculation model, and the scattering characteristics of cylindrical surfaces under the bistatic multiphoton detection system are analyzed in detail based on the closed model in Section 3. Finally, conclusions are presented in Section 4.

2. Modeling

The characteristics of multi-photon remote sensing come from the superposition of all uncertainties in the interaction between incident photons and atoms, which is reflected in QRCS [29]. For bistatic radar systems, we will mainly use Eq. (2) to derive the closed-form model of QRCS for a cylinder. Equation (2) is based on the assumption that the energy incident on the target is conserved with the scattered energy of the target without considering the influence of diffraction and absorption effects [14]. When M photons are incident on the target surface of N atoms and $N \to \infty$, the effect of multiphoton interacting with the same atom is negligible [29].

First, we give the schematic diagram of the cylindrical target detected by multi-photon bistatic radar, as shown in Fig. 1(a), where l is the length of the cylinder, r is the radius of the cylinder, $\lambda$ is the wavelength of the incident wave, the incident and scattering pitch angles are ${\theta _i}$ and ${\theta _s}$, and the incident and scattering azimuth angles are ${\varphi _i}$ and ${\varphi _s}$, respectively. The numerator of Eq. (2) determines all directional characteristics of QRCS, and the denominator mainly determines the overall amplitude characteristics of QRCS [16]. At the same time, the denominator is calculated by the numerator through the space integral, so the numerator is particularly critical. In the Cartesian coordinate system, vector decomposition is carried out in three-dimensional space for three-dimensional targets. The following vector decomposition ${\boldsymbol x^{\prime}} = {\boldsymbol x} + {\boldsymbol y} + {\boldsymbol z}$ is required, as shown in Fig. 1(b), where $\alpha$ is the azimuth angles of atoms on the cylindrical surface. And ${\boldsymbol k^{\prime}} - {\boldsymbol k = }{{\boldsymbol K}_x} + {{\boldsymbol K}_y} + {{\boldsymbol K}_z}$, where ${{\boldsymbol K}_x}$, ${{\boldsymbol K}_y}$, and ${{\boldsymbol K}_z}$ are the decompositions of subtraction of reflected wave vectors ${\boldsymbol k^{\prime}}$ and incident wave vectors ${\boldsymbol k}$ on the X-axis, Y-axis, and Z-axis in the Cartesian coordinate system, respectively [16,28]. Then in Eq. (2)

(3)$${\left|{\int\!\!\!\int {{e^{j({{\boldsymbol k^{\prime}} - {\boldsymbol k}} ){\boldsymbol x^{\prime}}}}dS} } \right|^{2M}}\textrm{ = }{\left|{\int\!\!\!\int\limits_s {{e^{({j{K_x}x + j{K_y}y + j{K_z}z} )}}ds} } \right|^{2M}}. $$

For bistatic quantum radar, ${K_x}\textrm{ = }k\sin ({{\theta_s}} )\cos ({{\varphi_s}} )- k\sin ({{\theta_i}} )\cos ({{\varphi_i}} )$, ${K_y}\textrm{ = }k\sin ({{\theta_s}} )\sin ({{\varphi_s}} )- k\sin ({{\theta_i}} )\sin ({{\varphi_i}} )$, and ${K_z}\textrm{ = }k\cos ({{\theta_s}} )- k\cos ({{\theta_i}} )$. According to the axisymmetric structure of the cylinder, as shown in Fig. 1, we consider the case of ${\varphi _i} = {\varphi _s} = {0^\mathrm{^\circ }}$, then ${K_x}\textrm{ = }k\sin ({{\theta_s}} )- k\sin ({{\theta_i}} )$, ${K_y}\textrm{ = }0$, and ${K_z}\textrm{ = }k\cos ({{\theta_s}} )- k\cos ({{\theta_i}} )$. Further,

