Abstract

We propose a parity-time (PT) symmetric fiber laser composed of two coupled ring cavities with gains and losses, which operates both in PT-symmetric and symmetry-broken regimes depending on the static phase shifts. We perform analytical and numerical analysis by the transfer matrix method taking into account gain saturation and predict laser bistability in the PT-symmetric regime in contrast to a symmetry-broken single-mode operation. In the PT-broken regime, the generation power counterintuitively increases with an increase of the cavity losses.

© 2018 Chinese Laser Press

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References

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  1. A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103, 093902 (2009).
    [Crossref]
  2. C. E. Ruter, 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]
  3. A. A. Zyablovsky, A. P. Vinogradov, A. A. Pukhov, A. V. Dorofeenko, and A. A. Lisyansky, “PT-symmetry in optics,” Phys. Usp. 57, 1063–1082 (2014).
    [Crossref]
  4. V. V. Konotop, J. K. Yang, and D. A. Zezyulin, “Nonlinear waves in PT-symmetric systems,” Rev. Mod. Phys. 88, 035002 (2016).
    [Crossref]
  5. S. V. Suchkov, A. A. Sukhorukov, J. H. Huang, S. V. Dmitriev, C. Lee, and Y. S. Kivshar, “Nonlinear switching and solitons in PT-symmetric photonic systems,” Laser Photon. Rev. 10, 177–213 (2016).
    [Crossref]
  6. 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]
  7. H. Hodaei, M. A. Miri, M. Heinrich, D. N. Christodoulides, and M. Khajavikhan, “Parity-time-symmetric microring lasers,” Science 346, 975–978 (2014).
    [Crossref]
  8. H. Hodaei, A. U. Hassan, W. E. Hayenga, M. A. Miri, D. N. Christodoulides, and M. Khajavikhan, “Dark-state lasers: mode management using exceptional points,” Opt. Lett. 41, 3049–3052 (2016).
    [Crossref]
  9. W. L. Liu, M. Li, R. S. Guzzon, E. J. Norberg, J. S. Parker, M. Z. Lu, L. A. Coldren, and J. P. Yao, “An integrated parity-time symmetric wavelength-tunable single-mode microring laser,” Nat. Commun. 8, 15389 (2017).
    [Crossref]
  10. H. Hodaei, M. A. Miri, A. U. Hassan, W. E. Hayenga, M. Heinrich, D. N. Christodoulides, and M. Khajavikhan, “Single mode lasing in transversely multi-moded PT-symmetric microring resonators,” Laser Photon. Rev. 10, 494–499 (2016).
    [Crossref]
  11. J. Ren, H. Hodaei, G. Harari, A. U. Hassan, W. Chow, M. Soltani, D. Christodoulides, and M. Khajavikhan, “Ultrasensitive micro-scale parity-time-symmetric ring laser gyroscope,” Opt. Lett. 42, 1556–1559 (2017).
    [Crossref]
  12. H. Hodaei, A. U. Hassan, S. Wittek, H. Garcia-Gracia, R. El Ganainy, D. N. Christodoulides, and M. Khajavikhan, “Enhanced sensitivity at higher-order exceptional points,” Nature 548, 187–191 (2017).
    [Crossref]
  13. M. Brandstetter, M. Liertzer, C. Deutsch, P. Klang, J. Schoberl, H. E. Tureci, G. Strasser, K. Unterrainer, and S. Rotter, “Reversing the pump dependence of a laser at an exceptional point,” Nat. Commun. 5, 4034 (2014).
    [Crossref]
  14. A. Pick, B. Zhen, O. D. Miller, C. W. Hsu, F. Hernandez, A. W. Rodriguez, M. Soljacic, and S. G. Johnson, “General theory of spontaneous emission near exceptional points,” Opt. Express 25, 12325–12348 (2017).
    [Crossref]
  15. K. H. Kim, M. S. Hwang, H. R. Kim, J. H. Choi, Y. S. No, and H. G. Park, “Direct observation of exceptional points in coupled photonic-crystal lasers with asymmetric optical gains,” Nat. Commun. 7, 13893 (2016).
    [Crossref]
  16. S. Longhi, “PT-symmetric mode-locking,” Opt. Lett. 41, 4518–4521 (2016).
    [Crossref]
  17. S. Longhi, “PT-symmetric laser absorber,” Phys. Rev. A 82, 031801 (2010).
    [Crossref]
  18. Z. J. Wong, Y. L. Xu, J. Kim, K. O’Brien, Y. Wang, L. Feng, and X. Zhang, “Lasing and anti-lasing in a single cavity,” Nat. Photonics 10, 796–801 (2016).
    [Crossref]
  19. A. K. Jahromi, A. U. Hassan, D. Christodoulides, and A. Abouraddy, “Statistical parity-time-symmetric lasing in an optical fiber network,” Nat. Commun. 8, 1359 (2017).
  20. A. Regensburger, C. Bersch, M. A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity-time synthetic photonic lattices,” Nature 488, 167–171 (2012).
    [Crossref]
  21. L. Ge and R. El Ganainy, “Nonlinear modal interactions in parity-time (PT) symmetric lasers,” Sci. Rep. 6, 24889 (2016).
    [Crossref]

