Skip to main content

Theory and Numerical Modelling of Parity-Time Symmetric Structures in Photonics: Introduction and Grating Structures in One Dimension

  • Chapter
  • First Online:
Recent Trends in Computational Photonics

Part of the book series: Springer Series in Optical Sciences ((SSOS,volume 204))

Abstract

A class of structures based on \(\mathcal {PT}\)-symmetric Bragg gratings in the presence of both gain and loss is studied. The basic concepts and properties of parity and time reversal in one-dimensional structures that possess idealised material properties are given. The impact of realistic material properties on the behaviour of these devices is then investigated. Further extension to include material non-linearity is used to study an innovative all-optical memory device.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. C.M. Bender, S. Boettcher, Real spectra in non-Hermitian Hamiltonians having PT symmetry. Phys. Rev. Lett. 80(24), 5243 (1998)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  2. C.M. Bender, D.C. Brody, H.F. Jones, Complex extension of quantum mechanics. Phys. Rev. Lett. 89(27), 270401 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  3. Q.H. Wang, S.Z. Chia, J.H. Zhang, PT symmetry as a generalization of Hermiticity. J. Phys. A Math. Theor. 43(29) (2010)

    Google Scholar 

  4. D. Dizdarevic, D. Dast, D. Haag, J. Main, H. Cartarius, G. Wunner, Cusp bifurcation in the eigenvalue spectrum of PT-symmetric Bose-Einstein condensates. Phys. Rev. A 91(3), 033636 (2015)

    Article  ADS  Google Scholar 

  5. R. Gutöhrlein, J. Schnabel, I. Iskandarov, H. Cartarius, J. Main, G. Wunner, Realizing PT-symmetric BEC subsystems in closed Hermitian systems. J. Phys. A Math. Theor. 48(33), 335302 (2015)

    Article  MATH  MathSciNet  Google Scholar 

  6. F. Single, H. Cartarius, G. Wunner, J. Main, Coupling approach for the realization of a PT-symmetric potential for a Bose-Einstein condensate in a double well. Phys. Rev. A 90(4), 042123 (2014)

    Article  ADS  Google Scholar 

  7. F. Bagarello, G. Pantano, Pseudo-Fermions in an electronic loss-gain circuit. Int. J. Theor. Phys. 52(12), 4507 (2013)

    Article  MATH  Google Scholar 

  8. J. Schindler, Z. Lin, J.M. Lee, H. Ramezani, F.M. Ellis, T. Kottos, PT-symmetric electronics. J. Phys. A Math. Theor. 45(44), 444029 (2012)

    Article  ADS  MATH  Google Scholar 

  9. J. Schindler, A. Li, M.C. Zheng, F.M. Ellis, T. Kottos, Experimental study of active LRC circuits with PT symmetries. Phys. Rev. A 84(4), 040101 (2011)

    Article  ADS  Google Scholar 

  10. C.M. Bender, B.K. Berntson, D. Parker, E. Samuel, Observation of PT phase transition in a simple mechanical system. Am. J. Phys. 81(3), 173 (2013)

    Article  ADS  Google Scholar 

  11. X. Zhu, H. Ramezani, C. Shi, J. Zhu, X. Zhang, PT-Symmetric acoustics. Phys. Rev. X 4(3), 031042 (2014)

    Google Scholar 

  12. R. Fleury, D. Sounas, A. Alù, An invisible acoustic sensor based on parity-time symmetry. Nat. Commun. 6, 5905 (2015)

    Article  ADS  Google Scholar 

  13. C. Poli, M. Bellec, U. Kuhl, F. Mortessagne, H. Schomerus, Selective enhancement of topologically induced interface states in a dielectric resonator chain. Nat. Commun. 6, 6710 (2015)

    Article  ADS  Google Scholar 

  14. S. Bittner, B. Dietz, U. Günther, H.L. Harney, M. Miski-Oglu, A. Richter, F. Schäfer, PT symmetry and spontaneous symmetry breaking in a microwave billiard. Phys. Rev. Lett. 108(2), 024101 (2012)

