Abstract

Starting from well-known absolute instruments that provide perfect imaging, we analyze a class of rotationally symmetric compact closed manifolds, namely, geodesic lenses. We demonstrate with a numerical method that light rays confined on geodesic lenses form closed trajectories, and that for optical waves, the spectrum of a geodesic lens is (at least approximately) degenerate and equidistant. Moreover, we fabricate two geodesic lenses in micrometer and millimeter scales and observe curved light rays along the geodesics. Our experimental setup may offer a new platform to investigate light propagation on curved surfaces.

© 2019 Chinese Laser Press

Full Article  |  PDF Article
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References

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  1. M. Born and E. Wolf, Principles of Optics: Electromagnetic Theoryof Propagation, Interference and Diffraction of Light, CUP Archive (2000).
  2. R. K. Luneburg and M. Herzberger, Mathematical Theory of Optics (University of California, 1964).
  3. T. Tyc, L. Herzánová, M. Šarbort, and K. Bering, “Absolute instruments and perfect imaging in geometrical optics,” New J. Phys. 13, 115004 (2011).
    [Crossref]
  4. T. Tyc and A. J. Danner, “Absolute optical instruments, classical superintegrability, and separability of the Hamilton-Jacobi equation,” Phys. Rev. A 96, 053838 (2017).
    [Crossref]
  5. J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966–3969 (2000).
    [Crossref]
  6. T. Tyc and A. Danner, “Frequency spectra of absolute optical instruments,” New J. Phys. 14, 085023 (2012).
    [Crossref]
  7. T. Tyc, H. Chen, A. Danner, and Y. Xu, “Invisible lenses with positive isotropic refractive index,” Phys. Rev. A 90, 053829 (2014).
    [Crossref]
  8. T. Tyc, “Spectra of absolute instruments from the WKB approximation,” New J. Phys. 15, 065005 (2013).
    [Crossref]
  9. K. Zuzaňáková and T. Tyc, “Scattering of waves by the invisible lens,” J. Opt. 19, 015601 (2016).
    [Crossref]
  10. R. Rinehart, “A solution of the problem of rapid scanning for radar antennae,” J. Appl. Phys. 19, 860–862 (1948).
    [Crossref]
  11. S. Cornbleet and P. Rinous, “Generalised formulas for equivalent geodesic and nonuniform refractive lenses,” IEE Proc. H-Microw. Opt. Antennas 128, 95 (1981).
    [Crossref]
  12. M. Šarbort and T. Tyc, “Spherical media and geodesic lenses in geometrical optics,” J. Opt. 14, 075705 (2012).
    [Crossref]
  13. S. Batz and U. Peschel, “Linear and nonlinear optics in curved space,” Phys. Rev. A 78, 043821 (2008).
    [Crossref]
  14. V. H. Schultheiss, S. Batz, A. Szameit, F. Dreisow, S. Nolte, A. Tünnermann, S. Longhi, and U. Peschel, “Optics in curved space,” Phys. Rev. Lett. 105, 143901 (2010).
    [Crossref]
  15. R. Bekenstein, J. Nemirovsky, I. Kaminer, and M. Segev, “Shape-preserving accelerating electromagnetic wavepackets in curved space,” Phys. Rev. X 4, 011038 (2014).
    [Crossref]
  16. A. Patsyk, M. A. Bandres, R. Bekenstein, and M. Segev, “Observation of accelerating wave packets in curved space,” Phys. Rev. X 8, 011001 (2018).
    [Crossref]
  17. V. H. Schultheiss, S. Batz, and U. Peschel, “Hanbury Brown and Twiss measurements in curved space,” Nat. Photonics 10, 106–110 (2016).
    [Crossref]
  18. R. Bekenstein, Y. Kabessa, Y. Sharabi, O. Tal, N. Engheta, G. Eisenstein, A. J. Agranat, and M. Segev, “Control of light by curved space in nanophotonic structures,” Nat. Photonics 11, 664–670 (2017).
    [Crossref]
  19. U. Leonhardt, “Perfect imaging without negative refraction,” New J. Phys. 11, 093040 (2009).
    [Crossref]
  20. J. Perczel, T. Tyc, and U. Leonhardt, “Invisibility cloaking without superluminal propagation,” New J. Phys. 13, 083007 (2011).
    [Crossref]
  21. U. Leonhardt and T. Philbin, Geometry and Light: The Science of Invisibility (Dover, 2010).
  22. A. L. Besse, Manifolds All of Whose Geodesics Are Closed (Springer, 2012).
  23. M. Šarbort, Non-Euclidean Geometry in Optics, Ph.D thesis (Masaryk University, 2013).
  24. C. Sheng, H. Liu, Y. Wang, S. Zhu, and D. Genov, “Trapping light by mimicking gravitational lensing,” Nat. Photonics 7, 902–906 (2013).
    [Crossref]
  25. C. Sheng, R. Bekenstein, H. Liu, S. Zhu, and M. Segev, “Wavefront shaping through emulated curved space in waveguide settings,” Nat. Commun. 7, 10747 (2016).
    [Crossref]
  26. D. Wang, C. Liu, H. Liu, J. Han, and S. Zhang, “Wave dynamics on toroidal surface,” Opt. Express 26, 17820–17829 (2018).
    [Crossref]

