Abstract

Goos-Hänchen (GH) effect is a fundamental phenomenon in optics. Here we demonstrate theoretically that the surface modes at Parity-time (PT) symmetric interfaces, can induce a giant GH shift at a specific incident angle. It is found that the amplitude of the GH shift can be tuned by adjusting the thickness of the bilayer, and as the thickness grows, its maximum value can go to infinity in theory. The physical mechanism behind this interesting feature is that the surface modes at PT interfaces are quasi-bound states in continuum (BICs), which lead to rapid variation in the phase of the scattered waves. Our work enriches the previous studies about GH effect in PT bilayer structures and provides a way in turn to explore the BICs in non-Hermitian photonic systems.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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    [Crossref]
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    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref]
  35. Y. Xu, J. Jiang, and H. Chen, “Stable lossless polaritons on non-Hermitian optical interfaces,” Phys. Rev. B 95(4), 041409 (2017).
    [Crossref]
  36. S. Savoia, G. Castaldi, V. Galdi, A. Alù, and N. Engheta, “Tunneling of obliquely incident waves through PT -symmetric epsilon-near-zero bilayers,” Phys. Rev. B Condens. Matter Mater. Phys. 89(8), 085105 (2014).
    [Crossref]
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    [Crossref]

2018 (1)

R. El-Ganainy, K. Makris, M. Khajavikhan, Z. Musslimani, S. Rotter, and D. Christodoulides, “Non-Hermitian physics and PT symmetry,” Nat. Phys. 14(1), 11–19 (2018).
[Crossref]

2017 (6)

W. Yu, H. Sun, and L. Gao, “Magnetic control of Goos-Hänchen shifts in a yttrium-iron-garnet film,” Sci. Rep. 7(1), 45866 (2017).
[Crossref] [PubMed]

L. Feng, R. El-Ganainy, and L. Ge, “Non-Hermitian photonics based on parity–time symmetry,” Nat. Photonics 11(12), 752–762 (2017).
[Crossref]

Y. Fu and Y. Xu, “Asymmetric effects in waveguide systems using PT symmetry and zero index metamaterials,” Sci. Rep. 7(1), 12476 (2017).
[Crossref] [PubMed]

Y. Fu, X. Zhang, Y. Xu, and H. Chen, “Design of zero index metamaterials with PT symmetry using epsilon-near-zero media with defects,” J. Appl. Phys. 121(9), 094503 (2017).
[Crossref]

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]

Y. Xu, J. Jiang, and H. Chen, “Stable lossless polaritons on non-Hermitian optical interfaces,” Phys. Rev. B 95(4), 041409 (2017).
[Crossref]

2016 (7)

C. W. Hsu, B. Zhen, A. D. Stone, J. Joannopoulos, and M. Soljačić, “Bound states in the continuum,” Nat. Rev. Mater. 1(9), 16048 (2016).
[Crossref]

V. J. Yallapragada, A. P. Ravishankar, G. L. Mulay, G. S. Agarwal, and V. G. Achanta, “Observation of giant Goos-Hänchen and angular shifts at designed metasurfaces,” Sci. Rep. 6(1), 19319 (2016).
[Crossref] [PubMed]

Y. Fan, N. Shen, F. Zhang, Z. Wei, H. Li, Q. Zhao, Q. Fu, P. Zhang, T. Koschny, and C. M. Soukoulis, “Electrically tunable Goos-Hänchen effect with graphene in the terahertz regime,” Adv. Opt. Mater. 4(11), 1824–1828 (2016).
[Crossref]

M. Merano, “Optical beam shifts in graphene and single-layer boron-nitride,” Opt. Lett. 41(24), 5780–5783 (2016).
[Crossref] [PubMed]

Z. Wong, Y. Xu, J. Kim, K. O’Brien, Y. Wang, L. Feng, and X. Zhang, “Lasing and anti-lasing in a single cavity,” Nat. Photonics 10(12), 796–801 (2016).
[Crossref]

Y. Fu, Y. Xu, and H. Chen, “Zero index metamaterials with PT symmetry in a waveguide system,” Opt. Express 24(2), 1648–1657 (2016).
[Crossref] [PubMed]

C. Xu, J. Xu, G. Song, C. Zhu, Y. Yang, and G. S. Agarwal, “Enhanced displacements in reflected beams at hyperbolic metamaterials,” Opt. Express 24(19), 21767–21776 (2016).
[Crossref] [PubMed]

2015 (2)

S. Savoia, G. Castaldi, V. Galdi, A. Alù, and N. Engheta, “PT-symmetry-induced wave confinement and guiding in ε -near-zero metamaterials,” Phys. Rev. B Condens. Matter Mater. Phys. 91(11), 115114 (2015).
[Crossref]

Y. Xu, C. T. Chan, and H. Chen, “Goos-Hänchen effect in epsilon-near-zero metamaterials,” Sci. Rep. 5(1), 8681 (2015).
[Crossref] [PubMed]

2014 (2)

X. Zhu, H. Ramezani, C. Shi, J. Zhu, and X. Zhang, “PT-symmetric acoustics,” Phys. Rev. X 4(3), 031042 (2014).
[Crossref]

S. Savoia, G. Castaldi, V. Galdi, A. Alù, and N. Engheta, “Tunneling of obliquely incident waves through PT -symmetric epsilon-near-zero bilayers,” Phys. Rev. B Condens. Matter Mater. Phys. 89(8), 085105 (2014).
[Crossref]

2013 (2)

L. Feng, Y. L. Xu, W. S. Fegadolli, M. H. Lu, J. E. 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]

G. Jayaswal, G. Mistura, and M. Merano, “Weak measurement of the Goos-Hänchen shift,” Opt. Lett. 38(8), 1232–1234 (2013).
[Crossref] [PubMed]

