Abstract

We theoretically investigate the Goos-Hänchen (GH) shifts of optical beam in a defective photonic crystal composed of dielectric multilayers and graphene. The system is non-Hermitian and possesses exceptional points (EPs) as the scattering matrix becomes defective at the zero points of reflection. The reflective wave at EPs experiences an abrupt phase change and there the eigenvalues of scattering matrix coalesce. The GH shifts are extremely large near EPs in parametric space composed of dielectric refractive index and incident angle. The positive and negative maxima of GH shifts could be as high as 103 times of the incident wavelength. The direction of GH shifts switches at EPs and the EPs position can be readily controlled by the chemical potential of graphene. Moreover, the GH shifts should remarkably change as the incident waves impinge on the structure from opposite directions. The study of GH shifts in the graphene incorporated multilayers may find great applications in highly sensitive sensors.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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    [Crossref] [PubMed]
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2018 (1)

Z. Hong, Q. Zhang, S. A. Rezvani, P. Lan, and P. Lu, “Tunable few-cycle pulses from a dual-chirped optical parametric amplifier pumped by broadband laser,” Opt. Laser Technol. 98, 169–177 (2018).
[Crossref]

2017 (7)

P. Ma and L. Gao, “Large and tunable lateral shifts in one-dimensional PT-symmetric layered structures,” Opt. Express 25(9), 9676–9688 (2017).
[Crossref] [PubMed]

M. P. Araujo, S. De Leo, and G. G. Maia, “Optimizing Weak Measurements to Detect Angular Deviations,” Ann. Phys. (Berlin) 529(9), 1600357 (2017).
[Crossref]

D. Zhao, Z. Q. Wang, H. Long, K. Wang, B. Wang, and P. X. Lu, “Optical bistability in defective photonic multilayers doped by graphene,” Opt. Quantum Electron. 49(4), 163 (2017).
[Crossref]

S. Ke, B. Wang, H. Long, K. Wang, and P. Lu, “Topological mode switching in a graphene doublet with exceptional points,” Opt. Quantum Electron. 49(6), 224 (2017).
[Crossref]

H. Huang, S. Ke, B. Wang, H. Long, K. Wang, and P. Lu, “Numerical study on plasmonic absorption enhancement by a rippled graphene sheet,” J. Lightwave Technol. 35(2), 320–324 (2017).
[Crossref]

Q. Liu, B. Wang, S. Ke, H. Long, K. Wang, and P. Lu, “Exceptional points in Fano-resonant graphene metamaterials,” Opt. Express 25(7), 7203–7212 (2017).
[Crossref] [PubMed]

S. L. Wang, B. Wang, C. Z. Qin, K. Wang, H. Long, and P. X. Lu, “Rabi oscillations of optical modes in a waveguide with dynamic modulation,” Opt. Quantum Electron. 49(11), 389 (2017).
[Crossref]

2016 (6)

2015 (2)

2014 (4)

L. Feng, X. Zhu, S. Yang, H. Zhu, P. Zhang, X. Yin, Y. Wang, and X. Zhang, “Demonstration of a large-scale optical exceptional point structure,” Opt. Express 22(2), 1760–1767 (2014).
[Crossref] [PubMed]

M. P. Araújo, S. A. Carvalho, and S. De Leo, “The asymmetric Goos–Hänchen effect,” J. Opt. 16(1), 015702 (2014).
[Crossref]

M. P. Araujo, S. A. Carvalho, and S. De Leo, “Maximal breaking of symmetry at critical angle and closed-form expression for angular deviations of the Snell law,” Phys. Rev. A 90(3), 033844 (2014).
[Crossref]

G. Jayaswal, G. Mistura, and M. Merano, “Observing angular deviations in light-beam reflection via weak measurements,” Opt. Lett. 39(21), 6257–6260 (2014).
[Crossref] [PubMed]

2013 (3)

K. Y. Bliokh and A. Aiello, “Goos-Hänchen and Imbert-Fedorov beam shifts: an overview,” J. Opt. 15(1), 014001 (2013).
[Crossref]

M. Abbas and S. Qamar, “Amplitude control of the goos-hänchen shift via a kerr nonlinearity,” Laser Phys. Lett. 11(1), 5201 (2013).

L. Feng, Y. L. Xu, W. S. Fegadolli, M. H. Lu, J. E. B. Oliveira, V. R. Almeida, Y. F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nat. Mater. 12(2), 108–113 (2013).
[Crossref] [PubMed]

2012 (4)

V. V. Konotop, V. S. Shchesnovich, and D. A. Zezyulin, “Giant amplification of modes in parity-time symmetric waveguides,” Phys. Lett. A 376(42-43), 2750–2753 (2012).
[Crossref]

C. Prajapati and D. Ranganathan, “Goos-Hanchen and Imbert-Fedorov shifts for Hermite-Gauss beams,” J. Opt. Soc. Am. A 29(7), 1377–1382 (2012).
[Crossref] [PubMed]

A. Aiello, “Goos-Hänchen and Imbert-Federov shifts: a novel perspective,” New J. Phys. 14(1), 013058 (2012).
[Crossref]

S. Thongrattanasiri, I. Silveiro, and G. D. A. F. Javier, “Plasmons in electrostatically doped graphene,” Appl. Phys. Lett. 100(20), 201105 (2012).
[Crossref]

2011 (4)

