Abstract

Recently, unidirectional invisibility has been demonstrated in parity-time (PT) symmetric periodic structures and has attracted great attention. Nevertheless, fabrication of a complex periodic structure may not be practically easy. In this paper, a simple two-layer non-PT-symmetric slab structure is proposed to realize unidirectional invisibility. We numerically show that in such conventional structure consisting of two slabs with different real parts of refractive indices, unidirectional invisibility can be achieved as proper imaginary parts of refractive indices and thicknesses of the slabs are satisfied. Moreover, the unidirectional invisibility can be converted to unidirectional reflection when the imaginary parts of the refractive indices are tuned to their odd symmetric forms.

© 2014 Optical Society of America

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  1. S. Longhi and G. Della Valle, “Photonic realization of PT-symmetric quantum field theories,” Phys. Rev. A 85(1), 012112 (2012).
    [Crossref]
  2. C. M. Bender and S. Boettcher, “Real spectra in non-Hermitian Hamiltonians having PT symmetry,” Phys. Rev. Lett. 80(24), 5243–5246 (1998).
    [Crossref]
  3. C. M. Bender, “Making sense of non-Hermitian Hamiltonians,” Rep. Prog. Phys. 70(6), 947–1018 (2007).
    [Crossref]
  4. N. Hatano and D. R. Nelson, “Localization transitions in non-Hermitian quantum mechanics,” Phys. Rev. Lett. 77(3), 570–573 (1996).
    [Crossref] [PubMed]
  5. O. Bendix, R. Fleischmann, T. Kottos, and B. Shapiro, “Exponentially fragile PT symmetry in lattices with localized eigenmodes,” Phys. Rev. Lett. 103(3), 030402 (2009).
    [Crossref] [PubMed]
  6. M. Hiller, T. Kottos, and A. Ossipov, “Bifurcations in resonance widths of an open Bose-Hubbard dimer,” Phys. Rev. A 73(6), 063625 (2006).
    [Crossref]
  7. R. El-Ganainy, K. G. Makris, D. N. Christodoulides, and Z. H. Musslimani, “Theory of coupled optical PT-symmetric structures,” Opt. Lett. 32(17), 2632–2634 (2007).
    [Crossref] [PubMed]
  8. S. Longhi, “Bloch oscillations in complex crystals with PT symmetry,” Phys. Rev. Lett. 103(12), 123601 (2009).
    [Crossref] [PubMed]
  9. O. Bendix, R. Fleischmann, T. Kottos, and B. Shapiro, “Optical structures with local PT-symmetry,” J. Phys. A 43(26), 265305 (2010).
    [Crossref]
  10. K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam dynamics in PT symmetric optical lattices,” Phys. Rev. Lett. 100(10), 103904 (2008).
    [Crossref] [PubMed]
  11. M. C. Zheng, D. N. Christodoulides, R. Fleischmann, and T. Kottos, “PT optical lattices and universality in beam dynamics,” Phys. Rev. A 82(1), 010103 (2010).
    [Crossref]
  12. A. Guo, G. J. Salamo, 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(9), 093902 (2009).
    [Crossref] [PubMed]
  13. S. Longhi, “PT-symmetric laser absorber,” Phys. Rev. A 82(3), 031801 (2010).
    [Crossref]
  14. 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]
  15. 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 (2012).
    [Crossref] [PubMed]
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  17. Y. Shen and G. P. Wang, “Gain-assisted time delay of plasmons in coupled metal ring resonator waveguides,” Opt. Express 17(15), 12807–12812 (2009), http://www.opticsinfobase.org/oe/abstract.cfm?uri=oe-17-15-12807 .
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  19. C. C. Baker, J. Heikenfeld, Z. Yu, and A. J. Steckl, “Optical amplification and electroluminescence at 1.54 μm in Er-doped zinc silicate germanate on silicon,” Appl. Phys. Lett. 84(9), 1462–1464 (2004).
    [Crossref]
  20. K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antenn. Propag. AP-14, 302–307 (1966).
  21. Y. Shen and G. P. Wang, “Optical bistability in metal gap waveguide nanocavities,” Opt. Express 16(12), 8421–8426 (2008), http://www.opticsinfobase.org/oe/abstract.cfm?uri=oe-16-12-8421 .
    [Crossref] [PubMed]

2012 (2)

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 (2012).
[Crossref] [PubMed]

S. Longhi and G. Della Valle, “Photonic realization of PT-symmetric quantum field theories,” Phys. Rev. A 85(1), 012112 (2012).
[Crossref]