2012 (2)

L. Ge, Y. D. Chong, and A. D. Stone, “Conservation relations and anisotropic transmission resonances in one-dimensional PT -symmetric photonic heterostructures,” Phys. Rev. A 85(2), 023802 (2012).
[Crossref]

I. V. Soboleva, V. V. Moskalenko, and A. A. Fedyanin, “Giant Goos-Hänchen effect and Fano resonance at photonic crystal surfaces,” Phys. Rev. Lett. 108(12), 123901 (2012).
[Crossref] [PubMed]

2011 (2)

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]

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]

2010 (2)

S. Longhi, “PT -symmetric laser absorber,” Phys. Rev. A 82(3), 031801 (2010).
[Crossref]

D. Gao and L. Gao, “Goos–Hänchen shift of the reflection from nonlinear nanocomposites with electric field tunability,” Appl. Phys. Lett. 97(4), 041903 (2010).
[Crossref]

2009 (1)

A. Mostafazadeh, “Spectral singularities of complex scattering potentials and infinite reflection and transmission coefficients at real energies,” Phys. Rev. Lett. 102(22), 220402 (2009).
[Crossref] [PubMed]

2007 (2)

P. T. Leung, C. W. Chen, and H. P. Chiang, “Large negative Goos-Hänchen shift at metal surfaces,” Opt. Commun. 276(2), 206–208 (2007).
[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]

2006 (1)

J. He, J. Yi, and S. He, “Giant negative Goos-Hänchen shifts for a photonic crystal with a negative effective index,” Opt. Express 14(7), 3024–3029 (2006).
[Crossref] [PubMed]

2005 (1)

L. G. Wang, H. Chen, and S. Y. Zhu, “Large negative Goos-Hänchen shift from a weakly absorbing dielectric slab,” Opt. Lett. 30(21), 2936–2938 (2005).
[Crossref] [PubMed]

2003 (2)

D. Felbacq, A. Moreau, and R. Smaâli, “Goos-Hänchen effect in the gaps of photonic crystals,” Opt. Lett. 28(18), 1633–1635 (2003).
[Crossref] [PubMed]

C. F. Li, “Negative lateral shift of a light beam transmitted through a dielectric slab and interaction of boundary effects,” Phys. Rev. Lett. 91(13), 133903 (2003).
[Crossref] [PubMed]

2002 (2)

H. M. Lai and S. W. Chan, “Large and negative Goos-Hänchen shift near the Brewster dip on reflection from weakly absorbing media,” Opt. Lett. 27(9), 680–682 (2002).
[Crossref] [PubMed]

P. R. Berman, “Goos-Hänchen shift in negatively refractive media,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 66(6 Pt 2), 067603 (2002).
[Crossref] [PubMed]

1982 (1)

W. Wild and C. Giles, “Goos-Hänchen shifts from absorbing media,” Phys. Rev. A 25(4), 2099–2101 (1982).
[Crossref]

1947 (1)

F. Goos and H. Hänchen, “Einneuer und fundamentaler Versuch zur Totalreflexion,” Ann. Phys. 436(7-8), 333–346 (1947).
[Crossref]

’t Hooft, G. W.

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]

Achanta, V. G.

V. J. Yallapragada, A. P. Ravishankar, G. L. Mulay, G. S. Agarwal, and V. G. Achanta, “Observation of giant Goos-Hänchen and angular shifts at designed metasurfaces,” Sci. Rep. 6(1), 19319 (2016).
[Crossref] [PubMed]

Agarwal, G. S.

V. J. Yallapragada, A. P. Ravishankar, G. L. Mulay, G. S. Agarwal, and V. G. Achanta, “Observation of giant Goos-Hänchen and angular shifts at designed metasurfaces,” Sci. Rep. 6(1), 19319 (2016).
[Crossref] [PubMed]

C. Xu, J. Xu, G. Song, C. Zhu, Y. Yang, and G. S. Agarwal, “Enhanced displacements in reflected beams at hyperbolic metamaterials,” Opt. Express 24(19), 21767–21776 (2016).
[Crossref] [PubMed]

Aiello, A.

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. 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]

Alù, A.

S. Savoia, G. Castaldi, V. Galdi, A. Alù, and N. Engheta, “PT-symmetry-induced wave confinement and guiding in ε -near-zero metamaterials,” Phys. Rev. B Condens. Matter Mater. Phys. 91(11), 115114 (2015).
[Crossref]

S. Savoia, G. Castaldi, V. Galdi, A. Alù, and N. Engheta, “Tunneling of obliquely incident waves through PT -symmetric epsilon-near-zero bilayers,” Phys. Rev. B Condens. Matter Mater. Phys. 89(8), 085105 (2014).
[Crossref]

Berman, P. R.

P. R. Berman, “Goos-Hänchen shift in negatively refractive media,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 66(6 Pt 2), 067603 (2002).
[Crossref] [PubMed]

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]

Castaldi, G.

S. Savoia, G. Castaldi, V. Galdi, A. Alù, and N. Engheta, “PT-symmetry-induced wave confinement and guiding in ε -near-zero metamaterials,” Phys. Rev. B Condens. Matter Mater. Phys. 91(11), 115114 (2015).
[Crossref]

S. Savoia, G. Castaldi, V. Galdi, A. Alù, and N. Engheta, “Tunneling of obliquely incident waves through PT -symmetric epsilon-near-zero bilayers,” Phys. Rev. B Condens. Matter Mater. Phys. 89(8), 085105 (2014).
[Crossref]

Chan, C. T.

Y. Xu, C. T. Chan, and H. Chen, “Goos-Hänchen effect in epsilon-near-zero metamaterials,” Sci. Rep. 5(1), 8681 (2015).
[Crossref] [PubMed]

Chan, S. W.

