Abstract

We consider the design and implementation of a two-dimensional metamaterial relay lens, conceptually formed by flattening a Maxwell fisheye lens—a perfect imaging device—through the use of a coordinate transformation. Because Maxwell’s equations are form-invariant under coordinate transformations, the specifications for the constitutive parameters of the device are obtained immediately in a procedure that has now become known as transformation optics. To obtain a more favorable implementation of the lens, we seek a quasi-conformal transformation optics transformation that minimizes the required anisotropy, such that the resulting lens can be formed using isotropic, dielectric-only media. We demonstrate a flattened Maxwell lens at microwave frequencies using a nonresonant metamaterial and confirm its focusing and broad bandwidth behavior. Such planar, dielectric-only structures can be readily implemented in infrared and optical waveguides.

© 2011 Optical Society of America

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References

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  1. D. Schurig, J. B. Pendry, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006).
    [CrossRef] [PubMed]
  2. D. Schurig, J. Mock, B. Justice, S. Cummer, J. Pendry, A. Starr, and D. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314, 977–980 (2006).
    [CrossRef] [PubMed]
  3. J. Li and J. B. Pendry, “Hiding under the carpet: a new strategy for cloaking,” Phys. Rev. Lett. 101, 1–4 (2008).
    [CrossRef]
  4. T. Zentgraf, G. Bartal, J. Valentine, J. Li, and X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater. 8, 568–571(2009).
    [CrossRef] [PubMed]
  5. C. B. Poitras, L. H. Gabriell, J. Cardenas, and M. Lipson, “Silicon nanostructure cloak operating at optical frequencies,” Nat. Photon. 3, 461–463 (2009).
    [CrossRef]
  6. E. W. Marchland, Gradient Index Optics (Academic, 1978).
  7. J. E. Eaton, “An extension of the Luneburg-type lenses,” Naval Res. Lab. 4110, 1–20 (1953).
  8. T. Tyc, U. Leonhardt, Y. G. Ma, and C. K. Ong, “An omnidirectional retroreflector based on the transmutation of dielectric singularities,” Nat. Mater. 8, 639–642 (2009).
    [CrossRef] [PubMed]
  9. D. Schurig, “An aberration-free lens with zero f-number,” New J. Phys. 10, 115034 (2008).
    [CrossRef]
  10. N. Kundtz and D. R. Smith, “Extreme-angle broadband metamaterial lens,” Nat. Mater. 9, 129–132 (2009).
    [CrossRef] [PubMed]
  11. N. B. Kundtz, D. R. Smith, Y. Urzhumov, and N. I. Landy, “Enhancing imaging systems using transformation optics,” Opt. Express 18, 21238–21251 (2010).
    [CrossRef] [PubMed]
  12. C. R. Simovski, “Bloch material parameters of magneto-dielectric metamaterials and the concept of Bloch lattices,” Metamaterials 1, 62–80 (2007).
    [CrossRef]
  13. D. R. Smith, “Analytic expressions for the constitutive parameters of magnetoelectric metamaterials,” Phys. Rev. 81, 036605(2009).
    [CrossRef]
  14. D. J. Robbins, J. B. Pendry, A. J. Holden, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microwave Theory Tech. 47, 2075–2084 (1999).
    [CrossRef]
  15. J. J. Mock, D. Schurig, and D. R. Smith, “Electric-field-coupled resonators for negative permittivity metamaterials,” Appl. Phys. Lett. 88, 041109 (2006).
    [CrossRef]
  16. U. Leonhardt, “Optical conformal mapping,” Science 312, 1777–1780 (2006).
    [CrossRef] [PubMed]
  17. N. Landy and W. Padilla, “Guiding light with conformal transformations,” Opt. Express 17, 14872–14879 (2009).
    [CrossRef] [PubMed]
  18. B. Justice, J. Mock, L. Guo, A. Degiron, D. Schurig, and D. R. Smith, “Spatial mapping of the internal and external electromagnetic fields of negative index metamaterials,” Opt. Express 14, 8694–8705 (2006).
    [CrossRef] [PubMed]
  19. T. Koschny, D. R. Smith, D. C. Vier, and C. M. Soukoulis, “Electromagnetic parameter retrieval from inhomogeneous metamaterials,” Phys. Rev. E 71, 036617 (2005).
    [CrossRef]
  20. V. D. Likar-Smiljanic, A. R. Djordjevic, R. M. Biljic, and T. K. Sarkar, “Wideband frequency-domain characterization of fr-4 and time-domain causality,” IEEE Trans. Electromagn. Compat. 43, 662–667 (2005).
    [CrossRef]