2017 (5)

W. L. Liu, M. Li, R. S. Guzzon, E. J. Norberg, J. S. Parker, M. Z. Lu, L. A. Coldren, and J. P. Yao, “An integrated parity-time symmetric wavelength-tunable single-mode microring laser,” Nat. Commun. 8, 15389 (2017).
[Crossref]

H. Hodaei, A. U. Hassan, S. Wittek, H. Garcia-Gracia, R. El Ganainy, D. N. Christodoulides, and M. Khajavikhan, “Enhanced sensitivity at higher-order exceptional points,” Nature 548, 187–191 (2017).
[Crossref]

A. K. Jahromi, A. U. Hassan, D. Christodoulides, and A. Abouraddy, “Statistical parity-time-symmetric lasing in an optical fiber network,” Nat. Commun. 8, 1359 (2017).

J. Ren, H. Hodaei, G. Harari, A. U. Hassan, W. Chow, M. Soltani, D. Christodoulides, and M. Khajavikhan, “Ultrasensitive micro-scale parity-time-symmetric ring laser gyroscope,” Opt. Lett. 42, 1556–1559 (2017).
[Crossref]

A. Pick, B. Zhen, O. D. Miller, C. W. Hsu, F. Hernandez, A. W. Rodriguez, M. Soljacic, and S. G. Johnson, “General theory of spontaneous emission near exceptional points,” Opt. Express 25, 12325–12348 (2017).
[Crossref]

2016 (8)

Z. J. Wong, Y. L. Xu, J. Kim, K. O’Brien, Y. Wang, L. Feng, and X. Zhang, “Lasing and anti-lasing in a single cavity,” Nat. Photonics 10, 796–801 (2016).
[Crossref]

L. Ge and R. El Ganainy, “Nonlinear modal interactions in parity-time (PT) symmetric lasers,” Sci. Rep. 6, 24889 (2016).
[Crossref]

K. H. Kim, M. S. Hwang, H. R. Kim, J. H. Choi, Y. S. No, and H. G. Park, “Direct observation of exceptional points in coupled photonic-crystal lasers with asymmetric optical gains,” Nat. Commun. 7, 13893 (2016).
[Crossref]

H. Hodaei, A. U. Hassan, W. E. Hayenga, M. A. Miri, D. N. Christodoulides, and M. Khajavikhan, “Dark-state lasers: mode management using exceptional points,” Opt. Lett. 41, 3049–3052 (2016).
[Crossref]

S. Longhi, “PT-symmetric mode-locking,” Opt. Lett. 41, 4518–4521 (2016).
[Crossref]

H. Hodaei, M. A. Miri, A. U. Hassan, W. E. Hayenga, M. Heinrich, D. N. Christodoulides, and M. Khajavikhan, “Single mode lasing in transversely multi-moded PT-symmetric microring resonators,” Laser Photon. Rev. 10, 494–499 (2016).
[Crossref]

V. V. Konotop, J. K. Yang, and D. A. Zezyulin, “Nonlinear waves in PT-symmetric systems,” Rev. Mod. Phys. 88, 035002 (2016).
[Crossref]

S. V. Suchkov, A. A. Sukhorukov, J. H. Huang, S. V. Dmitriev, C. Lee, and Y. S. Kivshar, “Nonlinear switching and solitons in PT-symmetric photonic systems,” Laser Photon. Rev. 10, 177–213 (2016).
[Crossref]

2014 (4)

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]

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

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

M. Brandstetter, M. Liertzer, C. Deutsch, P. Klang, J. Schoberl, H. E. Tureci, G. Strasser, K. Unterrainer, and S. Rotter, “Reversing the pump dependence of a laser at an exceptional point,” Nat. Commun. 5, 4034 (2014).
[Crossref]

2012 (1)

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

2010 (2)

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

C. E. Ruter, 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]

2009 (1)

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

Abouraddy, A.

A. K. Jahromi, A. U. Hassan, D. Christodoulides, and A. Abouraddy, “Statistical parity-time-symmetric lasing in an optical fiber network,” Nat. Commun. 8, 1359 (2017).

Aimez, V.

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

Bersch, C.

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

Brandstetter, M.

M. Brandstetter, M. Liertzer, C. Deutsch, P. Klang, J. Schoberl, H. E. Tureci, G. Strasser, K. Unterrainer, and S. Rotter, “Reversing the pump dependence of a laser at an exceptional point,” Nat. Commun. 5, 4034 (2014).
[Crossref]

Choi, J. H.

