We describe recent theoretical and experimental progress on making objects invisible to detection by electromagnetic waves. Ideas for devices that would once have seemed fanciful may now be at least approximately implemented physically using a new class of artificially structured materials called metamaterials. Maxwell's equations have transformation laws that allow for the design of electromagnetic material parameters that steer light around a hidden region, returning it to its original path on the far side. Not only would observers be unaware of the contents of the hidden region, they would not even be aware that something was being hidden. An object contained in the hidden region, which would have no shadow, is said to be cloaked. Proposals for, and even experimental implementations of, such cloaking devices have received the most attention, but other designs having striking effects on wave propagation are possible. All of these designs are initially based on the transformation laws of the equations that govern wave propagation but, due to the singular parameters that give rise to 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.

MSC codes

  1. 78A40
  2. 35P25
  3. 35R30


  1. cloaking
  2. transformation optics
  3. electromagnetic wormholes
  4. invisibility

Get full access to this article

View all available purchase options and get full access to this article.


A. Alu and N. Engheta, Achieving transparency with plasmonic and metamaterial coatings, Phys. Rev. E, 72 (2005), article 016623.
K. Astala and L. Päivärinta, Calderón's inverse conductivity problem in the plane, Ann. of Math., 163 (2006), pp. 265–299.
K. Astala, M. Lassas, and L. Päivärinta, Calderón's inverse problem for anisotropic conductivity in the plane, Comm. Partial Differential Equations, 30 (2005), pp. 207–224.
K. Astala, M. Lassas, and L. Päivärinta, Limits of visibility and invisibility for Calderón's inverse problem in the plane, in preparation.
M. Belishev and Y. Kurylev, To the reconstruction of a Riemannian manifold via its spectral data (B-method), Comm. Partial Differential Equations, 17 (1992), pp. 767–804.
Y. Benveniste and T. Miloh, Neutral inhomogeneities in conduction phenomena, J. Mech. Phys. Solids., 47 (1999), pp. 1873–1892.
Y. Berezanskii, The uniqueness theorem in the inverse problem of spectral analysis for the Schrödinger equation, Trudy Moskov. Mat. Obsch., 7 (1958), pp. 1–62 (in Russian).
L. Borcea, Electrical impedance tomography, Inverse Problems, 18 (2002), pp. R99–R136.
G. Bouchitte and D. Felbacq, Homogenization of a set of parallel fibers, Waves in Random Media, 7 (1997), pp. 1–12.
G. Bouchitte and D. Felbacq, Homogenization near resonances and artificial magnetism from dielectrics, C. R. Math. Acad. Sci. Paris, 339 (2004), pp. 377–382.
G. Bouchitte and D. Felbacq, Left handed media and homogenization of photonic crystals, Optics Lett., 30 (2005), pp. 1189–1191.
G. Bouchitte and D. Felbacq, Homogenization of wire mesh photonic crystals embedded in a medium with a negative permeability, Phys. Rev. Lett., 94 (2005), article 183902.
G. Bouchitte and D. Felbacq, Negative refraction in periodic and random photonic crystals, New J. Phys., 7 (2005), article 159.
R. Brown and R. Torres, Uniqueness in the inverse conductivity problem for conductivities with $3/2$ derivatives in $L^p, p>2n$, J. Fourier Anal. Appl., 9 (2003), pp. 1049–1056.
R. Brown and G. Uhlmann, Uniqueness in the inverse conductivity problem with less regular conductivities in two dimensions, Comm. Partial Differential Equations, 22 (1997), pp. 1009–10027.