H. M. Lai and S. W. Chan, “Large and negative Goos-Hänchen shift near the Brewster dip on reflection from weakly absorbing media,” Opt. Lett. 27(9), 680–682 (2002).
[Crossref] [PubMed]

Chen, C. W.

P. T. Leung, C. W. Chen, and H. P. Chiang, “Large negative Goos-Hänchen shift at metal surfaces,” Opt. Commun. 276(2), 206–208 (2007).
[Crossref]

Chen, H.

Y. Xu, J. Jiang, and H. Chen, “Stable lossless polaritons on non-Hermitian optical interfaces,” Phys. Rev. B 95(4), 041409 (2017).
[Crossref]

Y. Fu, X. Zhang, Y. Xu, and H. Chen, “Design of zero index metamaterials with PT symmetry using epsilon-near-zero media with defects,” J. Appl. Phys. 121(9), 094503 (2017).
[Crossref]

Y. Fu, Y. Xu, and H. Chen, “Zero index metamaterials with PT symmetry in a waveguide system,” Opt. Express 24(2), 1648–1657 (2016).
[Crossref] [PubMed]

Y. Xu, C. T. Chan, and H. Chen, “Goos-Hänchen effect in epsilon-near-zero metamaterials,” Sci. Rep. 5(1), 8681 (2015).
[Crossref] [PubMed]

L. G. Wang, H. Chen, and S. Y. Zhu, “Large negative Goos-Hänchen shift from a weakly absorbing dielectric slab,” Opt. Lett. 30(21), 2936–2938 (2005).
[Crossref] [PubMed]

Chen, Y. F.

L. Feng, Y. L. Xu, W. S. Fegadolli, M. H. Lu, J. E. 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]

Chiang, H. P.

P. T. Leung, C. W. Chen, and H. P. Chiang, “Large negative Goos-Hänchen shift at metal surfaces,” Opt. Commun. 276(2), 206–208 (2007).
[Crossref]

Chong, Y. D.

L. Ge, Y. D. Chong, and A. D. Stone, “Conservation relations and anisotropic transmission resonances in one-dimensional PT -symmetric photonic heterostructures,” Phys. Rev. A 85(2), 023802 (2012).
[Crossref]

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.

R. El-Ganainy, K. Makris, M. Khajavikhan, Z. Musslimani, S. Rotter, and D. Christodoulides, “Non-Hermitian physics and PT symmetry,” Nat. Phys. 14(1), 11–19 (2018).
[Crossref]

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]

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.

R. El-Ganainy, K. Makris, M. Khajavikhan, Z. Musslimani, S. Rotter, and D. Christodoulides, “Non-Hermitian physics and PT symmetry,” Nat. Phys. 14(1), 11–19 (2018).
[Crossref]

L. Feng, R. El-Ganainy, and L. Ge, “Non-Hermitian photonics based on parity–time symmetry,” Nat. Photonics 11(12), 752–762 (2017).
[Crossref]

Eliel, E. R.

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]

Engheta, N.

S. Savoia, G. Castaldi, V. Galdi, A. Alù, and N. Engheta, “PT-symmetry-induced wave confinement and guiding in ε -near-zero metamaterials,” Phys. Rev. B Condens. Matter Mater. Phys. 91(11), 115114 (2015).
[Crossref]

S. Savoia, G. Castaldi, V. Galdi, A. Alù, and N. Engheta, “Tunneling of obliquely incident waves through PT -symmetric epsilon-near-zero bilayers,” Phys. Rev. B Condens. Matter Mater. Phys. 89(8), 085105 (2014).
[Crossref]

Fan, Y.

Y. Fan, N. Shen, F. Zhang, Z. Wei, H. Li, Q. Zhao, Q. Fu, P. Zhang, T. Koschny, and C. M. Soukoulis, “Electrically tunable Goos-Hänchen effect with graphene in the terahertz regime,” Adv. Opt. Mater. 4(11), 1824–1828 (2016).
[Crossref]

Fedyanin, A. A.

I. V. Soboleva, V. V. Moskalenko, and A. A. Fedyanin, “Giant Goos-Hänchen effect and Fano resonance at photonic crystal surfaces,” Phys. Rev. Lett. 108(12), 123901 (2012).
[Crossref] [PubMed]

Fegadolli, W. S.

L. Feng, Y. L. Xu, W. S. Fegadolli, M. H. Lu, J. E. 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]

Felbacq, D.

D. Felbacq, A. Moreau, and R. Smaâli, “Goos-Hänchen effect in the gaps of photonic crystals,” Opt. Lett. 28(18), 1633–1635 (2003).
[Crossref] [PubMed]

Feng, L.

L. Feng, R. El-Ganainy, and L. Ge, “Non-Hermitian photonics based on parity–time symmetry,” Nat. Photonics 11(12), 752–762 (2017).
[Crossref]

Z. Wong, Y. Xu, J. Kim, K. O’Brien, Y. Wang, L. Feng, and X. Zhang, “Lasing and anti-lasing in a single cavity,” Nat. Photonics 10(12), 796–801 (2016).
[Crossref]

L. Feng, Y. L. Xu, W. S. Fegadolli, M. H. Lu, J. E. 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]

Fu, Q.

Y. Fan, N. Shen, F. Zhang, Z. Wei, H. Li, Q. Zhao, Q. Fu, P. Zhang, T. Koschny, and C. M. Soukoulis, “Electrically tunable Goos-Hänchen effect with graphene in the terahertz regime,” Adv. Opt. Mater. 4(11), 1824–1828 (2016).
[Crossref]

Fu, Y.