(4)$$\begin{aligned} {\left|{\int\!\!\!\int {{e^{j({{\boldsymbol k^{\prime}} - {\boldsymbol k}} ){\boldsymbol x^{\prime}}}}dS} } \right|^{2M}} &= \left|{\int_{ - \pi /2}^{\pi /2} {r{e^{({j{K_x}r\cos (\alpha )+ j{K_y}r\sin (\alpha )} )}}} } \right.{\left. {d\alpha \int_{ - l/2}^{l/2} {{e^{({j{K_z}z} )}}dz} } \right|^{2M}}\\ &= \left|{\int_{ - \frac{\pi }{2}}^{\frac{\pi }{2}} {r{e^{({j{K_x}r\cos \alpha } )}}d\alpha } } \right.{\left. {\int_{ - l/2}^{l/2} {{e^{({j{K_z}z} )}}dz} } \right|^{2M}}\\ &= {\left|{rl\sin c\left( {\frac{{{K_z}l}}{{2\pi }}} \right)\int_{ - \frac{\pi }{2}}^{\frac{\pi }{2}} {{e^{({j{K_x}r\cos \alpha } )}}d\alpha } } \right|^{2M}}\\ &= \left|{\pi rl\sin c\left( {kl\frac{{\cos {\theta_s} - \cos {\theta_i}}}{{2\pi }}} \right)} \right.{ {{J_0}({kr({\sin {\theta_s} - \sin {\theta_i}} )} )} |^{2M}} \end{aligned}, $$

where ${J_0}$ represents the zero-order Bessel function.

Fig. 1. The schematic diagram of cylindrical target detected by multi-photon bistatic radar. (a) incident vector and scattering vector. (b) position vector decomposition of atoms interacting with photons.

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Further, the denominator of Eq. (2) can be written as

(5)$$\pi \int_0^\pi {{{\left|{\pi rl\sin c\left( {kl\frac{{\cos \theta^{\prime}}}{{2\pi }}} \right){J_0}({kr\sin \theta^{\prime}} )} \right|}^{2M}}} \sin \theta ^{\prime}d\theta ^{\prime}\textrm{ = }{\lambda ^{M + 1}}{r^M}{l^{2M - 1}}\chi ({k,r,l,M} ), $$

where

(6)$$\chi ({k,r,l,M} )\textrm{ = }\pi \int_0^{\pi /2} {\left( {\frac{{{r^M}l}}{{{\lambda^{M + 1}}}}} \right)} {\left|{\pi \sin c\left( {\frac{{kl}}{{2\pi }}\cos \theta^{\prime}} \right){J_0}({kr\sin \theta^{\prime}} )} \right|^{2M}}d\theta ^{\prime}. $$

In addition, under the far-field conditions of remote sensing, there are

(7)$${A_ \bot }({{\theta_i},{\varphi_i}} )= 2rl|{\sin ({{\theta_i}} )} |. $$

Substituting Eqs. (4), (5) and (7) into Eq. (2), the closed-form expression of bistatic multi-photon QRCS for the cylindrical surface is

(8)$${\sigma _Q} = \frac{{8\pi }}{{\chi ({k,r,l,M} )}}\left( {\frac{{{r^{M + 1}}{l^2}}}{{{\lambda^{M + 1}}}}} \right)|{\sin {\theta_i}} |{\left|{\pi \sin c\left( {kl\frac{{\cos {\theta_s} - \cos {\theta_i}}}{{2\pi }}} \right){J_0}({kr({\sin {\theta_s} - \sin {\theta_i}} )} )} \right|^{2M}}. $$

Under monostatic illumination, ${\boldsymbol k^{\prime} = } - {\boldsymbol k}$. Then ${K_x}\textrm{ = 2}k\sin (\theta )\cos (\varphi )$, ${K_y}\textrm{ = 2}k\sin (\theta )\sin (\varphi )$, and ${K_z}\textrm{ = 2}k\cos (\theta )$ [16]. According to the above derivation process, the corresponding monostatic version can be obtained