O. Bruno and S. Lintner, Superlens-cloaking of small dielectric bodies in the quasistatic regime, J. Appl. Phys., 102 (2007), article 124502.
W. Cai, U. Chettiar, A. Kildishev, and V. Shalaev, Optical cloaking with metamaterials, Nature Photonics, 1 (2007), pp. 224–227.
W. Cai, U. Chettiar, A. Kildishev, G. Milton, and V. Shalaev, Non-magnetic Cloak without Reflection, preprint, 2007; available online from http://arxiv.org/abs/0707.3641v1.
A.P. Calderón, On an inverse boundary value problem, in Seminar on Numerical Analysis and Its Applications to Continuum Physics (Rio de Janeiro, 1980), Soc. Brasil. Mat., Rio de Janeiro, 1980, pp. 65–73.
H. Chen and C.T. Chan, Transformation media that rotate electromagnetic fields, Appl. Phys. Lett., 90 (2007), article 241105.
H. Chen and C.T. Chan, Acoustic cloaking in three dimensions using acoustic metamaterials, Appl. Phys. Lett., 91 (2007), article 183518.
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), article 241104(R).
H.-S. Chen, B.-I. Wu, B. Zhang, and J.A. Kong, Electromagnetic wave interactions with a metamaterial cloak, Phys. Rev. Lett., 99 (2007), article 063903.
M. Cheney, D. Isaacson, and J. C. Newell, Electrical impedance tomography, SIAM Rev., 41 (1999), pp. 85–101.
A. Cho, Hope dims for the perfect lens, ScienceNow, http://sciencenow.sciencemag.org/cgi/content/full/2007/423/2.
S. Cummer, B.-I. Popa, D. Schurig, D. Smith, and J. Pendry, Full-wave simulations of electromagnetic cloaking structures, Phys. Rev. E, 74 (2006), article 036621.
S. Cummer and D. Schurig, One path to acoustic cloaking, New J. Phys., 9 (2007), article 45.
S. Cummer, B. Popa, O. Schurig, D. Smith, J. Pendry, M. Rahm, and A. Starr, Scattering theory derivation of a 3D acoustic cloaking shell, Phys. Rev. Lett., 100 (2008), article 024301.
A. Einstein and N. Rosen, The particle problem in the general theory of relativity, Phys. Rev., 48 (1935), pp. 73–77.
G. Eleftheriades and K. Balmain, eds., Negative-Refraction Metamaterials, IEEE/Wiley, Hoboken, NJ, 2005.
N. Fang, H. Lee, C. Sun, and X. Zhang, Subdiffraction-limited optical imaging with a silver superlens, Science, 308 (2005), pp. 534–537.
A. Greenleaf, Y. Kurylev, M. Lassas, and G. Uhlmann, Full-wave invisibility of active devices at all frequencies, Comm. Math. Phys., 275 (2007), pp. 749–789.
A. Greenleaf, Y. Kurylev, M. Lassas, and G. Uhlmann, Electromagnetic wormholes and virtual magnetic monopoles from metamaterials, Phys. Rev. Lett., 99 (2007), article 183901.
A. Greenleaf, Y. Kurylev, M. Lassas, and G. Uhlmann, Improvement of cylindrical cloaking with the SHS lining, Optics Express, 15 (2007), pp. 12717–12734.
A. Greenleaf, Y. Kurylev, M. Lassas, and G. Uhlmann, Electromagnetic wormholes via handlebody constructions, Comm. Math. Phys., 281 (2008), pp. 369–385
A. Greenleaf, Y. Kurylev, M. Lassas, and G. Uhlmann, Comment on “Scattering theory derivation of a 3D acoustic cloaking shell," preprint, 2008; available online from http://arXiv.org/abs/0801.3279.
A. Greenleaf, M. Lassas, and G. Uhlmann, The Calderón problem for conormal potentials, I: Global uniqueness and reconstruction, Comm. Pure Appl. Math., 56 (2003), pp. 328–352.
A. Greenleaf, M. Lassas, and G. Uhlmann, Anisotropic conductivities that cannot detected in EIT, Physiolog. Meas. (special issue on Impedance Tomography), 24 (2003), pp. 413–420.
A. Greenleaf, M. Lassas, and G. Uhlmann, On nonuniqueness for Calderón's inverse problem, Math. Res. Lett., 10 (2003), pp. 685–693.
I. Hänninen, I. Lindell, and A. Sihvola, Realization of generalized soft-and-hard boundary, Prog. Electromag. Res., 64 (2006), pp. 317–333.