Y. Fu and Y. Xu, “Asymmetric effects in waveguide systems using PT symmetry and zero index metamaterials,” Sci. Rep. 7(1), 12476 (2017).
[Crossref] [PubMed]

Y. Fu, X. Zhang, Y. Xu, and H. Chen, “Design of zero index metamaterials with PT symmetry using epsilon-near-zero media with defects,” J. Appl. Phys. 121(9), 094503 (2017).
[Crossref]

Y. Fu, Y. Xu, and H. Chen, “Zero index metamaterials with PT symmetry in a waveguide system,” Opt. Express 24(2), 1648–1657 (2016).
[Crossref] [PubMed]

Galdi, V.

S. Savoia, G. Castaldi, V. Galdi, A. Alù, and N. Engheta, “PT-symmetry-induced wave confinement and guiding in ε -near-zero metamaterials,” Phys. Rev. B Condens. Matter Mater. Phys. 91(11), 115114 (2015).
[Crossref]

S. Savoia, G. Castaldi, V. Galdi, A. Alù, and N. Engheta, “Tunneling of obliquely incident waves through PT -symmetric epsilon-near-zero bilayers,” Phys. Rev. B Condens. Matter Mater. Phys. 89(8), 085105 (2014).
[Crossref]

Gao, D.

D. Gao and L. Gao, “Goos–Hänchen shift of the reflection from nonlinear nanocomposites with electric field tunability,” Appl. Phys. Lett. 97(4), 041903 (2010).
[Crossref]

Gao, L.

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]

W. Yu, H. Sun, and L. Gao, “Magnetic control of Goos-Hänchen shifts in a yttrium-iron-garnet film,” Sci. Rep. 7(1), 45866 (2017).
[Crossref] [PubMed]

D. Gao and L. Gao, “Goos–Hänchen shift of the reflection from nonlinear nanocomposites with electric field tunability,” Appl. Phys. Lett. 97(4), 041903 (2010).
[Crossref]

Ge, L.

L. Feng, R. El-Ganainy, and L. Ge, “Non-Hermitian photonics based on parity–time symmetry,” Nat. Photonics 11(12), 752–762 (2017).
[Crossref]

L. Ge, Y. D. Chong, and A. D. Stone, “Conservation relations and anisotropic transmission resonances in one-dimensional PT -symmetric photonic heterostructures,” Phys. Rev. A 85(2), 023802 (2012).
[Crossref]

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]

Giles, C.

W. Wild and C. Giles, “Goos-Hänchen shifts from absorbing media,” Phys. Rev. A 25(4), 2099–2101 (1982).
[Crossref]

Goos, F.

F. Goos and H. Hänchen, “Einneuer und fundamentaler Versuch zur Totalreflexion,” Ann. Phys. 436(7-8), 333–346 (1947).
[Crossref]

Hänchen, H.

F. Goos and H. Hänchen, “Einneuer und fundamentaler Versuch zur Totalreflexion,” Ann. Phys. 436(7-8), 333–346 (1947).
[Crossref]

He, J.

J. He, J. Yi, and S. He, “Giant negative Goos-Hänchen shifts for a photonic crystal with a negative effective index,” Opt. Express 14(7), 3024–3029 (2006).
[Crossref] [PubMed]

He, S.

J. He, J. Yi, and S. He, “Giant negative Goos-Hänchen shifts for a photonic crystal with a negative effective index,” Opt. Express 14(7), 3024–3029 (2006).
[Crossref] [PubMed]

Hsu, C. W.

C. W. Hsu, B. Zhen, A. D. Stone, J. Joannopoulos, and M. Soljačić, “Bound states in the continuum,” Nat. Rev. Mater. 1(9), 16048 (2016).
[Crossref]

Jayaswal, G.

G. Jayaswal, G. Mistura, and M. Merano, “Weak measurement of the Goos-Hänchen shift,” Opt. Lett. 38(8), 1232–1234 (2013).
[Crossref] [PubMed]

Jiang, J.

Y. Xu, J. Jiang, and H. Chen, “Stable lossless polaritons on non-Hermitian optical interfaces,” Phys. Rev. B 95(4), 041409 (2017).
[Crossref]

Joannopoulos, J.

C. W. Hsu, B. Zhen, A. D. Stone, J. Joannopoulos, and M. Soljačić, “Bound states in the continuum,” Nat. Rev. Mater. 1(9), 16048 (2016).
[Crossref]

Khajavikhan, M.

R. El-Ganainy, K. Makris, M. Khajavikhan, Z. Musslimani, S. Rotter, and D. Christodoulides, “Non-Hermitian physics and PT symmetry,” Nat. Phys. 14(1), 11–19 (2018).
[Crossref]

Kim, J.

Z. Wong, Y. Xu, J. Kim, K. O’Brien, Y. Wang, L. Feng, and X. Zhang, “Lasing and anti-lasing in a single cavity,” Nat. Photonics 10(12), 796–801 (2016).
[Crossref]

Koschny, T.

Y. Fan, N. Shen, F. Zhang, Z. Wei, H. Li, Q. Zhao, Q. Fu, P. Zhang, T. Koschny, and C. M. Soukoulis, “Electrically tunable Goos-Hänchen effect with graphene in the terahertz regime,” Adv. Opt. Mater. 4(11), 1824–1828 (2016).
[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.

H. M. Lai and S. W. Chan, “Large and negative Goos-Hänchen shift near the Brewster dip on reflection from weakly absorbing media,” Opt. Lett. 27(9), 680–682 (2002).
[Crossref] [PubMed]

Leung, P. T.

P. T. Leung, C. W. Chen, and H. P. Chiang, “Large negative Goos-Hänchen shift at metal surfaces,” Opt. Commun. 276(2), 206–208 (2007).
[Crossref]

Li, C. F.

C. F. Li, “Negative lateral shift of a light beam transmitted through a dielectric slab and interaction of boundary effects,” Phys. Rev. Lett. 91(13), 133903 (2003).
[Crossref] [PubMed]

Li, H.