(9)$${\sigma _Q} = \frac{{8\pi }}{{\chi ({k,r,l,M} )}}\left( {\frac{{{r^{M + 1}}{l^2}}}{{{\lambda^{M + 1}}}}} \right)|{\sin \theta } |{\left|{\pi \sin c\left( {kl\frac{{\cos \theta }}{\pi }} \right){J_0}({2kr\sin \theta } )} \right|^{2M}}. $$

3. Results and analysis

3.1 Verification

Since no quantum devices have sufficient performance to carry out corresponding remote sensing experiments, we compare the theoretical results of derived closed-form expression with those of numerical model in the Ref. [24,34] in the monostatic case. The incident photon wavelength is 0.03 m, the photon number is 1, $l = 6\lambda$, and $r = \lambda$. All data in the figures of this paper are presented in dB ($10{\log _{10}}({\cdot} )$), and they are all non-negative values. Figure 2 shows that the results of deduced closed-form expression are consistent with those of numerical calculation. Small differences at large angles are due to atomic sampling in numerical simulation, as studied in Ref. [16].

Fig. 2. The QRCS of the derived closed-form model and the numerical model in literature for a cylinder.

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3.2 Cylindrical curvature

Equation (4) shows that the QRCS for the cylinder and rectangular plate is very similar. However, the difference is also apparent. From the physical model, the cylinder has curvature in the azimuth direction, but the rectangular plate does not, which reflects that the QRCS of the atomic arrangement with the equal interval on the cylindrical surface is equivalent to that of the atomic arrangement with the unequal interval on the rectangular plate in the azimuth direction. QRCS will produce different responses to different atomic arrangements [16,35], which may be the fundamental reason for the difference between the QRCS of three-dimensional and planar targets. Figure 3 shows the equivalent diagram of multiphoton QRCS for a cylinder with homogeneous surface atoms and a rectangular plate with inhomogeneous surface atoms. From the process of deriving the closed-form expression, only X and Z terms play a role in the QRCS for a cylinder when ${\varphi _i} = {\varphi _s} = {0^\mathrm{^\circ }}$, so next, we investigate the QRCS for the rectangular target in the case of ${\varphi _i} = {\varphi _s} = {90^\mathrm{^\circ }}$. The rectangular plate has a length of l and a width of r, and the atomic arrangement intervals are equally arranged in the z-axis direction and according to $x = r\cos \alpha$ in the x-axis direction. $\alpha$ values homogeneously within the range of $({ - \pi /2,\pi /2} )$, which will lead to the inhomogeneous arrangement of atoms on the surface of the rectangular plate, as shown in the right side of Fig. 3. Unless otherwise specified, the interval of atoms on the target surface is arranged equally by default in most QRCS research. From the above discussion, we will see that QRCS for the rectangular plate with inhomogeneous arrangement of atoms is similar to that for a cylinder with inhomogeneous arrangement of atoms, as shown in Fig. 3.

Fig. 3. Equivalent schematic diagram of QRCS for a cylinder with homogeneous surface atoms and a rectangular plate with inhomogeneous surface atoms.

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From the derived closed-form expression, the curvature of the cylinder produces a $|{J_0}(kr({\sin {\theta_s} - \sin {\theta_i}} ))|^{2M}$ term, as shown in Fig. 4. The influence of cylindrical curvature is to modulate the pattern of Fig. 4 into the amplitude of QRCS for a rectangular plate. Figures 4(a) and (b) show that the modulation intensity differs for the different incident and scattering angles. With the increase of curvature radius, the number of sidelobes of the ${|{{J_0}({kr({\sin {\theta_s} - \sin {\theta_i}} )} )} |^{2M}}$ pattern increases, and the intensity difference increases accordingly, which is consistent with the result caused by the increase in the length of the rectangular plate.