A. Hoffman, L. Alekseyev, S. Howard, K. Frank, D. Wasserman, V. Podolsky, E. Narimanov, D. Sivco, and C. Gmachi, Negative refraction in semiconductor metamaterials, Nature Materials, 6 (2007), pp. 946–950.
D. Isaacson, J. Mueller, and S. Siltanen, eds., Special issue on electrical impedance tomography, Physiolog. Measurement, 24 (2003).
A. Jenkins, Metamaterials: Lost in space, Nature Photonics, 2 (2008), p. 11.
A. Kachalov and Y. Kurylev, Multidimensional inverse problem with incomplete boundary spectral data, Comm. Partial Differential Equations, 23 (1998), pp. 55–95.
A. Kachalov, Y. Kurylev, and M. Lassas, Inverse Boundary Spectral Problems, Chapman & Hall/CRC Monogr. Surv. Pure Appl. Math. 123, Chapman & Hall/CRC, Boca Raton, FL, 2001.
M. Kerker, Invisible bodies, J. Opt. Soc. Amer., 65 (1975), pp. 376–379.
P.-S. Kildal, Definition of artificially soft and hard surfaces for electromagnetic waves, Electron. Lett., 24 (1988), pp. 168–170.
P.-S. Kildal, Artificially soft and hard surfaces in electromagnetics, IEEE Trans. Antennas and Propagation, 38 (1990), pp. 1537–1544.
P.-S. Kildal, A. Kishk, and Z. Sipus, RF invisibility using metamaterials: Harry Potter's cloak or the Emperor's new clothes?, in Proceedings of the IEEE APS International Symposium, Hawaii, 2007, pp. 2361–2364.
A.V. Kildidhev and E.E. Narimanov, Impedance-Matched Hyperlens, preprint, 2007; available online from http://arxiv.org/abs/0708.3798v1.
A. Kildishev and V. Shalaev, Engineering space for light via transformation optics, Opt. Lett., 33 (2007), pp. 43–45.
T. Kilpeläinen, J. Kinnunen, and O. Martio, Sobolev spaces with zero boundary values on metric spaces, Potential Anal., 12 (2000), pp. 233–247.
R. Kohn, D. Onofrei, M. Vogelius, and M. Weinstein, Cloaking via change of variables for the Helmholtz equation, in preparation.
R. Kohn, H. Shen, M. Vogelius, and M. Weinstein, Cloaking via change of variables in electrical impedance tomography, Inverse Problems, 24 (2008), article 015016.
R. Kohn and S. Shipman, Magnetism and Homogenization of Micro-resonators, preprint, 2007; available online from http://arXiv.org/abs/0712.2210v1.
R. Kohn and M. Vogelius, Determining conductivity by boundary measurements II: Interior results, Comm. Pure Appl. Math., 38 (1985), pp. 643–667.
R. V. Kohn and M. Vogelius, Identification of an unknown conductivity by means of measurements at the boundary, in Inverse Problems, SIAM-AMS Proc. 14, AMS, Providence, RI, 1984, pp. 113–123.
V. Kolehmainen, M. Lassas, and P. Ola, The inverse conductivity problem with an imperfectly known boundary, SIAM J. Appl. Math., 66 (2005), pp. 365–383.
Y. Kurylev, Multidimensional inverse boundary problems by the BC-method: Groups of transformations and uniqueness results, Math. Comput. Modelling, 18 (1993), pp. 33–46.
Y. Kurylev, M. Lassas, and E. Somersalo, Maxwell's equations with a polarization independent wave velocity: Direct and inverse problems, J. Math. Pure Appl. 86 (2006), pp. 237–270.
M. Lassas and G. Uhlmann, Determining Riemannian manifold from boundary measurements, Ann. Sci. École Norm. Sup., 34 (2001), pp. 771–787.
M. Lassas, M. Taylor, and G. Uhlmann, The Dirichlet-to-Neumann map for complete Riemannian manifolds with boundary, Comm. Geom. Anal., 11 (2003), pp. 207–222.
R. Lavine and A. Nachman, The Faddeev–Lipmann–Schwinger equation in multidimensional quantum inverse scattering, in Inverse Problems: An Interdisciplinary Study (Montpellier, 1986), Academic Press, London, 1987, pp. 169–174.
J. Lee and G. Uhlmann, Determining anisotropic real-analytic conductivities by boundary measurements, Comm. Pure Appl. Math., 42 (1989), pp. 1097–1112.
