Abstract

The unique properties of optical waveguides electrically controlled by means of graphene layers are investigated. We demonstrate that, thanks to tunable losses induced by graphene layers, a careful design of silicon on silica ridge waveguides can be used to explore passive PT-symmetry breaking in directional couplers. We prove that the exceptional point of the system can be probed by varying the applied voltage and we thus propose very compact photonic structures which can be exploited to control coupling between waveguides and to tailor discrete diffraction in arrays.

© 2012 OSA

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  1. K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, “Electric field effect in atomically thin carbon films,” Science306, 666–669 (2004).
    [CrossRef] [PubMed]
  2. K. Kim, J. Y. Choi, T. Kim, S. H. Cho, and H. J. Chung, “A role for graphene in silicon-based semiconductor devices,” Nature (London)479, 338–344 (2011).
    [CrossRef]
  3. A. Vakil and N. Engheta, “Transformation optics using graphene,” Science332, 1291–1294 (2011).
    [CrossRef] [PubMed]
  4. F. Bonaccorso, Z. Sun, T. Hasan, and A. C. Ferrari, “Graphene photonics and optoelectronics,” Nat. Photon.4, 611–622 (2010).
    [CrossRef]
  5. M. Liu, X. Yin, E. Ulin-Avila, B. Geng, T. Zentgraf, L. Ju, F. Wang, and X. Zhang, “A graphene-based broadband optical modulator,” Nature (London)474, 64–67 (2011).
    [CrossRef]
  6. M. Liu, X. Yin, and X. Zhang, “Double-layer graphene optical modulator,” Nano Lett.12, 1482–1485 (2012).
    [CrossRef] [PubMed]
  7. M. Midrio, S. Boscolo, M. Moresco, M. Romagnoli, C. De Angelis, A. Locatelli, and A.-D. Capobianco, “Graphene-assisted critically-coupled optical ring modulator,” Opt. Express20, 23144–23155 (2012).
    [CrossRef]
  8. Q. Bao, H. Zhang, B. Wang, Z. Ni, C. Lim, Y. Wang, D. Y. Tang, and K. P. Loh, “Broadband graphene polarizer,” Nat. Photon.5, 411–415 (2011).
    [CrossRef]
  9. J. T. Kim and C. G. Choi, “Graphene-based polymer waveguide polarizer,” Opt. Express20, 3556–3562 (2012).
    [CrossRef] [PubMed]
  10. Z. Q. Li, E. A. Henriksen, Z. Jiang, Z. Hao, M. C. Martin, P. Kim, H. L. Stormer, and D. N. Basov, “Dirac charge dynamics in graphene by infrared spectroscopy,” Nat. Phys.4, 532–535 (2008).
    [CrossRef]
  11. S. Klaiman, U. Gunther, and N. Moiseyev, “Visualization of branch points in PT-symmetric waveguides,” Phys. Rev. Lett.101, 080402 (2008).
    [CrossRef] [PubMed]
  12. A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-symmetry breaking in complex optical potentials,” Phys. Rev. Lett.103, 093902 (2009).
    [CrossRef] [PubMed]
  13. S. Yu, G. X. Piao, D. R. Mason, S. In, and N. Park, “Spatiospectral separation of exceptional points in PT-symmetric optical potentials,” Phys. Rev. A86, 031802 (2012).
    [CrossRef]
  14. G. W. Hanson, “Dyadic Green’s functions for an anisotropic, non-local model of biased graphene,” IEEE Trans. Antennas Propagat.56, 747–757 (2008).
    [CrossRef]
  15. D. N. Christodoulides and R. I. Joseph, “Discrete self-focusing in nonlinear arrays of coupled waveguides,” Opt. Lett.13, 794–796 (1988).
    [CrossRef] [PubMed]
  16. T. Pertsch, T. Zentgraf, U. Peschel, A. Brauer, and F. Lederer, “Anomalous refraction and diffraction in discrete optical systems,” Phys. Rev. Lett.88, 093901 (2002).
    [CrossRef] [PubMed]

