Skip to Main Content

Bulletin of the American Mathematical Society

The Bulletin publishes expository articles on contemporary mathematical research, written in a way that gives insight to mathematicians who may not be experts in the particular topic. The Bulletin also publishes reviews of selected books in mathematics and short articles in the Mathematical Perspectives section, both by invitation only.

ISSN 1088-9485 (online) ISSN 0273-0979 (print)

The 2020 MCQ for Bulletin of the American Mathematical Society is 0.84.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

Invisibility and inverse problems
HTML articles powered by AMS MathViewer

by Allan Greenleaf, Yaroslav Kurylev, Matti Lassas and Gunther Uhlmann PDF
Bull. Amer. Math. Soc. 46 (2009), 55-97 Request permission

Abstract:

We describe recent theoretical and experimental progress on making objects invisible. Ideas for devices that would have once seemed fanciful may now be at least approximately realized physically, using a new class of artificially structured materials, metamaterials. The equations that govern a variety of wave phenomena, including electrostatics, electromagnetism, acoustics and quantum mechanics, have transformation laws under changes of variables which allow one to design material parameters that steer waves around a hidden region, returning them to their original path on the far side. Not only are observers unaware of the contents of the hidden region, they are not even aware that something is being hidden; the object, which casts no shadow, is said to be cloaked. Proposals for, and even experimental implementations of, such cloaking devices have received the most attention, but other devices having striking effects on wave propagation, unseen in nature, are also possible. These designs are initially based on the transformation laws of the relevant PDEs, but due to the singular transformations needed for the desired effects, care needs to be taken in formulating and analyzing physically meaningful solutions. We recount the recent history of the subject and discuss some of the mathematical and physical issues involved.
References
  • M. J. Ablowitz, D. Bar Yaacov, and A. S. Fokas, On the inverse scattering transform for the Kadomtsev-Petviashvili equation, Stud. Appl. Math. 69 (1983), no. 2, 135–143. MR 715426, DOI 10.1002/sapm1983692135
  • Grégoire Allaire, Homogenization and two-scale convergence, SIAM J. Math. Anal. 23 (1992), no. 6, 1482–1518. MR 1185639, DOI 10.1137/0523084
    • Allaire2 G. Allaire and A. Damlamian and U. Hornung, Two-scale convergence on periodic surfaces and applications, In A. Bourgeat, C. Carasso, S. Luckhaus and A. Mikelic (eds.), Mathematical Modelling of Flow through Porous Media, 15–25, Singapore, World Scientific, 1995. AE A. Alu and N. Engheta, Achieving transparency with plasmonic and metamaterial coatings, Phys. Rev. E 72, 016623 (2005)
  • Kari Astala and Lassi Päivärinta, Calderón’s inverse conductivity problem in the plane, Ann. of Math. (2) 163 (2006), no. 1, 265–299. MR 2195135, DOI 10.4007/annals.2006.163.265
  • Kari Astala, Lassi Päivärinta, and Matti Lassas, Calderón’s inverse problem for anisotropic conductivity in the plane, Comm. Partial Differential Equations 30 (2005), no. 1-3, 207–224. MR 2131051, DOI 10.1081/PDE-200044485
    • ALP2 K. Astala, M. Lassas, and L. Päivärinta, Limits of visibility and invisibility for Calderón’s inverse problem in the plane, in preparation.