    Article  ADS  Google Scholar 

  15. H.F. Jones, Analytic results for a PT -symmetric optical structure. J. Phys. A Math. Theor. 45(13), 135306 (2012)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  16. H. Ramezani, T. Kottos, R. El-Ganainy, D.N. Christodoulides, Unidirectional nonlinear PT-symmetric optical structures. Phys. Rev. A 82(4), 043803 (2010)

    Article  ADS  Google Scholar 

  17. Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, D.N. Christodoulides, Unidirectional invisibility induced by PT-symmetric periodic structures. Phys. Rev. Lett. 106(21), 213901 (2011)

    Article  ADS  Google Scholar 

  18. M. Kulishov, B. Kress, R. Slavík, Resonant cavities based on Parity-Time-symmetric diffractive gratings. Opt. Express 21(8), 68 (2013)

    Article  Google Scholar 

  19. S. Phang, A. Vukovic, H. Susanto, T.M. Benson, P. Sewell, Ultrafast optical switching using parity-time symmetric Bragg gratings. J. Opt. Soc. Am. B 30(11), 2984 (2013)

    Article  ADS  Google Scholar 

  20. S. Phang, A. Vukovic, H. Susanto, T.M. Benson, P. Sewell, Impact of dispersive and saturable gain/loss on bistability of nonlinear parity-time Bragg gratings. Opt. Lett. 39(9), 2603 (2014)

    Article  ADS  Google Scholar 

  21. S. Phang, A. Vukovic, T.M. Benson, H. Susanto, P. Sewell, A versatile all-optical parity-time signal processing device using a Bragg grating induced using positive and negative Kerr-nonlinearity. Opt. Quantum Electron. 47(1), 37 (2015)

    Article  Google Scholar 

  22. C.Y. Huang, R. Zhang, J.L. Han, J. Zheng, J.Q. Xu, Type-II perfect absorption and amplification modes with controllable bandwidth in combined PT-symmetric and conventional Bragg-grating structures. Phys. Rev. A 89(2), 023842 (2014)

    Article  ADS  Google Scholar 

  23. N.X.A. Rivolta, B. Maes, Diffractive switching by interference in a tailored PT-symmetric grating. J. Opt. Soc. Am. B 32(7), 1330 (2015)

    Article  ADS  Google Scholar 

  24. A. Regensburger, M.A. Miri, C. Bersch, J. Näger, G. Onishchukov, D.N. Christodoulides, U. Peschel, Observation of defect states in PT-symmetric optical lattices. Phys. Rev. Lett. 110(22), 223902 (2013)

    Article  ADS  Google Scholar 

  25. S. Longhi, Invisibility in PT-symmetric complex crystals. J. Phys. A Math. Theor. 44(48), 485302 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  26. K.G. Makris, R. El-Ganainy, D.N. Christodoulides, Beam dynamics in PT symmetric optical lattices. Phys. Rev. Lett. 100(10), 103904 (2008)

    Article  ADS  Google Scholar 

  27. H. Nolting, G. Sztefka, J. Čtyroký, in integrated Photonics Research, vol. 4930 (OSA, Boston, Massachusetts, 1996), pp. 76–80

    Google Scholar 

  28. A. Ruschhaupt, F. Delgado, J.G. Muga, Physical realization of -symmetric potential scattering in a planar slab waveguide. J. Phys. A. Math. Gen. 38(9), L171 (2005)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  29. M. Greenberg, M. Orenstein, Optical unidirectional devices by complex spatial single sideband perturbation. IEEE J. Quantum Electron. 41(7), 2005 (1013)

    Google Scholar 

  30. A.A. Sukhorukov, Z. Xu, Y.S. Kivshar, Nonlinear suppression of time reversals in PT-symmetric optical couplers. Phys. Rev. A 82(4), 043818 (2010)

    Article  ADS  Google Scholar 

  31. F. Nazari, M. Nazari, M.K. Moravvej-Farshi, A 2x2 spatial optical switch based on PT-symmetry. Opt. Lett. 36(22), 4368 (2011)