Fig. 4. ${|{{J_0}({kr({\sin {\theta_s} - \sin {\theta_i}} )} )} |^{2M}}$ with the change of incident and scattering angles: (a) $r = \lambda$, (b) $r = 2\lambda$.

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In order to observe the effect of ${|{{J_0}({kr({\sin {\theta_s} - \sin {\theta_i}} )} )} |^{2M}}$ term on QRCS, we normalize QRCS for a cylinder and a rectangular plate. In the case of ${\theta _s} = {\theta _i} + {180^\mathrm{^\circ }}$, the wavelength of the incident photon is set to 0.03 m, $l = 6\lambda$, $r = \lambda$, The length and width of the rectangular plate are set to $6\lambda$ and $2\lambda$, respectively. From ${|{{J_0}({kr({\sin {\theta_s} - \sin {\theta_i}} )} )} |^{2M}}$, we can easily see that the influence of the photon number on this term is exponential, and here we set the photon number to 1. The comparison chart of normalized QRCS between the cylinder and rectangular plate is shown in Fig. 5. Figure 5 intuitively shows that the influence of cylindrical curvature on QRCS is modulating the pattern of ${|{{J_0}({kr({\sin {\theta_s} - \sin {\theta_i}} )} )} |^{2M}}$ into the envelope of QRCS for a rectangular plate.

Fig. 5. The comparison chart of normalized QRCS between the cylinder and rectangular plate.

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3.3 Bistatic system

Among the characteristics of bistatic multiphoton QRCS for a cylinder, the characteristics brought by the bistatic system are of particular concern in this paper. The analysis of the bistatic system can also reflect the analytical advantages of the derived closed-form expression. Equation (8) shows that the influence of the bistatic system on QRCS is reflected in $\sin c\left( {kl\frac{{\cos {\theta_s} - \cos {\theta_i}}}{{2\pi }}} \right){J_0}({kr({\sin {\theta_s} - \sin {\theta_i}} )} )$ term. Based on the experimental parameters in Section 3.1, we set ${\theta _s} = {\theta _i} + {180^\mathrm{^\circ }}$ in monostatic QRCS and ${\theta _i} ={-} {90^\mathrm{^\circ }}$ in bistatic QRCS. In the monostatic system, $\sin c\left( {kl\frac{{\cos {\theta_s} - \cos {\theta_i}}}{{2\pi }}} \right){J_0}({kr({\sin {\theta_s} - \sin {\theta_i}} )} )$ is transformed into $\sin c\left( {kl\frac{{\cos {\theta_s}}}{\pi }} \right){J_0}({2kr\sin {\theta_s}} )$. Comparing $\sin c\left( {kl\frac{{\cos {\theta_s} - \cos {\theta_i}}}{{2\pi }}} \right){J_0}({kr({\sin {\theta_s} - \sin {\theta_i}} )} )$ in the bistatic system with $\sin c\left( {kl\frac{{\cos {\theta_s}}}{\pi }} \right){J_0}({2kr\sin {\theta_s}} )$ in the monostatic system, we can see that the difference lies in $\sin c$ term and ${J_0}$ term. The variable value of $\sin c$ term in the bistatic case is half that in the monostatic case, which will lead to the sidelobe width of the bistatic QRCS being twice that of the monostatic QRCS and the sidelobe number of the bistatic QRCS being half that of the monostatic QRCS. The influence of ${J_0}$ term has been discussed in detail in Section 3.2. The variable value of ${J_0}$ term in the bistatic case is half of that in the monostatic case, so the sidelobe number of amplitude modulation caused by ${J_0}$ term will also be reduced. Meanwhile, Fig. 4 shows that the amplitude change of ${J_0}$ term with the parameter change is small. The results in Fig. 6 verify the above analysis. Note that the monostatic system is a special case of the bistatic system. The significance of the monostatic QRCS curve in Fig. 6 lies in the target's scattering ability when the incident and scattering directions precisely coincide. On the other hand, the significance of the bistatic QRCS curve in Fig. 6 lies in the target's scattering ability for all directions under a fixed incident direction. The bistatic system is generally considered advantageous for detecting stealth targets, as stealth strategies for such targets typically focus on reducing the backscattering in the forward direction while weakening stealth in other scattering directions. As illustrated in Fig. 7, the derived multiphoton bistatic QRCS provides guidance on potential strong scattering directions for the target under various directions of photon incidence.