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional invisibility induced by PT-symmetric periodic structures,” Phys. Rev. Lett. 106(21), 213901 (2011).
[Crossref] [PubMed]

Y. D. Chong, L. Ge, and A. D. Stone, “PT-symmetry breaking and laser-absorber modes in optical scattering systems,” Phys. Rev. Lett. 106(9), 093902 (2011).
[Crossref] [PubMed]

J. C. Martinez and M. B. A. Jalil, “Theory of giant Faraday rotation and Goos-Hänchen shift in graphene,” Europhys. Lett. 96(2), 27008 (2011).
[Crossref]

S. Longhi, G. Della Valle, and K. Staliunas, “Goos-Hänchen shift in complex crystals,” Phys. Rev. A 84(4), 042119 (2011).
[Crossref]

2010 (1)

C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity–time symmetry in optics,” Nat. Phys. 6(3), 192–195 (2010).
[Crossref]

2009 (4)

A. Aiello, M. Merano, and J. P. Woerdman, “Brewster Cross Polarization,” Opt. Lett. 34(8), 1207–1209 (2009).
[Crossref] [PubMed]

M. Merano, A. Aiello, M. P. van Exter, and J. P. Woerdman, “Observing angular deviations in the specular reflection of a light beam,” Nat. Photonics 3(6), 337–340 (2009).
[Crossref]

A. Aiello, M. Merano, and J. P. Woerdman, “Duality between spatial and angular shift in optical reflection,” Phys. Rev. A 80(6), 061801 (2009).
[Crossref]

B. Zhao and L. Gao, “Temperature-dependent Goos-Hänchen shift on the interface of metal/dielectric composites,” Opt. Express 17(24), 21433–21441 (2009).
[Crossref] [PubMed]

2008 (1)

2007 (1)

2006 (2)

2005 (1)

2002 (1)

1998 (2)

B. M. Jost, A. A. R. Al-Rashed, and B. E. Saleh, “Observation of the Goos-Hänchen effect in a phase-conjugate mirror,” Phys. Rev. Lett. 81(11), 2233–2235 (1998).
[Crossref]

C. M. Bender and S. Boettcher, “Real spectra in non-Hermitian Hamiltonians having PT symmetry,” Phys. Rev. Lett. 80(24), 5243–5246 (1998).
[Crossref]

1947 (1)

F. Goos and H. Hänchen, “A new and fundamental experiment on total reflection,” Ann. Phys. (Leipzig) 1(7-8), 333–346 (1947).
[Crossref]

’t Hooft, G. W.

Abbas, M.

M. Abbas and S. Qamar, “Amplitude control of the goos-hänchen shift via a kerr nonlinearity,” Laser Phys. Lett. 11(1), 5201 (2013).

Aiello, A.

K. Y. Bliokh and A. Aiello, “Goos-Hänchen and Imbert-Fedorov beam shifts: an overview,” J. Opt. 15(1), 014001 (2013).
[Crossref]

A. Aiello, “Goos-Hänchen and Imbert-Federov shifts: a novel perspective,” New J. Phys. 14(1), 013058 (2012).
[Crossref]

M. Merano, A. Aiello, M. P. van Exter, and J. P. Woerdman, “Observing angular deviations in the specular reflection of a light beam,” Nat. Photonics 3(6), 337–340 (2009).
[Crossref]

A. Aiello, M. Merano, and J. P. Woerdman, “Duality between spatial and angular shift in optical reflection,” Phys. Rev. A 80(6), 061801 (2009).
[Crossref]

A. Aiello, M. Merano, and J. P. Woerdman, “Brewster Cross Polarization,” Opt. Lett. 34(8), 1207–1209 (2009).
[Crossref] [PubMed]

A. Aiello and J. P. Woerdman, “Role of beam propagation in Goos-Hänchen and Imbert-Fedorov shifts,” Opt. Lett. 33(13), 1437–1439 (2008).
[Crossref] [PubMed]

M. Merano, A. Aiello, G. W. ’t Hooft, M. P. van Exter, E. R. Eliel, and J. P. Woerdman, “Observation of Goos-Hänchen shifts in metallic reflection,” Opt. Express 15(24), 15928–15934 (2007).
[Crossref] [PubMed]

Almeida, V. R.

L. Feng, Y. L. Xu, W. S. Fegadolli, M. H. Lu, J. E. B. Oliveira, V. R. Almeida, Y. F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nat. Mater. 12(2), 108–113 (2013).
[Crossref] [PubMed]

Al-Rashed, A. A. R.

B. M. Jost, A. A. R. Al-Rashed, and B. E. Saleh, “Observation of the Goos-Hänchen effect in a phase-conjugate mirror,” Phys. Rev. Lett. 81(11), 2233–2235 (1998).
[Crossref]

Araujo, M. P.

M. P. Araujo, S. De Leo, and G. G. Maia, “Optimizing Weak Measurements to Detect Angular Deviations,” Ann. Phys. (Berlin) 529(9), 1600357 (2017).
[Crossref]

M. P. Araujo, S. De Leo, and G. G. Maia, “Closed–form expression for the Goos-Hänchen lateral displacement,” Phys. Rev. A 93(2), 023801 (2016).
[Crossref]

M. P. Araujo, S. A. Carvalho, and S. De Leo, “Maximal breaking of symmetry at critical angle and closed-form expression for angular deviations of the Snell law,” Phys. Rev. A 90(3), 033844 (2014).
[Crossref]

Araújo, M. P.

