Abstract

We investigate the properties of parity-time symmetric periodic photonic structures using Heesh-Shubnikov group theory. Classical group theory cannot be used to categorize the symmetry of the eigenmodes because the time-inversion operator is antiunitary. Fortunately, corepresentations of Heesh-Shubnikov groups have been developed to characterize the effect of antiunitary operators on eigenfunctions. Using the example structure of a one-dimensional photonic lattice, we identify the corepresentations of eigenmodes at both low and high symmetry points in the photonic band diagram. We find that thresholdless parity-time transitions are associated with particular classes of corepresentations. The approach is completely general and can be applied to parity-time symmetric photonic lattices of any dimension. The predictive power of this approach provides a powerful design tool for parity-time symmetric photonic device design.

© 2016 Optical Society of America

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References

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2016 (2)

A. Mock, “Parity-time symmetry breaking in two-dimensional photonic crystals: square lattice,” Phys. Rev. A 93, 063812 (2016).
[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] [PubMed]

2015 (7)

L. Ge, K. G. Makris, D. N. Christodoulides, and L. Feng, “Scattering in 𝒫𝒯- and ℛ𝒯-symmetric multimode waveguides: Generalized conservation laws and spontaneous symmetry breaking beyond one dimension,” Phys. Rev. A 92, 062135 (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,” Euro-physics Letters 112, 31003 (2015).
[Crossref]

L. Ge, “Parity-time symmetry in a flat-band system,” Phys. Rev. A 92, 052103 (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, 235310 (2015).
[Crossref]

S. Phang, A. Vukovic, S. C. Creagh, T. M. Benson, P. D. Sewell, and G. Gradoni, “Parity-time symmetric coupled micro resonators with a dispersive gain/loss,” Opt. Express 23, 11493–11507 (2015).
[Crossref] [PubMed]

X.-F. Zhu, “Defect states and exceptional point splitting in the band gaps of one-dimensional parity-time lattices,” Opt. Express 23, 22274–22284 (2015).
[Crossref] [PubMed]

2014 (11)

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

S. Longhi and L. Feng, “𝒫𝒯-symmetric microring laser-absorber,” Opt. Lett. 39, 5026–5029 (2014).
[Crossref] [PubMed]

H. Alaeian and J. A. Dionne, “Parity-time-symmetric plasmonic metamaterials,” Phys. Rev. A 89, 033829 (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,” Optics Communications 313, 139–145 (2014).
[Crossref]

V. Yannopapas, “Spontaneous 𝒫𝒯-symmetry breaking in complex frequency band structures,” Phys. Rev. A 89, 013808 (2014).
[Crossref]

M. Kulishov, B. Kress, and H. F. Jones, “Novel optical characteristics of a Fabry-Perot resonator with embedded 𝒫𝒯-symmetrical grating,” Optics Express 22, 23164–23181 (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] [PubMed]

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

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

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] [PubMed]

2013 (2)

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

A. Regensburger, M.-A. Miri, C. Bersch, J. Näger, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Observation of defect states in 𝒫𝒯-symmetric optical lattices,” Phys. Rev. Lett. 110, 223902 (2013).
[Crossref]

2012 (2)

A. Regensburger, C. Bersch, M.-A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity-time synthetic photonic lattices,” Nature 488, 167–171 (2012).
[Crossref] [PubMed]

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

2011 (3)

H. Benisty, A. Degiron, A. Lupu, A. D. Lustrac, S. Chénais, S. Forget, M. Besbes, G. Barbillon, S. Bruyant, Blaize Sylvain, and G. Lérondel, “Implementation of 𝒫𝒯 symmetric devices using plasmonics: principle and applications,” Opt. Express 19, 18004–18019 (2011).
[Crossref] [PubMed]

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

2010 (5)

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]

S. Longhi, “𝒫𝒯-symmetric laser absorber,” Phys. Rev. A 82, 031801(R) (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]

K. Zhou, Z. Guo, J. Wang, and S. Liu, “Defect modes in defective parity-time symmetric periodic complex potentials,” Opt. Lett. 35, 2928–2930 (2010).
[Crossref] [PubMed]

J. Čtyroký, Kuzmiak, Vladimír, and S. Eyderman, “Waveguide structures with antisymmetric gain/loss profile,” Opt. Express 18, 21585–21593 (2010).
[Crossref] [PubMed]

