Abstract

This work demonstrates how the crystal symmetry of photonic crystal defect waveguides interacts with simple, experimentally realizable parity-time ($ {\cal P}{\cal T} $) symmetric regions of chip-scale absorption and amplification to control the existence and location of exceptional points in the first Brillouin zone. Our analysis is based on Heesh–Shubnikov group theory and is generalizable to a large class of devices for which the symmetry groups can be identified. Transverse, longitudinal, and transverse–longitudinal hybrid $ {\cal P}{\cal T} $ symmetries are considered, and for each, a triangular lattice photonic crystal waveguide with lattice-aligned and lattice-shifted cladding orientations is analyzed. We find that various symmetry combinations produce either strictly real-valued or strictly complex-valued eigenfrequencies at the Brillouin zone boundary. We also show how symmetry can be used to predict $ {\cal P}{\cal T} $ transitions at accidental degeneracies in the waveguide bands. It is shown how symmetry can be used to design single-mode waveguides, and we discovered exceptional points whose propagation constants are highly sensitive to the non-Hermiticity factor.

© 2019 Optical Society of America

Full Article  |  PDF Article
OSA Recommended Articles
Quantum statistical signature of $ {\cal P}{\cal T} $PT symmetry breaking

Stefano Longhi
Opt. Lett. 45(6) 1591-1594 (2020)

Fano effect in a one-dimensional photonic lattice with side-coupled P T-symmetric non-Hermitian defects

Xue-Si Li, Piao-Piao Huang, Jing He, Lian-Lian Zhang, and Wei-Jiang Gong
Opt. Express 28(6) 8560-8573 (2020)

References

  • View by:
  • |
  • |
  • |

  1. C. M. Bender and S. Boettcher, “Real spectra in non-Hermitian Hamiltonians having PT symmetry,” Phys. Rev. Lett. 80, 5243–5246 (1998).
    [Crossref]
  2. C. M. Bender, S. Boettcher, and P. N. Meisinger, “PT-symmetric quantum mechanics,” J. Math. Phys. 40, 2201–2229 (1999).
    [Crossref]
  3. C. M. Bender, D. C. Brody, and H. F. Jones, “Complex extension of quantum mechanics,” Phys. Rev. Lett. 89, 270401 (2002).
    [Crossref]
  4. Ş. K. Özdemir, R. Stefan, F. Nori, and L. Yang, “Parity-time symmetry and exceptional points in photonics,” Nat. Mater. 18, 783 (2019).
    [Crossref]
  5. A. Mock, “Characterization of parity-time symmetry in photonic lattices using Heesh-Shubnikov group theory,” Opt. Express 24, 22693–22707 (2016).
    [Crossref]
  6. A. Mock, “Comprehensive understanding of parity-time transitions in PT symmetric photonic crystals with an antiunitary group theory,” Phys. Rev. A 95, 043803 (2017).
    [Crossref]
  7. 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, 11–19 (2018).
    [Crossref]
  8. P. Miao, Z. Zhang, J. Sun, W. Walasik, S. Longhi, N. M. Litchinitser, and L. Feng, “Orbital angular momentum microlaser,” Science 353, 464–467 (2016).
    [Crossref]
  9. 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, 213901 (2011).
    [Crossref]
  10. L. Feng, M. Ayache, J. Huang, Y.-L. Xu, M.-H. Lu, Y.-F. Chen, Y. Fainman, and A. Scherer, “Nonreciprocal light propagation in a silicon photonic circuit,” Science 333, 729–733 (2011).
    [Crossref]
  11. 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, 108–113 (2013).
    [Crossref]
  12. B. Peng, Ş. K. Özdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery micro cavities,” Nat. Phys. 10, 394–398 (2014).
    [Crossref]
  13. 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, 187–191 (2017).
    [Crossref]
  14. L. Feng, Z. J. Wong, R.-M. Ma, Y. Wang, and X. Zhang, “Single-mode laser by parity-time symmetry breaking,” Science 346, 972–975 (2014).
    [Crossref]
  15. H. Hodaei, M.-A. Miri, M. Heinrich, D. N. Christodoulides, and M. Khajavikhan, “Parity-time-symmetric microring lasers,” Science 346, 975–978 (2014).
    [Crossref]
  16. H. Hodaei, M.-A. Miri, A. U. Hassan, W. E. Hayenga, M. Heinrich, D. N. Christodoulides, and M. Khajavikhan, “Parity-time-symmetric coupled microring lasers operating around an exceptional point,” Opt. Lett. 40, 4955–4958 (2015).
    [Crossref]
  17. H. Hodaei, M.-A. Miri, A. U. Hassan, W. E. Hayenga, M. Heinrich, D. N. Christodoulides, and M. Khajavikhan, “Single mode lasing in transversely multi-moded PT-symmetric microring resonators,” Laser Photon. Rev. 10, 494–499 (2016).
    [Crossref]
  18. W. Liu, M. Li, R. S. Guzzon, E. J. Norberg, J. S. Parker, M. Lu, L. A. Coldren, and J. Yao, “An integrated parity-time symmetric wavelength-tunable single-mode microring laser,” Nat. Commun. 8, 15389 (2017).
    [Crossref]
  19. 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, 192–195 (2010).
    [Crossref]
  20. M. Kremer, T. Biesenthal, L. J. Maczewsky, M. Heinrich, R. Thomale, and A. Szameit, “Demonstration of a two-dimensional PT-symmetric crystal,” Nat. Commun. 10, 435 (2019).
    [Crossref]
  21. Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, “Optical solitons in PT periodic potentials,” Phys. Rev. Lett. 100, 030402 (2008).
    [Crossref]
  22. A. Szameit, M. C. Rechtsman, O. Bahat-Treidel, and M. Segev, “PT-symmetry in honeycomb photonic lattices,” Phys. Rev. A 84, 021806 (2011).
    [Crossref]
  23. J. Xie, Z. Su, W. Chen, G. Chen, J. Lv, D. Mihalache, and Y. He, “Defect solitons in two-dimensional photonic lattices with parity-time symmetry,” Opt. Commun. 313, 139–145 (2014).
    [Crossref]
  24. L. Ge and A. D. Stone, “Parity-time symmetry breaking beyond one dimension: the role of degeneracy,” Phys. Rev. X 4, 031011 (2014).
    [Crossref]
  25. H. Wang, S. Shi, X. Ren, X. Zhu, B. A. Malomed, D. Mihalache, and Y. He, “Two-dimensional solitons in triangular photonic lattices with parity-time symmetry,” Opt. Commun. 335, 146–152 (2015).
    [Crossref]
  26. K. S. Agarwal, R. K. Pathak, and Y. N. Joglekar, “Exactly solvable-symmetric models in two dimensions,” Europhys. Lett. 112, 31003 (2015).
    [Crossref]
  27. M. Turdeuev, M. Botey, I. Giden, R. Herrero, H. Kurt, E. Ozbay, and K. Staliunas, “Two-dimensional complex parity-time-symmetric photonic structures,” Phys. Rev. A 91, 023825 (2015).
    [Crossref]
  28. A. Cerjan, A. Raman, and S. Fan, “Exceptional contours and band structure design in parity-time symmetric photonic crystals,” Phys. Rev. Lett. 116, 203902 (2016).
    [Crossref]
  29. A. Cerjan and S. Fan, “Effects of non-uniform distributions of gain and loss in photonic crystals,” New J. Phys. 18, 125007 (2016).
    [Crossref]
  30. W. W. Ahmed, R. Herrero, M. Botey, and K. Staliunas, “Self-collimation in PT-symmetric crystals,” Phys. Rev. A 95, 053830 (2017).
    [Crossref]
  31. L. Ge, “Non-Hermitian lattices with a flat band and polynomial power increase,” Photon. Res. 6, A10–A17 (2018).
    [Crossref]
  32. R. El-Ganainy, K. G. Makris, D. N. Christodoulides, and Z. H. Musslimani, “Theory of coupled optical PT-symmetric structures,” Opt. Lett. 32, 2632–2634 (2007).
    [Crossref]
  33. A. Mock, “Engineering modes of PT symmetric photonic crystals,” in European Conference on Antennas and Propagation (2018), paper CS14.2.
  34. Y.-T. Fang, S.-F. Ye, and X.-X. Li, “Unique band coalescence and exceptional points from two-dimensional photonic crystal waveguide,” J. Opt. 21, 055103 (2019).
    [Crossref]
  35. S. G. Joannopoulos, J. D. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light, 2nd ed. (Princeton University, 2008).
  36. H.-G. Park, S.-H. Kim, S.-H. Kwon, Y.-G. Ju, H.-K. Yang, J.-H. Baek, S.-B. Kim, and Y.-H. Lee, “Electrically driven single-cell photonic crystal laser,” Science 305, 1444–1447 (2004).
    [Crossref]
  37. M. H. Shih, W. Kuang, A. Mock, M. Bagheri, E. H. Hwang, J. D. O’Brien, and P. D. Dapkus, “High-quality-factor photonic crystal heterostructure laser,” Appl. Phys. Lett. 89, 101104 (2006).
    [Crossref]
  38. L. Lu, A. Mock, T. Yang, M. H. Shih, E. H. Hwang, M. Bagheri, A. Stapleton, S. Farrell, J. D. O’Brien, and P. D. Dapkus, “120 µW peak output power from edge-emitting photonic crystal double heterostructure nanocavity lasers,” Appl. Phys. Lett. 94, 111101 (2009).
    [Crossref]
  39. L. Lu, A. Mock, E. H. Hwang, J. O’Brien, and P. D. Dapkus, “High-peak-power efficient edge-emitting photonic crystal nanocavity lasers,” Opt. Lett. 34, 2346–2648 (2009).
    [Crossref]
  40. S. Matsuo, K. Takeda, T. Sato, M. Notomi, A. Shinya, K. Nozaki, H. Taniyama, K. Hasebe, and T. Kakitsuka, “Room-temperature continuous-wave operation of lateral current injection wavelength-scale embedded active-region photonic-crystal laser,” Opt. Express 20, 3773–3780 (2012).
    [Crossref]
  41. L. Lu, A. Mock, M. Bagheri, E. H. Hwang, J. D. O’Brien, and P. D. Dapkus, “Double-heterostructure photonic crystal lasers with reduced threshold pump power and increased slope efficiency obtained by quantum well intermixing,” Opt. Express 16, 17342–17347 (2008).
    [Crossref]
  42. L. A. Coldren and S. W. Corzine, Diode Lasers and Photonic Integrated Circuits (Wiley, 1995).
  43. A. A. Zyablovsky, A. P. Vinogradov, A. V. Dorofeenko, A. A. Pukhov, and A. A. Lisyansky, “Causality and phase transitions in PT-symmetric optical systems,” Phys. Rev. A 89, 033808 (2014).
    [Crossref]
  44. D. M. Tsvetkov, V. A. Bushuev, V. V. Konotop, and B. I. Mantsyzov, “Broadband quasi-PT-symmetry sustained by inhomogeneous broadening of the spectral line,” Phys. Rev. A 98, 053844 (2018).
    [Crossref]
  45. K. Sakoda, Optical Properties of Photonic Crystals (Springer, 2001).
  46. K. Srinivasan and O. Painter, “Momentum space design of high-Q photonic crystal optical cavities,” Opt. Express 10, 670–684 (2002).
    [Crossref]
  47. K. Sakoda, “Proof of the universality of mode symmetries in creating photonic Dirac cones,” Opt. Express 20, 25181–25194 (2012).
    [Crossref]
  48. H. Heesh, “Zur Strukturtheorie der ebenen Symmetriegruppen,” Zeitschrift für Kristallographie-Cryst. Mater. 71, 95–102 (1929).
  49. A. V. Shubnikov, Symmetry and Antisymmetry of Finite Figures (Izd-vo Akademii nauk SSSR, 1951).
  50. A. V. Shubnikov and N. V. Belov, Colored Symmetry (Macmillan, 1964).
  51. E. P. Wigner, Group Theory and Its Application to the Quantum Mechanics of Atomic Spectra (Academic, 1959).
  52. A. P. Cracknell, Group Theory in Solid-State Physics (Taylor and Francis, 1975).
  53. A. P. Cracknell, Magnetism in Crystalline Materials (Pergamon, 1975).
  54. M. El-Batanouny and F. Wooten, Symmetry and Condensed Matter Physics: A Computational Approach (Cambridge University, 2008).
  55. G. Frobenius and I. Schur, Über die reellen Darstellungen der endlichen Gruppen (Sitzungsberichte Der Berliner Mathematischen Gesellschaft, 1906), pp. 186–208.
  56. J. O. Dimmock and R. G. Wheeler, “Symmetry properties of wave functions in magnetic crystals,” Phys. Rev. 127, 391–404 (1962).
    [Crossref]
  57. M. Tinkham, Group Theory and Quantum Mechanics (Dover Publications, 1964).
  58. R. Liboff, Primer for Point and Space Groups (Springer-Verlag, 2004).
  59. H. H. Benisty, C. Weisbuch, D. Labilloy, M. Rattier, C. J. M. Smith, T. F. Krauss, R. M. D. L. Rue, R. Houdré, U. Oesterle, C. Jouanin, and D. Cassagne, “Optical and confinement properties of two-dimensional photonic crystals,” J. Lightwave Technol. 17, 2063–2077 (1999).
    [Crossref]
  60. B. Zhen, C. W. Hsu, Y. Igarashi, L. Lu, I. Kaminer, A. Pick, S.-L. Chua, J. D. Joannopoulos, and M. Soljačić, “Spawning rings of exceptional points out of Dirac cones,” Nature 525, 354–358 (2015).
    [Crossref]
  61. A. Mock, L. Lu, and J. O’Brien, “Space group theory and Fourier space analysis of two-dimensional photonic crystal waveguides,” Phys. Rev. B 81, 155115 (2010).
    [Crossref]
  62. H. Benisty, “Modal analysis of optical guides with two-dimensional photonic band-gap boundaries,” J. Appl. Phys. 79, 7483–7492 (1996).
    [Crossref]
  63. W. Kuang and J. O’Brien, “Reducing the out-of-plane radiation loss of photonic crystal waveguides on high-index substrates,” Opt. Lett. 29, 860–862 (2004).
    [Crossref]
  64. G. Valerio, F. Ghasemifard, Z. Sipus, and O. Quevedo-Teruel, “Glide-symmetric all-metal holey metasurfaces for low-dispersive artificial materials: modeling and properties,” IEEE Trans. Microwave Theory Tech. 66, 3210–3223 (2018).
    [Crossref]
  65. F. Ghasemifard, M. Norgen, O. Quevedo-Teruel, and G. Valerio, “Analyzing glide-symmetric holey metasurfaces using a generalized Floquet theorem,” IEEE Access 6, 71743–71750 (2018).
    [Crossref]
  66. M. Bagheriasl, O. Quevedo-Teruel, and G. Valerio, “Bloch analysis of artificial lines and surfaces exhibiting glide symmetry,” IEEE Trans. Microwave Theory Tech. 67, 2618–2628 (2019).
    [Crossref]