2018 (2)

A. Patsyk, M. A. Bandres, R. Bekenstein, and M. Segev, “Observation of accelerating wave packets in curved space,” Phys. Rev. X 8, 011001 (2018).
[Crossref]

D. Wang, C. Liu, H. Liu, J. Han, and S. Zhang, “Wave dynamics on toroidal surface,” Opt. Express 26, 17820–17829 (2018).
[Crossref]

2017 (2)

R. Bekenstein, Y. Kabessa, Y. Sharabi, O. Tal, N. Engheta, G. Eisenstein, A. J. Agranat, and M. Segev, “Control of light by curved space in nanophotonic structures,” Nat. Photonics 11, 664–670 (2017).
[Crossref]

T. Tyc and A. J. Danner, “Absolute optical instruments, classical superintegrability, and separability of the Hamilton-Jacobi equation,” Phys. Rev. A 96, 053838 (2017).
[Crossref]

2016 (3)

K. Zuzaňáková and T. Tyc, “Scattering of waves by the invisible lens,” J. Opt. 19, 015601 (2016).
[Crossref]

V. H. Schultheiss, S. Batz, and U. Peschel, “Hanbury Brown and Twiss measurements in curved space,” Nat. Photonics 10, 106–110 (2016).
[Crossref]

C. Sheng, R. Bekenstein, H. Liu, S. Zhu, and M. Segev, “Wavefront shaping through emulated curved space in waveguide settings,” Nat. Commun. 7, 10747 (2016).
[Crossref]

2014 (2)

R. Bekenstein, J. Nemirovsky, I. Kaminer, and M. Segev, “Shape-preserving accelerating electromagnetic wavepackets in curved space,” Phys. Rev. X 4, 011038 (2014).
[Crossref]

T. Tyc, H. Chen, A. Danner, and Y. Xu, “Invisible lenses with positive isotropic refractive index,” Phys. Rev. A 90, 053829 (2014).
[Crossref]

2013 (2)

T. Tyc, “Spectra of absolute instruments from the WKB approximation,” New J. Phys. 15, 065005 (2013).
[Crossref]

C. Sheng, H. Liu, Y. Wang, S. Zhu, and D. Genov, “Trapping light by mimicking gravitational lensing,” Nat. Photonics 7, 902–906 (2013).
[Crossref]

2012 (2)

T. Tyc and A. Danner, “Frequency spectra of absolute optical instruments,” New J. Phys. 14, 085023 (2012).
[Crossref]

M. Šarbort and T. Tyc, “Spherical media and geodesic lenses in geometrical optics,” J. Opt. 14, 075705 (2012).
[Crossref]

2011 (2)

T. Tyc, L. Herzánová, M. Šarbort, and K. Bering, “Absolute instruments and perfect imaging in geometrical optics,” New J. Phys. 13, 115004 (2011).
[Crossref]

J. Perczel, T. Tyc, and U. Leonhardt, “Invisibility cloaking without superluminal propagation,” New J. Phys. 13, 083007 (2011).
[Crossref]

2010 (1)

