Abstract

The paper reports on the coupling of Parity-Time (PT)-symmetric whispering gallery resonators with realistic material and gain/loss models. Response of the PT system is analyzed for the case of low and high material and gain dispersion, and also for two practical scenarios when the pump frequency is not aligned with the resonant frequency of the desired whispering gallery mode and when there is imbalance in the gain/loss profile. The results show that the presence of dispersion and frequency misalignment causes skewness in frequency bifurcation and significant reduction of the PT breaking point, respectively. Finally, we demonstrate a lasing mode operation which occurs due to an early PT-breaking by increasing loss in a PT system with unbalanced gain and loss.

© 2015 Optical Society of America

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References

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  1. C. M. Bender, S. Boettcher, and P. N. Meisinger, “PT–symmetric quantum mechanics,” J. Math. Phys. 40, 2201 (1999).
    [Crossref]
  2. Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional invisibility induced by PT–iymmetric periodic structures,” Phys. Rev. Lett. 106, 213901 (2011).
    [Crossref]
  3. Y. D. Chong, L. Ge, and A. D. Stone, “PT–Symmetry breaking and laser-absorber modes in optical scattering systems,” Phys. Rev. Lett. 106, 093902 (2011).
    [Crossref]
  4. H. Benisty, C. Yan, A. Degiron, and A. Lupu, “Healing near-PT-symmetric structures to restore their characteristic singularities: analysis and examples,” J. Lightwave Technol. 30, 2675–2683 (2012).
    [Crossref]
  5. A. Lupu, H. Benisty, and A. Degiron, “Switching using PT symmetry in plasmonic systems: positive role of the losses,” Opt. Express 21, 192–195 (2013).
    [Crossref]
  6. F. Nazari, M. Nazari, and M. K. Moravvej-Farshi, “A 2 × 2 spatial optical switch based on PT–symmetry,” Opt. Lett. 36, 4368–4370 (2011).
    [Crossref] [PubMed]
  7. H. Hodaei, M.-A. Miri, M. Heinrich, D. N. Christodoulides, and M. Khajavikhan, “PT symmetric large area single mode DFB lasers,” in Proceedings of CLEO: 2014 (OSA, 2014), Vol. 1, paper FM1D.3.
  8. S. Longhi and L. Feng, “PT-symmetric microring laser-absorber,” Opt. Lett. 39, 5026–5029 (2014).
    [Crossref] [PubMed]
  9. S. Phang, A. Vukovic, H. Susanto, T. M. Benson, and P. Sewell, “Ultrafast optical switching using parity–time symmetric Bragg gratings,” J. Opt. Soc. Am. B 30, 2984–2991 (2013).
    [Crossref]
  10. S. Phang, A. Vukovic, T. M. Benson, H. Susanto, and 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, 37–47 (2015).
    [Crossref]
  11. M. Kulishov, B. Kress, and R. Slavk, “Resonant cavities based on Parity-Time-symmetric diffractive gratings,” Opt. Express 21, 68–70 (2013).
    [Crossref]
  12. H. F. Jones, “Analytic results for a PT -symmetric optical structure,” J. Phys. A-Math. Theor. 45, 135306 (2012).
    [Crossref]
  13. J. Čtyroký, V. Kuzmiak, and S. Eyderman, “Waveguide structures with antisymmetric gain/loss profile,” Opt. Express 18, 21585–21593 (2010).
    [Crossref] [PubMed]
  14. J. Čtyroký, “Dispersion properties of coupled waveguides with loss and gain: a full-vectorial analysis,” Opt. Quantum Electron. 46, 465–475 (2014).
    [Crossref]
  15. R. El-Ganainy, K. G. Makris, D. N. Christodoulides, and Z. H. Musslimani, “Theory of coupled optical PT-symmetric structures,” Opt. Lett. 32, 2632–2634 (2007).
    [Crossref] [PubMed]
  16. M. Greenberg and M. Orenstein, “Unidirectional complex grating assisted couplers,” Opt. Express 12, 4013–4018 (2004).
    [Crossref] [PubMed]
  17. S. Longhi, “PT-symmetric laser absorber,” Phys. Rev. A 82, 031801 (2010).
    [Crossref]
  18. H. Nolting, G. Sztefka, and J. Čtyroký, “Wave propagation in a waveguide with a balance of gain and loss,” in Integrated Photonics Research (OSA, 1996), pp. 76–80.
  19. S. Phang, A. Vukovic, H. Susanto, T. M. Benson, and P. Sewell, “Impact of dispersive and saturable gain/loss on bistability of nonlinear parity-time Bragg gratings,” Opt. Lett. 39, 2603–2606 (2014).
    [Crossref] [PubMed]
  20. C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nat. Phys. 6, 192–195 (2010).
    [Crossref]
  21. B. Peng, S. K. Özdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Paritytime-symmetric whispering-gallery microcavities,” Nat. Phys. 10, 1–5 (2014).
    [Crossref]
  22. B. Peng, S. K. Özdemir, S. Rotter, H. Yilmaz, M. Liertzer, F. Monifi, C. M. Bender, F. Nori, and L. Yang, “Loss-induced suppression and revival of lasing,” Science 346(6207), 328–332 (2014).
    [Crossref] [PubMed]
  23. A. Regensburger, M. Miri, and C. Bersch, “Observation of defect states in PT-symmetric optical lattices,” Phys. Rev. Lett. 110, 223902 (2013).
    [Crossref] [PubMed]
  24. L. Chang, X. Jiang, S. Hua, C. Yang, J. Wen, L. Jiang, G. Li, G. Wang, and M. Xiao, “Parity–time symmetry and variable optical isolation in active-passive-coupled microresonators,” Nat. Photonics 8, 524–529 (2014).
    [Crossref]
  25. L. Feng, Z. J. Wong, R.-M. Ma, Y. Wang, and X. Zhang, “Single-mode laser by parity-time symmetry breaking,” Science 346, 972–975 (2014).
    [Crossref] [PubMed]
  26. S. Phang, A. Vukovic, H. Susanto, T. M. Benson, and P. Sewell, “Practical limitation on operation of nonlinear parity-time Bragg gratings,” in META 2014 Conference (2014), pp. 270–275.
  27. S. C. Creagh and M. D. Finn, “Evanescent coupling between discs: a model for near-integrable tunnelling,” J. Phys. A. Math. Gen. 34, 3791–3801 (2001).
    [Crossref]
  28. J. Paul, C. Christopoulos, and D. W. P. Thomas, “Generalized material models in TLM.I. Materials with frequency-dependent properties,” IEEE Trans. Antennas Propag. 47, 1528–1534 (1999).
    [Crossref]
  29. C. Christopoulos, The Transmission-Line Modeling Method TLM (IEEE-Oxford University, 1995).
    [Crossref]
  30. L. D. Landau, J. S. Bell, M. J. Kearsley, L. P. Pitaevskii, E. M. Lifshitz, and J. B. Sykes, Electrodynamics of Continuous Media, 2. (Elsevier, 1984).
  31. A. A. Zyablovsky, A. P. Vinogradov, A. V. Dorofeenko, A. A. Pukhov, and A. A. Lisyansky, “Causality and phase transitions in PT-symmetric optical systems,” Phys. Rev. A 89, 033808 (2014).
    [Crossref]
  32. S. C. Hagness, R. M. Joseph, and A. Taflove, “Subpicosecond electrodynamics of distributed Bragg reflector microlasers: results from finite difference time domain simulations,” Radio Sci. 31, 931–941 (1996).
    [Crossref]
  33. S. V. Boriskina, “Spectrally engineered photonic molecules as optical sensors with enhanced sensitivity: a proposal and numerical analysis,” J. Opt. Soc. Am. B 23, 1565–1573 (2006).
    [Crossref]
  34. E. I. Smotrova and A. I. Nosich, “Optical coupling of an active microdisk to a passive one: effect on the lasing thresholds of the whispering-gallery supermodes,” Opt. Lett. 38, 2059–2061 (2013).
    [Crossref] [PubMed]
  35. E. Smotrova, A. Nosich, T. M. Benson, and P. Sewell, “Optical coupling of whispering-gallery modes of two identical microdisks and its effect on photonic molecule lasing,” IEEE J. Sel. Top. Quantum Electron. 12, 78–85 (2006).
    [Crossref]
  36. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (U.S. Department of Commerce, NIST, 1972).