  • H. Attouch, Variational convergence for functions and operators, Applicable Mathematics Series, Pitman (Advanced Publishing Program), Boston, MA, 1984. MR 773850
  • R. Beals and R. Coifman, Scattering, transformations spectrales et équations d’évolution non linéaire. II, Goulaouic-Meyer-Schwartz Seminar, 1981/1982, École Polytech., Palaiseau, 1982, pp. Exp. No. XXI, 9 (French). MR 671618
  • R. Beals and R. R. Coifman, Multidimensional inverse scatterings and nonlinear partial differential equations, Pseudodifferential operators and applications (Notre Dame, Ind., 1984) Proc. Sympos. Pure Math., vol. 43, Amer. Math. Soc., Providence, RI, 1985, pp. 45–70. MR 812283, DOI 10.1090/pspum/043/812283
  • Y. Benveniste and T. Miloh, Neutral inhomogeneities in conduction phenomena, J. Mech. Phys. Solids 47 (1999), no. 9, 1873–1892. MR 1695877, DOI 10.1016/S0022-5096(98)00127-6
  • Yu. M. Berezanski, The uniqueness theorem in the inverse problem of spectral analysis for the Schrödinger equation, Trudy Moskov. Mat. Obšč. 7 (1958), 1–62 (Russian). MR 0101377
  • Liliana Borcea, Electrical impedance tomography, Inverse Problems 18 (2002), no. 6, R99–R136. MR 1955896, DOI 10.1088/0266-5611/18/6/201
  • Russell M. Brown and Rodolfo H. Torres, Uniqueness in the inverse conductivity problem for conductivities with $3/2$ derivatives in $L^p,\ p>2n$, J. Fourier Anal. Appl. 9 (2003), no. 6, 563–574. MR 2026763, DOI 10.1007/s00041-003-0902-3
  • Russell M. Brown and Gunther A. Uhlmann, Uniqueness in the inverse conductivity problem for nonsmooth conductivities in two dimensions, Comm. Partial Differential Equations 22 (1997), no. 5-6, 1009–1027. MR 1452176, DOI 10.1080/03605309708821292
  • A. L. Bukhgeim, Recovering a potential from Cauchy data in the two-dimensional case, J. Inverse Ill-Posed Probl. 16 (2008), no. 1, 19–33. MR 2387648, DOI 10.1515/jiip.2008.002
    • Cai W. Cai, U. Chettiar, A. Kildishev, V. Shalaev, Optical cloaking with metamaterials, Nature Photonics 1 (2007), 224–227. Cai2 W. Cai, U. Chettiar, A. Kildishev, G. Milton and V. Shalaev, Non-magnetic cloak without reflection, arXiv:0707.3641 (2007).
  • Alberto-P. Calderón, On an inverse boundary value problem, Seminar on Numerical Analysis and its Applications to Continuum Physics (Rio de Janeiro, 1980) Soc. Brasil. Mat., Rio de Janeiro, 1980, pp. 65–73. MR 590275
  • Sagun Chanillo, A problem in electrical prospection and an $n$-dimensional Borg-Levinson theorem, Proc. Amer. Math. Soc. 108 (1990), no. 3, 761–767. MR 998731, DOI 10.1090/S0002-9939-1990-0998731-1
    • Ch1 H. Chen and C.T. Chan, Transformation media that rotate electromagnetic fields, Appl. Phys. Lett. 90 (2007), 241105. Ch2 H. Chen, Z. Liang, P. Yao, X. Jiang, H. Ma and C.T. Chan, Extending the bandwidth of electromagnetic cloaks, Phys. Rev. B, 76 (2007), 241104(R). Ch3 H. Chen and C.T. Chan, Acoustic cloaking in three dimensions using acoustic metamaterials, Appl. Phys. Lett. 91 (2007), 183518. CWZK H.-S. Chen, B.-I. Wu, B. Zhang and J.A. Kong, Electromagnetic wave interactions with a metamaterial cloak, Phys. Rev. Lett. 99 (2007), 063903.
  • Margaret Cheney, David Isaacson, and Jonathan C. Newell, Electrical impedance tomography, SIAM Rev. 41 (1999), no. 1, 85–101. MR 1669729, DOI 10.1137/S0036144598333613
  • Andrej Cherkaev, Variational methods for structural optimization, Applied Mathematical Sciences, vol. 140, Springer-Verlag, New York, 2000. MR 1763123, DOI 10.1007/978-1-4612-1188-4
    • CPSSP S. Cummer, B.-I. Popa, D. Schurig, D. Smith and J. Pendry, Full-wave simulations of electromagnetic cloaking structures, Phys. Rev. E 74, 036621 (2006). CuSc S. Cummer and D. Schurig, One path to acoustic cloaking, New Jour. Phys. 9 (2007), 45. Cu S. Cummer, et al., Scattering theory derivation of a 3D acoustic cloaking shell, Phys. Rev. Lett. 100 (2008), 024301.
  • Gianni Dal Maso, An introduction to $\Gamma$-convergence, Progress in Nonlinear Differential Equations and their Applications, vol. 8, Birkhäuser Boston, Inc., Boston, MA, 1993. MR 1201152, DOI 10.1007/978-1-4612-0327-8
    • Dolin L. S. Dolin, To the possibility of comparison of three-dimensional electromagnetic systems with nonuniform anisotropic filling, Izv. Vuzove, Radiofizika 4 (1961), 964-967.