2012 (4)

M. Liu, X. Yin, and X. Zhang, “Double-layer graphene optical modulator,” Nano Lett.12, 1482–1485 (2012).
[CrossRef] [PubMed]

S. Yu, G. X. Piao, D. R. Mason, S. In, and N. Park, “Spatiospectral separation of exceptional points in PT-symmetric optical potentials,” Phys. Rev. A86, 031802 (2012).
[CrossRef]

J. T. Kim and C. G. Choi, “Graphene-based polymer waveguide polarizer,” Opt. Express20, 3556–3562 (2012).
[CrossRef] [PubMed]

M. Midrio, S. Boscolo, M. Moresco, M. Romagnoli, C. De Angelis, A. Locatelli, and A.-D. Capobianco, “Graphene-assisted critically-coupled optical ring modulator,” Opt. Express20, 23144–23155 (2012).
[CrossRef]

2011 (4)

Q. Bao, H. Zhang, B. Wang, Z. Ni, C. Lim, Y. Wang, D. Y. Tang, and K. P. Loh, “Broadband graphene polarizer,” Nat. Photon.5, 411–415 (2011).
[CrossRef]

K. Kim, J. Y. Choi, T. Kim, S. H. Cho, and H. J. Chung, “A role for graphene in silicon-based semiconductor devices,” Nature (London)479, 338–344 (2011).
[CrossRef]

A. Vakil and N. Engheta, “Transformation optics using graphene,” Science332, 1291–1294 (2011).
[CrossRef] [PubMed]

M. Liu, X. Yin, E. Ulin-Avila, B. Geng, T. Zentgraf, L. Ju, F. Wang, and X. Zhang, “A graphene-based broadband optical modulator,” Nature (London)474, 64–67 (2011).
[CrossRef]

2010 (1)

F. Bonaccorso, Z. Sun, T. Hasan, and A. C. Ferrari, “Graphene photonics and optoelectronics,” Nat. Photon.4, 611–622 (2010).
[CrossRef]

2009 (1)

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-symmetry breaking in complex optical potentials,” Phys. Rev. Lett.103, 093902 (2009).
[CrossRef] [PubMed]

2008 (3)

G. W. Hanson, “Dyadic Green’s functions for an anisotropic, non-local model of biased graphene,” IEEE Trans. Antennas Propagat.56, 747–757 (2008).
[CrossRef]

Z. Q. Li, E. A. Henriksen, Z. Jiang, Z. Hao, M. C. Martin, P. Kim, H. L. Stormer, and D. N. Basov, “Dirac charge dynamics in graphene by infrared spectroscopy,” Nat. Phys.4, 532–535 (2008).
[CrossRef]

S. Klaiman, U. Gunther, and N. Moiseyev, “Visualization of branch points in PT-symmetric waveguides,” Phys. Rev. Lett.101, 080402 (2008).
[CrossRef] [PubMed]

2004 (1)

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, “Electric field effect in atomically thin carbon films,” Science306, 666–669 (2004).
[CrossRef] [PubMed]

2002 (1)

T. Pertsch, T. Zentgraf, U. Peschel, A. Brauer, and F. Lederer, “Anomalous refraction and diffraction in discrete optical systems,” Phys. Rev. Lett.88, 093901 (2002).
[CrossRef] [PubMed]

1988 (1)

Aimez, V.

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-symmetry breaking in complex optical potentials,” Phys. Rev. Lett.103, 093902 (2009).
[CrossRef] [PubMed]

Bao, Q.

Q. Bao, H. Zhang, B. Wang, Z. Ni, C. Lim, Y. Wang, D. Y. Tang, and K. P. Loh, “Broadband graphene polarizer,” Nat. Photon.5, 411–415 (2011).
[CrossRef]

Basov, D. N.

Z. Q. Li, E. A. Henriksen, Z. Jiang, Z. Hao, M. C. Martin, P. Kim, H. L. Stormer, and D. N. Basov, “Dirac charge dynamics in graphene by infrared spectroscopy,” Nat. Phys.4, 532–535 (2008).
[CrossRef]

Bonaccorso, F.

