Abstract

Optical gradient force in a parity-time (PT)-symmetric coupled-waveguide system is theoretically studied. We find that when the system evolves from PT-symmetric region to broken-PT-symmetric region, the normalized optical forces of the two eigenmodes decrease first and become the same when the exceptional point is reached. Besides, the optical force induced PT phase transition is demonstrated. It is worth noting that, when the system is in the broken-PT-symmetric region and the length of the waveguide is much longer than the propagation length of the lossy eigenmode, the total optical gradient force acting on the two waveguides will decrease with the decreasing of the gap. This work gives us a new understanding of integrated optomechanics by combining with PT symmetry.

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

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References

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  1. C. M. Bender, “Making sense of non-Hermitian Hamiltonians,” Rep. Prog. Phys. 70(6), 947–1018 (2007).
    [Crossref]
  2. C. M. Bender and S. Boettcher, “Real spectra in non-Hermitian Hamiltonians having PT symmetry,” Phys. Rev. Lett. 80(24), 5243–5246 (1998).
    [Crossref]
  3. L. Feng, R. El-Ganainy, and L. Ge, “Non-Hermitian photonics based on parity–time symmetry,” Nat. Photonics 11(12), 752–762 (2017).
    [Crossref]
  4. R. El-Ganainy, K. G. Makris, M. Khajavikhan, Z. H. Musslimani, S. Rotter, and D. N. Christodoulides, “Non-Hermitian physics and PT symmetry,” Nat. Phys. 14(1), 11–19 (2018).
    [Crossref]
  5. C. M. Bender, D. C. Brody, and H. F. Jones, “Complex Extension of Quantum Mechanics,” Phys. Rev. Lett. 89(27), 270401 (2002).
    [Crossref] [PubMed]
  6. S. Bittner, B. Dietz, U. Günther, H. L. Harney, M. Miski-Oglu, A. Richter, and F. Schäfer, “PT Symmetry and Spontaneous Symmetry Breaking in a Microwave Billiard,” Phys. Rev. Lett. 108(2), 024101 (2012).
    [Crossref] [PubMed]
  7. R. Fleury, D. Sounas, and A. Alù, “An invisible acoustic sensor based on parity-time symmetry,” Nat. Commun. 6(1), 5905 (2015).
    [Crossref] [PubMed]
  8. A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103(9), 093902 (2009).
    [Crossref] [PubMed]
  9. A. A. Zyablovsky, A. P. Vinogradov, A. A. Pukhov, A. V. Dorofeenko, and A. A. Lisyansky, “PT-symmetry in optics,” Phys.- Usp. 57(11), 1063–1082 (2014).
    [Crossref]
  10. C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity–time symmetry in optics,” Nat. Phys. 6(3), 192–195 (2010).
    [Crossref]
  11. Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional Invisibility Induced by PT-Symmetric Periodic Structures,” Phys. Rev. Lett. 106(21), 213901 (2011).
    [Crossref] [PubMed]
  12. B. Peng, S. K. Ozdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10(5), 394–398 (2014).
    [Crossref]
  13. L. Chang, X. Jiang, S. Hua, C. Yang, J. Wen, L. Jiang, G. Li, G. Wang, and M. Xiao, “Parity-time symmetry and variable optical isolation in active-passive-coupled microresonators,” Nat. Photonics 8(7), 524–529 (2014).
    [Crossref]
  14. S. Longhi, “Bloch Oscillations in Complex Crystals with PT Symmetry,” Phys. Rev. Lett. 103(12), 123601 (2009).
    [Crossref] [PubMed]
  15. M. Wimmer, A. Regensburger, M. A. Miri, C. Bersch, D. N. Christodoulides, and U. Peschel, “Observation of optical solitons in PT-symmetric lattices,” Nat. Commun. 6(1), 7782 (2015).
    [Crossref] [PubMed]
  16. L. Feng, Z. J. Wong, R. M. Ma, Y. Wang, and X. Zhang, “Single-mode laser by parity-time symmetry breaking,” Science 346(6212), 972–975 (2014).
    [Crossref] [PubMed]
  17. H. Hodaei, M. A. Miri, M. Heinrich, D. N. Christodoulides, and M. Khajavikhan, “Parity-time-symmetric microring lasers,” Science 346(6212), 975–978 (2014).
    [Crossref] [PubMed]
  18. L. Feng, Y. L. Xu, W. S. Fegadolli, M. H. Lu, J. E. B. Oliveira, V. R. Almeida, Y.-F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nat. Mater. 12(2), 108–113 (2013).
    [Crossref] [PubMed]
  19. D. Van Thourhout and J. Roels, “Optomechanical device actuation through the optical gradient force,” Nat. Photonics 4(4), 211–217 (2010).
    [Crossref]
  20. J. Ma and M. L. Povinelli, “Applications of optomechanical effects for on-chip manipulation of light signals,” Curr. Opin. Solid. St. M. 16(2), 82–90 (2012).
    [Crossref]
  21. X. Sun, J. Zheng, M. Poot, C. W. Wong, and H. X. Tang, “Femtogram Doubly Clamped Nanomechanical Resonators Embedded in a High-Q Two-Dimensional Photonic Crystal Nanocavity,” Nano Lett. 12(5), 2299–2305 (2012).
    [Crossref] [PubMed]
  22. H. Li, J. W. Noh, Y. Chen, and M. Li, “Enhanced optical forces in integrated hybrid plasmonic waveguides,” Opt. Express 21(10), 11839–11851 (2013).
    [Crossref] [PubMed]
  23. Z. Huang, K. Cui, G. Bai, X. Feng, F. Liu, W. Zhang, and Y. Huang, “High-mechanical-frequency characteristics of optomechanical crystal cavity with coupling waveguide,” Sci. Rep. 6(1), 34160 (2016).
    [Crossref] [PubMed]
  24. X. Xu, L. Shi, Y. Liu, Z. Wang, and X. Zhang, “Enhanced optical gradient forces between coupled graphene sheets,” Sci. Rep. 6(1), 28568 (2016).
    [Crossref] [PubMed]
  25. M. L. Povinelli, M. Lončar, M. Ibanescu, E. J. Smythe, S. G. Johnson, F. Capasso, and J. D. Joannopoulos, “Evanescent-wave bonding between optical waveguides,” Opt. Lett. 30(22), 3042–3044 (2005).
    [Crossref] [PubMed]
  26. J. Ma and M. L. Povinelli, “Mechanical Kerr nonlinearities due to bipolar optical forces between deformable silicon waveguides,” Opt. Express 19(11), 10102–10110 (2011).
    [Crossref] [PubMed]
  27. Y. W. Hu, Y. F. Xiao, Y. C. Liu, and Q. Gong, “Optomechanical sensing with on-chip microcavities,” Front. Phys. 8(5), 475–490 (2013).
    [Crossref]
  28. M. Bagheri, M. Poot, L. Fan, F. Marquardt, and H. X. Tang, “Photonic Cavity Synchronization of Nanomechanical Oscillators,” Phys. Rev. Lett. 111(21), 213902 (2013).
    [Crossref] [PubMed]
  29. Z. Wang, L. Shi, Y. Liu, X. Xu, and X. Zhang, “Optical nonreciprocity in asymmetric optomechanical couplers,” Sci. Rep. 5(1), 8657 (2015).
    [Crossref] [PubMed]
  30. K. Fang, J. Luo, A. Metelmann, M. H. Matheny, F. Marquardt, A. A. Clerk, and O. Painter, “Generalized non-reciprocity in an optomechanical circuit via synthetic magnetism and reservoir engineering,” Nat. Phys. 13(5), 465–471 (2017).
    [Crossref]
  31. E. Verhagen and A. Alù, “Optomechanical nonreciprocity,” Nat. Phys. 13(10), 922–924 (2017).
    [Crossref]
  32. L. Fan, C. L. Zou, M. Poot, R. Cheng, X. Guo, X. Han, and H. X. Tang, “Integrated optomechanical single-photon frequency shifter,” Nat. Photonics 10(12), 766–770 (2016).
    [Crossref]
  33. J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, 1998).
  34. M. Li, W. H. P. Pernice, C. Xiong, T. Baehr-Jones, M. Hochberg, and H. X. Tang, “Harnessing optical forces in integrated photonic circuits,” Nature 456(7221), 480–484 (2008).
    [Crossref] [PubMed]
  35. M. Li, W. H. P. Pernice, and H. X. Tang, “Tunable bipolar optical interactions between guided lightwaves,” Nat. Photonics 3(8), 464–468 (2009).
    [Crossref]
  36. J. Roels, I. De Vlaminck, L. Lagae, B. Maes, D. Van Thourhout, and R. Baets, “Tunable optical forces between nanophotonic waveguides,” Nat. Nanotechnol. 4(8), 510–513 (2009).
    [Crossref] [PubMed]
  37. Y. Lumer, Y. Plotnik, M. C. Rechtsman, and M. Segev, “Nonlinearly Induced PT transition in Photonic Systems,” Phys. Rev. Lett. 111(26), 263901 (2013).
    [Crossref] [PubMed]
  38. W. Walasik, C. Ma, and N. M. Litchinitser, “Nonlinear parity-time-symmetric transition in finite-size optical couplers,” Opt. Lett. 40(22), 5327–5330 (2015).
    [Crossref] [PubMed]
  39. W. P. Huang and J. Mu, “Complex coupled-mode theory for optical waveguides,” Opt. Express 17(21), 19134–19152 (2009).
    [Crossref] [PubMed]
  40. K. Ikeda, R. E. Saperstein, N. Alic, and Y. Fainman, “Thermal and Kerr nonlinear properties of plasma-deposited silicon nitride/ silicon dioxide waveguides,” Opt. Express 16(17), 12987–12994 (2008).
    [Crossref] [PubMed]
  41. C. J. Krückel, A. Fülöp, T. Klintberg, J. Bengtsson, P. A. Andrekson, and V. Torres-Company, “Linear and nonlinear characterization of low-stress high-confinement silicon-rich nitride waveguides,” Opt. Express 23(20), 25827–25837 (2015).
    [Crossref] [PubMed]

