Active manipulation of exterior electromagnetic fields by using surface sources

Onofrei, Daniel; Platt, Eric; Egarguin, Neil

doi:10.1090/qam/1567

Authors: Daniel Onofrei, Eric Platt and Neil Jerome A. Egarguin
Journal: Quart. Appl. Math. 78 (2020), 641-670
MSC (2010): Primary 35Q60, 45Q05
DOI: https://doi.org/10.1090/qam/1567
Published electronically: January 22, 2020
MathSciNet review: 4148822
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Abstract: In this paper, we establish a scheme for the active manipulation of electromagnetic fields in prescribed exterior regions using a surface source. We prove the existence of the necessary surface current (electric or magnetic) on a single source to approximate prescribed electromagnetic fields on given regions of space (bounded or possibly the far field). We provide two constructive schemes for the computation of the required surface currents: our first strategy makes use of the Debye representation results for the electromagnetic field and builds up on previous control results for scalar fields discussed in [J. Integral Equations Appl. 26 (2014), pp. 553–579]; the second strategy we propose makes use of integral electromagnetic representation results and follows theoretically from the first. We provide theoretical validation for both computational schemes and present supporting numerical simulations for the first strategy in several applied scenarios.

References

Milton Abramowitz and Irene A. Stegun, Handbook of mathematical functions with formulas, graphs, and mathematical tables, National Bureau of Standards Applied Mathematics Series, vol. 55, For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C., 1964. MR 0167642

M. Alian and H. Oraizi, Electromagnetic multiple pec object scattering using equivalence principle and addition theorem for spherical wave harmonics, IEEE Transactions on Antennas and Propagation 66 (2018), no. 11, 6233–6243.

Habib Ammari, Hyeonbae Kang, Hyundae Lee, Mikyoung Lim, and Sanghyeon Yu, Enhancement of near cloaking for the full Maxwell equations, SIAM J. Appl. Math. 73 (2013), no. 6, 2055–2076. MR 3127003, DOI https://doi.org/10.1137/120903610
Habib Ammari, Matias Ruiz, Sanghyeon Yu, and Hai Zhang, Mathematical analysis of plasmonic resonances for nanoparticles: the full Maxwell equations, J. Differential Equations 261 (2016), no. 6, 3615–3669. MR 3527640, DOI https://doi.org/10.1016/j.jde.2016.05.036
Habib Ammari, Michael S. Vogelius, and Darko Volkov, Asymptotic formulas for perturbations in the electromagnetic fields due to the presence of inhomogeneities of small diameter. II. The full Maxwell equations, J. Math. Pures Appl. (9) 80 (2001), no. 8, 769–814. MR 1860816, DOI https://doi.org/10.1016/S0021-7824%2801%2901217-X
Habib Ammari, Wei Wu, and Sanghyeon Yu, Double-negative electromagnetic metamaterials due to chirality, Quart. Appl. Math. 77 (2019), no. 1, 105–130. MR 3897921, DOI https://doi.org/10.1090/qam/1516
Thomas S. Angell and Andreas Kirsch, Optimization methods in electromagnetic radiation, Springer Monographs in Mathematics, Springer-Verlag, New York, 2004. MR 2026856
Andrii Anikushyn and Michael Pokojovy, Global well-posedness and exponential stability for heterogeneous anisotropic Maxwell’s equations under a nonlinear boundary feedback with delay, J. Math. Anal. Appl. 475 (2019), no. 1, 278–312. MR 3944321, DOI https://doi.org/10.1016/j.jmaa.2019.02.042

Constantine A. Balanis, Advanced engineering electromagnetics, Second Edition, Wiley Interscience, 2012.

Alain Bensoussan, Giuseppe Da Prato, Michel C. Delfour, and Sanjoy K. Mitter, Representation and control of infinite dimensional systems, 2nd ed., Systems & Control: Foundations & Applications, Birkhäuser Boston, Inc., Boston, MA, 2007. MR 2273323

Wenshan Cai and Vladimir Shalaev, Optical metamaterials: Fundamentals and applications, 1st ed., Springer-Verlag, 2010.