2011 (1)

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 (3)

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

O. Bendix, R. Fleischmann, T. Kottos, and B. Shapiro, “Optical structures with local PT-symmetry,” J. Phys. A 43(26), 265305 (2010).
[Crossref]

M. C. Zheng, D. N. Christodoulides, R. Fleischmann, and T. Kottos, “PT optical lattices and universality in beam dynamics,” Phys. Rev. A 82(1), 010103 (2010).
[Crossref]

2009 (4)

A. Guo, G. J. Salamo, 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(9), 093902 (2009).
[Crossref] [PubMed]

S. Longhi, “Bloch oscillations in complex crystals with PT symmetry,” Phys. Rev. Lett. 103(12), 123601 (2009).
[Crossref] [PubMed]

O. Bendix, R. Fleischmann, T. Kottos, and B. Shapiro, “Exponentially fragile PT symmetry in lattices with localized eigenmodes,” Phys. Rev. Lett. 103(3), 030402 (2009).
[Crossref] [PubMed]

Y. Shen and G. P. Wang, “Gain-assisted time delay of plasmons in coupled metal ring resonator waveguides,” Opt. Express 17(15), 12807–12812 (2009), http://www.opticsinfobase.org/oe/abstract.cfm?uri=oe-17-15-12807 .
[Crossref] [PubMed]

2008 (2)

Y. Shen and G. P. Wang, “Optical bistability in metal gap waveguide nanocavities,” Opt. Express 16(12), 8421–8426 (2008), http://www.opticsinfobase.org/oe/abstract.cfm?uri=oe-16-12-8421 .
[Crossref] [PubMed]

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam dynamics in PT symmetric optical lattices,” Phys. Rev. Lett. 100(10), 103904 (2008).
[Crossref] [PubMed]

2007 (2)

2006 (1)

M. Hiller, T. Kottos, and A. Ossipov, “Bifurcations in resonance widths of an open Bose-Hubbard dimer,” Phys. Rev. A 73(6), 063625 (2006).
[Crossref]

2004 (1)

C. C. Baker, J. Heikenfeld, Z. Yu, and A. J. Steckl, “Optical amplification and electroluminescence at 1.54 μm in Er-doped zinc silicate germanate on silicon,” Appl. Phys. Lett. 84(9), 1462–1464 (2004).
[Crossref]

1999 (1)

1998 (1)

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

1996 (1)

N. Hatano and D. R. Nelson, “Localization transitions in non-Hermitian quantum mechanics,” Phys. Rev. Lett. 77(3), 570–573 (1996).
[Crossref] [PubMed]

1966 (1)

K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antenn. Propag. AP-14, 302–307 (1966).

Aimez, V.

A. Guo, G. J. Salamo, 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(9), 093902 (2009).
[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 (2012).
[Crossref] [PubMed]

Baker, C. C.

C. C. Baker, J. Heikenfeld, Z. Yu, and A. J. Steckl, “Optical amplification and electroluminescence at 1.54 μm in Er-doped zinc silicate germanate on silicon,” Appl. Phys. Lett. 84(9), 1462–1464 (2004).
[Crossref]

Bender, C. M.

C. M. Bender, “Making sense of non-Hermitian Hamiltonians,” Rep. Prog. Phys. 70(6), 947–1018 (2007).
[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]

Bendix, O.

O. Bendix, R. Fleischmann, T. Kottos, and B. Shapiro, “Optical structures with local PT-symmetry,” J. Phys. A 43(26), 265305 (2010).
[Crossref]

O. Bendix, R. Fleischmann, T. Kottos, and B. Shapiro, “Exponentially fragile PT symmetry in lattices with localized eigenmodes,” Phys. Rev. Lett. 103(3), 030402 (2009).
[Crossref] [PubMed]

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]

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 (2012).
[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]

M. C. Zheng, D. N. Christodoulides, R. Fleischmann, and T. Kottos, “PT optical lattices and universality in beam dynamics,” Phys. Rev. A 82(1), 010103 (2010).
[Crossref]

A. Guo, G. J. Salamo, 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(9), 093902 (2009).
[Crossref] [PubMed]

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam dynamics in PT symmetric optical lattices,” Phys. Rev. Lett. 100(10), 103904 (2008).
[Crossref] [PubMed]

R. El-Ganainy, K. G. Makris, D. N. Christodoulides, and Z. H. Musslimani, “Theory of coupled optical PT-symmetric structures,” Opt. Lett. 32(17), 2632–2634 (2007).
[Crossref] [PubMed]

Della Valle, G.

