Closed-form expressions and analysis for the slumping effect of a cuboid in the scattering characteristics of quantum radar

Zhifu Tian; Di Wu; Tao Hu

doi:10.1364/OE.441100

1. Introduction

The ultra-low measurement limit and high detection sensitivity of quantum radars have attracted the interest of radar research [1–3]. Researchers have devoted themselves to the development of quantum devices [4–6] and constantly explore the possible characteristics of quantum radar [7–11]. The quantum radar cross-section (QRCS) is the basic parameter in the quantum radar equation and quantum scattering characteristics [12]. Analogous to the classical radar cross-section (CRCS), the QRCS is defined as

(1)$${\sigma _Q} = \mathop {\lim }\limits_{R \to \infty } 4\pi {R^2}\frac{{\left\langle {{{\hat{I}}_s}({{r_s},{r_d},t} )} \right\rangle }}{{\left\langle {{{\hat{I}}_i}({{r_s},t} )} \right\rangle }}, $$

where $\left\langle {{\hat{I}}_s(r_s,r_d,t)} \right\rangle$ and $\left\langle {{\hat{I}}_s(r_s,t)} \right\rangle $ represent the scattered and incident energy intensities, respectively, $r_s$ and $r_d$ are the positions of the quantum radar transmitter and receiver, respectively, and $R$ is the distance between the target and radar. Under the conditions of monostatic detection and ignoring diffraction and absorption, the scattered energy approximately equals the incident energy according to the energy conservation theorem. Further, the expression for the QRCS in numerical calculations is given by

(2)$${\sigma _Q} = 4\pi {A_ \bot }({\theta ,\varphi } )\mathop {\lim }\limits_{R \to \infty } \frac{{{{\left|{\sum\limits_{n = 1}^N {\exp ({jk\Delta {R_n}} )} } \right|}^2}}}{{\int\!\!\!\int {{{\left|{\sum\limits_{n = 1}^N {\exp ({jk\Delta {{R^{\prime}}_n}} )} } \right|}^2}\sin ({\theta^{\prime}} )d\theta ^{\prime}d\varphi ^{\prime}} }}, $$

where $A_\bot \left( {\theta ,\varphi } \right)$ is the total projected area of the cuboid perpendicular to the direction of the incident wave, $\Delta {R_n}$ is the distance between the radar and the atoms on the surface of the cuboid, and $k$ is the wavenumber.

Previous research objects for the QRCS have included circular, triangular, rectangular, and other flat objects [13–17]. These studies show that the QRCS has a main-lobe advantage under multi-photon irradiation [18]. However, few works give an accurate QRCS for three-dimensional objects [19–22], which limits our understanding. Due to the vast number of calculations and their variable accuracy, an accurate spatial presentation and additional analysis of the QRCS for three-dimensional objects have not been given. Numerical calculations have led to a new slumping effect in the QRCS of three-dimensional targets [23]. The slumping effect refers to the fact that the QRCS value of a three-dimensional target decreases significantly in the main lobe direction during the continuous irradiation of incident photon pulses from different directions. The first discovery of the slumping effect is reported in [23], which aroused our interest in the new quantum scattering characteristics, quantitative description of the new effect and so on. This phenomenon is believed to be due to drastic changes in the number of illuminated atoms, but the specific process and quantitative relationship between the number of illuminated atoms and degree of slumping are still unclear. In this paper, we try to verify the slumping effect and determine the quantitative relationship by analytic means.

This paper provides closed-form expressions for the QRCS over the complete viewing angle of a cuboid. The expressions allow mathematical analysis, and the calculation accuracy does not change with the simulation conditions. The performance of the cuboid under different view angles is analyzed in detail, and the QRCS of a cuboid with full three-dimensional visual angles is given for the first time. The slumping effect and the QRCS of a cuboid of different electrical sizes are further analyzed.