2019 (4)

Ş. K. Özdemir, R. Stefan, F. Nori, and L. Yang, “Parity-time symmetry and exceptional points in photonics,” Nat. Mater. 18, 783 (2019).
[Crossref]

M. Kremer, T. Biesenthal, L. J. Maczewsky, M. Heinrich, R. Thomale, and A. Szameit, “Demonstration of a two-dimensional PT-symmetric crystal,” Nat. Commun. 10, 435 (2019).
[Crossref]

Y.-T. Fang, S.-F. Ye, and X.-X. Li, “Unique band coalescence and exceptional points from two-dimensional photonic crystal waveguide,” J. Opt. 21, 055103 (2019).
[Crossref]

M. Bagheriasl, O. Quevedo-Teruel, and G. Valerio, “Bloch analysis of artificial lines and surfaces exhibiting glide symmetry,” IEEE Trans. Microwave Theory Tech. 67, 2618–2628 (2019).
[Crossref]

2018 (5)

D. M. Tsvetkov, V. A. Bushuev, V. V. Konotop, and B. I. Mantsyzov, “Broadband quasi-PT-symmetry sustained by inhomogeneous broadening of the spectral line,” Phys. Rev. A 98, 053844 (2018).
[Crossref]

G. Valerio, F. Ghasemifard, Z. Sipus, and O. Quevedo-Teruel, “Glide-symmetric all-metal holey metasurfaces for low-dispersive artificial materials: modeling and properties,” IEEE Trans. Microwave Theory Tech. 66, 3210–3223 (2018).
[Crossref]

F. Ghasemifard, M. Norgen, O. Quevedo-Teruel, and G. Valerio, “Analyzing glide-symmetric holey metasurfaces using a generalized Floquet theorem,” IEEE Access 6, 71743–71750 (2018).
[Crossref]

L. Ge, “Non-Hermitian lattices with a flat band and polynomial power increase,” Photon. Res. 6, A10–A17 (2018).
[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, 11–19 (2018).
[Crossref]

2017 (4)

A. Mock, “Comprehensive understanding of parity-time transitions in PT symmetric photonic crystals with an antiunitary group theory,” Phys. Rev. A 95, 043803 (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, 187–191 (2017).
[Crossref]

W. W. Ahmed, R. Herrero, M. Botey, and K. Staliunas, “Self-collimation in PT-symmetric crystals,” Phys. Rev. A 95, 053830 (2017).
[Crossref]

W. Liu, M. Li, R. S. Guzzon, E. J. Norberg, J. S. Parker, M. Lu, L. A. Coldren, and J. Yao, “An integrated parity-time symmetric wavelength-tunable single-mode microring laser,” Nat. Commun. 8, 15389 (2017).
[Crossref]

2016 (5)

A. Cerjan, A. Raman, and S. Fan, “Exceptional contours and band structure design in parity-time symmetric photonic crystals,” Phys. Rev. Lett. 116, 203902 (2016).
[Crossref]

A. Cerjan and S. Fan, “Effects of non-uniform distributions of gain and loss in photonic crystals,” New J. Phys. 18, 125007 (2016).
[Crossref]

H. Hodaei, M.-A. Miri, A. U. Hassan, W. E. Hayenga, M. Heinrich, D. N. Christodoulides, and M. Khajavikhan, “Single mode lasing in transversely multi-moded PT-symmetric microring resonators,” Laser Photon. Rev. 10, 494–499 (2016).
[Crossref]

P. Miao, Z. Zhang, J. Sun, W. Walasik, S. Longhi, N. M. Litchinitser, and L. Feng, “Orbital angular momentum microlaser,” Science 353, 464–467 (2016).
[Crossref]

A. Mock, “Characterization of parity-time symmetry in photonic lattices using Heesh-Shubnikov group theory,” Opt. Express 24, 22693–22707 (2016).
[Crossref]

2015 (5)

H. Hodaei, M.-A. Miri, A. U. Hassan, W. E. Hayenga, M. Heinrich, D. N. Christodoulides, and M. Khajavikhan, “Parity-time-symmetric coupled microring lasers operating around an exceptional point,” Opt. Lett. 40, 4955–4958 (2015).
[Crossref]

H. Wang, S. Shi, X. Ren, X. Zhu, B. A. Malomed, D. Mihalache, and Y. He, “Two-dimensional solitons in triangular photonic lattices with parity-time symmetry,” Opt. Commun. 335, 146–152 (2015).
[Crossref]

K. S. Agarwal, R. K. Pathak, and Y. N. Joglekar, “Exactly solvable-symmetric models in two dimensions,” Europhys. Lett. 112, 31003 (2015).
[Crossref]

M. Turdeuev, M. Botey, I. Giden, R. Herrero, H. Kurt, E. Ozbay, and K. Staliunas, “Two-dimensional complex parity-time-symmetric photonic structures,” Phys. Rev. A 91, 023825 (2015).
[Crossref]

B. Zhen, C. W. Hsu, Y. Igarashi, L. Lu, I. Kaminer, A. Pick, S.-L. Chua, J. D. Joannopoulos, and M. Soljačić, “Spawning rings of exceptional points out of Dirac cones,” Nature 525, 354–358 (2015).
[Crossref]

2014 (6)

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

J. Xie, Z. Su, W. Chen, G. Chen, J. Lv, D. Mihalache, and Y. He, “Defect solitons in two-dimensional photonic lattices with parity-time symmetry,” Opt. Commun. 313, 139–145 (2014).
[Crossref]

L. Ge and A. D. Stone, “Parity-time symmetry breaking beyond one dimension: the role of degeneracy,” Phys. Rev. X 4, 031011 (2014).
[Crossref]

L. Feng, Z. J. Wong, R.-M. Ma, Y. Wang, and X. Zhang, “Single-mode laser by parity-time symmetry breaking,” Science 346, 972–975 (2014).
[Crossref]

H. Hodaei, M.-A. Miri, M. Heinrich, D. N. Christodoulides, and M. Khajavikhan, “Parity-time-symmetric microring lasers,” Science 346, 975–978 (2014).
[Crossref]

B. Peng, Ş. K. Özdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery micro cavities,” Nat. Phys. 10, 394–398 (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, 108–113 (2013).
[Crossref]

2012 (2)

2011 (3)

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, 213901 (2011).
[Crossref]

L. Feng, M. Ayache, J. Huang, Y.-L. Xu, M.-H. Lu, Y.-F. Chen, Y. Fainman, and A. Scherer, “Nonreciprocal light propagation in a silicon photonic circuit,” Science 333, 729–733 (2011).
[Crossref]

A. Szameit, M. C. Rechtsman, O. Bahat-Treidel, and M. Segev, “PT-symmetry in honeycomb photonic lattices,” Phys. Rev. A 84, 021806 (2011).
[Crossref]

2010 (2)

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, 192–195 (2010).
[Crossref]

A. Mock, L. Lu, and J. O’Brien, “Space group theory and Fourier space analysis of two-dimensional photonic crystal waveguides,” Phys. Rev. B 81, 155115 (2010).
[Crossref]

2009 (2)

L. Lu, A. Mock, T. Yang, M. H. Shih, E. H. Hwang, M. Bagheri, A. Stapleton, S. Farrell, J. D. O’Brien, and P. D. Dapkus, “120 µW peak output power from edge-emitting photonic crystal double heterostructure nanocavity lasers,” Appl. Phys. Lett. 94, 111101 (2009).
[Crossref]

L. Lu, A. Mock, E. H. Hwang, J. O’Brien, and P. D. Dapkus, “High-peak-power efficient edge-emitting photonic crystal nanocavity lasers,” Opt. Lett. 34, 2346–2648 (2009).
[Crossref]

2008 (2)

2007 (1)

2006 (1)

M. H. Shih, W. Kuang, A. Mock, M. Bagheri, E. H. Hwang, J. D. O’Brien, and P. D. Dapkus, “High-quality-factor photonic crystal heterostructure laser,” Appl. Phys. Lett. 89, 101104 (2006).
[Crossref]

2004 (2)

W. Kuang and J. O’Brien, “Reducing the out-of-plane radiation loss of photonic crystal waveguides on high-index substrates,” Opt. Lett. 29, 860–862 (2004).
[Crossref]

H.-G. Park, S.-H. Kim, S.-H. Kwon, Y.-G. Ju, H.-K. Yang, J.-H. Baek, S.-B. Kim, and Y.-H. Lee, “Electrically driven single-cell photonic crystal laser,” Science 305, 1444–1447 (2004).
[Crossref]

2002 (2)

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

K. Srinivasan and O. Painter, “Momentum space design of high-Q photonic crystal optical cavities,” Opt. Express 10, 670–684 (2002).
[Crossref]

1999 (2)

1998 (1)

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

1996 (1)

H. Benisty, “Modal analysis of optical guides with two-dimensional photonic band-gap boundaries,” J. Appl. Phys. 79, 7483–7492 (1996).
[Crossref]

1962 (1)

J. O. Dimmock and R. G. Wheeler, “Symmetry properties of wave functions in magnetic crystals,” Phys. Rev. 127, 391–404 (1962).
[Crossref]

1929 (1)

H. Heesh, “Zur Strukturtheorie der ebenen Symmetriegruppen,” Zeitschrift für Kristallographie-Cryst. Mater. 71, 95–102 (1929).