    Article  ADS  Google Scholar 

  32. J. Čtyroký, V. Kuzmiak, S. Eyderman, Waveguide structures with antisymmetric gain/loss profile. Opt. Express 18(21), 21585 (2010)

    Article  ADS  Google Scholar 

  33. A. Lupu, H. Benisty, A. Degiron, Switching using PT symmetry in plasmonic systems: positive role of the losses. Opt. Express 21(18), 192 (2013)

    Article  Google Scholar 

  34. H. Benisty, A. Degiron, A. Lupu, A.D. Lustrac, S. Forget, M. Besbes, G. Barbillon, A. Bruyant, S. Blaize, G. Lérondel, Implementation of PT symmetric devices using plasmonics: principle and applications. Opt. Express 19(19), 3567 (2011)

    Article  Google Scholar 

  35. H. Alaeian, Ja. Dionne, Parity-time-symmetric plasmonic metamaterials. Phys. Rev. A 89(3), 033829 (2014)

    Google Scholar 

  36. B. Baum, H. Alaeian, J. Dionne, A parity-time symmetric coherent plasmonic absorber-amplifier. J. Appl. Phys. 117(063106), 063106 (2015)

    Article  ADS  Google Scholar 

  37. L. Feng, Z.J. Wong, Y. Wang, X. Zhang, R.M.R.M. Ma, Y. Wang, X. Zhang, R.M.R.M. Ma, Y. Wang, X. Zhang, Single-mode laser by parity-time symmetry breaking. Science 346(6212), 972 (2014)

    Article  ADS  Google Scholar 

  38. L. Chang, X. Jiang, S. Hua, C. Yang, J. Wen, L. Jiang, G. Li, G. Wang, M. Xiao, Parity-time symmetry and variable optical isolation in active-passive-coupled microresonators. Nat. Photon. 8(7), 524 (2014)

    Article  ADS  Google Scholar 

  39. S. Longhi, L. Feng, PT-symmetric microring laser-absorber. Opt. Lett. 39(17), 5026 (2014)

    Article  ADS  Google Scholar 

  40. B. Peng, K. Ozdemir, S. Rotter, H. Yilmaz, M. Liertzer, F. Monifi, C.M. Bender, F. Nori, L. Yang, Loss-induced suppression and revival of lasing. Science 346(6207), 328 (2014)

    Article  ADS  Google Scholar 

  41. S. Phang, A. Vukovic, S.C. Creagh, T.M. Benson, P.D. Sewell, G. Gradoni, Parity-time symmetric coupled microresonators with a dispersive gain/loss. Opt. Express 23(9), 11493 (2015)

    Article  ADS  Google Scholar 

  42. B. Peng, K. Özdemir, F. Lei, F. Monifi, M. Gianfreda, G.L. Long, S. Fan, F. Nori, C.M. Bender, L. Yang, Parity-time-symmetric whispering-gallery microcavities. Nat. Phys. 10(5), 394 (2014)

    Google Scholar 

  43. S. Phang, A. Vukovic, S.C. Creagh, P.D. Sewell, G. Gradoni, T.M. Benson, Localized single frequency lasing states in a finite parity-time symmetric resonator chain. Sci. Rep. 6(20499), 1 (2016)

    Google Scholar 

  44. N. Zettili, Quantum Mechanics: Concepts and Applications, 2nd edn. (Wiley, New York, NY, 2009)

    Google Scholar 

  45. A. Yariv, Quantum Electronics, 3rd edn. (Wiley, New York, NY, 1989)

    Google Scholar 

  46. C.M. Bender, Introduction to PT-symmetric quantum theory. Contemp. Phys. 46(4), 277 (2005)

    Article  ADS  Google Scholar 

  47. C.M. Bender, Making sense of non-Hermitian Hamiltonians. Rep. Prog. Phys. 70(6), 947 (2007)

    Article  ADS  MathSciNet  Google Scholar 

  48. A. Mostafazadeh, Invisibility and PT symmetry. Phys. Rev. A - At. Mol. Opt. Phys. 87(1), 012103 (2013)

    Google Scholar 

  49. I. Ge, Y.D. Chong, A.D. Stone, Conservation relations and anisotropic transmission resonances in one-dimensional PT-symmetric photonic heterostructures. Phys. Rev. A - At. Mol. Opt. Phys. 85(2), 1 (2012)