F. Bonaccorso, Z. Sun, T. Hasan, and A. C. Ferrari, “Graphene photonics and optoelectronics,” Nat. Photon.4, 611–622 (2010).
[CrossRef]

Boscolo, S.

Brauer, A.

T. Pertsch, T. Zentgraf, U. Peschel, A. Brauer, and F. Lederer, “Anomalous refraction and diffraction in discrete optical systems,” Phys. Rev. Lett.88, 093901 (2002).
[CrossRef] [PubMed]

Capobianco, A.-D.

Cho, S. H.

K. Kim, J. Y. Choi, T. Kim, S. H. Cho, and H. J. Chung, “A role for graphene in silicon-based semiconductor devices,” Nature (London)479, 338–344 (2011).
[CrossRef]

Choi, C. G.

Choi, J. Y.

K. Kim, J. Y. Choi, T. Kim, S. H. Cho, and H. J. Chung, “A role for graphene in silicon-based semiconductor devices,” Nature (London)479, 338–344 (2011).
[CrossRef]

Christodoulides, D. N.

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-symmetry breaking in complex optical potentials,” Phys. Rev. Lett.103, 093902 (2009).
[CrossRef] [PubMed]

D. N. Christodoulides and R. I. Joseph, “Discrete self-focusing in nonlinear arrays of coupled waveguides,” Opt. Lett.13, 794–796 (1988).
[CrossRef] [PubMed]

Chung, H. J.

K. Kim, J. Y. Choi, T. Kim, S. H. Cho, and H. J. Chung, “A role for graphene in silicon-based semiconductor devices,” Nature (London)479, 338–344 (2011).
[CrossRef]

De Angelis, C.

Dubonos, S. V.

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, “Electric field effect in atomically thin carbon films,” Science306, 666–669 (2004).
[CrossRef] [PubMed]

Duchesne, D.

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-symmetry breaking in complex optical potentials,” Phys. Rev. Lett.103, 093902 (2009).
[CrossRef] [PubMed]

Engheta, N.

A. Vakil and N. Engheta, “Transformation optics using graphene,” Science332, 1291–1294 (2011).
[CrossRef] [PubMed]

Ferrari, A. C.

F. Bonaccorso, Z. Sun, T. Hasan, and A. C. Ferrari, “Graphene photonics and optoelectronics,” Nat. Photon.4, 611–622 (2010).
[CrossRef]

Firsov, A. A.

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, “Electric field effect in atomically thin carbon films,” Science306, 666–669 (2004).
[CrossRef] [PubMed]

Geim, A. K.

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, “Electric field effect in atomically thin carbon films,” Science306, 666–669 (2004).
[CrossRef] [PubMed]

Geng, B.

M. Liu, X. Yin, E. Ulin-Avila, B. Geng, T. Zentgraf, L. Ju, F. Wang, and X. Zhang, “A graphene-based broadband optical modulator,” Nature (London)474, 64–67 (2011).
[CrossRef]

Grigorieva, I. V.

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, “Electric field effect in atomically thin carbon films,” Science306, 666–669 (2004).
[CrossRef] [PubMed]

Gunther, U.

S. Klaiman, U. Gunther, and N. Moiseyev, “Visualization of branch points in PT-symmetric waveguides,” Phys. Rev. Lett.101, 080402 (2008).
[CrossRef] [PubMed]

Guo, A.

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-symmetry breaking in complex optical potentials,” Phys. Rev. Lett.103, 093902 (2009).
[CrossRef] [PubMed]

Hanson, G. W.

G. W. Hanson, “Dyadic Green’s functions for an anisotropic, non-local model of biased graphene,” IEEE Trans. Antennas Propagat.56, 747–757 (2008).
[CrossRef]

Hao, Z.

Z. Q. Li, E. A. Henriksen, Z. Jiang, Z. Hao, M. C. Martin, P. Kim, H. L. Stormer, and D. N. Basov, “Dirac charge dynamics in graphene by infrared spectroscopy,” Nat. Phys.4, 532–535 (2008).
[CrossRef]

Hasan, T.