2018 (1)

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

2017 (3)

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

K. Fang, J. Luo, A. Metelmann, M. H. Matheny, F. Marquardt, A. A. Clerk, and O. Painter, “Generalized non-reciprocity in an optomechanical circuit via synthetic magnetism and reservoir engineering,” Nat. Phys. 13(5), 465–471 (2017).
[Crossref]

E. Verhagen and A. Alù, “Optomechanical nonreciprocity,” Nat. Phys. 13(10), 922–924 (2017).
[Crossref]

2016 (3)

L. Fan, C. L. Zou, M. Poot, R. Cheng, X. Guo, X. Han, and H. X. Tang, “Integrated optomechanical single-photon frequency shifter,” Nat. Photonics 10(12), 766–770 (2016).
[Crossref]

Z. Huang, K. Cui, G. Bai, X. Feng, F. Liu, W. Zhang, and Y. Huang, “High-mechanical-frequency characteristics of optomechanical crystal cavity with coupling waveguide,” Sci. Rep. 6(1), 34160 (2016).
[Crossref] [PubMed]

X. Xu, L. Shi, Y. Liu, Z. Wang, and X. Zhang, “Enhanced optical gradient forces between coupled graphene sheets,” Sci. Rep. 6(1), 28568 (2016).
[Crossref] [PubMed]

2015 (5)

Z. Wang, L. Shi, Y. Liu, X. Xu, and X. Zhang, “Optical nonreciprocity in asymmetric optomechanical couplers,” Sci. Rep. 5(1), 8657 (2015).
[Crossref] [PubMed]

R. Fleury, D. Sounas, and A. Alù, “An invisible acoustic sensor based on parity-time symmetry,” Nat. Commun. 6(1), 5905 (2015).
[Crossref] [PubMed]

M. Wimmer, A. Regensburger, M. A. Miri, C. Bersch, D. N. Christodoulides, and U. Peschel, “Observation of optical solitons in PT-symmetric lattices,” Nat. Commun. 6(1), 7782 (2015).
[Crossref] [PubMed]

W. Walasik, C. Ma, and N. M. Litchinitser, “Nonlinear parity-time-symmetric transition in finite-size optical couplers,” Opt. Lett. 40(22), 5327–5330 (2015).
[Crossref] [PubMed]

C. J. Krückel, A. Fülöp, T. Klintberg, J. Bengtsson, P. A. Andrekson, and V. Torres-Company, “Linear and nonlinear characterization of low-stress high-confinement silicon-rich nitride waveguides,” Opt. Express 23(20), 25827–25837 (2015).
[Crossref] [PubMed]

2014 (5)

L. Feng, Z. J. Wong, R. M. Ma, Y. Wang, and X. Zhang, “Single-mode laser by parity-time symmetry breaking,” Science 346(6212), 972–975 (2014).
[Crossref] [PubMed]

H. Hodaei, M. A. Miri, M. Heinrich, D. N. Christodoulides, and M. Khajavikhan, “Parity-time-symmetric microring lasers,” Science 346(6212), 975–978 (2014).
[Crossref] [PubMed]

A. A. Zyablovsky, A. P. Vinogradov, A. A. Pukhov, A. V. Dorofeenko, and A. A. Lisyansky, “PT-symmetry in optics,” Phys.- Usp. 57(11), 1063–1082 (2014).
[Crossref]

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

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

2013 (5)

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

Y. W. Hu, Y. F. Xiao, Y. C. Liu, and Q. Gong, “Optomechanical sensing with on-chip microcavities,” Front. Phys. 8(5), 475–490 (2013).
[Crossref]

M. Bagheri, M. Poot, L. Fan, F. Marquardt, and H. X. Tang, “Photonic Cavity Synchronization of Nanomechanical Oscillators,” Phys. Rev. Lett. 111(21), 213902 (2013).
[Crossref] [PubMed]

H. Li, J. W. Noh, Y. Chen, and M. Li, “Enhanced optical forces in integrated hybrid plasmonic waveguides,” Opt. Express 21(10), 11839–11851 (2013).
[Crossref] [PubMed]

Y. Lumer, Y. Plotnik, M. C. Rechtsman, and M. Segev, “Nonlinearly Induced PT transition in Photonic Systems,” Phys. Rev. Lett. 111(26), 263901 (2013).
[Crossref] [PubMed]

2012 (3)

J. Ma and M. L. Povinelli, “Applications of optomechanical effects for on-chip manipulation of light signals,” Curr. Opin. Solid. St. M. 16(2), 82–90 (2012).
[Crossref]

