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Abstract
Though much work has been done concerning quantum radar cross section (QRCS), the related question about quantum radar scattering characteristics of targets in the atmospheric medium has not been considered. Understanding this question is of primary significance in the military and civil fields of quantum radar. The main objective of the paper is to propose a new algorithm for the calculation of QRCS in homogeneous atmospheric medium (M-QRCS). Therefore, based on the chain of beam splitters proposed by M. Lanzagorta to describe homogeneous atmospheric medium, the photon attenuation model is established, the photon wave function is modified, and the M-QRCS equation is proposed. Furthermore, in order to obtain an accurate M-QRCS response, we carry out simulation experiments on a flat rectangular plate in the atmospheric medium composed of different atomic arrays. Based on this, we study the influence of the attenuation coefficient, temperature, and visibility on the intensity of the peak value of main lobe and side lobes of M-QRCS. In addition, it is worth noting that the numerical calculation method proposed in this paper is based on the interaction between photons and atoms on the target surface, so it is suitable for the calculation and simulation of M-QRCS for targets of any shape.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Quantum radar refers to a novel detection technology combining radar technology and quantum technology, which has received more and more attention in recent years [1–4]. It makes a full use of the quantum characteristics of electromagnetic waves to break through the performance limits of classical radar detection, and it has broad application prospects [5,6]. Quantum radar cross section (QRCS) is a fundamental parameter to determine the scattering characteristics of a target under quantum radar irradiation, which serves as a vital indicator in stealth target design [7].
The performance of quantum radar can be improved by entanglement and superposition. In this paper, the analyzed M-QRCS utilizes superposition originating from the photon interacting with the atoms on an object. When a photon impinges on an object, it interacts with all the atoms in the object through a quantum superposition. And in response to this interaction, each atom of the object emits a photon characterized by a wave function [8]. The wave functions add up to determine the "reflected wave". It is significant to note that the interaction process between photon and target follows quantum electrodynamics.
M. Lanzagorta first proposed the concept of QRCS, and gave the expression of QRCS based on atom-photon scattering process. The mathematical expression of QRCS for a monostatic radar when the signal has no energy loss in the propagation process is given by [9]:
Based on the above theory, on the one hand, for the case of single photon incidence, the research on monostatic and bistatic QRCS has mainly focused on the numerical calculation and simulation methods of typical geometrical targets, such as the flat rectangular plate and the flat circular plate, the influence of target size and photon frequency on the scattering results [11–16], and the calculation methods of electrically large targets [17]. In addition, the influence of target polarization on QRCS was studied by M. J. Brandsema, and the result shows that the influence of target on incident photons is reflected in the electric dipole moments of atoms [18]. Subsequently, the quantum scattering characteristics of a dihedral corner reflector and typical three-dimensional targets, such as a circular cone and a convex target, were further analyzed [19–26]. On the one hand, for the case of multiple photons incidence, a closed-form model of multiphoton QRCS of flat rectangular plate is proposed [27,28]. However, the calculation and analysis of QRCS of the target mentioned above do not consider the energy loss of signal photons during the transmission from radar to target, which is not conducive to the application of quantum radar in realistic scenarios. Therefore, the study of quantum radar target characteristics still has numerous problems to deal with, such as the analysis and detection of targets in the atmospheric medium. Specifically, the calculation of the QRCS in an atmospheric medium is to be solved.
Against the aforementioned background, in this paper, a method for calculating the monostatic M-QRCS of a target in the homogeneous atmospheric medium is proposed. Firstly, utilizing a chain of beam splitters adequately describes the homogeneous atmospheric medium. secondly, the quantized electromagnetic field equation in the atmospheric medium is obtained. Thirdly, the photon wave function is modified based on the quantized electromagnetic field in the atmospheric medium. Finally, the M-QRCS equation is obtained based on the modified photon wave function. Subsequently, the accuracy of the M-QRCS response is obtained by simulating of a flat rectangular plate target in the atmospheric medium. Based on this, the influence of the attenuation coefficient, visibility, and temperature on the energy of main lobe and side lobe of the M-QRCS is analyzed. These results are of great significance for further studying the quantum scattering characteristics of targets in the atmospheric medium.
2. M-QRCS equation of targets in the homogeneous atmospheric medium
2.1 Quantization of the electromagnetic field
The electromagnetic fields are made of elementary quantum excitations of momentum $\mathbf {k}$, frequency $\omega _{\mathbf {k}}$, and polarization $\lambda$ in a cavity of quantization volume $V$. The quantum fields can be expressed as:
The only term of the quantum electric field that contributes is the term $\boldsymbol {\hat {E}}^{\left ( + \right )}\left ( \mathbf {r},t \right )$ in Eq. (5) when it comes to the measurement process. The interaction model between a target in the homogeneous atmospheric medium and the monostatic quantum radar is shown in Fig. 1.