2010 (1)

2009 (6)

D. R. Smith, “Analytic expressions for the constitutive parameters of magnetoelectric metamaterials,” Phys. Rev. 81, 036605(2009).
[CrossRef]

N. Landy and W. Padilla, “Guiding light with conformal transformations,” Opt. Express 17, 14872–14879 (2009).
[CrossRef] [PubMed]

T. Zentgraf, G. Bartal, J. Valentine, J. Li, and X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater. 8, 568–571(2009).
[CrossRef] [PubMed]

C. B. Poitras, L. H. Gabriell, J. Cardenas, and M. Lipson, “Silicon nanostructure cloak operating at optical frequencies,” Nat. Photon. 3, 461–463 (2009).
[CrossRef]

T. Tyc, U. Leonhardt, Y. G. Ma, and C. K. Ong, “An omnidirectional retroreflector based on the transmutation of dielectric singularities,” Nat. Mater. 8, 639–642 (2009).
[CrossRef] [PubMed]

N. Kundtz and D. R. Smith, “Extreme-angle broadband metamaterial lens,” Nat. Mater. 9, 129–132 (2009).
[CrossRef] [PubMed]

2008 (2)

J. Li and J. B. Pendry, “Hiding under the carpet: a new strategy for cloaking,” Phys. Rev. Lett. 101, 1–4 (2008).
[CrossRef]

D. Schurig, “An aberration-free lens with zero f-number,” New J. Phys. 10, 115034 (2008).
[CrossRef]

2007 (1)

C. R. Simovski, “Bloch material parameters of magneto-dielectric metamaterials and the concept of Bloch lattices,” Metamaterials 1, 62–80 (2007).
[CrossRef]

2006 (5)

B. Justice, J. Mock, L. Guo, A. Degiron, D. Schurig, and D. R. Smith, “Spatial mapping of the internal and external electromagnetic fields of negative index metamaterials,” Opt. Express 14, 8694–8705 (2006).
[CrossRef] [PubMed]

D. Schurig, J. B. Pendry, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006).
[CrossRef] [PubMed]

D. Schurig, J. Mock, B. Justice, S. Cummer, J. Pendry, A. Starr, and D. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314, 977–980 (2006).
[CrossRef] [PubMed]

J. J. Mock, D. Schurig, and D. R. Smith, “Electric-field-coupled resonators for negative permittivity metamaterials,” Appl. Phys. Lett. 88, 041109 (2006).
[CrossRef]

U. Leonhardt, “Optical conformal mapping,” Science 312, 1777–1780 (2006).
[CrossRef] [PubMed]

2005 (2)

T. Koschny, D. R. Smith, D. C. Vier, and C. M. Soukoulis, “Electromagnetic parameter retrieval from inhomogeneous metamaterials,” Phys. Rev. E 71, 036617 (2005).
[CrossRef]

V. D. Likar-Smiljanic, A. R. Djordjevic, R. M. Biljic, and T. K. Sarkar, “Wideband frequency-domain characterization of fr-4 and time-domain causality,” IEEE Trans. Electromagn. Compat. 43, 662–667 (2005).
[CrossRef]

1999 (1)

D. J. Robbins, J. B. Pendry, A. J. Holden, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microwave Theory Tech. 47, 2075–2084 (1999).
[CrossRef]

1953 (1)

J. E. Eaton, “An extension of the Luneburg-type lenses,” Naval Res. Lab. 4110, 1–20 (1953).

Bartal, G.

T. Zentgraf, G. Bartal, J. Valentine, J. Li, and X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater. 8, 568–571(2009).
[CrossRef] [PubMed]

Biljic, R. M.

