Abstract

As invisibility cloaking has recently become experimental reality, it is interesting to explore ways to reveal remaining imperfections. In essence, the idea of most invisibility cloaks is to recover the optical path lengths without an object (to be made invisible) by a suitable arrangement around that object. Optical path length is proportional to the time of flight of a light ray or to the optical phase accumulated by a light wave. Thus, time-of-flight images provide a direct and intuitive tool for probing imperfections. Indeed, recent phase-sensitive experiments on the carpet cloak have already made early steps in this direction. In the macroscopic world, time-of-flight images could be measured directly by light detection and ranging (LIDAR). Here, we show calculated time-of-flight images of the conformal Gaussian carpet cloak, the conformal grating cloak, the cylindrical free-space cloak, and of the invisible sphere. All results are obtained by using a ray-velocity equation of motion derived from Fermat’s principle.

© 2011 OSA

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References

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  1. J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312(5781), 1780–1782 (2006).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  6. J. Li and J. B. Pendry, “Hiding under the carpet: a new strategy for cloaking,” Phys. Rev. Lett. 101(20), 203901 (2008).
    [CrossRef] [PubMed]
  7. R. Liu, C. Ji, J. J. Mock, J. Y. Chin, T. J. Cui, and D. R. Smith, “Broadband ground-plane cloak,” Science 323(5912), 366–369 (2009).
    [CrossRef] [PubMed]
  8. J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater. 8(7), 568–571 (2009).
    [CrossRef] [PubMed]
  9. L. H. Gabrielli, J. Cardenas, C. B. Poitras, and M. Lipson, “Silicon nanostructure cloak operating at optical frequencies,” Nat. Photonics 3(8), 461–463 (2009).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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  15. J. Fischer, T. Ergin, and M. Wegener, “Three-dimensional polarization-independent visible-frequency carpet invisibility cloak,” Opt. Lett. 36(11), 2059–2061 (2011).
    [CrossRef] [PubMed]
  16. M. Gharghi, C. Gladden, T. Zentgraf, Y. Liu, X. Yin, J. Valentine, and X. Zhang, “A carpet cloak for visible light,” Nano Lett. 11(7), 2825–2828 (2011).
    [CrossRef] [PubMed]
  17. T. Ergin, J. Fischer, and M. Wegener, “Optical phase cloaking of 700-nm light waves in the far field by a three-dimensional carpet cloak,” Phys. Rev. Lett. 107(17), 173901 (2011).
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2011 (6)

B. Zhang, Y. Luo, X. Liu, and G. Barbastathis, “Macroscopic invisibility cloak for visible light,” Phys. Rev. Lett. 106(3), 033901 (2011).
[CrossRef] [PubMed]

X. Chen, Y. Luo, J. Zhang, K. Jiang, J. B. Pendry, and S. Zhang, “Macroscopic invisibility cloaking of visible light,” Nat Commun 2, 176 (2011).
[CrossRef] [PubMed]

M. Gharghi, C. Gladden, T. Zentgraf, Y. Liu, X. Yin, J. Valentine, and X. Zhang, “A carpet cloak for visible light,” Nano Lett. 11(7), 2825–2828 (2011).
[CrossRef] [PubMed]

T. Ergin, J. Fischer, and M. Wegener, “Optical phase cloaking of 700-nm light waves in the far field by a three-dimensional carpet cloak,” Phys. Rev. Lett. 107(17), 173901 (2011).
[CrossRef]

J. C. Halimeh, R. Schmied, and M. Wegener, “Newtonian photorealistic ray tracing of grating cloaks and correlation-function-based cloaking-quality assessment,” Opt. Express 19(7), 6078–6092 (2011).
[CrossRef] [PubMed]

J. Fischer, T. Ergin, and M. Wegener, “Three-dimensional polarization-independent visible-frequency carpet invisibility cloak,” Opt. Lett. 36(11), 2059–2061 (2011).
[CrossRef] [PubMed]