Fig. 1. The interaction model between a target in the atmospheric medium and the monostatic quantum radar.
The quantization of the electromagnetic field in the atmospheric medium is much more complicated. However, M. Lanzagorta proposed that the atmospheric medium is adequately described by a chain of beam splitters, from which it is possible to obtain simplified quantized field equations. And the beam splitters chain is shown in Fig. 2 [9]. The operator $\hat {a}_{\text {in}}$ denotes the incident light in each beam splitter, and the operator $\hat {a}_{\text {s}}$ represents the light of scattered or absorbed by the atmospheric medium, and the operator $\hat {b}$ describes the possible contributions from the atmospheric medium to the light field, and the operator $\hat {a}_{\text {out}}$ represents the net output of light.
In this paper, we simulate the homogeneous atmospheric medium through the chain of beam splitters, and use the continuous mode quantization scheme to describe the relationship between the output and input fields of quantized light to establish a photon attenuation model. Based on the traversal of the chain of beam splitters, the annihilation quantum operator can be modified into the following formula (7) [9]:
Substituting Eq. (8) into Eq. (5) gives the following equation:
The photon wave function $\varPsi _{\gamma }$ is then defined as the expected value of the quantum electric field as it transitions from the 1 photon state (represented $\left | \gamma \right\rangle$) to the vacuum state $\left | 0 \right\rangle$ [9],
It is noticeable that this definition of the photon wave function is different from the usual one, however, Eq. (10) is adequate for our purpose. Based on the analysis of the quantized electric field in the atmospheric medium, Eq. (10) becomes:
2.2 Emitted photon state from an atom
The emitted photon state is obtained based on the Hamiltonian of the atom-field interaction. In the electric dipole approximation, the Hamiltonian of the interaction between the quantized electromagnetic field and the atom can be written as:
where $\mathbf {\hat {d}}$ is the electric dipole moment of the atom. For a two-level atom with upper energy state $\left | e \right\rangle$, lower energy state $\left | g \right\rangle$, and transition frequency $\omega _0$, the electric dipole moment of atom in the interaction picture is [8]:Substituting Eq. (8) into Eq. (3) gives $\boldsymbol {\hat {E}}_{\text {m}}\left ( \mathbf {r},t \right )$ as follows:
Therefore, according to the Eq. (13) and Eq. (15), Eq. (12) can be rewritten as:
Based on rotating wave approximation [8], Eq. (16) becomes:
Next, it is assumed that the initial atom is in the upper energy state $\left | e \right\rangle$ and all modes of the light field are in the vacuum state $\left | 0 \right\rangle$ (spontaneous radiation), then the state of the system at time $t$ is as follows:
Then, we use the Schrödinger equation:
Substituting Eq. (17) and Eq. (19) into Eq. (21), which can obtain two coupled differential formulas for the time varying probability amplitudes.
Then, substituting Eq. (24) into Eq. (22), we obtain:
Since the modes are so closely spaced, based on the Weisskopf-Wigner approximation [8], the summation can be converted into an integral by the following transformation:
Equation (18) can be rewritten as:
Therefore, $\left | \text {g}_{\mathbf {k},\lambda }^{*}\left ( \mathbf {r}_0 \right ) \right |^2$ can be written as:
According to the Eq. (26) and Eq. (28), Eq. (25) becomes:
The spectrum of light emitted by an atom is centered on the atomic transition frequency $\omega _0$. By making a substitution $\omega ^3\rightarrow \omega _{0}^{3}$ in the integration of frequency in the above formula. Therefore, the integration of frequency becomes:
Equation (29) now becomes:
Therefore, the solution of Eq. (31) is:
Substituting this solution into Eq. (24), we obtain:
When $\varGamma t\gg 1$, namely, $t\gg {1}/{\varGamma }$, the term $e^{i\left ( \omega _0-\omega _{\mathbf {k}} \right ) t-{\varGamma t}/{2}}$ tends to zero. Then, Eq. (34) is rewritten as:
Therefore, Eq. (19) becomes:
The $\left | g \right \rangle$ (lower energy state) represents the state of the atom and is of no use us, so we ignore it. And the remainder of the second term denotes the emitted photon state, called $\left | \gamma _0 \right \rangle$, which we can write as:
2.3 Photon wave function
We rewritten Eq. (38) as:
The Eq. (40) can be rewritten as:
Then, we transform the summation over $\mathbf {k}$ into an integral based on Eq. (26) by the following:
Since we are making a measurement at one particular angle only, the emitted photon wave function should be angle dependent. Therefore, we need to rewrite Eq. (42) to only be an integration over, which will obtain the angular distribution of the transformation amplitude.