V. D. Likar-Smiljanic, A. R. Djordjevic, R. M. Biljic, and T. K. Sarkar, “Wideband frequency-domain characterization of fr-4 and time-domain causality,” IEEE Trans. Electromagn. Compat. 43, 662–667 (2005).
[CrossRef]

Cardenas, J.

C. B. Poitras, L. H. Gabriell, J. Cardenas, and M. Lipson, “Silicon nanostructure cloak operating at optical frequencies,” Nat. Photon. 3, 461–463 (2009).
[CrossRef]

Cummer, S.

D. Schurig, J. Mock, B. Justice, S. Cummer, J. Pendry, A. Starr, and D. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314, 977–980 (2006).
[CrossRef] [PubMed]

Degiron, A.

Djordjevic, A. R.

V. D. Likar-Smiljanic, A. R. Djordjevic, R. M. Biljic, and T. K. Sarkar, “Wideband frequency-domain characterization of fr-4 and time-domain causality,” IEEE Trans. Electromagn. Compat. 43, 662–667 (2005).
[CrossRef]

Eaton, J. E.

J. E. Eaton, “An extension of the Luneburg-type lenses,” Naval Res. Lab. 4110, 1–20 (1953).

Gabriell, L. H.

C. B. Poitras, L. H. Gabriell, J. Cardenas, and M. Lipson, “Silicon nanostructure cloak operating at optical frequencies,” Nat. Photon. 3, 461–463 (2009).
[CrossRef]

Guo, L.

Holden, A. J.

D. J. Robbins, J. B. Pendry, A. J. Holden, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microwave Theory Tech. 47, 2075–2084 (1999).
[CrossRef]

Justice, B.

D. Schurig, J. Mock, B. Justice, S. Cummer, J. Pendry, A. Starr, and D. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314, 977–980 (2006).
[CrossRef] [PubMed]

B. Justice, J. Mock, L. Guo, A. Degiron, D. Schurig, and D. R. Smith, “Spatial mapping of the internal and external electromagnetic fields of negative index metamaterials,” Opt. Express 14, 8694–8705 (2006).
[CrossRef] [PubMed]

Koschny, T.

T. Koschny, D. R. Smith, D. C. Vier, and C. M. Soukoulis, “Electromagnetic parameter retrieval from inhomogeneous metamaterials,” Phys. Rev. E 71, 036617 (2005).
[CrossRef]

Kundtz, N.

N. Kundtz and D. R. Smith, “Extreme-angle broadband metamaterial lens,” Nat. Mater. 9, 129–132 (2009).
[CrossRef] [PubMed]

Kundtz, N. B.

Landy, N.

Landy, N. I.

Leonhardt, U.

T. Tyc, U. Leonhardt, Y. G. Ma, and C. K. Ong, “An omnidirectional retroreflector based on the transmutation of dielectric singularities,” Nat. Mater. 8, 639–642 (2009).
[CrossRef] [PubMed]

U. Leonhardt, “Optical conformal mapping,” Science 312, 1777–1780 (2006).
[CrossRef] [PubMed]

Li, J.

T. Zentgraf, G. Bartal, J. Valentine, J. Li, and X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater. 8, 568–571(2009).
[CrossRef] [PubMed]

J. Li and J. B. Pendry, “Hiding under the carpet: a new strategy for cloaking,” Phys. Rev. Lett. 101, 1–4 (2008).
[CrossRef]

Likar-Smiljanic, V. D.

V. D. Likar-Smiljanic, A. R. Djordjevic, R. M. Biljic, and T. K. Sarkar, “Wideband frequency-domain characterization of fr-4 and time-domain causality,” IEEE Trans. Electromagn. Compat. 43, 662–667 (2005).
[CrossRef]

Lipson, M.

C. B. Poitras, L. H. Gabriell, J. Cardenas, and M. Lipson, “Silicon nanostructure cloak operating at optical frequencies,” Nat. Photon. 3, 461–463 (2009).
[CrossRef]

Ma, Y. G.

T. Tyc, U. Leonhardt, Y. G. Ma, and C. K. Ong, “An omnidirectional retroreflector based on the transmutation of dielectric singularities,” Nat. Mater. 8, 639–642 (2009).
[CrossRef] [PubMed]

Marchland, E. W.