X. Sun, J. Zheng, M. Poot, C. W. Wong, and H. X. Tang, “Femtogram Doubly Clamped Nanomechanical Resonators Embedded in a High-Q Two-Dimensional Photonic Crystal Nanocavity,” Nano Lett. 12(5), 2299–2305 (2012).
[Crossref] [PubMed]

S. Bittner, B. Dietz, U. Günther, H. L. Harney, M. Miski-Oglu, A. Richter, and F. Schäfer, “PT Symmetry and Spontaneous Symmetry Breaking in a Microwave Billiard,” Phys. Rev. Lett. 108(2), 024101 (2012).
[Crossref] [PubMed]

2011 (2)

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional Invisibility Induced by PT-Symmetric Periodic Structures,” Phys. Rev. Lett. 106(21), 213901 (2011).
[Crossref] [PubMed]

J. Ma and M. L. Povinelli, “Mechanical Kerr nonlinearities due to bipolar optical forces between deformable silicon waveguides,” Opt. Express 19(11), 10102–10110 (2011).
[Crossref] [PubMed]

2010 (2)

D. Van Thourhout and J. Roels, “Optomechanical device actuation through the optical gradient force,” Nat. Photonics 4(4), 211–217 (2010).
[Crossref]

C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity–time symmetry in optics,” Nat. Phys. 6(3), 192–195 (2010).
[Crossref]

2009 (5)

S. Longhi, “Bloch Oscillations in Complex Crystals with PT Symmetry,” Phys. Rev. Lett. 103(12), 123601 (2009).
[Crossref] [PubMed]

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

M. Li, W. H. P. Pernice, and H. X. Tang, “Tunable bipolar optical interactions between guided lightwaves,” Nat. Photonics 3(8), 464–468 (2009).
[Crossref]

J. Roels, I. De Vlaminck, L. Lagae, B. Maes, D. Van Thourhout, and R. Baets, “Tunable optical forces between nanophotonic waveguides,” Nat. Nanotechnol. 4(8), 510–513 (2009).
[Crossref] [PubMed]

W. P. Huang and J. Mu, “Complex coupled-mode theory for optical waveguides,” Opt. Express 17(21), 19134–19152 (2009).
[Crossref] [PubMed]

2008 (2)

K. Ikeda, R. E. Saperstein, N. Alic, and Y. Fainman, “Thermal and Kerr nonlinear properties of plasma-deposited silicon nitride/ silicon dioxide waveguides,” Opt. Express 16(17), 12987–12994 (2008).
[Crossref] [PubMed]

M. Li, W. H. P. Pernice, C. Xiong, T. Baehr-Jones, M. Hochberg, and H. X. Tang, “Harnessing optical forces in integrated photonic circuits,” Nature 456(7221), 480–484 (2008).
[Crossref] [PubMed]

2007 (1)

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

2005 (1)

2002 (1)

C. M. Bender, D. C. Brody, and H. F. Jones, “Complex Extension of Quantum Mechanics,” Phys. Rev. Lett. 89(27), 270401 (2002).
[Crossref] [PubMed]

1998 (1)

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

Aimez, V.

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

Alic, N.

Almeida, V. R.

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

Alù, A.

E. Verhagen and A. Alù, “Optomechanical nonreciprocity,” Nat. Phys. 13(10), 922–924 (2017).
[Crossref]

R. Fleury, D. Sounas, and A. Alù, “An invisible acoustic sensor based on parity-time symmetry,” Nat. Commun. 6(1), 5905 (2015).
[Crossref] [PubMed]

Andrekson, P. A.

Baehr-Jones, T.

M. Li, W. H. P. Pernice, C. Xiong, T. Baehr-Jones, M. Hochberg, and H. X. Tang, “Harnessing optical forces in integrated photonic circuits,” Nature 456(7221), 480–484 (2008).
[Crossref] [PubMed]

Baets, R.

J. Roels, I. De Vlaminck, L. Lagae, B. Maes, D. Van Thourhout, and R. Baets, “Tunable optical forces between nanophotonic waveguides,” Nat. Nanotechnol. 4(8), 510–513 (2009).
[Crossref] [PubMed]

Bagheri, M.

M. Bagheri, M. Poot, L. Fan, F. Marquardt, and H. X. Tang, “Photonic Cavity Synchronization of Nanomechanical Oscillators,” Phys. Rev. Lett. 111(21), 213902 (2013).
[Crossref] [PubMed]

Bai, G.

Z. Huang, K. Cui, G. Bai, X. Feng, F. Liu, W. Zhang, and Y. Huang, “High-mechanical-frequency characteristics of optomechanical crystal cavity with coupling waveguide,” Sci. Rep. 6(1), 34160 (2016).
[Crossref] [PubMed]

Bender, C. M.

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

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

C. M. Bender, D. C. Brody, and H. F. Jones, “Complex Extension of Quantum Mechanics,” Phys. Rev. Lett. 89(27), 270401 (2002).
[Crossref] [PubMed]

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

Bengtsson, J.

Bersch, C.

M. Wimmer, A. Regensburger, M. A. Miri, C. Bersch, D. N. Christodoulides, and U. Peschel, “Observation of optical solitons in PT-symmetric lattices,” Nat. Commun. 6(1), 7782 (2015).
[Crossref] [PubMed]

Bittner, S.

S. Bittner, B. Dietz, U. Günther, H. L. Harney, M. Miski-Oglu, A. Richter, and F. Schäfer, “PT Symmetry and Spontaneous Symmetry Breaking in a Microwave Billiard,” Phys. Rev. Lett. 108(2), 024101 (2012).
[Crossref] [PubMed]

Boettcher, S.

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

Brody, D. C.

C. M. Bender, D. C. Brody, and H. F. Jones, “Complex Extension of Quantum Mechanics,” Phys. Rev. Lett. 89(27), 270401 (2002).
[Crossref] [PubMed]

Cao, H.

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional Invisibility Induced by PT-Symmetric Periodic Structures,” Phys. Rev. Lett. 106(21), 213901 (2011).
[Crossref] [PubMed]

Capasso, F.

Chang, L.

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

Chen, Y.

Chen, Y.-F.

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

Cheng, R.

L. Fan, C. L. Zou, M. Poot, R. Cheng, X. Guo, X. Han, and H. X. Tang, “Integrated optomechanical single-photon frequency shifter,” Nat. Photonics 10(12), 766–770 (2016).
[Crossref]

Christodoulides, D. N.

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

M. Wimmer, A. Regensburger, M. A. Miri, C. Bersch, D. N. Christodoulides, and U. Peschel, “Observation of optical solitons in PT-symmetric lattices,” Nat. Commun. 6(1), 7782 (2015).
[Crossref] [PubMed]

H. Hodaei, M. A. Miri, M. Heinrich, D. N. Christodoulides, and M. Khajavikhan, “Parity-time-symmetric microring lasers,” Science 346(6212), 975–978 (2014).
[Crossref] [PubMed]

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional Invisibility Induced by PT-Symmetric Periodic Structures,” Phys. Rev. Lett. 106(21), 213901 (2011).
[Crossref] [PubMed]

C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity–time symmetry in optics,” Nat. Phys. 6(3), 192–195 (2010).
[Crossref]

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

Clerk, A. A.