M. P. Araújo, S. A. Carvalho, and S. De Leo, “The asymmetric Goos–Hänchen effect,” J. Opt. 16(1), 015702 (2014).
[Crossref]

Bender, C. M.

C. M. Bender and S. Boettcher, “Real spectra in non-Hermitian Hamiltonians having PT symmetry,” Phys. Rev. Lett. 80(24), 5243–5246 (1998).
[Crossref]

Bliokh, K. Y.

K. Y. Bliokh and A. Aiello, “Goos-Hänchen and Imbert-Fedorov beam shifts: an overview,” J. Opt. 15(1), 014001 (2013).
[Crossref]

Boettcher, S.

C. M. Bender and S. Boettcher, “Real spectra in non-Hermitian Hamiltonians having PT symmetry,” Phys. Rev. Lett. 80(24), 5243–5246 (1998).
[Crossref]

Cao, H.

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional invisibility induced by PT-symmetric periodic structures,” Phys. Rev. Lett. 106(21), 213901 (2011).
[Crossref] [PubMed]

Carvalho, S. A.

O. J. S. Santana, S. A. Carvalho, S. De Leo, and L. E. E. de Araujo, “Weak measurement of the composite Goos-Hänchen shift in the critical region,” Opt. Lett. 41(16), 3884–3887 (2016).
[Crossref] [PubMed]

M. P. Araujo, S. A. Carvalho, and S. De Leo, “Maximal breaking of symmetry at critical angle and closed-form expression for angular deviations of the Snell law,” Phys. Rev. A 90(3), 033844 (2014).
[Crossref]

M. P. Araújo, S. A. Carvalho, and S. De Leo, “The asymmetric Goos–Hänchen effect,” J. Opt. 16(1), 015702 (2014).
[Crossref]

Chan, S. W.

Chen, H.

Chen, Y. F.

L. Feng, Y. L. Xu, W. S. Fegadolli, M. H. Lu, J. E. B. Oliveira, V. R. Almeida, Y. F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nat. Mater. 12(2), 108–113 (2013).
[Crossref] [PubMed]

Chong, Y. D.

Y. D. Chong, L. Ge, and A. D. Stone, “PT-symmetry breaking and laser-absorber modes in optical scattering systems,” Phys. Rev. Lett. 106(9), 093902 (2011).
[Crossref] [PubMed]

Christodoulides, D. N.

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional invisibility induced by PT-symmetric periodic structures,” Phys. Rev. Lett. 106(21), 213901 (2011).
[Crossref] [PubMed]

C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity–time symmetry in optics,” Nat. Phys. 6(3), 192–195 (2010).
[Crossref]

Chuang, Y. L.

Y. L. Chuang and R. K. Lee, “Giant Goos-Hänchen shift using PT symmetry,” Phys. Rev. A 92(1), 013815 (2015).
[Crossref]

de Araujo, L. E. E.

De Leo, S.

M. P. Araujo, S. De Leo, and G. G. Maia, “Optimizing Weak Measurements to Detect Angular Deviations,” Ann. Phys. (Berlin) 529(9), 1600357 (2017).
[Crossref]

M. P. Araujo, S. De Leo, and G. G. Maia, “Closed–form expression for the Goos-Hänchen lateral displacement,” Phys. Rev. A 93(2), 023801 (2016).
[Crossref]

O. J. S. Santana, S. A. Carvalho, S. De Leo, and L. E. E. de Araujo, “Weak measurement of the composite Goos-Hänchen shift in the critical region,” Opt. Lett. 41(16), 3884–3887 (2016).
[Crossref] [PubMed]

M. P. Araujo, S. A. Carvalho, and S. De Leo, “Maximal breaking of symmetry at critical angle and closed-form expression for angular deviations of the Snell law,” Phys. Rev. A 90(3), 033844 (2014).
[Crossref]

M. P. Araújo, S. A. Carvalho, and S. De Leo, “The asymmetric Goos–Hänchen effect,” J. Opt. 16(1), 015702 (2014).
[Crossref]

Della Valle, G.

S. Longhi, G. Della Valle, and K. Staliunas, “Goos-Hänchen shift in complex crystals,” Phys. Rev. A 84(4), 042119 (2011).
[Crossref]

Eichelkraut, T.

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional invisibility induced by PT-symmetric periodic structures,” Phys. Rev. Lett. 106(21), 213901 (2011).
[Crossref] [PubMed]

El-Ganainy, R.

C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity–time symmetry in optics,” Nat. Phys. 6(3), 192–195 (2010).
[Crossref]

Eliel, E. R.

Fegadolli, W. S.

L. Feng, Y. L. Xu, W. S. Fegadolli, M. H. Lu, J. E. B. Oliveira, V. R. Almeida, Y. F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nat. Mater. 12(2), 108–113 (2013).
[Crossref] [PubMed]

Feng, L.

L. Feng, X. Zhu, S. Yang, H. Zhu, P. Zhang, X. Yin, Y. Wang, and X. Zhang, “Demonstration of a large-scale optical exceptional point structure,” Opt. Express 22(2), 1760–1767 (2014).
[Crossref] [PubMed]

L. Feng, Y. L. Xu, W. S. Fegadolli, M. H. Lu, J. E. B. Oliveira, V. R. Almeida, Y. F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nat. Mater. 12(2), 108–113 (2013).
[Crossref] [PubMed]

Gao, L.

