Abstract

Asymmetric reflection in Bragg gratings and asymmetric diffraction in diffraction gratings are both linked to parity-time (PT) symmetry in non-Hermitian optics, but their direct relation has not been examined. To fill this gap, we first consider a PT-symmetric sinusoidal grating to compare the contrast of forward and backward reflectivities and the ratio of ±1-order diffraction efficiencies. Analytical and numerical results show that they change with identical tendencies and peaks at same positions in a wide parameter space, indicating thus an intrinsic link in both PT symmetric and PT broken phases. The underlying physics is found to be that the unbalanced coupling strengths between forward and backward reflected waves are identical to those between 0-order and ±1-order diffracted waves. We then consider a non-Hermitian grating dynamically induced in cold atomic lattices to include higher-order diffractions and corresponding reflections.Full numerical calculations show that the aforementioned findings hold also true in this complicated but practical grating, even in more general non-Hermitian cases beyond the exact PT symmetry.

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

Full Article  |  PDF Article
OSA Recommended Articles
Light propagation through a PT-symmetric photonic-crystal

Vladimir V. Konotop and Boris I. Mantsyzov
Opt. Express 24(23) 26146-26151 (2016)

Spectral discrete diffraction with non-Hermitian coupling

Shaolin Ke, Qingjie Liu, Dong Zhao, and Weiwei Liu
J. Opt. Soc. Am. B 35(10) 2387-2393 (2018)

General coupled mode theory in non-Hermitian waveguides

Jing Xu and Yuntian Chen
Opt. Express 23(17) 22619-22627 (2015)

References

  • View by:
  • |
  • |
  • |

  1. 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]
  2. Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, “Optical solitons in PT periodic potentials,” Phys. Rev. Lett. 100(3), 030402 (2008).
    [Crossref] [PubMed]
  3. 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]
  4. R. El-Ganainy, K. G. Makris, M. Khajavikhan, Z. H. Musslimani, S. Rotter, and D. N. Christodoulides, “Non-Hermitian physics and PT symmetry,” Nat. Phys. 14(1), 11–19 (2018).
    [Crossref]
  5. C. M. Bender and S. Boettcher, “Real spectra in non-Hermitian Hamiltonians having PT symmetry,” Phys. Rev. Lett. 80(24), 5243–5246 (1998).
    [Crossref]
  6. 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]
  7. C. M. Bender, D. C. Brody, and H. F. Jones, “Complex extension of quantum mechanics,” Phys. Rev. Lett. 89(27), 270401 (2002).
    [Crossref]
  8. A. Mostafazadeh, “Exact PT-symmetry is equivalent to Hermiticity,” J. Phys. A: Math. Gen. 36(25), 7081–7091 (2003).
    [Crossref]
  9. C. M. Bender, D. C. Brody, H. F. Jones, and B. K. Meister, “Faster than Hermitian quantum mechanics,” Phys. Rev. Lett. 98(4), 040403 (2007).
    [Crossref]
  10. C. M. Bender, S. Boettcher, and P. N. Meisinger, “PT-symmetric quantum mechanics,” J. Math. Phys. 40(5), 2201–2229 (1999).
    [Crossref]
  11. A. Ruschhaupt, F. Delgado, and J. G. Muga, “Physical realization of PT-symmetric potential scattering in a planar slab waveguide,” J. Phys. A: Math. Gen. 38(9), L171–L176 (2005).
    [Crossref]
  12. S. Longhi, “Optical realization of relativistic non-Hermitian quantum mechanics,” Phys. Rev. Lett. 105(1), 013903 (2010).
    [Crossref] [PubMed]
  13. L. Chang, X. Jiang, S. Hua, C. Yang, J. Wen, L. Jiang, G. Li, G. Wang, and M. Xiao, “Parity-time symmetry and variable optical isolation in active-passive-coupled microresonators,” Nat. Photonics 8(7), 524–529 (2014).
    [Crossref]
  14. B. Peng, S. K. Ozdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Nonreciprocal light transmission in parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10(5), 394–398 (2014).
    [Crossref]
  15. T. Eichelkraut, R. Heilmann, S. Weimann, S. Stützer, F. Dreisow, D. N. Christodoulides, S. Nolte, and A. Szameit, “Mobility transition from ballistic to diffusive transport in non-Hermitian lattices,” Nat. Commun. 4(1), 2533 (2013).
    [Crossref] [PubMed]
  16. H. Ramezani, T. Kottos, R. El-Ganainy, and D. N. Christodoulides, “Unidirectional nonlinear PT-symmetric optical structures,” Phys. Rev. A 82(4), 043803 (2010).
    [Crossref]
  17. X. Wang and J.-H. Wu, “Optical PT-symmetry and PT-antisymmetry in coherently driven atomic lattices,” Opt. Express 24(4), 4289–4298 (2016).
    [Crossref] [PubMed]
  18. J.-H. Wu, M. Artoni, and G. C. La Rocca, “Parity-time-antisymmetric atomic lattices without gain,” Phys. Rev. A 91(3), 033811 (2015).
    [Crossref]
  19. J.-H. Wu, M. Artoni, and G. C. La Rocca, “Non-Hermitian degeneracies and unidirectional reflectionless atomic lattices,” Phys. Rev. Lett. 113(12), 123004 (2014).
    [Crossref] [PubMed]
  20. 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]
  21. A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103(9), 093902 (2009).
    [Crossref] [PubMed]
  22. S. Weimann, M. Kremer, Y. Plotnik, Y. Lumer, S. Nolte, K. G. Makris, M. Segev, M. C. Rechtsman, and A. Szameit, “Topologically protected bound states in photonic parity-time-symmetric crystals,” Nat. Mater. 16(4), 433–438 (2017).
    [Crossref]
  23. H. Jing, S. K. Özdemir, X. Y. Lü, J. Zhang, L. Yang, and F. Nori, “PT-symmetric phonon laser,” Phys. Rev. Lett. 113(5), 053604 (2014).
    [Crossref] [PubMed]
  24. S. Longhi, “Parity-time symmetry meets photonics: a new twist in non-Hermitian optics,” EPL 120(6), 64001 (2017).
    [Crossref]
  25. E. Yang, Y. Lu, Y. Wang, Y. Dai, and P. Wang, “Unidirectional reflectionless phenomenon in periodic ternary layered material,” Opt. Express 24(13), 14311–14321 (2016).
    [Crossref] [PubMed]
  26. C. Y. Huang, R. Zhang, J. L. Han, J. Zheng, and J. Q. Xu, “Type-II perfect absorption and amplification modes with controllable bandwidth in combined PT-symmetric and conventional Bragg-grating structures,” Phys. Rev. A 89(2), 023842 (2014).
    [Crossref]
  27. B. Midya, “Supersymmetry-generated one-way-invisible PT-symmetric optical crystals,” Phys. Rev. A 89(3), 032116 (2014).
    [Crossref]
  28. S. Longhi, “Talbot self-imaging in PT-symmetric complex crystals,” Phys. Rev. A 90(4), 043827 (2014).
    [Crossref]
  29. J. T. Cole, K. G. Makris, Z. H. Musslimani, D. N. Christodoulides, and S. Rotter, “Twofold PT symmetry in doubly exponential optical lattices,” Phys. Rev. A 93(1), 013803 (2016).
    [Crossref]
  30. S. Nixon and J. Yang, “Light propagation in periodically modulated complex waveguides,” Phys. Rev. A 91(3), 033807 (2015).
    [Crossref]
  31. A. Sinha and R. Roychoudhury, “Coherent perfect absorption in PT symmetric optical lattice,” Int. J. Theor. Phys. 54(11), 3991–4004 (2015).
    [Crossref]
  32. S. Longhi, “Half-spectral unidirectional invisibility in non-Hermitian periodic optical structures,” Opt. Lett. 40(23), 5694–5697 (2015).
    [Crossref] [PubMed]
  33. Y.-M. Liu, F. Gao, C.-H. Fan, and J.-H. Wu, “Asymmetric light diffraction of an atomic grating with PT symmetry,” Opt. Lett. 42(21), 4283–4286 (2017).
    [Crossref] [PubMed]
  34. M. Kulishov, J. M. Laniel, N. Bélanger, J. Azaña, and D. V. Plant, “Nonreciprocal waveguide Bragg gratings,” Opt. Express 13(8), 3068–3078 (2005).
    [Crossref] [PubMed]
  35. T. Shui, W. X. Yang, S. Liu, L. Li, and Z. Zhu, “Asymmetric diffraction by atomic gratings with optical PT symmetry in the Raman-Nath regime,” Phys. Rev. A 97(3), 033819 (2018).
    [Crossref]
  36. Y. J. Rao, “Recent progress in applications of in-fibre Bragg grating sensors,” Opt. Lasers Eng. 31(4), 297–324 (1999).
    [Crossref]
  37. T. K. Gaylord and M. G. Moharam, “Analysis and applications of optical diffraction by gratings,” Proc. IEEE 73(5), 894–937 (1985).
    [Crossref]
  38. A. Mostafazadeh, “Generalized unitarity and reciprocity relations for PT-symmetric scattering potentials,” J. Phys. A: Math. Theor. 47(50), 505303 (2014).
    [Crossref]
  39. 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]
  40. X.-F. Zhu, Y.-G. Peng, and D.-G. Zhao, “Anisotropic reflection oscillation in periodic multilayer structures of parity-time symmetry,” Opt. Express 22(15), 18401–18411 (2014).
    [Crossref] [PubMed]
  41. M. Kulishov, H. F. Jones, and B. Kress, “Analysis of PT-symmetric volume gratings beyond the paraxial approximation,” Opt. Express 23(7), 9347–9362 (2015).
    [Crossref] [PubMed]
  42. M. Kulishov, H. F. Jones, and B. Kress, “Analysis of unidirectional non-paraxial invisibility of purely reflective PT-symmetric volume gratings,” Opt. Express 23(14), 18694–18711 (2015).
    [Crossref] [PubMed]
  43. A. Ruschhaupt, T. Dowdall, M. A. Simón, and J. G. Muga, “Asymmetric scattering by non-Hermitian potentials,” EPL 120(2), 20001 (2018).
    [Crossref]
  44. V. A. Bushuev, L. V. Dergacheva, and B. I. Mantsyzov, “Asymmetric pendulum effect and transparency change of PT-symmetric photonic crystals under dynamical Bragg diffraction beyond the paraxial approximation,” Phys. Rev. A 95(3), 033843 (2017).
    [Crossref]
  45. S. Longhi, “Invisibility in PT-symmetric complex crystals,” J. Phys. A: Math. Theor. 44(48), 485302 (2011).
    [Crossref]
  46. M. Kulishov and B. Kress, “Free space diffraction on active gratings with balanced phase and gain/loss modulations,” Opt. Express 20(28), 29319–29328 (2012).
    [Crossref]
  47. M. V. Berry, “Lop-sided diffraction by absorbing crystals,” J. Phys. A: Math. Gen. 31(15), 3493–3502 (1998).
    [Crossref]
  48. M. V. Berry and D. H. J. O’Dell, “Diffraction by volume gratings with imaginary potentials,” J. Phys. A: Math. Gen. 31(8), 2093–2101 (1998).
    [Crossref]
  49. 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]
  50. H. Y. Ling, Y. Q. Li, and M. Xiao, “Electromagnetically induced grating: homogeneously broadened medium,” Phys. Rev. A 57(2), 1338–1344 (1998).
    [Crossref]
  51. M. Artoni, G. La Rocca, and F. Bassani, “Resonantly absorbing one-dimensional photonic crystals,” Phys. Rev. E 72(4), 046604 (2005).
    [Crossref]
  52. Y. Zhang, Y.-M. Liu, X.-D. Tian, T.-Y. Zheng, and J.-H. Wu, “Tunable high-order photonic band gaps of ultraviolet light in cold atoms,” Phys. Rev. A 91(1), 013826 (2015).
    [Crossref]
  53. Y. Zhang, Y.-M. Liu, T.-Y. Zheng, and J.-H. Wu, “Light reflector, amplifier, and splitter based on gain-assisted photonic band gaps,” Phys. Rev. A 94(1), 013836 (2016).
    [Crossref]
  54. F. Gao, J.-R. Dong, Y.-M. Liu, Y. Zhang, and J.-H. Wu, “Multiple PT symmetry and tunable scattering behaviors in a heterojunction cavity,” J. Opt. Soc. Am. B 35(9), 2075–2080 (2018).
    [Crossref]
  55. L. L. Sánchez-Soto, J. J. Monzón, A. G. Barriuso, and J. F. Cariũena, “The transfer matrix: a geometrical perspective,” Phys. Rep. 513(4), 191–227 (2012).
    [Crossref]