Y. Fan, N. Shen, F. Zhang, Z. Wei, H. Li, Q. Zhao, Q. Fu, P. Zhang, T. Koschny, and C. M. Soukoulis, “Electrically tunable Goos-Hänchen effect with graphene in the terahertz regime,” Adv. Opt. Mater. 4(11), 1824–1828 (2016).
[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]

Longhi, S.

S. Longhi, “PT -symmetric laser absorber,” Phys. Rev. A 82(3), 031801 (2010).
[Crossref]

Lu, M. H.

L. Feng, Y. L. Xu, W. S. Fegadolli, M. H. Lu, J. E. 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]

Ma, P.

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]

Makris, K.

R. El-Ganainy, K. Makris, M. Khajavikhan, Z. Musslimani, S. Rotter, and D. Christodoulides, “Non-Hermitian physics and PT symmetry,” Nat. Phys. 14(1), 11–19 (2018).
[Crossref]

Merano, M.

M. Merano, “Optical beam shifts in graphene and single-layer boron-nitride,” Opt. Lett. 41(24), 5780–5783 (2016).
[Crossref] [PubMed]

G. Jayaswal, G. Mistura, and M. Merano, “Weak measurement of the Goos-Hänchen shift,” Opt. Lett. 38(8), 1232–1234 (2013).
[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]

Mistura, G.

G. Jayaswal, G. Mistura, and M. Merano, “Weak measurement of the Goos-Hänchen shift,” Opt. Lett. 38(8), 1232–1234 (2013).
[Crossref] [PubMed]

Moreau, A.

D. Felbacq, A. Moreau, and R. Smaâli, “Goos-Hänchen effect in the gaps of photonic crystals,” Opt. Lett. 28(18), 1633–1635 (2003).
[Crossref] [PubMed]

Moskalenko, V. V.

I. V. Soboleva, V. V. Moskalenko, and A. A. Fedyanin, “Giant Goos-Hänchen effect and Fano resonance at photonic crystal surfaces,” Phys. Rev. Lett. 108(12), 123901 (2012).
[Crossref] [PubMed]

Mostafazadeh, A.

A. Mostafazadeh, “Spectral singularities of complex scattering potentials and infinite reflection and transmission coefficients at real energies,” Phys. Rev. Lett. 102(22), 220402 (2009).
[Crossref] [PubMed]

Mulay, G. L.

V. J. Yallapragada, A. P. Ravishankar, G. L. Mulay, G. S. Agarwal, and V. G. Achanta, “Observation of giant Goos-Hänchen and angular shifts at designed metasurfaces,” Sci. Rep. 6(1), 19319 (2016).
[Crossref] [PubMed]

Musslimani, Z.

R. El-Ganainy, K. Makris, M. Khajavikhan, Z. Musslimani, S. Rotter, and D. Christodoulides, “Non-Hermitian physics and PT symmetry,” Nat. Phys. 14(1), 11–19 (2018).
[Crossref]

O’Brien, K.

Z. Wong, Y. Xu, J. Kim, K. O’Brien, Y. Wang, L. Feng, and X. Zhang, “Lasing and anti-lasing in a single cavity,” Nat. Photonics 10(12), 796–801 (2016).
[Crossref]

Oliveira, J. E.

L. Feng, Y. L. Xu, W. S. Fegadolli, M. H. Lu, J. E. 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]

Ramezani, H.

X. Zhu, H. Ramezani, C. Shi, J. Zhu, and X. Zhang, “PT-symmetric acoustics,” Phys. Rev. X 4(3), 031042 (2014).
[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]

Ravishankar, A. P.

V. J. Yallapragada, A. P. Ravishankar, G. L. Mulay, G. S. Agarwal, and V. G. Achanta, “Observation of giant Goos-Hänchen and angular shifts at designed metasurfaces,” Sci. Rep. 6(1), 19319 (2016).
[Crossref] [PubMed]

Rotter, S.

R. El-Ganainy, K. Makris, M. Khajavikhan, Z. Musslimani, S. Rotter, and D. Christodoulides, “Non-Hermitian physics and PT symmetry,” Nat. Phys. 14(1), 11–19 (2018).
[Crossref]

Savoia, S.

S. Savoia, G. Castaldi, V. Galdi, A. Alù, and N. Engheta, “PT-symmetry-induced wave confinement and guiding in ε -near-zero metamaterials,” Phys. Rev. B Condens. Matter Mater. Phys. 91(11), 115114 (2015).
[Crossref]

S. Savoia, G. Castaldi, V. Galdi, A. Alù, and N. Engheta, “Tunneling of obliquely incident waves through PT -symmetric epsilon-near-zero bilayers,” Phys. Rev. B Condens. Matter Mater. Phys. 89(8), 085105 (2014).
[Crossref]

Scherer, A.

L. Feng, Y. L. Xu, W. S. Fegadolli, M. H. Lu, J. E. 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]

Shen, N.

Y. Fan, N. Shen, F. Zhang, Z. Wei, H. Li, Q. Zhao, Q. Fu, P. Zhang, T. Koschny, and C. M. Soukoulis, “Electrically tunable Goos-Hänchen effect with graphene in the terahertz regime,” Adv. Opt. Mater. 4(11), 1824–1828 (2016).
[Crossref]

Shi, C.

X. Zhu, H. Ramezani, C. Shi, J. Zhu, and X. Zhang, “PT-symmetric acoustics,” Phys. Rev. X 4(3), 031042 (2014).
[Crossref]

Smaâli, R.

D. Felbacq, A. Moreau, and R. Smaâli, “Goos-Hänchen effect in the gaps of photonic crystals,” Opt. Lett. 28(18), 1633–1635 (2003).
[Crossref] [PubMed]

Soboleva, I. V.