Ge, L.

Y. D. Chong, L. Ge, and A. D. Stone, “PT-symmetry breaking and laser-absorber modes in optical scattering systems,” Phys. Rev. Lett. 106(9), 093902 (2011).
[Crossref] [PubMed]

Goos, F.

F. Goos and H. Hänchen, “A new and fundamental experiment on total reflection,” Ann. Phys. (Leipzig) 1(7-8), 333–346 (1947).
[Crossref]

Hänchen, H.

F. Goos and H. Hänchen, “A new and fundamental experiment on total reflection,” Ann. Phys. (Leipzig) 1(7-8), 333–346 (1947).
[Crossref]

He, J.

He, S.

Hong, Z.

Z. Hong, Q. Zhang, S. A. Rezvani, P. Lan, and P. Lu, “Tunable few-cycle pulses from a dual-chirped optical parametric amplifier pumped by broadband laser,” Opt. Laser Technol. 98, 169–177 (2018).
[Crossref]

Huang, H.

Jalil, M. B. A.

J. C. Martinez and M. B. A. Jalil, “Theory of giant Faraday rotation and Goos-Hänchen shift in graphene,” Europhys. Lett. 96(2), 27008 (2011).
[Crossref]

Javier, G. D. A. F.

S. Thongrattanasiri, I. Silveiro, and G. D. A. F. Javier, “Plasmons in electrostatically doped graphene,” Appl. Phys. Lett. 100(20), 201105 (2012).
[Crossref]

Jayaswal, G.

Jost, B. M.

B. M. Jost, A. A. R. Al-Rashed, and B. E. Saleh, “Observation of the Goos-Hänchen effect in a phase-conjugate mirror,” Phys. Rev. Lett. 81(11), 2233–2235 (1998).
[Crossref]

Ke, S.

Kip, D.

C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity–time symmetry in optics,” Nat. Phys. 6(3), 192–195 (2010).
[Crossref]

Konotop, V. V.

V. V. Konotop, V. S. Shchesnovich, and D. A. Zezyulin, “Giant amplification of modes in parity-time symmetric waveguides,” Phys. Lett. A 376(42-43), 2750–2753 (2012).
[Crossref]

Kottos, T.

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional invisibility induced by PT-symmetric periodic structures,” Phys. Rev. Lett. 106(21), 213901 (2011).
[Crossref] [PubMed]

Lai, H. M.

Lan, P.

Z. Hong, Q. Zhang, S. A. Rezvani, P. Lan, and P. Lu, “Tunable few-cycle pulses from a dual-chirped optical parametric amplifier pumped by broadband laser,” Opt. Laser Technol. 98, 169–177 (2018).
[Crossref]

Lee, R. K.

Y. L. Chuang and R. K. Lee, “Giant Goos-Hänchen shift using PT symmetry,” Phys. Rev. A 92(1), 013815 (2015).
[Crossref]

Lin, Z.

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional invisibility induced by PT-symmetric periodic structures,” Phys. Rev. Lett. 106(21), 213901 (2011).
[Crossref] [PubMed]

Liu, Q.

Long, H.

Q. Liu, B. Wang, S. Ke, H. Long, K. Wang, and P. Lu, “Exceptional points in Fano-resonant graphene metamaterials,” Opt. Express 25(7), 7203–7212 (2017).
[Crossref] [PubMed]

H. Huang, S. Ke, B. Wang, H. Long, K. Wang, and P. Lu, “Numerical study on plasmonic absorption enhancement by a rippled graphene sheet,” J. Lightwave Technol. 35(2), 320–324 (2017).
[Crossref]

S. Ke, B. Wang, H. Long, K. Wang, and P. Lu, “Topological mode switching in a graphene doublet with exceptional points,” Opt. Quantum Electron. 49(6), 224 (2017).
[Crossref]

D. Zhao, Z. Q. Wang, H. Long, K. Wang, B. Wang, and P. X. Lu, “Optical bistability in defective photonic multilayers doped by graphene,” Opt. Quantum Electron. 49(4), 163 (2017).
[Crossref]

S. L. Wang, B. Wang, C. Z. Qin, K. Wang, H. Long, and P. X. Lu, “Rabi oscillations of optical modes in a waveguide with dynamic modulation,” Opt. Quantum Electron. 49(11), 389 (2017).
[Crossref]

S. Ke, B. Wang, C. Qin, H. Long, K. Wang, and P. Lu, “Exceptional points and asymmetric mode switching in plasmonic waveguides,” J. Lightwave Technol. 34(22), 5258–5262 (2016).
[Crossref]

C. Qin, B. Wang, H. Long, K. Wang, and P. Lu, “Nonreciprocal phase shift and mode modulation in dynamic graphene waveguides,” J. Lightwave Technol. 34(16), 3877–3883 (2016).

Longhi, S.

S. Longhi, G. Della Valle, and K. Staliunas, “Goos-Hänchen shift in complex crystals,” Phys. Rev. A 84(4), 042119 (2011).
[Crossref]

Lu, M. H.

L. Feng, Y. L. Xu, W. S. Fegadolli, M. H. Lu, J. E. B. Oliveira, V. R. Almeida, Y. F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nat. Mater. 12(2), 108–113 (2013).
[Crossref] [PubMed]

Lu, P.