2015 (1)

S. Phang, A. Vukovic, T. M. Benson, H. Susanto, and 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, 37–47 (2015).
[Crossref]

2014 (8)

J. Čtyroký, “Dispersion properties of coupled waveguides with loss and gain: a full-vectorial analysis,” Opt. Quantum Electron. 46, 465–475 (2014).
[Crossref]

S. Phang, A. Vukovic, H. Susanto, T. M. Benson, and P. Sewell, “Impact of dispersive and saturable gain/loss on bistability of nonlinear parity-time Bragg gratings,” Opt. Lett. 39, 2603–2606 (2014).
[Crossref] [PubMed]

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

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

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

B. Peng, S. K. Özdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Paritytime-symmetric whispering-gallery microcavities,” Nat. Phys. 10, 1–5 (2014).
[Crossref]

B. Peng, S. K. Özdemir, S. Rotter, H. Yilmaz, M. Liertzer, F. Monifi, C. M. Bender, F. Nori, and L. Yang, “Loss-induced suppression and revival of lasing,” Science 346(6207), 328–332 (2014).
[Crossref] [PubMed]

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

2013 (5)

A. Regensburger, M. Miri, and C. Bersch, “Observation of defect states in PT-symmetric optical lattices,” Phys. Rev. Lett. 110, 223902 (2013).
[Crossref] [PubMed]

E. I. Smotrova and A. I. Nosich, “Optical coupling of an active microdisk to a passive one: effect on the lasing thresholds of the whispering-gallery supermodes,” Opt. Lett. 38, 2059–2061 (2013).
[Crossref] [PubMed]

S. Phang, A. Vukovic, H. Susanto, T. M. Benson, and P. Sewell, “Ultrafast optical switching using parity–time symmetric Bragg gratings,” J. Opt. Soc. Am. B 30, 2984–2991 (2013).
[Crossref]

A. Lupu, H. Benisty, and A. Degiron, “Switching using PT symmetry in plasmonic systems: positive role of the losses,” Opt. Express 21, 192–195 (2013).
[Crossref]

M. Kulishov, B. Kress, and R. Slavk, “Resonant cavities based on Parity-Time-symmetric diffractive gratings,” Opt. Express 21, 68–70 (2013).
[Crossref]

2012 (2)

2011 (3)

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

Y. D. Chong, L. Ge, and A. D. Stone, “PT–Symmetry breaking and laser-absorber modes in optical scattering systems,” Phys. Rev. Lett. 106, 093902 (2011).
[Crossref]

F. Nazari, M. Nazari, and M. K. Moravvej-Farshi, “A 2 × 2 spatial optical switch based on PT–symmetry,” Opt. Lett. 36, 4368–4370 (2011).
[Crossref] [PubMed]

