Abstract

We study bulk optical modes in media with a parity-time symmetric permittivity tensor via the k-surface approach. We show that k-surfaces support multiple exceptional points (EPs), and the off-axial EPs can be manipulated by tuning the tensor elements. They merge into diabolic points if they have opposite handednesses, and annihilate each other otherwise. The underlying physical mechanisms and the potential applications are discussed.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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    [Crossref]
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  4. L. Feng, R. El-Ganainy, and L. Ge, “Non-Hermitian photonics based on parity-time symmetry,” Nat. Photonics 11(12), 752–762 (2017).
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  5. R. El-Ganainy, K. G. Makris, M. Khajavikhan, Z. H. Musslimani, S. Rotter, and D. N. Christodoulides, “Non-Hermitian physics and $\mathcal {PT}$PT symmetry,” Nat. Phys. 14(1), 11–19 (2018).
    [Crossref]
  6. V. V. Konotop, J. Yang, and D. A. Zezyulin, “Nonlinear waves in $\mathcal {PT}$PT-symmetric systems,” Rev. Mod. Phys. 88(3), 035002 (2016).
    [Crossref]
  7. S. Klaiman, U. Gunther, and N. Moiseyev, “Visualization of branch points in $\mathcal {PT}$PT-symmetric waveguides,” Phys. Rev. Lett. 101(8), 080402 (2008).
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  9. 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]
  10. W. Wan, Y. Chong, L. Ge, H. Noh, A. D. Stone, and H. Cao, “Time-reversed lasing and interferometric control of absorption,” Science 331(6019), 889–892 (2011).
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  11. S. Longhi, “$\mathcal {PT}$PT-symmetric laser absorber,” Phys. Rev. A 82(3), 031801 (2010).
    [Crossref]
  12. 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]
  13. M. Lawrence, N. Xu, X. Zhang, L. Cong, J. Han, W. Zhang, and S. Zhang, “Manifestation of $\mathcal {PT}$PT symmetry breaking in polarization space with terahertz metasurfaces,” Phys. Rev. Lett. 113(9), 093901 (2014).
    [Crossref]
  14. J. Gear, F. Liu, S. T. Chu, S. Rotter, and J. Li, “Parity-time symmetry from stacking purely dielectric and magnetic slabs,” Phys. Rev. A 91(3), 033825 (2015).
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  15. W. Wang, L. Q. Wang, R. D. Xue, H. L. Chen, R. P. Guo, Y. Liu, and J. Chen, “Unidirectional excitation of radiative-loss-free surface plasmon polaritons in $\mathcal {PT}$PT-symmetric systems,” Phys. Rev. Lett. 119(7), 077401 (2017).
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  21. S. A. H. Gangaraj and F. Monticone, “Topological waveguiding near an exceptional point: defect-immune, slow-light, and loss-immune propagation,” Phys. Rev. Lett. 121(9), 093901 (2018).
    [Crossref]
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    [Crossref]
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    [Crossref]
  24. A. Pick, B. Zhen, O. D. Miller, C. W. Hsu, F. Hernandez, A. W. Rodriguez, M. Soljacic, and S. G. Johnson, “General theory of spontaneous emission near exceptional points,” Opt. Express 25(11), 12325 (2017).
    [Crossref]
  25. M.-A. Miri and A. Alù, “Exceptional points in optics and photonics,” Science 363(6422), eaar7709 (2019).
    [Crossref]
  26. Y. Li and C. Argyropoulos, “Exceptional points and spectral singularities in active epsilon-near-zero plasmonic waveguides,” Phys. Rev. B 99(7), 075413 (2019).
    [Crossref]
  27. W. D. Heiss and H. L. Harney, “The chirality of exceptional points,” Eur. Phys. J. D 17(2), 149–151 (2001).
    [Crossref]
  28. X. F. Zhu, H. Ramezani, C. Shi, J. Zhu, and X. Zhang, “$\mathcal {PT}$PT-symmetric acoustics,” Phys. Rev. X 4(3), 031042 (2014).
    [Crossref]
  29. S. Richter, H. G. Zirnstein, J. Zuniga-Perez, E. Kruger, C. Deparis, L. Trefflich, C. Sturm, B. Rosenow, M. Grundmann, and R. Schmidt-Grund, “Voigt exceptional points in an anisotropic ZnO-based planar microcavity: square-root topology, polarization vortices, and circularity,” Phys. Rev. Lett. 123(22), 227401 (2019).
    [Crossref]
  30. M. V. Berry and M. R. Dennis, “The optical singularities of birefringent dichroic chiral crystals,” Proc. R. Soc. London, Ser. A 459(2033), 1261–1292 (2003).
    [Crossref]
  31. C. Sturm and M. Grundmann, “Singular optical axes in biaxial crystals and analysis of their spectral dispersion effects in β-Ga2O3,” Phys. Rev. A 93(5), 053839 (2016).
    [Crossref]
  32. M. Grundmann, C. Sturm, C. Kranert, S. Richter, R. Schmidt-Grund, C. Deparis, and J. Zuniga-Perez, “Optically anisotropic media: new approaches to the dielectric function, singular axes, microcavity modes and Raman scattering intensities,” Phys. Status Solidi RRL 11(1), 1600295 (2017).
    [Crossref]
  33. K. Ding, Z. Q. Zhang, and C. T. Chan, “Coalescence of exceptional points and phase diagrams for one-dimensional PT-symmetric photonic crystals,” Phys. Rev. B 92(23), 235310 (2015).
    [Crossref]
  34. Y. R. Zhang, Z. Z. Zhang, J. Q. Yuan, M. Kang, and J. Chen, “High-order exceptional points in non-Hermitian Moire lattices,” Front. Phys. 14(5), 53603 (2019).
    [Crossref]