Lu, P. X.

S. L. Wang, B. Wang, C. Z. Qin, K. Wang, H. Long, and P. X. Lu, “Rabi oscillations of optical modes in a waveguide with dynamic modulation,” Opt. Quantum Electron. 49(11), 389 (2017).
[Crossref]

D. Zhao, Z. Q. Wang, H. Long, K. Wang, B. Wang, and P. X. Lu, “Optical bistability in defective photonic multilayers doped by graphene,” Opt. Quantum Electron. 49(4), 163 (2017).
[Crossref]

Ma, P.

Maia, G. G.

M. P. Araujo, S. De Leo, and G. G. Maia, “Optimizing Weak Measurements to Detect Angular Deviations,” Ann. Phys. (Berlin) 529(9), 1600357 (2017).
[Crossref]

M. P. Araujo, S. De Leo, and G. G. Maia, “Closed–form expression for the Goos-Hänchen lateral displacement,” Phys. Rev. A 93(2), 023801 (2016).
[Crossref]

Makris, K. G.

C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity–time symmetry in optics,” Nat. Phys. 6(3), 192–195 (2010).
[Crossref]

Martinez, J. C.

J. C. Martinez and M. B. A. Jalil, “Theory of giant Faraday rotation and Goos-Hänchen shift in graphene,” Europhys. Lett. 96(2), 27008 (2011).
[Crossref]

Merano, M.

Mistura, G.

Oliveira, J. E. B.

L. Feng, Y. L. Xu, W. S. Fegadolli, M. H. Lu, J. E. B. Oliveira, V. R. Almeida, Y. F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nat. Mater. 12(2), 108–113 (2013).
[Crossref] [PubMed]

Prajapati, C.

Qamar, S.

M. Abbas and S. Qamar, “Amplitude control of the goos-hänchen shift via a kerr nonlinearity,” Laser Phys. Lett. 11(1), 5201 (2013).

Qin, C.

Qin, C. Z.

S. L. Wang, B. Wang, C. Z. Qin, K. Wang, H. Long, and P. X. Lu, “Rabi oscillations of optical modes in a waveguide with dynamic modulation,” Opt. Quantum Electron. 49(11), 389 (2017).
[Crossref]

Ramezani, H.

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional invisibility induced by PT-symmetric periodic structures,” Phys. Rev. Lett. 106(21), 213901 (2011).
[Crossref] [PubMed]

Ranganathan, D.

Rezvani, S. A.

Z. Hong, Q. Zhang, S. A. Rezvani, P. Lan, and P. Lu, “Tunable few-cycle pulses from a dual-chirped optical parametric amplifier pumped by broadband laser,” Opt. Laser Technol. 98, 169–177 (2018).
[Crossref]

Rüter, C. E.

C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity–time symmetry in optics,” Nat. Phys. 6(3), 192–195 (2010).
[Crossref]

Saleh, B. E.

B. M. Jost, A. A. R. Al-Rashed, and B. E. Saleh, “Observation of the Goos-Hänchen effect in a phase-conjugate mirror,” Phys. Rev. Lett. 81(11), 2233–2235 (1998).
[Crossref]

Santana, O. J. S.

Scherer, A.

L. Feng, Y. L. Xu, W. S. Fegadolli, M. H. Lu, J. E. B. Oliveira, V. R. Almeida, Y. F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nat. Mater. 12(2), 108–113 (2013).
[Crossref] [PubMed]

Segev, M.

C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity–time symmetry in optics,” Nat. Phys. 6(3), 192–195 (2010).
[Crossref]

Shchesnovich, V. S.

V. V. Konotop, V. S. Shchesnovich, and D. A. Zezyulin, “Giant amplification of modes in parity-time symmetric waveguides,” Phys. Lett. A 376(42-43), 2750–2753 (2012).
[Crossref]

Silveiro, I.

S. Thongrattanasiri, I. Silveiro, and G. D. A. F. Javier, “Plasmons in electrostatically doped graphene,” Appl. Phys. Lett. 100(20), 201105 (2012).
[Crossref]

Staliunas, K.

S. Longhi, G. Della Valle, and K. Staliunas, “Goos-Hänchen shift in complex crystals,” Phys. Rev. A 84(4), 042119 (2011).
[Crossref]

Stone, A. D.

Y. D. Chong, L. Ge, and A. D. Stone, “PT-symmetry breaking and laser-absorber modes in optical scattering systems,” Phys. Rev. Lett. 106(9), 093902 (2011).
[Crossref] [PubMed]

Thongrattanasiri, S.

S. Thongrattanasiri, I. Silveiro, and G. D. A. F. Javier, “Plasmons in electrostatically doped graphene,” Appl. Phys. Lett. 100(20), 201105 (2012).
[Crossref]

van Exter, M. P.

M. Merano, A. Aiello, M. P. van Exter, and J. P. Woerdman, “Observing angular deviations in the specular reflection of a light beam,” Nat. Photonics 3(6), 337–340 (2009).
[Crossref]

M. Merano, A. Aiello, G. W. ’t Hooft, M. P. van Exter, E. R. Eliel, and J. P. Woerdman, “Observation of Goos-Hänchen shifts in metallic reflection,” Opt. Express 15(24), 15928–15934 (2007).
[Crossref] [PubMed]

Wang, B.