Agarwal, K. S.

K. S. Agarwal, R. K. Pathak, and Y. N. Joglekar, “Exactly solvable-symmetric models in two dimensions,” Europhys. Lett. 112, 31003 (2015).
[Crossref]

Ahmed, W. W.

W. W. Ahmed, R. Herrero, M. Botey, and K. Staliunas, “Self-collimation in PT-symmetric crystals,” Phys. Rev. A 95, 053830 (2017).
[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, 108–113 (2013).
[Crossref]

Ayache, M.

L. Feng, M. Ayache, J. Huang, Y.-L. Xu, M.-H. Lu, Y.-F. Chen, Y. Fainman, and A. Scherer, “Nonreciprocal light propagation in a silicon photonic circuit,” Science 333, 729–733 (2011).
[Crossref]

Baek, J.-H.

H.-G. Park, S.-H. Kim, S.-H. Kwon, Y.-G. Ju, H.-K. Yang, J.-H. Baek, S.-B. Kim, and Y.-H. Lee, “Electrically driven single-cell photonic crystal laser,” Science 305, 1444–1447 (2004).
[Crossref]

Bagheri, M.

L. Lu, A. Mock, T. Yang, M. H. Shih, E. H. Hwang, M. Bagheri, A. Stapleton, S. Farrell, J. D. O’Brien, and P. D. Dapkus, “120 µW peak output power from edge-emitting photonic crystal double heterostructure nanocavity lasers,” Appl. Phys. Lett. 94, 111101 (2009).
[Crossref]

L. Lu, A. Mock, M. Bagheri, E. H. Hwang, J. D. O’Brien, and P. D. Dapkus, “Double-heterostructure photonic crystal lasers with reduced threshold pump power and increased slope efficiency obtained by quantum well intermixing,” Opt. Express 16, 17342–17347 (2008).
[Crossref]

M. H. Shih, W. Kuang, A. Mock, M. Bagheri, E. H. Hwang, J. D. O’Brien, and P. D. Dapkus, “High-quality-factor photonic crystal heterostructure laser,” Appl. Phys. Lett. 89, 101104 (2006).
[Crossref]

Bagheriasl, M.

M. Bagheriasl, O. Quevedo-Teruel, and G. Valerio, “Bloch analysis of artificial lines and surfaces exhibiting glide symmetry,” IEEE Trans. Microwave Theory Tech. 67, 2618–2628 (2019).
[Crossref]

Bahat-Treidel, O.

A. Szameit, M. C. Rechtsman, O. Bahat-Treidel, and M. Segev, “PT-symmetry in honeycomb photonic lattices,” Phys. Rev. A 84, 021806 (2011).
[Crossref]

Belov, N. V.

A. V. Shubnikov and N. V. Belov, Colored Symmetry (Macmillan, 1964).

Bender, C. M.

B. Peng, Ş. K. Özdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery micro cavities,” Nat. Phys. 10, 394–398 (2014).
[Crossref]

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

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

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

Benisty, H.

H. Benisty, “Modal analysis of optical guides with two-dimensional photonic band-gap boundaries,” J. Appl. Phys. 79, 7483–7492 (1996).
[Crossref]

Benisty, H. H.

Biesenthal, T.

M. Kremer, T. Biesenthal, L. J. Maczewsky, M. Heinrich, R. Thomale, and A. Szameit, “Demonstration of a two-dimensional PT-symmetric crystal,” Nat. Commun. 10, 435 (2019).
[Crossref]

Boettcher, S.

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

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

Botey, M.

W. W. Ahmed, R. Herrero, M. Botey, and K. Staliunas, “Self-collimation in PT-symmetric crystals,” Phys. Rev. A 95, 053830 (2017).
[Crossref]

M. Turdeuev, M. Botey, I. Giden, R. Herrero, H. Kurt, E. Ozbay, and K. Staliunas, “Two-dimensional complex parity-time-symmetric photonic structures,” Phys. Rev. A 91, 023825 (2015).
[Crossref]

Brody, D. C.

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

Bushuev, V. A.

D. M. Tsvetkov, V. A. Bushuev, V. V. Konotop, and B. I. Mantsyzov, “Broadband quasi-PT-symmetry sustained by inhomogeneous broadening of the spectral line,” Phys. Rev. A 98, 053844 (2018).
[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, 213901 (2011).
[Crossref]

Cassagne, D.

Cerjan, A.

A. Cerjan, A. Raman, and S. Fan, “Exceptional contours and band structure design in parity-time symmetric photonic crystals,” Phys. Rev. Lett. 116, 203902 (2016).
[Crossref]

A. Cerjan and S. Fan, “Effects of non-uniform distributions of gain and loss in photonic crystals,” New J. Phys. 18, 125007 (2016).
[Crossref]

Chen, G.

J. Xie, Z. Su, W. Chen, G. Chen, J. Lv, D. Mihalache, and Y. He, “Defect solitons in two-dimensional photonic lattices with parity-time symmetry,” Opt. Commun. 313, 139–145 (2014).
[Crossref]

Chen, W.

J. Xie, Z. Su, W. Chen, G. Chen, J. Lv, D. Mihalache, and Y. He, “Defect solitons in two-dimensional photonic lattices with parity-time symmetry,” Opt. Commun. 313, 139–145 (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, 108–113 (2013).
[Crossref]

L. Feng, M. Ayache, J. Huang, Y.-L. Xu, M.-H. Lu, Y.-F. Chen, Y. Fainman, and A. Scherer, “Nonreciprocal light propagation in a silicon photonic circuit,” Science 333, 729–733 (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 PT symmetry,” Nat. Phys. 14, 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, 187–191 (2017).
[Crossref]

H. Hodaei, M.-A. Miri, A. U. Hassan, W. E. Hayenga, M. Heinrich, D. N. Christodoulides, and M. Khajavikhan, “Single mode lasing in transversely multi-moded PT-symmetric microring resonators,” Laser Photon. Rev. 10, 494–499 (2016).
[Crossref]

H. Hodaei, M.-A. Miri, A. U. Hassan, W. E. Hayenga, M. Heinrich, D. N. Christodoulides, and M. Khajavikhan, “Parity-time-symmetric coupled microring lasers operating around an exceptional point,” Opt. Lett. 40, 4955–4958 (2015).
[Crossref]

H. Hodaei, M.-A. Miri, M. Heinrich, D. N. Christodoulides, and M. Khajavikhan, “Parity-time-symmetric microring lasers,” Science 346, 975–978 (2014).
[Crossref]

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional invisibility induced by PT-symmetric periodic structures,” Phys. Rev. Lett. 106, 213901 (2011).
[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, 192–195 (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, 030402 (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, 2632–2634 (2007).
[Crossref]

Chua, S.-L.

B. Zhen, C. W. Hsu, Y. Igarashi, L. Lu, I. Kaminer, A. Pick, S.-L. Chua, J. D. Joannopoulos, and M. Soljačić, “Spawning rings of exceptional points out of Dirac cones,” Nature 525, 354–358 (2015).
[Crossref]

Coldren, L. A.

W. Liu, M. Li, R. S. Guzzon, E. J. Norberg, J. S. Parker, M. Lu, L. A. Coldren, and J. Yao, “An integrated parity-time symmetric wavelength-tunable single-mode microring laser,” Nat. Commun. 8, 15389 (2017).
[Crossref]

L. A. Coldren and S. W. Corzine, Diode Lasers and Photonic Integrated Circuits (Wiley, 1995).

Corzine, S. W.

L. A. Coldren and S. W. Corzine, Diode Lasers and Photonic Integrated Circuits (Wiley, 1995).

Cracknell, A. P.

A. P. Cracknell, Group Theory in Solid-State Physics (Taylor and Francis, 1975).

A. P. Cracknell, Magnetism in Crystalline Materials (Pergamon, 1975).

Dapkus, P. D.

L. Lu, A. Mock, T. Yang, M. H. Shih, E. H. Hwang, M. Bagheri, A. Stapleton, S. Farrell, J. D. O’Brien, and P. D. Dapkus, “120 µW peak output power from edge-emitting photonic crystal double heterostructure nanocavity lasers,” Appl. Phys. Lett. 94, 111101 (2009).
[Crossref]

L. Lu, A. Mock, E. H. Hwang, J. O’Brien, and P. D. Dapkus, “High-peak-power efficient edge-emitting photonic crystal nanocavity lasers,” Opt. Lett. 34, 2346–2648 (2009).
[Crossref]

L. Lu, A. Mock, M. Bagheri, E. H. Hwang, J. D. O’Brien, and P. D. Dapkus, “Double-heterostructure photonic crystal lasers with reduced threshold pump power and increased slope efficiency obtained by quantum well intermixing,” Opt. Express 16, 17342–17347 (2008).
[Crossref]

M. H. Shih, W. Kuang, A. Mock, M. Bagheri, E. H. Hwang, J. D. O’Brien, and P. D. Dapkus, “High-quality-factor photonic crystal heterostructure laser,” Appl. Phys. Lett. 89, 101104 (2006).
[Crossref]

Dimmock, J. O.

J. O. Dimmock and R. G. Wheeler, “Symmetry properties of wave functions in magnetic crystals,” Phys. Rev. 127, 391–404 (1962).
[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 PT-symmetric optical systems,” Phys. Rev. A 89, 033808 (2014).
[Crossref]

Eichelkraut, T.

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

El-Batanouny, M.

M. El-Batanouny and F. Wooten, Symmetry and Condensed Matter Physics: A Computational Approach (Cambridge University, 2008).

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, 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, 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, 192–195 (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, 030402 (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, 2632–2634 (2007).
[Crossref]

Fainman, Y.

L. Feng, M. Ayache, J. Huang, Y.-L. Xu, M.-H. Lu, Y.-F. Chen, Y. Fainman, and A. Scherer, “Nonreciprocal light propagation in a silicon photonic circuit,” Science 333, 729–733 (2011).
[Crossref]

Fan, S.

A. Cerjan, A. Raman, and S. Fan, “Exceptional contours and band structure design in parity-time symmetric photonic crystals,” Phys. Rev. Lett. 116, 203902 (2016).
[Crossref]

A. Cerjan and S. Fan, “Effects of non-uniform distributions of gain and loss in photonic crystals,” New J. Phys. 18, 125007 (2016).
[Crossref]

B. Peng, Ş. K. Özdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery micro cavities,” Nat. Phys. 10, 394–398 (2014).
[Crossref]

Fang, Y.-T.

Y.-T. Fang, S.-F. Ye, and X.-X. Li, “Unique band coalescence and exceptional points from two-dimensional photonic crystal waveguide,” J. Opt. 21, 055103 (2019).
[Crossref]

Farrell, S.

L. Lu, A. Mock, T. Yang, M. H. Shih, E. H. Hwang, M. Bagheri, A. Stapleton, S. Farrell, J. D. O’Brien, and P. D. Dapkus, “120 µW peak output power from edge-emitting photonic crystal double heterostructure nanocavity lasers,” Appl. Phys. Lett. 94, 111101 (2009).
[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, 108–113 (2013).
[Crossref]

Feng, L.

P. Miao, Z. Zhang, J. Sun, W. Walasik, S. Longhi, N. M. Litchinitser, and L. Feng, “Orbital angular momentum microlaser,” Science 353, 464–467 (2016).
[Crossref]

L. Feng, Z. J. Wong, R.-M. Ma, Y. Wang, and X. Zhang, “Single-mode laser by parity-time symmetry breaking,” Science 346, 972–975 (2014).
[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, 108–113 (2013).
[Crossref]

L. Feng, M. Ayache, J. Huang, Y.-L. Xu, M.-H. Lu, Y.-F. Chen, Y. Fainman, and A. Scherer, “Nonreciprocal light propagation in a silicon photonic circuit,” Science 333, 729–733 (2011).
[Crossref]

Frobenius, G.

G. Frobenius and I. Schur, Über die reellen Darstellungen der endlichen Gruppen (Sitzungsberichte Der Berliner Mathematischen Gesellschaft, 1906), pp. 186–208.

Garcia-Gracia, H.

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, 187–191 (2017).
[Crossref]

Ge, L.