Sebastien Clauzier, Said M. Mikki, and Yahia M. M. Antar, Design of near-field synthesis arrays through global optimization, IEEE Trans. Antennas and Propagation 63 (2015), no. 1, 151–165. MR 3301535, DOI https://doi.org/10.1109/TAP.2014.2367536
David L. Colton and Rainer Kress, Integral equation methods in scattering theory, Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1983. A Wiley-Interscience Publication. MR 700400
David Colton and Rainer Kress, Inverse acoustic and electromagnetic scattering theory, 3rd ed., Applied Mathematical Sciences, vol. 93, Springer, New York, 2013. MR 2986407
Patrick Courilleau and Thierry Horsin Molinaro, On the controllability for Maxwell’s equations in specific media, C. R. Math. Acad. Sci. Paris 341 (2005), no. 11, 665–668 (English, with English and French summaries). MR 2183345, DOI https://doi.org/10.1016/j.crma.2005.10.011

Richard V. Craster and Sebastien Guenneau, Acoustic metamaterials: Negative refraction, imaging, lensing and cloaking, Springer, 2013.

Steven A Cummer and David Schurig, One path to acoustic cloaking, New Journal of Physics 9 (2007), no. 3.

M. Darbas, O. Goubet, and S. Lohrengel, Exact boundary controllability of the second-order Maxwell system: theory and numerical simulation, Comput. Math. Appl. 63 (2012), no. 7, 1212–1237. MR 2900096, DOI https://doi.org/10.1016/j.camwa.2011.12.046
Marion Darbas and Stephanie Lohrengel, Numerical reconstruction of small perturbations in the electromagnetic coefficients of a dielectric material, J. Comput. Math. 32 (2014), no. 1, 21–38. MR 3164915, DOI https://doi.org/10.4208/jcm.1309-m4378
Anthony J. Devaney, Nonradiating surface sources, J. Opt. Soc. Amer. A 21 (2004), no. 11, 2216–2222. MR 2122596, DOI https://doi.org/10.1364/JOSAA.21.002216
Anthony J. Devaney, Mathematical foundations of imaging, tomography and wavefield inversion, Cambridge University Press, Cambridge, 2012. MR 2975719

A. Doicu, T. Wriedt, and I. Eremin, Acoustic and electromagnetic scattering analysis using discrete sources, Springer Monographs in Mathematics, London: Academic Press, 2000.

L. Dolin, On the possibility of comparison of three-dimensional electromagnetic systems with nonuniform anisotropic filling, Izv. Vyssh. Uchebn. Zaved. Radiofiz. 4 (196101), 964–967.

Junjie Du, Shiyang Liu, and Zhifang Lin, Broadband optical cloak and illusion created by the low order active sources, Opt. Express 20 (2012), no. 8, 8608–8617.

Neil Jerome A. Egarguin, Daniel Onofrei, and Eric Platt, Sensitivity analysis for the active manipulation of helmholtz fields in 3d, Inverse Problems in Science and Engineering (2018).

Eng Swee Siah, M. Sasena, J. L. Volakis, P. Y. Papalambros, and R. W. Wiese, Fast parameter optimization of large-scale electromagnetic objects using direct with kriging metamodeling, IEEE Transactions on Microwave Theory and Techniques 52 (2004), no. 1, 276–285.

Nader Engheta and Richard W Ziolkowski, Metamaterials: Physics and engineering explorations, Wiley-Interscience, 2006.

Ariel Epstein and George V. Eleftheriades, Huygens’ metasurfaces via the equivalence principle: design and applications, J. Opt. Soc. Am. B 33 (2016), no. 2, A31–A50.

Feng Ling, Chao-Fu Wang, and Jian-Ming Jin, An efficient algorithm for analyzing large-scale microstrip structures using adaptive integral method combined with discrete complex-image method, IEEE Transactions on Microwave Theory and Techniques 48 (2000), no. 5, 832–839.

Allan Greenleaf, Matti Lassas, and Gunther Uhlmann, Anisotropic conductivities that cannot be detected by eit., Physiological measurement 24 2 (2003), 413–419.