F. Bonaccorso, Z. Sun, T. Hasan, and A. C. Ferrari, “Graphene photonics and optoelectronics,” Nat. Photon.4, 611–622 (2010).
[CrossRef]

Henriksen, E. A.

Z. Q. Li, E. A. Henriksen, Z. Jiang, Z. Hao, M. C. Martin, P. Kim, H. L. Stormer, and D. N. Basov, “Dirac charge dynamics in graphene by infrared spectroscopy,” Nat. Phys.4, 532–535 (2008).
[CrossRef]

In, S.

S. Yu, G. X. Piao, D. R. Mason, S. In, and N. Park, “Spatiospectral separation of exceptional points in PT-symmetric optical potentials,” Phys. Rev. A86, 031802 (2012).
[CrossRef]

Jiang, D.

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, “Electric field effect in atomically thin carbon films,” Science306, 666–669 (2004).
[CrossRef] [PubMed]

Jiang, Z.

Z. Q. Li, E. A. Henriksen, Z. Jiang, Z. Hao, M. C. Martin, P. Kim, H. L. Stormer, and D. N. Basov, “Dirac charge dynamics in graphene by infrared spectroscopy,” Nat. Phys.4, 532–535 (2008).
[CrossRef]

Joseph, R. I.

Ju, L.

M. Liu, X. Yin, E. Ulin-Avila, B. Geng, T. Zentgraf, L. Ju, F. Wang, and X. Zhang, “A graphene-based broadband optical modulator,” Nature (London)474, 64–67 (2011).
[CrossRef]

Kim, J. T.

Kim, K.

K. Kim, J. Y. Choi, T. Kim, S. H. Cho, and H. J. Chung, “A role for graphene in silicon-based semiconductor devices,” Nature (London)479, 338–344 (2011).
[CrossRef]

Kim, P.

Z. Q. Li, E. A. Henriksen, Z. Jiang, Z. Hao, M. C. Martin, P. Kim, H. L. Stormer, and D. N. Basov, “Dirac charge dynamics in graphene by infrared spectroscopy,” Nat. Phys.4, 532–535 (2008).
[CrossRef]

Kim, T.

K. Kim, J. Y. Choi, T. Kim, S. H. Cho, and H. J. Chung, “A role for graphene in silicon-based semiconductor devices,” Nature (London)479, 338–344 (2011).
[CrossRef]

Klaiman, S.

S. Klaiman, U. Gunther, and N. Moiseyev, “Visualization of branch points in PT-symmetric waveguides,” Phys. Rev. Lett.101, 080402 (2008).
[CrossRef] [PubMed]

Lederer, F.

T. Pertsch, T. Zentgraf, U. Peschel, A. Brauer, and F. Lederer, “Anomalous refraction and diffraction in discrete optical systems,” Phys. Rev. Lett.88, 093901 (2002).
[CrossRef] [PubMed]

Li, Z. Q.

Z. Q. Li, E. A. Henriksen, Z. Jiang, Z. Hao, M. C. Martin, P. Kim, H. L. Stormer, and D. N. Basov, “Dirac charge dynamics in graphene by infrared spectroscopy,” Nat. Phys.4, 532–535 (2008).
[CrossRef]

Lim, C.

Q. Bao, H. Zhang, B. Wang, Z. Ni, C. Lim, Y. Wang, D. Y. Tang, and K. P. Loh, “Broadband graphene polarizer,” Nat. Photon.5, 411–415 (2011).
[CrossRef]

Liu, M.

M. Liu, X. Yin, and X. Zhang, “Double-layer graphene optical modulator,” Nano Lett.12, 1482–1485 (2012).
[CrossRef] [PubMed]

M. Liu, X. Yin, E. Ulin-Avila, B. Geng, T. Zentgraf, L. Ju, F. Wang, and X. Zhang, “A graphene-based broadband optical modulator,” Nature (London)474, 64–67 (2011).
[CrossRef]

Locatelli, A.