  • David Dos Santos Ferreira, Carlos E. Kenig, Johannes Sjöstrand, and Gunther Uhlmann, Determining a magnetic Schrödinger operator from partial Cauchy data, Comm. Math. Phys. 271 (2007), no. 2, 467–488. MR 2287913, DOI 10.1007/s00220-006-0151-9
    • ER A. Einstein and N. Rosen, The particle problem in the general theory of relativity, Physical Review 48 (1935), 73. Elef G. Eleftheriades and K. Balmain, eds., Negative-Refraction Metamaterials, IEEE/Wiley, Hoboken (2005).
  • L. D. Faddeev, The inverse problem in the quantum theory of scattering. II, Current problems in mathematics, Vol. 3 (Russian), Akad. Nauk SSSR Vsesojuz. Inst. Naučn. i Tehn. Informacii, Moscow, 1974, pp. 93–180, 259. (loose errata) (Russian). MR 0523015
  • Allan Greenleaf, Yaroslav Kurylev, Matti Lassas, and Gunther Uhlmann, Full-wave invisibility of active devices at all frequencies, Comm. Math. Phys. 275 (2007), no. 3, 749–789. MR 2336363, DOI 10.1007/s00220-007-0311-6
    • GKLU2 A. Greenleaf, Y. Kurylev, M. Lassas, G. Uhlmann: Electromagnetic wormholes and virtual magnetic monopoles from metamaterials, Phys. Rev. Lett. 99 (2007), 183901. GKLU3 A. Greenleaf, Y. Kurylev, M. Lassas, G. Uhlmann: Improvement of cylindrical cloaking with the SHS lining, Optics Express 15 (2007), 12717–12734. GKLU4 A. Greenleaf, Y. Kurylev, M. Lassas, G. Uhlmann, Electromagnetic wormholes via handlebody constructions, Comm. Math. Phys. 281 (2008), 369–385. GKLU5 A. Greenleaf, Y. Kurylev, M. Lassas, G. Uhlmann, Comment on “Scattering theory derivation of a 3D acoustic cloaking shell”, arXiv:0801.3279 (2008). GKLU6 A. Greenleaf, Y. Kurylev, M. Lassas, G. Uhlmann, Isotropic transformation optics: Approximate acoustic and quantum cloaking, New Journal of Physics (to appear); arXiv:0806.0085 (2008). GKLU7 A. Greenleaf, Y. Kurylev, M. Lassas, G. Uhlmann, Approximate quantum cloaking and almost trapped states, Phys. Rev. Lett. (to appear); arXiv:0806.0368 (2008). GKLU8 A. Greenleaf, Y. Kurylev, M. Lassas, G. Uhlmann, Approximate quantum and acoustic cloaking, in preparation.
  • Allan Greenleaf, Matti Lassas, and Gunther Uhlmann, The Calderón problem for conormal potentials. I. Global uniqueness and reconstruction, Comm. Pure Appl. Math. 56 (2003), no. 3, 328–352. MR 1941812, DOI 10.1002/cpa.10061
    • GLU2 A. Greenleaf, M. Lassas, and G. Uhlmann, Anisotropic conductivities that cannot be detected in EIT, Physiolog. Meas. (special issue on Impedance Tomography) 24 (2003), 413–420.
  • Allan Greenleaf, Matti Lassas, and Gunther Uhlmann, On nonuniqueness for Calderón’s inverse problem, Math. Res. Lett. 10 (2003), no. 5-6, 685–693. MR 2024725, DOI 10.4310/MRL.2003.v10.n5.a11
    • GS C. Guillarmou, A. Sa Barreto, Inverse problems for Einstein manifolds, arXiv:0710.1136v1 (2007)
  • Gennadi Henkin and Vincent Michel, On the explicit reconstruction of a Riemann surface from its Dirichlet-Neumann operator, Geom. Funct. Anal. 17 (2007), no. 1, 116–155. MR 2306654, DOI 10.1007/s00039-006-0590-7
    • HLS I. Hänninen, I. Lindell and A. Sihvola, Realization of generalized soft-and-hard boundary, Prog. Electromag. Res. PIER 64 (2006), 317. Ho D. Holder, Electrical Impedance Tomography, Institute of Physics Publishing, Bristol and Philadelphia (2005). Hoff A. Hoffman, et al., Negative refraction in semiconductor metamaterials, Nature Materials, doi:10.1038/nmat2033 (14 Oct 2007). HIMS D. Isaacson, J. Mueller and S. Siltanen, Special issue on electrical impedance tomography of Physiolog. Meas. 24, 2003.