2009 (3)

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

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

A. Mostafazadeh, “Resonance phenomenon related to spectral singularities, complex barrier potential, and resonating waveguides,” Phys. Rev. A 80, 032711 (2009).
[Crossref]

2008 (2)

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]

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

2007 (1)

2002 (1)

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

1999 (1)

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

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]

1991 (1)

M. Plihal, A. Shambrook, A. A. Maradudin, and P. Sheng, “Two-dimensional photonic band structures,” Opt. Commun. 80, 199–204 (1991).
[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-Crystalline Materials 71, 95–102 (1929).

Agarwal, K. S.

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

Aimez, V.

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

Alaeian, H.

H. Alaeian and J. A. Dionne, “Parity-time-symmetric plasmonic metamaterials,” Phys. Rev. A 89, 033829 (2014).
[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. Materials 12, 108–113 (2013).
[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]

Barbillon, G.

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.

Benson, T. M.

Bersch, C.

A. Regensburger, M.-A. Miri, C. Bersch, J. Näger, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Observation of defect states in 𝒫𝒯-symmetric optical lattices,” Phys. Rev. Lett. 110, 223902 (2013).
[Crossref]

A. Regensburger, C. Bersch, M.-A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity-time synthetic photonic lattices,” Nature 488, 167–171 (2012).
[Crossref] [PubMed]

Besbes, M.

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]

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]

Bruyant, S.

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]

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] [PubMed]

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, 235310 (2015).
[Crossref]

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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]

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]

Regensburger, A.

A. Regensburger, M.-A. Miri, C. Bersch, J. Näger, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Observation of defect states in 𝒫𝒯-symmetric optical lattices,” Phys. Rev. Lett. 110, 223902 (2013).
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A. Regensburger, C. Bersch, M.-A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity-time synthetic photonic lattices,” Nature 488, 167–171 (2012).
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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]

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).
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[Crossref]

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A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of 𝒫𝒯-symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103, 093902 (2009).
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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. Materials 12, 108–113 (2013).
[Crossref]

Schur, I.

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

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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).
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M. Plihal, A. Shambrook, A. A. Maradudin, and P. Sheng, “Two-dimensional photonic band structures,” Opt. Commun. 80, 199–204 (1991).
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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).
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A. V. Shubnikov and N. V. Belov, Colored Symmetry (Macmillan, 1964).

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

L. Ge, Y. D. Chong, and A. D. Stone, “Conservation relations and anisotropic transmission resonances in one-dimensional 𝒫𝒯-symmetric photonic heterostructures,” Phys. Rev. A 85, 023802 (2012).
[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,” Optics Communications 313, 139–145 (2014).
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Sylvain, Blaize

Szameit, A.

A. Szameit, M. C. Rechtsman, O. Bahat-Treidel, and M. Segev, “𝒫𝒯-symmetry in honeycomb photonic lattices,” Phys. Rev. A 84, 021806 (2011).
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M. Tinkham, Group Theory and Quantum Mechanics (Dover Publications, Inc., 1964).

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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).
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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).
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L. Chang, X. Jiang, S. Hua, C. Yang, J. Wen, L. Jiang, G. Li, G. Wang, and M. Xiao, “Parity-time symmetry and variable optical isolation in active-passive-coupled microresonators,” Nat. Photonics 8, 524–529 (2014).
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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,” Optics Communications 313, 139–145 (2014).
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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. Materials 12, 108–113 (2013).
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L. Chang, X. Jiang, S. Hua, C. Yang, J. Wen, L. Jiang, G. Li, G. Wang, and M. Xiao, “Parity-time symmetry and variable optical isolation in active-passive-coupled microresonators,” Nat. Photonics 8, 524–529 (2014).
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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).
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Zhou, K.