Thorkild B. Hansen and Arthur D. Yaghjian, Plane-wave theory of time-domain fields, IEEE Press Series on Electromagnetic Wave Theory, IEEE Press, New York, 1999. Near-field scanning applications. MR 1767418
M. Hubenthal and D. Onofrei, Sensitivity analysis for active control of the Helmholtz equation, Appl. Numer. Math. 106 (2016), 1–23. MR 3499954, DOI https://doi.org/10.1016/j.apnum.2016.03.003
S. S. Krigman and C. E. Wayne, Boundary controllability of Maxwell’s equations with nonzero conductivity inside a cube. I. Spectral controllability, J. Math. Anal. Appl. 329 (2007), no. 2, 1375–1396. MR 2294900, DOI https://doi.org/10.1016/j.jmaa.2006.06.101
S. S. Krigman and C. E. Wayne, Boundary controllability of Maxwell’s equations with nonzero conductivity inside a cube. II. Lack of exact controllability and controllability for very smooth solutions, J. Math. Anal. Appl. 329 (2007), no. 2, 1355–1374. MR 2296930, DOI https://doi.org/10.1016/j.jmaa.2006.06.102
John E. Lagnese, Exact boundary controllability of Maxwell’s equations in a general region, SIAM J. Control Optim. 27 (1989), no. 2, 374–388. MR 984833, DOI https://doi.org/10.1137/0327019
Irena Lasiecka and Roberto Triggiani, Control theory for partial differential equations: continuous and approximation theories. I, Encyclopedia of Mathematics and its Applications, vol. 74, Cambridge University Press, Cambridge, 2000. Abstract parabolic systems. MR 1745475
Shung Wu Lee, Johannes Boersma, Chak Lam Law, and Georges A. Deschamps, Singularity in Green’s function and its numerical evaluation, IEEE Trans. Antennas and Propagation 28 (1980), no. 3, 311–317. MR 574002, DOI https://doi.org/10.1109/TAP.1980.1142342
J.-L. Lions, Contrôle optimal de systèmes gouvernés par des équations aux dérivées partielles, Dunod, Paris; Gauthier-Villars, Paris, 1968 (French). Avant propos de P. Lelong. MR 0244606

J. Ma, Z. Nie, and X. Sun, Efficient modeling of large-scale electromagnetic well-logging problems using an improved nonconformal fem-ddm, IEEE Transactions on Geoscience and Remote Sensing 52 (2014), no. 3, 1825–1833.

Qian Ma, Zhong Mei, Shou Kui Zhu, Tian Yu Jin, and Tie Jun Cui, Experiments on active cloaking and illusion for laplace equation, Physical review letters 111 (201310), 173901.

Edwin A. Marengo and Anthony J. Devaney, The inverse source problem of electromagnetics: linear inversion formulation and minimum energy solution, IEEE Trans. Antennas and Propagation 47 (1999), no. 2, 410–412. MR 1686310, DOI https://doi.org/10.1109/8.761085
Edwin A. Marengo, Anthony J. Devaney, and Fred K. Gruber, Inverse source problem with reactive power constraint, IEEE Trans. Antennas and Propagation 52 (2004), no. 6, 1586–1595. MR 2071404, DOI https://doi.org/10.1109/TAP.2004.829408
Edwin A. Marengo and Richard W. Ziolkowski, Nonradiating and minimum energy sources and their fields: generalized source inversion theory and applications, IEEE Trans. Antennas and Propagation 48 (2000), no. 10, 1553–1562. MR 1809079, DOI https://doi.org/10.1109/8.899672
Said M. Mikki and Yahia M. M. Antar, A theory of antenna electromagnetic near field—Part II, IEEE Trans. Antennas and Propagation 59 (2011), no. 12, 4706–4724. MR 2907047, DOI https://doi.org/10.1109/TAP.2011.2165500

R. Mittra and R.K. Gordon, Radar scattering from bodies of revolution using an efficient partial differential equation algorithm, IEEE Transactions on Antennas and Propagation 37 (1989), no. 5, 538–545.

R. Mittra and O. Ramahi, Absorbing boundary conditions for the direct solution of partial differential equations arising in electromagnetic scattering problems, Progress In Electromagnetics Research 02 (1990), 133–173.

Shaun N. Mosley, Alternative potentials for the electromagnetic field, J. Math. Phys. 39 (1998), no. 5, 2702–2713. MR 1621454, DOI https://doi.org/10.1063/1.532415
Serge Nicaise, Exact boundary controllability of Maxwell’s equations in heterogeneous media and an application to an inverse source problem, SIAM J. Control Optim. 38 (2000), no. 4, 1145–1170. MR 1760064, DOI https://doi.org/10.1137/S0363012998344373

J. Niitsuma, Generalized integral formulation of electromagnetic cartesian multipole moments, Journal of Electromagnetic Waves and Applications (2013), 1–9.