2010 (3)

J. Čtyroký, V. Kuzmiak, and S. Eyderman, “Waveguide structures with antisymmetric gain/loss profile,” Opt. Express 18, 21585–21593 (2010).
[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, 192–195 (2010).
[Crossref]

S. Longhi, “PT-symmetric laser absorber,” Phys. Rev. A 82, 031801 (2010).
[Crossref]

2007 (1)

2006 (2)

S. V. Boriskina, “Spectrally engineered photonic molecules as optical sensors with enhanced sensitivity: a proposal and numerical analysis,” J. Opt. Soc. Am. B 23, 1565–1573 (2006).
[Crossref]

E. Smotrova, A. Nosich, T. M. Benson, and P. Sewell, “Optical coupling of whispering-gallery modes of two identical microdisks and its effect on photonic molecule lasing,” IEEE J. Sel. Top. Quantum Electron. 12, 78–85 (2006).
[Crossref]

2004 (1)

2001 (1)

S. C. Creagh and M. D. Finn, “Evanescent coupling between discs: a model for near-integrable tunnelling,” J. Phys. A. Math. Gen. 34, 3791–3801 (2001).
[Crossref]

1999 (2)

J. Paul, C. Christopoulos, and D. W. P. Thomas, “Generalized material models in TLM.I. Materials with frequency-dependent properties,” IEEE Trans. Antennas Propag. 47, 1528–1534 (1999).
[Crossref]

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

1996 (1)

S. C. Hagness, R. M. Joseph, and A. Taflove, “Subpicosecond electrodynamics of distributed Bragg reflector microlasers: results from finite difference time domain simulations,” Radio Sci. 31, 931–941 (1996).
[Crossref]

Abramowitz, M.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (U.S. Department of Commerce, NIST, 1972).

Bell, J. S.

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

Bender, C. M.

B. Peng, S. K. Özdemir, S. Rotter, H. Yilmaz, M. Liertzer, F. Monifi, C. M. Bender, F. Nori, and L. Yang, “Loss-induced suppression and revival of lasing,” Science 346(6207), 328–332 (2014).
[Crossref] [PubMed]

B. Peng, S. K. Özdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Paritytime-symmetric whispering-gallery microcavities,” Nat. Phys. 10, 1–5 (2014).
[Crossref]

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

Benisty, H.

A. Lupu, H. Benisty, and A. Degiron, “Switching using PT symmetry in plasmonic systems: positive role of the losses,” Opt. Express 21, 192–195 (2013).
[Crossref]

H. Benisty, C. Yan, A. Degiron, and A. Lupu, “Healing near-PT-symmetric structures to restore their characteristic singularities: analysis and examples,” J. Lightwave Technol. 30, 2675–2683 (2012).
[Crossref]

Benson, T. M.

S. Phang, A. Vukovic, T. M. Benson, H. Susanto, and 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, 37–47 (2015).
[Crossref]

S. Phang, A. Vukovic, H. Susanto, T. M. Benson, and P. Sewell, “Impact of dispersive and saturable gain/loss on bistability of nonlinear parity-time Bragg gratings,” Opt. Lett. 39, 2603–2606 (2014).
[Crossref] [PubMed]

S. Phang, A. Vukovic, H. Susanto, T. M. Benson, and P. Sewell, “Ultrafast optical switching using parity–time symmetric Bragg gratings,” J. Opt. Soc. Am. B 30, 2984–2991 (2013).
[Crossref]

E. Smotrova, A. Nosich, T. M. Benson, and P. Sewell, “Optical coupling of whispering-gallery modes of two identical microdisks and its effect on photonic molecule lasing,” IEEE J. Sel. Top. Quantum Electron. 12, 78–85 (2006).
[Crossref]

S. Phang, A. Vukovic, H. Susanto, T. M. Benson, and P. Sewell, “Practical limitation on operation of nonlinear parity-time Bragg gratings,” in META 2014 Conference (2014), pp. 270–275.

Bersch, C.

A. Regensburger, M. Miri, and C. Bersch, “Observation of defect states in PT-symmetric optical lattices,” Phys. Rev. Lett. 110, 223902 (2013).
[Crossref] [PubMed]

Boettcher, S.

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

Boriskina, S. V.

Cao, H.

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

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, 524–529 (2014).
[Crossref]

Chong, Y. D.

Y. D. Chong, L. Ge, and A. D. Stone, “PT–Symmetry breaking and laser-absorber modes in optical scattering systems,” Phys. Rev. Lett. 106, 093902 (2011).
[Crossref]

Christodoulides, D. N.

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

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

R. El-Ganainy, K. G. Makris, D. N. Christodoulides, and Z. H. Musslimani, “Theory of coupled optical PT-symmetric structures,” Opt. Lett. 32, 2632–2634 (2007).
[Crossref] [PubMed]

H. Hodaei, M.-A. Miri, M. Heinrich, D. N. Christodoulides, and M. Khajavikhan, “PT symmetric large area single mode DFB lasers,” in Proceedings of CLEO: 2014 (OSA, 2014), Vol. 1, paper FM1D.3.

Christopoulos, C.

J. Paul, C. Christopoulos, and D. W. P. Thomas, “Generalized material models in TLM.I. Materials with frequency-dependent properties,” IEEE Trans. Antennas Propag. 47, 1528–1534 (1999).
[Crossref]

C. Christopoulos, The Transmission-Line Modeling Method TLM (IEEE-Oxford University, 1995).
[Crossref]

Creagh, S. C.