E. W. Marchland, Gradient Index Optics (Academic, 1978).

Mock, J.

D. Schurig, J. Mock, B. Justice, S. Cummer, J. Pendry, A. Starr, and D. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314, 977–980 (2006).
[CrossRef] [PubMed]

B. Justice, J. Mock, L. Guo, A. Degiron, D. Schurig, and D. R. Smith, “Spatial mapping of the internal and external electromagnetic fields of negative index metamaterials,” Opt. Express 14, 8694–8705 (2006).
[CrossRef] [PubMed]

Mock, J. J.

J. J. Mock, D. Schurig, and D. R. Smith, “Electric-field-coupled resonators for negative permittivity metamaterials,” Appl. Phys. Lett. 88, 041109 (2006).
[CrossRef]

Ong, C. K.

T. Tyc, U. Leonhardt, Y. G. Ma, and C. K. Ong, “An omnidirectional retroreflector based on the transmutation of dielectric singularities,” Nat. Mater. 8, 639–642 (2009).
[CrossRef] [PubMed]

Padilla, W.

Pendry, J.

D. Schurig, J. Mock, B. Justice, S. Cummer, J. Pendry, A. Starr, and D. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314, 977–980 (2006).
[CrossRef] [PubMed]

Pendry, J. B.

J. Li and J. B. Pendry, “Hiding under the carpet: a new strategy for cloaking,” Phys. Rev. Lett. 101, 1–4 (2008).
[CrossRef]

D. Schurig, J. B. Pendry, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006).
[CrossRef] [PubMed]

D. J. Robbins, J. B. Pendry, A. J. Holden, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microwave Theory Tech. 47, 2075–2084 (1999).
[CrossRef]

Poitras, C. B.

C. B. Poitras, L. H. Gabriell, J. Cardenas, and M. Lipson, “Silicon nanostructure cloak operating at optical frequencies,” Nat. Photon. 3, 461–463 (2009).
[CrossRef]

Robbins, D. J.

D. J. Robbins, J. B. Pendry, A. J. Holden, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microwave Theory Tech. 47, 2075–2084 (1999).
[CrossRef]

Sarkar, T. K.

V. D. Likar-Smiljanic, A. R. Djordjevic, R. M. Biljic, and T. K. Sarkar, “Wideband frequency-domain characterization of fr-4 and time-domain causality,” IEEE Trans. Electromagn. Compat. 43, 662–667 (2005).
[CrossRef]

Schurig, D.

D. Schurig, “An aberration-free lens with zero f-number,” New J. Phys. 10, 115034 (2008).
[CrossRef]

D. Schurig, J. B. Pendry, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006).
[CrossRef] [PubMed]

D. Schurig, J. Mock, B. Justice, S. Cummer, J. Pendry, A. Starr, and D. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314, 977–980 (2006).
[CrossRef] [PubMed]

B. Justice, J. Mock, L. Guo, A. Degiron, D. Schurig, and D. R. Smith, “Spatial mapping of the internal and external electromagnetic fields of negative index metamaterials,” Opt. Express 14, 8694–8705 (2006).
[CrossRef] [PubMed]

J. J. Mock, D. Schurig, and D. R. Smith, “Electric-field-coupled resonators for negative permittivity metamaterials,” Appl. Phys. Lett. 88, 041109 (2006).
[CrossRef]

Simovski, C. R.

C. R. Simovski, “Bloch material parameters of magneto-dielectric metamaterials and the concept of Bloch lattices,” Metamaterials 1, 62–80 (2007).
[CrossRef]

Smith, D.

D. Schurig, J. Mock, B. Justice, S. Cummer, J. Pendry, A. Starr, and D. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314, 977–980 (2006).
[CrossRef] [PubMed]

Smith, D. R.