2010 (6)

A. J. Danner, “Visualizing invisibility: metamaterials-based optical devices in natural environments,” Opt. Express 18(4), 3332–3337 (2010).
[CrossRef] [PubMed]

A. Akbarzadeh and A. J. Danner, “Generalization of ray tracing in a linear inhomogeneous anisotropic medium: a coordinate-free approach,” J. Opt. Soc. Am. A 27(12), 2558–2562 (2010).
[CrossRef] [PubMed]

R. Schmied, J. C. Halimeh, and M. Wegener, “Conformal carpet and grating cloaks,” Opt. Express 18(23), 24361–24367 (2010).
[CrossRef] [PubMed]

H. Chen, C. T. Chan, and P. Sheng, “Transformation optics and metamaterials,” Nat. Mater. 9(5), 387–396 (2010).
[CrossRef] [PubMed]

T. Ergin, N. Stenger, P. Brenner, J. B. Pendry, and M. Wegener, “Three-dimensional invisibility cloak at optical wavelengths,” Science 328(5976), 337–339 (2010).
[CrossRef] [PubMed]

H. F. Ma and T. J. Cui, “Three-dimensional broadband ground-plane cloak made of metamaterials,” Nat. Commun. 1(3), 1–6 (2010).
[CrossRef] [PubMed]

2009 (5)

R. Liu, C. Ji, J. J. Mock, J. Y. Chin, T. J. Cui, and D. R. Smith, “Broadband ground-plane cloak,” Science 323(5912), 366–369 (2009).
[CrossRef] [PubMed]

J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater. 8(7), 568–571 (2009).
[CrossRef] [PubMed]

L. H. Gabrielli, J. Cardenas, C. B. Poitras, and M. Lipson, “Silicon nanostructure cloak operating at optical frequencies,” Nat. Photonics 3(8), 461–463 (2009).
[CrossRef]

U. Leonhardt and T. Tyc, “Broadband invisibility by non-Euclidean cloaking,” Science 323(5910), 110–112 (2009).
[CrossRef] [PubMed]

J. H. Lee, J. Blair, V. A. Tamma, Q. Wu, S. J. Rhee, C. J. Summers, and W. Park, “Direct visualization of optical frequency invisibility cloak based on silicon nanorod array,” Opt. Express 17(15), 12922–12928 (2009).
[CrossRef] [PubMed]

2008 (1)

J. Li and J. B. Pendry, “Hiding under the carpet: a new strategy for cloaking,” Phys. Rev. Lett. 101(20), 203901 (2008).
[CrossRef] [PubMed]

2006 (3)

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312(5781), 1780–1782 (2006).
[CrossRef] [PubMed]

U. Leonhardt, “Optical conformal mapping,” Science 312(5781), 1777–1780 (2006).
[CrossRef] [PubMed]

D. Schurig, J. B. Pendry, and D. R. Smith, “Calculation of material properties and ray tracing in transformation media,” Opt. Express 14(21), 9794–9804 (2006).
[CrossRef] [PubMed]

Akbarzadeh, A.

Barbastathis, G.

B. Zhang, Y. Luo, X. Liu, and G. Barbastathis, “Macroscopic invisibility cloak for visible light,” Phys. Rev. Lett. 106(3), 033901 (2011).
[CrossRef] [PubMed]

Bartal, G.

J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater. 8(7), 568–571 (2009).
[CrossRef] [PubMed]

Blair, J.

Brenner, P.

T. Ergin, N. Stenger, P. Brenner, J. B. Pendry, and M. Wegener, “Three-dimensional invisibility cloak at optical wavelengths,” Science 328(5976), 337–339 (2010).
[CrossRef] [PubMed]

Cardenas, J.

L. H. Gabrielli, J. Cardenas, C. B. Poitras, and M. Lipson, “Silicon nanostructure cloak operating at optical frequencies,” Nat. Photonics 3(8), 461–463 (2009).
[CrossRef]

Chan, C. T.