    Google Scholar 

  50. Y.D. Chong, L. Ge, A.D. Stone, PT-symmetry breaking and laser-absorber modes in optical scattering systems. Phys. Rev. Lett. 106(9), 093902 (2011)

    Article  ADS  Google Scholar 

  51. S. Longhi, PT-symmetric laser absorber. Phys. Rev. A 82(3), 031801 (2010)

    Article  ADS  Google Scholar 

  52. A. Mostafazadeh, Spectral singularities of complex scattering potentials and infinite reflection and transmission coefficients at real energies. Phys. Rev. Lett. 102(22), 220402 (2009)

    Article  ADS  Google Scholar 

  53. F.A. Benson, T.M. Benson, Fields, Waves and Transmission Lines (Springer, Amsterdam, 1991)

    Book  MATH  Google Scholar 

  54. S. Ramo, J.R. Whinnery, T.V. Duzer, Fields and Waves in Communication Electronics 3rd edn. (Wiley, New York, 1999)

    Google Scholar 

  55. D.M. Pozar, Microwave Engineering, 4th edn. (Wiley, New York, NY, 2011)

    Google Scholar 

  56. H.A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, New Jersey, 1983)

    Google Scholar 

  57. R.E. Collin, Field Theory of Guided Waves, 2nd edn. (IEEE Press, New York, NY, 1991)

    MATH  Google Scholar 

  58. D. Jalas, A. Petrov, M. Eich, W. Freude, S. Fan, Z. Yu, R. Baets, M. Popović, A. Melloni, J.D. Joannopoulos, M. Vanwolleghem, C.R. Doerr, H. Renner, What is - and what is not - an optical isolator. Nat. Photon. 7(8), 579 (2013)

    Article  ADS  Google Scholar 

  59. K. Iizuka, Elements of Photonics, vol. II (Wiley, New York, NY, 2002)

    Book  Google Scholar 

  60. B.E.A. Saleh, M.C. Teich, Fundamentals of Photonics, 2nd edn. (Wiley, New York, NY, 2007)

    Google Scholar 

  61. J.M. Liu, Photonic Devices (Cambridge University Press, Cambridge, 2005)

    Book  Google Scholar 

  62. S.C. Hagness, R.M. Joseph, A. Taflove, Subpicosecond electrodynamics of distributed Bragg reflector microlasers: Results from finite difference time domain simulations. Radio Sci. 31(4), 931 (1996)

    Article  ADS  Google Scholar 

  63. A.E. Siegman, Lasers (University Science Book, Palo Alto, CA, 1986)

    Google Scholar 

  64. M. Robertson, Private Communication, October 2015

    Google Scholar 

  65. L.D. Landau, J.S. Bell, M.J. Kearsley, L.P. Pitaevskii, E.M. Lifshitz, J.B. Sykes, Electrodynamics of Continuous Media, 2nd edn. (Elsevier, London, 1984)

    Google Scholar 

  66. A.A. Zyablovsky, A.P. Vinogradov, A.V. Dorofeenko, A.A. Pukhov, A.A. Lisyansky, Causality and phase transitions in PT-symmetric optical systems. Phys. Rev. A 89(3), 033808 (2014)

    Article  ADS  Google Scholar 

  67. W. Hoefer, The transmission-line matrix method-theory and applications. IEEE Trans. Microw. Theory Tech. 33(10), 882 (1985)

    Article  ADS  Google Scholar 

  68. C. Christopoulos, The Transmission-Line Modeling Method TLM (IEEE Press, Piscataway, 1995)

    Book  Google Scholar 

  69. J. Paul, Modelling of general electromagnetic material properties in TLM. Ph.D. thesis University of Nottingham (1998)