Loh, K. P.

Q. Bao, H. Zhang, B. Wang, Z. Ni, C. Lim, Y. Wang, D. Y. Tang, and K. P. Loh, “Broadband graphene polarizer,” Nat. Photon.5, 411–415 (2011).
[CrossRef]

Martin, M. C.

Z. Q. Li, E. A. Henriksen, Z. Jiang, Z. Hao, M. C. Martin, P. Kim, H. L. Stormer, and D. N. Basov, “Dirac charge dynamics in graphene by infrared spectroscopy,” Nat. Phys.4, 532–535 (2008).
[CrossRef]

Mason, D. R.

S. Yu, G. X. Piao, D. R. Mason, S. In, and N. Park, “Spatiospectral separation of exceptional points in PT-symmetric optical potentials,” Phys. Rev. A86, 031802 (2012).
[CrossRef]

Midrio, M.

Moiseyev, N.

S. Klaiman, U. Gunther, and N. Moiseyev, “Visualization of branch points in PT-symmetric waveguides,” Phys. Rev. Lett.101, 080402 (2008).
[CrossRef] [PubMed]

Morandotti, R.

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-symmetry breaking in complex optical potentials,” Phys. Rev. Lett.103, 093902 (2009).
[CrossRef] [PubMed]

Moresco, M.

Morozov, S. V.

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, “Electric field effect in atomically thin carbon films,” Science306, 666–669 (2004).
[CrossRef] [PubMed]

Ni, Z.

Q. Bao, H. Zhang, B. Wang, Z. Ni, C. Lim, Y. Wang, D. Y. Tang, and K. P. Loh, “Broadband graphene polarizer,” Nat. Photon.5, 411–415 (2011).
[CrossRef]

Novoselov, K. S.

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, “Electric field effect in atomically thin carbon films,” Science306, 666–669 (2004).
[CrossRef] [PubMed]

Park, N.

S. Yu, G. X. Piao, D. R. Mason, S. In, and N. Park, “Spatiospectral separation of exceptional points in PT-symmetric optical potentials,” Phys. Rev. A86, 031802 (2012).
[CrossRef]

Pertsch, T.

T. Pertsch, T. Zentgraf, U. Peschel, A. Brauer, and F. Lederer, “Anomalous refraction and diffraction in discrete optical systems,” Phys. Rev. Lett.88, 093901 (2002).
[CrossRef] [PubMed]

Peschel, U.

T. Pertsch, T. Zentgraf, U. Peschel, A. Brauer, and F. Lederer, “Anomalous refraction and diffraction in discrete optical systems,” Phys. Rev. Lett.88, 093901 (2002).
[CrossRef] [PubMed]

Piao, G. X.

S. Yu, G. X. Piao, D. R. Mason, S. In, and N. Park, “Spatiospectral separation of exceptional points in PT-symmetric optical potentials,” Phys. Rev. A86, 031802 (2012).
[CrossRef]

Romagnoli, M.

Salamo, G. J.

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-symmetry breaking in complex optical potentials,” Phys. Rev. Lett.103, 093902 (2009).
[CrossRef] [PubMed]

Siviloglou, G. A.

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-symmetry breaking in complex optical potentials,” Phys. Rev. Lett.103, 093902 (2009).
[CrossRef] [PubMed]

Stormer, H. L.

Z. Q. Li, E. A. Henriksen, Z. Jiang, Z. Hao, M. C. Martin, P. Kim, H. L. Stormer, and D. N. Basov, “Dirac charge dynamics in graphene by infrared spectroscopy,” Nat. Phys.4, 532–535 (2008).
[CrossRef]

Sun, Z.

F. Bonaccorso, Z. Sun, T. Hasan, and A. C. Ferrari, “Graphene photonics and optoelectronics,” Nat. Photon.4, 611–622 (2010).
[CrossRef]

Tang, D. Y.

