Non-Hermitian Optical Waveguide Couplers

  • Sergey V. Suchkov
  • Andrey A. Sukhorukov
  • Yuri S. Kivshar
Part of the Springer Tracts in Modern Physics book series (STMP, volume 280)


We discuss the PT symmetry effects in non-Hermitian waveguiding geometries starting from a classical example of a two-core coupler with gain and loss. We demonstrate that a nonlinear response can break the PT symmetry in a coupler, and discuss the regimes of parametric amplification and nonlocality associated with such systems. Then, we analyse non-Hermitical trimers and also a PT-symmetric system embedded into an array of waveguides. Finally, we demonstrate the existence of nontrivial modes in non-Hermitian waveguiding structures with asymmetric layers of gain and loss.



The authors acknowledge support by the Australian Research Council (ARC), including Discovery Project DP160100619.


  1. 1.
    Bender, C.M., Boettcher, S.: Real spectra in non-Hermitian hamiltonians having PT symmetry. Phys. Rev. Lett. 80, 5243–5246 (1998)ADSMathSciNetCrossRefGoogle Scholar
  2. 2.
    El Ganainy, R., Makris, K.G., Christodoulides, D.N., Musslimani, Z.H.: Theory of coupled optical PT-symmetric structures. Opt. Lett. 32, 2632–2634 (2007)ADSCrossRefGoogle Scholar
  3. 3.
    Makris, K.G., El Ganainy, R., Christodoulides, D.N., Musslimani, Z.H.: Beam dynamics in PT symmetric optical lattices. Phys. Rev. Lett. 100, 103904–4 (2008)ADSCrossRefGoogle Scholar
  4. 4.
    Ruter, C.E., Makris, K.G., El Ganainy, R., Christodoulides, D.N., Segev, M., Kip, D.: Observation of parity-time symmetry in optics. Nat. Phys. 6, 192–195 (2010)CrossRefGoogle Scholar
  5. 5.
    Feng, L., Wong, Z.J., Ma, R.M., Wang, Y., Zhang, X.: Single-mode laser by parity-time symmetry breaking. Science 346, 972–975 (2014)ADSCrossRefGoogle Scholar
  6. 6.
    Hodaei, H., Miri, M.A., Heinrich, M., Christodoulides, D.N., Khajavikhan, M.: Parity-time-symmetric microring lasers. Science 346, 975–978 (2014)ADSCrossRefGoogle Scholar
  7. 7.
    Kato, T.: Perturbation Theory for Linear Operators, 2nd edn. Springer, Berlin (1995)CrossRefGoogle Scholar
  8. 8.
    Klaiman, S., Guenther, U., Moiseyev, N.: Visualization of branch points in PT-symmetric waveguides. Phys. Rev. Lett. 101, 080402–4 (2008)ADSMathSciNetCrossRefGoogle Scholar
  9. 9.
    Heiss, W.D.: Exceptional points of non-hermitian operators. J. Phys. A 37, 2455–2464 (2004)ADSMathSciNetCrossRefGoogle Scholar
  10. 10.
    Muller, M., Rotter, I.: Exceptional points in open quantum systems. J. Phys. A 41, 244018–15 (2008)ADSMathSciNetCrossRefGoogle Scholar
  11. 11.
    Mostafazadeh, A.: Self-dual spectral singularities and coherent perfect absorbing lasers without PT-symmetry. J. Phys. A 45, 444024–10 (2012)ADSMathSciNetCrossRefGoogle Scholar
  12. 12.
    Mostafazadeh, A.: Pseudo-hermiticity versus PT symmetry: the necessary condition for the reality of the spectrum of a non-Hermitian Hamiltonian. J. Math. Phys. 43, 205–214 (2002)ADSMathSciNetCrossRefGoogle Scholar