2. Closed-form expressions

The calculated projected area is an important part of the QRCS. Its importance is more obvious when calculating the QRCS of three-dimensional objects. For a large plane composite object, like a cuboid, we directly use the panel-based external normal vector method to calculate the projected area under single-photon irradiation. The panels of the cuboid are numbered as 1–6 in Fig. 1. The unit normal vectors of panels 1–6 are ${k_{1 - 6}}$, respectively, and their coordinates are (1,0,0), (0,1,0), (0,0,1), (-1,0,0), (0,-1,0), and (0,0,-1). At the same time, the unit vector of the incident wave direction is ${k_i} = ({ - \cos (\varphi )\cos (\theta ), - \cos (\varphi )\sin (\theta ), - \sin (\varphi )} )$. To distinguish the illuminated and shaded areas, we introduce the contribution factor G, which is equal to 1 for illuminated areas and is 0 for shaded areas. Then, the decision formula of the contribution factor for each panel is given by

(3)$$\left\{ {\begin{array}{c} {G = 1,\begin{array}{c} {} \end{array}{k_i} \cdot {k_j} \lt 0}\\ {G = 0,\begin{array}{c} {} \end{array}{k_i} \cdot {k_j} \ge 0} \end{array}} \right.,\begin{array}{c} {} \end{array}j = 1,2,3,4,5,6.$$

According to the contribution factor, the projected area is

(4)$${A_ \bot }({\theta ,\varphi } )= \sum\limits_i {({{G_i}{A_{i \bot }}({\theta ,\varphi } )} )}. $$

Fig. 1. Schematic of the vector decomposition for the interference distance. The pitch and azimuth angles of the radar relative to the cuboid are $\varphi$ and $\theta$, respectively. The ${x_i}^\prime$ are the position vectors of the atoms on the surface of the ${i^{th}}$ panel, ${x_i}$ are the vectors for the atoms on the surface of the ${i^{th}}$ panel to the radar, and d is the vector from the radar to the origin.

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Further, substituting Eq. (4) into Eq. (2) and converting the sum into an integral, we have

(5)$${\sigma _Q} = 4\pi \sum\limits_i {({{G_i}{A_{i \bot }}({\theta ,\varphi } )} )} \frac{{{{\left|{\sum\limits_i {{G_i}\int\!\!\!\int\limits_{{s_i}} {\exp ({jK{x_i}^\prime } )ds} } } \right|}^2}}}{{\int\!\!\!\int {{{\left|{\sum\limits_i {{G_i}\int\!\!\!\int\limits_{{s_i}} {\exp ({jK{x_i}^\prime } )ds} } } \right|}^2}\sin ({\theta^{\prime}} )d\theta ^{\prime}d\varphi ^{\prime}} }}, $$

where K is the difference between the scattered and incident wave vectors. The results of the QRCS are determined from the spatial interference of the scattered probability waves. Different interference distances are the key to produce various interference patterns. In Fig. 1, the interference distances of the scattered waves for the three illuminated panels have the following decomposition:

(6)$$x_1 = d + x_1^{\prime} ,\,\,x_2 = d + x_2^{\prime} ,\,\,x_3 = d + x_3^{\prime}. $$

Scattering from the same object has the constant term $\exp ({jKd} )$, so the main reason for different interference distances lies in the various position vectors of the scattering points on the panels. The symmetry of a cuboid allows limited view angles of $\theta \in ({0,{{90}^ \circ }} )$ and $\varphi \in ({0,{{90}^ \circ }} )$, and the illuminated surfaces are always panels 1, 2, and 3. Eliminating the same constant term in Eq. (5) transforms the numerator into

(7)$$\begin{aligned} \sum\limits_i {\int\!\!\!\int\limits_{{s_i}} {\exp ({jK{x_i}} )ds} } &= \int_{ - a}^0 {\int_{ - c}^0 {\exp ({j{K_y}y + j{K_z}z} )dydz} } \\ &+ \int_{ - b}^0 {\int_{ - c}^0 {\exp ({j{K_x}x + j{K_z}z} )dxdz} } + \int_{ - a}^0 {\int_{ - b}^0 {\exp ({j{K_x}x + j{K_y}y} )dxdy} } \end{aligned}, $$

where ${K_x} = 2k\cos (\varphi )\cos (\theta )$, ${K_y} = 2k\cos (\varphi )\sin (\theta )$, and ${K_z} = 2k\sin (\varphi )$. The integration results of the first, second, and third terms in Eq. (7) are then