K. Fang, J. Luo, A. Metelmann, M. H. Matheny, F. Marquardt, A. A. Clerk, and O. Painter, “Generalized non-reciprocity in an optomechanical circuit via synthetic magnetism and reservoir engineering,” Nat. Phys. 13(5), 465–471 (2017).
[Crossref]

Cui, K.

Z. Huang, K. Cui, G. Bai, X. Feng, F. Liu, W. Zhang, and Y. Huang, “High-mechanical-frequency characteristics of optomechanical crystal cavity with coupling waveguide,” Sci. Rep. 6(1), 34160 (2016).
[Crossref] [PubMed]

De Vlaminck, I.

J. Roels, I. De Vlaminck, L. Lagae, B. Maes, D. Van Thourhout, and R. Baets, “Tunable optical forces between nanophotonic waveguides,” Nat. Nanotechnol. 4(8), 510–513 (2009).
[Crossref] [PubMed]

Dietz, B.

S. Bittner, B. Dietz, U. Günther, H. L. Harney, M. Miski-Oglu, A. Richter, and F. Schäfer, “PT Symmetry and Spontaneous Symmetry Breaking in a Microwave Billiard,” Phys. Rev. Lett. 108(2), 024101 (2012).
[Crossref] [PubMed]

Dorofeenko, A. V.

A. A. Zyablovsky, A. P. Vinogradov, A. A. Pukhov, A. V. Dorofeenko, and A. A. Lisyansky, “PT-symmetry in optics,” Phys.- Usp. 57(11), 1063–1082 (2014).
[Crossref]

Duchesne, D.

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

Eichelkraut, T.

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional Invisibility Induced by PT-Symmetric Periodic Structures,” Phys. Rev. Lett. 106(21), 213901 (2011).
[Crossref] [PubMed]

El-Ganainy, R.

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

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

C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity–time symmetry in optics,” Nat. Phys. 6(3), 192–195 (2010).
[Crossref]

Fainman, Y.

Fan, L.

L. Fan, C. L. Zou, M. Poot, R. Cheng, X. Guo, X. Han, and H. X. Tang, “Integrated optomechanical single-photon frequency shifter,” Nat. Photonics 10(12), 766–770 (2016).
[Crossref]

M. Bagheri, M. Poot, L. Fan, F. Marquardt, and H. X. Tang, “Photonic Cavity Synchronization of Nanomechanical Oscillators,” Phys. Rev. Lett. 111(21), 213902 (2013).
[Crossref] [PubMed]

Fan, S.

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

Fang, K.

K. Fang, J. Luo, A. Metelmann, M. H. Matheny, F. Marquardt, A. A. Clerk, and O. Painter, “Generalized non-reciprocity in an optomechanical circuit via synthetic magnetism and reservoir engineering,” Nat. Phys. 13(5), 465–471 (2017).
[Crossref]

Fegadolli, W. S.

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

Feng, L.

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

L. Feng, Z. J. Wong, R. M. Ma, Y. Wang, and X. Zhang, “Single-mode laser by parity-time symmetry breaking,” Science 346(6212), 972–975 (2014).
[Crossref] [PubMed]

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

Feng, X.

Z. Huang, K. Cui, G. Bai, X. Feng, F. Liu, W. Zhang, and Y. Huang, “High-mechanical-frequency characteristics of optomechanical crystal cavity with coupling waveguide,” Sci. Rep. 6(1), 34160 (2016).
[Crossref] [PubMed]

Fleury, R.

R. Fleury, D. Sounas, and A. Alù, “An invisible acoustic sensor based on parity-time symmetry,” Nat. Commun. 6(1), 5905 (2015).
[Crossref] [PubMed]

Fülöp, A.

Ge, L.

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

Gianfreda, M.

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

Gong, Q.

Y. W. Hu, Y. F. Xiao, Y. C. Liu, and Q. Gong, “Optomechanical sensing with on-chip microcavities,” Front. Phys. 8(5), 475–490 (2013).
[Crossref]

Günther, U.

S. Bittner, B. Dietz, U. Günther, H. L. Harney, M. Miski-Oglu, A. Richter, and F. Schäfer, “PT Symmetry and Spontaneous Symmetry Breaking in a Microwave Billiard,” Phys. Rev. Lett. 108(2), 024101 (2012).
[Crossref] [PubMed]

Guo, A.

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

Guo, X.

L. Fan, C. L. Zou, M. Poot, R. Cheng, X. Guo, X. Han, and H. X. Tang, “Integrated optomechanical single-photon frequency shifter,” Nat. Photonics 10(12), 766–770 (2016).
[Crossref]

Han, X.

L. Fan, C. L. Zou, M. Poot, R. Cheng, X. Guo, X. Han, and H. X. Tang, “Integrated optomechanical single-photon frequency shifter,” Nat. Photonics 10(12), 766–770 (2016).
[Crossref]

Harney, H. L.

S. Bittner, B. Dietz, U. Günther, H. L. Harney, M. Miski-Oglu, A. Richter, and F. Schäfer, “PT Symmetry and Spontaneous Symmetry Breaking in a Microwave Billiard,” Phys. Rev. Lett. 108(2), 024101 (2012).
[Crossref] [PubMed]

Heinrich, M.

H. Hodaei, M. A. Miri, M. Heinrich, D. N. Christodoulides, and M. Khajavikhan, “Parity-time-symmetric microring lasers,” Science 346(6212), 975–978 (2014).
[Crossref] [PubMed]

Hochberg, M.

M. Li, W. H. P. Pernice, C. Xiong, T. Baehr-Jones, M. Hochberg, and H. X. Tang, “Harnessing optical forces in integrated photonic circuits,” Nature 456(7221), 480–484 (2008).
[Crossref] [PubMed]

Hodaei, H.

H. Hodaei, M. A. Miri, M. Heinrich, D. N. Christodoulides, and M. Khajavikhan, “Parity-time-symmetric microring lasers,” Science 346(6212), 975–978 (2014).
[Crossref] [PubMed]

Hu, Y. W.

Y. W. Hu, Y. F. Xiao, Y. C. Liu, and Q. Gong, “Optomechanical sensing with on-chip microcavities,” Front. Phys. 8(5), 475–490 (2013).
[Crossref]

Hua, S.

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

Huang, W. P.

Huang, Y.

Z. Huang, K. Cui, G. Bai, X. Feng, F. Liu, W. Zhang, and Y. Huang, “High-mechanical-frequency characteristics of optomechanical crystal cavity with coupling waveguide,” Sci. Rep. 6(1), 34160 (2016).
[Crossref] [PubMed]

Huang, Z.

Z. Huang, K. Cui, G. Bai, X. Feng, F. Liu, W. Zhang, and Y. Huang, “High-mechanical-frequency characteristics of optomechanical crystal cavity with coupling waveguide,” Sci. Rep. 6(1), 34160 (2016).
[Crossref] [PubMed]

Ibanescu, M.

Ikeda, K.

Jiang, L.

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

Jiang, X.

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

Joannopoulos, J. D.

Johnson, S. G.

Jones, H. F.

C. M. Bender, D. C. Brody, and H. F. Jones, “Complex Extension of Quantum Mechanics,” Phys. Rev. Lett. 89(27), 270401 (2002).
[Crossref] [PubMed]

Khajavikhan, M.