L. Ge, “Non-Hermitian lattices with a flat band and polynomial power increase,” Photon. Res. 6, A10–A17 (2018).
[Crossref]

L. Ge and A. D. Stone, “Parity-time symmetry breaking beyond one dimension: the role of degeneracy,” Phys. Rev. X 4, 031011 (2014).
[Crossref]

Ghasemifard, F.

G. Valerio, F. Ghasemifard, Z. Sipus, and O. Quevedo-Teruel, “Glide-symmetric all-metal holey metasurfaces for low-dispersive artificial materials: modeling and properties,” IEEE Trans. Microwave Theory Tech. 66, 3210–3223 (2018).
[Crossref]

F. Ghasemifard, M. Norgen, O. Quevedo-Teruel, and G. Valerio, “Analyzing glide-symmetric holey metasurfaces using a generalized Floquet theorem,” IEEE Access 6, 71743–71750 (2018).
[Crossref]

Gianfreda, M.

B. Peng, Ş. K. Özdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery micro cavities,” Nat. Phys. 10, 394–398 (2014).
[Crossref]

Giden, I.

M. Turdeuev, M. Botey, I. Giden, R. Herrero, H. Kurt, E. Ozbay, and K. Staliunas, “Two-dimensional complex parity-time-symmetric photonic structures,” Phys. Rev. A 91, 023825 (2015).
[Crossref]

Guzzon, R. S.

W. Liu, M. Li, R. S. Guzzon, E. J. Norberg, J. S. Parker, M. Lu, L. A. Coldren, and J. Yao, “An integrated parity-time symmetric wavelength-tunable single-mode microring laser,” Nat. Commun. 8, 15389 (2017).
[Crossref]

Hasebe, K.

Hassan, A. U.

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, 187–191 (2017).
[Crossref]

H. Hodaei, M.-A. Miri, A. U. Hassan, W. E. Hayenga, M. Heinrich, D. N. Christodoulides, and M. Khajavikhan, “Single mode lasing in transversely multi-moded PT-symmetric microring resonators,” Laser Photon. Rev. 10, 494–499 (2016).
[Crossref]

H. Hodaei, M.-A. Miri, A. U. Hassan, W. E. Hayenga, M. Heinrich, D. N. Christodoulides, and M. Khajavikhan, “Parity-time-symmetric coupled microring lasers operating around an exceptional point,” Opt. Lett. 40, 4955–4958 (2015).
[Crossref]

Hayenga, W. E.

H. Hodaei, M.-A. Miri, A. U. Hassan, W. E. Hayenga, M. Heinrich, D. N. Christodoulides, and M. Khajavikhan, “Single mode lasing in transversely multi-moded PT-symmetric microring resonators,” Laser Photon. Rev. 10, 494–499 (2016).
[Crossref]

H. Hodaei, M.-A. Miri, A. U. Hassan, W. E. Hayenga, M. Heinrich, D. N. Christodoulides, and M. Khajavikhan, “Parity-time-symmetric coupled microring lasers operating around an exceptional point,” Opt. Lett. 40, 4955–4958 (2015).
[Crossref]

He, Y.

H. Wang, S. Shi, X. Ren, X. Zhu, B. A. Malomed, D. Mihalache, and Y. He, “Two-dimensional solitons in triangular photonic lattices with parity-time symmetry,” Opt. Commun. 335, 146–152 (2015).
[Crossref]

J. Xie, Z. Su, W. Chen, G. Chen, J. Lv, D. Mihalache, and Y. He, “Defect solitons in two-dimensional photonic lattices with parity-time symmetry,” Opt. Commun. 313, 139–145 (2014).
[Crossref]

Heesh, H.

H. Heesh, “Zur Strukturtheorie der ebenen Symmetriegruppen,” Zeitschrift für Kristallographie-Cryst. Mater. 71, 95–102 (1929).

Heinrich, M.

M. Kremer, T. Biesenthal, L. J. Maczewsky, M. Heinrich, R. Thomale, and A. Szameit, “Demonstration of a two-dimensional PT-symmetric crystal,” Nat. Commun. 10, 435 (2019).
[Crossref]

H. Hodaei, M.-A. Miri, A. U. Hassan, W. E. Hayenga, M. Heinrich, D. N. Christodoulides, and M. Khajavikhan, “Single mode lasing in transversely multi-moded PT-symmetric microring resonators,” Laser Photon. Rev. 10, 494–499 (2016).
[Crossref]

H. Hodaei, M.-A. Miri, A. U. Hassan, W. E. Hayenga, M. Heinrich, D. N. Christodoulides, and M. Khajavikhan, “Parity-time-symmetric coupled microring lasers operating around an exceptional point,” Opt. Lett. 40, 4955–4958 (2015).
[Crossref]

H. Hodaei, M.-A. Miri, M. Heinrich, D. N. Christodoulides, and M. Khajavikhan, “Parity-time-symmetric microring lasers,” Science 346, 975–978 (2014).
[Crossref]

Herrero, R.

W. W. Ahmed, R. Herrero, M. Botey, and K. Staliunas, “Self-collimation in PT-symmetric crystals,” Phys. Rev. A 95, 053830 (2017).
[Crossref]

M. Turdeuev, M. Botey, I. Giden, R. Herrero, H. Kurt, E. Ozbay, and K. Staliunas, “Two-dimensional complex parity-time-symmetric photonic structures,” Phys. Rev. A 91, 023825 (2015).
[Crossref]

Hodaei, H.

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, 187–191 (2017).
[Crossref]

H. Hodaei, M.-A. Miri, A. U. Hassan, W. E. Hayenga, M. Heinrich, D. N. Christodoulides, and M. Khajavikhan, “Single mode lasing in transversely multi-moded PT-symmetric microring resonators,” Laser Photon. Rev. 10, 494–499 (2016).
[Crossref]

H. Hodaei, M.-A. Miri, A. U. Hassan, W. E. Hayenga, M. Heinrich, D. N. Christodoulides, and M. Khajavikhan, “Parity-time-symmetric coupled microring lasers operating around an exceptional point,” Opt. Lett. 40, 4955–4958 (2015).
[Crossref]

H. Hodaei, M.-A. Miri, M. Heinrich, D. N. Christodoulides, and M. Khajavikhan, “Parity-time-symmetric microring lasers,” Science 346, 975–978 (2014).
[Crossref]

Houdré, R.

Hsu, C. W.

B. Zhen, C. W. Hsu, Y. Igarashi, L. Lu, I. Kaminer, A. Pick, S.-L. Chua, J. D. Joannopoulos, and M. Soljačić, “Spawning rings of exceptional points out of Dirac cones,” Nature 525, 354–358 (2015).
[Crossref]

Huang, J.

L. Feng, M. Ayache, J. Huang, Y.-L. Xu, M.-H. Lu, Y.-F. Chen, Y. Fainman, and A. Scherer, “Nonreciprocal light propagation in a silicon photonic circuit,” Science 333, 729–733 (2011).
[Crossref]

Hwang, E. H.

L. Lu, A. Mock, E. H. Hwang, J. O’Brien, and P. D. Dapkus, “High-peak-power efficient edge-emitting photonic crystal nanocavity lasers,” Opt. Lett. 34, 2346–2648 (2009).
[Crossref]

L. Lu, A. Mock, T. Yang, M. H. Shih, E. H. Hwang, M. Bagheri, A. Stapleton, S. Farrell, J. D. O’Brien, and P. D. Dapkus, “120 µW peak output power from edge-emitting photonic crystal double heterostructure nanocavity lasers,” Appl. Phys. Lett. 94, 111101 (2009).
[Crossref]

L. Lu, A. Mock, M. Bagheri, E. H. Hwang, J. D. O’Brien, and P. D. Dapkus, “Double-heterostructure photonic crystal lasers with reduced threshold pump power and increased slope efficiency obtained by quantum well intermixing,” Opt. Express 16, 17342–17347 (2008).
[Crossref]

M. H. Shih, W. Kuang, A. Mock, M. Bagheri, E. H. Hwang, J. D. O’Brien, and P. D. Dapkus, “High-quality-factor photonic crystal heterostructure laser,” Appl. Phys. Lett. 89, 101104 (2006).
[Crossref]

Igarashi, Y.

B. Zhen, C. W. Hsu, Y. Igarashi, L. Lu, I. Kaminer, A. Pick, S.-L. Chua, J. D. Joannopoulos, and M. Soljačić, “Spawning rings of exceptional points out of Dirac cones,” Nature 525, 354–358 (2015).
[Crossref]

Joannopoulos, J. D.

B. Zhen, C. W. Hsu, Y. Igarashi, L. Lu, I. Kaminer, A. Pick, S.-L. Chua, J. D. Joannopoulos, and M. Soljačić, “Spawning rings of exceptional points out of Dirac cones,” Nature 525, 354–358 (2015).
[Crossref]

Joannopoulos, S. G.

S. G. Joannopoulos, J. D. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light, 2nd ed. (Princeton University, 2008).

Joglekar, Y. N.

K. S. Agarwal, R. K. Pathak, and Y. N. Joglekar, “Exactly solvable-symmetric models in two dimensions,” Europhys. Lett. 112, 31003 (2015).
[Crossref]

Johnson, J. D.

S. G. Joannopoulos, J. D. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light, 2nd ed. (Princeton University, 2008).

Jones, H. F.

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

Jouanin, C.

Ju, Y.-G.

H.-G. Park, S.-H. Kim, S.-H. Kwon, Y.-G. Ju, H.-K. Yang, J.-H. Baek, S.-B. Kim, and Y.-H. Lee, “Electrically driven single-cell photonic crystal laser,” Science 305, 1444–1447 (2004).
[Crossref]

Kakitsuka, T.

Kaminer, I.

B. Zhen, C. W. Hsu, Y. Igarashi, L. Lu, I. Kaminer, A. Pick, S.-L. Chua, J. D. Joannopoulos, and M. Soljačić, “Spawning rings of exceptional points out of Dirac cones,” Nature 525, 354–358 (2015).
[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, 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, 187–191 (2017).
[Crossref]

H. Hodaei, M.-A. Miri, A. U. Hassan, W. E. Hayenga, M. Heinrich, D. N. Christodoulides, and M. Khajavikhan, “Single mode lasing in transversely multi-moded PT-symmetric microring resonators,” Laser Photon. Rev. 10, 494–499 (2016).
[Crossref]

H. Hodaei, M.-A. Miri, A. U. Hassan, W. E. Hayenga, M. Heinrich, D. N. Christodoulides, and M. Khajavikhan, “Parity-time-symmetric coupled microring lasers operating around an exceptional point,” Opt. Lett. 40, 4955–4958 (2015).
[Crossref]

H. Hodaei, M.-A. Miri, M. Heinrich, D. N. Christodoulides, and M. Khajavikhan, “Parity-time-symmetric microring lasers,” Science 346, 975–978 (2014).
[Crossref]

Kim, S.-B.

H.-G. Park, S.-H. Kim, S.-H. Kwon, Y.-G. Ju, H.-K. Yang, J.-H. Baek, S.-B. Kim, and Y.-H. Lee, “Electrically driven single-cell photonic crystal laser,” Science 305, 1444–1447 (2004).
[Crossref]

Kim, S.-H.

H.-G. Park, S.-H. Kim, S.-H. Kwon, Y.-G. Ju, H.-K. Yang, J.-H. Baek, S.-B. Kim, and Y.-H. Lee, “Electrically driven single-cell photonic crystal laser,” Science 305, 1444–1447 (2004).
[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, 192–195 (2010).
[Crossref]

Konotop, V. V.

D. M. Tsvetkov, V. A. Bushuev, V. V. Konotop, and B. I. Mantsyzov, “Broadband quasi-PT-symmetry sustained by inhomogeneous broadening of the spectral line,” Phys. Rev. A 98, 053844 (2018).
[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, 213901 (2011).
[Crossref]

Krauss, T. F.

Kremer, M.

M. Kremer, T. Biesenthal, L. J. Maczewsky, M. Heinrich, R. Thomale, and A. Szameit, “Demonstration of a two-dimensional PT-symmetric crystal,” Nat. Commun. 10, 435 (2019).
[Crossref]

Kuang, W.