2018 (4)

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

T. Shui, W. X. Yang, S. Liu, L. Li, and Z. Zhu, “Asymmetric diffraction by atomic gratings with optical PT symmetry in the Raman-Nath regime,” Phys. Rev. A 97(3), 033819 (2018).
[Crossref]

A. Ruschhaupt, T. Dowdall, M. A. Simón, and J. G. Muga, “Asymmetric scattering by non-Hermitian potentials,” EPL 120(2), 20001 (2018).
[Crossref]

F. Gao, J.-R. Dong, Y.-M. Liu, Y. Zhang, and J.-H. Wu, “Multiple PT symmetry and tunable scattering behaviors in a heterojunction cavity,” J. Opt. Soc. Am. B 35(9), 2075–2080 (2018).
[Crossref]

2017 (4)

V. A. Bushuev, L. V. Dergacheva, and B. I. Mantsyzov, “Asymmetric pendulum effect and transparency change of PT-symmetric photonic crystals under dynamical Bragg diffraction beyond the paraxial approximation,” Phys. Rev. A 95(3), 033843 (2017).
[Crossref]

S. Longhi, “Parity-time symmetry meets photonics: a new twist in non-Hermitian optics,” EPL 120(6), 64001 (2017).
[Crossref]

S. Weimann, M. Kremer, Y. Plotnik, Y. Lumer, S. Nolte, K. G. Makris, M. Segev, M. C. Rechtsman, and A. Szameit, “Topologically protected bound states in photonic parity-time-symmetric crystals,” Nat. Mater. 16(4), 433–438 (2017).
[Crossref]

Y.-M. Liu, F. Gao, C.-H. Fan, and J.-H. Wu, “Asymmetric light diffraction of an atomic grating with PT symmetry,” Opt. Lett. 42(21), 4283–4286 (2017).
[Crossref] [PubMed]

2016 (4)

J. T. Cole, K. G. Makris, Z. H. Musslimani, D. N. Christodoulides, and S. Rotter, “Twofold PT symmetry in doubly exponential optical lattices,” Phys. Rev. A 93(1), 013803 (2016).
[Crossref]

E. Yang, Y. Lu, Y. Wang, Y. Dai, and P. Wang, “Unidirectional reflectionless phenomenon in periodic ternary layered material,” Opt. Express 24(13), 14311–14321 (2016).
[Crossref] [PubMed]

X. Wang and J.-H. Wu, “Optical PT-symmetry and PT-antisymmetry in coherently driven atomic lattices,” Opt. Express 24(4), 4289–4298 (2016).
[Crossref] [PubMed]

Y. Zhang, Y.-M. Liu, T.-Y. Zheng, and J.-H. Wu, “Light reflector, amplifier, and splitter based on gain-assisted photonic band gaps,” Phys. Rev. A 94(1), 013836 (2016).
[Crossref]

2015 (7)

Y. Zhang, Y.-M. Liu, X.-D. Tian, T.-Y. Zheng, and J.-H. Wu, “Tunable high-order photonic band gaps of ultraviolet light in cold atoms,” Phys. Rev. A 91(1), 013826 (2015).
[Crossref]

M. Kulishov, H. F. Jones, and B. Kress, “Analysis of PT-symmetric volume gratings beyond the paraxial approximation,” Opt. Express 23(7), 9347–9362 (2015).
[Crossref] [PubMed]

M. Kulishov, H. F. Jones, and B. Kress, “Analysis of unidirectional non-paraxial invisibility of purely reflective PT-symmetric volume gratings,” Opt. Express 23(14), 18694–18711 (2015).
[Crossref] [PubMed]

J.-H. Wu, M. Artoni, and G. C. La Rocca, “Parity-time-antisymmetric atomic lattices without gain,” Phys. Rev. A 91(3), 033811 (2015).
[Crossref]

S. Nixon and J. Yang, “Light propagation in periodically modulated complex waveguides,” Phys. Rev. A 91(3), 033807 (2015).
[Crossref]

A. Sinha and R. Roychoudhury, “Coherent perfect absorption in PT symmetric optical lattice,” Int. J. Theor. Phys. 54(11), 3991–4004 (2015).
[Crossref]

S. Longhi, “Half-spectral unidirectional invisibility in non-Hermitian periodic optical structures,” Opt. Lett. 40(23), 5694–5697 (2015).
[Crossref] [PubMed]

2014 (9)

H. Jing, S. K. Özdemir, X. Y. Lü, J. Zhang, L. Yang, and F. Nori, “PT-symmetric phonon laser,” Phys. Rev. Lett. 113(5), 053604 (2014).
[Crossref] [PubMed]

J.-H. Wu, M. Artoni, and G. C. La Rocca, “Non-Hermitian degeneracies and unidirectional reflectionless atomic lattices,” Phys. Rev. Lett. 113(12), 123004 (2014).
[Crossref] [PubMed]

C. Y. Huang, R. Zhang, J. L. Han, J. Zheng, and J. Q. Xu, “Type-II perfect absorption and amplification modes with controllable bandwidth in combined PT-symmetric and conventional Bragg-grating structures,” Phys. Rev. A 89(2), 023842 (2014).
[Crossref]

B. Midya, “Supersymmetry-generated one-way-invisible PT-symmetric optical crystals,” Phys. Rev. A 89(3), 032116 (2014).
[Crossref]