I. V. Soboleva, V. V. Moskalenko, and A. A. Fedyanin, “Giant Goos-Hänchen effect and Fano resonance at photonic crystal surfaces,” Phys. Rev. Lett. 108(12), 123901 (2012).
[Crossref] [PubMed]

Soljacic, M.

C. W. Hsu, B. Zhen, A. D. Stone, J. Joannopoulos, and M. Soljačić, “Bound states in the continuum,” Nat. Rev. Mater. 1(9), 16048 (2016).
[Crossref]

Song, G.

C. Xu, J. Xu, G. Song, C. Zhu, Y. Yang, and G. S. Agarwal, “Enhanced displacements in reflected beams at hyperbolic metamaterials,” Opt. Express 24(19), 21767–21776 (2016).
[Crossref] [PubMed]

Soukoulis, C. M.

Y. Fan, N. Shen, F. Zhang, Z. Wei, H. Li, Q. Zhao, Q. Fu, P. Zhang, T. Koschny, and C. M. Soukoulis, “Electrically tunable Goos-Hänchen effect with graphene in the terahertz regime,” Adv. Opt. Mater. 4(11), 1824–1828 (2016).
[Crossref]

Stone, A. D.

C. W. Hsu, B. Zhen, A. D. Stone, J. Joannopoulos, and M. Soljačić, “Bound states in the continuum,” Nat. Rev. Mater. 1(9), 16048 (2016).
[Crossref]

L. Ge, Y. D. Chong, and A. D. Stone, “Conservation relations and anisotropic transmission resonances in one-dimensional PT -symmetric photonic heterostructures,” Phys. Rev. A 85(2), 023802 (2012).
[Crossref]

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]

Sun, H.

W. Yu, H. Sun, and L. Gao, “Magnetic control of Goos-Hänchen shifts in a yttrium-iron-garnet film,” Sci. Rep. 7(1), 45866 (2017).
[Crossref] [PubMed]

van Exter, M. P.

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, L. G.

L. G. Wang, H. Chen, and S. Y. Zhu, “Large negative Goos-Hänchen shift from a weakly absorbing dielectric slab,” Opt. Lett. 30(21), 2936–2938 (2005).
[Crossref] [PubMed]

Wang, Y.

Z. Wong, Y. Xu, J. Kim, K. O’Brien, Y. Wang, L. Feng, and X. Zhang, “Lasing and anti-lasing in a single cavity,” Nat. Photonics 10(12), 796–801 (2016).
[Crossref]

Wei, Z.

Y. Fan, N. Shen, F. Zhang, Z. Wei, H. Li, Q. Zhao, Q. Fu, P. Zhang, T. Koschny, and C. M. Soukoulis, “Electrically tunable Goos-Hänchen effect with graphene in the terahertz regime,” Adv. Opt. Mater. 4(11), 1824–1828 (2016).
[Crossref]

Wild, W.

W. Wild and C. Giles, “Goos-Hänchen shifts from absorbing media,” Phys. Rev. A 25(4), 2099–2101 (1982).
[Crossref]

Woerdman, J. P.

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]

Wong, Z.

Z. Wong, Y. Xu, J. Kim, K. O’Brien, Y. Wang, L. Feng, and X. Zhang, “Lasing and anti-lasing in a single cavity,” Nat. Photonics 10(12), 796–801 (2016).
[Crossref]

Xu, C.

C. Xu, J. Xu, G. Song, C. Zhu, Y. Yang, and G. S. Agarwal, “Enhanced displacements in reflected beams at hyperbolic metamaterials,” Opt. Express 24(19), 21767–21776 (2016).
[Crossref] [PubMed]

Xu, J.

C. Xu, J. Xu, G. Song, C. Zhu, Y. Yang, and G. S. Agarwal, “Enhanced displacements in reflected beams at hyperbolic metamaterials,” Opt. Express 24(19), 21767–21776 (2016).
[Crossref] [PubMed]

Xu, Y.

Y. Xu, J. Jiang, and H. Chen, “Stable lossless polaritons on non-Hermitian optical interfaces,” Phys. Rev. B 95(4), 041409 (2017).
[Crossref]

Y. Fu, X. Zhang, Y. Xu, and H. Chen, “Design of zero index metamaterials with PT symmetry using epsilon-near-zero media with defects,” J. Appl. Phys. 121(9), 094503 (2017).
[Crossref]

Y. Fu and Y. Xu, “Asymmetric effects in waveguide systems using PT symmetry and zero index metamaterials,” Sci. Rep. 7(1), 12476 (2017).
[Crossref] [PubMed]

Z. Wong, Y. Xu, J. Kim, K. O’Brien, Y. Wang, L. Feng, and X. Zhang, “Lasing and anti-lasing in a single cavity,” Nat. Photonics 10(12), 796–801 (2016).
[Crossref]

Y. Fu, Y. Xu, and H. Chen, “Zero index metamaterials with PT symmetry in a waveguide system,” Opt. Express 24(2), 1648–1657 (2016).
[Crossref] [PubMed]

Y. Xu, C. T. Chan, and H. Chen, “Goos-Hänchen effect in epsilon-near-zero metamaterials,” Sci. Rep. 5(1), 8681 (2015).
[Crossref] [PubMed]

Xu, Y. L.

L. Feng, Y. L. Xu, W. S. Fegadolli, M. H. Lu, J. E. 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]

Yallapragada, V. J.

V. J. Yallapragada, A. P. Ravishankar, G. L. Mulay, G. S. Agarwal, and V. G. Achanta, “Observation of giant Goos-Hänchen and angular shifts at designed metasurfaces,” Sci. Rep. 6(1), 19319 (2016).
[Crossref] [PubMed]

Yang, Y.