(8)$$\int_{ - a}^0 {\int_{ - c}^0 {\exp ({j{K_y}y + j{K_z}z} )dydz} } = \exp \left( {jk\left( {\frac{a}{2} + \frac{c}{2}} \right)} \right)ac\frac{{\sin ({ka\cos (\varphi )\sin (\theta )} )}}{{ka\cos (\varphi )\sin (\theta )}}\frac{{\sin ({kc\sin (\varphi )} )}}{{kc\sin (\varphi )}}, $$

(9)$$\int_{ - b}^0 {\int_{ - c}^0 {\exp ({j{K_x}x + j{K_z}z} )dxdz} } = \exp \left( {jk\left( {\frac{b}{2} + \frac{c}{2}} \right)} \right)bc\frac{{\sin ({kb\cos (\varphi )\cos (\theta )} )}}{{kb\cos (\varphi )\cos (\theta )}}\frac{{\sin ({kc\sin (\varphi )} )}}{{kc\sin (\varphi )}}, $$

(10)$$\begin{aligned} \int_{ - a}^0 {\int_{ - b}^0 {\exp ({j{K_x}x + j{K_y}y} )dxdy} } &= \exp \left( {jk\left( {\frac{b}{2} + \frac{a}{2}} \right)} \right)\\ & ab\frac{{\sin ({kb\cos (\varphi )\cos (\theta )} )}}{{kb\cos (\varphi )\cos (\theta )}}\frac{{\sin ({ka\cos (\varphi )\sin (\theta )} )}}{{ka\cos (\varphi )\sin (\theta )}} \end{aligned}. $$

Substituting Eq. (7) into Eq. (5) transforms the denominator into

(11)$$\begin{aligned} &\int\!\!\!\int {{{\left|{\sum\limits_i {{G_i}\int\!\!\!\int\limits_{{s_i}} {\exp ({jK{x_i}} )ds} } } \right|}^2}\sin ({\theta^{\prime}} )d\theta ^{\prime}d\varphi ^{\prime}} \textrm{ = }\int\!\!\!\int {\left|{\int_{ - a}^0 {\int_{ - c}^0 {\exp ({j{K_y}y + j{K_z}z} )dydz} } } \right.} \\ &\quad{\left. {\begin{array}{ccc} {}&{}&{} \end{array} + \int_{ - b}^0 {\int_{ - c}^0 {\exp ({j{K_x}x + j{K_z}z} )dxdz} } + \int_{ - a}^0 {\int_{ - b}^0 {\exp ({j{K_x}x + j{K_y}y} )dxdy} } } \right|^2}\sin ({\theta^{\prime}} )d\theta ^{\prime}d\varphi ^{\prime} \end{aligned}. $$

Observing Eqs. (8)–(10) indicates the phases of the Sinc functions in the integration of the three formulas are orthogonal. Thus, the integration results of the cross multiplying terms are much smaller than the sum of the square terms in Eq. (11). Therefore, we can make the following approximation:

(12)$$\begin{aligned} &\int\!\!\!\int {{{{\bigg |}{\sum\limits_i {{G_i}\int\!\!\!\int\limits_{{s_i}} {\exp ({jK{x_i}} )ds} } } {\bigg |}}^2}\sin ({\theta^{\prime}} )d\theta ^{\prime}d\varphi ^{\prime}} \approx \int\!\!\!\int {{\bigg (} {{{{\bigg |}{\int_{ - a}^0 {\int_{ - c}^0 {\exp ({j{K_y}y + j{K_z}z} )dydz} } } {\bigg |}}^2}} } \\ &\quad\quad + {{{{\bigg |}{\int_{ - b}^0 {\int_{ - c}^0 {\exp ({j{K_x}x + j{K_z}z} )dxdz} } } {\bigg |}}^2} + {{{\bigg |}{\int_{ - a}^0 {\int_{ - b}^0 {\exp ({j{K_x}x + j{K_y}y} )dxdy} } } {\bigg |}}^2}} {\bigg )}\sin ({\theta^{\prime}} )d\theta ^{\prime}d\varphi ^{\prime} \end{aligned}. $$

Furthermore, Eq. (12) is equal to ${\lambda ^2}({ac\chi ({k,a,c} )+ bc\chi ({k,b,c} )+ ab\chi ({k,a,b} )} )$[9], where $\lambda$ is the wavelength and the integral terms $\chi ({k,a,c} )$, $\chi ({k,b,c} )$, and $\chi ({k,a,b} )$ approach a constant 1 under high-frequency incident waves [9,17].Therefore, we write Eq. (5) as:

(13)$$\begin{aligned} &{\sigma _Q} = 4\pi \sum\limits_i {({{G_i}{A_{i \bot }}({\theta ,\varphi } )} )} \left|{\exp \left( {jk\left( {\frac{a}{2} + \frac{c}{2}} \right)} \right)ac\frac{{\sin ({ka\cos (\varphi )\sin (\theta )} )}}{{ka\cos (\varphi )\sin (\theta )}}\frac{{\sin ({kc\sin (\varphi )} )}}{{kc\sin (\varphi )}}} \right.\\ &\textrm{ + }\exp \left( {jk\left( {\frac{b}{2} + \frac{c}{2}} \right)} \right)bc\frac{{\sin ({kb\cos (\varphi )\cos (\theta )} )}}{{kb\cos (\varphi )\cos (\theta )}}\frac{{\sin ({kc\sin (\varphi )} )}}{{kc\sin (\varphi )}}\textrm{ + }\exp \left( {jk\left( {\frac{b}{2} + \frac{a}{2}} \right)} \right)\\ &{\left. {ab\frac{{\sin ({kb\cos (\varphi )\cos (\theta )} )}}{{kb\cos (\varphi )\cos (\theta )}}\frac{{\sin ({ka\cos (\varphi )\sin (\theta )} )}}{{ka\cos (\varphi )\sin (\theta )}}} \right|^\textrm{2}}{({{\lambda^2}({ac + bc + ab} )} )^{\textrm{ - 1}}} \end{aligned}, $$

where the exponential term is associated with the established coordinate system. Next, we attain closed-form expressions for the cuboid under two special view angles.

When $\varphi \textrm{ = }{\textrm{0}^ \circ }$ and $\theta \in ({0,{{90}^ \circ }} )$ or when $\varphi \in ({0,{{90}^ \circ }} )$ and $\theta \textrm{ = }{\textrm{0}^ \circ }$, the illuminated surfaces are always two parts of the cuboid. Taking $\varphi \textrm{ = }{\textrm{0}^ \circ }$ and $\theta \in ({0,{{90}^ \circ }} )$ as an example, only the first and second summation terms need to be considered in Eqs. (4), (7), and (12). The QRCS of the cuboid can be obtained as:

(14)$$\begin{aligned} {\sigma _Q} &= 4\pi ({ac\cos (\theta )+ bc\sin (\theta )} )\left|{\exp \left( {jk\left( {\frac{a}{2} + \frac{c}{2}} \right)} \right)ac\frac{{\sin ({ka\sin (\theta )} )}}{{ka\sin (\theta )}}} \right.\\ &\quad{\left. {\textrm{ + }\exp \left( {jk\left( {\frac{b}{2} + \frac{c}{2}} \right)} \right)bc\frac{{\sin ({kb\cos (\theta )} )}}{{kb\cos (\theta )}}} \right|^2}{({{\lambda^2}({ac + bc} )} )^{\textrm{ - 1}}} \end{aligned}. $$

For $\varphi \textrm{ = }{\textrm{0}^ \circ }$ and $\theta \textrm{ = }{\textrm{0}^ \circ }$, there is only one illuminated surface. Thus, the QRCS of a cuboid can be obtained as:

(15)$${\sigma _Q} = \frac{{4\pi {{({ac} )}^2}}}{{{\lambda ^2}}}. $$

The symmetry of the cuboid in Eqs. (13)–(15) provides complete closed-form expressions for the monostatic QRCS under single-photon pulses irradiation. For multiple photons, Eq. (7) becomes the power of twice the number of photons. To simplify the analysis, the following results and analyses are all under single-photon pulses irradiation conditions.

3. Results and analysis

There are no current experiments due to limited quantum sources and single-quantum detectors with unsatisfactory performances in the microwave frequency band. Simulations can well represent the results of the closed-form expressions. In the simulations, $a = b = c = 1\begin{array}{c} {} \end{array}m$, $\lambda = 0.25\begin{array}{c} {} \end{array}m$, and the atomic distance is 0.04 times the wavelength.

