# Simulation and Analysis of Electromagnetic Scattering from Anisotropic Plasma-Coated Electrically Large and Complex Targets

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. The Scattered Fields for An Infinite Plasma-Coated PEC Slab

#### 2.1. The Plane Wave Spectrum Representation of the Field in the Plasmas

#### 2.2. The Total Scattered Fields in the Spectral Domain

#### 2.3. Propagation Analysis of Each Secondary Scattered Field

- ${\tilde{\stackrel{\rightharpoonup}{E}}}_{r}^{0}$ is the primary reflection field of the EM wave incident on the surface of the plasma, $\stackrel{=}{M}$ is the corresponding reflection matrix;
- ${\tilde{\stackrel{\rightharpoonup}{E}}}_{t}^{1}$ is the initial transmission field of the EM wave incident into the plasma layer, ${\stackrel{=}{M}}_{1}$ is the corresponding transmission matrix;
- ${\tilde{\stackrel{\rightharpoonup}{E}}}_{r1}^{1}$ is the reflected field after the initial transmission field is incident on the bottom PEC boundary, ${\stackrel{=}{M}}_{2}$ is the reflection matrix formed by ${\tilde{\stackrel{\rightharpoonup}{E}}}_{r1}^{1}$ and ${\tilde{\stackrel{\rightharpoonup}{E}}}_{t}^{1}$;
- ${\tilde{\stackrel{\rightharpoonup}{E}}}_{t1}^{0}$ is the transmission field transmitted into the air after one reflection in the layer, ${\stackrel{=}{M}}_{3}$ is the transmission matrix formed by ${\tilde{\stackrel{\rightharpoonup}{E}}}_{t1}^{0}$ and ${\tilde{\stackrel{\rightharpoonup}{E}}}_{r1}^{1}$;
- ${\tilde{\stackrel{\rightharpoonup}{E}}}_{r2}^{1}$ is the reflection field that is reflected back to the medium on the upper surface after one reflection in the layer, ${\stackrel{=}{M}}_{4}$ is the reflection matrix formed by ${\tilde{\stackrel{\rightharpoonup}{E}}}_{r2}^{1}$ and ${\tilde{\stackrel{\rightharpoonup}{E}}}_{r1}^{1}$;