V. H. Schultheiss, S. Batz, A. Szameit, F. Dreisow, S. Nolte, A. Tünnermann, S. Longhi, and U. Peschel, “Optics in curved space,” Phys. Rev. Lett. 105, 143901 (2010).
[Crossref]

2009 (1)

U. Leonhardt, “Perfect imaging without negative refraction,” New J. Phys. 11, 093040 (2009).
[Crossref]

2008 (1)

S. Batz and U. Peschel, “Linear and nonlinear optics in curved space,” Phys. Rev. A 78, 043821 (2008).
[Crossref]

2000 (1)

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966–3969 (2000).
[Crossref]

1981 (1)

S. Cornbleet and P. Rinous, “Generalised formulas for equivalent geodesic and nonuniform refractive lenses,” IEE Proc. H-Microw. Opt. Antennas 128, 95 (1981).
[Crossref]

1948 (1)

R. Rinehart, “A solution of the problem of rapid scanning for radar antennae,” J. Appl. Phys. 19, 860–862 (1948).
[Crossref]

Agranat, A. J.

R. Bekenstein, Y. Kabessa, Y. Sharabi, O. Tal, N. Engheta, G. Eisenstein, A. J. Agranat, and M. Segev, “Control of light by curved space in nanophotonic structures,” Nat. Photonics 11, 664–670 (2017).
[Crossref]

Bandres, M. A.

A. Patsyk, M. A. Bandres, R. Bekenstein, and M. Segev, “Observation of accelerating wave packets in curved space,” Phys. Rev. X 8, 011001 (2018).
[Crossref]

Batz, S.

V. H. Schultheiss, S. Batz, and U. Peschel, “Hanbury Brown and Twiss measurements in curved space,” Nat. Photonics 10, 106–110 (2016).
[Crossref]

V. H. Schultheiss, S. Batz, A. Szameit, F. Dreisow, S. Nolte, A. Tünnermann, S. Longhi, and U. Peschel, “Optics in curved space,” Phys. Rev. Lett. 105, 143901 (2010).
[Crossref]

S. Batz and U. Peschel, “Linear and nonlinear optics in curved space,” Phys. Rev. A 78, 043821 (2008).
[Crossref]

Bekenstein, R.

A. Patsyk, M. A. Bandres, R. Bekenstein, and M. Segev, “Observation of accelerating wave packets in curved space,” Phys. Rev. X 8, 011001 (2018).
[Crossref]

R. Bekenstein, Y. Kabessa, Y. Sharabi, O. Tal, N. Engheta, G. Eisenstein, A. J. Agranat, and M. Segev, “Control of light by curved space in nanophotonic structures,” Nat. Photonics 11, 664–670 (2017).
[Crossref]

C. Sheng, R. Bekenstein, H. Liu, S. Zhu, and M. Segev, “Wavefront shaping through emulated curved space in waveguide settings,” Nat. Commun. 7, 10747 (2016).
[Crossref]

R. Bekenstein, J. Nemirovsky, I. Kaminer, and M. Segev, “Shape-preserving accelerating electromagnetic wavepackets in curved space,” Phys. Rev. X 4, 011038 (2014).
[Crossref]

Bering, K.

T. Tyc, L. Herzánová, M. Šarbort, and K. Bering, “Absolute instruments and perfect imaging in geometrical optics,” New J. Phys. 13, 115004 (2011).
[Crossref]

Besse, A. L.

A. L. Besse, Manifolds All of Whose Geodesics Are Closed (Springer, 2012).

Born, M.

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theoryof Propagation, Interference and Diffraction of Light, CUP Archive (2000).

Chen, H.

T. Tyc, H. Chen, A. Danner, and Y. Xu, “Invisible lenses with positive isotropic refractive index,” Phys. Rev. A 90, 053829 (2014).
[Crossref]

Cornbleet, S.

S. Cornbleet and P. Rinous, “Generalised formulas for equivalent geodesic and nonuniform refractive lenses,” IEE Proc. H-Microw. Opt. Antennas 128, 95 (1981).
[Crossref]

Danner, A.