S. Longhi, “Talbot self-imaging in PT-symmetric complex crystals,” Phys. Rev. A 90(4), 043827 (2014).
[Crossref]

L. Chang, X. Jiang, S. Hua, C. Yang, J. Wen, L. Jiang, G. Li, G. Wang, and M. Xiao, “Parity-time symmetry and variable optical isolation in active-passive-coupled microresonators,” Nat. Photonics 8(7), 524–529 (2014).
[Crossref]

B. Peng, S. K. Ozdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Nonreciprocal light transmission in parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10(5), 394–398 (2014).
[Crossref]

A. Mostafazadeh, “Generalized unitarity and reciprocity relations for PT-symmetric scattering potentials,” J. Phys. A: Math. Theor. 47(50), 505303 (2014).
[Crossref]

X.-F. Zhu, Y.-G. Peng, and D.-G. Zhao, “Anisotropic reflection oscillation in periodic multilayer structures of parity-time symmetry,” Opt. Express 22(15), 18401–18411 (2014).
[Crossref] [PubMed]

2013 (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 (2013).
[Crossref]

T. Eichelkraut, R. Heilmann, S. Weimann, S. Stützer, F. Dreisow, D. N. Christodoulides, S. Nolte, and A. Szameit, “Mobility transition from ballistic to diffusive transport in non-Hermitian lattices,” Nat. Commun. 4(1), 2533 (2013).
[Crossref] [PubMed]

2012 (2)

M. Kulishov and B. Kress, “Free space diffraction on active gratings with balanced phase and gain/loss modulations,” Opt. Express 20(28), 29319–29328 (2012).
[Crossref]

L. L. Sánchez-Soto, J. J. Monzón, A. G. Barriuso, and J. F. Cariũena, “The transfer matrix: a geometrical perspective,” Phys. Rep. 513(4), 191–227 (2012).
[Crossref]

2011 (2)

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]

S. Longhi, “Invisibility in PT-symmetric complex crystals,” J. Phys. A: Math. Theor. 44(48), 485302 (2011).
[Crossref]

2010 (3)

H. Ramezani, T. Kottos, R. El-Ganainy, and D. N. Christodoulides, “Unidirectional nonlinear PT-symmetric optical structures,” Phys. Rev. A 82(4), 043803 (2010).
[Crossref]

S. Longhi, “Optical realization of relativistic non-Hermitian quantum mechanics,” Phys. Rev. Lett. 105(1), 013903 (2010).
[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]

2009 (2)

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103(9), 093902 (2009).
[Crossref] [PubMed]

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]

2008 (2)

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]

Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, “Optical solitons in PT periodic potentials,” Phys. Rev. Lett. 100(3), 030402 (2008).
[Crossref] [PubMed]

2007 (2)

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]

C. M. Bender, D. C. Brody, H. F. Jones, and B. K. Meister, “Faster than Hermitian quantum mechanics,” Phys. Rev. Lett. 98(4), 040403 (2007).
[Crossref]

2005 (3)

A. Ruschhaupt, F. Delgado, and J. G. Muga, “Physical realization of PT-symmetric potential scattering in a planar slab waveguide,” J. Phys. A: Math. Gen. 38(9), L171–L176 (2005).
[Crossref]

M. Kulishov, J. M. Laniel, N. Bélanger, J. Azaña, and D. V. Plant, “Nonreciprocal waveguide Bragg gratings,” Opt. Express 13(8), 3068–3078 (2005).
[Crossref] [PubMed]

M. Artoni, G. La Rocca, and F. Bassani, “Resonantly absorbing one-dimensional photonic crystals,” Phys. Rev. E 72(4), 046604 (2005).
[Crossref]

2003 (1)

A. Mostafazadeh, “Exact PT-symmetry is equivalent to Hermiticity,” J. Phys. A: Math. Gen. 36(25), 7081–7091 (2003).
[Crossref]

2002 (1)

C. M. Bender, D. C. Brody, and H. F. Jones, “Complex extension of quantum mechanics,” Phys. Rev. Lett. 89(27), 270401 (2002).
[Crossref]

1999 (2)

C. M. Bender, S. Boettcher, and P. N. Meisinger, “PT-symmetric quantum mechanics,” J. Math. Phys. 40(5), 2201–2229 (1999).
[Crossref]

Y. J. Rao, “Recent progress in applications of in-fibre Bragg grating sensors,” Opt. Lasers Eng. 31(4), 297–324 (1999).
[Crossref]

1998 (4)

H. Y. Ling, Y. Q. Li, and M. Xiao, “Electromagnetically induced grating: homogeneously broadened medium,” Phys. Rev. A 57(2), 1338–1344 (1998).
[Crossref]

M. V. Berry, “Lop-sided diffraction by absorbing crystals,” J. Phys. A: Math. Gen. 31(15), 3493–3502 (1998).
[Crossref]

M. V. Berry and D. H. J. O’Dell, “Diffraction by volume gratings with imaginary potentials,” J. Phys. A: Math. Gen. 31(8), 2093–2101 (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]

1985 (1)

T. K. Gaylord and M. G. Moharam, “Analysis and applications of optical diffraction by gratings,” Proc. IEEE 73(5), 894–937 (1985).
[Crossref]

Aimez, V.

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103(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 (2013).
[Crossref]

Artoni, M.

J.-H. Wu, M. Artoni, and G. C. La Rocca, “Parity-time-antisymmetric atomic lattices without gain,” Phys. Rev. A 91(3), 033811 (2015).
[Crossref]

J.-H. Wu, M. Artoni, and G. C. La Rocca, “Non-Hermitian degeneracies and unidirectional reflectionless atomic lattices,” Phys. Rev. Lett. 113(12), 123004 (2014).
[Crossref] [PubMed]

M. Artoni, G. La Rocca, and F. Bassani, “Resonantly absorbing one-dimensional photonic crystals,” Phys. Rev. E 72(4), 046604 (2005).
[Crossref]

Azaña, J.

Barriuso, A. G.

L. L. Sánchez-Soto, J. J. Monzón, A. G. Barriuso, and J. F. Cariũena, “The transfer matrix: a geometrical perspective,” Phys. Rep. 513(4), 191–227 (2012).
[Crossref]

Bassani, F.

M. Artoni, G. La Rocca, and F. Bassani, “Resonantly absorbing one-dimensional photonic crystals,” Phys. Rev. E 72(4), 046604 (2005).
[Crossref]

Bélanger, N.

Bender, C. M.

B. Peng, S. K. Ozdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Nonreciprocal light transmission in parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10(5), 394–398 (2014).
[Crossref]

C. M. Bender, D. C. Brody, H. F. Jones, and B. K. Meister, “Faster than Hermitian quantum mechanics,” Phys. Rev. Lett. 98(4), 040403 (2007).
[Crossref]

C. M. Bender, D. C. Brody, and H. F. Jones, “Complex extension of quantum mechanics,” Phys. Rev. Lett. 89(27), 270401 (2002).
[Crossref]

C. M. Bender, S. Boettcher, and P. N. Meisinger, “PT-symmetric quantum mechanics,” J. Math. Phys. 40(5), 2201–2229 (1999).
[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]

Berry, M. V.

M. V. Berry, “Lop-sided diffraction by absorbing crystals,” J. Phys. A: Math. Gen. 31(15), 3493–3502 (1998).
[Crossref]

M. V. Berry and D. H. J. O’Dell, “Diffraction by volume gratings with imaginary potentials,” J. Phys. A: Math. Gen. 31(8), 2093–2101 (1998).
[Crossref]

Boettcher, S.

C. M. Bender, S. Boettcher, and P. N. Meisinger, “PT-symmetric quantum mechanics,” J. Math. Phys. 40(5), 2201–2229 (1999).
[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]

Brody, D. C.

C. M. Bender, D. C. Brody, H. F. Jones, and B. K. Meister, “Faster than Hermitian quantum mechanics,” Phys. Rev. Lett. 98(4), 040403 (2007).
[Crossref]

C. M. Bender, D. C. Brody, and H. F. Jones, “Complex extension of quantum mechanics,” Phys. Rev. Lett. 89(27), 270401 (2002).
[Crossref]

Bushuev, V. A.

V. A. Bushuev, L. V. Dergacheva, and B. I. Mantsyzov, “Asymmetric pendulum effect and transparency change of PT-symmetric photonic crystals under dynamical Bragg diffraction beyond the paraxial approximation,” Phys. Rev. A 95(3), 033843 (2017).
[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]

Cariuena, J. F.

L. L. Sánchez-Soto, J. J. Monzón, A. G. Barriuso, and J. F. Cariũena, “The transfer matrix: a geometrical perspective,” Phys. Rep. 513(4), 191–227 (2012).
[Crossref]

Chang, L.

L. Chang, X. Jiang, S. Hua, C. Yang, J. Wen, L. Jiang, G. Li, G. Wang, and M. Xiao, “Parity-time symmetry and variable optical isolation in active-passive-coupled microresonators,” Nat. Photonics 8(7), 524–529 (2014).
[Crossref]

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]

Christodoulides, D. N.