- The first reflection on the outer surface

- 2.
- The reflection of the substrate in the plasma layer

- 3.
- Transmission from the plasma layer to the upper half-space

## 3. Physical Optical Solution of Electrically Large Plasma-Coated Targets

#### 3.1. Surface EM Field of an Infinite Plasma-Coated PEC Slab

#### 3.2. PO Solution of Plasma-Coated Targets

## 4. Numerical Simulations and Discussion

#### 4.1. Validation and Analysis

#### 4.2. The Scattering Characteristics and Analysis of a Plasma-Coated Infinite PEC Plate

#### 4.3. Scattering Characteristics and Analysis of Plasma-Coated Complex Targets

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Hartunian, R.A.; Stewart, G.E.; Fergason, S.D.; Curtiss, T.J.; Seibold, R.W. Causes and Mitigation of Radio Frequency (RF) Blackout During Reentry of Reusable Launch Vehicles; Aerospace Corporation: El Segundo, CA, USA, 2007. [Google Scholar]
- Kushwaha, M.; Halevi, P. Magnetoplasma modes in thin films in the Faraday configuration. Phys. Rev. B Condens. Matter
**1987**, 35, 3879–3889. [Google Scholar] [CrossRef] [PubMed] - Kushwaha, M.; Halevi, P. Magnetoplasmons in thin films in the Voigt configuration. Phys. Rev. B Condens. Matter
**1987**, 36, 5960–5967. [Google Scholar] [CrossRef] - Kushwaha, M.; Halevi, P. Magnetoplasmons in thin films in the perpendicular configuration. Phys. Rev. B Condens. Matter
**1989**, 38, 12428–12435. [Google Scholar] [CrossRef] [PubMed] - Shi, J.; Gao, Y.; Wang, J.; Yuan, Z.; Ling, Y. Electromagnetic Reflection of Conductive Plane Covered with Magnetized Inhomogeneous Plasma. Int. J. Infrared Millim. Waves
**2001**, 22, 1167–1175. [Google Scholar] [CrossRef] - Zhang, J.; Liu, Z. Electromagnetic Reflection from Conductive Plate Coated with Nonuniform Plasma. Int. J. Infrared Millim. Waves
**2007**, 28, 71–78. [Google Scholar] [CrossRef] - Schneider, J.; Hudson, S. A finite-difference time-domain method applied to anisotropic material. IEEE Trans. Antennas Propag.
**1993**, 41, 994–999. [Google Scholar] [CrossRef] - Heald, M.A.; Wharton, C.B.; Furth, H.P. Plasma diagnostics with microwaves. Phys. Today
**1965**, 18, 72–74. [Google Scholar] [CrossRef] - Vidmar, R.J. On the use of atmospheric pressure plasmas as electromagnetic reflectors and absorbers. IEEE Trans. Plasma Sci.
**1990**, 18, 733–741. [Google Scholar] [CrossRef] - Cheng, G.; Liu, L. Direct finite-difference analysis of the electromagnetic-wave propagation in inhomogeneous plasma. IEEE Trans. Plasma Sci.
**2010**, 38, 3109–3115. [Google Scholar] [CrossRef] - Geng, Y.; Qiu, C. Extended Mie Theory for a Gyrotropic-Coated Conducting Sphere: An Analytical Approach. IEEE Trans. Antennas Propag.
**2011**, 59, 4364–4368. [Google Scholar] [CrossRef] - Song, Y.; Tse, C.; Qiu, C. Electromagnetic Scattering by a Gyrotropic-Coated Conducting Sphere Illuminated From Arbitrary Spatial Angles. IEEE Trans. Antennas Propag.
**2013**, 61, 3381–3386. [Google Scholar] [CrossRef] - Ghaffar, A.; Yaqoob, M.Z.; Alkanhal, M.; Sharif, M.; Naqvi, Q. Electromagnetic scattering from anisotropic plasma-coated perfect electromagnetic conductor cylinders. AEU-Int. J. Electron. Commun.
**2014**, 68, 767–772. [Google Scholar] [CrossRef] - Soudais, P.; Steve, H.; Dubois, F. Scattering from several test-objects computed by 3-D hybrid IE/PDE methods. IEEE Trans. Antennas Propag.
**1999**, 47, 646–653. [Google Scholar] [CrossRef] - Graglia, R.; Uslenghi, P.; Zich, R. Moment Method with Isoporametric Element for Three-Dimensional A nisotropic Scatterers. Proc. IEEE
**1989**, 77, 750–760. [Google Scholar] [CrossRef] - Yuan, J.; Niu, Z.; Gu, C. Electromagnetic Scattering by Arbitrarily Shaped PEC Targets Coated with Anisotropic Media Using Equivalent Dipole-Moment Method. J. Infrared Millim. Terahertz Waves
**2010**, 31, 744–752. [Google Scholar] [CrossRef] - Chung, S.S.M. FDTD simulations on radar cross sections of metal cone and plasma covered metal cone. Vacuum
**2012**, 86, 970–984. [Google Scholar] [CrossRef] - Liu, S.; Zhong, S. Analysis of backscattering RCS of targets coated with parabolic distribution and time-varying plasma media. Optik-Int. J. Light Electron. Opt.