  • Carlos E. Kenig, Johannes Sjöstrand, and Gunther Uhlmann, The Calderón problem with partial data, Ann. of Math. (2) 165 (2007), no. 2, 567–591. MR 2299741, DOI 10.4007/annals.2007.165.567
    • J A. Jenkins, Metamaterials: Lost in space, Nature Photonics 2, 11–11 (01 Jan 2008).
  • Alexander Katchalov, Yaroslav Kurylev, and Matti Lassas, Inverse boundary spectral problems, Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, vol. 123, Chapman & Hall/CRC, Boca Raton, FL, 2001. MR 1889089, DOI 10.1201/9781420036220
  • Tosio Kato, Perturbation theory for linear operators, 2nd ed., Grundlehren der Mathematischen Wissenschaften, Band 132, Springer-Verlag, Berlin-New York, 1976. MR 0407617
    • Ke M. Kerker, Invisible bodies, J. Opt. Soc. Am. 65 (1975), 376. Ki1 P.-S. Kildal, Definition of artificially soft and hard surfaces for electromagnetic waves, Electron. Lett. 24 (1988), 168–170. Ki2 P.-S. Kildal, Artificially soft and hard surfaces in electromagnetics, IEEE Trans. Ant. and Prop. 38, no. 10, 1537–1544 (1990). Ki3 P.-S. Kildal, A. Kishk, Z. Sipus, RF invisibility using metamaterials: Harry Potter’s cloak or the Emperor’s new clothes?, IEEE APS Int. Symp., Hawai, June, 2007.
  • Tero Kilpeläinen, Juha Kinnunen, and Olli Martio, Sobolev spaces with zero boundary values on metric spaces, Potential Anal. 12 (2000), no. 3, 233–247. MR 1752853, DOI 10.1023/A:1008601220456
    • talkVogelius R. Kohn, D. Onofrei, M. Vogelius and M. Weinstein, Cloaking via change of variables for the Helmholtz equation, in preparation.
  • R. V. Kohn, H. Shen, M. S. Vogelius, and M. I. Weinstein, Cloaking via change of variables in electric impedance tomography, Inverse Problems 24 (2008), no. 1, 015016, 21. MR 2384775, DOI 10.1088/0266-5611/24/1/015016
    • Ko R. Kohn and S. Shipman, Magnetism and homogenization of micro-resonators, arXiv:0712.2210v1 (2007).
  • Robert V. Kohn and Michael Vogelius, Identification of an unknown conductivity by means of measurements at the boundary, Inverse problems (New York, 1983) SIAM-AMS Proc., vol. 14, Amer. Math. Soc., Providence, RI, 1984, pp. 113–123. MR 773707, DOI 10.1002/cpa.3160370302
  • Ville Kolehmainen, Matti Lassas, and Petri Ola, The inverse conductivity problem with an imperfectly known boundary, SIAM J. Appl. Math. 66 (2005), no. 2, 365–383. MR 2203860, DOI 10.1137/040612737
  • Ya. V. Kurylëv, Multi-dimensional inverse boundary problems by BC-method: groups of transformations and uniqueness results, Math. Comput. Modelling 18 (1993), no. 1, 33–45. MR 1245191, DOI 10.1016/0895-7177(93)90077-C
  • Yaroslav Kurylev, Matti Lassas, and Erkki Somersalo, Maxwell’s equations with a polarization independent wave velocity: direct and inverse problems, J. Math. Pures Appl. (9) 86 (2006), no. 3, 237–270 (English, with English and French summaries). MR 2257731, DOI 10.1016/j.matpur.2006.01.008
    • LSMSP N. I. Landy, S. Sajuyigbe, J. J. Mock, D. R. Smith, and W. J. Padilla, Perfect metamaterial absorber, Phys. Rev. Lett. 100, (2008) 207402.