2019 (4)

M.-A. Miri and A. Alù, “Exceptional points in optics and photonics,” Science 363(6422), eaar7709 (2019).
[Crossref]

Y. Li and C. Argyropoulos, “Exceptional points and spectral singularities in active epsilon-near-zero plasmonic waveguides,” Phys. Rev. B 99(7), 075413 (2019).
[Crossref]

S. Richter, H. G. Zirnstein, J. Zuniga-Perez, E. Kruger, C. Deparis, L. Trefflich, C. Sturm, B. Rosenow, M. Grundmann, and R. Schmidt-Grund, “Voigt exceptional points in an anisotropic ZnO-based planar microcavity: square-root topology, polarization vortices, and circularity,” Phys. Rev. Lett. 123(22), 227401 (2019).
[Crossref]

Y. R. Zhang, Z. Z. Zhang, J. Q. Yuan, M. Kang, and J. Chen, “High-order exceptional points in non-Hermitian Moire lattices,” Front. Phys. 14(5), 53603 (2019).
[Crossref]

2018 (4)

T. Goldzak, A. A. Mailybaev, and N. Moiseyev, “Light stops at exceptional points,” Phys. Rev. Lett. 120(1), 013901 (2018).
[Crossref]

S. A. H. Gangaraj and F. Monticone, “Topological waveguiding near an exceptional point: defect-immune, slow-light, and loss-immune propagation,” Phys. Rev. Lett. 121(9), 093901 (2018).
[Crossref]

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

H. Shen, B. Zhen, and L. Fu, “Topological band theory for non-Hermitian Hamiltonians,” Phys. Rev. Lett. 120(14), 146402 (2018).
[Crossref]

2017 (7)

H. Hodaei, A. U. Hassan, S. Wittek, H. Garcia-Gracia, R. El-Ganainy, D. N. Christodoulides, and M. Khajavikhan, “Enhanced sensitivity at higher-order exceptional points,” Nature 548(7666), 187–191 (2017).
[Crossref]

W. Chen, S. K. Ozdemir, G. Zhao, J. Wiersig, and L. Yang, “Exceptional points enhanced sensing in an optical microcavity,” Nature 548(7666), 192–196 (2017).
[Crossref]

W. Wang, L. Q. Wang, R. D. Xue, H. L. Chen, R. P. Guo, Y. Liu, and J. Chen, “Unidirectional excitation of radiative-loss-free surface plasmon polaritons in $\mathcal {PT}$PT-symmetric systems,” Phys. Rev. Lett. 119(7), 077401 (2017).
[Crossref]

S. Assawaworrarit, X. Yu, and S. Fan, “Robust wireless power transfer using a nonlinear parity-time-symmetric circuit,” Nature 546(7658), 387–390 (2017).
[Crossref]