M. H. Shih, W. Kuang, A. Mock, M. Bagheri, E. H. Hwang, J. D. O’Brien, and P. D. Dapkus, “High-quality-factor photonic crystal heterostructure laser,” Appl. Phys. Lett. 89, 101104 (2006).
[Crossref]

W. Kuang and J. O’Brien, “Reducing the out-of-plane radiation loss of photonic crystal waveguides on high-index substrates,” Opt. Lett. 29, 860–862 (2004).
[Crossref]

Kurt, H.

M. Turdeuev, M. Botey, I. Giden, R. Herrero, H. Kurt, E. Ozbay, and K. Staliunas, “Two-dimensional complex parity-time-symmetric photonic structures,” Phys. Rev. A 91, 023825 (2015).
[Crossref]

Kwon, S.-H.

H.-G. Park, S.-H. Kim, S.-H. Kwon, Y.-G. Ju, H.-K. Yang, J.-H. Baek, S.-B. Kim, and Y.-H. Lee, “Electrically driven single-cell photonic crystal laser,” Science 305, 1444–1447 (2004).
[Crossref]

Labilloy, D.

Lee, Y.-H.

H.-G. Park, S.-H. Kim, S.-H. Kwon, Y.-G. Ju, H.-K. Yang, J.-H. Baek, S.-B. Kim, and Y.-H. Lee, “Electrically driven single-cell photonic crystal laser,” Science 305, 1444–1447 (2004).
[Crossref]

Lei, F.

B. Peng, Ş. K. Özdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery micro cavities,” Nat. Phys. 10, 394–398 (2014).
[Crossref]

Li, M.

W. Liu, M. Li, R. S. Guzzon, E. J. Norberg, J. S. Parker, M. Lu, L. A. Coldren, and J. Yao, “An integrated parity-time symmetric wavelength-tunable single-mode microring laser,” Nat. Commun. 8, 15389 (2017).
[Crossref]

Li, X.-X.

Y.-T. Fang, S.-F. Ye, and X.-X. Li, “Unique band coalescence and exceptional points from two-dimensional photonic crystal waveguide,” J. Opt. 21, 055103 (2019).
[Crossref]

Liboff, R.

R. Liboff, Primer for Point and Space Groups (Springer-Verlag, 2004).

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, 213901 (2011).
[Crossref]

Lisyansky, A. A.

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

Litchinitser, N. M.

P. Miao, Z. Zhang, J. Sun, W. Walasik, S. Longhi, N. M. Litchinitser, and L. Feng, “Orbital angular momentum microlaser,” Science 353, 464–467 (2016).
[Crossref]

Liu, W.

W. Liu, M. Li, R. S. Guzzon, E. J. Norberg, J. S. Parker, M. Lu, L. A. Coldren, and J. Yao, “An integrated parity-time symmetric wavelength-tunable single-mode microring laser,” Nat. Commun. 8, 15389 (2017).
[Crossref]

Long, G. L.

B. Peng, Ş. K. Özdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery micro cavities,” Nat. Phys. 10, 394–398 (2014).
[Crossref]

Longhi, S.

P. Miao, Z. Zhang, J. Sun, W. Walasik, S. Longhi, N. M. Litchinitser, and L. Feng, “Orbital angular momentum microlaser,” Science 353, 464–467 (2016).
[Crossref]

Lu, L.

B. Zhen, C. W. Hsu, Y. Igarashi, L. Lu, I. Kaminer, A. Pick, S.-L. Chua, J. D. Joannopoulos, and M. Soljačić, “Spawning rings of exceptional points out of Dirac cones,” Nature 525, 354–358 (2015).
[Crossref]

A. Mock, L. Lu, and J. O’Brien, “Space group theory and Fourier space analysis of two-dimensional photonic crystal waveguides,” Phys. Rev. B 81, 155115 (2010).
[Crossref]

L. Lu, A. Mock, T. Yang, M. H. Shih, E. H. Hwang, M. Bagheri, A. Stapleton, S. Farrell, J. D. O’Brien, and P. D. Dapkus, “120 µW peak output power from edge-emitting photonic crystal double heterostructure nanocavity lasers,” Appl. Phys. Lett. 94, 111101 (2009).
[Crossref]

L. Lu, A. Mock, E. H. Hwang, J. O’Brien, and P. D. Dapkus, “High-peak-power efficient edge-emitting photonic crystal nanocavity lasers,” Opt. Lett. 34, 2346–2648 (2009).
[Crossref]

L. Lu, A. Mock, M. Bagheri, E. H. Hwang, J. D. O’Brien, and P. D. Dapkus, “Double-heterostructure photonic crystal lasers with reduced threshold pump power and increased slope efficiency obtained by quantum well intermixing,” Opt. Express 16, 17342–17347 (2008).
[Crossref]

Lu, M.

W. Liu, M. Li, R. S. Guzzon, E. J. Norberg, J. S. Parker, M. Lu, L. A. Coldren, and J. Yao, “An integrated parity-time symmetric wavelength-tunable single-mode microring laser,” Nat. Commun. 8, 15389 (2017).
[Crossref]

Lu, M.-H.

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

L. Feng, M. Ayache, J. Huang, Y.-L. Xu, M.-H. Lu, Y.-F. Chen, Y. Fainman, and A. Scherer, “Nonreciprocal light propagation in a silicon photonic circuit,” Science 333, 729–733 (2011).
[Crossref]

Lv, J.

J. Xie, Z. Su, W. Chen, G. Chen, J. Lv, D. Mihalache, and Y. He, “Defect solitons in two-dimensional photonic lattices with parity-time symmetry,” Opt. Commun. 313, 139–145 (2014).
[Crossref]

Ma, R.-M.

L. Feng, Z. J. Wong, R.-M. Ma, Y. Wang, and X. Zhang, “Single-mode laser by parity-time symmetry breaking,” Science 346, 972–975 (2014).
[Crossref]

Maczewsky, L. J.

M. Kremer, T. Biesenthal, L. J. Maczewsky, M. Heinrich, R. Thomale, and A. Szameit, “Demonstration of a two-dimensional PT-symmetric crystal,” Nat. Commun. 10, 435 (2019).
[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, 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, 192–195 (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, 030402 (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, 2632–2634 (2007).
[Crossref]

Malomed, B. A.

H. Wang, S. Shi, X. Ren, X. Zhu, B. A. Malomed, D. Mihalache, and Y. He, “Two-dimensional solitons in triangular photonic lattices with parity-time symmetry,” Opt. Commun. 335, 146–152 (2015).
[Crossref]

Mantsyzov, B. I.

D. M. Tsvetkov, V. A. Bushuev, V. V. Konotop, and B. I. Mantsyzov, “Broadband quasi-PT-symmetry sustained by inhomogeneous broadening of the spectral line,” Phys. Rev. A 98, 053844 (2018).
[Crossref]

Matsuo, S.

Meade, R. D.

S. G. Joannopoulos, J. D. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light, 2nd ed. (Princeton University, 2008).

Meisinger, P. N.

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

Miao, P.

P. Miao, Z. Zhang, J. Sun, W. Walasik, S. Longhi, N. M. Litchinitser, and L. Feng, “Orbital angular momentum microlaser,” Science 353, 464–467 (2016).
[Crossref]

Mihalache, D.

H. Wang, S. Shi, X. Ren, X. Zhu, B. A. Malomed, D. Mihalache, and Y. He, “Two-dimensional solitons in triangular photonic lattices with parity-time symmetry,” Opt. Commun. 335, 146–152 (2015).
[Crossref]

J. Xie, Z. Su, W. Chen, G. Chen, J. Lv, D. Mihalache, and Y. He, “Defect solitons in two-dimensional photonic lattices with parity-time symmetry,” Opt. Commun. 313, 139–145 (2014).
[Crossref]

Miri, M.-A.

H. Hodaei, M.-A. Miri, A. U. Hassan, W. E. Hayenga, M. Heinrich, D. N. Christodoulides, and M. Khajavikhan, “Single mode lasing in transversely multi-moded PT-symmetric microring resonators,” Laser Photon. Rev. 10, 494–499 (2016).
[Crossref]

H. Hodaei, M.-A. Miri, A. U. Hassan, W. E. Hayenga, M. Heinrich, D. N. Christodoulides, and M. Khajavikhan, “Parity-time-symmetric coupled microring lasers operating around an exceptional point,” Opt. Lett. 40, 4955–4958 (2015).
[Crossref]

H. Hodaei, M.-A. Miri, M. Heinrich, D. N. Christodoulides, and M. Khajavikhan, “Parity-time-symmetric microring lasers,” Science 346, 975–978 (2014).
[Crossref]

Mock, A.

A. Mock, “Comprehensive understanding of parity-time transitions in PT symmetric photonic crystals with an antiunitary group theory,” Phys. Rev. A 95, 043803 (2017).
[Crossref]

A. Mock, “Characterization of parity-time symmetry in photonic lattices using Heesh-Shubnikov group theory,” Opt. Express 24, 22693–22707 (2016).
[Crossref]

A. Mock, L. Lu, and J. O’Brien, “Space group theory and Fourier space analysis of two-dimensional photonic crystal waveguides,” Phys. Rev. B 81, 155115 (2010).
[Crossref]

L. Lu, A. Mock, T. Yang, M. H. Shih, E. H. Hwang, M. Bagheri, A. Stapleton, S. Farrell, J. D. O’Brien, and P. D. Dapkus, “120 µW peak output power from edge-emitting photonic crystal double heterostructure nanocavity lasers,” Appl. Phys. Lett. 94, 111101 (2009).
[Crossref]

L. Lu, A. Mock, E. H. Hwang, J. O’Brien, and P. D. Dapkus, “High-peak-power efficient edge-emitting photonic crystal nanocavity lasers,” Opt. Lett. 34, 2346–2648 (2009).
[Crossref]

L. Lu, A. Mock, M. Bagheri, E. H. Hwang, J. D. O’Brien, and P. D. Dapkus, “Double-heterostructure photonic crystal lasers with reduced threshold pump power and increased slope efficiency obtained by quantum well intermixing,” Opt. Express 16, 17342–17347 (2008).
[Crossref]

M. H. Shih, W. Kuang, A. Mock, M. Bagheri, E. H. Hwang, J. D. O’Brien, and P. D. Dapkus, “High-quality-factor photonic crystal heterostructure laser,” Appl. Phys. Lett. 89, 101104 (2006).
[Crossref]

A. Mock, “Engineering modes of PT symmetric photonic crystals,” in European Conference on Antennas and Propagation (2018), paper CS14.2.

Monifi, F.

B. Peng, Ş. K. Özdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery micro cavities,” Nat. Phys. 10, 394–398 (2014).
[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, 11–19 (2018).
[Crossref]

Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, “Optical solitons in PT periodic potentials,” Phys. Rev. Lett. 100, 030402 (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, 2632–2634 (2007).
[Crossref]

Norberg, E. J.

W. Liu, M. Li, R. S. Guzzon, E. J. Norberg, J. S. Parker, M. Lu, L. A. Coldren, and J. Yao, “An integrated parity-time symmetric wavelength-tunable single-mode microring laser,” Nat. Commun. 8, 15389 (2017).
[Crossref]

Norgen, M.

F. Ghasemifard, M. Norgen, O. Quevedo-Teruel, and G. Valerio, “Analyzing glide-symmetric holey metasurfaces using a generalized Floquet theorem,” IEEE Access 6, 71743–71750 (2018).
[Crossref]

Nori, F.

Ş. K. Özdemir, R. Stefan, F. Nori, and L. Yang, “Parity-time symmetry and exceptional points in photonics,” Nat. Mater. 18, 783 (2019).
[Crossref]

B. Peng, Ş. K. Özdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery micro cavities,” Nat. Phys. 10, 394–398 (2014).
[Crossref]

Notomi, M.

Nozaki, K.

O’Brien, J.

A. Mock, L. Lu, and J. O’Brien, “Space group theory and Fourier space analysis of two-dimensional photonic crystal waveguides,” Phys. Rev. B 81, 155115 (2010).
[Crossref]

L. Lu, A. Mock, E. H. Hwang, J. O’Brien, and P. D. Dapkus, “High-peak-power efficient edge-emitting photonic crystal nanocavity lasers,” Opt. Lett. 34, 2346–2648 (2009).
[Crossref]

W. Kuang and J. O’Brien, “Reducing the out-of-plane radiation loss of photonic crystal waveguides on high-index substrates,” Opt. Lett. 29, 860–862 (2004).
[Crossref]

O’Brien, J. D.