  13. 13.
    Chen, Y.J., Snyder, A.W., Payne, D.N.: Twin core nonlinear couplers with gain and loss. IEEE J. Quantum Electron. 28, 239–245 (1992)ADSCrossRefGoogle Scholar
  14. 14.
    Malomed, B.A., Peng, G.D., Chu, P.L.: Nonlinear-optical amplifier based on a dual-core fiber. Opt. Lett. 21, 330–332 (1996)ADSCrossRefGoogle Scholar
  15. 15.
    Ramezani, H., Kottos, T., El Ganainy, R., Christodoulides, D.N.: Unidirectional nonlinear PT-symmetric optical structures. Phys. Rev. A 82, 043803–6 (2010)ADSCrossRefGoogle Scholar
  16. 16.
    Sukhorukov, A.A., Xu, Z.Y., Kivshar, Y.S.: Nonlinear suppression of time reversals in PT-symmetric optical couplers. Phys. Rev. A 82, 043818–5 (2010)ADSCrossRefGoogle Scholar
  17. 17.
    Kevrekidis, P.G., Pelinovsky, D.E., Tyugin, D.Y.: Nonlinear dynamics in PT-symmetric lattices. J. Phys. A 46, 365201–17 (2013)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Barashenkov, I.V.: Hamiltonian formulation of the standard PT-symmetric nonlinear Schrödinger dimer. Phys. Rev. A 90, 045802–4 (2014)ADSCrossRefGoogle Scholar
  19. 19.
    Barashenkov, I.V., Jackson, G.S., Flach, S.: Blow-up regimes in the PT-symmetric coupler and the actively coupled dimer. Phys. Rev. A 88, 053817–8 (2013)ADSCrossRefGoogle Scholar
  20. 20.
    Pickton, J., Susanto, H.: Integrability of PT-symmetric dimers. Phys. Rev. A 88, 063840–8 (2013)ADSCrossRefGoogle Scholar
  21. 21.
    Barashenkov, I.V., Pelinovsky, D.E., Dubard, P.: Dimer with gain and loss: integrability and PT-symmetry restoration. J. Phys. A 48, 325201–28 (2015)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Lupu, A., Benisty, H., Degiron, A.: Using optical PT-symmetry for switching applications. Photonics Nanostruct. Fundam. Appl. 12, 305–311 (2014)ADSCrossRefGoogle Scholar
  23. 23.
    Schindler, J., Li, A., Zheng, M.C., Ellis, F.M., Kottos, T.: Experimental study of active LRC circuits with PT symmetries. Phys. Rev. A 84, 040101–5 (2011)ADSCrossRefGoogle Scholar
  24. 24.
    Cuevas, J., Kevrekidis, P.G., Saxena, A., Khare, A.: PT-symmetric dimer of coupled nonlinear oscillators. Phys. Rev. A 88, 032108–11 (2013)ADSCrossRefGoogle Scholar
  25. 25.
    Duanmu, M., Li, K., Horne, R.L., Kevrekidis, P.G., Whitaker, N.: Linear and nonlinear parity-time-symmetric oligomers: a dynamical systems analysis. Philos. Trans. R. Soc. A 371, 20120171–19 (2013)ADSMathSciNetCrossRefGoogle Scholar
  26. 26.
    Miroshnichenko, A.E., Malomed, B.A., Kivshar, Y.S.: Nonlinearly PT-symmetric systems: spontaneous symmetry breaking and transmission resonances. Phys. Rev. A 84, 012123–4 (2011)ADSCrossRefGoogle Scholar
  27. 27.
    Zezyulin, D.A., Kartashov, Y.V., Konotop, V.V.: Stability of solitons in PT-symmetric nonlinear potentials. Europhys. Lett. 96, 64003–6 (2011)ADSCrossRefGoogle Scholar
  28. 28.
    Longhi, S.: Phase transitions in wick-rotated PT-symmetric optics. Ann. Phys. 360, 150–160 (2015)CrossRefGoogle Scholar
  29. 29.