Furthermore, we let $\omega _0=ck_0$ and use the residue theorem to calculate as follows:
Thus, the Eq. (42) can be rewritten as:
The sum of the polarization vectors in the Eq. (45) can be written as [18]:
In the above equation, $\mathcal {E}_0$ is as follows:
where the term $\mathbf {r}-\mathbf {r}_0$ represents the distance from the atom to an observation point, we let $\mathbf {r}-\mathbf {r}_0=\varDelta \boldsymbol {R}_0$, therefore, the Eq. (11) can be written as:This equation represents the photon wave function. For the case of multiple atoms, we write as $\mathbf {r} _j$, where the subscript $j$ is an index that describes the atom we are analyzing. Therefore, we would replace $\varDelta \boldsymbol {R}_0$ with $\varDelta \boldsymbol {R}_j$, which the Eq. (50) can be rewrite as:
2.4 Deriving the M-QRCS equation
In analogy to the classical radar cross section, the equation of QRCS was proposed by Lanzagorta [9], which defines the QRCS as:
The expectation value of the scattered intensity $\left\langle \hat {I}_{sq}\left ( \mathbf {r}_s,\mathbf {r}_d,t \right ) \right \rangle$ by a photon is reflected by $N$ atoms, is given by [29]:
Ignoring the effect of absorption, energy conservation in the optical regime requires that all the incident energy has to be reflected in some direction. Therefore, incident intensity will be equal to the scattered intensity density, integrated over all viewing angles [29,30]:
In the case that the wavelength of the incident photon is much smaller than the target dimensions, which the incident intensity on the surface is uniform. The total incident energy on the target is given by:
Assuming that all of the incident energy through the homogeneous atmospheric medium is scattered by the perfectly reflecting object surface to all parts of the space, Eq. (55) can be further written as:
We assume the conservation of energy in the optical regime, the $\sigma _Q$ for a monostatic quantum radar can be approximated by:
Finally, the equation of M-QRCS is given by:
The M-QRCS response is based solely on how the incident photon couples with the atoms in the target. Therefore, we assume that the atoms in the object have independently distributed, random dipole moment orientations, then the term $\left | \left ( \mathbf {d}_{eg}\cdot \mathbf {e}_{\mathbf {k}} \right ) \right |^2$ becomes a constant. The Eq. (58) can be rewritten as:
3. Simulation and analysis
At present, there are no quantum devices with sufficient performance to conduct corresponding detection experiments. We give the relevant results by simulating. Therefore, in this section, we carry out the simulation experiments of the target M-QRCS with a flat rectangular plate in the homogeneous atmospheric medium. Moreover, we assume that this plate is a perfectly reflecting object. The geometry of the monostatic quantum radar-medium-target system is shown in Fig. 3. In addition, typical examples of the atmospheric attenuation caused by fog or clouds are given in Table 1 for different visibility and temperature conditions [9]. In addition, $\lambda =3.2$ cm is inside the radar X-band (8-12GHz, 2.5-3.75 cm). According to Table 1, in this specific regime, we can find that the attenuation coefficient decreases as the temperature increases when visibility remains constant.
Table 1. Attenuation coefficient $\chi$ ($\text {km}^{-1}$) caused by clouds on different temperature ($\lambda =3.2 \text {cm}$).
3.1 Different atomic arrays of the target
In order to obtain an accurate M-QRCS response, there must be an adequate number of "atoms", otherwise the response will be incorrect and highly aliased. Therefore, we consider a rectangular plate target of the same dimensions, but with different interatomic distances and number of atoms, that is, different atomic arrays. The size of the rectangular plate is $1\text {m}\times 1\text {m}$. The considered cases are arrays of $100 \times 100$, $150 \times 150$, $500 \times 500$, $800 \times 800$, and $1000 \times 1000$ atoms, respectively. The wavelength of a photon is $\lambda = 3.2$ cm, the visibility is V = 30m, and temperature is T = 0$^{\circ }$C. For the case of a single photon incident, the simulation results are shown in Fig. 4.
Fig. 4. Plots of rectangular plate target M-QRCS for arrays of $100 \times 100$, $150 \times 150$, $500 \times 500$, $800 \times 800$, and $1000 \times 1000$ atoms, respectively.
The simulation results show that the M-QRCS response of a plate target made of 10000 atoms and a single quantum radar signal photon is highly aliased and incorrect in Fig. 4. It is significant to note that this M-QRCS equation is applicable to the calculation of the M-QRCS of two-dimensional or three-dimensional targets in the homogeneous atmospheric medium. However, as can be seen from Fig. 5, the M-QRCS simulation of a target made of $850 \times 850$ atoms (722500 atoms) took over 4 hours on a i5-10210U CPU @ 1.6GHz Intel computer. Furthermore, we discover that the time required in the simulation experiment is proportional to the number of atoms. As a consequence, the target size and atomic composition are limited by the available computational resources. For complex targets, more atoms are involved in computer simulation, which leads to an increase in computational effort. In the future, for M-QRCS simulation of complex targets such as aircraft, we need to improve not only the algorithm but also the performance of hardware devices.