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

J. T. Cole, K. G. Makris, Z. H. Musslimani, D. N. Christodoulides, and S. Rotter, “Twofold PT symmetry in doubly exponential optical lattices,” Phys. Rev. A 93(1), 013803 (2016).
[Crossref]

T. Eichelkraut, R. Heilmann, S. Weimann, S. Stützer, F. Dreisow, D. N. Christodoulides, S. Nolte, and A. Szameit, “Mobility transition from ballistic to diffusive transport in non-Hermitian lattices,” Nat. Commun. 4(1), 2533 (2013).
[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]

H. Ramezani, T. Kottos, R. El-Ganainy, and D. N. Christodoulides, “Unidirectional nonlinear PT-symmetric optical structures,” Phys. Rev. A 82(4), 043803 (2010).
[Crossref]

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]

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103(9), 093902 (2009).
[Crossref] [PubMed]

Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, “Optical solitons in PT periodic potentials,” Phys. Rev. Lett. 100(3), 030402 (2008).
[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]

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]

Cole, J. T.

J. T. Cole, K. G. Makris, Z. H. Musslimani, D. N. Christodoulides, and S. Rotter, “Twofold PT symmetry in doubly exponential optical lattices,” Phys. Rev. A 93(1), 013803 (2016).
[Crossref]

Dai, Y.

Delgado, F.

A. Ruschhaupt, F. Delgado, and J. G. Muga, “Physical realization of PT-symmetric potential scattering in a planar slab waveguide,” J. Phys. A: Math. Gen. 38(9), L171–L176 (2005).
[Crossref]

Dergacheva, L. V.

V. A. Bushuev, L. V. Dergacheva, and B. I. Mantsyzov, “Asymmetric pendulum effect and transparency change of PT-symmetric photonic crystals under dynamical Bragg diffraction beyond the paraxial approximation,” Phys. Rev. A 95(3), 033843 (2017).
[Crossref]

Dong, J.-R.

Dowdall, T.

A. Ruschhaupt, T. Dowdall, M. A. Simón, and J. G. Muga, “Asymmetric scattering by non-Hermitian potentials,” EPL 120(2), 20001 (2018).
[Crossref]

Dreisow, F.

T. Eichelkraut, R. Heilmann, S. Weimann, S. Stützer, F. Dreisow, D. N. Christodoulides, S. Nolte, and A. Szameit, “Mobility transition from ballistic to diffusive transport in non-Hermitian lattices,” Nat. Commun. 4(1), 2533 (2013).
[Crossref] [PubMed]

Duchesne, D.

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103(9), 093902 (2009).
[Crossref] [PubMed]

Eichelkraut, T.

T. Eichelkraut, R. Heilmann, S. Weimann, S. Stützer, F. Dreisow, D. N. Christodoulides, S. Nolte, and A. Szameit, “Mobility transition from ballistic to diffusive transport in non-Hermitian lattices,” Nat. Commun. 4(1), 2533 (2013).
[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]

El-Ganainy, R.

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

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]

H. Ramezani, T. Kottos, R. El-Ganainy, and D. N. Christodoulides, “Unidirectional nonlinear PT-symmetric optical structures,” Phys. Rev. A 82(4), 043803 (2010).
[Crossref]

Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, “Optical solitons in PT periodic potentials,” Phys. Rev. Lett. 100(3), 030402 (2008).
[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]

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]

Fan, C.-H.

Fan, S.

B. Peng, S. K. Ozdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Nonreciprocal light transmission in parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10(5), 394–398 (2014).
[Crossref]

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]

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 (2013).
[Crossref]

Gao, F.

Gaylord, T. K.

T. K. Gaylord and M. G. Moharam, “Analysis and applications of optical diffraction by gratings,” Proc. IEEE 73(5), 894–937 (1985).
[Crossref]

Gianfreda, M.

B. Peng, S. K. Ozdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Nonreciprocal light transmission in parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10(5), 394–398 (2014).
[Crossref]

Guo, A.

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103(9), 093902 (2009).
[Crossref] [PubMed]

Han, J. L.

C. Y. Huang, R. Zhang, J. L. Han, J. Zheng, and J. Q. Xu, “Type-II perfect absorption and amplification modes with controllable bandwidth in combined PT-symmetric and conventional Bragg-grating structures,” Phys. Rev. A 89(2), 023842 (2014).
[Crossref]

Heilmann, R.

T. Eichelkraut, R. Heilmann, S. Weimann, S. Stützer, F. Dreisow, D. N. Christodoulides, S. Nolte, and A. Szameit, “Mobility transition from ballistic to diffusive transport in non-Hermitian lattices,” Nat. Commun. 4(1), 2533 (2013).
[Crossref] [PubMed]

Hua, S.

L. Chang, X. Jiang, S. Hua, C. Yang, J. Wen, L. Jiang, G. Li, G. Wang, and M. Xiao, “Parity-time symmetry and variable optical isolation in active-passive-coupled microresonators,” Nat. Photonics 8(7), 524–529 (2014).
[Crossref]

Huang, C. Y.

C. Y. Huang, R. Zhang, J. L. Han, J. Zheng, and J. Q. Xu, “Type-II perfect absorption and amplification modes with controllable bandwidth in combined PT-symmetric and conventional Bragg-grating structures,” Phys. Rev. A 89(2), 023842 (2014).
[Crossref]

Jiang, L.

L. Chang, X. Jiang, S. Hua, C. Yang, J. Wen, L. Jiang, G. Li, G. Wang, and M. Xiao, “Parity-time symmetry and variable optical isolation in active-passive-coupled microresonators,” Nat. Photonics 8(7), 524–529 (2014).
[Crossref]

Jiang, X.

L. Chang, X. Jiang, S. Hua, C. Yang, J. Wen, L. Jiang, G. Li, G. Wang, and M. Xiao, “Parity-time symmetry and variable optical isolation in active-passive-coupled microresonators,” Nat. Photonics 8(7), 524–529 (2014).
[Crossref]

Jing, H.

H. Jing, S. K. Özdemir, X. Y. Lü, J. Zhang, L. Yang, and F. Nori, “PT-symmetric phonon laser,” Phys. Rev. Lett. 113(5), 053604 (2014).
[Crossref] [PubMed]

Jones, H. F.

M. Kulishov, H. F. Jones, and B. Kress, “Analysis of unidirectional non-paraxial invisibility of purely reflective PT-symmetric volume gratings,” Opt. Express 23(14), 18694–18711 (2015).
[Crossref] [PubMed]

M. Kulishov, H. F. Jones, and B. Kress, “Analysis of PT-symmetric volume gratings beyond the paraxial approximation,” Opt. Express 23(7), 9347–9362 (2015).
[Crossref] [PubMed]

C. M. Bender, D. C. Brody, H. F. Jones, and B. K. Meister, “Faster than Hermitian quantum mechanics,” Phys. Rev. Lett. 98(4), 040403 (2007).
[Crossref]

C. M. Bender, D. C. Brody, and H. F. Jones, “Complex extension of quantum mechanics,” Phys. Rev. Lett. 89(27), 270401 (2002).
[Crossref]

Khajavikhan, M.

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

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]

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]

H. Ramezani, T. Kottos, R. El-Ganainy, and D. N. Christodoulides, “Unidirectional nonlinear PT-symmetric optical structures,” Phys. Rev. A 82(4), 043803 (2010).
[Crossref]

Kremer, M.

S. Weimann, M. Kremer, Y. Plotnik, Y. Lumer, S. Nolte, K. G. Makris, M. Segev, M. C. Rechtsman, and A. Szameit, “Topologically protected bound states in photonic parity-time-symmetric crystals,” Nat. Mater. 16(4), 433–438 (2017).
[Crossref]

Kress, B.

Kulishov, M.

Laniel, J. M.

Lei, F.

B. Peng, S. K. Ozdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Nonreciprocal light transmission in parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10(5), 394–398 (2014).
[Crossref]

Li, G.

L. Chang, X. Jiang, S. Hua, C. Yang, J. Wen, L. Jiang, G. Li, G. Wang, and M. Xiao, “Parity-time symmetry and variable optical isolation in active-passive-coupled microresonators,” Nat. Photonics 8(7), 524–529 (2014).
[Crossref]

Li, L.

T. Shui, W. X. Yang, S. Liu, L. Li, and Z. Zhu, “Asymmetric diffraction by atomic gratings with optical PT symmetry in the Raman-Nath regime,” Phys. Rev. A 97(3), 033819 (2018).
[Crossref]

Li, Y. Q.

H. Y. Ling, Y. Q. Li, and M. Xiao, “Electromagnetically induced grating: homogeneously broadened medium,” Phys. Rev. A 57(2), 1338–1344 (1998).
[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]

Ling, H. Y.

H. Y. Ling, Y. Q. Li, and M. Xiao, “Electromagnetically induced grating: homogeneously broadened medium,” Phys. Rev. A 57(2), 1338–1344 (1998).
[Crossref]

Liu, S.

