Hostname: page-component-77c89778f8-vpsfw Total loading time: 0 Render date: 2024-07-20T04:31:39.012Z Has data issue: false hasContentIssue false
  • English
  • Français

On isotropic cloaking and interior transmission eigenvalue problems

Part of: Equations of mathematical physics and other areas of application Elliptic equations and systems Partial differential equations, boundary value problems Spectral theory and eigenvalue problems Optics, electromagnetic theory - General Miscellaneous topics - Partial differential equations

Published online by Cambridge University Press:  22 May 2017

XIA JI  and
HONGYU LIU
XIA JI
Affiliation:
LSEC, Institute of Computational Mathematics, Chinese Academy of Sciences, Beijing, 100190, P. R. China email: jixia@lsec.cc.ac.cn
HONGYU LIU
Affiliation:
Department of Mathematics, Hong Kong Baptist University, Kowloon Tong, Hong Kong SAR, P. R. China HKBU Institute of Research and Continuing Education, Virtual University Park, Shenzhen, P. R. China email: hongyu.liuip@gmail.com
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper is concerned with the invisibility cloaking in acoustic wave scattering from a new perspective. We are especially interested in achieving the invisibility cloaking by completely regular and isotropic mediums. It is shown that an interior transmission eigenvalue problem arises in our study, which is the one considered theoretically in Cakoni et al. (Transmission eigenvalues for inhomogeneous media containing obstacles, Inverse Problems and Imaging, 6 (2012), 373–398). Based on such an observation, we propose a cloaking scheme that takes a three-layer structure including a cloaked region, a lossy layer and a cloaking shell. The target medium in the cloaked region can be arbitrary but regular, whereas the mediums in the lossy layer and the cloaking shell are both regular and isotropic. We establish that if a certain non-transparency condition is satisfied, then there exists an infinite set of incident waves such that the cloaking device is nearly invisible under the corresponding wave interrogation. The set of waves is generated from the Herglotz approximation of the associated interior transmission eigenfunctions. We provide both theoretical and numerical justifications.

Keywords

Acoustic wave scatteringinvisibility cloakingisotropic and regularinterior transmission eigenvalues