U. Leonhardt, Optical conformal mapping, Science, 312 (2006), pp. 1777–1780.
U. Leonhardt and T. Philbin, General relativity in electrical engineering, New J. Phys., 8 (2006), article 247.
I. Lindell, Generalized soft-and-hard surface, IEEE Trans. Antennas and Propagation, 50 (2002), pp. 926–929.
N. Liu, H. Guo, L. Fu, S. Kaiser, H. Schweizer, and H. Giessen, Three-dimensional photonic metamaterials at optical frequencies, Nature Materials, 7 (2008), pp. 31–37.
D. Miller, On perfect cloaking, Opt. Express., 14 (2006), pp. 12457–12466.
G. Milton, New metamaterials with macroscopic behavior outside that of continuum elastodynamics, New J. Phys., 9 (2007), article 359.
G. Milton, M. Briane, and J. Willis, On cloaking for elasticity and physical equations with a transformation invariant form, New J. Phys., 8 (2006), article 248.
G. Milton and N.-A. Nicorovici, On the cloaking effects associated with anomalous localized resonance, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 462 (2006), pp. 3027–3059.
G. Milton, N.-A. Nicorovici, and R. McPhedran, Opaque perfect lenses, Phys. B, to appear.
A. Nachman, Global uniqueness for a two-dimensional inverse boundary value problem, Ann. of Math., 143 (1996), pp. 71–96.
N. Nicorovici, R. McPhedran, and G. Milton, Optical and dielectric properties of partially resonant composites, Phys. Rev. B, 49 (1994), pp. 8479–8482.
N. Nicorovici, G. Milton, R. McPhedran, and L. Botten, Quasistatic cloaking of two-dimensional polarizable discrete systems by anomalous resonance, Opt. Express, 15 (2007), pp. 6314–6323.
T. Ochiai, U. Leonhardt, and J. Nacher, A Novel Design of Dielectric Perfect Invisibility Devices, preprint, 2007; available online from http://arxiv.org/abs/0711.1122.
P. Ola, L. Päivärinta, and E. Somersalo, An inverse boundary value problem in electrodynamics, Duke Math. J., 70 (1993), pp. 617–653.
L. Päivärinta, A. Panchenko, and G. Uhlmann, Complex geometrical optics for Lipschitz conductivities, Rev. Mat. Iberoam., 19 (2003), pp. 57–72.
J.B. Pendry, D. Schurig, and D.R. Smith, Controlling electromagnetic fields, Science, 312 (2006), pp. 1780–1782.
J.B. Pendry, D. Schurig, and D.R. Smith, Calculation of material properties and ray tracing in transformation media, Opt. Express., 14 (2006), pp. 9794–9804.
J. Pendry, Negative refraction makes a perfect lens, Phys. Rev. Lett., 8 (2000), pp. 3966–3969.
Physorg.com, The Mathematics of Cloaking, http://www.physorg.com/news86358402.html (Dec. 26, 2006).
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), article 063903.
M. Rahm, D. Schurig, D. Roberts, S. Cummer, D. Smith, and J. Pendry, Design of electromagnetic cloaks and concentrators using form-invariant coordinate transformations of Maxwell's equations, Photonics and Nanostructures, 6 (2008), pp. 87–95.
M. Rahm, D. Schurig, D.A. Roberts, S.A. Cummer, D.R. Smith, and J. Pendry, Design of Electromagnetic Cloaks and Concentrators Using Form-Invariant Coordinate Transformations of Maxwell's Equations, preprint, 2007; available online from http://arxiv.org/abs/0706.2452v1.
Z. Ruan, M. Yan, C. Neff, and M. Qiu, Ideal cylindrical cloak: Perfect but sensitive to tiny perturbations, Phys. Rev. Lett., 99 (2007), article 113903.
D. Schurig, J. Mock, B. Justice, S. Cummer, J. Pendry, A. Starr, and D. Smith, Metamaterial electromagnetic cloak at microwave frequencies, Science, 314 (2006), pp. 977–980.
D. Schurig, J. Pendry, and D.R. Smith, Transformation-designed optical elements, Opt. Express., 15 (2007), pp. 14772–14782.