    Moreira, F.C., Abdullaev, F.K., Konotop, V.V., Yulin, A.V.: Localized modes in χ (2) media with PT-symmetric localized potential. Phys. Rev. A 86, 053815–7 (2012)ADSCrossRefGoogle Scholar
  30. 30.
    Li, K., Zezyulin, D.A., Kevrekidis, P.G., Konotop, V.V., Abdullaev, F.K.: PT-symmetric coupler with χ (2) nonlinearity. Phys. Rev. A 88, 053820–11 (2013)ADSCrossRefGoogle Scholar
  31. 31.
    Abdullaev, F.K., Umarov, B.A.: Exact solitonic solutions for optical media with χ (2) nonlinearity and PT-symmetric potentials. J. Phys. Conf. Ser. 553, 012001–6 (2014)CrossRefGoogle Scholar
  32. 32.
    Boyd, R.W.: Nonlinear Optics, 3rd edn. Academic, San Diego (2008)Google Scholar
  33. 33.
    Antonosyan, D.A., Solntsev, A.S., Sukhorukov, A.A.: Parity-time anti-symmetric parametric amplifier. Opt. Lett. 40, 4575–4578 (2015)ADSCrossRefGoogle Scholar
  34. 34.
    Jones, H.F.: Scattering from localized non-hermitian potentials. Phys. Rev. D 76, 125003–5 (2007)ADSCrossRefGoogle Scholar
  35. 35.
    Znojil, M.: Scattering theory using smeared non-Hermitian potentials. Phys. Rev. D 80, 045009–12 (2009)ADSMathSciNetCrossRefGoogle Scholar
  36. 36.
    Dmitriev, S.V., Sukhorukov, A.A., Kivshar, Y.S.: Binary parity-time-symmetric nonlinear lattices with balanced gain and loss. Opt. Lett. 35, 2976–2978 (2010)ADSCrossRefGoogle Scholar
  37. 37.
    Zheng, M.C., Christodoulides, D.N., Fleischmann, R., Kottos, T.: PT optical lattices and universality in beam dynamics. Phys. Rev. A 82, 010103–4 (2010)ADSCrossRefGoogle Scholar
  38. 38.
    Sukhorukov, A.A., Dmitriev, S.V., Suchkov, S.V., Kivshar, Y.S.: Nonlocality in PT-symmetric waveguide arrays with gain and loss. Opt. Lett. 37, 2148–2150 (2012)ADSCrossRefGoogle Scholar
  39. 39.
    Longhi, S., Cannata, F., Ventura, A.: Spontaneous PT symmetry breaking in dirac-kronig-penney crystals. Phys. Rev. B 84, 235131–8 (2011)ADSCrossRefGoogle Scholar
  40. 40.
    Moiseyev, N.: Crossing rule for a PT-symmetric two-level time-periodic system. Phys. Rev. A 83, 052125–5 (2011)ADSCrossRefGoogle Scholar
  41. 41.
    Joglekar, Y.N., Marathe, R., Durganandini, P., Pathak, R.K.: PT spectroscopy of the Rabi problem. Phys. Rev. A 90, 040101–4 (2014)ADSCrossRefGoogle Scholar
  42. 42.
    Gong, J.B., Wang, Q.H.: Stabilizing non-Hermitian systems by periodic driving. Phys. Rev. A 91, 042135–6 (2015)ADSCrossRefGoogle Scholar
  43. 43.
    Driben, R., Malomed, B.A.: Stability of solitons in parity-time-symmetric couplers. Opt. Lett. 36, 4323–4325 (2011)ADSCrossRefGoogle Scholar
  44. 44.
    Driben, R., Malomed, B.A.: Stabilization of solitons in PT models with supersymmetry by periodic management. Europhys. Lett. 96, 51001–5 (2011)ADSCrossRefGoogle Scholar
  45. 45.
    Horne, R.L., Cuevas, J., Kevrekidis, P.G., Whitaker, N., Abdullaev, F.K., Frantzeskakis, D.J.: PT-symmetry management in oligomer systems. J. Phys. A 46, 485101–19 (2013)MathSciNetCrossRefGoogle Scholar
  46. 46.