L. Lu, A. Mock, T. Yang, M. H. Shih, E. H. Hwang, M. Bagheri, A. Stapleton, S. Farrell, J. D. O’Brien, and P. D. Dapkus, “120 µW peak output power from edge-emitting photonic crystal double heterostructure nanocavity lasers,” Appl. Phys. Lett. 94, 111101 (2009).
[Crossref]

L. Lu, A. Mock, M. Bagheri, E. H. Hwang, J. D. O’Brien, and P. D. Dapkus, “Double-heterostructure photonic crystal lasers with reduced threshold pump power and increased slope efficiency obtained by quantum well intermixing,” Opt. Express 16, 17342–17347 (2008).
[Crossref]

M. H. Shih, W. Kuang, A. Mock, M. Bagheri, E. H. Hwang, J. D. O’Brien, and P. D. Dapkus, “High-quality-factor photonic crystal heterostructure laser,” Appl. Phys. Lett. 89, 101104 (2006).
[Crossref]

Oesterle, U.

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, 108–113 (2013).
[Crossref]

Ozbay, E.

M. Turdeuev, M. Botey, I. Giden, R. Herrero, H. Kurt, E. Ozbay, and K. Staliunas, “Two-dimensional complex parity-time-symmetric photonic structures,” Phys. Rev. A 91, 023825 (2015).
[Crossref]

Özdemir, S. K.

Ş. K. Özdemir, R. Stefan, F. Nori, and L. Yang, “Parity-time symmetry and exceptional points in photonics,” Nat. Mater. 18, 783 (2019).
[Crossref]

B. Peng, Ş. K. Özdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery micro cavities,” Nat. Phys. 10, 394–398 (2014).
[Crossref]

Painter, O.

Park, H.-G.

H.-G. Park, S.-H. Kim, S.-H. Kwon, Y.-G. Ju, H.-K. Yang, J.-H. Baek, S.-B. Kim, and Y.-H. Lee, “Electrically driven single-cell photonic crystal laser,” Science 305, 1444–1447 (2004).
[Crossref]

Parker, J. S.

W. Liu, M. Li, R. S. Guzzon, E. J. Norberg, J. S. Parker, M. Lu, L. A. Coldren, and J. Yao, “An integrated parity-time symmetric wavelength-tunable single-mode microring laser,” Nat. Commun. 8, 15389 (2017).
[Crossref]

Pathak, R. K.

K. S. Agarwal, R. K. Pathak, and Y. N. Joglekar, “Exactly solvable-symmetric models in two dimensions,” Europhys. Lett. 112, 31003 (2015).
[Crossref]

Peng, B.

B. Peng, Ş. K. Özdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery micro cavities,” Nat. Phys. 10, 394–398 (2014).
[Crossref]

Pick, A.

B. Zhen, C. W. Hsu, Y. Igarashi, L. Lu, I. Kaminer, A. Pick, S.-L. Chua, J. D. Joannopoulos, and M. Soljačić, “Spawning rings of exceptional points out of Dirac cones,” Nature 525, 354–358 (2015).
[Crossref]

Pukhov, A. A.

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

Quevedo-Teruel, O.

M. Bagheriasl, O. Quevedo-Teruel, and G. Valerio, “Bloch analysis of artificial lines and surfaces exhibiting glide symmetry,” IEEE Trans. Microwave Theory Tech. 67, 2618–2628 (2019).
[Crossref]

F. Ghasemifard, M. Norgen, O. Quevedo-Teruel, and G. Valerio, “Analyzing glide-symmetric holey metasurfaces using a generalized Floquet theorem,” IEEE Access 6, 71743–71750 (2018).
[Crossref]

G. Valerio, F. Ghasemifard, Z. Sipus, and O. Quevedo-Teruel, “Glide-symmetric all-metal holey metasurfaces for low-dispersive artificial materials: modeling and properties,” IEEE Trans. Microwave Theory Tech. 66, 3210–3223 (2018).
[Crossref]

Raman, A.

A. Cerjan, A. Raman, and S. Fan, “Exceptional contours and band structure design in parity-time symmetric photonic crystals,” Phys. Rev. Lett. 116, 203902 (2016).
[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, 213901 (2011).
[Crossref]

Rattier, M.

Rechtsman, M. C.

A. Szameit, M. C. Rechtsman, O. Bahat-Treidel, and M. Segev, “PT-symmetry in honeycomb photonic lattices,” Phys. Rev. A 84, 021806 (2011).
[Crossref]

Ren, X.

H. Wang, S. Shi, X. Ren, X. Zhu, B. A. Malomed, D. Mihalache, and Y. He, “Two-dimensional solitons in triangular photonic lattices with parity-time symmetry,” Opt. Commun. 335, 146–152 (2015).
[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, 11–19 (2018).
[Crossref]

Rue, R. M. D. L.

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, 192–195 (2010).
[Crossref]

Sakoda, K.

Sato, T.

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, 108–113 (2013).
[Crossref]

L. Feng, M. Ayache, J. Huang, Y.-L. Xu, M.-H. Lu, Y.-F. Chen, Y. Fainman, and A. Scherer, “Nonreciprocal light propagation in a silicon photonic circuit,” Science 333, 729–733 (2011).
[Crossref]

Schur, I.

G. Frobenius and I. Schur, Über die reellen Darstellungen der endlichen Gruppen (Sitzungsberichte Der Berliner Mathematischen Gesellschaft, 1906), pp. 186–208.

Segev, M.

A. Szameit, M. C. Rechtsman, O. Bahat-Treidel, and M. Segev, “PT-symmetry in honeycomb photonic lattices,” Phys. Rev. A 84, 021806 (2011).
[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, 192–195 (2010).
[Crossref]

Shi, S.

H. Wang, S. Shi, X. Ren, X. Zhu, B. A. Malomed, D. Mihalache, and Y. He, “Two-dimensional solitons in triangular photonic lattices with parity-time symmetry,” Opt. Commun. 335, 146–152 (2015).
[Crossref]

Shih, M. H.

L. Lu, A. Mock, T. Yang, M. H. Shih, E. H. Hwang, M. Bagheri, A. Stapleton, S. Farrell, J. D. O’Brien, and P. D. Dapkus, “120 µW peak output power from edge-emitting photonic crystal double heterostructure nanocavity lasers,” Appl. Phys. Lett. 94, 111101 (2009).
[Crossref]

M. H. Shih, W. Kuang, A. Mock, M. Bagheri, E. H. Hwang, J. D. O’Brien, and P. D. Dapkus, “High-quality-factor photonic crystal heterostructure laser,” Appl. Phys. Lett. 89, 101104 (2006).
[Crossref]

Shinya, A.

Shubnikov, A. V.

A. V. Shubnikov, Symmetry and Antisymmetry of Finite Figures (Izd-vo Akademii nauk SSSR, 1951).

A. V. Shubnikov and N. V. Belov, Colored Symmetry (Macmillan, 1964).

Sipus, Z.

G. Valerio, F. Ghasemifard, Z. Sipus, and O. Quevedo-Teruel, “Glide-symmetric all-metal holey metasurfaces for low-dispersive artificial materials: modeling and properties,” IEEE Trans. Microwave Theory Tech. 66, 3210–3223 (2018).
[Crossref]

Smith, C. J. M.

Soljacic, M.

B. Zhen, C. W. Hsu, Y. Igarashi, L. Lu, I. Kaminer, A. Pick, S.-L. Chua, J. D. Joannopoulos, and M. Soljačić, “Spawning rings of exceptional points out of Dirac cones,” Nature 525, 354–358 (2015).
[Crossref]

Srinivasan, K.

Staliunas, K.

W. W. Ahmed, R. Herrero, M. Botey, and K. Staliunas, “Self-collimation in PT-symmetric crystals,” Phys. Rev. A 95, 053830 (2017).
[Crossref]

M. Turdeuev, M. Botey, I. Giden, R. Herrero, H. Kurt, E. Ozbay, and K. Staliunas, “Two-dimensional complex parity-time-symmetric photonic structures,” Phys. Rev. A 91, 023825 (2015).
[Crossref]

Stapleton, A.

L. Lu, A. Mock, T. Yang, M. H. Shih, E. H. Hwang, M. Bagheri, A. Stapleton, S. Farrell, J. D. O’Brien, and P. D. Dapkus, “120 µW peak output power from edge-emitting photonic crystal double heterostructure nanocavity lasers,” Appl. Phys. Lett. 94, 111101 (2009).
[Crossref]

Stefan, R.

Ş. K. Özdemir, R. Stefan, F. Nori, and L. Yang, “Parity-time symmetry and exceptional points in photonics,” Nat. Mater. 18, 783 (2019).
[Crossref]

Stone, A. D.

L. Ge and A. D. Stone, “Parity-time symmetry breaking beyond one dimension: the role of degeneracy,” Phys. Rev. X 4, 031011 (2014).
[Crossref]

Su, Z.

J. Xie, Z. Su, W. Chen, G. Chen, J. Lv, D. Mihalache, and Y. He, “Defect solitons in two-dimensional photonic lattices with parity-time symmetry,” Opt. Commun. 313, 139–145 (2014).
[Crossref]

Sun, J.

P. Miao, Z. Zhang, J. Sun, W. Walasik, S. Longhi, N. M. Litchinitser, and L. Feng, “Orbital angular momentum microlaser,” Science 353, 464–467 (2016).
[Crossref]

Szameit, A.

M. Kremer, T. Biesenthal, L. J. Maczewsky, M. Heinrich, R. Thomale, and A. Szameit, “Demonstration of a two-dimensional PT-symmetric crystal,” Nat. Commun. 10, 435 (2019).
[Crossref]

A. Szameit, M. C. Rechtsman, O. Bahat-Treidel, and M. Segev, “PT-symmetry in honeycomb photonic lattices,” Phys. Rev. A 84, 021806 (2011).
[Crossref]

Takeda, K.

Taniyama, H.

Thomale, R.

M. Kremer, T. Biesenthal, L. J. Maczewsky, M. Heinrich, R. Thomale, and A. Szameit, “Demonstration of a two-dimensional PT-symmetric crystal,” Nat. Commun. 10, 435 (2019).
[Crossref]

Tinkham, M.

M. Tinkham, Group Theory and Quantum Mechanics (Dover Publications, 1964).

Tsvetkov, D. M.

D. M. Tsvetkov, V. A. Bushuev, V. V. Konotop, and B. I. Mantsyzov, “Broadband quasi-PT-symmetry sustained by inhomogeneous broadening of the spectral line,” Phys. Rev. A 98, 053844 (2018).
[Crossref]

Turdeuev, M.

M. Turdeuev, M. Botey, I. Giden, R. Herrero, H. Kurt, E. Ozbay, and K. Staliunas, “Two-dimensional complex parity-time-symmetric photonic structures,” Phys. Rev. A 91, 023825 (2015).
[Crossref]

Valerio, G.

M. Bagheriasl, O. Quevedo-Teruel, and G. Valerio, “Bloch analysis of artificial lines and surfaces exhibiting glide symmetry,” IEEE Trans. Microwave Theory Tech. 67, 2618–2628 (2019).
[Crossref]

F. Ghasemifard, M. Norgen, O. Quevedo-Teruel, and G. Valerio, “Analyzing glide-symmetric holey metasurfaces using a generalized Floquet theorem,” IEEE Access 6, 71743–71750 (2018).
[Crossref]

G. Valerio, F. Ghasemifard, Z. Sipus, and O. Quevedo-Teruel, “Glide-symmetric all-metal holey metasurfaces for low-dispersive artificial materials: modeling and properties,” IEEE Trans. Microwave Theory Tech. 66, 3210–3223 (2018).
[Crossref]

Vinogradov, A. P.

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

Walasik, W.

P. Miao, Z. Zhang, J. Sun, W. Walasik, S. Longhi, N. M. Litchinitser, and L. Feng, “Orbital angular momentum microlaser,” Science 353, 464–467 (2016).
[Crossref]

Wang, H.