T. Tyc, H. Chen, A. Danner, and Y. Xu, “Invisible lenses with positive isotropic refractive index,” Phys. Rev. A 90, 053829 (2014).
[Crossref]

T. Tyc and A. Danner, “Frequency spectra of absolute optical instruments,” New J. Phys. 14, 085023 (2012).
[Crossref]

Danner, A. J.

T. Tyc and A. J. Danner, “Absolute optical instruments, classical superintegrability, and separability of the Hamilton-Jacobi equation,” Phys. Rev. A 96, 053838 (2017).
[Crossref]

Dreisow, F.

V. H. Schultheiss, S. Batz, A. Szameit, F. Dreisow, S. Nolte, A. Tünnermann, S. Longhi, and U. Peschel, “Optics in curved space,” Phys. Rev. Lett. 105, 143901 (2010).
[Crossref]

Eisenstein, G.

R. Bekenstein, Y. Kabessa, Y. Sharabi, O. Tal, N. Engheta, G. Eisenstein, A. J. Agranat, and M. Segev, “Control of light by curved space in nanophotonic structures,” Nat. Photonics 11, 664–670 (2017).
[Crossref]

Engheta, N.

R. Bekenstein, Y. Kabessa, Y. Sharabi, O. Tal, N. Engheta, G. Eisenstein, A. J. Agranat, and M. Segev, “Control of light by curved space in nanophotonic structures,” Nat. Photonics 11, 664–670 (2017).
[Crossref]

Genov, D.

C. Sheng, H. Liu, Y. Wang, S. Zhu, and D. Genov, “Trapping light by mimicking gravitational lensing,” Nat. Photonics 7, 902–906 (2013).
[Crossref]

Han, J.

Herzánová, L.

T. Tyc, L. Herzánová, M. Šarbort, and K. Bering, “Absolute instruments and perfect imaging in geometrical optics,” New J. Phys. 13, 115004 (2011).
[Crossref]

Herzberger, M.

R. K. Luneburg and M. Herzberger, Mathematical Theory of Optics (University of California, 1964).

Kabessa, Y.

R. Bekenstein, Y. Kabessa, Y. Sharabi, O. Tal, N. Engheta, G. Eisenstein, A. J. Agranat, and M. Segev, “Control of light by curved space in nanophotonic structures,” Nat. Photonics 11, 664–670 (2017).
[Crossref]

Kaminer, I.

R. Bekenstein, J. Nemirovsky, I. Kaminer, and M. Segev, “Shape-preserving accelerating electromagnetic wavepackets in curved space,” Phys. Rev. X 4, 011038 (2014).
[Crossref]

Leonhardt, U.

J. Perczel, T. Tyc, and U. Leonhardt, “Invisibility cloaking without superluminal propagation,” New J. Phys. 13, 083007 (2011).
[Crossref]

U. Leonhardt, “Perfect imaging without negative refraction,” New J. Phys. 11, 093040 (2009).
[Crossref]

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

Liu, C.

Liu, H.

D. Wang, C. Liu, H. Liu, J. Han, and S. Zhang, “Wave dynamics on toroidal surface,” Opt. Express 26, 17820–17829 (2018).
[Crossref]

C. Sheng, R. Bekenstein, H. Liu, S. Zhu, and M. Segev, “Wavefront shaping through emulated curved space in waveguide settings,” Nat. Commun. 7, 10747 (2016).
[Crossref]

C. Sheng, H. Liu, Y. Wang, S. Zhu, and D. Genov, “Trapping light by mimicking gravitational lensing,” Nat. Photonics 7, 902–906 (2013).
[Crossref]

Longhi, S.

V. H. Schultheiss, S. Batz, A. Szameit, F. Dreisow, S. Nolte, A. Tünnermann, S. Longhi, and U. Peschel, “Optics in curved space,” Phys. Rev. Lett. 105, 143901 (2010).
[Crossref]

Luneburg, R. K.

R. K. Luneburg and M. Herzberger, Mathematical Theory of Optics (University of California, 1964).

Nemirovsky, J.

R. Bekenstein, J. Nemirovsky, I. Kaminer, and M. Segev, “Shape-preserving accelerating electromagnetic wavepackets in curved space,” Phys. Rev. X 4, 011038 (2014).
[Crossref]

Nolte, S.