H. Chen, C. T. Chan, and P. Sheng, “Transformation optics and metamaterials,” Nat. Mater. 9(5), 387–396 (2010).
[CrossRef] [PubMed]

Chen, H.

H. Chen, C. T. Chan, and P. Sheng, “Transformation optics and metamaterials,” Nat. Mater. 9(5), 387–396 (2010).
[CrossRef] [PubMed]

Chen, X.

X. Chen, Y. Luo, J. Zhang, K. Jiang, J. B. Pendry, and S. Zhang, “Macroscopic invisibility cloaking of visible light,” Nat Commun 2, 176 (2011).
[CrossRef] [PubMed]

Chin, J. Y.

R. Liu, C. Ji, J. J. Mock, J. Y. Chin, T. J. Cui, and D. R. Smith, “Broadband ground-plane cloak,” Science 323(5912), 366–369 (2009).
[CrossRef] [PubMed]

Cui, T. J.

H. F. Ma and T. J. Cui, “Three-dimensional broadband ground-plane cloak made of metamaterials,” Nat. Commun. 1(3), 1–6 (2010).
[CrossRef] [PubMed]

R. Liu, C. Ji, J. J. Mock, J. Y. Chin, T. J. Cui, and D. R. Smith, “Broadband ground-plane cloak,” Science 323(5912), 366–369 (2009).
[CrossRef] [PubMed]

Danner, A. J.

Ergin, T.

T. Ergin, J. Fischer, and M. Wegener, “Optical phase cloaking of 700-nm light waves in the far field by a three-dimensional carpet cloak,” Phys. Rev. Lett. 107(17), 173901 (2011).
[CrossRef]

J. Fischer, T. Ergin, and M. Wegener, “Three-dimensional polarization-independent visible-frequency carpet invisibility cloak,” Opt. Lett. 36(11), 2059–2061 (2011).
[CrossRef] [PubMed]

T. Ergin, N. Stenger, P. Brenner, J. B. Pendry, and M. Wegener, “Three-dimensional invisibility cloak at optical wavelengths,” Science 328(5976), 337–339 (2010).
[CrossRef] [PubMed]

Fischer, J.

J. Fischer, T. Ergin, and M. Wegener, “Three-dimensional polarization-independent visible-frequency carpet invisibility cloak,” Opt. Lett. 36(11), 2059–2061 (2011).
[CrossRef] [PubMed]

T. Ergin, J. Fischer, and M. Wegener, “Optical phase cloaking of 700-nm light waves in the far field by a three-dimensional carpet cloak,” Phys. Rev. Lett. 107(17), 173901 (2011).
[CrossRef]

Gabrielli, L. H.

L. H. Gabrielli, J. Cardenas, C. B. Poitras, and M. Lipson, “Silicon nanostructure cloak operating at optical frequencies,” Nat. Photonics 3(8), 461–463 (2009).
[CrossRef]

Gharghi, M.

M. Gharghi, C. Gladden, T. Zentgraf, Y. Liu, X. Yin, J. Valentine, and X. Zhang, “A carpet cloak for visible light,” Nano Lett. 11(7), 2825–2828 (2011).
[CrossRef] [PubMed]

Gladden, C.

M. Gharghi, C. Gladden, T. Zentgraf, Y. Liu, X. Yin, J. Valentine, and X. Zhang, “A carpet cloak for visible light,” Nano Lett. 11(7), 2825–2828 (2011).
[CrossRef] [PubMed]

Halimeh, J. C.

Ji, C.

R. Liu, C. Ji, J. J. Mock, J. Y. Chin, T. J. Cui, and D. R. Smith, “Broadband ground-plane cloak,” Science 323(5912), 366–369 (2009).
[CrossRef] [PubMed]

Jiang, K.