H. Wang, S. Shi, X. Ren, X. Zhu, B. A. Malomed, D. Mihalache, and Y. He, “Two-dimensional solitons in triangular photonic lattices with parity-time symmetry,” Opt. Commun. 335, 146–152 (2015).
[Crossref]

Wang, Y.

L. Feng, Z. J. Wong, R.-M. Ma, Y. Wang, and X. Zhang, “Single-mode laser by parity-time symmetry breaking,” Science 346, 972–975 (2014).
[Crossref]

Weisbuch, C.

Wheeler, R. G.

J. O. Dimmock and R. G. Wheeler, “Symmetry properties of wave functions in magnetic crystals,” Phys. Rev. 127, 391–404 (1962).
[Crossref]

Wigner, E. P.

E. P. Wigner, Group Theory and Its Application to the Quantum Mechanics of Atomic Spectra (Academic, 1959).

Winn, J. N.

S. G. Joannopoulos, J. D. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light, 2nd ed. (Princeton University, 2008).

Wittek, S.

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, 187–191 (2017).
[Crossref]

Wong, Z. J.

L. Feng, Z. J. Wong, R.-M. Ma, Y. Wang, and X. Zhang, “Single-mode laser by parity-time symmetry breaking,” Science 346, 972–975 (2014).
[Crossref]

Wooten, F.

M. El-Batanouny and F. Wooten, Symmetry and Condensed Matter Physics: A Computational Approach (Cambridge University, 2008).

Xie, J.

J. Xie, Z. Su, W. Chen, G. Chen, J. Lv, D. Mihalache, and Y. He, “Defect solitons in two-dimensional photonic lattices with parity-time symmetry,” Opt. Commun. 313, 139–145 (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, 108–113 (2013).
[Crossref]

L. Feng, M. Ayache, J. Huang, Y.-L. Xu, M.-H. Lu, Y.-F. Chen, Y. Fainman, and A. Scherer, “Nonreciprocal light propagation in a silicon photonic circuit,” Science 333, 729–733 (2011).
[Crossref]

Yang, H.-K.

H.-G. Park, S.-H. Kim, S.-H. Kwon, Y.-G. Ju, H.-K. Yang, J.-H. Baek, S.-B. Kim, and Y.-H. Lee, “Electrically driven single-cell photonic crystal laser,” Science 305, 1444–1447 (2004).
[Crossref]

Yang, L.

Ş. K. Özdemir, R. Stefan, F. Nori, and L. Yang, “Parity-time symmetry and exceptional points in photonics,” Nat. Mater. 18, 783 (2019).
[Crossref]

B. Peng, Ş. K. Özdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery micro cavities,” Nat. Phys. 10, 394–398 (2014).
[Crossref]

Yang, T.

L. Lu, A. Mock, T. Yang, M. H. Shih, E. H. Hwang, M. Bagheri, A. Stapleton, S. Farrell, J. D. O’Brien, and P. D. Dapkus, “120 µW peak output power from edge-emitting photonic crystal double heterostructure nanocavity lasers,” Appl. Phys. Lett. 94, 111101 (2009).
[Crossref]

Yao, J.

W. Liu, M. Li, R. S. Guzzon, E. J. Norberg, J. S. Parker, M. Lu, L. A. Coldren, and J. Yao, “An integrated parity-time symmetric wavelength-tunable single-mode microring laser,” Nat. Commun. 8, 15389 (2017).
[Crossref]

Ye, S.-F.

Y.-T. Fang, S.-F. Ye, and X.-X. Li, “Unique band coalescence and exceptional points from two-dimensional photonic crystal waveguide,” J. Opt. 21, 055103 (2019).
[Crossref]

Zhang, X.

L. Feng, Z. J. Wong, R.-M. Ma, Y. Wang, and X. Zhang, “Single-mode laser by parity-time symmetry breaking,” Science 346, 972–975 (2014).
[Crossref]

Zhang, Z.

P. Miao, Z. Zhang, J. Sun, W. Walasik, S. Longhi, N. M. Litchinitser, and L. Feng, “Orbital angular momentum microlaser,” Science 353, 464–467 (2016).
[Crossref]

Zhen, B.

B. Zhen, C. W. Hsu, Y. Igarashi, L. Lu, I. Kaminer, A. Pick, S.-L. Chua, J. D. Joannopoulos, and M. Soljačić, “Spawning rings of exceptional points out of Dirac cones,” Nature 525, 354–358 (2015).
[Crossref]

Zhu, X.

H. Wang, S. Shi, X. Ren, X. Zhu, B. A. Malomed, D. Mihalache, and Y. He, “Two-dimensional solitons in triangular photonic lattices with parity-time symmetry,” Opt. Commun. 335, 146–152 (2015).
[Crossref]

Zyablovsky, A. A.

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

Appl. Phys. Lett. (2)

M. H. Shih, W. Kuang, A. Mock, M. Bagheri, E. H. Hwang, J. D. O’Brien, and P. D. Dapkus, “High-quality-factor photonic crystal heterostructure laser,” Appl. Phys. Lett. 89, 101104 (2006).
[Crossref]

L. Lu, A. Mock, T. Yang, M. H. Shih, E. H. Hwang, M. Bagheri, A. Stapleton, S. Farrell, J. D. O’Brien, and P. D. Dapkus, “120 µW peak output power from edge-emitting photonic crystal double heterostructure nanocavity lasers,” Appl. Phys. Lett. 94, 111101 (2009).
[Crossref]

Europhys. Lett. (1)

K. S. Agarwal, R. K. Pathak, and Y. N. Joglekar, “Exactly solvable-symmetric models in two dimensions,” Europhys. Lett. 112, 31003 (2015).
[Crossref]

IEEE Access (1)

F. Ghasemifard, M. Norgen, O. Quevedo-Teruel, and G. Valerio, “Analyzing glide-symmetric holey metasurfaces using a generalized Floquet theorem,” IEEE Access 6, 71743–71750 (2018).
[Crossref]

IEEE Trans. Microwave Theory Tech. (2)

M. Bagheriasl, O. Quevedo-Teruel, and G. Valerio, “Bloch analysis of artificial lines and surfaces exhibiting glide symmetry,” IEEE Trans. Microwave Theory Tech. 67, 2618–2628 (2019).
[Crossref]

G. Valerio, F. Ghasemifard, Z. Sipus, and O. Quevedo-Teruel, “Glide-symmetric all-metal holey metasurfaces for low-dispersive artificial materials: modeling and properties,” IEEE Trans. Microwave Theory Tech. 66, 3210–3223 (2018).
[Crossref]

J. Appl. Phys. (1)

H. Benisty, “Modal analysis of optical guides with two-dimensional photonic band-gap boundaries,” J. Appl. Phys. 79, 7483–7492 (1996).
[Crossref]

J. Lightwave Technol. (1)

J. Math. Phys. (1)

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

J. Opt. (1)

Y.-T. Fang, S.-F. Ye, and X.-X. Li, “Unique band coalescence and exceptional points from two-dimensional photonic crystal waveguide,” J. Opt. 21, 055103 (2019).
[Crossref]

Laser Photon. Rev. (1)

H. Hodaei, M.-A. Miri, A. U. Hassan, W. E. Hayenga, M. Heinrich, D. N. Christodoulides, and M. Khajavikhan, “Single mode lasing in transversely multi-moded PT-symmetric microring resonators,” Laser Photon. Rev. 10, 494–499 (2016).
[Crossref]

Nat. Commun. (2)

W. Liu, M. Li, R. S. Guzzon, E. J. Norberg, J. S. Parker, M. Lu, L. A. Coldren, and J. Yao, “An integrated parity-time symmetric wavelength-tunable single-mode microring laser,” Nat. Commun. 8, 15389 (2017).
[Crossref]

M. Kremer, T. Biesenthal, L. J. Maczewsky, M. Heinrich, R. Thomale, and A. Szameit, “Demonstration of a two-dimensional PT-symmetric crystal,” Nat. Commun. 10, 435 (2019).
[Crossref]

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, 108–113 (2013).
[Crossref]

Ş. K. Özdemir, R. Stefan, F. Nori, and L. Yang, “Parity-time symmetry and exceptional points in photonics,” Nat. Mater. 18, 783 (2019).
[Crossref]

Nat. Phys. (3)

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, 11–19 (2018).
[Crossref]

B. Peng, Ş. K. Özdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery micro cavities,” Nat. Phys. 10, 394–398 (2014).
[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, 192–195 (2010).
[Crossref]

Nature (2)

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, 187–191 (2017).
[Crossref]

B. Zhen, C. W. Hsu, Y. Igarashi, L. Lu, I. Kaminer, A. Pick, S.-L. Chua, J. D. Joannopoulos, and M. Soljačić, “Spawning rings of exceptional points out of Dirac cones,” Nature 525, 354–358 (2015).
[Crossref]

New J. Phys. (1)

A. Cerjan and S. Fan, “Effects of non-uniform distributions of gain and loss in photonic crystals,” New J. Phys. 18, 125007 (2016).
[Crossref]

Opt. Commun. (2)

J. Xie, Z. Su, W. Chen, G. Chen, J. Lv, D. Mihalache, and Y. He, “Defect solitons in two-dimensional photonic lattices with parity-time symmetry,” Opt. Commun. 313, 139–145 (2014).
[Crossref]

H. Wang, S. Shi, X. Ren, X. Zhu, B. A. Malomed, D. Mihalache, and Y. He, “Two-dimensional solitons in triangular photonic lattices with parity-time symmetry,” Opt. Commun. 335, 146–152 (2015).
[Crossref]

Opt. Express (5)

Opt. Lett. (4)

Photon. Res. (1)

Phys. Rev. (1)

J. O. Dimmock and R. G. Wheeler, “Symmetry properties of wave functions in magnetic crystals,” Phys. Rev. 127, 391–404 (1962).
[Crossref]

Phys. Rev. A (6)

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

D. M. Tsvetkov, V. A. Bushuev, V. V. Konotop, and B. I. Mantsyzov, “Broadband quasi-PT-symmetry sustained by inhomogeneous broadening of the spectral line,” Phys. Rev. A 98, 053844 (2018).
[Crossref]

A. Szameit, M. C. Rechtsman, O. Bahat-Treidel, and M. Segev, “PT-symmetry in honeycomb photonic lattices,” Phys. Rev. A 84, 021806 (2011).
[Crossref]

W. W. Ahmed, R. Herrero, M. Botey, and K. Staliunas, “Self-collimation in PT-symmetric crystals,” Phys. Rev. A 95, 053830 (2017).
[Crossref]

M. Turdeuev, M. Botey, I. Giden, R. Herrero, H. Kurt, E. Ozbay, and K. Staliunas, “Two-dimensional complex parity-time-symmetric photonic structures,” Phys. Rev. A 91, 023825 (2015).
[Crossref]

A. Mock, “Comprehensive understanding of parity-time transitions in PT symmetric photonic crystals with an antiunitary group theory,” Phys. Rev. A 95, 043803 (2017).
[Crossref]

Phys. Rev. B (1)

A. Mock, L. Lu, and J. O’Brien, “Space group theory and Fourier space analysis of two-dimensional photonic crystal waveguides,” Phys. Rev. B 81, 155115 (2010).
[Crossref]

Phys. Rev. Lett. (5)

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

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

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

A. Cerjan, A. Raman, and S. Fan, “Exceptional contours and band structure design in parity-time symmetric photonic crystals,” Phys. Rev. Lett. 116, 203902 (2016).
[Crossref]

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

Phys. Rev. X (1)

L. Ge and A. D. Stone, “Parity-time symmetry breaking beyond one dimension: the role of degeneracy,” Phys. Rev. X 4, 031011 (2014).
[Crossref]

Science (5)

H.-G. Park, S.-H. Kim, S.-H. Kwon, Y.-G. Ju, H.-K. Yang, J.-H. Baek, S.-B. Kim, and Y.-H. Lee, “Electrically driven single-cell photonic crystal laser,” Science 305, 1444–1447 (2004).
[Crossref]

L. Feng, Z. J. Wong, R.-M. Ma, Y. Wang, and X. Zhang, “Single-mode laser by parity-time symmetry breaking,” Science 346, 972–975 (2014).
[Crossref]

H. Hodaei, M.-A. Miri, M. Heinrich, D. N. Christodoulides, and M. Khajavikhan, “Parity-time-symmetric microring lasers,” Science 346, 975–978 (2014).
[Crossref]

L. Feng, M. Ayache, J. Huang, Y.-L. Xu, M.-H. Lu, Y.-F. Chen, Y. Fainman, and A. Scherer, “Nonreciprocal light propagation in a silicon photonic circuit,” Science 333, 729–733 (2011).
[Crossref]

P. Miao, Z. Zhang, J. Sun, W. Walasik, S. Longhi, N. M. Litchinitser, and L. Feng, “Orbital angular momentum microlaser,” Science 353, 464–467 (2016).
[Crossref]

Zeitschrift für Kristallographie-Cryst. Mater. (1)

H. Heesh, “Zur Strukturtheorie der ebenen Symmetriegruppen,” Zeitschrift für Kristallographie-Cryst. Mater. 71, 95–102 (1929).