S. C. Creagh and M. D. Finn, “Evanescent coupling between discs: a model for near-integrable tunnelling,” J. Phys. A. Math. Gen. 34, 3791–3801 (2001).
[Crossref]

Ctyroký, J.

J. Čtyroký, “Dispersion properties of coupled waveguides with loss and gain: a full-vectorial analysis,” Opt. Quantum Electron. 46, 465–475 (2014).
[Crossref]

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

H. Nolting, G. Sztefka, and J. Čtyroký, “Wave propagation in a waveguide with a balance of gain and loss,” in Integrated Photonics Research (OSA, 1996), pp. 76–80.

Degiron, A.

A. Lupu, H. Benisty, and A. Degiron, “Switching using PT symmetry in plasmonic systems: positive role of the losses,” Opt. Express 21, 192–195 (2013).
[Crossref]

H. Benisty, C. Yan, A. Degiron, and A. Lupu, “Healing near-PT-symmetric structures to restore their characteristic singularities: analysis and examples,” J. Lightwave Technol. 30, 2675–2683 (2012).
[Crossref]

Dorofeenko, A. V.

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

Eichelkraut, T.

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

El-Ganainy, R.

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

R. El-Ganainy, K. G. Makris, D. N. Christodoulides, and Z. H. Musslimani, “Theory of coupled optical PT-symmetric structures,” Opt. Lett. 32, 2632–2634 (2007).
[Crossref] [PubMed]

Eyderman, S.

Fan, S.

B. Peng, S. K. Özdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Paritytime-symmetric whispering-gallery microcavities,” Nat. Phys. 10, 1–5 (2014).
[Crossref]

Feng, L.

S. Longhi and L. Feng, “PT-symmetric microring laser-absorber,” Opt. Lett. 39, 5026–5029 (2014).
[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, 972–975 (2014).
[Crossref] [PubMed]

Finn, M. D.

S. C. Creagh and M. D. Finn, “Evanescent coupling between discs: a model for near-integrable tunnelling,” J. Phys. A. Math. Gen. 34, 3791–3801 (2001).
[Crossref]

Ge, L.

Y. D. Chong, L. Ge, and A. D. Stone, “PT–Symmetry breaking and laser-absorber modes in optical scattering systems,” Phys. Rev. Lett. 106, 093902 (2011).
[Crossref]

Gianfreda, M.

B. Peng, S. K. Özdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Paritytime-symmetric whispering-gallery microcavities,” Nat. Phys. 10, 1–5 (2014).
[Crossref]

Greenberg, M.

Hagness, S. C.

S. C. Hagness, R. M. Joseph, and A. Taflove, “Subpicosecond electrodynamics of distributed Bragg reflector microlasers: results from finite difference time domain simulations,” Radio Sci. 31, 931–941 (1996).
[Crossref]

Heinrich, M.

H. Hodaei, M.-A. Miri, M. Heinrich, D. N. Christodoulides, and M. Khajavikhan, “PT symmetric large area single mode DFB lasers,” in Proceedings of CLEO: 2014 (OSA, 2014), Vol. 1, paper FM1D.3.

Hodaei, H.

H. Hodaei, M.-A. Miri, M. Heinrich, D. N. Christodoulides, and M. Khajavikhan, “PT symmetric large area single mode DFB lasers,” in Proceedings of CLEO: 2014 (OSA, 2014), Vol. 1, paper FM1D.3.

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, 524–529 (2014).
[Crossref]

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, 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, 524–529 (2014).
[Crossref]

Jones, H. F.

H. F. Jones, “Analytic results for a PT -symmetric optical structure,” J. Phys. A-Math. Theor. 45, 135306 (2012).
[Crossref]

Joseph, R. M.

S. C. Hagness, R. M. Joseph, and A. Taflove, “Subpicosecond electrodynamics of distributed Bragg reflector microlasers: results from finite difference time domain simulations,” Radio Sci. 31, 931–941 (1996).
[Crossref]

Kearsley, M. J.

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

Khajavikhan, M.

H. Hodaei, M.-A. Miri, M. Heinrich, D. N. Christodoulides, and M. Khajavikhan, “PT symmetric large area single mode DFB lasers,” in Proceedings of CLEO: 2014 (OSA, 2014), Vol. 1, paper FM1D.3.

Kip, D.

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

Kottos, T.

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

Kress, B.

M. Kulishov, B. Kress, and R. Slavk, “Resonant cavities based on Parity-Time-symmetric diffractive gratings,” Opt. Express 21, 68–70 (2013).
[Crossref]

Kulishov, M.

M. Kulishov, B. Kress, and R. Slavk, “Resonant cavities based on Parity-Time-symmetric diffractive gratings,” Opt. Express 21, 68–70 (2013).
[Crossref]

Kuzmiak, V.

Landau, L. D.

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

Lei, F.

B. Peng, S. K. Özdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Paritytime-symmetric whispering-gallery microcavities,” Nat. Phys. 10, 1–5 (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, 524–529 (2014).
[Crossref]

Liertzer, M.

B. Peng, S. K. Özdemir, S. Rotter, H. Yilmaz, M. Liertzer, F. Monifi, C. M. Bender, F. Nori, and L. Yang, “Loss-induced suppression and revival of lasing,” Science 346(6207), 328–332 (2014).
[Crossref] [PubMed]

Lifshitz, E. M.

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

Lin, Z.

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

Lisyansky, A. A.

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

Long, G. L.