**2013**, 124, 6850–6852. [Google Scholar] [CrossRef] - Xu, L.; Yuan, N. FDTD Formulations for Scattering From 3-D Anisotropic Magnetized Plasma Objects. IEEE Antennas Wirel. Propag. Lett.
**2006**, 5, 335–338. [Google Scholar] [CrossRef] - Sheng, X.; Peng, Z. Analysis of scattering by large objects with off-diagonally anisotropic material using finite element-boundary integral-multilevel fast multipole algorithm. IET Microw. Antennas Propag.
**2010**, 4, 492–500. [Google Scholar] [CrossRef] - Wanjun, S.; Hou, Z. RCS Prediction of Objects Coated by Magnetized Plasma Via Scale Model With FDTD. IEEE Trans. Microw. Theory Tech.
**2017**, 65, 1939–1945. [Google Scholar] [CrossRef] - Dan, L.; Tong, C.; Jiao, W. RCS simulation of plasma-coated targets modeled by NURBS surfaces. In Proceedings of the 2009 Asia Pacific Microwave Conference, Singapore, 7–10 December 2009. [Google Scholar]
- Liu, S.; Guo, L. Analyzing the Electromagnetic Scattering Characteristics for 3-D Inhomogeneous Plasma Sheath Based on PO Method. IEEE Trans. Plasma Sci.
**2016**, 44, 2838–2843. [Google Scholar] [CrossRef] - Yu, Q.; Cong, Z.; He, Z.; Ding, D.; Chen, R. Study on Electromagnetic Scattering Characteristic of Hypervelocity Model with SBR Method. In Proceedings of the 2018 IEEE International Conference on Computational Electromagnetics (ICCEM), Chengdu, China, 26–28 March 2018. [Google Scholar]
- Bian, Z.; Li, J.; Guo, L.; Luo, X. Analyzing the Electromagnetic Scattering Characteristics of a Hypersonic Vehicle Based on the Inhomogeneity Zonal Medium Model. IEEE Trans. Antennas Propag.
**2021**, 69, 971–982. [Google Scholar] [CrossRef] - Li, J.; Bao, H.; Ding, D. Analysis for Scattering of Non-homogeneous Medium by Time Domain Volume Shooting and Bouncing Rays. Appl. Comput. Electromagn. Soc. J.
**2021**, 36, 245–251. [Google Scholar] [CrossRef] - Yang, B.; Chen, R.; He, Z.; Yin, H. A Tetrahedral Meshed SBR method for RCS of plasma-coated cavity. In Proceedings of the 2019 International Applied Computational Electromagnetics Society Symposium-China (ACES), Nanjing, China, 8–11 August 2019; pp. 1–3. [Google Scholar] [CrossRef]
- Platzman, P.M.; Ozaki, H.T. Scattering of Electromagnetic Waves from an Infinitely Long Magnetized Cylindrical Plasma. J. Appl. Phys.
**1960**, 31, 1597–1601. [Google Scholar] [CrossRef] - Yeh, K.C.; Liu, C.H. Theory of ionospheric waves. IEEE Trans. Plasma Sci.
**1972**, 1, 42. [Google Scholar] [CrossRef] - Yao, J.; He, S.; Li, C.; Yin, H.; Wang, C.; Zhu, G. An asymptotic solution of the scattering from a biaxial electric anisotropic slab with a PEC substrate. J. Electromagn. Waves Appl.
**2013**, 27, 1534–1549. [Google Scholar] [CrossRef] - Yao, J.J.; He, S.Y.; Zhang, Y.H.; Yin, H.C.; Wang, C.; Zhu, G.Q. Evaluation of Scattering from Electrically Large and Complex PEC Target Coated with Uniaxial Electric Anisotropic Medium Layer Based on Asymptotic Solution in Spectral Domain. IEEE Trans. Antennas Propag.
**2014**, 62, 2175–2186. [Google Scholar] [CrossRef] - Wait, J.R. Some boundary value problems involving plasma media. J. Res. Natl. Bur. Stand.
**1961**, 65, 137–150. [Google Scholar] [CrossRef] - Elking, D.M.; Roedder, J.M.; Car, D.D.; Alspach, S.D. A review of high-frequency radar cross section analysis capabilities at McDonnell Douglas Aerospace. IEEE Antennas Propag. Mag.
**1995**, 37, 33–42. [Google Scholar] [CrossRef] - Gordon, W. Far-field approximations to the Kirchoff-Helmholtz representations of scattered fields. IEEE Trans. Antennas Propag.
**1975**, 23, 590–592. [Google Scholar] [CrossRef] - Titchener, J.B.; Willis, J.R. The reflection of electromagnetic waves from stratified anisotropic media. IEEE Trans. Antennas Propag.
**1991**, 39, 35–39. [Google Scholar] [CrossRef]