MSC classification

Primary: 35J57: Boundary value problems for second-order elliptic systems 35P25: Scattering theory 35Q60: PDEs in connection with optics and electromagnetic theory 78A46: Inverse scattering problems
Secondary: 35R30: Inverse problems 65N25: Eigenvalue problems 65N30: Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods
Type
Papers
Information
European Journal of Applied Mathematics , Volume 29 , Issue 2 , April 2018 , pp. 253 - 280
Copyright
Copyright © Cambridge University Press 2017 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Alu, A. & Engheta, N. (2005) Achieving transparency with plasmonic and metamaterial coatings. Phys. Rev. E 72, 016623.CrossRefGoogle ScholarPubMed
[2] Ammari, H., Ciraolo, G., Kang, H., Lee, H. & Milton, G. (2013) Spectral theory of a Neumann–Poincaré-type operator and analysis of cloaking due to anomalous localized resonance. Arch. Ration. Mech. Anal. 208, 667–692.CrossRefGoogle Scholar
[3] Ammari, H., Kang, H., Lee, H. & Lim, M. (2013) Enhancement of near-cloaking using generalized polarization tensors vanishing structures. Part I: The conductivity problem. Comm. Math. Phys. 317, 253–266.CrossRefGoogle Scholar
[4] Ammari, H., Kang, H., Lee, H. & Lim, M. (2013) Enhancement of near-cloaking. Part II: The Helmholtz equation. Comm. Math. Phys. 317, 485–502.CrossRefGoogle Scholar
[5] Ammari, H., Kang, H., Lee, H. & Lim, M. (2013) Enhancement of near cloaking for the full Maxwell equations. SIAM J. Appl. Math. 73, 2055–2076.CrossRefGoogle Scholar
[6] Ando, K., Kang, H. & Liu, H. (2016) Plasmon resonance with finite frequencies: A validation of the quasi-static approximation for diametrically small inclusions. SIAM J. Appl. Math. 76 (2), 731–749.CrossRefGoogle Scholar
[7] Bonnet-Ben Dhia, A., Chesnel, L. & Nazarov, S. A. (2015) Non-scattering wavenumbers and far field invisibility for a finite set of incident/scattering directions. Inverse Problems 31 (4), 045006.CrossRefGoogle Scholar
[8] Bao, G. & Liu, H. (2014) Nearly cloaking the full Maxwell equations. SIAM J. Appl. Math. 74, 724–742.CrossRefGoogle Scholar
[9] Bao, G., Liu, H. & Zou, J. (2014) Nearly cloaking the full Maxwell equations: Cloaking active contents with general conducting layers. J. Math. Pures Appl. 101 (9), 716–733.CrossRefGoogle Scholar
[10] Blåsten, E. & Liu, H. On corners scattering stably and stable shape determination by a single far-field pattern, arXiv:1611.03647.Google Scholar
[11] Blåsten, E., Päivärinta, L. & Sylvester, J. (2014) Corners always scatter. Comm. Math. Phys. 331 (2), 725–753.CrossRefGoogle Scholar
[12] Cakoni, F. & Colton, D. (2014) A Qualitative Approach to Inverse Scattering Theory, Springer, New York.CrossRefGoogle Scholar
[13] Cakoni, F., Cossonniére, A. & Haddar, H. (2012) Transmission eigenvalues for inhomogeneous media containing obstacles. Inverse Problems and Imaging 6, 373–398.CrossRefGoogle Scholar
[14] Cakoni, F., Colton, D. & Gintides, D. (2010) The interior transmission eigenvalue problem. SIAM J. Math. Anal. 42, 2912–2921.CrossRefGoogle Scholar
[15] Cakoni, F., Gintides, D. & Haddar, H. (2010) The existence of an infinite discrete set of transmission eigenvalues. SIAM J. Math. Anal. 42, 237–255.CrossRefGoogle Scholar
[16] Cakoni, F. & Kirsch, A. (2010) On the interior transmission eigenvalue problem. Int. J. Comput. Sci. Math. 3, 142–167.CrossRefGoogle Scholar
[17] Chen, H. & Chan, C. T. (2010) Acoustic cloaking and transformation acoustics. J. Phys. D: Appl. Phys. 43, 113001.CrossRefGoogle Scholar
[18] Colton, D., Kirsch, A. & Päivärinta, L. (1989) Far-field patterns for acoustic waves in an inhomogeneous medium. SIAM J. Math. Anal. 20, 1472–1483.CrossRefGoogle Scholar
[19] Colton, D. & Kress, R. (1998) Inverse Acoustic and Electromagnetic Scattering Theory, 2nd ed., Springer-Verlag, Berlin.CrossRefGoogle Scholar
[20] Colton, D. & Monk, P. (1988) The inverse scattering problem for time-harmonic acoustic waves in an inhomogeneous medium. Quart. J. Mech. Apple. Math. 41, 97–125.CrossRefGoogle Scholar
[21] Colton, D., Monk, P. & Sun, J. (2010) Analytical and computational methods for transmission eigenvalues. Inverse Problems 26, 045011.CrossRefGoogle Scholar
[22] Elschner, J. & Hu, G. (2015) Corners and edges always scatter. Inverse Problems 31 (1), 015003.CrossRefGoogle Scholar
[23] Elschner, J. & Hu, G. Acoustic scattering from corners, edges and circular cones, arXiv:1603.05186.Google Scholar