B. Peng, S. K. Özdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Paritytime-symmetric whispering-gallery microcavities,” Nat. Phys. 10, 1–5 (2014).
[Crossref]

Longhi, S.

Lupu, A.

A. Lupu, H. Benisty, and A. Degiron, “Switching using PT symmetry in plasmonic systems: positive role of the losses,” Opt. Express 21, 192–195 (2013).
[Crossref]

H. Benisty, C. Yan, A. Degiron, and A. Lupu, “Healing near-PT-symmetric structures to restore their characteristic singularities: analysis and examples,” J. Lightwave Technol. 30, 2675–2683 (2012).
[Crossref]

Ma, R.-M.

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

Makris, K. G.

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

R. El-Ganainy, K. G. Makris, D. N. Christodoulides, and Z. H. Musslimani, “Theory of coupled optical PT-symmetric structures,” Opt. Lett. 32, 2632–2634 (2007).
[Crossref] [PubMed]

Meisinger, P. N.

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

Miri, M.

A. Regensburger, M. Miri, and C. Bersch, “Observation of defect states in PT-symmetric optical lattices,” Phys. Rev. Lett. 110, 223902 (2013).
[Crossref] [PubMed]

Miri, M.-A.

H. Hodaei, M.-A. Miri, M. Heinrich, D. N. Christodoulides, and M. Khajavikhan, “PT symmetric large area single mode DFB lasers,” in Proceedings of CLEO: 2014 (OSA, 2014), Vol. 1, paper FM1D.3.

Monifi, F.

B. Peng, S. K. Özdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Paritytime-symmetric whispering-gallery microcavities,” Nat. Phys. 10, 1–5 (2014).
[Crossref]

B. Peng, S. K. Özdemir, S. Rotter, H. Yilmaz, M. Liertzer, F. Monifi, C. M. Bender, F. Nori, and L. Yang, “Loss-induced suppression and revival of lasing,” Science 346(6207), 328–332 (2014).
[Crossref] [PubMed]

Moravvej-Farshi, M. K.

Musslimani, Z. H.

Nazari, F.

Nazari, M.

Nolting, H.

H. Nolting, G. Sztefka, and J. Čtyroký, “Wave propagation in a waveguide with a balance of gain and loss,” in Integrated Photonics Research (OSA, 1996), pp. 76–80.

Nori, F.

B. Peng, S. K. Özdemir, S. Rotter, H. Yilmaz, M. Liertzer, F. Monifi, C. M. Bender, F. Nori, and L. Yang, “Loss-induced suppression and revival of lasing,” Science 346(6207), 328–332 (2014).
[Crossref] [PubMed]

B. Peng, S. K. Özdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Paritytime-symmetric whispering-gallery microcavities,” Nat. Phys. 10, 1–5 (2014).
[Crossref]

Nosich, A.

E. Smotrova, A. Nosich, T. M. Benson, and P. Sewell, “Optical coupling of whispering-gallery modes of two identical microdisks and its effect on photonic molecule lasing,” IEEE J. Sel. Top. Quantum Electron. 12, 78–85 (2006).
[Crossref]

Nosich, A. I.

Orenstein, M.

Özdemir, S. K.

B. Peng, S. K. Özdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Paritytime-symmetric whispering-gallery microcavities,” Nat. Phys. 10, 1–5 (2014).
[Crossref]

B. Peng, S. K. Özdemir, S. Rotter, H. Yilmaz, M. Liertzer, F. Monifi, C. M. Bender, F. Nori, and L. Yang, “Loss-induced suppression and revival of lasing,” Science 346(6207), 328–332 (2014).
[Crossref] [PubMed]

Paul, J.

J. Paul, C. Christopoulos, and D. W. P. Thomas, “Generalized material models in TLM.I. Materials with frequency-dependent properties,” IEEE Trans. Antennas Propag. 47, 1528–1534 (1999).
[Crossref]

Peng, B.

B. Peng, S. K. Özdemir, S. Rotter, H. Yilmaz, M. Liertzer, F. Monifi, C. M. Bender, F. Nori, and L. Yang, “Loss-induced suppression and revival of lasing,” Science 346(6207), 328–332 (2014).
[Crossref] [PubMed]

B. Peng, S. K. Özdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Paritytime-symmetric whispering-gallery microcavities,” Nat. Phys. 10, 1–5 (2014).
[Crossref]

Phang, S.

S. Phang, A. Vukovic, T. M. Benson, H. Susanto, and 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, 37–47 (2015).
[Crossref]

S. Phang, A. Vukovic, H. Susanto, T. M. Benson, and P. Sewell, “Impact of dispersive and saturable gain/loss on bistability of nonlinear parity-time Bragg gratings,” Opt. Lett. 39, 2603–2606 (2014).
[Crossref] [PubMed]

S. Phang, A. Vukovic, H. Susanto, T. M. Benson, and P. Sewell, “Ultrafast optical switching using parity–time symmetric Bragg gratings,” J. Opt. Soc. Am. B 30, 2984–2991 (2013).
[Crossref]

S. Phang, A. Vukovic, H. Susanto, T. M. Benson, and P. Sewell, “Practical limitation on operation of nonlinear parity-time Bragg gratings,” in META 2014 Conference (2014), pp. 270–275.

Pitaevskii, L. P.

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

Pukhov, A. A.

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

Ramezani, H.

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

Regensburger, A.

A. Regensburger, M. Miri, and C. Bersch, “Observation of defect states in PT-symmetric optical lattices,” Phys. Rev. Lett. 110, 223902 (2013).
[Crossref] [PubMed]

Rotter, S.