K. H. Kim, M. S. Hwang, H. R. Kim, J. H. Choi, Y. S. No, and H. G. Park, “Direct observation of exceptional points in coupled photonic-crystal lasers with asymmetric optical gains,” Nat. Commun. 7, 13893 (2016).
[Crossref]

Chow, W.

Christodoulides, D.

J. Ren, H. Hodaei, G. Harari, A. U. Hassan, W. Chow, M. Soltani, D. Christodoulides, and M. Khajavikhan, “Ultrasensitive micro-scale parity-time-symmetric ring laser gyroscope,” Opt. Lett. 42, 1556–1559 (2017).
[Crossref]

A. K. Jahromi, A. U. Hassan, D. Christodoulides, and A. Abouraddy, “Statistical parity-time-symmetric lasing in an optical fiber network,” Nat. Commun. 8, 1359 (2017).

Christodoulides, D. N.

H. Hodaei, A. U. Hassan, S. Wittek, H. Garcia-Gracia, R. El Ganainy, D. N. Christodoulides, and M. Khajavikhan, “Enhanced sensitivity at higher-order exceptional points,” Nature 548, 187–191 (2017).
[Crossref]

H. Hodaei, M. A. Miri, A. U. Hassan, W. E. Hayenga, M. Heinrich, D. N. Christodoulides, and M. Khajavikhan, “Single mode lasing in transversely multi-moded PT-symmetric microring resonators,” Laser Photon. Rev. 10, 494–499 (2016).
[Crossref]

H. Hodaei, A. U. Hassan, W. E. Hayenga, M. A. Miri, D. N. Christodoulides, and M. Khajavikhan, “Dark-state lasers: mode management using exceptional points,” Opt. Lett. 41, 3049–3052 (2016).
[Crossref]

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

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

C. E. Ruter, 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]

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

Coldren, L. A.

W. L. Liu, M. Li, R. S. Guzzon, E. J. Norberg, J. S. Parker, M. Z. Lu, L. A. Coldren, and J. P. Yao, “An integrated parity-time symmetric wavelength-tunable single-mode microring laser,” Nat. Commun. 8, 15389 (2017).
[Crossref]

Deutsch, C.

M. Brandstetter, M. Liertzer, C. Deutsch, P. Klang, J. Schoberl, H. E. Tureci, G. Strasser, K. Unterrainer, and S. Rotter, “Reversing the pump dependence of a laser at an exceptional point,” Nat. Commun. 5, 4034 (2014).
[Crossref]

Dmitriev, S. V.

S. V. Suchkov, A. A. Sukhorukov, J. H. Huang, S. V. Dmitriev, C. Lee, and Y. S. Kivshar, “Nonlinear switching and solitons in PT-symmetric photonic systems,” Laser Photon. Rev. 10, 177–213 (2016).
[Crossref]

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, 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, 093902 (2009).
[Crossref]

El Ganainy, R.

H. Hodaei, A. U. Hassan, S. Wittek, H. Garcia-Gracia, R. El Ganainy, D. N. Christodoulides, and M. Khajavikhan, “Enhanced sensitivity at higher-order exceptional points,” Nature 548, 187–191 (2017).
[Crossref]

L. Ge and R. El Ganainy, “Nonlinear modal interactions in parity-time (PT) symmetric lasers,” Sci. Rep. 6, 24889 (2016).
[Crossref]

C. E. Ruter, 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]

Feng, L.

Z. J. Wong, Y. L. Xu, J. Kim, K. O’Brien, Y. Wang, L. Feng, and X. Zhang, “Lasing and anti-lasing in a single cavity,” Nat. Photonics 10, 796–801 (2016).
[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]

Garcia-Gracia, H.

H. Hodaei, A. U. Hassan, S. Wittek, H. Garcia-Gracia, R. El Ganainy, D. N. Christodoulides, and M. Khajavikhan, “Enhanced sensitivity at higher-order exceptional points,” Nature 548, 187–191 (2017).
[Crossref]

Ge, L.

L. Ge and R. El Ganainy, “Nonlinear modal interactions in parity-time (PT) symmetric lasers,” Sci. Rep. 6, 24889 (2016).
[Crossref]

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, 093902 (2009).
[Crossref]

Guzzon, R. S.

W. L. Liu, M. Li, R. S. Guzzon, E. J. Norberg, J. S. Parker, M. Z. Lu, L. A. Coldren, and J. P. Yao, “An integrated parity-time symmetric wavelength-tunable single-mode microring laser,” Nat. Commun. 8, 15389 (2017).
[Crossref]

Harari, G.

Hassan, A. U.