X. Chen, Y. Luo, J. Zhang, K. Jiang, J. B. Pendry, and S. Zhang, “Macroscopic invisibility cloaking of visible light,” Nat Commun 2, 176 (2011).
[CrossRef] [PubMed]

Lee, J. H.

Leonhardt, U.

U. Leonhardt and T. Tyc, “Broadband invisibility by non-Euclidean cloaking,” Science 323(5910), 110–112 (2009).
[CrossRef] [PubMed]

U. Leonhardt, “Optical conformal mapping,” Science 312(5781), 1777–1780 (2006).
[CrossRef] [PubMed]

Li, J.

J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater. 8(7), 568–571 (2009).
[CrossRef] [PubMed]

J. Li and J. B. Pendry, “Hiding under the carpet: a new strategy for cloaking,” Phys. Rev. Lett. 101(20), 203901 (2008).
[CrossRef] [PubMed]

Lipson, M.

L. H. Gabrielli, J. Cardenas, C. B. Poitras, and M. Lipson, “Silicon nanostructure cloak operating at optical frequencies,” Nat. Photonics 3(8), 461–463 (2009).
[CrossRef]

Liu, R.

R. Liu, C. Ji, J. J. Mock, J. Y. Chin, T. J. Cui, and D. R. Smith, “Broadband ground-plane cloak,” Science 323(5912), 366–369 (2009).
[CrossRef] [PubMed]

Liu, X.

B. Zhang, Y. Luo, X. Liu, and G. Barbastathis, “Macroscopic invisibility cloak for visible light,” Phys. Rev. Lett. 106(3), 033901 (2011).
[CrossRef] [PubMed]

Liu, Y.

M. Gharghi, C. Gladden, T. Zentgraf, Y. Liu, X. Yin, J. Valentine, and X. Zhang, “A carpet cloak for visible light,” Nano Lett. 11(7), 2825–2828 (2011).
[CrossRef] [PubMed]

Luo, Y.

B. Zhang, Y. Luo, X. Liu, and G. Barbastathis, “Macroscopic invisibility cloak for visible light,” Phys. Rev. Lett. 106(3), 033901 (2011).
[CrossRef] [PubMed]

X. Chen, Y. Luo, J. Zhang, K. Jiang, J. B. Pendry, and S. Zhang, “Macroscopic invisibility cloaking of visible light,” Nat Commun 2, 176 (2011).
[CrossRef] [PubMed]

Ma, H. F.

H. F. Ma and T. J. Cui, “Three-dimensional broadband ground-plane cloak made of metamaterials,” Nat. Commun. 1(3), 1–6 (2010).
[CrossRef] [PubMed]

Mock, J. J.

R. Liu, C. Ji, J. J. Mock, J. Y. Chin, T. J. Cui, and D. R. Smith, “Broadband ground-plane cloak,” Science 323(5912), 366–369 (2009).
[CrossRef] [PubMed]

Park, W.

Pendry, J. B.

X. Chen, Y. Luo, J. Zhang, K. Jiang, J. B. Pendry, and S. Zhang, “Macroscopic invisibility cloaking of visible light,” Nat Commun 2, 176 (2011).
[CrossRef] [PubMed]

T. Ergin, N. Stenger, P. Brenner, J. B. Pendry, and M. Wegener, “Three-dimensional invisibility cloak at optical wavelengths,” Science 328(5976), 337–339 (2010).
[CrossRef] [PubMed]

J. Li and J. B. Pendry, “Hiding under the carpet: a new strategy for cloaking,” Phys. Rev. Lett. 101(20), 203901 (2008).
[CrossRef] [PubMed]

D. Schurig, J. B. Pendry, and D. R. Smith, “Calculation of material properties and ray tracing in transformation media,” Opt. Express 14(21), 9794–9804 (2006).
[CrossRef] [PubMed]

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312(5781), 1780–1782 (2006).
[CrossRef] [PubMed]

Poitras, C. B.