Andrew N. Norris, Feruza A. Amirkulova, and William J. Parnell, Source amplitudes for active exterior cloaking, Inverse Problems 28 (2012), no. 10, 105002, 20. MR 2974016, DOI https://doi.org/10.1088/0266-5611/28/10/105002
Michael O’Neil, A generalized Debye source approach to electromagnetic scattering in layered media, J. Math. Phys. 55 (2014), no. 1, 012901, 16. MR 3390430, DOI https://doi.org/10.1063/1.4862747
Daniel Onofrei, Active manipulation of fields modeled by the Helmholtz equation, J. Integral Equations Appl. 26 (2014), no. 4, 553–579. MR 3299831, DOI https://doi.org/10.1216/JIE-2014-26-4-553
D. Onofrei and E. Platt, On the synthesis of acoustic sources with controllable near fields, Wave Motion 77 (2018), 12–27. MR 3754438, DOI https://doi.org/10.1016/j.wavemoti.2017.10.004
J. B. Pendry, D. Schurig, and D. R. Smith, Controlling electromagnetic fields, Science 312 (2006), no. 5781, 1780–1782. MR 2237570, DOI https://doi.org/10.1126/science.1125907

Vicente Romero-Garcia and Anne-Christine Hladky-Hennion, Fundamentals and applications of acoustic metamaterials: From seismic to radio frequency, Wiley-Interscience, 2019.

E. G. Peter Rowe, Decomposition of vector fields by scalar potentials, J. Phys. A 12 (1979), no. 1, 145–150. MR 518538
S. S. Krigman, Exact boundary controllability of Maxwell’s equations with weak conductivity in the heterogeneous medium inside a general domain, Discrete Contin. Dyn. Syst. Dynamical systems and differential equations. Proceedings of the 6th AIMS International Conference, suppl. (2007), 590–601. MR 2409895

Kazuaki Sakoda, Electromagnetic metamaterials: Modern insights into macroscopic electromagnetic fields, Springer Series in Materials Science, Springer, 2019.

M. Selvanayagam and G. V. Eleftheriades, An active electromagnetic cloak using the equivalence principle, IEEE Antennas and Wireless Propagation Letters 11 (2012), 1226–1229.

Michael Selvanayagam and George V. Eleftheriades, Experimental demonstration of active electromagnetic cloaking, Phys. Rev. X 3 (2013), 041011.

Valery P. Smyshlyaev, The high-frequency diffraction of electromagnetic waves by cones of arbitrary cross sections, SIAM J. Appl. Math. 53 (1993), no. 3, 670–688. MR 1218379, DOI https://doi.org/10.1137/0153034
Ting Su, Lei Du, and Rushan Chen, Electromagnetic scattering for multiple PEC bodies of revolution using equivalence principle algorithm, IEEE Trans. Antennas and Propagation 62 (2014), no. 5, 2736–2744. MR 3209123, DOI https://doi.org/10.1109/TAP.2014.2308522

T. Su, J. Wang, J. Chen, R. Chen, and W. Wang, Fast computation of em scattering from an uav swarm, 2019 IEEE international conference on computational electromagnetics (ICCEM), 2019, pp. 1–3.

Wei Wei, Hong-Ming Yin, and Juming Tang, An optimal control problem for microwave heating, Nonlinear Anal. 75 (2012), no. 4, 2024–2036. MR 2870896, DOI https://doi.org/10.1016/j.na.2011.10.003
Calvin H. Wilcox, Debye potentials, J. Math. Mech. 6 (1957), 167–201. MR 0087462, DOI https://doi.org/10.1512/iumj.1957.6.56007

Y. Xu, H. Yang, R. Shen, L. Zhu, and X. Huang, Scattering analysis of multiobject electromagnetic systems using stepwise method of moment, IEEE Transactions on Antennas and Propagation 67 (2019), no. 3, 1740–1747.

A.D. Yaghjian, Electric dyadic green’s functions in the source region, Proceedings of the IEEE 68 (1980), no. 2, 248–263.

Ran Zhao, Zhi Xiang Huang, Yongpin P. Chen, and Jun Hu, Solving electromagnetic scattering from complex composite objects with domain decomposition method based on hybrid surface integral equations, Eng. Anal. Bound. Elem. 85 (2017), 99–104. MR 3719416, DOI https://doi.org/10.1016/j.enganabound.2017.09.014