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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).
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[Crossref]

Nat. Photonics (1)

L. Chang, X. Jiang, S. Hua, C. Yang, J. Wen, L. Jiang, G. Li, G. Wang, and M. Xiao, “Parity-time symmetry and variable optical isolation in active-passive-coupled microresonators,” Nat. Photonics 8, 524–529 (2014).
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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).
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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 (1)

A. Regensburger, C. Bersch, M.-A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity-time synthetic photonic lattices,” Nature 488, 167–171 (2012).
[Crossref] [PubMed]

Opt. Commun. (2)

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]

M. Plihal, A. Shambrook, A. A. Maradudin, and P. Sheng, “Two-dimensional photonic band structures,” Opt. Commun. 80, 199–204 (1991).
[Crossref]

Opt. Express (5)

Opt. Lett. (3)

Optics Communications (1)

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,” Optics Communications 313, 139–145 (2014).
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Optics Express (1)

M. Kulishov, B. Kress, and H. F. Jones, “Novel optical characteristics of a Fabry-Perot resonator with embedded 𝒫𝒯-symmetrical grating,” Optics Express 22, 23164–23181 (2014).
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Phys. Rev. (1)

J. O. Dimmock and R. G. Wheeler, “Symmetry properties of wave functions in magnetic crystals,” Phys. Rev. 127, 391–404 (1962).
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Phys. Rev. A (9)

H. Alaeian and J. A. Dionne, “Parity-time-symmetric plasmonic metamaterials,” Phys. Rev. A 89, 033829 (2014).
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L. Ge, K. G. Makris, D. N. Christodoulides, and L. Feng, “Scattering in 𝒫𝒯- and ℛ𝒯-symmetric multimode waveguides: Generalized conservation laws and spontaneous symmetry breaking beyond one dimension,” Phys. Rev. A 92, 062135 (2015).
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V. Yannopapas, “Spontaneous 𝒫𝒯-symmetry breaking in complex frequency band structures,” Phys. Rev. A 89, 013808 (2014).
[Crossref]

L. Ge, “Parity-time symmetry in a flat-band system,” Phys. Rev. A 92, 052103 (2015).
[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]

L. Ge, Y. D. Chong, and A. D. Stone, “Conservation relations and anisotropic transmission resonances in one-dimensional 𝒫𝒯-symmetric photonic heterostructures,” Phys. Rev. A 85, 023802 (2012).
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S. Longhi, “𝒫𝒯-symmetric laser absorber,” Phys. Rev. A 82, 031801(R) (2010).
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A. Mostafazadeh, “Resonance phenomenon related to spectral singularities, complex barrier potential, and resonating waveguides,” Phys. Rev. A 80, 032711 (2009).
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A. Mock, “Parity-time symmetry breaking in two-dimensional photonic crystals: square lattice,” Phys. Rev. A 93, 063812 (2016).
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Phys. Rev. B (2)

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, 235310 (2015).
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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).
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Phys. Rev. Lett. (9)

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).
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A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of 𝒫𝒯-symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103, 093902 (2009).
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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).
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C. M. Bender and S. Boettcher, “Real spectra in non-Hermitian Hamiltonians having 𝒫𝒯 symmetry,” Phys. Rev. Lett. 80, 5243–5246 (1998).
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A. Regensburger, M.-A. Miri, C. Bersch, J. Näger, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Observation of defect states in 𝒫𝒯-symmetric optical lattices,” Phys. Rev. Lett. 110, 223902 (2013).
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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).
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K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam dynamics in 𝒫𝒯 symmetric optical lattices,” Phys. Rev. Lett. 100, 103904 (2008).
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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).

Science (2)

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).
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H. Hodaei, M.-A. Miri, M. Heinrich, D. N. Christodoulides, and M. Khajavikhan, “Parity-time-symmetric microring lasers,” Science 346, 975–978 (2014).
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Other (11)

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

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

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

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

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[Crossref]

V. Heine, Group Theory in Quantum Mechanics (Pergamon Press, 1960).

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

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

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

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

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Figures (6)