V. H. Schultheiss, S. Batz, A. Szameit, F. Dreisow, S. Nolte, A. Tünnermann, S. Longhi, and U. Peschel, “Optics in curved space,” Phys. Rev. Lett. 105, 143901 (2010).
[Crossref]

Patsyk, A.

A. Patsyk, M. A. Bandres, R. Bekenstein, and M. Segev, “Observation of accelerating wave packets in curved space,” Phys. Rev. X 8, 011001 (2018).
[Crossref]

Pendry, J. B.

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966–3969 (2000).
[Crossref]

Perczel, J.

J. Perczel, T. Tyc, and U. Leonhardt, “Invisibility cloaking without superluminal propagation,” New J. Phys. 13, 083007 (2011).
[Crossref]

Peschel, U.

V. H. Schultheiss, S. Batz, and U. Peschel, “Hanbury Brown and Twiss measurements in curved space,” Nat. Photonics 10, 106–110 (2016).
[Crossref]

V. H. Schultheiss, S. Batz, A. Szameit, F. Dreisow, S. Nolte, A. Tünnermann, S. Longhi, and U. Peschel, “Optics in curved space,” Phys. Rev. Lett. 105, 143901 (2010).
[Crossref]

S. Batz and U. Peschel, “Linear and nonlinear optics in curved space,” Phys. Rev. A 78, 043821 (2008).
[Crossref]

Philbin, T.

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

Rinehart, R.

R. Rinehart, “A solution of the problem of rapid scanning for radar antennae,” J. Appl. Phys. 19, 860–862 (1948).
[Crossref]

Rinous, P.

S. Cornbleet and P. Rinous, “Generalised formulas for equivalent geodesic and nonuniform refractive lenses,” IEE Proc. H-Microw. Opt. Antennas 128, 95 (1981).
[Crossref]

Šarbort, M.

M. Šarbort and T. Tyc, “Spherical media and geodesic lenses in geometrical optics,” J. Opt. 14, 075705 (2012).
[Crossref]

T. Tyc, L. Herzánová, M. Šarbort, and K. Bering, “Absolute instruments and perfect imaging in geometrical optics,” New J. Phys. 13, 115004 (2011).
[Crossref]

M. Šarbort, Non-Euclidean Geometry in Optics, Ph.D thesis (Masaryk University, 2013).

Schultheiss, V. H.

V. H. Schultheiss, S. Batz, and U. Peschel, “Hanbury Brown and Twiss measurements in curved space,” Nat. Photonics 10, 106–110 (2016).
[Crossref]

V. H. Schultheiss, S. Batz, A. Szameit, F. Dreisow, S. Nolte, A. Tünnermann, S. Longhi, and U. Peschel, “Optics in curved space,” Phys. Rev. Lett. 105, 143901 (2010).
[Crossref]

Segev, M.

A. Patsyk, M. A. Bandres, R. Bekenstein, and M. Segev, “Observation of accelerating wave packets in curved space,” Phys. Rev. X 8, 011001 (2018).
[Crossref]

R. Bekenstein, Y. Kabessa, Y. Sharabi, O. Tal, N. Engheta, G. Eisenstein, A. J. Agranat, and M. Segev, “Control of light by curved space in nanophotonic structures,” Nat. Photonics 11, 664–670 (2017).
[Crossref]

C. Sheng, R. Bekenstein, H. Liu, S. Zhu, and M. Segev, “Wavefront shaping through emulated curved space in waveguide settings,” Nat. Commun. 7, 10747 (2016).
[Crossref]

R. Bekenstein, J. Nemirovsky, I. Kaminer, and M. Segev, “Shape-preserving accelerating electromagnetic wavepackets in curved space,” Phys. Rev. X 4, 011038 (2014).
[Crossref]

Sharabi, Y.

R. Bekenstein, Y. Kabessa, Y. Sharabi, O. Tal, N. Engheta, G. Eisenstein, A. J. Agranat, and M. Segev, “Control of light by curved space in nanophotonic structures,” Nat. Photonics 11, 664–670 (2017).
[Crossref]

Sheng, C.