L. H. Gabrielli, J. Cardenas, C. B. Poitras, and M. Lipson, “Silicon nanostructure cloak operating at optical frequencies,” Nat. Photonics 3(8), 461–463 (2009).
[CrossRef]

Rhee, S. J.

Schmied, R.

Schurig, D.

Sheng, P.

H. Chen, C. T. Chan, and P. Sheng, “Transformation optics and metamaterials,” Nat. Mater. 9(5), 387–396 (2010).
[CrossRef] [PubMed]

Smith, D. R.

R. Liu, C. Ji, J. J. Mock, J. Y. Chin, T. J. Cui, and D. R. Smith, “Broadband ground-plane cloak,” Science 323(5912), 366–369 (2009).
[CrossRef] [PubMed]

D. Schurig, J. B. Pendry, and D. R. Smith, “Calculation of material properties and ray tracing in transformation media,” Opt. Express 14(21), 9794–9804 (2006).
[CrossRef] [PubMed]

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312(5781), 1780–1782 (2006).
[CrossRef] [PubMed]

Stenger, N.

T. Ergin, N. Stenger, P. Brenner, J. B. Pendry, and M. Wegener, “Three-dimensional invisibility cloak at optical wavelengths,” Science 328(5976), 337–339 (2010).
[CrossRef] [PubMed]

Summers, C. J.

Tamma, V. A.

Tyc, T.

U. Leonhardt and T. Tyc, “Broadband invisibility by non-Euclidean cloaking,” Science 323(5910), 110–112 (2009).
[CrossRef] [PubMed]

Valentine, J.

M. Gharghi, C. Gladden, T. Zentgraf, Y. Liu, X. Yin, J. Valentine, and X. Zhang, “A carpet cloak for visible light,” Nano Lett. 11(7), 2825–2828 (2011).
[CrossRef] [PubMed]

J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater. 8(7), 568–571 (2009).
[CrossRef] [PubMed]

Wegener, M.

Wu, Q.

Yin, X.

M. Gharghi, C. Gladden, T. Zentgraf, Y. Liu, X. Yin, J. Valentine, and X. Zhang, “A carpet cloak for visible light,” Nano Lett. 11(7), 2825–2828 (2011).
[CrossRef] [PubMed]

Zentgraf, T.

M. Gharghi, C. Gladden, T. Zentgraf, Y. Liu, X. Yin, J. Valentine, and X. Zhang, “A carpet cloak for visible light,” Nano Lett. 11(7), 2825–2828 (2011).
[CrossRef] [PubMed]

J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater. 8(7), 568–571 (2009).
[CrossRef] [PubMed]

Zhang, B.

B. Zhang, Y. Luo, X. Liu, and G. Barbastathis, “Macroscopic invisibility cloak for visible light,” Phys. Rev. Lett. 106(3), 033901 (2011).
[CrossRef] [PubMed]

Zhang, J.

X. Chen, Y. Luo, J. Zhang, K. Jiang, J. B. Pendry, and S. Zhang, “Macroscopic invisibility cloaking of visible light,” Nat Commun 2, 176 (2011).
[CrossRef] [PubMed]

Zhang, S.

X. Chen, Y. Luo, J. Zhang, K. Jiang, J. B. Pendry, and S. Zhang, “Macroscopic invisibility cloaking of visible light,” Nat Commun 2, 176 (2011).
[CrossRef] [PubMed]

Zhang, X.