Q. Bao, H. Zhang, B. Wang, Z. Ni, C. Lim, Y. Wang, D. Y. Tang, and K. P. Loh, “Broadband graphene polarizer,” Nat. Photon.5, 411–415 (2011).
[CrossRef]

Ulin-Avila, E.

M. Liu, X. Yin, E. Ulin-Avila, B. Geng, T. Zentgraf, L. Ju, F. Wang, and X. Zhang, “A graphene-based broadband optical modulator,” Nature (London)474, 64–67 (2011).
[CrossRef]

Vakil, A.

A. Vakil and N. Engheta, “Transformation optics using graphene,” Science332, 1291–1294 (2011).
[CrossRef] [PubMed]

Volatier-Ravat, M.

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-symmetry breaking in complex optical potentials,” Phys. Rev. Lett.103, 093902 (2009).
[CrossRef] [PubMed]

Wang, B.

Q. Bao, H. Zhang, B. Wang, Z. Ni, C. Lim, Y. Wang, D. Y. Tang, and K. P. Loh, “Broadband graphene polarizer,” Nat. Photon.5, 411–415 (2011).
[CrossRef]

Wang, F.

M. Liu, X. Yin, E. Ulin-Avila, B. Geng, T. Zentgraf, L. Ju, F. Wang, and X. Zhang, “A graphene-based broadband optical modulator,” Nature (London)474, 64–67 (2011).
[CrossRef]

Wang, Y.

Q. Bao, H. Zhang, B. Wang, Z. Ni, C. Lim, Y. Wang, D. Y. Tang, and K. P. Loh, “Broadband graphene polarizer,” Nat. Photon.5, 411–415 (2011).
[CrossRef]

Yin, X.

M. Liu, X. Yin, and X. Zhang, “Double-layer graphene optical modulator,” Nano Lett.12, 1482–1485 (2012).
[CrossRef] [PubMed]

M. Liu, X. Yin, E. Ulin-Avila, B. Geng, T. Zentgraf, L. Ju, F. Wang, and X. Zhang, “A graphene-based broadband optical modulator,” Nature (London)474, 64–67 (2011).
[CrossRef]

Yu, S.

S. Yu, G. X. Piao, D. R. Mason, S. In, and N. Park, “Spatiospectral separation of exceptional points in PT-symmetric optical potentials,” Phys. Rev. A86, 031802 (2012).
[CrossRef]

Zentgraf, T.

M. Liu, X. Yin, E. Ulin-Avila, B. Geng, T. Zentgraf, L. Ju, F. Wang, and X. Zhang, “A graphene-based broadband optical modulator,” Nature (London)474, 64–67 (2011).
[CrossRef]

T. Pertsch, T. Zentgraf, U. Peschel, A. Brauer, and F. Lederer, “Anomalous refraction and diffraction in discrete optical systems,” Phys. Rev. Lett.88, 093901 (2002).
[CrossRef] [PubMed]

Zhang, H.

Q. Bao, H. Zhang, B. Wang, Z. Ni, C. Lim, Y. Wang, D. Y. Tang, and K. P. Loh, “Broadband graphene polarizer,” Nat. Photon.5, 411–415 (2011).
[CrossRef]

Zhang, X.

M. Liu, X. Yin, and X. Zhang, “Double-layer graphene optical modulator,” Nano Lett.12, 1482–1485 (2012).
[CrossRef] [PubMed]

M. Liu, X. Yin, E. Ulin-Avila, B. Geng, T. Zentgraf, L. Ju, F. Wang, and X. Zhang, “A graphene-based broadband optical modulator,” Nature (London)474, 64–67 (2011).
[CrossRef]

Zhang, Y.