S. Longhi and G. Della Valle, “Photonic realization of PT-symmetric quantum field theories,” Phys. Rev. A 85(1), 012112 (2012).
[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.

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam dynamics in PT symmetric optical lattices,” Phys. Rev. Lett. 100(10), 103904 (2008).
[Crossref] [PubMed]

R. El-Ganainy, K. G. Makris, D. N. Christodoulides, and Z. H. Musslimani, “Theory of coupled optical PT-symmetric structures,” Opt. Lett. 32(17), 2632–2634 (2007).
[Crossref] [PubMed]

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 (2012).
[Crossref] [PubMed]

Feng, 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 (2012).
[Crossref] [PubMed]

Fleischmann, R.

O. Bendix, R. Fleischmann, T. Kottos, and B. Shapiro, “Optical structures with local PT-symmetry,” J. Phys. A 43(26), 265305 (2010).
[Crossref]

M. C. Zheng, D. N. Christodoulides, R. Fleischmann, and T. Kottos, “PT optical lattices and universality in beam dynamics,” Phys. Rev. A 82(1), 010103 (2010).
[Crossref]

O. Bendix, R. Fleischmann, T. Kottos, and B. Shapiro, “Exponentially fragile PT symmetry in lattices with localized eigenmodes,” Phys. Rev. Lett. 103(3), 030402 (2009).
[Crossref] [PubMed]

Guo, A.

A. Guo, G. J. Salamo, 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(9), 093902 (2009).
[Crossref] [PubMed]

Hatano, N.

N. Hatano and D. R. Nelson, “Localization transitions in non-Hermitian quantum mechanics,” Phys. Rev. Lett. 77(3), 570–573 (1996).
[Crossref] [PubMed]

Heikenfeld, J.

C. C. Baker, J. Heikenfeld, Z. Yu, and A. J. Steckl, “Optical amplification and electroluminescence at 1.54 μm in Er-doped zinc silicate germanate on silicon,” Appl. Phys. Lett. 84(9), 1462–1464 (2004).
[Crossref]

Hiller, M.

M. Hiller, T. Kottos, and A. Ossipov, “Bifurcations in resonance widths of an open Bose-Hubbard dimer,” Phys. Rev. A 73(6), 063625 (2006).
[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]

M. C. Zheng, D. N. Christodoulides, R. Fleischmann, and T. Kottos, “PT optical lattices and universality in beam dynamics,” Phys. Rev. A 82(1), 010103 (2010).
[Crossref]

O. Bendix, R. Fleischmann, T. Kottos, and B. Shapiro, “Optical structures with local PT-symmetry,” J. Phys. A 43(26), 265305 (2010).
[Crossref]

O. Bendix, R. Fleischmann, T. Kottos, and B. Shapiro, “Exponentially fragile PT symmetry in lattices with localized eigenmodes,” Phys. Rev. Lett. 103(3), 030402 (2009).
[Crossref] [PubMed]

M. Hiller, T. Kottos, and A. Ossipov, “Bifurcations in resonance widths of an open Bose-Hubbard dimer,” Phys. Rev. A 73(6), 063625 (2006).
[Crossref]

Lee, R. K.

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 and G. Della Valle, “Photonic realization of PT-symmetric quantum field theories,” Phys. Rev. A 85(1), 012112 (2012).
[Crossref]

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

S. Longhi, “Bloch oscillations in complex crystals with PT symmetry,” Phys. Rev. Lett. 103(12), 123601 (2009).
[Crossref] [PubMed]

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 (2012).
[Crossref] [PubMed]

Makris, K. G.

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam dynamics in PT symmetric optical lattices,” Phys. Rev. Lett. 100(10), 103904 (2008).
[Crossref] [PubMed]

R. El-Ganainy, K. G. Makris, D. N. Christodoulides, and Z. H. Musslimani, “Theory of coupled optical PT-symmetric structures,” Opt. Lett. 32(17), 2632–2634 (2007).
[Crossref] [PubMed]

Musslimani, Z. H.

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam dynamics in PT symmetric optical lattices,” Phys. Rev. Lett. 100(10), 103904 (2008).
[Crossref] [PubMed]

R. El-Ganainy, K. G. Makris, D. N. Christodoulides, and Z. H. Musslimani, “Theory of coupled optical PT-symmetric structures,” Opt. Lett. 32(17), 2632–2634 (2007).
[Crossref] [PubMed]

Nelson, D. R.