Fig. 6. Bistatic QRCS and monostatic QRCS.

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Fig. 7. Bistatic QRCS when ${\theta _i} ={-} {90^\mathrm{^\circ }}$, $- {75^\mathrm{^\circ }}$, and $- {60^\mathrm{^\circ }}$, respectively.

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Based on experimental parameters in Section 3.1, we set ${\theta _i} ={-} {90^\mathrm{^\circ }}$, $- {75^\mathrm{^\circ }}$, and $- {60^\mathrm{^\circ }}$, respectively, and the results of bistatic QRCS are shown in Fig. 7. Figure 7 shows that the mirror direction of bistatic QRCS for a cylinder has strong scattering. Moreover, we find that the sidelobe of one side of the scattering angle near cylindrical surface is enhanced, while the sidelobe of the other side is suppressed, which is not reported in bistatic QRCS for the planar target, so we continue to investigate the ${J_0}$ term as shown in Fig. 8. Figure 8 shows that when ${\theta _i} ={-} {90^\mathrm{^\circ }}$, $- {75^\mathrm{^\circ }}$, and $- {60^\mathrm{^\circ }}$, respectively, the bistatic QRCS for the cylinder is all enhanced far from the center. When ${\theta _i} ={-} {90^\mathrm{^\circ }}$, the left and right fifth sidelobe are mainly enhanced, when ${\theta _i} ={-} {75^\mathrm{^\circ }}$, the left fourth fifth sidelobe and the right sixth seventh sidelobe are mainly enhanced, and when ${\theta _i} ={-} {60^\mathrm{^\circ }}$, the left second third sidelobe and the right seventh eighth sidelobe are mainly enhanced, which results in that the intensity of bistatic QRCS for the cylinder at the same scattering angle on the left side of the main lobe increases with the increase of incident angle, while the intensity at the same scattering angle on the right side decreases with the increase of incident angle. This enhancement or suppression comes from the joint action of ${J_0}$ term and sidelobe intensity. The results indicate that the bistatic multiphoton radar is beneficial in detecting the cylindrical target from the direction near the cylindrical surface.

Fig. 8. The curves of ${|{{J_0}({kr({\sin {\theta_s} - \sin {\theta_i}} )} )} |^{2M}}$ when ${\theta _i} ={-} {90^\mathrm{^\circ }}$, $- {75^\mathrm{^\circ }}$, and $- {60^\mathrm{^\circ }}$, respectively.

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3.4 Cylindrical radius

The closed-form expression states that cylinders with different radius will produce different scattering responses when irradiated by multiple photons under bistatic system. According to the analysis of Eq. (8), all the influencing terms of r are $\chi ({k,r,l,M} )$, ${|{{J_0}({kr({\sin {\theta_s} - \sin {\theta_i}} )} )} |^{2M}}$, and ${r^{M + 1}}$. Previous analysis in Section 3.2 indicates that the influence of ${|{{J_0}({kr({\sin {\theta_s} - \sin {\theta_i}} )} )} |^{2M}}$ on the amplitude of QRCS is in the form of fluctuation. We fix $M = 2$, $\lambda = 0.03m$ and $l = 0.18m$, and the relationship between r and $\chi ({k,r,l,M} )$ at this time is shown in Fig. 9. Figure 9 shows that $\chi ({k,r,l,M} )$ tends to be a constant in the case of an electrically large radius. Therefore, under the condition that other parameters are fixed, the influence of r on the bistatic multiphoton QRCS for a cylinder is mainly shown as the influence of ${r^{M + 1}}$. So the overall amplitude of QRCS increases with the increase of radius shown in Fig. 10, which has an exponential relationship with the photon number. We can also see the influence of ${|{{J_0}({kr({\sin {\theta_s} - \sin {\theta_i}} )} )} |^{2M}}$ on the envelope from the position of the magenta circle in Fig. 10.