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), pp. 3356–3358
G. Shvets, Metamaterials add an extra dimension, Nature Materials, 7 (2008), pp. 7–8.
D. Smith and J. Pendry, Homogenization of metamaterials by field averaging, J. Opt. Soc. Amer. B Opt. Phys., 23 (2006), pp. 391–403.
I. Smolyaninov, Y. Hung, and C. Davis, Electromagnetic Cloaking in the Visible Frequency Range, preprint, 2007; available online from http://arxiv.org/abs/0709.2862v2.
Z. Sun and G. Uhlmann, Anisotropic inverse problems in two dimensions, Inverse Problems, 19 (2003), pp. 1001–1010.
J. Sylvester, An anisotropic inverse boundary value problem, Comm. Pure Appl. Math., 43 (1990), pp. 201–232.
J. Sylvester and G. Uhlmann, A global uniqueness theorem for an inverse boundary value problem, Ann. of Math., 125 (1987), pp. 153–169.
K. Tsakmakidis and O. Hess, Optics: Watch your back, Nature, 451 (2008), article 27.
G. Uhlmann, Developments in inverse problems since Calderón's foundational paper, in Harmonic Analysis and Partial Differential Equations, M. Christ, C. Kenig, and C. Sadosky, eds., University of Chicago Press, Chicago, 1999, pp. 295–345 .
G. Uhlmann, Scattering by a metric, in Encyclopedia on Scattering, R. Pike and P. Sabatier, eds., Academic Press, London, New York, 2002, pp. 1668–1677.
V.G. Veselago, The electrodynamics of substances with simultaneously negative values of $\epsilon$ and $\mu$, Soviet Phys. Uspekhi, 10 (1968), pp. 509–514.
R. Walser, Metamaterials: An introduction, in Introduction to Complex Mediums for Electromagnetics and Optics, W.S. Weiglhofer and A. Lakhtakia, eds., SPIE Press, Bellingham, WA, 2003, pp. 295–316.
A. Ward and J. Pendry, Refraction and geometry in Maxwell's equations, J. Modern Opt., 43 (1996), pp. 773–793.
R. Weder, A Rigorous Time-Domain Analysis of Full-Wave Electromagnetic Cloaking (Invisibility), preprint, 2008; available online from http://arxiv.org/abs/0704.0248v4.
R. Weder, A rigorous analysis of high order electromagnetic invisibility cloaks, J. Phys. A, 41 (2008), article 065207.
R. Weder, The Boundary Conditions for Electromagnetic Invisibility Cloaks, preprint, 2008; available online from http://arxiv.org/abs/0801.3611v2.
B. Wood and J. Pendry, Metamaterials at zero frequency, J. Phys. Condens. Matter, 19 (2007), article 076208.
A. Yaghjian and S. Maci, Alternative derivation of electromagnetic cloaks and concentrators, New J. Phys., 10 (2008), article 115022.
M. Yan, Z. Ruan, and M. Qiu, Scattering characteristics of simplified cylindrical invisibility cloaks, Opt. Express., 15 (2007), pp. 17772–17782.
B. Zhang, H. Chen, B.-I. Wu, Y. Luo, L. Ran, and J. A. Kong, Response of a cylindrical invisibility cloak to electromagnetic waves, Phys. Rev. B, 76 (2007), article 121101(R).
B. Zhang, H. Chen, B.-I. Wu, and J. A. Kong, Extraordinary surface voltage effect in the invisibility cloak with an active device inside, Phys. Rev. Lett., 100 (2008), article 063904.
F. Zolla, S, Guenneau, A. Nicolet, and J. Pendry, Electromagnetic analysis of cylindrical invisibility cloaks and the mirage effect, Opt. Lett., 32 (2007), pp. 1069–1071.

Information & Authors


Published In

cover image SIAM Review
SIAM Review
Pages: 3 - 33
ISSN (online): 1095-7200


Submitted: 27 February 2008
Accepted: 5 September 2008
Published online: 5 February 2009

MSC codes

  1. 78A40
  2. 35P25
  3. 35R30


  1. cloaking
  2. transformation optics
  3. electromagnetic wormholes
  4. invisibility



Metrics & Citations



If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.

Cited By







Copy the content Link

Share with email

Email a colleague

Share on social media