  • Matti Lassas and Gunther Uhlmann, On determining a Riemannian manifold from the Dirichlet-to-Neumann map, Ann. Sci. École Norm. Sup. (4) 34 (2001), no. 5, 771–787 (English, with English and French summaries). MR 1862026, DOI 10.1016/S0012-9593(01)01076-X
  • Matti Lassas, Michael Taylor, and Gunther Uhlmann, The Dirichlet-to-Neumann map for complete Riemannian manifolds with boundary, Comm. Anal. Geom. 11 (2003), no. 2, 207–221. MR 2014876, DOI 10.4310/CAG.2003.v11.n2.a2
  • Richard B. Lavine and Adrian I. Nachman, The Faddeev-Lippmann-Schwinger equation in multidimensional quantum inverse scattering, Inverse problems: an interdisciplinary study (Montpellier, 1986) Adv. Electron. Electron Phys., Suppl. 19, Academic Press, London, 1987, pp. 169–174. MR 1005570
  • John M. Lee and Gunther Uhlmann, Determining anisotropic real-analytic conductivities by boundary measurements, Comm. Pure Appl. Math. 42 (1989), no. 8, 1097–1112. MR 1029119, DOI 10.1002/cpa.3160420804
  • Ulf Leonhardt, Optical conformal mapping, Science 312 (2006), no. 5781, 1777–1780. MR 2237569, DOI 10.1126/science.1126493
    • LeP U. Leonhardt and T. Philbin, General relativity in electrical engineering, New J. Phys. 8 (2006), 247; doi:10.1088/1367-2630/8/10/247. NJP U. Leonhardt and D. Smith, eds., Focus issue on Cloaking and Transformation Optics, New Jour. Phys. (2008). LeT U. Leonhardt and T. Tyc, Superantenna made of transformation media, arXiv:0806.0070v1 (2008)
  • Ismo V. Lindell, Generalized soft-and-hard surface, IEEE Trans. Antennas and Propagation 50 (2002), no. 7, 926–929. MR 1929635, DOI 10.1109/TAP.2002.800698
    • Liu N. Liu, H. Guo, L. Fu, S. Kaiser, H. Schweizer and H. Giessen, Three-dimensional photonic metamaterials at optical frequencies, Nature Materials 7 (2008), 31–37. Mil D. Miller, On perfect cloaking, Opt. Exp. 14 (2006), 12457–12466. Miltbook G. Milton, The Theory of Composites, Camb. U. Pr., 2001. Milt G. Milton, New metamaterials with macroscopic behavior outside that of continuum elastodynamics, New Jour. Phys. 9 (2007), 359. MBW G. Milton, M. Briane, and J. Willis, On cloaking for elasticity and physical equations with a transformation invariant form, New J. Phys. 8 (2006), 248.
  • Graeme W. Milton and Nicolae-Alexandru P. Nicorovici, On the cloaking effects associated with anomalous localized resonance, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 462 (2006), no. 2074, 3027–3059. MR 2263683, DOI 10.1098/rspa.2006.1715
  • Adrian I. Nachman, Reconstructions from boundary measurements, Ann. of Math. (2) 128 (1988), no. 3, 531–576. MR 970610, DOI 10.2307/1971435
  • Adrian I. Nachman, Global uniqueness for a two-dimensional inverse boundary value problem, Ann. of Math. (2) 143 (1996), no. 1, 71–96. MR 1370758, DOI 10.2307/2118653
  • Adrian I. Nachman and Mark J. Ablowitz, A multidimensional inverse-scattering method, Stud. Appl. Math. 71 (1984), no. 3, 243–250. MR 769078, DOI 10.1002/sapm1984713243
  • Adrian Nachman, John Sylvester, and Gunther Uhlmann, An $n$-dimensional Borg-Levinson theorem, Comm. Math. Phys. 115 (1988), no. 4, 595–605. MR 933457
  • R. G. Novikov, A multidimensional inverse spectral problem for the equation $-\Delta \psi +(v(x)-Eu(x))\psi =0$, Funktsional. Anal. i Prilozhen. 22 (1988), no. 4, 11–22, 96 (Russian); English transl., Funct. Anal. Appl. 22 (1988), no. 4, 263–272 (1989). MR 976992, DOI 10.1007/BF01077418
    • Norris A. Norris, Acoustic cloaking theory, Proc. Royal Soc. A, doi:10.1098/rspa.2008.0076 (2008).