S. L. Wang, B. Wang, C. Z. Qin, K. Wang, H. Long, and P. X. Lu, “Rabi oscillations of optical modes in a waveguide with dynamic modulation,” Opt. Quantum Electron. 49(11), 389 (2017).
[Crossref]

H. Huang, S. Ke, B. Wang, H. Long, K. Wang, and P. Lu, “Numerical study on plasmonic absorption enhancement by a rippled graphene sheet,” J. Lightwave Technol. 35(2), 320–324 (2017).
[Crossref]

S. Ke, B. Wang, H. Long, K. Wang, and P. Lu, “Topological mode switching in a graphene doublet with exceptional points,” Opt. Quantum Electron. 49(6), 224 (2017).
[Crossref]

D. Zhao, Z. Q. Wang, H. Long, K. Wang, B. Wang, and P. X. Lu, “Optical bistability in defective photonic multilayers doped by graphene,” Opt. Quantum Electron. 49(4), 163 (2017).
[Crossref]

Q. Liu, B. Wang, S. Ke, H. Long, K. Wang, and P. Lu, “Exceptional points in Fano-resonant graphene metamaterials,” Opt. Express 25(7), 7203–7212 (2017).
[Crossref] [PubMed]

S. Ke, B. Wang, C. Qin, H. Long, K. Wang, and P. Lu, “Exceptional points and asymmetric mode switching in plasmonic waveguides,” J. Lightwave Technol. 34(22), 5258–5262 (2016).
[Crossref]

C. Qin, B. Wang, H. Long, K. Wang, and P. Lu, “Nonreciprocal phase shift and mode modulation in dynamic graphene waveguides,” J. Lightwave Technol. 34(16), 3877–3883 (2016).

Wang, K.

S. L. Wang, B. Wang, C. Z. Qin, K. Wang, H. Long, and P. X. Lu, “Rabi oscillations of optical modes in a waveguide with dynamic modulation,” Opt. Quantum Electron. 49(11), 389 (2017).
[Crossref]

S. Ke, B. Wang, H. Long, K. Wang, and P. Lu, “Topological mode switching in a graphene doublet with exceptional points,” Opt. Quantum Electron. 49(6), 224 (2017).
[Crossref]

D. Zhao, Z. Q. Wang, H. Long, K. Wang, B. Wang, and P. X. Lu, “Optical bistability in defective photonic multilayers doped by graphene,” Opt. Quantum Electron. 49(4), 163 (2017).
[Crossref]

Q. Liu, B. Wang, S. Ke, H. Long, K. Wang, and P. Lu, “Exceptional points in Fano-resonant graphene metamaterials,” Opt. Express 25(7), 7203–7212 (2017).
[Crossref] [PubMed]

H. Huang, S. Ke, B. Wang, H. Long, K. Wang, and P. Lu, “Numerical study on plasmonic absorption enhancement by a rippled graphene sheet,” J. Lightwave Technol. 35(2), 320–324 (2017).
[Crossref]

C. Qin, B. Wang, H. Long, K. Wang, and P. Lu, “Nonreciprocal phase shift and mode modulation in dynamic graphene waveguides,” J. Lightwave Technol. 34(16), 3877–3883 (2016).

S. Ke, B. Wang, C. Qin, H. Long, K. Wang, and P. Lu, “Exceptional points and asymmetric mode switching in plasmonic waveguides,” J. Lightwave Technol. 34(22), 5258–5262 (2016).
[Crossref]

Wang, L. G.

Wang, S. L.

S. L. Wang, B. Wang, C. Z. Qin, K. Wang, H. Long, and P. X. Lu, “Rabi oscillations of optical modes in a waveguide with dynamic modulation,” Opt. Quantum Electron. 49(11), 389 (2017).
[Crossref]

Wang, Y.

Wang, Z. Q.

D. Zhao, Z. Q. Wang, H. Long, K. Wang, B. Wang, and P. X. Lu, “Optical bistability in defective photonic multilayers doped by graphene,” Opt. Quantum Electron. 49(4), 163 (2017).
[Crossref]

Woerdman, J. P.

Xu, Y. L.

L. Feng, Y. L. Xu, W. S. Fegadolli, M. H. Lu, J. E. B. Oliveira, V. R. Almeida, Y. F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nat. Mater. 12(2), 108–113 (2013).
[Crossref] [PubMed]

Yang, S.

Yi, J.

Yin, X.

Zezyulin, D. A.

V. V. Konotop, V. S. Shchesnovich, and D. A. Zezyulin, “Giant amplification of modes in parity-time symmetric waveguides,” Phys. Lett. A 376(42-43), 2750–2753 (2012).
[Crossref]

Zhang, P.

Zhang, Q.

Z. Hong, Q. Zhang, S. A. Rezvani, P. Lan, and P. Lu, “Tunable few-cycle pulses from a dual-chirped optical parametric amplifier pumped by broadband laser,” Opt. Laser Technol. 98, 169–177 (2018).
[Crossref]

Zhang, X.

Zhao, B.

Zhao, D.

D. Zhao, Z. Q. Wang, H. Long, K. Wang, B. Wang, and P. X. Lu, “Optical bistability in defective photonic multilayers doped by graphene,” Opt. Quantum Electron. 49(4), 163 (2017).
[Crossref]

Zhu, H.