C. Sheng, R. Bekenstein, H. Liu, S. Zhu, and M. Segev, “Wavefront shaping through emulated curved space in waveguide settings,” Nat. Commun. 7, 10747 (2016).
[Crossref]

C. Sheng, H. Liu, Y. Wang, S. Zhu, and D. Genov, “Trapping light by mimicking gravitational lensing,” Nat. Photonics 7, 902–906 (2013).
[Crossref]

Szameit, A.

V. H. Schultheiss, S. Batz, A. Szameit, F. Dreisow, S. Nolte, A. Tünnermann, S. Longhi, and U. Peschel, “Optics in curved space,” Phys. Rev. Lett. 105, 143901 (2010).
[Crossref]

Tal, O.

R. Bekenstein, Y. Kabessa, Y. Sharabi, O. Tal, N. Engheta, G. Eisenstein, A. J. Agranat, and M. Segev, “Control of light by curved space in nanophotonic structures,” Nat. Photonics 11, 664–670 (2017).
[Crossref]

Tünnermann, A.

V. H. Schultheiss, S. Batz, A. Szameit, F. Dreisow, S. Nolte, A. Tünnermann, S. Longhi, and U. Peschel, “Optics in curved space,” Phys. Rev. Lett. 105, 143901 (2010).
[Crossref]

Tyc, T.

T. Tyc and A. J. Danner, “Absolute optical instruments, classical superintegrability, and separability of the Hamilton-Jacobi equation,” Phys. Rev. A 96, 053838 (2017).
[Crossref]

K. Zuzaňáková and T. Tyc, “Scattering of waves by the invisible lens,” J. Opt. 19, 015601 (2016).
[Crossref]

T. Tyc, H. Chen, A. Danner, and Y. Xu, “Invisible lenses with positive isotropic refractive index,” Phys. Rev. A 90, 053829 (2014).
[Crossref]

T. Tyc, “Spectra of absolute instruments from the WKB approximation,” New J. Phys. 15, 065005 (2013).
[Crossref]

T. Tyc and A. Danner, “Frequency spectra of absolute optical instruments,” New J. Phys. 14, 085023 (2012).
[Crossref]

M. Šarbort and T. Tyc, “Spherical media and geodesic lenses in geometrical optics,” J. Opt. 14, 075705 (2012).
[Crossref]

T. Tyc, L. Herzánová, M. Šarbort, and K. Bering, “Absolute instruments and perfect imaging in geometrical optics,” New J. Phys. 13, 115004 (2011).
[Crossref]

J. Perczel, T. Tyc, and U. Leonhardt, “Invisibility cloaking without superluminal propagation,” New J. Phys. 13, 083007 (2011).
[Crossref]

Wang, D.

Wang, Y.

C. Sheng, H. Liu, Y. Wang, S. Zhu, and D. Genov, “Trapping light by mimicking gravitational lensing,” Nat. Photonics 7, 902–906 (2013).
[Crossref]

Wolf, E.

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theoryof Propagation, Interference and Diffraction of Light, CUP Archive (2000).

Xu, Y.

T. Tyc, H. Chen, A. Danner, and Y. Xu, “Invisible lenses with positive isotropic refractive index,” Phys. Rev. A 90, 053829 (2014).
[Crossref]

Zhang, S.

Zhu, S.

C. Sheng, R. Bekenstein, H. Liu, S. Zhu, and M. Segev, “Wavefront shaping through emulated curved space in waveguide settings,” Nat. Commun. 7, 10747 (2016).
[Crossref]

C. Sheng, H. Liu, Y. Wang, S. Zhu, and D. Genov, “Trapping light by mimicking gravitational lensing,” Nat. Photonics 7, 902–906 (2013).
[Crossref]

Zuzanáková, K.