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

H. Hodaei, M. A. Miri, M. Heinrich, D. N. Christodoulides, and M. Khajavikhan, “Parity-time-symmetric microring lasers,” Science 346(6212), 975–978 (2014).
[Crossref] [PubMed]

Kip, D.

C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity–time symmetry in optics,” Nat. Phys. 6(3), 192–195 (2010).
[Crossref]

Klintberg, T.

Kottos, T.

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional Invisibility Induced by PT-Symmetric Periodic Structures,” Phys. Rev. Lett. 106(21), 213901 (2011).
[Crossref] [PubMed]

Krückel, C. J.

Lagae, L.

J. Roels, I. De Vlaminck, L. Lagae, B. Maes, D. Van Thourhout, and R. Baets, “Tunable optical forces between nanophotonic waveguides,” Nat. Nanotechnol. 4(8), 510–513 (2009).
[Crossref] [PubMed]

Lei, F.

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

Li, G.

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

Li, H.

Li, M.

H. Li, J. W. Noh, Y. Chen, and M. Li, “Enhanced optical forces in integrated hybrid plasmonic waveguides,” Opt. Express 21(10), 11839–11851 (2013).
[Crossref] [PubMed]

M. Li, W. H. P. Pernice, and H. X. Tang, “Tunable bipolar optical interactions between guided lightwaves,” Nat. Photonics 3(8), 464–468 (2009).
[Crossref]

M. Li, W. H. P. Pernice, C. Xiong, T. Baehr-Jones, M. Hochberg, and H. X. Tang, “Harnessing optical forces in integrated photonic circuits,” Nature 456(7221), 480–484 (2008).
[Crossref] [PubMed]

Lin, Z.

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional Invisibility Induced by PT-Symmetric Periodic Structures,” Phys. Rev. Lett. 106(21), 213901 (2011).
[Crossref] [PubMed]

Lisyansky, A. A.

A. A. Zyablovsky, A. P. Vinogradov, A. A. Pukhov, A. V. Dorofeenko, and A. A. Lisyansky, “PT-symmetry in optics,” Phys.- Usp. 57(11), 1063–1082 (2014).
[Crossref]

Litchinitser, N. M.

Liu, F.

Z. Huang, K. Cui, G. Bai, X. Feng, F. Liu, W. Zhang, and Y. Huang, “High-mechanical-frequency characteristics of optomechanical crystal cavity with coupling waveguide,” Sci. Rep. 6(1), 34160 (2016).
[Crossref] [PubMed]

Liu, Y.

X. Xu, L. Shi, Y. Liu, Z. Wang, and X. Zhang, “Enhanced optical gradient forces between coupled graphene sheets,” Sci. Rep. 6(1), 28568 (2016).
[Crossref] [PubMed]

Z. Wang, L. Shi, Y. Liu, X. Xu, and X. Zhang, “Optical nonreciprocity in asymmetric optomechanical couplers,” Sci. Rep. 5(1), 8657 (2015).
[Crossref] [PubMed]

Liu, Y. C.

Y. W. Hu, Y. F. Xiao, Y. C. Liu, and Q. Gong, “Optomechanical sensing with on-chip microcavities,” Front. Phys. 8(5), 475–490 (2013).
[Crossref]

Loncar, M.

Long, G. L.

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

Longhi, S.

S. Longhi, “Bloch Oscillations in Complex Crystals with PT Symmetry,” Phys. Rev. Lett. 103(12), 123601 (2009).
[Crossref] [PubMed]

Lu, M. H.

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

Lumer, Y.

Y. Lumer, Y. Plotnik, M. C. Rechtsman, and M. Segev, “Nonlinearly Induced PT transition in Photonic Systems,” Phys. Rev. Lett. 111(26), 263901 (2013).
[Crossref] [PubMed]

Luo, J.

K. Fang, J. Luo, A. Metelmann, M. H. Matheny, F. Marquardt, A. A. Clerk, and O. Painter, “Generalized non-reciprocity in an optomechanical circuit via synthetic magnetism and reservoir engineering,” Nat. Phys. 13(5), 465–471 (2017).
[Crossref]

Ma, C.

Ma, J.

J. Ma and M. L. Povinelli, “Applications of optomechanical effects for on-chip manipulation of light signals,” Curr. Opin. Solid. St. M. 16(2), 82–90 (2012).
[Crossref]

J. Ma and M. L. Povinelli, “Mechanical Kerr nonlinearities due to bipolar optical forces between deformable silicon waveguides,” Opt. Express 19(11), 10102–10110 (2011).
[Crossref] [PubMed]

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(6212), 972–975 (2014).
[Crossref] [PubMed]

Maes, B.

J. Roels, I. De Vlaminck, L. Lagae, B. Maes, D. Van Thourhout, and R. Baets, “Tunable optical forces between nanophotonic waveguides,” Nat. Nanotechnol. 4(8), 510–513 (2009).
[Crossref] [PubMed]

Makris, K. G.

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

C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity–time symmetry in optics,” Nat. Phys. 6(3), 192–195 (2010).
[Crossref]

Marquardt, F.

K. Fang, J. Luo, A. Metelmann, M. H. Matheny, F. Marquardt, A. A. Clerk, and O. Painter, “Generalized non-reciprocity in an optomechanical circuit via synthetic magnetism and reservoir engineering,” Nat. Phys. 13(5), 465–471 (2017).
[Crossref]

M. Bagheri, M. Poot, L. Fan, F. Marquardt, and H. X. Tang, “Photonic Cavity Synchronization of Nanomechanical Oscillators,” Phys. Rev. Lett. 111(21), 213902 (2013).
[Crossref] [PubMed]

Matheny, M. H.

K. Fang, J. Luo, A. Metelmann, M. H. Matheny, F. Marquardt, A. A. Clerk, and O. Painter, “Generalized non-reciprocity in an optomechanical circuit via synthetic magnetism and reservoir engineering,” Nat. Phys. 13(5), 465–471 (2017).
[Crossref]

Metelmann, A.

K. Fang, J. Luo, A. Metelmann, M. H. Matheny, F. Marquardt, A. A. Clerk, and O. Painter, “Generalized non-reciprocity in an optomechanical circuit via synthetic magnetism and reservoir engineering,” Nat. Phys. 13(5), 465–471 (2017).
[Crossref]

Miri, M. A.

M. Wimmer, A. Regensburger, M. A. Miri, C. Bersch, D. N. Christodoulides, and U. Peschel, “Observation of optical solitons in PT-symmetric lattices,” Nat. Commun. 6(1), 7782 (2015).
[Crossref] [PubMed]

H. Hodaei, M. A. Miri, M. Heinrich, D. N. Christodoulides, and M. Khajavikhan, “Parity-time-symmetric microring lasers,” Science 346(6212), 975–978 (2014).
[Crossref] [PubMed]

Miski-Oglu, M.

S. Bittner, B. Dietz, U. Günther, H. L. Harney, M. Miski-Oglu, A. Richter, and F. Schäfer, “PT Symmetry and Spontaneous Symmetry Breaking in a Microwave Billiard,” Phys. Rev. Lett. 108(2), 024101 (2012).
[Crossref] [PubMed]

Monifi, F.

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

Morandotti, R.

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

Mu, J.

Musslimani, Z. H.

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

Noh, J. W.

Nori, F.

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

Oliveira, J. E. B.

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

Ozdemir, S. K.