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

A. Pick, B. Zhen, O. D. Miller, C. W. Hsu, F. Hernandez, A. W. Rodriguez, M. Soljacic, and S. G. Johnson, “General theory of spontaneous emission near exceptional points,” Opt. Express 25(11), 12325 (2017).
[Crossref]

M. Grundmann, C. Sturm, C. Kranert, S. Richter, R. Schmidt-Grund, C. Deparis, and J. Zuniga-Perez, “Optically anisotropic media: new approaches to the dielectric function, singular axes, microcavity modes and Raman scattering intensities,” Phys. Status Solidi RRL 11(1), 1600295 (2017).
[Crossref]

2016 (3)

C. Sturm and M. Grundmann, “Singular optical axes in biaxial crystals and analysis of their spectral dispersion effects in β-Ga2O3,” Phys. Rev. A 93(5), 053839 (2016).
[Crossref]

Z. Lin, A. Pick, M. Loncar, and A. W. Rodriguez, “Enhanced spontaneous emission at third-order Dirac exceptional points in inverse-designed photonic crystals,” Phys. Rev. Lett. 117(10), 107402 (2016).
[Crossref]

V. V. Konotop, J. Yang, and D. A. Zezyulin, “Nonlinear waves in $\mathcal {PT}$PT-symmetric systems,” Rev. Mod. Phys. 88(3), 035002 (2016).
[Crossref]

2015 (3)

H. Cao and J. Wiersig, “Dielectric microcavities: model systems for wave chaos and non-Hermitian physics,” Rev. Mod. Phys. 87(1), 61–111 (2015).
[Crossref]

J. Gear, F. Liu, S. T. Chu, S. Rotter, and J. Li, “Parity-time symmetry from stacking purely dielectric and magnetic slabs,” Phys. Rev. A 91(3), 033825 (2015).
[Crossref]

K. Ding, Z. Q. Zhang, and C. T. Chan, “Coalescence of exceptional points and phase diagrams for one-dimensional PT-symmetric photonic crystals,” Phys. Rev. B 92(23), 235310 (2015).
[Crossref]

2014 (3)

A. A. Zyablovsky, A. P. Vinogradov, A. V. Dorofeenko, A. A. Pukhov, and A. A. Lisyansky, “Causality and phase transitions in $\mathcal {PT}$PT-symmetric optical systems,” Phys. Rev. A 89(3), 033808 (2014).
[Crossref]

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

M. Lawrence, N. Xu, X. Zhang, L. Cong, J. Han, W. Zhang, and S. Zhang, “Manifestation of $\mathcal {PT}$PT symmetry breaking in polarization space with terahertz metasurfaces,” Phys. Rev. Lett. 113(9), 093901 (2014).
[Crossref]

2013 (1)

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

2011 (1)

W. Wan, Y. Chong, L. Ge, H. Noh, A. D. Stone, and H. Cao, “Time-reversed lasing and interferometric control of absorption,” Science 331(6019), 889–892 (2011).
[Crossref]

2010 (2)

S. Longhi, “$\mathcal {PT}$PT-symmetric laser absorber,” Phys. Rev. A 82(3), 031801 (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]

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 $\mathcal {PT}$PT-symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103(9), 093902 (2009).
[Crossref]

S. Longhi, “Quantum-optical analogies using photonic structures,” Laser Photonics Rev. 3(3), 243–261 (2009).
[Crossref]

2008 (1)

S. Klaiman, U. Gunther, and N. Moiseyev, “Visualization of branch points in $\mathcal {PT}$PT-symmetric waveguides,” Phys. Rev. Lett. 101(8), 080402 (2008).
[Crossref]

2007 (1)

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

2003 (1)

M. V. Berry and M. R. Dennis, “The optical singularities of birefringent dichroic chiral crystals,” Proc. R. Soc. London, Ser. A 459(2033), 1261–1292 (2003).
[Crossref]

2001 (1)

W. D. Heiss and H. L. Harney, “The chirality of exceptional points,” Eur. Phys. J. D 17(2), 149–151 (2001).
[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 $\mathcal {PT}$PT-symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103(9), 093902 (2009).
[Crossref]

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]

Alù, A.

M.-A. Miri and A. Alù, “Exceptional points in optics and photonics,” Science 363(6422), eaar7709 (2019).
[Crossref]

Argyropoulos, C.