K. Zuzaňáková and T. Tyc, “Scattering of waves by the invisible lens,” J. Opt. 19, 015601 (2016).
[Crossref]

IEE Proc. H-Microw. Opt. Antennas (1)

S. Cornbleet and P. Rinous, “Generalised formulas for equivalent geodesic and nonuniform refractive lenses,” IEE Proc. H-Microw. Opt. Antennas 128, 95 (1981).
[Crossref]

J. Appl. Phys. (1)

R. Rinehart, “A solution of the problem of rapid scanning for radar antennae,” J. Appl. Phys. 19, 860–862 (1948).
[Crossref]

J. Opt. (2)

M. Šarbort and T. Tyc, “Spherical media and geodesic lenses in geometrical optics,” J. Opt. 14, 075705 (2012).
[Crossref]

K. Zuzaňáková and T. Tyc, “Scattering of waves by the invisible lens,” J. Opt. 19, 015601 (2016).
[Crossref]

Nat. Commun. (1)

C. Sheng, R. Bekenstein, H. Liu, S. Zhu, and M. Segev, “Wavefront shaping through emulated curved space in waveguide settings,” Nat. Commun. 7, 10747 (2016).
[Crossref]

Nat. Photonics (3)

C. Sheng, H. Liu, Y. Wang, S. Zhu, and D. Genov, “Trapping light by mimicking gravitational lensing,” Nat. Photonics 7, 902–906 (2013).
[Crossref]

V. H. Schultheiss, S. Batz, and U. Peschel, “Hanbury Brown and Twiss measurements in curved space,” Nat. Photonics 10, 106–110 (2016).
[Crossref]

R. Bekenstein, Y. Kabessa, Y. Sharabi, O. Tal, N. Engheta, G. Eisenstein, A. J. Agranat, and M. Segev, “Control of light by curved space in nanophotonic structures,” Nat. Photonics 11, 664–670 (2017).
[Crossref]

New J. Phys. (5)

U. Leonhardt, “Perfect imaging without negative refraction,” New J. Phys. 11, 093040 (2009).
[Crossref]

J. Perczel, T. Tyc, and U. Leonhardt, “Invisibility cloaking without superluminal propagation,” New J. Phys. 13, 083007 (2011).
[Crossref]

T. Tyc, “Spectra of absolute instruments from the WKB approximation,” New J. Phys. 15, 065005 (2013).
[Crossref]

T. Tyc and A. Danner, “Frequency spectra of absolute optical instruments,” New J. Phys. 14, 085023 (2012).
[Crossref]

T. Tyc, L. Herzánová, M. Šarbort, and K. Bering, “Absolute instruments and perfect imaging in geometrical optics,” New J. Phys. 13, 115004 (2011).
[Crossref]

Opt. Express (1)

Phys. Rev. A (3)

T. Tyc and A. J. Danner, “Absolute optical instruments, classical superintegrability, and separability of the Hamilton-Jacobi equation,” Phys. Rev. A 96, 053838 (2017).
[Crossref]

T. Tyc, H. Chen, A. Danner, and Y. Xu, “Invisible lenses with positive isotropic refractive index,” Phys. Rev. A 90, 053829 (2014).
[Crossref]

S. Batz and U. Peschel, “Linear and nonlinear optics in curved space,” Phys. Rev. A 78, 043821 (2008).
[Crossref]

Phys. Rev. Lett. (2)

V. H. Schultheiss, S. Batz, A. Szameit, F. Dreisow, S. Nolte, A. Tünnermann, S. Longhi, and U. Peschel, “Optics in curved space,” Phys. Rev. Lett. 105, 143901 (2010).
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Phys. Rev. X (2)

R. Bekenstein, J. Nemirovsky, I. Kaminer, and M. Segev, “Shape-preserving accelerating electromagnetic wavepackets in curved space,” Phys. Rev. X 4, 011038 (2014).
[Crossref]

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[Crossref]

Other (5)

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Figures (8)