T. Shui, W. X. Yang, S. Liu, L. Li, and Z. Zhu, “Asymmetric diffraction by atomic gratings with optical PT symmetry in the Raman-Nath regime,” Phys. Rev. A 97(3), 033819 (2018).
[Crossref]

Liu, Y.-M.

F. Gao, J.-R. Dong, Y.-M. Liu, Y. Zhang, and J.-H. Wu, “Multiple PT symmetry and tunable scattering behaviors in a heterojunction cavity,” J. Opt. Soc. Am. B 35(9), 2075–2080 (2018).
[Crossref]

Y.-M. Liu, F. Gao, C.-H. Fan, and J.-H. Wu, “Asymmetric light diffraction of an atomic grating with PT symmetry,” Opt. Lett. 42(21), 4283–4286 (2017).
[Crossref] [PubMed]

Y. Zhang, Y.-M. Liu, T.-Y. Zheng, and J.-H. Wu, “Light reflector, amplifier, and splitter based on gain-assisted photonic band gaps,” Phys. Rev. A 94(1), 013836 (2016).
[Crossref]

Y. Zhang, Y.-M. Liu, X.-D. Tian, T.-Y. Zheng, and J.-H. Wu, “Tunable high-order photonic band gaps of ultraviolet light in cold atoms,” Phys. Rev. A 91(1), 013826 (2015).
[Crossref]

Long, G. L.

B. Peng, S. K. Ozdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Nonreciprocal light transmission in parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10(5), 394–398 (2014).
[Crossref]

Longhi, S.

S. Longhi, “Parity-time symmetry meets photonics: a new twist in non-Hermitian optics,” EPL 120(6), 64001 (2017).
[Crossref]

S. Longhi, “Half-spectral unidirectional invisibility in non-Hermitian periodic optical structures,” Opt. Lett. 40(23), 5694–5697 (2015).
[Crossref] [PubMed]

S. Longhi, “Talbot self-imaging in PT-symmetric complex crystals,” Phys. Rev. A 90(4), 043827 (2014).
[Crossref]

S. Longhi, “Invisibility in PT-symmetric complex crystals,” J. Phys. A: Math. Theor. 44(48), 485302 (2011).
[Crossref]

S. Longhi, “Optical realization of relativistic non-Hermitian quantum mechanics,” Phys. Rev. Lett. 105(1), 013903 (2010).
[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 (2013).
[Crossref]

Lu, Y.

Lü, X. Y.

H. Jing, S. K. Özdemir, X. Y. Lü, J. Zhang, L. Yang, and F. Nori, “PT-symmetric phonon laser,” Phys. Rev. Lett. 113(5), 053604 (2014).
[Crossref] [PubMed]

Lumer, Y.

S. Weimann, M. Kremer, Y. Plotnik, Y. Lumer, S. Nolte, K. G. Makris, M. Segev, M. C. Rechtsman, and A. Szameit, “Topologically protected bound states in photonic parity-time-symmetric crystals,” Nat. Mater. 16(4), 433–438 (2017).
[Crossref]

Makris, K. G.

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

S. Weimann, M. Kremer, Y. Plotnik, Y. Lumer, S. Nolte, K. G. Makris, M. Segev, M. C. Rechtsman, and A. Szameit, “Topologically protected bound states in photonic parity-time-symmetric crystals,” Nat. Mater. 16(4), 433–438 (2017).
[Crossref]

J. T. Cole, K. G. Makris, Z. H. Musslimani, D. N. Christodoulides, and S. Rotter, “Twofold PT symmetry in doubly exponential optical lattices,” Phys. Rev. A 93(1), 013803 (2016).
[Crossref]

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]

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]

Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, “Optical solitons in PT periodic potentials,” Phys. Rev. Lett. 100(3), 030402 (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]

Mantsyzov, B. I.

V. A. Bushuev, L. V. Dergacheva, and B. I. Mantsyzov, “Asymmetric pendulum effect and transparency change of PT-symmetric photonic crystals under dynamical Bragg diffraction beyond the paraxial approximation,” Phys. Rev. A 95(3), 033843 (2017).
[Crossref]

Meisinger, P. N.

C. M. Bender, S. Boettcher, and P. N. Meisinger, “PT-symmetric quantum mechanics,” J. Math. Phys. 40(5), 2201–2229 (1999).
[Crossref]

Meister, B. K.

C. M. Bender, D. C. Brody, H. F. Jones, and B. K. Meister, “Faster than Hermitian quantum mechanics,” Phys. Rev. Lett. 98(4), 040403 (2007).
[Crossref]

Midya, B.

B. Midya, “Supersymmetry-generated one-way-invisible PT-symmetric optical crystals,” Phys. Rev. A 89(3), 032116 (2014).
[Crossref]

Moharam, M. G.

T. K. Gaylord and M. G. Moharam, “Analysis and applications of optical diffraction by gratings,” Proc. IEEE 73(5), 894–937 (1985).
[Crossref]

Monifi, F.

B. Peng, S. K. Ozdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Nonreciprocal light transmission in parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10(5), 394–398 (2014).
[Crossref]

Monzón, J. J.

L. L. Sánchez-Soto, J. J. Monzón, A. G. Barriuso, and J. F. Cariũena, “The transfer matrix: a geometrical perspective,” Phys. Rep. 513(4), 191–227 (2012).
[Crossref]

Morandotti, R.

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103(9), 093902 (2009).
[Crossref] [PubMed]

Mostafazadeh, A.

A. Mostafazadeh, “Generalized unitarity and reciprocity relations for PT-symmetric scattering potentials,” J. Phys. A: Math. Theor. 47(50), 505303 (2014).
[Crossref]

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]

A. Mostafazadeh, “Exact PT-symmetry is equivalent to Hermiticity,” J. Phys. A: Math. Gen. 36(25), 7081–7091 (2003).
[Crossref]

Muga, J. G.

A. Ruschhaupt, T. Dowdall, M. A. Simón, and J. G. Muga, “Asymmetric scattering by non-Hermitian potentials,” EPL 120(2), 20001 (2018).
[Crossref]

A. Ruschhaupt, F. Delgado, and J. G. Muga, “Physical realization of PT-symmetric potential scattering in a planar slab waveguide,” J. Phys. A: Math. Gen. 38(9), L171–L176 (2005).
[Crossref]

Musslimani, Z. H.

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

J. T. Cole, K. G. Makris, Z. H. Musslimani, D. N. Christodoulides, and S. Rotter, “Twofold PT symmetry in doubly exponential optical lattices,” Phys. Rev. A 93(1), 013803 (2016).
[Crossref]

Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, “Optical solitons in PT periodic potentials,” Phys. Rev. Lett. 100(3), 030402 (2008).
[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]

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]

Nixon, S.

S. Nixon and J. Yang, “Light propagation in periodically modulated complex waveguides,” Phys. Rev. A 91(3), 033807 (2015).
[Crossref]

Nolte, S.

S. Weimann, M. Kremer, Y. Plotnik, Y. Lumer, S. Nolte, K. G. Makris, M. Segev, M. C. Rechtsman, and A. Szameit, “Topologically protected bound states in photonic parity-time-symmetric crystals,” Nat. Mater. 16(4), 433–438 (2017).
[Crossref]

T. Eichelkraut, R. Heilmann, S. Weimann, S. Stützer, F. Dreisow, D. N. Christodoulides, S. Nolte, and A. Szameit, “Mobility transition from ballistic to diffusive transport in non-Hermitian lattices,” Nat. Commun. 4(1), 2533 (2013).
[Crossref] [PubMed]

Nori, F.

B. Peng, S. K. Ozdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Nonreciprocal light transmission in parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10(5), 394–398 (2014).
[Crossref]

H. Jing, S. K. Özdemir, X. Y. Lü, J. Zhang, L. Yang, and F. Nori, “PT-symmetric phonon laser,” Phys. Rev. Lett. 113(5), 053604 (2014).
[Crossref] [PubMed]

O’Dell, D. H. J.

M. V. Berry and D. H. J. O’Dell, “Diffraction by volume gratings with imaginary potentials,” J. Phys. A: Math. Gen. 31(8), 2093–2101 (1998).
[Crossref]

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]

Ozdemir, S. K.

B. Peng, S. K. Ozdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Nonreciprocal light transmission in parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10(5), 394–398 (2014).
[Crossref]

Özdemir, S. K.

H. Jing, S. K. Özdemir, X. Y. Lü, J. Zhang, L. Yang, and F. Nori, “PT-symmetric phonon laser,” Phys. Rev. Lett. 113(5), 053604 (2014).
[Crossref] [PubMed]

Peng, B.

B. Peng, S. K. Ozdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Nonreciprocal light transmission in parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10(5), 394–398 (2014).
[Crossref]

Peng, Y.-G.

Plant, D. V.

Plotnik, Y.