C. Xu, J. Xu, G. Song, C. Zhu, Y. Yang, and G. S. Agarwal, “Enhanced displacements in reflected beams at hyperbolic metamaterials,” Opt. Express 24(19), 21767–21776 (2016).
[Crossref] [PubMed]

Yi, J.

J. He, J. Yi, and S. He, “Giant negative Goos-Hänchen shifts for a photonic crystal with a negative effective index,” Opt. Express 14(7), 3024–3029 (2006).
[Crossref] [PubMed]

Yu, W.

W. Yu, H. Sun, and L. Gao, “Magnetic control of Goos-Hänchen shifts in a yttrium-iron-garnet film,” Sci. Rep. 7(1), 45866 (2017).
[Crossref] [PubMed]

Zhang, F.

Y. Fan, N. Shen, F. Zhang, Z. Wei, H. Li, Q. Zhao, Q. Fu, P. Zhang, T. Koschny, and C. M. Soukoulis, “Electrically tunable Goos-Hänchen effect with graphene in the terahertz regime,” Adv. Opt. Mater. 4(11), 1824–1828 (2016).
[Crossref]

Zhang, P.

Y. Fan, N. Shen, F. Zhang, Z. Wei, H. Li, Q. Zhao, Q. Fu, P. Zhang, T. Koschny, and C. M. Soukoulis, “Electrically tunable Goos-Hänchen effect with graphene in the terahertz regime,” Adv. Opt. Mater. 4(11), 1824–1828 (2016).
[Crossref]

Zhang, X.

Y. Fu, X. Zhang, Y. Xu, and H. Chen, “Design of zero index metamaterials with PT symmetry using epsilon-near-zero media with defects,” J. Appl. Phys. 121(9), 094503 (2017).
[Crossref]

Z. Wong, Y. Xu, J. Kim, K. O’Brien, Y. Wang, L. Feng, and X. Zhang, “Lasing and anti-lasing in a single cavity,” Nat. Photonics 10(12), 796–801 (2016).
[Crossref]

X. Zhu, H. Ramezani, C. Shi, J. Zhu, and X. Zhang, “PT-symmetric acoustics,” Phys. Rev. X 4(3), 031042 (2014).
[Crossref]

Zhao, Q.

Y. Fan, N. Shen, F. Zhang, Z. Wei, H. Li, Q. Zhao, Q. Fu, P. Zhang, T. Koschny, and C. M. Soukoulis, “Electrically tunable Goos-Hänchen effect with graphene in the terahertz regime,” Adv. Opt. Mater. 4(11), 1824–1828 (2016).
[Crossref]

Zhen, B.

C. W. Hsu, B. Zhen, A. D. Stone, J. Joannopoulos, and M. Soljačić, “Bound states in the continuum,” Nat. Rev. Mater. 1(9), 16048 (2016).
[Crossref]

Zhu, C.

C. Xu, J. Xu, G. Song, C. Zhu, Y. Yang, and G. S. Agarwal, “Enhanced displacements in reflected beams at hyperbolic metamaterials,” Opt. Express 24(19), 21767–21776 (2016).
[Crossref] [PubMed]

Zhu, J.

X. Zhu, H. Ramezani, C. Shi, J. Zhu, and X. Zhang, “PT-symmetric acoustics,” Phys. Rev. X 4(3), 031042 (2014).
[Crossref]

Zhu, S. Y.

L. G. Wang, H. Chen, and S. Y. Zhu, “Large negative Goos-Hänchen shift from a weakly absorbing dielectric slab,” Opt. Lett. 30(21), 2936–2938 (2005).
[Crossref] [PubMed]

Zhu, X.

X. Zhu, H. Ramezani, C. Shi, J. Zhu, and X. Zhang, “PT-symmetric acoustics,” Phys. Rev. X 4(3), 031042 (2014).
[Crossref]

Adv. Opt. Mater. (1)

Y. Fan, N. Shen, F. Zhang, Z. Wei, H. Li, Q. Zhao, Q. Fu, P. Zhang, T. Koschny, and C. M. Soukoulis, “Electrically tunable Goos-Hänchen effect with graphene in the terahertz regime,” Adv. Opt. Mater. 4(11), 1824–1828 (2016).
[Crossref]

Ann. Phys. (1)

F. Goos and H. Hänchen, “Einneuer und fundamentaler Versuch zur Totalreflexion,” Ann. Phys. 436(7-8), 333–346 (1947).
[Crossref]

Appl. Phys. Lett. (1)

D. Gao and L. Gao, “Goos–Hänchen shift of the reflection from nonlinear nanocomposites with electric field tunability,” Appl. Phys. Lett. 97(4), 041903 (2010).
[Crossref]

J. Appl. Phys. (1)

Y. Fu, X. Zhang, Y. Xu, and H. Chen, “Design of zero index metamaterials with PT symmetry using epsilon-near-zero media with defects,” J. Appl. Phys. 121(9), 094503 (2017).
[Crossref]

Nat. Mater. (1)

L. Feng, Y. L. Xu, W. S. Fegadolli, M. H. Lu, J. E. 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 (2)

L. Feng, R. El-Ganainy, and L. Ge, “Non-Hermitian photonics based on parity–time symmetry,” Nat. Photonics 11(12), 752–762 (2017).
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Z. Wong, Y. Xu, J. Kim, K. O’Brien, Y. Wang, L. Feng, and X. Zhang, “Lasing and anti-lasing in a single cavity,” Nat. Photonics 10(12), 796–801 (2016).
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Nat. Phys. (1)

R. El-Ganainy, K. Makris, M. Khajavikhan, Z. Musslimani, S. Rotter, and D. Christodoulides, “Non-Hermitian physics and PT symmetry,” Nat. Phys. 14(1), 11–19 (2018).
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Nat. Rev. Mater. (1)