N. B. Kundtz, D. R. Smith, Y. Urzhumov, and N. I. Landy, “Enhancing imaging systems using transformation optics,” Opt. Express 18, 21238–21251 (2010).
[CrossRef] [PubMed]

D. R. Smith, “Analytic expressions for the constitutive parameters of magnetoelectric metamaterials,” Phys. Rev. 81, 036605(2009).
[CrossRef]

N. Kundtz and D. R. Smith, “Extreme-angle broadband metamaterial lens,” Nat. Mater. 9, 129–132 (2009).
[CrossRef] [PubMed]

D. Schurig, J. B. Pendry, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006).
[CrossRef] [PubMed]

J. J. Mock, D. Schurig, and D. R. Smith, “Electric-field-coupled resonators for negative permittivity metamaterials,” Appl. Phys. Lett. 88, 041109 (2006).
[CrossRef]

B. Justice, J. Mock, L. Guo, A. Degiron, D. Schurig, and D. R. Smith, “Spatial mapping of the internal and external electromagnetic fields of negative index metamaterials,” Opt. Express 14, 8694–8705 (2006).
[CrossRef] [PubMed]

T. Koschny, D. R. Smith, D. C. Vier, and C. M. Soukoulis, “Electromagnetic parameter retrieval from inhomogeneous metamaterials,” Phys. Rev. E 71, 036617 (2005).
[CrossRef]

Soukoulis, C. M.

T. Koschny, D. R. Smith, D. C. Vier, and C. M. Soukoulis, “Electromagnetic parameter retrieval from inhomogeneous metamaterials,” Phys. Rev. E 71, 036617 (2005).
[CrossRef]

Starr, A.

D. Schurig, J. Mock, B. Justice, S. Cummer, J. Pendry, A. Starr, and D. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314, 977–980 (2006).
[CrossRef] [PubMed]

Stewart, W. J.

D. J. Robbins, J. B. Pendry, A. J. Holden, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microwave Theory Tech. 47, 2075–2084 (1999).
[CrossRef]

Tyc, T.

T. Tyc, U. Leonhardt, Y. G. Ma, and C. K. Ong, “An omnidirectional retroreflector based on the transmutation of dielectric singularities,” Nat. Mater. 8, 639–642 (2009).
[CrossRef] [PubMed]

Urzhumov, Y.

Valentine, J.

T. Zentgraf, G. Bartal, J. Valentine, J. Li, and X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater. 8, 568–571(2009).
[CrossRef] [PubMed]

Vier, D. C.

T. Koschny, D. R. Smith, D. C. Vier, and C. M. Soukoulis, “Electromagnetic parameter retrieval from inhomogeneous metamaterials,” Phys. Rev. E 71, 036617 (2005).
[CrossRef]

Zentgraf, T.

T. Zentgraf, G. Bartal, J. Valentine, J. Li, and X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater. 8, 568–571(2009).
[CrossRef] [PubMed]

Zhang, X.

T. Zentgraf, G. Bartal, J. Valentine, J. Li, and X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater. 8, 568–571(2009).
[CrossRef] [PubMed]

Appl. Phys. Lett. (1)

J. J. Mock, D. Schurig, and D. R. Smith, “Electric-field-coupled resonators for negative permittivity metamaterials,” Appl. Phys. Lett. 88, 041109 (2006).
[CrossRef]

IEEE Trans. Electromagn. Compat. (1)

V. D. Likar-Smiljanic, A. R. Djordjevic, R. M. Biljic, and T. K. Sarkar, “Wideband frequency-domain characterization of fr-4 and time-domain causality,” IEEE Trans. Electromagn. Compat. 43, 662–667 (2005).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

D. J. Robbins, J. B. Pendry, A. J. Holden, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microwave Theory Tech. 47, 2075–2084 (1999).
[CrossRef]

Metamaterials (1)

C. R. Simovski, “Bloch material parameters of magneto-dielectric metamaterials and the concept of Bloch lattices,” Metamaterials 1, 62–80 (2007).
[CrossRef]

Nat. Mater. (3)

N. Kundtz and D. R. Smith, “Extreme-angle broadband metamaterial lens,” Nat. Mater. 9, 129–132 (2009).
[CrossRef] [PubMed]

T. Zentgraf, G. Bartal, J. Valentine, J. Li, and X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater. 8, 568–571(2009).
[CrossRef] [PubMed]

T. Tyc, U. Leonhardt, Y. G. Ma, and C. K. Ong, “An omnidirectional retroreflector based on the transmutation of dielectric singularities,” Nat. Mater. 8, 639–642 (2009).
[CrossRef] [PubMed]