S. Weimann, M. Kremer, Y. Plotnik, Y. Lumer, S. Nolte, K. G. Makris, M. Segev, M. C. Rechtsman, and A. Szameit, “Topologically protected bound states in photonic parity-time-symmetric crystals,” Nat. Mater. 16(4), 433–438 (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]

H. Ramezani, T. Kottos, R. El-Ganainy, and D. N. Christodoulides, “Unidirectional nonlinear PT-symmetric optical structures,” Phys. Rev. A 82(4), 043803 (2010).
[Crossref]

Rao, Y. J.

Y. J. Rao, “Recent progress in applications of in-fibre Bragg grating sensors,” Opt. Lasers Eng. 31(4), 297–324 (1999).
[Crossref]

Rechtsman, M. C.

S. Weimann, M. Kremer, Y. Plotnik, Y. Lumer, S. Nolte, K. G. Makris, M. Segev, M. C. Rechtsman, and A. Szameit, “Topologically protected bound states in photonic parity-time-symmetric crystals,” Nat. Mater. 16(4), 433–438 (2017).
[Crossref]

Rocca, G. C. La

J.-H. Wu, M. Artoni, and G. C. La Rocca, “Parity-time-antisymmetric atomic lattices without gain,” Phys. Rev. A 91(3), 033811 (2015).
[Crossref]

J.-H. Wu, M. Artoni, and G. C. La Rocca, “Non-Hermitian degeneracies and unidirectional reflectionless atomic lattices,” Phys. Rev. Lett. 113(12), 123004 (2014).
[Crossref] [PubMed]

Rocca, G. La

M. Artoni, G. La Rocca, and F. Bassani, “Resonantly absorbing one-dimensional photonic crystals,” Phys. Rev. E 72(4), 046604 (2005).
[Crossref]

Rotter, S.

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

J. T. Cole, K. G. Makris, Z. H. Musslimani, D. N. Christodoulides, and S. Rotter, “Twofold PT symmetry in doubly exponential optical lattices,” Phys. Rev. A 93(1), 013803 (2016).
[Crossref]

Roychoudhury, R.

A. Sinha and R. Roychoudhury, “Coherent perfect absorption in PT symmetric optical lattice,” Int. J. Theor. Phys. 54(11), 3991–4004 (2015).
[Crossref]

Ruschhaupt, A.

A. Ruschhaupt, T. Dowdall, M. A. Simón, and J. G. Muga, “Asymmetric scattering by non-Hermitian potentials,” EPL 120(2), 20001 (2018).
[Crossref]

A. Ruschhaupt, F. Delgado, and J. G. Muga, “Physical realization of PT-symmetric potential scattering in a planar slab waveguide,” J. Phys. A: Math. Gen. 38(9), L171–L176 (2005).
[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]

Salamo, G. J.

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103(9), 093902 (2009).
[Crossref] [PubMed]

Sánchez-Soto, L. L.

L. L. Sánchez-Soto, J. J. Monzón, A. G. Barriuso, and J. F. Cariũena, “The transfer matrix: a geometrical perspective,” Phys. Rep. 513(4), 191–227 (2012).
[Crossref]

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]

Segev, M.

S. Weimann, M. Kremer, Y. Plotnik, Y. Lumer, S. Nolte, K. G. Makris, M. Segev, M. C. Rechtsman, and A. Szameit, “Topologically protected bound states in photonic parity-time-symmetric crystals,” Nat. Mater. 16(4), 433–438 (2017).
[Crossref]

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]

Shui, T.

T. Shui, W. X. Yang, S. Liu, L. Li, and Z. Zhu, “Asymmetric diffraction by atomic gratings with optical PT symmetry in the Raman-Nath regime,” Phys. Rev. A 97(3), 033819 (2018).
[Crossref]

Simón, M. A.

A. Ruschhaupt, T. Dowdall, M. A. Simón, and J. G. Muga, “Asymmetric scattering by non-Hermitian potentials,” EPL 120(2), 20001 (2018).
[Crossref]

Sinha, A.

A. Sinha and R. Roychoudhury, “Coherent perfect absorption in PT symmetric optical lattice,” Int. J. Theor. Phys. 54(11), 3991–4004 (2015).
[Crossref]

Siviloglou, G. A.

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103(9), 093902 (2009).
[Crossref] [PubMed]

Stützer, S.

T. Eichelkraut, R. Heilmann, S. Weimann, S. Stützer, F. Dreisow, D. N. Christodoulides, S. Nolte, and A. Szameit, “Mobility transition from ballistic to diffusive transport in non-Hermitian lattices,” Nat. Commun. 4(1), 2533 (2013).
[Crossref] [PubMed]

Szameit, A.

S. Weimann, M. Kremer, Y. Plotnik, Y. Lumer, S. Nolte, K. G. Makris, M. Segev, M. C. Rechtsman, and A. Szameit, “Topologically protected bound states in photonic parity-time-symmetric crystals,” Nat. Mater. 16(4), 433–438 (2017).
[Crossref]

T. Eichelkraut, R. Heilmann, S. Weimann, S. Stützer, F. Dreisow, D. N. Christodoulides, S. Nolte, and A. Szameit, “Mobility transition from ballistic to diffusive transport in non-Hermitian lattices,” Nat. Commun. 4(1), 2533 (2013).
[Crossref] [PubMed]

Tian, X.-D.

Y. Zhang, Y.-M. Liu, X.-D. Tian, T.-Y. Zheng, and J.-H. Wu, “Tunable high-order photonic band gaps of ultraviolet light in cold atoms,” Phys. Rev. A 91(1), 013826 (2015).
[Crossref]

Volatier-Ravat, M.

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103(9), 093902 (2009).
[Crossref] [PubMed]

Wang, G.

L. Chang, X. Jiang, S. Hua, C. Yang, J. Wen, L. Jiang, G. Li, G. Wang, and M. Xiao, “Parity-time symmetry and variable optical isolation in active-passive-coupled microresonators,” Nat. Photonics 8(7), 524–529 (2014).
[Crossref]

Wang, P.

Wang, X.

Wang, Y.

Weimann, S.

S. Weimann, M. Kremer, Y. Plotnik, Y. Lumer, S. Nolte, K. G. Makris, M. Segev, M. C. Rechtsman, and A. Szameit, “Topologically protected bound states in photonic parity-time-symmetric crystals,” Nat. Mater. 16(4), 433–438 (2017).
[Crossref]

T. Eichelkraut, R. Heilmann, S. Weimann, S. Stützer, F. Dreisow, D. N. Christodoulides, S. Nolte, and A. Szameit, “Mobility transition from ballistic to diffusive transport in non-Hermitian lattices,” Nat. Commun. 4(1), 2533 (2013).
[Crossref] [PubMed]

Wen, J.

L. Chang, X. Jiang, S. Hua, C. Yang, J. Wen, L. Jiang, G. Li, G. Wang, and M. Xiao, “Parity-time symmetry and variable optical isolation in active-passive-coupled microresonators,” Nat. Photonics 8(7), 524–529 (2014).
[Crossref]

Wu, J.-H.

F. Gao, J.-R. Dong, Y.-M. Liu, Y. Zhang, and J.-H. Wu, “Multiple PT symmetry and tunable scattering behaviors in a heterojunction cavity,” J. Opt. Soc. Am. B 35(9), 2075–2080 (2018).
[Crossref]

Y.-M. Liu, F. Gao, C.-H. Fan, and J.-H. Wu, “Asymmetric light diffraction of an atomic grating with PT symmetry,” Opt. Lett. 42(21), 4283–4286 (2017).
[Crossref] [PubMed]

X. Wang and J.-H. Wu, “Optical PT-symmetry and PT-antisymmetry in coherently driven atomic lattices,” Opt. Express 24(4), 4289–4298 (2016).
[Crossref] [PubMed]

Y. Zhang, Y.-M. Liu, T.-Y. Zheng, and J.-H. Wu, “Light reflector, amplifier, and splitter based on gain-assisted photonic band gaps,” Phys. Rev. A 94(1), 013836 (2016).
[Crossref]

Y. Zhang, Y.-M. Liu, X.-D. Tian, T.-Y. Zheng, and J.-H. Wu, “Tunable high-order photonic band gaps of ultraviolet light in cold atoms,” Phys. Rev. A 91(1), 013826 (2015).
[Crossref]

J.-H. Wu, M. Artoni, and G. C. La Rocca, “Parity-time-antisymmetric atomic lattices without gain,” Phys. Rev. A 91(3), 033811 (2015).
[Crossref]

J.-H. Wu, M. Artoni, and G. C. La Rocca, “Non-Hermitian degeneracies and unidirectional reflectionless atomic lattices,” Phys. Rev. Lett. 113(12), 123004 (2014).
[Crossref] [PubMed]

Xiao, M.

L. Chang, X. Jiang, S. Hua, C. Yang, J. Wen, L. Jiang, G. Li, G. Wang, and M. Xiao, “Parity-time symmetry and variable optical isolation in active-passive-coupled microresonators,” Nat. Photonics 8(7), 524–529 (2014).
[Crossref]

H. Y. Ling, Y. Q. Li, and M. Xiao, “Electromagnetically induced grating: homogeneously broadened medium,” Phys. Rev. A 57(2), 1338–1344 (1998).
[Crossref]

Xu, J. Q.