Fig. 5. The time required for M-QRCS response of a plate target with the same dimensions and different number of atoms in the simulation experiment.
3.2 Comparison of QRCS and M-QRCS
The QRCS equation in the lossless environment and the M-QRCS equation in the homogeneous atmospheric medium are, respectively, Eq. (1) and Eq. (59). For the flat rectangular plate target, whose simulation results are shown in Fig. 6. The relevant parameters in the simulation experiment are $\lambda =3.2$ cm, T = 0$^{\circ }$C, $\chi = 0.046 \text {km}^{-1}$, and V = 30m, respectively.
Fig. 6. QRCS response of a rectangular plate ($800 \times 800$ atoms) $1\text {m}\times 1\text {m}$ in dimensions, with incident photon wavelength of 3.2 cm and attenuation coefficient of 0.046 $\text {km}^{-1}$ for the atmospheric medium.
The simulation results show that the QRCS of the plate target in the lossless environment has a similar waveform structure to the M-QRCS of the target in the atmospheric medium. However, the peak value of the M-QRCS of the plate targets in the atmospheric medium along the specular direction decreases, the width of the main lobe becomes narrower, and the side lobes decrease. It shows that the atmospheric medium affects the target detection performance of quantum radar. The absorption and scattering effects of the atmospheric medium on photons result in a lower probability of photon detection of the target.
3.3 Analysis of influencing factors of M-QRCS
We analyze factors that can influence the intensity of the peak value of the main lobe and side lobes. Specifically, the considered factors include the attenuation coefficient of medium, visibility, and temperature in this section. Among these factors, atmospheric visibility is the maximum distance at which a person with normal vision can identify an object of a certain size from the background (sky or ground). The rectangular plate size for the simulation is $a\times b=1\text {m}\times 1\text {m}$, and the wavelength of the signal photon is $\lambda =3.2$ cm. Figure 7 and Fig. 8 show how the M-QRCS of a rectangular plate target in the atmospheric medium varies with the attenuation coefficient, visibility, and temperature.
Fig. 7. The response of the M-QRCS of rectangular plate ($800 \times 800$ atoms) $1\text {m}\times 1\text {m}$ in dimensions with respect to range for $\lambda =3.2$ cm and T=0$^{\circ }$C in different attenuation coefficient.
Fig. 8. The response of the M-QRCS of rectangular plate ($800 \times 800$ atoms) $1\text {m}\times 1\text {m}$ in dimensions with respect to range for $\lambda =3.2$ cm. (a) M-QRCS response of target with different temperature (Visibility = 30m). (b) M-QRCS response of target with different visibility (T = 15$^{\circ }$C ).
It can be seen from Fig. 7 and Fig. 8(b) that when the temperature remains unchanged, as the attenuation coefficient increases (visibility decreases), the width of the main lobe of the M-QRCS of the plate target in the atmospheric medium along the specular direction becomes narrower and the side lobes decrease. Moreover, it can be seen from Fig. 8(a) that when the visibility is kept constant, as the temperature increases, the width of the main lobe of the M-QRCS becomes wider, the peak value of the main lobe increases, and the side lobes increase. Therefore, the photon emission and detection strategy should be selected according to the realistic situation of the quantum radar-medium-target system in order to obtain more information about the target in the atmospheric medium utilizing quantum radar.
4. Conclusion
In conclusion, we establish the attenuation model of the signal photon in the medium based on a chain of beam splitters that can adequately describe the homogeneous atmospheric medium proposed by M. Lanzagorta. Using this model, we propose the M-QRCS algorithm for targets in the homogeneous atmospheric medium. Then, in order to obtain an accurate M-QRCS response, we have carried out simulation experiments with a flat rectangular plate target of the same size but with different atomic arrays. Moreover, we compared the QRCS of a plate in the nondestructive environment with the M-QRCS of plate in the homogeneous atmospheric media. In addition, on this basis, we analyze the influence of the attenuation coefficient, visibility, and temperature on the main lobe and side lobe of the M-QRCS of the plate. The simulation results show that the photon emission and detection strategy should be selected according to the realistic scenarios of the quantum radar and target in the atmospheric medium in order to obtain more information about the target. Directions of future work include the calculation of M-QRCS for typical three-dimensional targets, complex targets in atmospheric medium, exploring the use of shock waves for quantum radar target detection, and trying to solve more problems related to target scattering characteristics, such as how to obtain more information about the target by identifying the states of photons.
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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