B. Peng, S. K. Özdemir, S. Rotter, H. Yilmaz, M. Liertzer, F. Monifi, C. M. Bender, F. Nori, and L. Yang, “Loss-induced suppression and revival of lasing,” Science 346(6207), 328–332 (2014).
[Crossref] [PubMed]

Rüter, C. E.

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

Segev, M.

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

Sewell, P.

S. Phang, A. Vukovic, T. M. Benson, H. Susanto, and 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, 37–47 (2015).
[Crossref]

S. Phang, A. Vukovic, H. Susanto, T. M. Benson, and P. Sewell, “Impact of dispersive and saturable gain/loss on bistability of nonlinear parity-time Bragg gratings,” Opt. Lett. 39, 2603–2606 (2014).
[Crossref] [PubMed]

S. Phang, A. Vukovic, H. Susanto, T. M. Benson, and P. Sewell, “Ultrafast optical switching using parity–time symmetric Bragg gratings,” J. Opt. Soc. Am. B 30, 2984–2991 (2013).
[Crossref]

E. Smotrova, A. Nosich, T. M. Benson, and P. Sewell, “Optical coupling of whispering-gallery modes of two identical microdisks and its effect on photonic molecule lasing,” IEEE J. Sel. Top. Quantum Electron. 12, 78–85 (2006).
[Crossref]

S. Phang, A. Vukovic, H. Susanto, T. M. Benson, and P. Sewell, “Practical limitation on operation of nonlinear parity-time Bragg gratings,” in META 2014 Conference (2014), pp. 270–275.

Slavk, R.

M. Kulishov, B. Kress, and R. Slavk, “Resonant cavities based on Parity-Time-symmetric diffractive gratings,” Opt. Express 21, 68–70 (2013).
[Crossref]

Smotrova, E.

E. Smotrova, A. Nosich, T. M. Benson, and P. Sewell, “Optical coupling of whispering-gallery modes of two identical microdisks and its effect on photonic molecule lasing,” IEEE J. Sel. Top. Quantum Electron. 12, 78–85 (2006).
[Crossref]

Smotrova, E. I.

Stegun, I. A.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (U.S. Department of Commerce, NIST, 1972).

Stone, A. D.

Y. D. Chong, L. Ge, and A. D. Stone, “PT–Symmetry breaking and laser-absorber modes in optical scattering systems,” Phys. Rev. Lett. 106, 093902 (2011).
[Crossref]

Susanto, H.

S. Phang, A. Vukovic, T. M. Benson, H. Susanto, and 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, 37–47 (2015).
[Crossref]

S. Phang, A. Vukovic, H. Susanto, T. M. Benson, and P. Sewell, “Impact of dispersive and saturable gain/loss on bistability of nonlinear parity-time Bragg gratings,” Opt. Lett. 39, 2603–2606 (2014).
[Crossref] [PubMed]

S. Phang, A. Vukovic, H. Susanto, T. M. Benson, and P. Sewell, “Ultrafast optical switching using parity–time symmetric Bragg gratings,” J. Opt. Soc. Am. B 30, 2984–2991 (2013).
[Crossref]

S. Phang, A. Vukovic, H. Susanto, T. M. Benson, and P. Sewell, “Practical limitation on operation of nonlinear parity-time Bragg gratings,” in META 2014 Conference (2014), pp. 270–275.

Sykes, J. B.

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

Sztefka, G.

H. Nolting, G. Sztefka, and J. Čtyroký, “Wave propagation in a waveguide with a balance of gain and loss,” in Integrated Photonics Research (OSA, 1996), pp. 76–80.

Taflove, A.

S. C. Hagness, R. M. Joseph, and A. Taflove, “Subpicosecond electrodynamics of distributed Bragg reflector microlasers: results from finite difference time domain simulations,” Radio Sci. 31, 931–941 (1996).
[Crossref]

Thomas, D. W. P.

J. Paul, C. Christopoulos, and D. W. P. Thomas, “Generalized material models in TLM.I. Materials with frequency-dependent properties,” IEEE Trans. Antennas Propag. 47, 1528–1534 (1999).
[Crossref]

Vinogradov, A. P.

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

Vukovic, A.

S. Phang, A. Vukovic, T. M. Benson, H. Susanto, and 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, 37–47 (2015).
[Crossref]

S. Phang, A. Vukovic, H. Susanto, T. M. Benson, and P. Sewell, “Impact of dispersive and saturable gain/loss on bistability of nonlinear parity-time Bragg gratings,” Opt. Lett. 39, 2603–2606 (2014).
[Crossref] [PubMed]

S. Phang, A. Vukovic, H. Susanto, T. M. Benson, and P. Sewell, “Ultrafast optical switching using parity–time symmetric Bragg gratings,” J. Opt. Soc. Am. B 30, 2984–2991 (2013).
[Crossref]

S. Phang, A. Vukovic, H. Susanto, T. M. Benson, and P. Sewell, “Practical limitation on operation of nonlinear parity-time Bragg gratings,” in META 2014 Conference (2014), pp. 270–275.

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, 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, 972–975 (2014).
[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, 524–529 (2014).
[Crossref]

Wong, Z. J.

L. Feng, Z. J. Wong, R.-M. Ma, Y. Wang, and X. Zhang, “Single-mode laser by parity-time symmetry breaking,” Science 346, 972–975 (2014).
[Crossref] [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, 524–529 (2014).
[Crossref]

Yan, C.

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, 524–529 (2014).
[Crossref]

Yang, L.