    D’Ambroise, J., Malomed, B.A., Kevrekidis, P.G.: Quasi-energies, parametric resonances, and stability limits in ac-driven PT-symmetric systems. Chaos 24, 023136–10 (2014)ADSMathSciNetCrossRefGoogle Scholar
  47. 47.
    Battelli, F., Diblik, J., Feckan, M., Pickton, J., Pospisil, M., Susanto, H.: Dynamics of generalized PT-symmetric dimers with time-periodic gain-loss. Nonlinear Dynam. 81, 353–371 (2015)MathSciNetCrossRefGoogle Scholar
  48. 48.
    Martinez, A.J., Molina, M.I., Turitsyn, S.K., Kivshar, Y.S.: Nonlinear multicore waveguiding structures with balanced gain and loss. Phys. Rev. A 91, 023822–8 (2015)ADSMathSciNetCrossRefGoogle Scholar
  49. 49.
    Liu, J.B., Xie, X.T., Shan, C.J., Liu, T.K., Lee, R.K., Wu, Y.: Optical bistability in nonlinear periodical structures with PT-symmetric potential. Laser Phys. 25, 015102–5 (2015)ADSGoogle Scholar
  50. 50.
    Greenberg, M., Orenstein, M.: Irreversible coupling by use of dissipative optics. Opt. Lett. 29, 451–453 (2004)ADSCrossRefGoogle Scholar
  51. 51.
    Greenberg, M., Orenstein, M.: Unidirectional complex gratings assisted couplers. Opt. Express 12, 4013–4018 (2004)ADSCrossRefGoogle Scholar
  52. 52.
    Greenberg, M., Orenstein, M.: Optical unidirectional devices by complex spatial single sideband perturbation. IEEE J. Quantum Electron. 41, 1013–1023 (2005)ADSCrossRefGoogle Scholar
  53. 53.
    West, B.R., Plant, D.V.: Transfer matrix analysis of the unidirectional grating-assisted codirectional coupler. Appl. Opt. 46, 8052–8060 (2007)ADSCrossRefGoogle Scholar
  54. 54.
    Luo, X.B., Huang, J.H., Zhong, H.H., Qin, X.Z., Xie, Q.T., Kivshar, Y.S., Lee, C.H.: Pseudo-parity-time symmetry in optical systems. Phys. Rev. Lett. 110, 243902–5 (2013)ADSCrossRefGoogle Scholar
  55. 55.
    Yuce, C.: Pseudo PT symmetric lattice. Eur. Phys. J. D 69, 11–5 (2015)ADSCrossRefGoogle Scholar
  56. 56.
    Konotop, V.V., Zezyulin, D.A.: Stochastic parity-time-symmetric coupler. Opt. Lett. 39, 1223–1226 (2014)ADSCrossRefGoogle Scholar
  57. 57.
    Guo, A., Salamo, G.J., Duchesne, D., Morandotti, R., Volatier-Ravat, M., Aimez, V., Siviloglou, G.A., Christodoulides, D.N.: Observation of PT-symmetry breaking in complex optical potentials. Phys. Rev. Lett. 103, 093902–4 (2009)ADSCrossRefGoogle Scholar
  58. 58.
    Suchkov, S.V., Fotsa-Ngaffo, F., Kenfack-Jiotsa, A., Tikeng, A.D., Kofane, T.C., Kivshar, Y.S., Sukhorukov, A.A.: Non-Hermitian trimers: PT-symmetry versus pseudo-Hermiticity. New J. Phys. 18, 065005–9 (2016)ADSCrossRefGoogle Scholar
  59. 59.
    Lin, Z., Ramezani, H., Eichelkraut, T., Kottos, T., Cao, H., Christodoulides, D.N.: Unidirectional invisibility induced by PT-symmetric periodic structures. Phys. Rev. Lett. 106, 213901–4 (2011)ADSCrossRefGoogle Scholar
  60. 60.
    Dmitriev, S.V., Suchkov, S.V., Sukhorukov, A.A., Kivshar, Y.S.: Scattering of linear and nonlinear waves in a waveguide array with a PT-symmetric defect. Phys. Rev. A 84, 013833–5 (2011)ADSCrossRefGoogle Scholar
  61. 61.
    Mostafazadeh, A.: Invisibility and PT symmetry. Phys. Rev. A 87, 012103–8 (2013)ADSCrossRefGoogle Scholar