M. Gharghi, C. Gladden, T. Zentgraf, Y. Liu, X. Yin, J. Valentine, and X. Zhang, “A carpet cloak for visible light,” Nano Lett. 11(7), 2825–2828 (2011).
[CrossRef] [PubMed]

J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater. 8(7), 568–571 (2009).
[CrossRef] [PubMed]

J. Opt. Soc. Am. A (1)

Nano Lett. (1)

M. Gharghi, C. Gladden, T. Zentgraf, Y. Liu, X. Yin, J. Valentine, and X. Zhang, “A carpet cloak for visible light,” Nano Lett. 11(7), 2825–2828 (2011).
[CrossRef] [PubMed]

Nat Commun (1)

X. Chen, Y. Luo, J. Zhang, K. Jiang, J. B. Pendry, and S. Zhang, “Macroscopic invisibility cloaking of visible light,” Nat Commun 2, 176 (2011).
[CrossRef] [PubMed]

Nat. Commun. (1)

H. F. Ma and T. J. Cui, “Three-dimensional broadband ground-plane cloak made of metamaterials,” Nat. Commun. 1(3), 1–6 (2010).
[CrossRef] [PubMed]

Nat. Mater. (2)

H. Chen, C. T. Chan, and P. Sheng, “Transformation optics and metamaterials,” Nat. Mater. 9(5), 387–396 (2010).
[CrossRef] [PubMed]

J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater. 8(7), 568–571 (2009).
[CrossRef] [PubMed]

Nat. Photonics (1)

L. H. Gabrielli, J. Cardenas, C. B. Poitras, and M. Lipson, “Silicon nanostructure cloak operating at optical frequencies,” Nat. Photonics 3(8), 461–463 (2009).
[CrossRef]

Opt. Express (5)

Opt. Lett. (1)

Phys. Rev. Lett. (3)

J. Li and J. B. Pendry, “Hiding under the carpet: a new strategy for cloaking,” Phys. Rev. Lett. 101(20), 203901 (2008).
[CrossRef] [PubMed]

B. Zhang, Y. Luo, X. Liu, and G. Barbastathis, “Macroscopic invisibility cloak for visible light,” Phys. Rev. Lett. 106(3), 033901 (2011).
[CrossRef] [PubMed]

T. Ergin, J. Fischer, and M. Wegener, “Optical phase cloaking of 700-nm light waves in the far field by a three-dimensional carpet cloak,” Phys. Rev. Lett. 107(17), 173901 (2011).
[CrossRef]

Science (5)

U. Leonhardt and T. Tyc, “Broadband invisibility by non-Euclidean cloaking,” Science 323(5910), 110–112 (2009).
[CrossRef] [PubMed]

R. Liu, C. Ji, J. J. Mock, J. Y. Chin, T. J. Cui, and D. R. Smith, “Broadband ground-plane cloak,” Science 323(5912), 366–369 (2009).
[CrossRef] [PubMed]

T. Ergin, N. Stenger, P. Brenner, J. B. Pendry, and M. Wegener, “Three-dimensional invisibility cloak at optical wavelengths,” Science 328(5976), 337–339 (2010).
[CrossRef] [PubMed]

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312(5781), 1780–1782 (2006).
[CrossRef] [PubMed]

U. Leonhardt, “Optical conformal mapping,” Science 312(5781), 1777–1780 (2006).
[CrossRef] [PubMed]

Other (3)

U. Leonhardt and T. G. Philbin, Geometry and Light: The Science of Invisibility (Dover, Mineola, 2010).

L. D. Landau, E. M. Lifshitz, and L. P. Pitaevskii, Electrodynamics of Continuous Media, Vol. 8 (Butterworth-Heinemann, Oxford, 1984).

M. Born and E. Wolf, Principles of Optics, 7th. ed. (University Press, Cambridge, 1999).

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Figures (7)

Fig. 1
Fig. 1

Ray-tracing results for the Gaussian carpet cloak [6, 18, 19]. The scenery has previously been introduced [19]: a model is standing in front of a book shelf and is looking at her reflection in a mirror. (a) Rendered image without Gaussian bump in the mirror. (b) Rendered image with bump. (c) Rendered image with bump and with cloak. (d) Relative time-of-flight (TOF) image corresponding to the pixel-wise time delay between (c) and (a). The parameters correspond to those that we have previously used in Fig. 4 of Ref. 19. All ray-tracing calculations shown in this paper neglect the effects of dispersion. Thus, the color in panel (c) is merely for illustration.