Zhu, S. Y.

Zhu, X.

Zhu, X. F.

Ann. Phys. (Berlin) (1)

M. P. Araujo, S. De Leo, and G. G. Maia, “Optimizing Weak Measurements to Detect Angular Deviations,” Ann. Phys. (Berlin) 529(9), 1600357 (2017).
[Crossref]

Ann. Phys. (Leipzig) (1)

F. Goos and H. Hänchen, “A new and fundamental experiment on total reflection,” Ann. Phys. (Leipzig) 1(7-8), 333–346 (1947).
[Crossref]

Appl. Phys. Lett. (1)

S. Thongrattanasiri, I. Silveiro, and G. D. A. F. Javier, “Plasmons in electrostatically doped graphene,” Appl. Phys. Lett. 100(20), 201105 (2012).
[Crossref]

Europhys. Lett. (1)

J. C. Martinez and M. B. A. Jalil, “Theory of giant Faraday rotation and Goos-Hänchen shift in graphene,” Europhys. Lett. 96(2), 27008 (2011).
[Crossref]

J. Lightwave Technol. (3)

J. Opt. (2)

M. P. Araújo, S. A. Carvalho, and S. De Leo, “The asymmetric Goos–Hänchen effect,” J. Opt. 16(1), 015702 (2014).
[Crossref]

K. Y. Bliokh and A. Aiello, “Goos-Hänchen and Imbert-Fedorov beam shifts: an overview,” J. Opt. 15(1), 014001 (2013).
[Crossref]

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

Laser Phys. Lett. (1)

M. Abbas and S. Qamar, “Amplitude control of the goos-hänchen shift via a kerr nonlinearity,” Laser Phys. Lett. 11(1), 5201 (2013).

Nat. Mater. (1)

L. Feng, Y. L. Xu, W. S. Fegadolli, M. H. Lu, J. E. B. Oliveira, V. R. Almeida, Y. F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nat. Mater. 12(2), 108–113 (2013).
[Crossref] [PubMed]

Nat. Photonics (1)

M. Merano, A. Aiello, M. P. van Exter, and J. P. Woerdman, “Observing angular deviations in the specular reflection of a light beam,” Nat. Photonics 3(6), 337–340 (2009).
[Crossref]

Nat. Phys. (1)

C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity–time symmetry in optics,” Nat. Phys. 6(3), 192–195 (2010).
[Crossref]

New J. Phys. (1)

A. Aiello, “Goos-Hänchen and Imbert-Federov shifts: a novel perspective,” New J. Phys. 14(1), 013058 (2012).
[Crossref]

Opt. Express (7)

Opt. Laser Technol. (1)

Z. Hong, Q. Zhang, S. A. Rezvani, P. Lan, and P. Lu, “Tunable few-cycle pulses from a dual-chirped optical parametric amplifier pumped by broadband laser,” Opt. Laser Technol. 98, 169–177 (2018).
[Crossref]

Opt. Lett. (8)

Opt. Quantum Electron. (3)

D. Zhao, Z. Q. Wang, H. Long, K. Wang, B. Wang, and P. X. Lu, “Optical bistability in defective photonic multilayers doped by graphene,” Opt. Quantum Electron. 49(4), 163 (2017).
[Crossref]

S. Ke, B. Wang, H. Long, K. Wang, and P. Lu, “Topological mode switching in a graphene doublet with exceptional points,” Opt. Quantum Electron. 49(6), 224 (2017).
[Crossref]

S. L. Wang, B. Wang, C. Z. Qin, K. Wang, H. Long, and P. X. Lu, “Rabi oscillations of optical modes in a waveguide with dynamic modulation,” Opt. Quantum Electron. 49(11), 389 (2017).
[Crossref]

Phys. Lett. A (1)

V. V. Konotop, V. S. Shchesnovich, and D. A. Zezyulin, “Giant amplification of modes in parity-time symmetric waveguides,” Phys. Lett. A 376(42-43), 2750–2753 (2012).
[Crossref]

Phys. Rev. A (6)

M. Merano, “Fresnel coefficients of a two-dimensional atomic crystal,” Phys. Rev. A 93(1), 013832 (2016).
[Crossref]

Y. L. Chuang and R. K. Lee, “Giant Goos-Hänchen shift using PT symmetry,” Phys. Rev. A 92(1), 013815 (2015).
[Crossref]

S. Longhi, G. Della Valle, and K. Staliunas, “Goos-Hänchen shift in complex crystals,” Phys. Rev. A 84(4), 042119 (2011).
[Crossref]

M. P. Araujo, S. De Leo, and G. G. Maia, “Closed–form expression for the Goos-Hänchen lateral displacement,” Phys. Rev. A 93(2), 023801 (2016).
[Crossref]

A. Aiello, M. Merano, and J. P. Woerdman, “Duality between spatial and angular shift in optical reflection,” Phys. Rev. A 80(6), 061801 (2009).
[Crossref]

M. P. Araujo, S. A. Carvalho, and S. De Leo, “Maximal breaking of symmetry at critical angle and closed-form expression for angular deviations of the Snell law,” Phys. Rev. A 90(3), 033844 (2014).
[Crossref]

Phys. Rev. Lett. (4)

B. M. Jost, A. A. R. Al-Rashed, and B. E. Saleh, “Observation of the Goos-Hänchen effect in a phase-conjugate mirror,” Phys. Rev. Lett. 81(11), 2233–2235 (1998).
[Crossref]