Y. Li and C. Argyropoulos, “Exceptional points and spectral singularities in active epsilon-near-zero plasmonic waveguides,” Phys. Rev. B 99(7), 075413 (2019).
[Crossref]

Assawaworrarit, S.

S. Assawaworrarit, X. Yu, and S. Fan, “Robust wireless power transfer using a nonlinear parity-time-symmetric circuit,” Nature 546(7658), 387–390 (2017).
[Crossref]

Bender, C. M.

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

Berry, M. V.

M. V. Berry and M. R. Dennis, “The optical singularities of birefringent dichroic chiral crystals,” Proc. R. Soc. London, Ser. A 459(2033), 1261–1292 (2003).
[Crossref]

Cao, H.

H. Cao and J. Wiersig, “Dielectric microcavities: model systems for wave chaos and non-Hermitian physics,” Rev. Mod. Phys. 87(1), 61–111 (2015).
[Crossref]

W. Wan, Y. Chong, L. Ge, H. Noh, A. D. Stone, and H. Cao, “Time-reversed lasing and interferometric control of absorption,” Science 331(6019), 889–892 (2011).
[Crossref]

Chan, C. T.

K. Ding, Z. Q. Zhang, and C. T. Chan, “Coalescence of exceptional points and phase diagrams for one-dimensional PT-symmetric photonic crystals,” Phys. Rev. B 92(23), 235310 (2015).
[Crossref]

Chen, H. L.

W. Wang, L. Q. Wang, R. D. Xue, H. L. Chen, R. P. Guo, Y. Liu, and J. Chen, “Unidirectional excitation of radiative-loss-free surface plasmon polaritons in $\mathcal {PT}$PT-symmetric systems,” Phys. Rev. Lett. 119(7), 077401 (2017).
[Crossref]

Chen, J.

Y. R. Zhang, Z. Z. Zhang, J. Q. Yuan, M. Kang, and J. Chen, “High-order exceptional points in non-Hermitian Moire lattices,” Front. Phys. 14(5), 53603 (2019).
[Crossref]

W. Wang, L. Q. Wang, R. D. Xue, H. L. Chen, R. P. Guo, Y. Liu, and J. Chen, “Unidirectional excitation of radiative-loss-free surface plasmon polaritons in $\mathcal {PT}$PT-symmetric systems,” Phys. Rev. Lett. 119(7), 077401 (2017).
[Crossref]

Chen, W.

W. Chen, S. K. Ozdemir, G. Zhao, J. Wiersig, and L. Yang, “Exceptional points enhanced sensing in an optical microcavity,” Nature 548(7666), 192–196 (2017).
[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]

Chong, Y.

W. Wan, Y. Chong, L. Ge, H. Noh, A. D. Stone, and H. Cao, “Time-reversed lasing and interferometric control of absorption,” Science 331(6019), 889–892 (2011).
[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 $\mathcal {PT}$PT symmetry,” Nat. Phys. 14(1), 11–19 (2018).
[Crossref]

H. Hodaei, A. U. Hassan, S. Wittek, H. Garcia-Gracia, R. El-Ganainy, D. N. Christodoulides, and M. Khajavikhan, “Enhanced sensitivity at higher-order exceptional points,” Nature 548(7666), 187–191 (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]

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

Chu, S. T.

J. Gear, F. Liu, S. T. Chu, S. Rotter, and J. Li, “Parity-time symmetry from stacking purely dielectric and magnetic slabs,” Phys. Rev. A 91(3), 033825 (2015).
[Crossref]

Cong, L.

M. Lawrence, N. Xu, X. Zhang, L. Cong, J. Han, W. Zhang, and S. Zhang, “Manifestation of $\mathcal {PT}$PT symmetry breaking in polarization space with terahertz metasurfaces,” Phys. Rev. Lett. 113(9), 093901 (2014).
[Crossref]

Dennis, M. R.

M. V. Berry and M. R. Dennis, “The optical singularities of birefringent dichroic chiral crystals,” Proc. R. Soc. London, Ser. A 459(2033), 1261–1292 (2003).
[Crossref]

Deparis, C.