B. Peng, S. K. Özdemir, S. Rotter, H. Yilmaz, M. Liertzer, F. Monifi, C. M. Bender, F. Nori, and L. Yang, “Loss-induced suppression and revival of lasing,” Science 346(6207), 328–332 (2014).
[Crossref] [PubMed]

B. Peng, S. K. Özdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Paritytime-symmetric whispering-gallery microcavities,” Nat. Phys. 10, 1–5 (2014).
[Crossref]

Yilmaz, H.

B. Peng, S. K. Özdemir, S. Rotter, H. Yilmaz, M. Liertzer, F. Monifi, C. M. Bender, F. Nori, and L. Yang, “Loss-induced suppression and revival of lasing,” Science 346(6207), 328–332 (2014).
[Crossref] [PubMed]

Zhang, X.

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

Zyablovsky, A. A.

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

IEEE J. Sel. Top. Quantum Electron. (1)

E. Smotrova, A. Nosich, T. M. Benson, and P. Sewell, “Optical coupling of whispering-gallery modes of two identical microdisks and its effect on photonic molecule lasing,” IEEE J. Sel. Top. Quantum Electron. 12, 78–85 (2006).
[Crossref]

IEEE Trans. Antennas Propag. (1)

J. Paul, C. Christopoulos, and D. W. P. Thomas, “Generalized material models in TLM.I. Materials with frequency-dependent properties,” IEEE Trans. Antennas Propag. 47, 1528–1534 (1999).
[Crossref]

J. Lightwave Technol. (1)

J. Math. Phys. (1)

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

J. Opt. Soc. Am. B (2)

J. Phys. A-Math. Theor. (1)

H. F. Jones, “Analytic results for a PT -symmetric optical structure,” J. Phys. A-Math. Theor. 45, 135306 (2012).
[Crossref]

J. Phys. A. Math. Gen. (1)

S. C. Creagh and M. D. Finn, “Evanescent coupling between discs: a model for near-integrable tunnelling,” J. Phys. A. Math. Gen. 34, 3791–3801 (2001).
[Crossref]

Nat. Photonics (1)

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

Nat. Phys. (2)

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

B. Peng, S. K. Özdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Paritytime-symmetric whispering-gallery microcavities,” Nat. Phys. 10, 1–5 (2014).
[Crossref]

Opt. Express (4)

M. Kulishov, B. Kress, and R. Slavk, “Resonant cavities based on Parity-Time-symmetric diffractive gratings,” Opt. Express 21, 68–70 (2013).
[Crossref]

A. Lupu, H. Benisty, and A. Degiron, “Switching using PT symmetry in plasmonic systems: positive role of the losses,” Opt. Express 21, 192–195 (2013).
[Crossref]

M. Greenberg and M. Orenstein, “Unidirectional complex grating assisted couplers,” Opt. Express 12, 4013–4018 (2004).
[Crossref] [PubMed]

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

Opt. Lett. (5)

Opt. Quantum Electron. (2)

S. Phang, A. Vukovic, T. M. Benson, H. Susanto, and 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, 37–47 (2015).
[Crossref]

J. Čtyroký, “Dispersion properties of coupled waveguides with loss and gain: a full-vectorial analysis,” Opt. Quantum Electron. 46, 465–475 (2014).
[Crossref]

Phys. Rev. A (2)

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

S. Longhi, “PT-symmetric laser absorber,” Phys. Rev. A 82, 031801 (2010).
[Crossref]

Phys. Rev. Lett. (3)

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

Y. D. Chong, L. Ge, and A. D. Stone, “PT–Symmetry breaking and laser-absorber modes in optical scattering systems,” Phys. Rev. Lett. 106, 093902 (2011).
[Crossref]

A. Regensburger, M. Miri, and C. Bersch, “Observation of defect states in PT-symmetric optical lattices,” Phys. Rev. Lett. 110, 223902 (2013).
[Crossref] [PubMed]

Radio Sci. (1)

S. C. Hagness, R. M. Joseph, and A. Taflove, “Subpicosecond electrodynamics of distributed Bragg reflector microlasers: results from finite difference time domain simulations,” Radio Sci. 31, 931–941 (1996).
[Crossref]

Science (2)

B. Peng, S. K. Özdemir, S. Rotter, H. Yilmaz, M. Liertzer, F. Monifi, C. M. Bender, F. Nori, and L. Yang, “Loss-induced suppression and revival of lasing,” Science 346(6207), 328–332 (2014).
[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, 972–975 (2014).
[Crossref] [PubMed]

Other (6)

S. Phang, A. Vukovic, H. Susanto, T. M. Benson, and P. Sewell, “Practical limitation on operation of nonlinear parity-time Bragg gratings,” in META 2014 Conference (2014), pp. 270–275.

C. Christopoulos, The Transmission-Line Modeling Method TLM (IEEE-Oxford University, 1995).
[Crossref]

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

H. Hodaei, M.-A. Miri, M. Heinrich, D. N. Christodoulides, and M. Khajavikhan, “PT symmetric large area single mode DFB lasers,” in Proceedings of CLEO: 2014 (OSA, 2014), Vol. 1, paper FM1D.3.

H. Nolting, G. Sztefka, and J. Čtyroký, “Wave propagation in a waveguide with a balance of gain and loss,” in Integrated Photonics Research (OSA, 1996), pp. 76–80.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (U.S. Department of Commerce, NIST, 1972).