Fig. 9. Curve of $\chi ({k,r,l,M} )$-radius wavelength ratio.

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Fig. 10. Bistatic multiphoton QRCS for cylinders with different radius.

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3.5 Cylindrical length

The influence of cylindrical length on QRCS for a cylinder should be similar to that of rectangular length on QRCS for a rectangular plate. According to Eq. (8), the sidelobe structure comes from $\sin c\left( {kl\frac{{\cos {\theta_s} - \cos {\theta_i}}}{{2\pi }}} \right)$ term. The sidelobe number of bistatic QRCS for a cylinder is $2l/\lambda - 2$. Two groups of experiments are carried out under photon numbers of 1 and 4. $\lambda = 0.03m$, $r = 0.03m$, and l are set as $0.09m$, $0.18m$, and $0.36m$, respectively. The results are shown in Fig. 11. Figure 11(a) shows that l affects the sidelobe of the QRCS curve and has almost no effect on the envelope, which is the conclusion under the condition of single photon. Figure 11(b) shows that the overall sidelobe amplitude of QRCS increases with the decrease of l under multi-photon conditions, which has not been reported before. Figures 11(a) and (b) demonstrate that this character only exists in the multiphoton situation, which provides an essential reference for multi-photon sensing. Figure 11(b) shows that using a low-brightness photon source, a cylinder with a smaller length has a wider range of increasing scattering, which means that under the same photon number condition, a cylinder with a small length has higher visibility for bistatic multiphoton system. Moreover, the small target is the difficulty of remote sensing.

Fig. 11. Bistatic multiphoton QRCS for cylinders with different length: (a) the photon number is 1, (b) the photon number is 4.

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3.6 Photon number

The superposed state of a small number of photons interacting with a large number of atoms on the target surface is the fundamental reason for the characteristics of multiphoton QRCS [30]. The deduced closed-form expression demonstrates that the influence of photon number is exponential, which leads to the stretching of QRCS within the whole scattering angle when the photon number increases. In the experiments, we fixed $\lambda = 0.03m$, $r = 0.03m$, and $l = 0.18m$ and set the photon number to 1, 2, and 4 shown in Fig. 12(a). Figure 12(a) verifies the conclusion of Eq. (8) on the photon number. The mainlobe and sidelobe of multiphoton QRCS become sharper, and the peak difference increases with the increase in photon number, which means that the energy scattered in the mirror direction is more concentrated in multiphoton remote sensing. When ${\theta _i} ={-} {90^\mathrm{^\circ }}$,we can observe a sharpening of the main beam at the cost of side lobe suppression in Fig. 12(a). However, the reported amplification of the main beam was never observed, which shows the difference between the QRCS for the cylinder and the previously reported QRCS for the rectangular plate [28,30,33]. This difference is reflected in the ${|{{J_0}({kr({\sin {\theta_s} - \sin {\theta_i}} )} )} |^{2M}}$ term introduced by the curvature of the cylinder. The pattern of ${|{{J_0}({kr({\sin {\theta_s} - \sin {\theta_i}} )} )} |^{2M}}$ in Section 3.2 shows that when the main lobe is located at the position of ${\theta _i} ={-} {90^\mathrm{^\circ }}$ and ${\theta _s} ={-} {90^\mathrm{^\circ }}$, the main lobe will be suppressed, and this suppression is exponentially related to the photon number. Meanwhile, when the main lobe is located at the positions of ${\theta _i} ={-} {30^\mathrm{^\circ }}$ and ${\theta _s} ={-} {30^\mathrm{^\circ }}$, the main lobe will be amplified, and this amplification is also exponentially related to the photon number, as shown in Fig. 12(b). Figure 12(c) also manifests that when utilizing the main lobe amplification effect, we should note that the larger the number of incident photons, the larger the amplitude of the amplification, but the smaller the angle range of the amplification. Of course, from Fig. 4, there is a specific range of transmitter and receiver positions for amplifying or suppressing the main lobe, rather than only a few single angular positions. Therefore, if we only develop a monostatic radar system with multiphoton sources, we will not be able to make use of the amplification effect of mainlobe under multi-photon sensing for the cylindrical structure of aerospace targets. With the multi-static multiphoton system, we can enhance the main lobe in different scattering angles by different incident angles to improve the overall detection performance through information fusion.