S. Richter, H. G. Zirnstein, J. Zuniga-Perez, E. Kruger, C. Deparis, L. Trefflich, C. Sturm, B. Rosenow, M. Grundmann, and R. Schmidt-Grund, “Voigt exceptional points in an anisotropic ZnO-based planar microcavity: square-root topology, polarization vortices, and circularity,” Phys. Rev. Lett. 123(22), 227401 (2019).
[Crossref]

M. Grundmann, C. Sturm, C. Kranert, S. Richter, R. Schmidt-Grund, C. Deparis, and J. Zuniga-Perez, “Optically anisotropic media: new approaches to the dielectric function, singular axes, microcavity modes and Raman scattering intensities,” Phys. Status Solidi RRL 11(1), 1600295 (2017).
[Crossref]

Ding, K.

K. Ding, Z. Q. Zhang, and C. T. Chan, “Coalescence of exceptional points and phase diagrams for one-dimensional PT-symmetric photonic crystals,” Phys. Rev. B 92(23), 235310 (2015).
[Crossref]

Dorofeenko, A. V.

A. A. Zyablovsky, A. P. Vinogradov, A. V. Dorofeenko, A. A. Pukhov, and A. A. Lisyansky, “Causality and phase transitions in $\mathcal {PT}$PT-symmetric optical systems,” Phys. Rev. A 89(3), 033808 (2014).
[Crossref]

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 $\mathcal {PT}$PT-symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103(9), 093902 (2009).
[Crossref]

El-Ganainy, R.

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

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

H. Hodaei, A. U. Hassan, S. Wittek, H. Garcia-Gracia, R. El-Ganainy, D. N. Christodoulides, and M. Khajavikhan, “Enhanced sensitivity at higher-order exceptional points,” Nature 548(7666), 187–191 (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]

Fan, S.

S. Assawaworrarit, X. Yu, and S. Fan, “Robust wireless power transfer using a nonlinear parity-time-symmetric circuit,” Nature 546(7658), 387–390 (2017).
[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, R. El-Ganainy, and L. Ge, “Non-Hermitian photonics based on parity-time symmetry,” Nat. Photonics 11(12), 752–762 (2017).
[Crossref]

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]

Fu, L.

H. Shen, B. Zhen, and L. Fu, “Topological band theory for non-Hermitian Hamiltonians,” Phys. Rev. Lett. 120(14), 146402 (2018).
[Crossref]

Gangaraj, S. A. H.

S. A. H. Gangaraj and F. Monticone, “Topological waveguiding near an exceptional point: defect-immune, slow-light, and loss-immune propagation,” Phys. Rev. Lett. 121(9), 093901 (2018).
[Crossref]

Garcia-Gracia, H.

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S. Richter, H. G. Zirnstein, J. Zuniga-Perez, E. Kruger, C. Deparis, L. Trefflich, C. Sturm, B. Rosenow, M. Grundmann, and R. Schmidt-Grund, “Voigt exceptional points in an anisotropic ZnO-based planar microcavity: square-root topology, polarization vortices, and circularity,” Phys. Rev. Lett. 123(22), 227401 (2019).
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M. Grundmann, C. Sturm, C. Kranert, S. Richter, R. Schmidt-Grund, C. Deparis, and J. Zuniga-Perez, “Optically anisotropic media: new approaches to the dielectric function, singular axes, microcavity modes and Raman scattering intensities,” Phys. Status Solidi RRL 11(1), 1600295 (2017).
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Nat. Photonics (1)

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Nat. Phys. (2)

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Nature (3)

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Opt. Express (1)

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A. A. Zyablovsky, A. P. Vinogradov, A. V. Dorofeenko, A. A. Pukhov, and A. A. Lisyansky, “Causality and phase transitions in $\mathcal {PT}$PT-symmetric optical systems,” Phys. Rev. A 89(3), 033808 (2014).
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S. Richter, H. G. Zirnstein, J. Zuniga-Perez, E. Kruger, C. Deparis, L. Trefflich, C. Sturm, B. Rosenow, M. Grundmann, and R. Schmidt-Grund, “Voigt exceptional points in an anisotropic ZnO-based planar microcavity: square-root topology, polarization vortices, and circularity,” Phys. Rev. Lett. 123(22), 227401 (2019).
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M. Grundmann, C. Sturm, C. Kranert, S. Richter, R. Schmidt-Grund, C. Deparis, and J. Zuniga-Perez, “Optically anisotropic media: new approaches to the dielectric function, singular axes, microcavity modes and Raman scattering intensities,” Phys. Status Solidi RRL 11(1), 1600295 (2017).
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Figures (6)