Other (13)

A. V. Shubnikov, Symmetry and Antisymmetry of Finite Figures (Izd-vo Akademii nauk SSSR, 1951).

A. V. Shubnikov and N. V. Belov, Colored Symmetry (Macmillan, 1964).

E. P. Wigner, Group Theory and Its Application to the Quantum Mechanics of Atomic Spectra (Academic, 1959).

A. P. Cracknell, Group Theory in Solid-State Physics (Taylor and Francis, 1975).

A. P. Cracknell, Magnetism in Crystalline Materials (Pergamon, 1975).

M. El-Batanouny and F. Wooten, Symmetry and Condensed Matter Physics: A Computational Approach (Cambridge University, 2008).

G. Frobenius and I. Schur, Über die reellen Darstellungen der endlichen Gruppen (Sitzungsberichte Der Berliner Mathematischen Gesellschaft, 1906), pp. 186–208.

K. Sakoda, Optical Properties of Photonic Crystals (Springer, 2001).

M. Tinkham, Group Theory and Quantum Mechanics (Dover Publications, 1964).

R. Liboff, Primer for Point and Space Groups (Springer-Verlag, 2004).

S. G. Joannopoulos, J. D. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light, 2nd ed. (Princeton University, 2008).

L. A. Coldren and S. W. Corzine, Diode Lasers and Photonic Integrated Circuits (Wiley, 1995).

A. Mock, “Engineering modes of PT symmetric photonic crystals,” in European Conference on Antennas and Propagation (2018), paper CS14.2.

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 (8)

Fig. 1.
Fig. 1. (a) Schematic depiction of a photonic crystal defect waveguide formed by removing a row of air holes in a triangular lattice photonic crystal. Gray regions have a refractive index $ n = 3.1 $. Magenta region has a refractive index $ {n_ - } = 3.1 - i0.025 $ and corresponds to wave amplification. Green region has a refractive index $ {n_ + } = 3.1 + i0.025 $ and corresponds to wave absorption. (b) Same as (a) but the spatial orientation of the amplifying and absorbing regions is changed as shown. (c) Side view of the geometry in (a). Lossless top and bottom cladding regions are shown with refractive index $ {n_c} \lt 3.1 $. A portion of the top cladding region is replaced by an absorbing material with refractive index $ {n_a} = {n_c} + i{n_\textit{ca}} $, where the imaginary part is tuned to impart absorption in the waveguide core equal in magnitude to the amplification. (d) Side view of the geometry in (a). The geometry is a free-standing semiconductor membrane. The absorption is induced by spatially selective doping. In both (c) and (d), the gain region is pumped optically from above.
Fig. 2.
Fig. 2. (a) Geometry schematic showing a lattice-aligned photonic crystal waveguide geometry superimposed on an absorption–amplification profile implementing transverse $ {\cal P}{\cal T} $ symmetry. The lattice constant is $ a $, and the supercell width is $ \Lambda $. Circles denote air holes with radii $ r = 0.3a $ perforating a semiconductor with refractive index $ n = 3.1 $. Magenta region has refractive index $ {n_ - } = 3.1 - i0.025 $, and green region has refractive index $ {n_ + } = 3.1 + i0.025 $. The yellow cross indicates origin about which symmetry operations are performed. (b) Photonic band diagram. Shaded regions represent photonic crystal cladding modes. Blue and red lines represent even-like and odd-like photonic crystal waveguide modes, respectively. Green line denotes a third-order waveguide band. (c) Detail of boxed region in (b). Brown curves represent the dispersion of the Hermitian photonic crystal waveguide ($ {n_i}(x,y) = 0 $). Imaginary part of eigenfrequencies is shown. (d) Spatial field distributions corresponding to points labeled 1 and 2 in (b). (e) Spatial field distributions corresponding to points labeled 3 and 4 in (c).
Fig. 3.
Fig. 3. (a) Geometry schematic showing a lattice-shifted photonic crystal waveguide geometry superimposed on an absorption–amplification profile with transverse $ {\cal P}{\cal T} $ symmetry. The photonic crystal defect waveguide has one side of the lattice shifted $ a/2 $ with respect to the other side along the waveguide propagation direction. The yellow cross indicates origin about which symmetry operations are performed. Other parameters are the same as those in Fig. 2(a). (b) Photonic band diagram. Shaded regions represent photonic crystal cladding modes. Blue and red lines represent even-like and odd-like photonic crystal waveguide modes, respectively. (c) Detail of boxed region in (b). Brown curves represent the dispersion of the Hermitian photonic crystal waveguide ($ {n_i}(x,y) = 0 $). Imaginary part of eigenfrequencies is shown. (d) Spatial field distributions corresponding to points labeled 1 and 2 in (c).
Fig. 4.
Fig. 4. (a) Geometry schematic showing longitudinally periodic amplifying and absorbing regions superimposed on a lattice-aligned photonic crystal waveguide geometry. The yellow cross indicates origin about which symmetry operations are performed. Other parameters are the same as Fig. 2(a). (b) Photonic band diagram. Shaded regions represent photonic crystal cladding modes. Blue and red lines represent even-like and odd-like photonic crystal waveguide modes, respectively. (c) Photonic band diagram from Fig. 2(b) showing how the narrowing of the Brillouin zone caused by the enlarged unit cell depicted in (a) causes band folding at $ \beta \Lambda ^\prime /\pi = 0.5 $. $ \Lambda ^\prime $ is the period of the geometry shown in Fig. 2(a) [$ \Lambda ^\prime = \Lambda /2 $, where $ \Lambda $ is depicted in (a)]. (d) Detail of boxed region in (b). Imaginary part of eigenfrequencies is shown. Cyan curve (bottom) is associated with blue curve (top), and magenta curve (bottom) is associated with red curve (top). (e) Spatial field distributions corresponding to points labeled 1 and 2 in (d).
Fig. 5.
Fig. 5. (a) Geometry schematic showing longitudinally periodic amplifying and absorbing regions superimposed on a lattice-shifted photonic crystal waveguide geometry. The photonic crystal lattice on either side of the waveguide is shifted by $ a/2 $ along the waveguide propagation direction. The yellow cross indicates origin about which symmetry operations are performed. Other parameters are the same as Fig. 2(a). (b) Photonic band diagram. Shaded regions represent photonic crystal cladding modes. Blue and red lines represent even-like and odd-like photonic crystal waveguide modes, respectively. (c) Detail of boxed region in (b). Imaginary part of eigenfrequencies is shown. Cyan curve (bottom) is associated with blue curve (top), and magenta curve (bottom) is associated with red curve (top). (d) Spatial field distributions corresponding to points labeled 1 and 2 in (c).
Fig. 6.
Fig. 6. (a) Geometry schematic showing hybrid transverse–longitudinal periodic amplifying and absorbing regions superimposed on a lattice-aligned photonic crystal waveguide geometry. The yellow cross indicates origin about which symmetry operations are performed. Other parameters are the same as Fig. 2(a). (b) Photonic band diagram. Shaded regions represent photonic crystal cladding modes. Blue and red lines represent even-like and odd-like photonic crystal waveguide modes, respectively. (c) Detail of right boxed region in (b). (d) Detail of left boxed region in (b).
Fig. 7.
Fig. 7. (a) Geometry schematic showing hybrid transverse–longitudinal periodic amplifying and absorbing regions superimposed on a lattice-shifted photonic crystal waveguide geometry. The photonic crystal lattice on either side of the waveguide is shifted by $ a/2 $ along the waveguide propagation direction. The yellow cross indicates origin about which symmetry operations are performed. Other parameters are the same as Fig. 2(a). (b) Photonic band diagram. Shaded regions represent photonic crystal cladding modes. Blue and red lines represent even-like and odd-like photonic crystal waveguide modes, respectively. (c) Detail of boxed region in (b). Imaginary part of eigenfrequencies is shown. Cyan curve (bottom) is associated with blue curve (top), and magenta curve (bottom) is associated with red curve (top). (d) Spatial field distributions corresponding to points labeled 1 and 2 in (c).
Fig. 8.
Fig. 8. Geometry schematics depicting $ {\cal P}{\cal T} $ symmetry arrangements that produce type (b) Heesh–Shubnikov corepresentations defined by Eq. (1) for the (a) transverse, (c) longitudinal, and (e) hybrid transverse–longitudinal amplifying and absorbing regions. In (a), the holes nearest the waveguide core are replaced by two smaller holes arranged on a diagonal. The small holes have radii of $ r = 0.2a $ and are offset from the original hole location by $ 0.15\sqrt 2 a $ along a 45° diagonal. In (c) and (e), the row of holes bordering the top of the waveguide core are shifted along $ x $ by $ 0.2a $. Other parameters are the same as Fig. 2(a). (b), (d), (f) Respective photonic band diagrams for the geometries shown in (a), (c), (e). Shaded regions represent photonic crystal cladding modes. Blue and red lines represent even-like and odd-like photonic crystal waveguide modes, respectively. Insets depict the imaginary part of the frequency bands near $ \beta \Lambda /\pi = 1 $. Cyan and magenta curves correspond to blue and red curves, respectively. In (b), the imaginary part of the blue band terminating near $ \omega a/(2\pi c) = 0.22 $ is depicted.

Tables (5)

Tables Icon

Table 1. Character Table for C 4 v a

Tables Icon

Table 2. Summary of Location of Real (R) and Complex-Valued (C) Frequencies in the First Brillouin Zone of the P T Symmetric Waveguide Geometries Studied in This Worka

Tables Icon

Table 3. Summary of Symmetry Operators in Little Groups at β = 0 , 0 < β Λ < π , and β Λ = π for the Three Amplifying and Absorbing Regions Shown in Fig. 8

Tables Icon

Table 4. Character Table for Group C s

Tables Icon

Table 5. Character Table for Group C 2 v

Equations (8)

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

B W χ ( B 2 ) = { n T y p e ( a ) , n T y p e ( b ) , 0 T y p e ( c ) ,
× [ 1 ε r ( x , y ) × H 0 ( ω 0 , x , y ) ] = ω 0 2 c 2 H 0 ( ω 0 , x , y ) ,
× [ 1 ε r ( x , y ) + i τ G ( x , y ) × H ( ω , x , y ) ] = ω 2 c 2 H ( ω , x , y ) ,
× { 1 ε r ( x , y ) [ 1 i τ G ( x , y ) ε r ( x , y ) ] × H ( ω , x , y ) } = ω 2 c 2 H ( ω , x , y ) ,
ω 0 2 a 1 ( ω ) i τ κ a 2 ( ω ) = ω 2 a 1 ( ω ) ,
i τ κ a 1 ( ω ) + ω 0 2 a 2 ( ω ) = ω 2 a 2 ( ω ) .
κ = c 2 G ( x , y ) ε r 2 ( x , y ) [ × H 0 e ( ω 0 , x , y ) ] [ × H 0 ( ω 0 , x , y ) ] d x d y .
c 2 G ( x , y ) ε r 2 ( x , y ) [ × H 0 e ( ω 0 , x , y ) ] [ × H 0 e ( ω 0 , x , y ) ] d x d y = 0 ,

Metrics