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

Painter, O.

K. Fang, J. Luo, A. Metelmann, M. H. Matheny, F. Marquardt, A. A. Clerk, and O. Painter, “Generalized non-reciprocity in an optomechanical circuit via synthetic magnetism and reservoir engineering,” Nat. Phys. 13(5), 465–471 (2017).
[Crossref]

Peng, B.

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

Pernice, W. H. P.

M. Li, W. H. P. Pernice, and H. X. Tang, “Tunable bipolar optical interactions between guided lightwaves,” Nat. Photonics 3(8), 464–468 (2009).
[Crossref]

M. Li, W. H. P. Pernice, C. Xiong, T. Baehr-Jones, M. Hochberg, and H. X. Tang, “Harnessing optical forces in integrated photonic circuits,” Nature 456(7221), 480–484 (2008).
[Crossref] [PubMed]

Peschel, U.

M. Wimmer, A. Regensburger, M. A. Miri, C. Bersch, D. N. Christodoulides, and U. Peschel, “Observation of optical solitons in PT-symmetric lattices,” Nat. Commun. 6(1), 7782 (2015).
[Crossref] [PubMed]

Plotnik, Y.

Y. Lumer, Y. Plotnik, M. C. Rechtsman, and M. Segev, “Nonlinearly Induced PT transition in Photonic Systems,” Phys. Rev. Lett. 111(26), 263901 (2013).
[Crossref] [PubMed]

Poot, M.

L. Fan, C. L. Zou, M. Poot, R. Cheng, X. Guo, X. Han, and H. X. Tang, “Integrated optomechanical single-photon frequency shifter,” Nat. Photonics 10(12), 766–770 (2016).
[Crossref]

M. Bagheri, M. Poot, L. Fan, F. Marquardt, and H. X. Tang, “Photonic Cavity Synchronization of Nanomechanical Oscillators,” Phys. Rev. Lett. 111(21), 213902 (2013).
[Crossref] [PubMed]

X. Sun, J. Zheng, M. Poot, C. W. Wong, and H. X. Tang, “Femtogram Doubly Clamped Nanomechanical Resonators Embedded in a High-Q Two-Dimensional Photonic Crystal Nanocavity,” Nano Lett. 12(5), 2299–2305 (2012).
[Crossref] [PubMed]

Povinelli, M. L.

Pukhov, A. A.

A. A. Zyablovsky, A. P. Vinogradov, A. A. Pukhov, A. V. Dorofeenko, and A. A. Lisyansky, “PT-symmetry in optics,” Phys.- Usp. 57(11), 1063–1082 (2014).
[Crossref]

Ramezani, H.

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional Invisibility Induced by PT-Symmetric Periodic Structures,” Phys. Rev. Lett. 106(21), 213901 (2011).
[Crossref] [PubMed]

Rechtsman, M. C.

Y. Lumer, Y. Plotnik, M. C. Rechtsman, and M. Segev, “Nonlinearly Induced PT transition in Photonic Systems,” Phys. Rev. Lett. 111(26), 263901 (2013).
[Crossref] [PubMed]

Regensburger, A.

M. Wimmer, A. Regensburger, M. A. Miri, C. Bersch, D. N. Christodoulides, and U. Peschel, “Observation of optical solitons in PT-symmetric lattices,” Nat. Commun. 6(1), 7782 (2015).
[Crossref] [PubMed]

Richter, A.

S. Bittner, B. Dietz, U. Günther, H. L. Harney, M. Miski-Oglu, A. Richter, and F. Schäfer, “PT Symmetry and Spontaneous Symmetry Breaking in a Microwave Billiard,” Phys. Rev. Lett. 108(2), 024101 (2012).
[Crossref] [PubMed]

Roels, J.

D. Van Thourhout and J. Roels, “Optomechanical device actuation through the optical gradient force,” Nat. Photonics 4(4), 211–217 (2010).
[Crossref]

J. Roels, I. De Vlaminck, L. Lagae, B. Maes, D. Van Thourhout, and R. Baets, “Tunable optical forces between nanophotonic waveguides,” Nat. Nanotechnol. 4(8), 510–513 (2009).
[Crossref] [PubMed]

Rotter, S.

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

Rüter, C. E.

C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity–time symmetry in optics,” Nat. Phys. 6(3), 192–195 (2010).
[Crossref]

Salamo, G. J.

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

Saperstein, R. E.

Schäfer, F.

S. Bittner, B. Dietz, U. Günther, H. L. Harney, M. Miski-Oglu, A. Richter, and F. Schäfer, “PT Symmetry and Spontaneous Symmetry Breaking in a Microwave Billiard,” Phys. Rev. Lett. 108(2), 024101 (2012).
[Crossref] [PubMed]

Scherer, A.

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

Segev, M.

Y. Lumer, Y. Plotnik, M. C. Rechtsman, and M. Segev, “Nonlinearly Induced PT transition in Photonic Systems,” Phys. Rev. Lett. 111(26), 263901 (2013).
[Crossref] [PubMed]

C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity–time symmetry in optics,” Nat. Phys. 6(3), 192–195 (2010).
[Crossref]

Shi, L.

X. Xu, L. Shi, Y. Liu, Z. Wang, and X. Zhang, “Enhanced optical gradient forces between coupled graphene sheets,” Sci. Rep. 6(1), 28568 (2016).
[Crossref] [PubMed]

Z. Wang, L. Shi, Y. Liu, X. Xu, and X. Zhang, “Optical nonreciprocity in asymmetric optomechanical couplers,” Sci. Rep. 5(1), 8657 (2015).
[Crossref] [PubMed]

Siviloglou, G. A.

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

Smythe, E. J.

Sounas, D.

R. Fleury, D. Sounas, and A. Alù, “An invisible acoustic sensor based on parity-time symmetry,” Nat. Commun. 6(1), 5905 (2015).
[Crossref] [PubMed]

Sun, X.

X. Sun, J. Zheng, M. Poot, C. W. Wong, and H. X. Tang, “Femtogram Doubly Clamped Nanomechanical Resonators Embedded in a High-Q Two-Dimensional Photonic Crystal Nanocavity,” Nano Lett. 12(5), 2299–2305 (2012).
[Crossref] [PubMed]

Tang, H. X.

L. Fan, C. L. Zou, M. Poot, R. Cheng, X. Guo, X. Han, and H. X. Tang, “Integrated optomechanical single-photon frequency shifter,” Nat. Photonics 10(12), 766–770 (2016).
[Crossref]

M. Bagheri, M. Poot, L. Fan, F. Marquardt, and H. X. Tang, “Photonic Cavity Synchronization of Nanomechanical Oscillators,” Phys. Rev. Lett. 111(21), 213902 (2013).
[Crossref] [PubMed]

X. Sun, J. Zheng, M. Poot, C. W. Wong, and H. X. Tang, “Femtogram Doubly Clamped Nanomechanical Resonators Embedded in a High-Q Two-Dimensional Photonic Crystal Nanocavity,” Nano Lett. 12(5), 2299–2305 (2012).
[Crossref] [PubMed]

M. Li, W. H. P. Pernice, and H. X. Tang, “Tunable bipolar optical interactions between guided lightwaves,” Nat. Photonics 3(8), 464–468 (2009).
[Crossref]

M. Li, W. H. P. Pernice, C. Xiong, T. Baehr-Jones, M. Hochberg, and H. X. Tang, “Harnessing optical forces in integrated photonic circuits,” Nature 456(7221), 480–484 (2008).
[Crossref] [PubMed]

Torres-Company, V.