Fig. 1.
Fig. 1. A schematic of the (a) real and (c) imaginary parts of the $k$-surfaces with parameters $\varepsilon _w=4$, $\gamma =2$ and $\varepsilon _d=2$. (b) and (c) show the projected contours at chosen $k_z$ values. EPs are labeled out by arrows. In our $k$-surface approach the lengths along the $\hat {\boldsymbol{k}}_0$ direction is proportional to Re{n} and Im{n}, respectively.
Fig. 2.
Fig. 2. Projected contours of the (a) real and (b) imaginary components of the $k$-surfaces in the $k_x=\pm k_y$ plane. EPs are labeled out. In the $k_x=-k_y$ plane Im{n}=0 because Re{n} has two values and the system is within the conserved $\mathcal {PT}$ phase.
Fig. 3.
Fig. 3. Projected contours of the real and imaginary components of the $k$-surfaces in the $k_x=\pm k_y$ plane for (a) $\varepsilon _d=3.5$, (b) $\varepsilon _d=4$, and (c) $\varepsilon _d=5$, respectively. When $\varepsilon _d=4$ the imaginary component is not shown because it is zero for all $k_x=\pm k_y$.
Fig. 4.
Fig. 4. $k$-surfaces when $\varepsilon _d=8/3$. EP$_\alpha$ and EP$_\beta$ merge into a diabolic point in the $\hat {\boldsymbol{k}}_0=2^{-1/2}[1,\pm 1,0]$ directions.
Fig. 5.
Fig. 5. A schematic of the (a) real and (c) imaginary parts of $k$-surfaces when $\varepsilon _d=15$. (b) and (c) show the projected contours at chosen $k_z$ values. EPs are labeled out by arrows.
Fig. 6.
Fig. 6. Refraction properties of EP mode into free space from a $xy$ interface. (a) the refractive index $n$ of EP, (b) its angle with respect to the $z$ axis (in the unit of radian), (c) the polarization of refracted field, and (d) the overall transmission coefficient in field intensity.

Equations (30)

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ε ¯ ¯ = ε 0 ( ε w j γ γ 0 γ ε w + j γ 0 0 0 ε d ) .
k × E = ω B ,
k × H = ω D .
k = n ω c k ^ 0 ,
k ^ 0 × ( k ^ 0 × ε ¯ ¯ 1 D ) = 1 n 2 D .
k D = 0 ,
k ^ 0 = [ k x , k y , k z ] T ,
( M 11 M 12 M 13 M 21 M 22 M 23 M 31 M 32 M 33 ) ( D x D y D z ) = ε w 2 n 2 ( D x D y D z ) ,
M 11 = ( ε w + i γ ) ( k y 2 + k z 2 ) + γ k x k y ,
M 12 = ( ε w i γ ) k x k y γ ( k y 2 + k z 2 ) ,
M 13 = ε w 2 ε d k x k z ,
M 21 = ( ε w + i γ ) k x k y γ ( k x 2 + k z 2 ) ,
M 22 = ( ε w i γ ) ( k x 2 + k z 2 ) + γ k x k y ,
M 23 = ε w 2 ε d k y k z ,
M 31 = ( ε w + i γ ) k x k z + γ k y k z ,
M 32 = ( ε w i γ ) k y k z + γ k x k z ,
M 33 = ε w 2 ε d ( k x 2 + k y 2 ) .
I m { M 11 + M 22 + M 33 } = 0 ,
k x 2 = k y 2 .
M [ 001 ] = ( ε w + i γ γ 0 γ ε w i γ 0 0 0 0 ) .
n [ 001 ] = ε w ,
D [ 001 ] = 2 1 / 2 [ 1 , i , 0 ] T .
M [ 1 ± 10 ] = 1 2 ( ( ε w + i γ ) ± γ ( ε w i γ ) γ 0 ( ε w + i γ ) γ ( ε w i γ ) ± γ 0 0 0 2 ε w 2 / ε d ) .
n [ 1 ± 10 ] o = ε d ,
D [ 1 ± 10 ] o = [ 0 , 0 , 1 ] T .
n [ 1 ± 10 ] e = ε w 2 ε w ± γ ,
D [ 1 ± 10 ] e = 1 2 [ 1 , 1 , 0 ] T .
S 3 = 2 Im { D θ D ϕ } .
ε d = ε w 2 ε w ± κ .
ε d = ε w .