**Figure 8.**The perpendicular and parallel vectors ${\widehat{e}}_{\perp}^{i},{\widehat{e}}_{//}^{i}$ and ${\widehat{e}}_{\perp}^{r},{\widehat{e}}_{//}^{r}$.

**Figure 9.**The reflection coefficients of the plasma-coated slab with a PEC substrate: (

**a**) against the angle $\theta $ incidence $d=0.1\lambda $; (

**b**) against the slab’s thickness $d$ $\phi ={45}^{\circ}$.

**Figure 10.**Comparison of reflection coefficients of infinite PEC slab coated with different media: (

**a**) plasma coating; (

**b**) uniaxial media coating at $\phi ={0}^{\circ}$; (

**c**) uniaxial media coating at $\phi ={45}^{\circ}$.

**Figure 11.**Amplitude and phase of mode fields in lossless plasma coating: (

**a**) $\left|{R}^{0}\right|$; (

**b**) $\angle {R}^{0}$; (

**c**) $\left|R\right|$; (

**d**) $\angle R$.

**Figure 12.**Amplitude and phase of mode fields in lossy plasma coating: (

**a**) $\left|{R}^{0}\right|$; (

**b**) $\angle {R}^{0}$; (

**c**) $\left|R\right|$; (

**d**) $\angle R$.

**Figure 13.**The amplitude of mode fields in lossless plasma coating: (

**a**) $\left|{R}^{0}\right|$; (

**b**) $\left|{T}^{0}\right|$; (

**c**) $\left|R\right|$.

**Figure 14.**The phase of mode fields in lossless plasma coating: (

**a**) $\angle {R}^{0}$; (

**b**) $\angle {T}^{0}$; (

**c**) $\angle R$.

**Figure 15.**The amplitude of mode fields in lossy plasma coating: (

**a**) $\left|{R}^{0}\right|$; (

**b**) $\left|{T}^{0}\right|$; (

**c**) $\left|R\right|$.

**Figure 16.**The phase of mode fields in lossy plasma coating: (

**a**) $\angle {R}^{0}$; (

**b**) $\angle {T}^{0}$; (

**c**) $\angle R$.

**Figure 17.**A $4{\lambda}_{0}\times 5{\lambda}_{0}$ plate coated with a plasma layer and its monostatic RCSs against angle $\theta $. The layer’s thickness is $d=0.1{\lambda}_{0}$. The incident angle is $\phi ={0}^{\circ}$. The relative permittivity tensor elements are ${\epsilon}_{1}=6-3j,{\epsilon}_{2}=-5j,{\epsilon}_{3}=4-0.7j$.

**Figure 19.**Monostatic RCSs from a PEC cube coated with (

**a**) lossless and (

**b**) lossy plasma layer. The relative permittivity tensor elements of the lossless and lossy plasma, respectively, are ${\epsilon}_{1}=6,{\epsilon}_{2}=-5j,{\epsilon}_{3}=4$ and ${\epsilon}_{1}=6-3j,{\epsilon}_{2}=-5j,{\epsilon}_{3}=4-0.7j$; the incident angle is $\phi ={0}^{\circ}$.

**Figure 20.**Monostatic RCSs from a PEC cylinder coated with a plasma layer in (

**a**) $xoy$ and (

**b**) $xoz$ planes. The relative permittivity tensor elements of the lossless and lossy plasma, respectively, are ${\epsilon}_{1}=10,{\epsilon}_{2}=-5j,{\epsilon}_{3}=3$ and ${\epsilon}_{1}=10-2j,{\epsilon}_{2}=-5j,{\epsilon}_{3}=3-4j$.

**Figure 22.**Monostatic RCSs of the plasma-coated missile in (

**a**) $xoy$ and (

**b**) $xoz$ planes. The relative permittivity tensor elements of the lossless and lossy plasma, respectively, are ${\epsilon}_{1}=10,{\epsilon}_{2}=-5j,{\epsilon}_{3}=3$ and ${\epsilon}_{1}=10-2j,{\epsilon}_{2}=-5j,{\epsilon}_{3}=3-4j$.

**Figure 24.**Monostatic RCSs of the plasma-coated aircraft in three observation planes: (

**a**) $xoy$, (

**b**) $xoz$, and (

**c**) $yoz$. The relative permittivity tensor elements of the lossless and lossy plasma, respectively, are ${\epsilon}_{1}=10,{\epsilon}_{2}=-5j,{\epsilon}_{3}=3$ and ${\epsilon}_{1}=10-2j,{\epsilon}_{2}=-5j,{\epsilon}_{3}=3-4j$.

**Table 1.**The CPU time and required memory of Figure 17.

Method | The Proposed Method | HFSS |
---|---|---|

CPU time | 6.17 s | 308 s |

Required memory | 30.79 MB | 6.26 GB |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Rao, Z.; Zhu, G.; He, S.; Li, C.; Yang, Z.; Liu, J.
Simulation and Analysis of Electromagnetic Scattering from Anisotropic Plasma-Coated Electrically Large and Complex Targets. *Remote Sens.* **2022**, *14*, 764.
https://doi.org/10.3390/rs14030764

**AMA Style**

Rao Z, Zhu G, He S, Li C, Yang Z, Liu J.
Simulation and Analysis of Electromagnetic Scattering from Anisotropic Plasma-Coated Electrically Large and Complex Targets. *Remote Sensing*. 2022; 14(3):764.
https://doi.org/10.3390/rs14030764

**Chicago/Turabian Style**

Rao, Zhenmin, Guoqiang Zhu, Siyuan He, Chao Li, Zewang Yang, and Jian Liu.
2022. "Simulation and Analysis of Electromagnetic Scattering from Anisotropic Plasma-Coated Electrically Large and Complex Targets" *Remote Sensing* 14, no. 3: 764.
https://doi.org/10.3390/rs14030764