The results of the closed-form expressions in Fig. 2 are consistent with that of the numerical calculations for a cube with a side of 1m illuminated with the 0.25m wavelength photon per pulse [23]. Some slight differences may be caused by setting the atomic distance on the surface of a cuboid in the numerical calculations, and the atomic distance significantly influences the side-lobe of the QRCS. In addition, when $\varphi \textrm{ = }{\textrm{0}^ \circ }$ and $\theta$ approaches 0°, there is a 3 dB slumping in the QRCS. The slumping in QRCS curves has been reported in work of former people [23]. The quantitative relationship and process of the slumping need to be further determined. Equations (13) and (14) are continuous, indicating we can take their limits. As $\varphi \to {0^{\circ }}$ and $\theta \to {0^{\circ }}$, Eqs. (13) and (14) become

(16)$$\begin{aligned} {\sigma _Q} &= 4\pi ac\left|{\exp \left( {jk\left( {\frac{a}{2} + \frac{c}{2}} \right)} \right)ac} \right.\textrm{ + }\exp \left( {jk\left( {\frac{b}{2} + \frac{c}{2}} \right)} \right)\frac{{c\lambda }}{{2\pi }}\sin ({kb} )\\ &\quad{\left. {\textrm{ + }\exp \left( {jk\left( {\frac{b}{2} + \frac{a}{2}} \right)} \right)\frac{{a\lambda }}{{2\pi }}\sin ({kb} )} \right|^\textrm{2}}{({{\lambda^2}({ac + bc + ab} )} )^{\textrm{ - 1}}} \end{aligned}, $$

(17)$${\sigma _Q} = 4\pi ac{\left|{\exp \left( {jk\left( {\frac{a}{2} + \frac{c}{2}} \right)} \right)ac\textrm{ + }\exp \left( {jk\left( {\frac{b}{2} + \frac{c}{2}} \right)} \right)\frac{{c\lambda }}{{2\pi }}\sin ({kb} )} \right|^\textrm{2}}{({{\lambda^2}({ac + bc} )} )^{\textrm{ - 1}}}. $$

Fig. 2. Curves of the QRCS for the closed-form expressions and numerical calculations of a cuboid ($\varphi \textrm{ = }{90^{\circ }}$).

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When the minimum of a, b, and c is greater than several wavelengths, we always have $ac \gg \frac{{a\lambda }}{{2\pi }}\sin ({kb} )$ and $ac \gg \frac{{c\lambda }}{{2\pi }}\sin ({kb} )$. Therefore, Eqs. (16) and (17) can be further simplified as:

(18)$${\sigma _Q} = \frac{{4\pi {{({ac} )}^2}}}{{{\lambda ^2}}}\frac{{ac}}{{({ac + bc + ab} )}}, $$

(19)$${\sigma _Q} = \frac{{4\pi {{({ac} )}^2}}}{{{\lambda ^2}}}\frac{{ac}}{{({ac + bc} )}}. $$

Comparing Eqs. (15) and (19) indicates the slumping for the QRCS shown in Fig. 2 is due to the denominator. Specifically, when $\varphi \textrm{ = }{\textrm{0}^ \circ }$ and $\theta \to {0^{\circ }}$, the number of illuminated atoms is halved, which directly leads to the 3dB drop in the integral results of the denominator while the numerator remained nearly unchanged during this process.

Three cuboid panels are illuminated when $\theta \in ({0,{{90}^ \circ }} )$ and $\varphi \in ({0,{{90}^ \circ }} )$. Comparing Eqs. (15), (18), and (19) indicates that as $\theta \to 0,\begin{array}{c} {} \end{array}\varphi \to 0$, the denominator of Eq. (18) is three times that of Eq. (15), while the numerator remains nearly unchanged. Thus, the slumping at this time is approximately 4.77 dB, as shown in Figs. 3(a) and 3(c). When $\varphi \textrm{ = }{\textrm{0}^ \circ }$ and $\theta \in ({0,{{90}^ \circ }} )$ or $\varphi \in ({0,{{90}^ \circ }} )$ and $\theta \textrm{ = }{\textrm{0}^ \circ }$, the denominator of Eq. (19) is twice that of Eq. (15), so the slumping is about 3 dB, as shown in Figs. 3(a) and 3(b). Equations (15), (18), and (19) show the quantitative relationship between the degree of slumping and the number of illuminated atoms. The above analysis results from closed expressions are consistent with those reported in [23]. By analyzing the closed expressions in detail, we can conclude that the magnitude of the slumping is inversely proportional to the area of illuminated panels. The area of illuminated panels is the representation of the number of illuminated atoms after integral mathematical processing in the closed-form expressions. It also means that with the change of the incident angle of photon pulses, the number of illuminated atoms on the target surface changes, which is the cause of the slumping effect. The quantum interference effect loses the original balance because the number of illuminated atoms on the target surface changes. This balance is determined by the physical basis of quantum interference, that is, the number of possible interactions between a photon and atoms.