Fig. 1.
Fig. 1. AIs (upper row) and corresponding GLs (lower row) with rotational symmetry. In AIs, the center of each lens is marked with S. The position vector is denoted with r. The angle parameter θ is omitted for simplification because of the rotational symmetry of AIs and GLs. Contour plots show a refractive index profile of n(r). The presented AIs are (a) Maxwell’s fish-eye lens, (b) generalized Maxwell’s fish-eye lens with M=5, (c) extended invisible lens, and (d) inverse invisible lens. The corresponding geodesic lenses are (a) sphere, (b) spindle with M=5, (c) Tannery’s pear, and (d) truncated Tannery’s pear. On GLs, the axis of rotational symmetry is denoted by a dashed line, which connects north (N) pole and south (S) pole. ρ is the radial coordinate, and h(ρ) is the length measured along the meridian from N pole on the geodesic surface. Light rays starting from point A form closed trajectories shown in different colors. S poles are mapped from the centers S of AIs, while N poles correspond to the infinities of AIs. Dashed black circles of AIs are places with refractive index of unity at radius of r0, which are equivalent to those of GLs.
Fig. 2.
Fig. 2. Real part of the modes with different indices N and m: (a) ψ22 on sphere, (b) ψ50 on spindle, (c) ψ22 on Tannery’s pear, and (d) ψ22 on truncated Tannery’s pear.
Fig. 3.
Fig. 3. Experimental setup and sample description. (a) Schematic of the observation and coupling scheme of the light to the geodesic lens. A laser beam is coupled to a 3D curved waveguide from the top and excites the rare-earth ions in the waveguide. The emitted fluorescent light at 615 nm is then collected by a CCD camera. (b) 3D curved waveguide morphology captured by a CCD camera when illuminated by white light. (c) Scanning electron microscope image of the 3D curved surface around the coupling grating (red dashed box) before spin coating. The cross structure corresponds to that displayed in (b) and is used to couple laser beams into the waveguide. (d) Scanning electron microscope image of the 3D curved surface with larger scale (blue dashed box) before spin coating. Based on this figure, one can get the accurate parameters of the 3D curved surface.
Fig. 4.
Fig. 4. Optical measurements and fitting results of light rays in a spindle with M=5. (a) Fitting the shape of micro-structured metallic needle waveguide with a spindle. (b) Optical measurements on micro-structured metallic needle waveguide. (c) Enlarged drawing nearby the coupling source. (d) Fitting the light trajectory of micro-structured metallic needle waveguide with a spindle.
Fig. 5.
Fig. 5. Sample fabrication process. (a) Position of straight silver wire, movement console (MC), and hydrogen flame. A straight silver wire is fixed on MC1 and MC2 and then put on a hydrogen flame at a proper position. (b) Metallic wire fusion process. The ends of the straight silver wire are pulled at speeds v1 and v2, and the silver wire gradually becomes tapered. (c) Two obtained metallic cones. The silver wire breaks into two metallic cones at some point.
Fig. 6.
Fig. 6. (a) Sketch of a metallic waveguide. (b) Cross section of the metallic waveguide.
Fig. 7.
Fig. 7. (a) Sketch of the coupling grating (yellow boxes). (b) SEM image of the metallic curved surface and the coupling grating (in red dashed box) before spin-coating process. (c) Grating coupling process and optical measurement of the sample. The yellow boxes in (a) show the coupling grating, the graded blue spot represents the exciting beam, and the blue arrows show the direction of the laser beam propagating in the waveguide. The coupling grating in (a) corresponds to the red dashed box in (b), which is fabricated before spin-coating process. The red dashed box region in (b) corresponds to the grating in (c), which is indicated by the red arrows.
Fig. 8.
Fig. 8. Optical measurements and fitting results of light rays on a sphere. (a) CCD camera picture of micro-structured sphere waveguide. (b) Light trajectory on micro-structured sphere waveguide. (c) Fitting the light trajectory with a sphere.

Tables (1)

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Table 1. Description of Four AIs and Corresponding GLs with Spectrum

Equations (9)

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(r/r0)n3/2+(r/r0)n1/22=0,
ρ=n(r)r,dh=n(r)dr,
d2xλdξ2+Γμνλdxμdξdxνdξ=0,
Γμνλ=((000ρ(h)ρ(h))μν(0ρ(h)ρ(h)ρ(h)ρ(h)0)μν)λ,
{h(ξ)ρ[h(ξ)]ρ[h(ξ)]θ(ξ)θ(ξ)=0,θ(ξ)+2ρ[h(ξ)]ρ[h(ξ)]h(ξ)θ(ξ)=0.
˜2ψ+ω2c2ψ=0,
ψ(h,φ)=R(h)eimφ,
2Rh2+1ρρhRh+(ω2c2m2h2)R=0.
ωNm=a·sNm+b,