Nat. Photon. (1)

C. B. Poitras, L. H. Gabriell, J. Cardenas, and M. Lipson, “Silicon nanostructure cloak operating at optical frequencies,” Nat. Photon. 3, 461–463 (2009).
[CrossRef]

Naval Res. Lab. (1)

J. E. Eaton, “An extension of the Luneburg-type lenses,” Naval Res. Lab. 4110, 1–20 (1953).

New J. Phys. (1)

D. Schurig, “An aberration-free lens with zero f-number,” New J. Phys. 10, 115034 (2008).
[CrossRef]

Opt. Express (3)

Phys. Rev. (1)

D. R. Smith, “Analytic expressions for the constitutive parameters of magnetoelectric metamaterials,” Phys. Rev. 81, 036605(2009).
[CrossRef]

Phys. Rev. E (1)

T. Koschny, D. R. Smith, D. C. Vier, and C. M. Soukoulis, “Electromagnetic parameter retrieval from inhomogeneous metamaterials,” Phys. Rev. E 71, 036617 (2005).
[CrossRef]

Phys. Rev. Lett. (1)

J. Li and J. B. Pendry, “Hiding under the carpet: a new strategy for cloaking,” Phys. Rev. Lett. 101, 1–4 (2008).
[CrossRef]

Science (3)

D. Schurig, J. B. Pendry, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006).
[CrossRef] [PubMed]

D. Schurig, J. Mock, B. Justice, S. Cummer, J. Pendry, A. Starr, and D. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314, 977–980 (2006).
[CrossRef] [PubMed]

U. Leonhardt, “Optical conformal mapping,” Science 312, 1777–1780 (2006).
[CrossRef] [PubMed]

Other (1)

E. W. Marchland, Gradient Index Optics (Academic, 1978).

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Figures (7)

Fig. 1
Fig. 1

Ray trace through a perfect relay lens.

Fig. 2
Fig. 2

The permittivity distribution of the transformed Maxwell lens in (a) the virtual space and (b) the physical space. The red-blue transition shows the boundary between indexes above unity, realizable with nonresonant metamaterials, and indexes below unity, realizable only with resonant metamaterials.

Fig. 3
Fig. 3

(a), (b), and (c) Show full-wave finite element simulation results for the truncated, ϵ z z , permittivity distribution. (a) Shows the source located at the center of the lens, (b) shows the source located just inside the truncation, and (c) shows the source located in the truncated region, but still on the boundary of the transformed lens. This qualitatively shows the effect of truncating the index profile as the source in (c) is poorly focused through the lens with much of its energy being radiated away. (d),(e), and (f) Show the experimentally measured electric field in the transformed Maxwell lens. (d) Shows the field for the source located at the center of the flattened lens boundary. (e) and (f) Show the source located at the edge of the flattened boundary at 12 and 8 Ghz , respectively. In all cases the source is located at the lower boundary.

Fig. 4
Fig. 4

Normalized intensity of the focal spots of the full permittivity profile (red, solid line) and the truncated permittivity profile (blue, dashed line). (a), (b), and (c) Correspond to the source positions in Figs. 3a, 3b, 3c, respectively. The focal spots for the full index profile and the truncated profile agree well for (a) and (b), where the source is inside the truncation. For (c), where the source is just outside the truncation, the magnitude of the truncated permittivity focal spot is reduced compared to the full permittivity focal spot.

Fig. 5
Fig. 5

The two types of unit cells used to implement the Maxwell and their effective epsilon in the z direction. (a) Unit cell for the nonresonant electric dipole composed of copper on FR4 in air. (b) Unit cell for the air slot in FR4.

Fig. 6
Fig. 6

The fabricated lens.

Fig. 7
Fig. 7

Experimentally measured field intensity along the object and image planes for three sources. The position axis of the sources has been mirrored across x = 0 so that images lie over their respective sources.

Equations (4)

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n ( r ) = n 0 1 + ( r / a ) 2 ,
ϵ i j = A i i A j j Det ( A ) ϵ i j ,
μ i j = A i i A j j Det ( A ) μ i j ,
ϵ z z = { ϵ z z ϵ z z 1 1 ϵ z z < 1 .

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