C. W. Hsu, B. Zhen, A. D. Stone, J. Joannopoulos, and M. Soljačić, “Bound states in the continuum,” Nat. Rev. Mater. 1(9), 16048 (2016).
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Opt. Commun. (1)

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Opt. Express (5)

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C. Xu, J. Xu, G. Song, C. Zhu, Y. Yang, and G. S. Agarwal, “Enhanced displacements in reflected beams at hyperbolic metamaterials,” Opt. Express 24(19), 21767–21776 (2016).
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P. Ma and L. Gao, “Large and tunable lateral shifts in one-dimensional PT-symmetric layered structures,” Opt. Express 25(9), 9676–9688 (2017).
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Y. Fu, Y. Xu, and H. Chen, “Zero index metamaterials with PT symmetry in a waveguide system,” Opt. Express 24(2), 1648–1657 (2016).
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Phys. Rev. B (1)

Y. Xu, J. Jiang, and H. Chen, “Stable lossless polaritons on non-Hermitian optical interfaces,” Phys. Rev. B 95(4), 041409 (2017).
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Phys. Rev. B Condens. Matter Mater. Phys. (2)

S. Savoia, G. Castaldi, V. Galdi, A. Alù, and N. Engheta, “Tunneling of obliquely incident waves through PT -symmetric epsilon-near-zero bilayers,” Phys. Rev. B Condens. Matter Mater. Phys. 89(8), 085105 (2014).
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S. Savoia, G. Castaldi, V. Galdi, A. Alù, and N. Engheta, “PT-symmetry-induced wave confinement and guiding in ε -near-zero metamaterials,” Phys. Rev. B Condens. Matter Mater. Phys. 91(11), 115114 (2015).
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Sci. Rep. (4)

Y. Fu and Y. Xu, “Asymmetric effects in waveguide systems using PT symmetry and zero index metamaterials,” Sci. Rep. 7(1), 12476 (2017).
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Y. Xu, C. T. Chan, and H. Chen, “Goos-Hänchen effect in epsilon-near-zero metamaterials,” Sci. Rep. 5(1), 8681 (2015).
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Figures (6)

Fig. 1
Fig. 1 (a) The schematics of a PT-symmetric bilayer structure. A linearly polarized light is incident on the bilayer structure from loss side indicated by the black arrows, or from the gain side marked by the red arrows. Both loss and gain layers with an identical thickness of d meet PT symmetry about z = 0. (b) The parameter space for surface states at PT-symmetric interfaces with infinite thickness. The red dashed line indicates the position where β s w = k 0 . The tiny circle indicate the parameters of ε l = 1.5 + 0.5 i and ε g = 1.5 0.5 i . (c) The dispersion relationship. For a given working frequency ω 0 , a surface mode with β s w < k 0 is in the continuum.
Fig. 2
Fig. 2 The bound states at finite PT interfaces. (a) The determinant value for different β and d when γ = 0.02 . (b) The determinant value for different β and γ when d = 0.6 λ . (c) The surface states based on Eq. (1) for infinite d. The inset shows the simulated field pattern for a surface state when γ = 0.02 . Here ε r = 0.001 in (a)-(c). The black dashed curves in (a) and (b) denote the solutions of surface states.
Fig. 3
Fig. 3 GH-shifts of transmitted wave in PT symmetric bilayers with different loss/gain for TM polarization (a) and TE polarization (b).
Fig. 4
Fig. 4 (a) The happening angles θ G H of the giant GH shift for different γ in Fig. 3(a) and the pseudo-Brewster angles θ p B (red hollow circles) in a loss medium ( ε r = 0.001 + i γ ) with a thickness of 1.2 λ . (b) The eigenvalues of S-matrix for γ = 0.02 and d = 0.6 λ . (c) The reflection, transmission and transmission phase for γ = 0.02 and d = 0.6 λ . (d) The enlarged drawing for γ = 0.02 in Fig. 3(a).
Fig. 5
Fig. 5 (a) The giant GH shift vs the thickness d for fixed material parameters. From right to left, the peaks of the giant GH shifts are located at 32.15 , 29.27 , 27.83 and 27.11 , with corresponding values of L 3.5 λ , 10 λ , 25 λ and 68 λ . (b) The reflectance | r L | 2 vs thickness d and the incident angle. Here ε l = 0.001 + 0.02 i and ε g = 0.001 0.02 i . (c) The reflectance | r L | 2 vs the incident angle for a fixed thickness of d = 2.0 λ . (d) Transmission phase vs the incident angle for different thickness of d = 0.4 λ , 0.6 λ , 1.0 λ , 1.2 λ and 2.0 λ , which are shown by black, red, green, blue, cyan and pink curves, respectively.
Fig. 6
Fig. 6 GH effect and BICs in the case of ε l = 1.5 + 0.5 i and ε g = 1.5 0.5 i . (a) The numerically simulated field pattern of the surface modes at the PT-bilayer interface. In simulations, the thickness of each layer is d = 3.0 λ and the source (see the five star) is mimicked by using a tiny circle with a current of 1A. (b)-(d) are the analytically calculated GH shifts, the reflection | r l | 2 and the transmission phase for different thicknesses. In (b), as d increases, the center point for maximum value goes to θ i n = 66.00 . For d = 2.4 λ and 3.0 λ , the displayed data are real data divided by 5 and 50, respectively.

Equations (4)

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β s w = β = ε k 0 1 2 cos τ x ^ ,
M = [ cos ( k l , z d ) i 1 q l sin ( k l , z d ) i q l sin ( k l , z d ) cos ( k l , z d ) ] [ cos ( k g , z d ) i 1 q g sin ( k g , z d ) i q g sin ( k g , z d ) cos ( k g , z d ) ] ,
t L = t G = t = 1 M 22 , r L = M 21 M 22 , r G = M 12 M 22 .
L r , t = λ 2 π d ϕ r , t d θ i n .

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