N. Hatano and D. R. Nelson, “Localization transitions in non-Hermitian quantum mechanics,” Phys. Rev. Lett. 77(3), 570–573 (1996).
[Crossref] [PubMed]

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 (2012).
[Crossref] [PubMed]

Ossipov, A.

M. Hiller, T. Kottos, and A. Ossipov, “Bifurcations in resonance widths of an open Bose-Hubbard dimer,” Phys. Rev. A 73(6), 063625 (2006).
[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]

Salamo, G. J.

A. Guo, G. J. Salamo, 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(9), 093902 (2009).
[Crossref] [PubMed]

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 (2012).
[Crossref] [PubMed]

A. Yariv, Y. Xu, R. K. Lee, and A. Scherer, “Coupled-resonator optical waveguide: a proposal and analysis,” Opt. Lett. 24(11), 711–713 (1999).
[Crossref] [PubMed]

Shapiro, B.

O. Bendix, R. Fleischmann, T. Kottos, and B. Shapiro, “Optical structures with local PT-symmetry,” J. Phys. A 43(26), 265305 (2010).
[Crossref]

O. Bendix, R. Fleischmann, T. Kottos, and B. Shapiro, “Exponentially fragile PT symmetry in lattices with localized eigenmodes,” Phys. Rev. Lett. 103(3), 030402 (2009).
[Crossref] [PubMed]

Shen, Y.

Siviloglou, G. A.

A. Guo, G. J. Salamo, 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(9), 093902 (2009).
[Crossref] [PubMed]

Steckl, A. J.

C. C. Baker, J. Heikenfeld, Z. Yu, and A. J. Steckl, “Optical amplification and electroluminescence at 1.54 μm in Er-doped zinc silicate germanate on silicon,” Appl. Phys. Lett. 84(9), 1462–1464 (2004).
[Crossref]

Volatier-Ravat, M.

A. Guo, G. J. Salamo, 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(9), 093902 (2009).
[Crossref] [PubMed]

Wang, G. P.

Xu, Y.

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 (2012).
[Crossref] [PubMed]

Yariv, A.

Yee, K. S.

K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antenn. Propag. AP-14, 302–307 (1966).

Yu, Z.

C. C. Baker, J. Heikenfeld, Z. Yu, and A. J. Steckl, “Optical amplification and electroluminescence at 1.54 μm in Er-doped zinc silicate germanate on silicon,” Appl. Phys. Lett. 84(9), 1462–1464 (2004).
[Crossref]

Zheng, M. C.

M. C. Zheng, D. N. Christodoulides, R. Fleischmann, and T. Kottos, “PT optical lattices and universality in beam dynamics,” Phys. Rev. A 82(1), 010103 (2010).
[Crossref]

Appl. Phys. Lett. (1)

C. C. Baker, J. Heikenfeld, Z. Yu, and A. J. Steckl, “Optical amplification and electroluminescence at 1.54 μm in Er-doped zinc silicate germanate on silicon,” Appl. Phys. Lett. 84(9), 1462–1464 (2004).
[Crossref]

IEEE Trans. Antenn. Propag. (1)

K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antenn. Propag. AP-14, 302–307 (1966).

J. Phys. A (1)

O. Bendix, R. Fleischmann, T. Kottos, and B. Shapiro, “Optical structures with local PT-symmetry,” J. Phys. A 43(26), 265305 (2010).
[Crossref]

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 (2012).
[Crossref] [PubMed]

Opt. Express (2)

Opt. Lett. (2)

Phys. Rev. A (4)

S. Longhi and G. Della Valle, “Photonic realization of PT-symmetric quantum field theories,” Phys. Rev. A 85(1), 012112 (2012).
[Crossref]

M. C. Zheng, D. N. Christodoulides, R. Fleischmann, and T. Kottos, “PT optical lattices and universality in beam dynamics,” Phys. Rev. A 82(1), 010103 (2010).
[Crossref]

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

M. Hiller, T. Kottos, and A. Ossipov, “Bifurcations in resonance widths of an open Bose-Hubbard dimer,” Phys. Rev. A 73(6), 063625 (2006).
[Crossref]

Phys. Rev. Lett. (7)

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]

A. Guo, G. J. Salamo, 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(9), 093902 (2009).
[Crossref] [PubMed]

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

N. Hatano and D. R. Nelson, “Localization transitions in non-Hermitian quantum mechanics,” Phys. Rev. Lett. 77(3), 570–573 (1996).
[Crossref] [PubMed]