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, “Electric field effect in atomically thin carbon films,” Science306, 666–669 (2004).
[CrossRef] [PubMed]

IEEE Trans. Antennas Propagat. (1)

G. W. Hanson, “Dyadic Green’s functions for an anisotropic, non-local model of biased graphene,” IEEE Trans. Antennas Propagat.56, 747–757 (2008).
[CrossRef]

Nano Lett. (1)

M. Liu, X. Yin, and X. Zhang, “Double-layer graphene optical modulator,” Nano Lett.12, 1482–1485 (2012).
[CrossRef] [PubMed]

Nat. Photon. (2)

F. Bonaccorso, Z. Sun, T. Hasan, and A. C. Ferrari, “Graphene photonics and optoelectronics,” Nat. Photon.4, 611–622 (2010).
[CrossRef]

Q. Bao, H. Zhang, B. Wang, Z. Ni, C. Lim, Y. Wang, D. Y. Tang, and K. P. Loh, “Broadband graphene polarizer,” Nat. Photon.5, 411–415 (2011).
[CrossRef]

Nat. Phys. (1)

Z. Q. Li, E. A. Henriksen, Z. Jiang, Z. Hao, M. C. Martin, P. Kim, H. L. Stormer, and D. N. Basov, “Dirac charge dynamics in graphene by infrared spectroscopy,” Nat. Phys.4, 532–535 (2008).
[CrossRef]

Nature (London) (2)

M. Liu, X. Yin, E. Ulin-Avila, B. Geng, T. Zentgraf, L. Ju, F. Wang, and X. Zhang, “A graphene-based broadband optical modulator,” Nature (London)474, 64–67 (2011).
[CrossRef]

K. Kim, J. Y. Choi, T. Kim, S. H. Cho, and H. J. Chung, “A role for graphene in silicon-based semiconductor devices,” Nature (London)479, 338–344 (2011).
[CrossRef]

Opt. Express (2)

Opt. Lett. (1)

Phys. Rev. A (1)

S. Yu, G. X. Piao, D. R. Mason, S. In, and N. Park, “Spatiospectral separation of exceptional points in PT-symmetric optical potentials,” Phys. Rev. A86, 031802 (2012).
[CrossRef]

Phys. Rev. Lett. (3)

T. Pertsch, T. Zentgraf, U. Peschel, A. Brauer, and F. Lederer, “Anomalous refraction and diffraction in discrete optical systems,” Phys. Rev. Lett.88, 093901 (2002).
[CrossRef] [PubMed]

S. Klaiman, U. Gunther, and N. Moiseyev, “Visualization of branch points in PT-symmetric waveguides,” Phys. Rev. Lett.101, 080402 (2008).
[CrossRef] [PubMed]

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-symmetry breaking in complex optical potentials,” Phys. Rev. Lett.103, 093902 (2009).
[CrossRef] [PubMed]

Science (2)

A. Vakil and N. Engheta, “Transformation optics using graphene,” Science332, 1291–1294 (2011).
[CrossRef] [PubMed]

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, “Electric field effect in atomically thin carbon films,” Science306, 666–669 (2004).
[CrossRef] [PubMed]

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Figures (5)

Fig. 1
Fig. 1

(a) Schematic view of the waveguide structure, with a detail of the central region with graphene layers. (b) Losses of the single waveguide (in dB/μm) when graphene is in OFF state (null voltage), and x-component of the electric field of the TE-like mode (inset).

Fig. 2
Fig. 2

Schematic view of the 300-nm-gap coupler (left), with the electric field of the low-(center) and high-loss mode (right) at 1530 nm. Graphene layers are in ON-OFF states.

Fig. 3
Fig. 3

(a) Losses of low- (red line) and high-loss mode (blue line) from mode solver (solid line) and CMT (dashed-dotted line). (b) Normalized attenuation constant of low- (red line with squares) and high-loss mode (blue line with circles) vs. normalized attenuation constant of the single waveguide at λ = 1530 nm. The vertical thin line indicates α = αmax.

Fig. 4
Fig. 4

Field amplitude in (a) first and (b) second waveguide of the coupler. Graphene layers are in ON-ON (black line), and ON-OFF (red line) states.

Fig. 5
Fig. 5

Discrete diffraction along the array. (a) All the graphene layers are in ON state. (b) Only graphene layers inside the central waveguide are in ON state.

Equations (1)

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d d z [ A 1 A 2 ] = i [ β + i α 1 C C * β + i α 2 ] [ A 1 A 2 ] ,

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