    Google Scholar 

  70. J. Paul, C. Christopoulos, D. Thomas, Generalized material models in TLM - part III: Materials with nonlinear properties. IEEE Trans. Antennas Propag. 50(7), 997 (2002)

    Article  ADS  Google Scholar 

  71. J. Paul, C. Christopoulos, D. Thomas, Generalized material models in TLM - part I: Materials with frequency-dependent properties. IEEE Trans. Antennas Propag. 47(10), 1528 (1999)

    Article  ADS  Google Scholar 

  72. V. Janyani, A. Vukovic, J. Paul, The development of TLM models for nonlinear optics. Microw. Rev. 10(1), 35 (2004)

    Google Scholar 

  73. V. Janyani, A. Vukovic, J.D. Paul, P. Sewell, T.M. Benson, Time domain simulation in photonics: A comparison of nonlinear dispersive polarisation models. Opt. Quantum Electron. 37(1–3), 3 (2005)

    Article  Google Scholar 

  74. X. Meng, P. Sewell, A. Vukovic, H.G. Dantanarayana, T.M. Benson, Efficient broadband simulations for thin optical structures. Opt. Quantum Electron. 45(4), 343 (2013)

    Article  Google Scholar 

  75. X. Meng, P. Sewell, S. Phang, A. Vukovic, T.M. Benson, Modeling curved carbon fiber composite (CFC) structures in the transmission-line modeling (TLM) method. IEEE Trans. Electromagn. Comp. 57(3), 384 (2015)

    Article  Google Scholar 

  76. R. Joseph, A. Taflove, FDTD Maxwell’s equations models for nonlinear electrodynamics and optics. IEEE Trans. Antennas Propag. 45(3), 364 (1997)

    Article  ADS  Google Scholar 

  77. Kh.A. Fatkhulla, V.V. Konotop, Nonlinear Waves: Classical and Quantum Aspects NATO Science Series II: Mathematics, Physics and Chemistry, vol. 153 (Springer, Netherlands 2005). https://doi.org/10.1007/1-4020-2190-9

  78. C. Conti, A. Di Falco, G. Assanto, Optical parametric oscillations in isotropic photonic crystals. Opt. Express 12(5), 823 (2004)

    Article  ADS  Google Scholar 

  79. V. Janyani, J. Paul, A. Vukovic, T. Benson, P. Sewell, TLM modelling of nonlinear optical effects in fibre Bragg gratings. IEE Proc. Optoelectron. 151(4), 185 (2004)

    Article  Google Scholar 

  80. E.V. Kazantseva, A.I. Maimistov, J.G. Caputo, Reduced Maxwell-Duffing description of extremely short pulses in nonresonant media. Phys. Rev. E 71(5), 056622 (2005)

    Article  ADS  Google Scholar 

  81. W.H. Press, S.A. Teukolsky, W.T. Vetterling, B.P. Flannery, Numerical Recipes in C++: The Art of Scientific Computing, 2nd edn. (Cambridge University Press, Cambridge, 2002)

    MATH  Google Scholar 

  82. A. Suryanto, G. Evan, M. Hammer, H.J.W.M. Hoekstra, A finite element scheme to study the nonlinear optical response of a finite grating without and with defect. Opt. Quantum Electron. 35(1997), 313 (2003)

    Google Scholar 

  83. L. Brzozowski, E. Sargent, Optical signal processing using nonlinear distributed feedback structures. IEEE J. Quantum Electron. 36(5), 550 (2000)

    Article  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to S. Phang .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2017 Springer International Publishing AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Phang, S. et al. (2017). Theory and Numerical Modelling of Parity-Time Symmetric Structures in Photonics: Introduction and Grating Structures in One Dimension. In: Agrawal, A., Benson, T., De La Rue, R., Wurtz, G. (eds) Recent Trends in Computational Photonics. Springer Series in Optical Sciences, vol 204. Springer, Cham. https://doi.org/10.1007/978-3-319-55438-9_6

Download citation

Publish with us

Policies and ethics