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Figures (9)

Fig. 1
Fig. 1 Schematic of two coupled cylindrical microresonators or radius a and separated by a distance g. Microresonators with gain and loss are denoted by μRG and μRL, respectively.
Fig. 2
Fig. 2 Frequency bifurcation of PT-coupled microresonator with a balanced gain (γG = −γ0) and loss (γL = γ0) as a function of gain/loss parameter at the peak of pumping beam γ0 = ωσ n (ωσ) for three different dispersion parameters, (a,b) ωστ = 0, (c,d) ωστ = 212 and (e,f) ωστ = 0.7
Fig. 3
Fig. 3 (a) Impact of dispersion to the real part of material at atomic transitional angular frequency ωσ due to the presence of gain and loss for different dispersion parameters; (b) Contrast between the real part of eigenfrequencies of PT-coupled microresonators for two different gain/loss parameter, i.e. γ0 = 7.5 rad/ps for (7,2) and 2.54 rad/ps for the (10,1) mode as function of dispersion parameter τ.
Fig. 4
Fig. 4 Frequency bifurcation of coupled microresonators with balanced gain and loss as function of gain/loss parameters γσ, for two different atomic transitional frequencies ωσ = 2π(f0+δ) with δ = 0.1 and 0.1 THz.
Fig. 5
Fig. 5 Complex eigenfrequency in a PT-coupled microresonator system with variable gain and fixed loss shown as a function of gain parameter G|, dispersion parameter ωστ = 212 [32] and shown for 3 different fixed loss value, i.e. γL = 5.565, 6.4281, and 7.291 rad/ps.
Fig. 6
Fig. 6 (a) Spatial electric field distribution of the coupled microresonators operated in the (7,2) mode. The black dashed line denotes the monitor line. The temporal evolution (b) and spectra (c) of the field on the monitor line are shown in the absence of gain and loss.
Fig. 7
Fig. 7 The temporal evolution and spectra of the field on the monitor line are shown for different gain/loss parameters, (a,b) for γ0 = 4.3 rad/ps and (c,d) for γ0 = 7.5 rad/ps with a negligible dispersion parameter using the TLM method.
Fig. 8
Fig. 8 Temporal and spectra of electric field along the monitor line for coupled PT microresonators with balanced gain and loss parameters operated for (7,2) mode with practical dispersion parameters ωστ = 212 [32] and for two different gain/loss parameter, i.e. (a,b) γ0 = 4.3 rad/ps and (c,d) γ0 = 7.5 rad/ps.
Fig. 9
Fig. 9 Temporal and spectra of electric field along the monitor line for coupled PT microresonators with unbalanced gain and loss operated for (7,2) mode with practical dispersion parameters ωστ = 212 [32], i.e. (a,b) γL = 5.565 rad/ps, (c,d) γL = 6.4281 rad/ps and (e,f) γL = 7.291 rad/ps while the gain parameter is kept constant at γG = 7.053 rad/ps.

Equations (27)

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ε r ( ω ) = ε j σ 0 2 ε 0 ω ( 1 1 + j ( ω + ω σ ) τ + 1 1 + j ( ω ω σ ) τ ) .
a ψ L n = F m L ψ L ,
F m L = z J m ( z ) J m ( z ) and z = n L k a ,
G 0 ( x , x ) = j 4 H 0 ( k | x x | ) ,
0 = B G + B L ( G 0 ( x , x ) ψ ( x ) n G 0 ( x , x ) n ψ ( x ) ) d s
ψ G = m α m G e j m θ G and ψ L = m α m L e j m θ L ,
ψ G n = m 1 a F m G α m G e j m θ G and ψ G n = m 1 a F m G α m G e j m θ G .
D G α G + C G L α L = 0 C L G α G + D L α L = 0.
α G = ( α m G α m + 1 G ) and α L = ( α m L α m + 1 L )
D m m G , L = J m ( u ) H m ( u ) ( F m G , L u H m ( u ) H m ( u ) ) , where u = k a ,
C l m G L = J l ( u ) H l + m ( w ) J m ( u ) ( F m L u L J m ( u ) J m ( u ) ) ,
a ˜ m L = J m ( u ) ( F m L u J m ( u ) J m ( u ) ) α m L
D ˜ G α ˜ G + C ˜ α ˜ L = 0 C ˜ α ˜ G + D ˜ L α ˜ L = 0 ,
D ˜ m m G , L = j H m ( u ) F m G , L u H m ( u ) J m ( u ) F m G , L u J m ( u ) , where u = k a ,
C ˜ l m = j H l + m ( w ) ,
j H m ( u ) Y m ( u ) and j H l + m ( u ) Y l + m ( u ) ,
( D ˜ L ) * D ˜ G and C ˜ * C ˜
n G n L .
ψ ± ψ G ± ψ L ,
M ( α ˜ m m G α ˜ m m L ) = 0 , where M = ( D ˜ m m G C ˜ m m C ˜ m m D ˜ m m L ) .
0 = det M = D ˜ m m G D ˜ m m L C ˜ m m 2 .
D m m 0 = 1 2 ( D ˜ m m G + D ˜ m m L ) and D m m I = 1 2 j ( D ˜ m m G D ˜ m m L )
ω 1 , 2 = ω 0 ± Δ ω 0 2 +
D m m 0 ( ω 0 ) = 0.
0 = det M = Δ ω 0 2 D m m 0 ( ω 0 ) 2 + D m m I ( ω 0 ) 2 C ˜ m m ( ω 0 ) 2 +
Δ ω 0 2 = C ˜ m m ( ω 0 ) 2 D m m I ( ω 0 ) 2 D m m 0 ( ω 0 ) ,
C ˜ m m ( ω 0 ) 2 = D m m I ( ω 0 ) 2

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