Van Thourhout, D.

D. Van Thourhout and J. Roels, “Optomechanical device actuation through the optical gradient force,” Nat. Photonics 4(4), 211–217 (2010).
[Crossref]

J. Roels, I. De Vlaminck, L. Lagae, B. Maes, D. Van Thourhout, and R. Baets, “Tunable optical forces between nanophotonic waveguides,” Nat. Nanotechnol. 4(8), 510–513 (2009).
[Crossref] [PubMed]

Verhagen, E.

E. Verhagen and A. Alù, “Optomechanical nonreciprocity,” Nat. Phys. 13(10), 922–924 (2017).
[Crossref]

Vinogradov, A. P.

A. A. Zyablovsky, A. P. Vinogradov, A. A. Pukhov, A. V. Dorofeenko, and A. A. Lisyansky, “PT-symmetry in optics,” Phys.- Usp. 57(11), 1063–1082 (2014).
[Crossref]

Volatier-Ravat, M.

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

Walasik, W.

Wang, G.

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

Wang, Y.

L. Feng, Z. J. Wong, R. M. Ma, Y. Wang, and X. Zhang, “Single-mode laser by parity-time symmetry breaking,” Science 346(6212), 972–975 (2014).
[Crossref] [PubMed]

Wang, Z.

X. Xu, L. Shi, Y. Liu, Z. Wang, and X. Zhang, “Enhanced optical gradient forces between coupled graphene sheets,” Sci. Rep. 6(1), 28568 (2016).
[Crossref] [PubMed]

Z. Wang, L. Shi, Y. Liu, X. Xu, and X. Zhang, “Optical nonreciprocity in asymmetric optomechanical couplers,” Sci. Rep. 5(1), 8657 (2015).
[Crossref] [PubMed]

Wen, J.

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

Wimmer, M.

M. Wimmer, A. Regensburger, M. A. Miri, C. Bersch, D. N. Christodoulides, and U. Peschel, “Observation of optical solitons in PT-symmetric lattices,” Nat. Commun. 6(1), 7782 (2015).
[Crossref] [PubMed]

Wong, C. W.

X. Sun, J. Zheng, M. Poot, C. W. Wong, and H. X. Tang, “Femtogram Doubly Clamped Nanomechanical Resonators Embedded in a High-Q Two-Dimensional Photonic Crystal Nanocavity,” Nano Lett. 12(5), 2299–2305 (2012).
[Crossref] [PubMed]

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(6212), 972–975 (2014).
[Crossref] [PubMed]

Xiao, M.

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

Xiao, Y. F.

Y. W. Hu, Y. F. Xiao, Y. C. Liu, and Q. Gong, “Optomechanical sensing with on-chip microcavities,” Front. Phys. 8(5), 475–490 (2013).
[Crossref]

Xiong, C.

M. Li, W. H. P. Pernice, C. Xiong, T. Baehr-Jones, M. Hochberg, and H. X. Tang, “Harnessing optical forces in integrated photonic circuits,” Nature 456(7221), 480–484 (2008).
[Crossref] [PubMed]

Xu, X.

X. Xu, L. Shi, Y. Liu, Z. Wang, and X. Zhang, “Enhanced optical gradient forces between coupled graphene sheets,” Sci. Rep. 6(1), 28568 (2016).
[Crossref] [PubMed]

Z. Wang, L. Shi, Y. Liu, X. Xu, and X. Zhang, “Optical nonreciprocity in asymmetric optomechanical couplers,” Sci. Rep. 5(1), 8657 (2015).
[Crossref] [PubMed]

Xu, Y. L.

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

Yang, C.

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

Yang, L.

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

Zhang, W.

Z. Huang, K. Cui, G. Bai, X. Feng, F. Liu, W. Zhang, and Y. Huang, “High-mechanical-frequency characteristics of optomechanical crystal cavity with coupling waveguide,” Sci. Rep. 6(1), 34160 (2016).
[Crossref] [PubMed]

Zhang, X.

X. Xu, L. Shi, Y. Liu, Z. Wang, and X. Zhang, “Enhanced optical gradient forces between coupled graphene sheets,” Sci. Rep. 6(1), 28568 (2016).
[Crossref] [PubMed]

Z. Wang, L. Shi, Y. Liu, X. Xu, and X. Zhang, “Optical nonreciprocity in asymmetric optomechanical couplers,” Sci. Rep. 5(1), 8657 (2015).
[Crossref] [PubMed]

L. Feng, Z. J. Wong, R. M. Ma, Y. Wang, and X. Zhang, “Single-mode laser by parity-time symmetry breaking,” Science 346(6212), 972–975 (2014).
[Crossref] [PubMed]

Zheng, J.

X. Sun, J. Zheng, M. Poot, C. W. Wong, and H. X. Tang, “Femtogram Doubly Clamped Nanomechanical Resonators Embedded in a High-Q Two-Dimensional Photonic Crystal Nanocavity,” Nano Lett. 12(5), 2299–2305 (2012).
[Crossref] [PubMed]

Zou, C. L.

L. Fan, C. L. Zou, M. Poot, R. Cheng, X. Guo, X. Han, and H. X. Tang, “Integrated optomechanical single-photon frequency shifter,” Nat. Photonics 10(12), 766–770 (2016).
[Crossref]

Zyablovsky, A. A.

A. A. Zyablovsky, A. P. Vinogradov, A. A. Pukhov, A. V. Dorofeenko, and A. A. Lisyansky, “PT-symmetry in optics,” Phys.- Usp. 57(11), 1063–1082 (2014).
[Crossref]

Curr. Opin. Solid. St. M. (1)

J. Ma and M. L. Povinelli, “Applications of optomechanical effects for on-chip manipulation of light signals,” Curr. Opin. Solid. St. M. 16(2), 82–90 (2012).
[Crossref]

Front. Phys. (1)

Y. W. Hu, Y. F. Xiao, Y. C. Liu, and Q. Gong, “Optomechanical sensing with on-chip microcavities,” Front. Phys. 8(5), 475–490 (2013).
[Crossref]

Nano Lett. (1)

X. Sun, J. Zheng, M. Poot, C. W. Wong, and H. X. Tang, “Femtogram Doubly Clamped Nanomechanical Resonators Embedded in a High-Q Two-Dimensional Photonic Crystal Nanocavity,” Nano Lett. 12(5), 2299–2305 (2012).
[Crossref] [PubMed]

Nat. Commun. (2)

R. Fleury, D. Sounas, and A. Alù, “An invisible acoustic sensor based on parity-time symmetry,” Nat. Commun. 6(1), 5905 (2015).
[Crossref] [PubMed]

M. Wimmer, A. Regensburger, M. A. Miri, C. Bersch, D. N. Christodoulides, and U. Peschel, “Observation of optical solitons in PT-symmetric lattices,” Nat. Commun. 6(1), 7782 (2015).
[Crossref] [PubMed]

Nat. Mater. (1)

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

Nat. Nanotechnol. (1)

J. Roels, I. De Vlaminck, L. Lagae, B. Maes, D. Van Thourhout, and R. Baets, “Tunable optical forces between nanophotonic waveguides,” Nat. Nanotechnol. 4(8), 510–513 (2009).
[Crossref] [PubMed]