  • T. Ochiai, U. Leonhardt, and J. C. Nacher, A novel design of dielectric perfect invisibility devices, J. Math. Phys. 49 (2008), no. 3, 032903, 13. MR 2406797, DOI 10.1063/1.2889717
  • Petri Ola, Lassi Päivärinta, and Erkki Somersalo, An inverse boundary value problem in electrodynamics, Duke Math. J. 70 (1993), no. 3, 617–653. MR 1224101, DOI 10.1215/S0012-7094-93-07014-7
  • Lassi Päivärinta, Alexander Panchenko, and Gunther Uhlmann, Complex geometrical optics solutions for Lipschitz conductivities, Rev. Mat. Iberoamericana 19 (2003), no. 1, 57–72. MR 1993415, DOI 10.4171/RMI/338
  • J. B. Pendry, D. Schurig, and D. R. Smith, Controlling electromagnetic fields, Science 312 (2006), no. 5781, 1780–1782. MR 2237570, DOI 10.1126/science.1125907
    • PSS2 J.B. Pendry, D. Schurig, and D.R. Smith, Calculation of material properties and ray tracing in transformation media, Opt. Exp. 14 (2006) 9794. Podc Physorg.com, The Mathematics of Cloaking, http:// www.physorg.com/news86358402.html (Dec. 26, 2006). Ra1 M. Rahm, D. Schurig, D. Roberts, S. Cummer, D. Smith, J. Pendry, Design of electromagnetic cloaks and concentrators using form-invariant coordinate transformations of Maxwell’s equations,, Photonics and Nanostructures, 6 (2008), 87–95. Ra2 M. Rahm, S. Cummer, D. Schurig, J. Pendry and D. Smith, Optical design of reflectionless complex media by finite embedded coordinate transformations, Phys. Rev. Lett. 100 (2008), 063903. RYNQ Z. Ruan, M. Yan, C. Neff and M. Qiu, Ideal cylindrical cloak: Perfect but sensitive to tiny perturbations, Phys. Rev. Lett. 99 (2007), 113903. Sc D. Schurig, J. Mock, B. Justice, S. Cummer, J. Pendry, A. Starr, and D. Smith, Metamaterial electromagnetic cloak at microwave frequencies, Science 314 (2006), 977–980. Sh V. Shalaev, W. Cai, U. Chettiar, H.-K. Yuan, A. Sarychev, V. Drachev, and A. Kildishev, Negative index of refraction in optical metamaterials Opt. Lett. 30 (2005), 3356–3358 SPR D. Schurig, J. Pendry, D. R. Smith, Transformation-designed optical elements Optics Express 15 (2007), 14772–14782. Shv G. Shvets, Metamaterials add an extra dimension, Nature Materials 7 (2008), 7–8. SP D. Smith and J. Pendry, Homogenization of metamaterials by field averaging, Jour. Opt. Soc. Am. B 23 (2006), 391–403. smol I. Smolyaninov, Y. Hung and C. Davis, Electromagnetic cloaking in the visible frequency range, arXiv:0709.2862v2 (2007).
  • Ziqi Sun and Gunther Uhlmann, Anisotropic inverse problems in two dimensions, Inverse Problems 19 (2003), no. 5, 1001–1010. MR 2024685, DOI 10.1088/0266-5611/19/5/301
  • John Sylvester, An anisotropic inverse boundary value problem, Comm. Pure Appl. Math. 43 (1990), no. 2, 201–232. MR 1038142, DOI 10.1002/cpa.3160430203
  • John Sylvester and Gunther Uhlmann, A uniqueness theorem for an inverse boundary value problem in electrical prospection, Comm. Pure Appl. Math. 39 (1986), no. 1, 91–112. MR 820341, DOI 10.1002/cpa.3160390106
  • John Sylvester and Gunther Uhlmann, A global uniqueness theorem for an inverse boundary value problem, Ann. of Math. (2) 125 (1987), no. 1, 153–169. MR 873380, DOI 10.2307/1971291
    • T K. Tsakmakidis and O. Hess, Optics: Watch your back, Nature 451, 27 (3 January 2008), doi:10.1038/451027a.