Fig. 1
Fig. 1 (a) Schematic diagram showing the one-dimensional ���� symmetric photonic lattice. Regions labeled n1 provide gain (n1 = nr + ini), and regions labeled n2 provide loss (n2 = nrini) for positive nr and ni. Point and space group symmetry operations are labeled. (b) Photonic band diagram calculated using the plane wave expansion method. Dashed black line: empty lattice band diagram. Insets depict the imaginary part of the frequency for modes with ����-degeneracy. λ0 is the vacuum wavelength. Corepresentation labels are defined in the text in Sections 2.1–2.3.
Fig. 2
Fig. 2 Magnetic field (Hz (x)) spatial distribution in ���� symmetric 1D lattice at k = 0. H1 corresponds to à 1 labeled in Fig. 1(b), and H2 corresponds to à 2 +. Transformed fields are shown to verify the characters in Table 1.
Fig. 3
Fig. 3 Magnetic field (Hz (x)) spatial distribution in ���� symmetric 1D lattice at k = π/Λ (point c in Fig. 1(b)). Hg corresponds to gain mode labeled c,g in Fig. 1(b), and Hl corresponds to the loss mode labeled c,l. Transformed fields are shown to verify the characters in Table 4.
Fig. 4
Fig. 4 Magnetic field (Hz (x)) spatial distribution in ���� symmetric 1D lattice at k = 0.8(π/Λ) (points a and b in Fig. 1(b)). Transformed fields are shown to verify the characters in Table 5.
Fig. 5
Fig. 5 Photonic band diagram for the 1D photonic lattice shown in Fig. 1(a) with n = 2 ± i0.7. (a) Real part of the frequencies. (b) Imaginary part of the frequencies.
Fig. 6
Fig. 6 Magnetic field (Hz (x)) spatial distribution in ���� symmetric 1D lattice with ni = 0.70 at k = 0.8(π/Λ) (point marked by a circle in Fig. 5). Transformed fields are shown to illustrate that mode symmetry differs from that predicted by group theory.

Tables (11)

Tables Icon

Table 1 Corepresentations of k=0.

Tables Icon

Table 2 Character table of C2v point group along with results of Dimmock and Wheeler test (α) and corepresentation type (Correp.).

Tables Icon

Table 3 Corepresentations for the unitary operators of k=π.

Tables Icon

Table 4 Corepresentations of the antiunitary operators of k=π.

Tables Icon

Table 5 Corepresentations of k=0.8(π/Λ).

Tables Icon

Table 6 Multiplication table for k=0.

Tables Icon

Table 7 Multiplication table for unitary subgroup of k=0.

Tables Icon

Table 8 Multiplication table for k=π.

Tables Icon

Table 9 Multiplication table for unitary subgroup of k=π. It is isomorphic to C2v (2mm).

Tables Icon

Table 10 Determination of the matrix elements for the Type (c) corepresentations of the unitary operators in k=π.

Tables Icon

Table 11 Determination of the matrix elements for the Type (c) corepresentations of the antiunitary operators in k=π.

Equations (12)

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

× [ 1 ( r ) × H ( r ) ] = Ξ H ( r ) = ( ω c ) 2 H ( r )
[ 𝒫 𝒯 , Ξ ] = 0 ,
Ξ 𝒫 𝒯 H ( r ) = 𝒫 𝒯 Ξ H ( r ) = 𝒫 𝒯 ( ω c ) 2 H ( r ) = ( ω * c ) 2 𝒫 𝒯 H ( r ) .
B 𝒲 χ ( B 2 ) = { n Type ( a ) , n Type ( b ) , 0 Type ( c ) ,
Γ i ( R ) = ( Δ ( R ) 0 0 Δ * ( S 1 R S ) )
Γ i ( R ) = ( 0 Δ * ( A 1 R ) Δ ( R A ) 0 ) .
U = [ 0 e i θ e i θ 0 ]
R [ H g k , i ( x ) H l k , i ( x ) ] = [ r 11 0 0 r 22 ] [ H g k , i ( x ) H l k , i ( x ) ] = [ r 11 H g k , i ( x ) r 22 H l k , i ( x ) ]
A [ H g k , i ( x ) H l k , i ( x ) ] = [ 0 a 12 a 21 0 ] [ H g k , i ( x ) H l k , i ( x ) ] = [ a 12 H l k , i ( x ) a 21 H g k , i ( x ) ] .
μ = [ 0 1 1 0 ]
{ R | t } { S | t } = { R S | R t + t } .
Γ i ( e ξ ) = Γ i ( ξ ) = Δ i ( e ) β Γ i ( e ¯ ξ ) = Γ i ( ξ ¯ ) = Δ i ( e ¯ ) β Γ i ( m ξ ) = Γ i ( μ ¯ ) = Δ i ( m ) β Γ i ( m ¯ ξ ) = Γ i ( μ ) = Δ i ( m ¯ ) β

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