C. M. Bender and S. Boettcher, “Real spectra in non-Hermitian Hamiltonians having PT symmetry,” Phys. Rev. Lett. 80(24), 5243–5246 (1998).
[Crossref]

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional invisibility induced by PT-symmetric periodic structures,” Phys. Rev. Lett. 106(21), 213901 (2011).
[Crossref] [PubMed]

Y. D. Chong, L. Ge, and A. D. Stone, “PT-symmetry breaking and laser-absorber modes in optical scattering systems,” Phys. Rev. Lett. 106(9), 093902 (2011).
[Crossref] [PubMed]

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Figures (10)

Fig. 1
Fig. 1 Schematic of non-Hermitian dielectric multilayers. For the primitive unit-cell layer A, B, graphene C and dielectric D, the thicknesses are d = 0.2 μm and the graphene thickness dc = 1 nm. The refractive indices of dielectrics A and B are na = 2.2, nb = 1.8 and nd = n′ + in″ with n″ = 0.05.
Fig. 2
Fig. 2 (a) The calculated spectra of T (the blue curve), R1 (the red curve) and R2 (the green curve) as a function of incident angle θ for dielectric refractive index n′ = 2.0. R1 and R2 are the reflectance of light incident from the left and right, respectively. (b) Electric fields intensity (|Ez|2) distribution of defective mode.
Fig. 3
Fig. 3 (a) Reflection spectra R1 in parametric space of incident angle and dielectric refractive index. EP1 locating at θEP1 = 23.978° and nEP1 = 2.316. (b) Real part and imaginary part of eigenvalues for scattering matrix vary with incident angle as n′ = nEP1. (c) Reflection spectra R2 and EP2 locating at θEP2 = 22.978° and nEP2 = 2.101. (d) Real part and imaginary part of eigenvalues for scattering matrix as n′ = nEP2. The black arrows point to the positions of EPs in (a) and (c). Light is incident from the left for (a) and (b), while incident from the right for (c) and (d).
Fig. 4
Fig. 4 (a) Phase of reflection coefficient in the vicinity of EP1. (b) Phase curve versus incident angle for several specific refractive indices around EP1. (c) Phase of reflection coefficient in the vicinity of EP2. (d) Phase curve versus incident angle for several specific refractive indices around EP2. Light is incident from the left for (a) and (b), while incident from the right for (c) and (d).
Fig. 5
Fig. 5 (a) GH shifts versus incident angle for dielectric refractive index n′ < nEP1. (b) GH shifts versus incident angle for dielectric refractive index n′ > nEP1. (c) Maxima of GH shifts varying with dielectric refractive index around nEP1 (in blue dot line). Incident angle of position in parametric space for maxima versus dielectric refractive index (in red dot line). (d) GH shifts in the vicinity of EP1. The GH shifts are negative in the part I, while positive in the part II. It has rescaled the results by taking logarithm log10|Δ1/ λ | for clarity.
Fig. 6
Fig. 6 (a) GH shifts versus incident angle for dielectric refractive index n′ < nEP2. (b) GH shifts versus incident angle for dielectric refractive index n′ > nEP2. (c) Maxima of GH shifts varying with dielectric refractive index around nEP2 (in blue dot line). Incident angle of position in parametric space for maxima versus dielectric refractive index (in red dot line). (d) GH shifts in the vicinity of EP2. The GH shifts are positive in the part I, while negative in the part II. It has rescaled the results by taking logarithm log10|Δ2/λ| for clarity.
Fig. 7
Fig. 7 (a) GH shifts versus dielectric refractive index for light incident from the left. (b) GH shifts versus dielectric refractive index for light incident from the right. The curves are in red, green and blue corresponding to incident angles θ = 23.5°, 24.0° and 24.5°, respectively.
Fig. 8
Fig. 8 (a) Position of zero-reflectance point modifying by graphene chemical potential in parametric space of incident angle and refractive index. Light is incident from the left. Refractive index is shown in blue and incident angle is shown in red. (b) Reflectance varying with incident angle and chemical potential. (c) GH shifts versus graphene chemical potential for three incident angles of θ = 23.85° (in red), 23.90° (in green) and 23.95° (in blue). (d) GH shifts in parametric space of incident angle and chemical potential. The refractive index n′ = 2.316 and it has rescaled the values by taking logarithm for clarity in (b)-(d).
Fig. 9
Fig. 9 (a) Electric field amplitude distribution of Gaussian beam with a finite width as θ = 24.8° and n′ = 2.46. (b) Profile of electric field at the incident interface. (c) Influence of width on GH shifts for Gaussian beam as θ = 22.3° and n′ = 2.29. Simulation of GH shifts for Gaussian beam with different widths (in blue dotted line) and result predicted in theory (in red dotted line).
Fig. 10
Fig. 10 (a) Sensitivity coefficient of GH shifts in parametric space of dielectric refractive index and incident angle around EP1. It has rescaled the values by taking logarithm log10|dΔ/(λdn′) | for clarity. (b) Sensitivity coefficient of GH shifts tuned by dielectric refractive index at incident angles θ = 23°, 24° and 25°, respectively.

Equations (2)

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S = [ t r 1 r 2 t ]
Δ 1 , 2 = λ 2 π d φ 1 , 2 d θ ,

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