Fig. 2
Fig. 2

As Fig. 1 but for the grating cloak [18, 19]. (a) Rendered image without grating in the mirror. (b) Rendered image with grating. (c) Rendered image with grating and with cloak. (d) Relative time-of-flight (TOF) image corresponding to the pixel-wise time delay between (c) and (a). The parameters correspond to those that we have previously used in Fig. 3 of Ref. 19.

Fig. 3
Fig. 3

(a) Illustration of the scenery used for the ray-tracing calculations for all free-space invisibilities in Figs. 4-7. A virtual point camera (the “eye”) looks at a model standing in front of a book shelf. The point camera is at a distance of 100 cm from the middle of the model’s eyes and at the same height above the floor with a field of view (FOV) of 42° vertically and 50° horizontally, emulating a human focal FOV. The axis of the metal cylinder and of the concentric cylindrical cloak in Figs. 4-6 as well as the center of the invisible sphere in Fig. 7 are at a distance of 45.5 cm from the middle between the model’s eyes and at the same height above the floor. (b) and (c) depict the raw images used as input for the ray-tracing calculations.

Fig. 4
Fig. 4

Ray-tracing results for the exact cylindrical free-space cloak [1]. In this scenery, which is illustrated in Fig. 3, a virtual point-camera looks at a model standing in front of a book shelf. (a) Rendered image without metal cylinder (compare Fig. 1(a)). (b) Rendered image with horizontal metal cylinder in front of model. Reflections from this cylinder show the other side of the room. (c) Rendered image with metal cylinder and with cylindrical cloak. (d) Relative time-of-flight (TOF) image corresponding to the pixel-wise time delay between (c) and (a). Note the units of attoseconds (1 as = 10−18 s) here rather than nanoseconds (1 ns = 10−9 s) as in Figs. 1(d), 2(d), 5(d), 6(d), and 7(c).

Fig. 5
Fig. 5

Ray-tracing results for the truncated cylindrical free-space cloak (compare with ideal cylindrical free-space cloak shown in Fig. 4). Here, tensor components below 0.1 are set to 0.1 and tensor components larger than 10 are set to 10.

Fig. 6
Fig. 6

Ray-tracing results for the cylindrical free-space cloak as in Fig. 4, but all components (13) of the ideal permittivity and permeability tensors are multiplied with a factor of 0.95 mimicking a 5% fabrication error.

Fig. 7
Fig. 7

Ray-tracing results for the invisible sphere. The scenery is as in Figs. 4-6. (a) Rendered image without the sphere. (b) Rendered image with invisible sphere. (c) Relative time-of-flight (TOF) image corresponding to the pixel-wise time delay between (b) and (a).

Equations (13)

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w= 1 2 ( D E + B H )= 1 2 ( ε 0 ( ε E ) E + μ 0 ( μ H ) H ),
ε ( r )= μ ( r ),
δ t 1 t 2 k v dt=0,
v = S w = E × H w ,
k = ω w D × B ,
δ t 1 t 2 ω w ( D × B ) v dt= δ t 1 t 2 ω w ( ( ε 0 ε E )×( μ 0 ε H ) ) v dt=0.
δ t 1 t 2 ( | ε   |  ε 1 ( E × H w ) ) v  dt = t 1 t 2 ( | ε   |  ε   1 v ) v  dt =0,
M ( r )=| ε | ε 1 =| μ | μ 1 ,
δ t 1 t 2 v ( M ( r ) v )dt=0 =:δ t 1 t 2 L( r , v ,t)dt.
L r i d dt L v i =0,
d v dt = M 1 2 ( v ( M ) v 2( ( M ) v ) v ).
ε r = μ r = ra a ; ε Θ = μ Θ = r ra ; ε z = μ z = ( b ba ) 2 ra a .
n = R r n + ( R r n ) 2 1 .

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