C. Y. Huang, R. Zhang, J. L. Han, J. Zheng, and J. Q. Xu, “Type-II perfect absorption and amplification modes with controllable bandwidth in combined PT-symmetric and conventional Bragg-grating structures,” Phys. Rev. A 89(2), 023842 (2014).
[Crossref]

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]

Yang, C.

L. Chang, X. Jiang, S. Hua, C. Yang, J. Wen, L. Jiang, G. Li, G. Wang, and M. Xiao, “Parity-time symmetry and variable optical isolation in active-passive-coupled microresonators,” Nat. Photonics 8(7), 524–529 (2014).
[Crossref]

Yang, E.

Yang, J.

S. Nixon and J. Yang, “Light propagation in periodically modulated complex waveguides,” Phys. Rev. A 91(3), 033807 (2015).
[Crossref]

Yang, L.

H. Jing, S. K. Özdemir, X. Y. Lü, J. Zhang, L. Yang, and F. Nori, “PT-symmetric phonon laser,” Phys. Rev. Lett. 113(5), 053604 (2014).
[Crossref] [PubMed]

B. Peng, S. K. Ozdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Nonreciprocal light transmission in parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10(5), 394–398 (2014).
[Crossref]

Yang, W. X.

T. Shui, W. X. Yang, S. Liu, L. Li, and Z. Zhu, “Asymmetric diffraction by atomic gratings with optical PT symmetry in the Raman-Nath regime,” Phys. Rev. A 97(3), 033819 (2018).
[Crossref]

Zhang, J.

H. Jing, S. K. Özdemir, X. Y. Lü, J. Zhang, L. Yang, and F. Nori, “PT-symmetric phonon laser,” Phys. Rev. Lett. 113(5), 053604 (2014).
[Crossref] [PubMed]

Zhang, R.

C. Y. Huang, R. Zhang, J. L. Han, J. Zheng, and J. Q. Xu, “Type-II perfect absorption and amplification modes with controllable bandwidth in combined PT-symmetric and conventional Bragg-grating structures,” Phys. Rev. A 89(2), 023842 (2014).
[Crossref]

Zhang, Y.

F. Gao, J.-R. Dong, Y.-M. Liu, Y. Zhang, and J.-H. Wu, “Multiple PT symmetry and tunable scattering behaviors in a heterojunction cavity,” J. Opt. Soc. Am. B 35(9), 2075–2080 (2018).
[Crossref]

Y. Zhang, Y.-M. Liu, T.-Y. Zheng, and J.-H. Wu, “Light reflector, amplifier, and splitter based on gain-assisted photonic band gaps,” Phys. Rev. A 94(1), 013836 (2016).
[Crossref]

Y. Zhang, Y.-M. Liu, X.-D. Tian, T.-Y. Zheng, and J.-H. Wu, “Tunable high-order photonic band gaps of ultraviolet light in cold atoms,” Phys. Rev. A 91(1), 013826 (2015).
[Crossref]

Zhao, D.-G.

Zheng, J.

C. Y. Huang, R. Zhang, J. L. Han, J. Zheng, and J. Q. Xu, “Type-II perfect absorption and amplification modes with controllable bandwidth in combined PT-symmetric and conventional Bragg-grating structures,” Phys. Rev. A 89(2), 023842 (2014).
[Crossref]

Zheng, T.-Y.

Y. Zhang, Y.-M. Liu, T.-Y. Zheng, and J.-H. Wu, “Light reflector, amplifier, and splitter based on gain-assisted photonic band gaps,” Phys. Rev. A 94(1), 013836 (2016).
[Crossref]

Y. Zhang, Y.-M. Liu, X.-D. Tian, T.-Y. Zheng, and J.-H. Wu, “Tunable high-order photonic band gaps of ultraviolet light in cold atoms,” Phys. Rev. A 91(1), 013826 (2015).
[Crossref]

Zhu, X.-F.

Zhu, Z.

T. Shui, W. X. Yang, S. Liu, L. Li, and Z. Zhu, “Asymmetric diffraction by atomic gratings with optical PT symmetry in the Raman-Nath regime,” Phys. Rev. A 97(3), 033819 (2018).
[Crossref]

EPL (2)

S. Longhi, “Parity-time symmetry meets photonics: a new twist in non-Hermitian optics,” EPL 120(6), 64001 (2017).
[Crossref]

A. Ruschhaupt, T. Dowdall, M. A. Simón, and J. G. Muga, “Asymmetric scattering by non-Hermitian potentials,” EPL 120(2), 20001 (2018).
[Crossref]

Int. J. Theor. Phys. (1)

A. Sinha and R. Roychoudhury, “Coherent perfect absorption in PT symmetric optical lattice,” Int. J. Theor. Phys. 54(11), 3991–4004 (2015).
[Crossref]

J. Math. Phys. (1)

C. M. Bender, S. Boettcher, and P. N. Meisinger, “PT-symmetric quantum mechanics,” J. Math. Phys. 40(5), 2201–2229 (1999).
[Crossref]

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

J. Phys. A: Math. Gen. (4)

M. V. Berry, “Lop-sided diffraction by absorbing crystals,” J. Phys. A: Math. Gen. 31(15), 3493–3502 (1998).
[Crossref]

M. V. Berry and D. H. J. O’Dell, “Diffraction by volume gratings with imaginary potentials,” J. Phys. A: Math. Gen. 31(8), 2093–2101 (1998).
[Crossref]

A. Ruschhaupt, F. Delgado, and J. G. Muga, “Physical realization of PT-symmetric potential scattering in a planar slab waveguide,” J. Phys. A: Math. Gen. 38(9), L171–L176 (2005).
[Crossref]

A. Mostafazadeh, “Exact PT-symmetry is equivalent to Hermiticity,” J. Phys. A: Math. Gen. 36(25), 7081–7091 (2003).
[Crossref]

J. Phys. A: Math. Theor. (2)

A. Mostafazadeh, “Generalized unitarity and reciprocity relations for PT-symmetric scattering potentials,” J. Phys. A: Math. Theor. 47(50), 505303 (2014).
[Crossref]

S. Longhi, “Invisibility in PT-symmetric complex crystals,” J. Phys. A: Math. Theor. 44(48), 485302 (2011).
[Crossref]

Nat. Commun. (1)

T. Eichelkraut, R. Heilmann, S. Weimann, S. Stützer, F. Dreisow, D. N. Christodoulides, S. Nolte, and A. Szameit, “Mobility transition from ballistic to diffusive transport in non-Hermitian lattices,” Nat. Commun. 4(1), 2533 (2013).
[Crossref] [PubMed]

Nat. Mater. (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 (2013).
[Crossref]

S. Weimann, M. Kremer, Y. Plotnik, Y. Lumer, S. Nolte, K. G. Makris, M. Segev, M. C. Rechtsman, and A. Szameit, “Topologically protected bound states in photonic parity-time-symmetric crystals,” Nat. Mater. 16(4), 433–438 (2017).
[Crossref]

Nat. Photonics (1)

L. Chang, X. Jiang, S. Hua, C. Yang, J. Wen, L. Jiang, G. Li, G. Wang, and M. Xiao, “Parity-time symmetry and variable optical isolation in active-passive-coupled microresonators,” Nat. Photonics 8(7), 524–529 (2014).
[Crossref]

Nat. Phys. (3)

B. Peng, S. K. Ozdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Nonreciprocal light transmission in parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10(5), 394–398 (2014).
[Crossref]

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

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]

Opt. Express (7)

Opt. Lasers Eng. (1)

Y. J. Rao, “Recent progress in applications of in-fibre Bragg grating sensors,” Opt. Lasers Eng. 31(4), 297–324 (1999).
[Crossref]

Opt. Lett. (3)

Phys. Rep. (1)

L. L. Sánchez-Soto, J. J. Monzón, A. G. Barriuso, and J. F. Cariũena, “The transfer matrix: a geometrical perspective,” Phys. Rep. 513(4), 191–227 (2012).
[Crossref]

Phys. Rev. A (12)

H. Y. Ling, Y. Q. Li, and M. Xiao, “Electromagnetically induced grating: homogeneously broadened medium,” Phys. Rev. A 57(2), 1338–1344 (1998).
[Crossref]

Y. Zhang, Y.-M. Liu, X.-D. Tian, T.-Y. Zheng, and J.-H. Wu, “Tunable high-order photonic band gaps of ultraviolet light in cold atoms,” Phys. Rev. A 91(1), 013826 (2015).
[Crossref]

Y. Zhang, Y.-M. Liu, T.-Y. Zheng, and J.-H. Wu, “Light reflector, amplifier, and splitter based on gain-assisted photonic band gaps,” Phys. Rev. A 94(1), 013836 (2016).
[Crossref]

V. A. Bushuev, L. V. Dergacheva, and B. I. Mantsyzov, “Asymmetric pendulum effect and transparency change of PT-symmetric photonic crystals under dynamical Bragg diffraction beyond the paraxial approximation,” Phys. Rev. A 95(3), 033843 (2017).
[Crossref]