Fig. 3. (a) Three-dimensional panoramagram of the QRCS for the cuboid; the Z coordinate of the highlighted point is the QRCS when $\varphi \textrm{ = }{\textrm{0}^ \circ }$ and $\theta \textrm{ = }{\textrm{0}^ \circ }$. (b) Three-dimensional panoramagram of the QRCS for the cuboid; the Z coordinate of the highlighted point is the QRCS when $\varphi \in ({0,{{90}^ \circ }} )$ and $\theta \textrm{ = }{\textrm{0}^ \circ }$. (c) Three-dimensional panoramagram of the QRCS for the cuboid; the Z coordinate of the highlighted point is the QRCS when $\theta \in ({0,{{90}^ \circ }} )$ and $\varphi \in ({0,{{90}^ \circ }} )$.

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We can get more information when the different electrical size ratio of a cuboid panel changes. In Fig. 4, $a = 2\begin{array}{c} {} \end{array}m,\begin{array}{c} {} \end{array}b = 1\begin{array}{c} {} \end{array}m,\begin{array}{c} {} \end{array}c = 1\begin{array}{c} {} \end{array}m$, and $\lambda \textrm{ = }0.25\begin{array}{c} {} \end{array}m$. As $\theta \to {0^{\circ }}$, the QRCS has a slumping of 1.76 dB, and as $\theta \to {90^{\circ }}$, the QRCS has a slumping of 4.77 dB. Note that the 4.77dB in Fig. 3 is caused by the number of the illuminated panels of a cube from three to one, and the 4.77dB in Fig. 4 is caused by the number of the illuminated panels of a cuboid from two to one. The reason why the values are the same is that the area of the two illuminated surfaces in Fig. 4 are consistent with those of the three illuminated surfaces in Fig. 3, which leads to the consistent magnitude of the slumping. This shows that the magnitude of the slumping effect has nothing to do with the structure of the targets composed of panels, but is directly related to the total area of illuminated panels. At the same time, it can be predicted that the slumping effect will also exist in other three-dimensional targets with sharp changes in the number of illuminated atoms on their surfaces. For complex three-dimensional targets which are not composed of panels, the magnitude of the slumping effect may also be related to the target structure.

Fig. 4. Curve of the cuboid with $a = 2\begin{array}{c} {} \end{array}m,\begin{array}{c} {} \end{array}b = 1\begin{array}{c} {} \end{array}m,\begin{array}{c} {} \end{array}c = 1\begin{array}{c} {} \end{array}m$, and $\lambda \textrm{ = }0.25\begin{array}{c} {} \end{array}m$ ($\varphi \textrm{ = }{90^{\circ }}$).

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When the electrical sizes of the panels are inconsistent, the observed slumping effect is small on the side of the panel with a large electrical size. In contrast, the side with a small electrical size is greatly affected by changes in the number of illuminated atoms. This difference increases with the electrical size difference of the panel. To reduce the quantum scattering intensity in the expected direction, the ratio of the target sizes requires special design. The slumping effect also provides a design concept for quantum stealth.

4. Conclusion

In conclusion, we derived closed-form expressions for the QRCS of a cuboid. The quantitative relationship between the observed slumping effect and the number of illuminated atoms is determined using the derived expressions. The influence of the different electrical sizes is analyzed. In the future, designs of verification experiments need to be performed for validation.

Funding

National Key Research and Development Program of China (2020-063-00); National Natural Science Foundation of China (62001162).

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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Closed-form expressions and analysis for the slumping effect of a cuboid in the scattering characteristics of quantum radar

Abstract

1. Introduction

2. Closed-form expressions

3. Results and analysis

4. Conclusion

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (4)

Equations (19)

Optics Express