  • Roy Pike and Pierre Sabatier (eds.), Scattering. Vol. 1, 2, Academic Press, Inc., San Diego, CA, 2002. Scattering and inverse scattering in pure and applied science. MR 1878885
  • Gunther Uhlmann, Developments in inverse problems since Calderón’s foundational paper, Harmonic analysis and partial differential equations (Chicago, IL, 1996) Chicago Lectures in Math., Univ. Chicago Press, Chicago, IL, 1999, pp. 295–345. MR 1743870
  • Gunther Uhlmann, Inverse boundary value problems and applications, Astérisque 207 (1992), 6, 153–211. Méthodes semi-classiques, Vol. 1 (Nantes, 1991). MR 1205179
  • Gunther Uhlmann and Jenn-Nan Wang, Complex spherical waves for the elasticity system and probing of inclusions, SIAM J. Math. Anal. 38 (2007), no. 6, 1967–1980. MR 2299437, DOI 10.1137/060651434
  • Gunther Uhlmann and András Vasy, Low-energy inverse problems in three-body scattering, Inverse Problems 18 (2002), no. 3, 719–736. MR 1910198, DOI 10.1088/0266-5611/18/3/313
    • walser R. Walser, in: W.S. Weiglhofer and A. Lakhtakia (Eds.), Introduction to Complex Mediums for Electromagnetics and Optics, S Press, Bellingham, WA, USA, 2003.
  • A. J. Ward and J. B. Pendry, Refraction and geometry in Maxwell’s equations, J. Modern Opt. 43 (1996), no. 4, 773–793. MR 1390260, DOI 10.1080/095003496155878
    • W1 R. Weder, A rigorous time-domain analysis of full-wave electromagnetic cloaking (Invisibility), arXiv:0704.0248 (2007). W2 R. Weder, A rigorous analysis of high order electromagnetic invisibility cloaks, Jour. Phys. A: Math. Theor. 41 (2008), 065207. W3 R. Weder, The boundary conditions for electromagnetic invisibility cloaks, arXiv:0801.3611 (2008). wood B. Wood and J. Pendry, Metamaterials at zero frequency, Jour. Phys.: Condens Matter 19 (2007), 076208. Ya A. Yaghjian and S. Maci, Alternative derivation of electromagnetic cloaks and concentrators, arXiv:0710.2933 (2007). YRQ M. Yan, Z. Ruan and M. Qiu, Scattering characteristics of simplified cylindrical invisibility cloaks, Opt. Exp. 15 (2007), 17772. Zhang1 B. Zhang, et al., Response of a cylindrical invisibility cloak to electromagnetic waves, Phys. Rev. B 76 (2007), 121101(R). Zhang2 B. Zhang, et al., Extraordinary surface voltage effect in the invisibility cloak with an active device inside, Phys. Rev. Lett. 100 (2008), 063904. Zhang S. Zhang, D. Genov, C. Sun and X. Zhang, Cloaking of matter waves, Phys. Rev. Lett. 100 (2008), 123002. Z F. Zolla, S, Guenneau, A. Nicolet and J. Pendry, Electromagnetic analysis of cylindrical invisibility cloaks and the mirage effect, Opt. Lett. 32 (2007), 1069–1071.
Similar Articles
Additional Information
  • Allan Greenleaf
  • Affiliation: Department of Mathematics, University of Rochester, Rochester, New York 14627
  • Yaroslav Kurylev
  • Affiliation: Department of Mathematics, University College London, Gower Street, London, WC1E 5BT, United Kingdom
  • Matti Lassas
  • Affiliation: Institute of Mathematics, Helsinki University of Technology, P.O. Box 1100, Helsinki 02015, Finland
  • Gunther Uhlmann
  • Affiliation: Department of Mathematics, University of Washington, Seattle, Washington 98195
  • MR Author ID: 175790
  • Received by editor(s): June 30, 2008
  • Published electronically: October 14, 2008
  • Additional Notes: The first author was partially supported by NSF grant DMS-0551894.
    The third author was partially supported by Academy of Finland CoE Project 213476.
    The fourth author was partially supported by the NSF and a Walker Family Endowed Professorship.
  • © Copyright 2008 American Mathematical Society
  • Journal: Bull. Amer. Math. Soc. 46 (2009), 55-97
  • MSC (2000): Primary 35R30, 78A46; Secondary 58J05, 78A10
  • DOI: https://doi.org/10.1090/S0273-0979-08-01232-9
  • MathSciNet review: 2457072