Nat. Photonics (5)

L. Fan, C. L. Zou, M. Poot, R. Cheng, X. Guo, X. Han, and H. X. Tang, “Integrated optomechanical single-photon frequency shifter,” Nat. Photonics 10(12), 766–770 (2016).
[Crossref]

D. Van Thourhout and J. Roels, “Optomechanical device actuation through the optical gradient force,” Nat. Photonics 4(4), 211–217 (2010).
[Crossref]

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

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

M. Li, W. H. P. Pernice, and H. X. Tang, “Tunable bipolar optical interactions between guided lightwaves,” Nat. Photonics 3(8), 464–468 (2009).
[Crossref]

Nat. Phys. (5)

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

C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity–time symmetry in optics,” Nat. Phys. 6(3), 192–195 (2010).
[Crossref]

K. Fang, J. Luo, A. Metelmann, M. H. Matheny, F. Marquardt, A. A. Clerk, and O. Painter, “Generalized non-reciprocity in an optomechanical circuit via synthetic magnetism and reservoir engineering,” Nat. Phys. 13(5), 465–471 (2017).
[Crossref]

E. Verhagen and A. Alù, “Optomechanical nonreciprocity,” Nat. Phys. 13(10), 922–924 (2017).
[Crossref]

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

Nature (1)

M. Li, W. H. P. Pernice, C. Xiong, T. Baehr-Jones, M. Hochberg, and H. X. Tang, “Harnessing optical forces in integrated photonic circuits,” Nature 456(7221), 480–484 (2008).
[Crossref] [PubMed]

Opt. Express (5)

Opt. Lett. (2)

Phys. Rev. Lett. (8)

M. Bagheri, M. Poot, L. Fan, F. Marquardt, and H. X. Tang, “Photonic Cavity Synchronization of Nanomechanical Oscillators,” Phys. Rev. Lett. 111(21), 213902 (2013).
[Crossref] [PubMed]

Y. Lumer, Y. Plotnik, M. C. Rechtsman, and M. Segev, “Nonlinearly Induced PT transition in Photonic Systems,” Phys. Rev. Lett. 111(26), 263901 (2013).
[Crossref] [PubMed]

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional Invisibility Induced by PT-Symmetric Periodic Structures,” Phys. Rev. Lett. 106(21), 213901 (2011).
[Crossref] [PubMed]

S. Longhi, “Bloch Oscillations in Complex Crystals with PT Symmetry,” Phys. Rev. Lett. 103(12), 123601 (2009).
[Crossref] [PubMed]

C. M. Bender, D. C. Brody, and H. F. Jones, “Complex Extension of Quantum Mechanics,” Phys. Rev. Lett. 89(27), 270401 (2002).
[Crossref] [PubMed]

S. Bittner, B. Dietz, U. Günther, H. L. Harney, M. Miski-Oglu, A. Richter, and F. Schäfer, “PT Symmetry and Spontaneous Symmetry Breaking in a Microwave Billiard,” Phys. Rev. Lett. 108(2), 024101 (2012).
[Crossref] [PubMed]

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

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

Phys.- Usp. (1)

A. A. Zyablovsky, A. P. Vinogradov, A. A. Pukhov, A. V. Dorofeenko, and A. A. Lisyansky, “PT-symmetry in optics,” Phys.- Usp. 57(11), 1063–1082 (2014).
[Crossref]

Rep. Prog. Phys. (1)

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

Sci. Rep. (3)

Z. Wang, L. Shi, Y. Liu, X. Xu, and X. Zhang, “Optical nonreciprocity in asymmetric optomechanical couplers,” Sci. Rep. 5(1), 8657 (2015).
[Crossref] [PubMed]

Z. Huang, K. Cui, G. Bai, X. Feng, F. Liu, W. Zhang, and Y. Huang, “High-mechanical-frequency characteristics of optomechanical crystal cavity with coupling waveguide,” Sci. Rep. 6(1), 34160 (2016).
[Crossref] [PubMed]

X. Xu, L. Shi, Y. Liu, Z. Wang, and X. Zhang, “Enhanced optical gradient forces between coupled graphene sheets,” Sci. Rep. 6(1), 28568 (2016).
[Crossref] [PubMed]

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(6212), 972–975 (2014).
[Crossref] [PubMed]

H. Hodaei, M. A. Miri, M. Heinrich, D. N. Christodoulides, and M. Khajavikhan, “Parity-time-symmetric microring lasers,” Science 346(6212), 975–978 (2014).
[Crossref] [PubMed]

Other (1)

J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, 1998).

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Figures (6)

Fig. 1
Fig. 1 PT-symmetric coupled-waveguide structure. (a) Schematic of the coupler structure. It is formed by two released waveguides with balanced gain (blue) and loss (purple). (b) Cross sections of the two freestanding waveguides. g is the gap between the two waveguides.
Fig. 2
Fig. 2 Effective refractive indices and the normalized optical forces in the PT-symmetric coupled-waveguide system. (a) Real (solid lines) and imaginary (dashed lines) parts of the effective refractive indices of |1 and |2 with g = 450 nm. (c) and (e) are effective refractive indices of the eigenmodes versus the gap with ε i = 0.0368 and ε i = 0, respectively. When ε i = 0, the eigenmodes become symmetric and antisymmetric supermodes. (b), (d) and (f) are corresponding normalized optical forces, respectively.
Fig. 3
Fig. 3 Normalized electric fields of the two eigenmodes in an electromagnetic oscillation period. (a) |S=( e iωt , e iωt ). (b) | AS=( e iωt , e iωt+π ). (c) | 1,2=( e iωt , e i(ωt±π/4) ), the directions of the electric fields in the yellow region are opposite.
Fig. 4
Fig. 4 Eigenmode field distributions of the PT-symmetric coupler structure. (a-b) |1 and |2 when PT symmetry is unbroken with g = 425 nm and ε i = 0.0368. (c-d) |1 and |2 when PT symmetry is broken with g = 475 nm and ε i = 0.0368.
Fig. 5
Fig. 5 (a) Optical force induced deflection of the DCB at different input powers. Without (solid lines) and with (dashed lines) optical Kerr nonlinearity considered. (b) Normalized optical forces of the two eigenmodes as a function of the position along the DCB. (c) Output powers of the two waveguides versus the input power with symmetric mode incident.
Fig. 6
Fig. 6 Fn with different initial phase differences versus the gap when the length of the DCB is 300 μm. (a) In the PT-symmetric coupler structure. (b) In the conventional Hermitian coupler structure. The insets show the enlarged view.

Equations (8)

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

F= s T ds
T ij =ε E i E j +μ H i H j δ ij (ε E 2 +μ H 2 )/2
N m d a m dz =i β m a m i n=1 κ mn ' a n i γ k | a m | 2 a m
κ mn ' = ω ε 0 4 s ( n ˜ 2 n 2 )( E tn E tm n 2 n ˜ 2 E zn E zm ) ds
E= a s (z) e i φ s (z) E s + a as (z) e i φ as (z) E as
F(z)= a s 2 (z) f n.s + a as 2 (z) f n.as + a s (z) a as (z) f n,b
YI d 4 u(z) d z 4 =F(z)
F n = 0 L F(z)dz P in

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