[24] Greenleaf, A., Kurylev, Y., Lassas, M., Leonhardt, U. & Uhlmann, G. (2012) Cloaked electromagnetic, acoustic, and quantum amplifiers via transformation optics. Proc. Natl. Acad. Sci. (PNAS) 109, 10169–10174.CrossRefGoogle ScholarPubMed
[25] Greenleaf, A., Kurylev, Y., Lassas, M. & Uhlmann, G. (2007) Improvement of cylindrical cloaking with SHS lining. Opt. Exp. 15, 12717–12734.CrossRefGoogle ScholarPubMed
[26] Greenleaf, A., Kurylev, Y., Lassas, M. & Uhlmann, G. (2011) Approximate quantum and acoustic cloaking. J. Spectr. Theory 1, 27–80.CrossRefGoogle Scholar
[27] Greenleaf, A., Kurylev, Y., Lassas, M. & Uhlmann, G. (2008) Isotropic transformation optics: Approximate acoustic and quantum cloaking. New J. Phys. 10, 115024.CrossRefGoogle Scholar
[28] Greenleaf, A., Kurylev, Y., Lassas, M. & Uhlmann, G. (2009) Invisibility and inverse prolems. Bull. A. M. S. 46, 55–97.CrossRefGoogle Scholar
[29] Greenleaf, A., Kurylev, Y., Lassas, M. & Uhlmann, G. (2009) Cloaking devices, electromagnetic wormholes and transformation optics. SIAM Rev. 51, 3–33.CrossRefGoogle Scholar
[30] Greenleaf, A., Lassas, M. & Uhlmann, G. (2003) Anisotropic conductivities that cannot be detected by EIT. Physiolog. Meas. 24, 413.CrossRefGoogle ScholarPubMed
[31] Greenleaf, A., Lassas, M. & Uhlmann, G. (2003) On nonuniqueness for Calderón's inverse problem. Math. Res. Lett. 10, 685–693.CrossRefGoogle Scholar
[32] Kettunen, H., Lassas, M. & Ola, P. (2014) On absence and existence of the anomalous localized resonace without the quasi-static approximation, arXiv:1406.6224v2.Google Scholar
[33] Kohn, R. V., Lu, J., Schweizer, B. & Weinstein, M. I. (2014) A variational perspective on cloaking by anomalous localized resonance. Comm. Math. Phys. 328, 1–27.CrossRefGoogle Scholar
[34] Kohn, R., Onofrei, O., Vogelius, M. & Weinstein, M. (2010) Cloaking via change of variables for the Helmholtz equation. Comm. Pure Appl. Math. 63, 973–1016.CrossRefGoogle Scholar
[35] Kirsch, A. (2009) On the existence of transmission eigenvalues. Inverse Probl. Imaging 3, 155–172.CrossRefGoogle Scholar
[36] Kirsch, A. & Päivärinta, L. (1998) On recovering obstacles inside inhomogeneities. Math. Methods Appl. Sci. 21 (7), 619–651.3.0.CO;2-P>CrossRefGoogle Scholar
[37] Leonhardt, U. (2006) Optical conformal mapping. Science 312, 1777–1780.CrossRefGoogle ScholarPubMed
[38] Li, H., Li, J. & Liu, H. (2015) On quasi-static cloaking due to anomalous localized resonance in ℝ3. SIAM J. Appl. Math. 75, 1245–1260.CrossRefGoogle Scholar
[39] Li, J., Liu, H., Rondi, L. & Uhlmann, G. (2015) Regularized transformation-optics cloaking for the Helmholtz equation: From partial cloak to full cloak. Comm. Math. Phys. 335, 671–712.CrossRefGoogle Scholar
[40] Liu, H. (2009) Virtual reshaping and invisibility in obstacle scattering. Inverse Problems 25, 045006.CrossRefGoogle Scholar
[41] Liu, H. (2013) On near-cloak in acoustic scattering. J. Differ. Equ. 254, 1230–1246.CrossRefGoogle Scholar
[42] Liu, H. & Sun, H., (20) Enhanced near-cloak by FSH lining. J. Math. Pures Appl. 99 (9), 17–42.CrossRefGoogle Scholar
[43] McLean, W. (2000) Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press, Cambridge.Google Scholar
[44] Milton, G. W. & Nicorovici, N.-A. P. (2006) On the cloaking effects associated with anomalous localized resonance. Proc. Roy. Soc. Lond. A 462, 3027–3095.Google Scholar
[45] Nédélec, J. C. (2001) Acoustic and Electromagnetic Equations: Integral Representations for Harmonic Problems, Springer-Verlag, New York.CrossRefGoogle Scholar
[46] Norris, A. N. (2008) Acoustic cloaking theory. Proc. R. Soc. Lond. A 464, 2411–2434.Google Scholar
[47] Päivärinta, L. & Sylvester, J., (2008) Transmission eigenvalues. SIAM J. Math. Anal. 40, 738–753.CrossRefGoogle Scholar
[48] Pendry, J. B., Schurig, D. & Smith, D. R. (2006) Controlling electromagnetic fields. Science 312, 1780–1782.CrossRefGoogle ScholarPubMed
[49] Sun, J. (2011) Iterative methods for transmission eigenvalues. SIAM J. Numer. Anal. 49, 1860–1874.CrossRefGoogle Scholar
[50] Uhlmann, G. (2009) Visibility and invisibility. In: ICIAM 07–6th International Congress on Industrial and Applied Mathematics, Eur. Math. Soc., Zürich, pp. 381–408.CrossRefGoogle Scholar
[51] Weck, N. (2004) Approximation by Herglotz wave functions. Math. Methods Appl. Sci. 27, 155–162.CrossRefGoogle Scholar
[52] Wloka, J. (1987) Partial Dierential Equations, Cambridge University Press, Cambridge.CrossRefGoogle Scholar