T. Shui, W. X. Yang, S. Liu, L. Li, and Z. Zhu, “Asymmetric diffraction by atomic gratings with optical PT symmetry in the Raman-Nath regime,” Phys. Rev. A 97(3), 033819 (2018).
[Crossref]

C. Y. Huang, R. Zhang, J. L. Han, J. Zheng, and J. Q. Xu, “Type-II perfect absorption and amplification modes with controllable bandwidth in combined PT-symmetric and conventional Bragg-grating structures,” Phys. Rev. A 89(2), 023842 (2014).
[Crossref]

B. Midya, “Supersymmetry-generated one-way-invisible PT-symmetric optical crystals,” Phys. Rev. A 89(3), 032116 (2014).
[Crossref]

S. Longhi, “Talbot self-imaging in PT-symmetric complex crystals,” Phys. Rev. A 90(4), 043827 (2014).
[Crossref]

J. T. Cole, K. G. Makris, Z. H. Musslimani, D. N. Christodoulides, and S. Rotter, “Twofold PT symmetry in doubly exponential optical lattices,” Phys. Rev. A 93(1), 013803 (2016).
[Crossref]

S. Nixon and J. Yang, “Light propagation in periodically modulated complex waveguides,” Phys. Rev. A 91(3), 033807 (2015).
[Crossref]

J.-H. Wu, M. Artoni, and G. C. La Rocca, “Parity-time-antisymmetric atomic lattices without gain,” Phys. Rev. A 91(3), 033811 (2015).
[Crossref]

H. Ramezani, T. Kottos, R. El-Ganainy, and D. N. Christodoulides, “Unidirectional nonlinear PT-symmetric optical structures,” Phys. Rev. A 82(4), 043803 (2010).
[Crossref]

Phys. Rev. E (1)

M. Artoni, G. La Rocca, and F. Bassani, “Resonantly absorbing one-dimensional photonic crystals,” Phys. Rev. E 72(4), 046604 (2005).
[Crossref]

Phys. Rev. Lett. (11)

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]

J.-H. Wu, M. Artoni, and G. C. La Rocca, “Non-Hermitian degeneracies and unidirectional reflectionless atomic lattices,” Phys. Rev. Lett. 113(12), 123004 (2014).
[Crossref] [PubMed]

S. Longhi, “Optical realization of relativistic non-Hermitian quantum mechanics,” Phys. Rev. Lett. 105(1), 013903 (2010).
[Crossref] [PubMed]

Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, “Optical solitons in PT periodic potentials,” Phys. Rev. Lett. 100(3), 030402 (2008).
[Crossref] [PubMed]

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]

C. M. Bender, D. C. Brody, H. F. Jones, and B. K. Meister, “Faster than Hermitian quantum mechanics,” Phys. Rev. Lett. 98(4), 040403 (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]

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]

C. M. Bender, D. C. Brody, and H. F. Jones, “Complex extension of quantum mechanics,” Phys. Rev. Lett. 89(27), 270401 (2002).
[Crossref]

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103(9), 093902 (2009).
[Crossref] [PubMed]

H. Jing, S. K. Özdemir, X. Y. Lü, J. Zhang, L. Yang, and F. Nori, “PT-symmetric phonon laser,” Phys. Rev. Lett. 113(5), 053604 (2014).
[Crossref] [PubMed]

Proc. IEEE (1)

T. K. Gaylord and M. G. Moharam, “Analysis and applications of optical diffraction by gratings,” Proc. IEEE 73(5), 894–937 (1985).
[Crossref]

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (6)

Fig. 1
Fig. 1 Schematic of a P T-symmetric periodic structure. It works as a Bragg grating as light is incident along the x-axis, viz. A ( L / 2 ) and B ( L / 2 ); as a diffraction grating as light is incident along the z-axis, viz. S0.
Fig. 2
Fig. 2 The comparison between R c o n ( K / 2 ) (blue solid line) and η1 (red dashed line) with n 0 = 1.0, n 1 = 0.01, and k 2 = 10 K. The grating has a thickness l z = 5 a along the z-axis.
Fig. 3
Fig. 3 Ratio η1 against ratio n 2 / n 1 and thickness lz with relevant parameters being the same as in Fig. 2.
Fig. 4
Fig. 4 An ensemble of cold   87 R b atoms loaded into an optical lattice ( l e f t) and driven into the four-level N configuration ( r i g h t). Relevant parameters used in the simulations are: Γ e g = Γ b g = Γ e a = Γ b a = 5.9 MHz, d e l t a c = δ d = 0, δ p = 6.25 MHz, Ω c 0 = 4.5 MHz, Ω d = 2.0 MHz, d = 0.2 a, N 0 = 5.0 × 10 12 cm   3, and d g e = 2.0 × 10 29 Cm.
Fig. 5
Fig. 5 R c o n (blue solid line) and ηm (red dashed line) for different reflection and diffraction orders are compared in the first row. Re[χ] (green solid line) and Im[χ] (orange dashed line), corresponding to the extreme values of R c o n and ηm, are shown in the second row with δ Ω c = 2.15 MHz in (a2) and 2.15 MHz in (a3); δ Ω c = 1.15 MHz in (b2) and 1.15 MHz in (b3); δ Ω c = 0.5 MHz in (c2) and 0.5 MHz in (c3); δ Ω c = 1.45 MHzin (c4) and 1.45 MHz in (c5).
Fig. 6
Fig. 6 Normalized light intensity for different diffraction orders corresponding to a few specific angles. We have used δ Ω c = 2.15 MHz in (a1) and 2.15 MHz in (a2) corresponding to the two extrema of η1; δ Ω c = 1.15 MHz in (b1) and 1.15 MHz in (b2) corresponding to the two extrema of η2; δ Ω c = 0.5 MHz in (c1), 0.5 MHz in (c2), 1.45 MHz in (c3), and 1.45 MHz in (c4) corresponding to the four extrema of η3.

Tables (1)

Tables Icon

Table 1 The similarities between asymmetric reflection and asymmetric diffraction. Δ k x is the change of the x-component wave number.

Equations (20)

Equations on this page are rendered with MathJax. Learn more.

n ( x ) = n 0 + n 1 c o s ( K x ) + i n 2 s i n ( K x ) .
n 2 ( x ) n 0 2 + 2 n 0 [ n 1 c o s ( K x ) + i n 2 s i n ( K x ) ] n 0 2 + n 0 [ κ + e i K x + κ e i K x ] ,
2 x 2 E 1 + k 1 2 n 2 ( x ) E 1 = 0.
E 1 ( x ) = A ( x ) e i K 2 x + B ( x ) e i K 2 x ,
A ( x ) x i π 2 a κ + B ( x ) = 0 , B ( x ) x i π 2 a κ A ( x ) = 0.
[ 2 x 2 + 2 z 2 ] E 2 + k 2 2 n 2 ( x ) E 2 = 0.
E 2 ( x , z ) = m = + S m ( z ) e i [ m K x + k 2 n 0 z ] .
d d z S m ( z ) + i m 2 K 2 2 k 2 n 0 S m ( z ) i k 2 2 [ κ S m + 1 ( z ) + κ + S m 1 ( z ) ] = 0.
S 1 ( z ) z + i K 2 2 k 2 n 0 S 1 ( z ) i k 2 2 κ + S 0 ( z ) = 0 , S 1 ( z ) z + i K 2 2 k 2 n 0 S 1 ( z ) i k 2 2 κ S 0 ( z ) = 0.
R c o n = R R R L M i n [ R R , R L ] ,
R c o n = ( n 1 + n 2 ) 2 ( n 1 n 2 ) 2 M i n [ ( n 1 + n 2 ) 2 , ( n 1 n 2 ) 2 ] .
η m = L o g 10 | S m | 2 | S m | 2 ,
N i ( x ) = N 0 e ( x x i ) 2 d 2 ,
Ω c = Ω c 0 + δ Ω c s i n [ 2 π ( x x i ) / a ] ,
H I = [ δ p | e e | + ( δ p δ d ) | a a | + δ c | b b | ] [ Ω p | e g | + Ω d | e a | + Ω c | b g | + h . c . ] ,
T p ( x ) = e i 2 π l z χ ( x ) / λ p ,
T p ( θ ) = a / 2 a / 2 T p ( x ) e i 2 π x R s i n θ d x ,
I p ( θ ) = | T p ( θ ) | 2 s i n 2 ( M π R s i n θ ) M 2 s i n 2 ( π R s i n θ ) ,
M j = 1 4 n j [ ( n j + 1 ) 2 e i k L j n j ( n j 1 ) 2 e i k L j n j ( n j 1 ) 2 e i k L j n j ( n j 1 ) 2 e i k L j n j ( n j 1 ) 2 e i k L j n j ( n j 1 ) 2 e i k L j n j ( n j + 1 ) 2 e i k L j n j ( n j 1 ) 2 e i k L j n j ] ,
M t r a n = j M j .

Metrics