Fig. 12. Bistatic multiphoton QRCS for cylinders illuminated by photons with different photon numbers and different incident angle: (a) ${\theta _i} ={-} {90^\mathrm{^\circ }}$, $M = 1,2,4$, (b) ${\theta _i} ={-} {30^\mathrm{^\circ }}$, $M = 1,2,4$, (c) ${\theta _i} ={-} {30^\mathrm{^\circ }}$, $M = 1,2,4,10,40$.

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4. Conclusion

In this paper, the cylindrical surface, the main structure of typical aircraft, especially missiles, is taken as the concrete research object. We extend the plane integral in the Fourier model to the space surface integral and obtain the closed-form expression of bistatic multiphoton QRCS for a cylinder. Using the derived expression, we conclude that QRCS for the three-dimensional target is equivalent to QRCS for a planar target with inhomogeneous atomic arrangement intervals. Then, we also discuss the bistatic QRCS between planar and cylindrical targets. The influence of cylindrical height, cylindrical radius, and incident photon number on bistatic multiphoton QRCS is analyzed. Furthermore, some conclusions are summarized as follows:

1) The cylindrical curvature makes bistatic multi-photon QRCS for the cylindrical target equal to bistatic multi-photon QRCS for the rectangular plate, plus a fluctuating envelope determined by curvature, incident, and scattering angle. This envelope makes the cylindrical target show the amplification of QRCS near the target surface at the scattering angle side compared with the planar target in the bistatic system, which is the scattering characteristic of the cylindrical target in the bistatic system.
2) The bistatic radar system has more detection advantages than the monostatic radar system because (a) The monostatic system is a special case of the bistatic system, and a transmit-receive separation system, by deploying receivers in different directions, is advantageous for receiving scattering echoes from various directions, significantly enhancing the probability of target detection. (b) Combining multi-photon and bistatic characteristics can enhance QRCS in various scattering angles, which cannot be achieved by the single bistatic system or classical radar system.
3) The multi-photon emission signal enables the quantum radar to obtain higher scattering intensity than the classical radar system in the main lobe range. The multi-photon QRCS for a cylinder with a smaller length is more significant, which is the scattering characteristic of the cylindrical target in the multi-photon system.

This paper provides a novel estimation method for calculating QRCS for targets with complex boundary conditions and a new understanding of QRCS for targets with three-dimensional convex surfaces and obtains some useful conclusions. Further work will study the performance of QRCS for convex surface targets with more curvature dimensions in different radar systems and design the verification experiment of QRCS for typical calibrators in quantum sensing.

Funding

Young Scientists Fund of the National Natural Science Foundation of China (62001162); National Basic Research Program of China (2020-063-00).

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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Analysis of bistatic multiphoton quantum radar cross section for the cylindrical surface

Abstract

1. Introduction

2. Modeling

3. Results and analysis

3.1 Verification

3.2 Cylindrical curvature

3.3 Bistatic system

3.4 Cylindrical radius

3.5 Cylindrical length

3.6 Photon number

4. Conclusion

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (12)

Equations (9)

Optics Express