O. Bendix, R. Fleischmann, T. Kottos, and B. Shapiro, “Exponentially fragile PT symmetry in lattices with localized eigenmodes,” Phys. Rev. Lett. 103(3), 030402 (2009).
[Crossref] [PubMed]

S. Longhi, “Bloch oscillations in complex crystals with PT symmetry,” Phys. Rev. Lett. 103(12), 123601 (2009).
[Crossref] [PubMed]

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam dynamics in PT symmetric optical lattices,” Phys. Rev. Lett. 100(10), 103904 (2008).
[Crossref] [PubMed]

Rep. Prog. Phys. (1)

C. M. Bender, “Making sense of non-Hermitian Hamiltonians,” Rep. Prog. Phys. 70(6), 947–1018 (2007).
[Crossref]

Other (1)

E. D. Palik, Handbook of Optical Constants of Solids (Academic, 1985).

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Figures (5)

Fig. 1
Fig. 1

(a) n 2 derived based on Eq. (5) for unidirectional invisibility of one-dimensional two-layer slab in (b) inset as m = 10, 20, 30 and n 1 = 1.444 . Groups (i) and (ii) correspond to solutions of unidirectional invisibility for left and right incidence, respectively; at singular point A, reflectionlessness happens for both right and left incidences. Based on n 2 , the corresponding effective length (b) L 2 can be obtained from Eq. (5). It is shown that no matter n 2 belongs to group (i) or group (ii) in (a), the same n 2 ' will correspond to a same L 2 .

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Fig. 2
Fig. 2

Reflections depend on n 2 ' of structure with n 2 and corresponding L 2 located at m = 10 in Fig. 1(a) (i) and Fig. 1(b) respectively. Reflections for right and left incidences, i.e., | r R | 2 and | r L | 2 , are illustrated by the blue dotted and red circle curves respectively. The results on linear scale are plotted in inset.

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Fig. 3
Fig. 3

Normalized intensity distribution along z direction when 1550 nm light from (a) left and (b) right is incident on the structure composed of Si and active material ZSG. For 1550 nm, n 1 of Si is approximately 1.444, and n 2 of ZSG is n 2 ' = 1.7514. n 2 ' ' , l 2 , and l 1 are set as −0.03034, 4.6373 μm, and 3.4886 μm respectively, according to curves of m = 10 in Fig. 1(a) (i) and Fig. 1(b) and it demonstrates the realization of unidirectional invisibility for instance. In (a) and (b), pink dashed lines denote the left and right unidirectional incident planes respectively.

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Fig. 4
Fig. 4

Reflections depending on structural parameter n 2 ' with n 2 and corresponding L 2 located at m = 10 in Fig. 1(a) (ii) and Fig. 1(b) respectively. Reflections for right and left incidence, i.e., | r R | 2 and | r L | 2 , are illustrated by the blue dotted and red circle curves respectively. The results on linear scale are plotted in the inset.

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Fig. 5
Fig. 5

Normalized intensity distribution along z direction when 1550 nm light from (a) left and (b) right is incident on the structure composed of Si and active material ZSG. For 1550 nm, n 1 of Si is approximately 1.444, and n 2 of ZSG is n 2 ' = 1.7514. n 2 ' ' , l 2 , and l 1 are set as 0.03034, 4.6373 μm, and 3.4886μm respectively according to curves of m = 10 in Fig. 1(a) (i) and Fig. 1(b) which demonstrate the realization of unidirectional invisibility for instance. In (a) and (b), pink dashed lines denote the left and right unidirectional incident planes respectively.

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Equations (5)

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M i = [ cos δ i − j η i sin δ i   − j sin δ i / η i cos δ i ]   ( i = 1 , 2 ) ,
δ 1 =2pπ+π/2, (p=0,1,2...),
r R r L = ( n 2 / n 1 − n 1 / n 2 )sin δ 2 +j( n 1 / n 0 − n 0 / n 1 )cos δ 2 ( n 1 / n 2 − n 2 / n 1 )sin δ 2 +j( n 1 / n 0 − n 0 / n 1 )cos δ 2 .
e j2 L 2 n 2 = ±( n 2 2 − n 1 2 )+ n 2 ( n 1 2 −1) ±( n 2 2 − n 1 2 )− n 2 ( n 1 2 −1) .
L 2 = 1 j2 n 2 [log ±( n 2 2 − n 1 2 )+ n 2 ( n 1 2 −1) ±( n 2 2 − n 1 2 )− n 2 ( n 1 2 −1) +j2mπ], (m=0, 1, 2...) .

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