  62. 62.
    Feng, L., Xu, Y.L., Fegadolli, W.S., Lu, M.H., Oliveira, J.E.B., Almeida, V.R., Chen, Y.F., Scherer, A.: Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies. Nat. Mater. 12, 108–113 (2013)ADSCrossRefGoogle Scholar
  63. 63.
    Li, K., Kevrekidis, P.G.: PT-symmetric oligomers: analytical solutions, linear stability, and nonlinear dynamics. Phys. Rev. E 83, 066608–7 (2011)ADSCrossRefGoogle Scholar
  64. 64.
    Li, K., Kevrekidis, P.G., Frantzeskakis, D.J., Ruter, C.E., Kip, D.: Revisiting the PT-symmetric trimer: bifurcations, ghost states and associated dynamics. J. Phys. A 46, 375304–12 (2013)MathSciNetCrossRefGoogle Scholar
  65. 65.
    Siegman, A.E.: Propagating modes in gain-guided optical fibers. J. Opt. Soc. Am. A 20, 1617–1628 (2003)ADSCrossRefGoogle Scholar
  66. 66.
    Dastmalchi, B., Tassin, P., Koschny, T., Soukoulis, C.M.: A new perspective on plasmonics: confinement and propagation length of surface plasmons for different materials and geometries. Adv. Opt. Mater. 4, 177–184 (2016)CrossRefGoogle Scholar
  67. 67.
    Boltasseva, A., Atwater, H.A.: Low-loss plasmonic metamaterials. Science 331, 290–291 (2011)ADSCrossRefGoogle Scholar
  68. 68.
    Stockman, M.I.: Spaser action, loss compensation, and stability in plasmonic systems with gain. Phys. Rev. Lett. 106, 156802–4 (2011)ADSCrossRefGoogle Scholar
  69. 69.
    Wuestner, S., Pusch, A., Tsakmakidis, K.L., Hamm, J.M., Hess, O.: Overcoming losses with gain in a negative refractive index metamaterial. Phys. Rev. Lett. 105, 127401–4 (2010)ADSCrossRefGoogle Scholar
  70. 70.
    Fang, A., Koschny, T., Soukoulis, C.M.: Self-consistent calculations of loss-compensated fishnet metamaterials. Phys. Rev. B 82, 121102–4 (2010)ADSCrossRefGoogle Scholar
  71. 71.
    Lupu, A., Benisty, H., Degiron, A.: Switching using PT symmetry in plasmonic systems: positive role of the losses. Opt. Express 21, 21651–21668 (2013)ADSCrossRefGoogle Scholar
  72. 72.
    Alaeian, H., Dionne, J.A.: Non-Hermitian nanophotonic and plasmonic waveguides. Phys. Rev. B 89, 075136–9 (2014)ADSCrossRefGoogle Scholar
  73. 73.
    Savoia, S., Castaldi, G., Galdi, V.: Non-Hermiticity-induced wave confinement and guiding in loss-gain-loss three-layer systems. Phys. Rev. A 94, 043838–10 (2016)ADSCrossRefGoogle Scholar
  74. 74.
    Walasik, W., Ma, C.C., Litchinitser, N.M.: Dissimilar directional couplers showing PT-symmetric-like behavior. New J. Phys. 19, 075002–8 (2017)ADSCrossRefGoogle Scholar
  75. 75.
    Turitsyna, E.G., Shadrivov, I.V., Kivshar, Y.S.: Guided modes in non-Hermitian optical waveguides. Phys. Rev. A 96, 033824–4 (2017)ADSCrossRefGoogle Scholar

Copyright information

© Springer Nature Singapore Pte Ltd. 2018

Authors and Affiliations

  • Sergey V. Suchkov
    • 1
  • Andrey A. Sukhorukov
    • 1
  • Yuri S. Kivshar
    • 1
  1. 1.Nonlinear Physics Centre, Research School of Physics and EngineeringAustralian National UniversityCanberraAustralia

Personalised recommendations