J. Ren, H. Hodaei, G. Harari, A. U. Hassan, W. Chow, M. Soltani, D. Christodoulides, and M. Khajavikhan, “Ultrasensitive micro-scale parity-time-symmetric ring laser gyroscope,” Opt. Lett. 42, 1556–1559 (2017).
[Crossref]

H. Hodaei, A. U. Hassan, S. Wittek, H. Garcia-Gracia, R. El Ganainy, D. N. Christodoulides, and M. Khajavikhan, “Enhanced sensitivity at higher-order exceptional points,” Nature 548, 187–191 (2017).
[Crossref]

A. K. Jahromi, A. U. Hassan, D. Christodoulides, and A. Abouraddy, “Statistical parity-time-symmetric lasing in an optical fiber network,” Nat. Commun. 8, 1359 (2017).

H. Hodaei, M. A. Miri, A. U. Hassan, W. E. Hayenga, M. Heinrich, D. N. Christodoulides, and M. Khajavikhan, “Single mode lasing in transversely multi-moded PT-symmetric microring resonators,” Laser Photon. Rev. 10, 494–499 (2016).
[Crossref]

H. Hodaei, A. U. Hassan, W. E. Hayenga, M. A. Miri, D. N. Christodoulides, and M. Khajavikhan, “Dark-state lasers: mode management using exceptional points,” Opt. Lett. 41, 3049–3052 (2016).
[Crossref]

Hayenga, W. E.

H. Hodaei, A. U. Hassan, W. E. Hayenga, M. A. Miri, D. N. Christodoulides, and M. Khajavikhan, “Dark-state lasers: mode management using exceptional points,” Opt. Lett. 41, 3049–3052 (2016).
[Crossref]

H. Hodaei, M. A. Miri, A. U. Hassan, W. E. Hayenga, M. Heinrich, D. N. Christodoulides, and M. Khajavikhan, “Single mode lasing in transversely multi-moded PT-symmetric microring resonators,” Laser Photon. Rev. 10, 494–499 (2016).
[Crossref]

Heinrich, M.

H. Hodaei, M. A. Miri, A. U. Hassan, W. E. Hayenga, M. Heinrich, D. N. Christodoulides, and M. Khajavikhan, “Single mode lasing in transversely multi-moded PT-symmetric microring resonators,” Laser Photon. Rev. 10, 494–499 (2016).
[Crossref]

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

Hernandez, F.

Hodaei, H.

J. Ren, H. Hodaei, G. Harari, A. U. Hassan, W. Chow, M. Soltani, D. Christodoulides, and M. Khajavikhan, “Ultrasensitive micro-scale parity-time-symmetric ring laser gyroscope,” Opt. Lett. 42, 1556–1559 (2017).
[Crossref]

H. Hodaei, A. U. Hassan, S. Wittek, H. Garcia-Gracia, R. El Ganainy, D. N. Christodoulides, and M. Khajavikhan, “Enhanced sensitivity at higher-order exceptional points,” Nature 548, 187–191 (2017).
[Crossref]

H. Hodaei, A. U. Hassan, W. E. Hayenga, M. A. Miri, D. N. Christodoulides, and M. Khajavikhan, “Dark-state lasers: mode management using exceptional points,” Opt. Lett. 41, 3049–3052 (2016).
[Crossref]

H. Hodaei, M. A. Miri, A. U. Hassan, W. E. Hayenga, M. Heinrich, D. N. Christodoulides, and M. Khajavikhan, “Single mode lasing in transversely multi-moded PT-symmetric microring resonators,” Laser Photon. Rev. 10, 494–499 (2016).
[Crossref]

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

Hsu, C. W.

Huang, J. H.

S. V. Suchkov, A. A. Sukhorukov, J. H. Huang, S. V. Dmitriev, C. Lee, and Y. S. Kivshar, “Nonlinear switching and solitons in PT-symmetric photonic systems,” Laser Photon. Rev. 10, 177–213 (2016).
[Crossref]

Hwang, M. S.

K. H. Kim, M. S. Hwang, H. R. Kim, J. H. Choi, Y. S. No, and H. G. Park, “Direct observation of exceptional points in coupled photonic-crystal lasers with asymmetric optical gains,” Nat. Commun. 7, 13893 (2016).
[Crossref]

Jahromi, A. K.

A. K. Jahromi, A. U. Hassan, D. Christodoulides, and A. Abouraddy, “Statistical parity-time-symmetric lasing in an optical fiber network,” Nat. Commun. 8, 1359 (2017).

Johnson, S. G.

Khajavikhan, M.

H. Hodaei, A. U. Hassan, S. Wittek, H. Garcia-Gracia, R. El Ganainy, D. N. Christodoulides, and M. Khajavikhan, “Enhanced sensitivity at higher-order exceptional points,” Nature 548, 187–191 (2017).
[Crossref]

J. Ren, H. Hodaei, G. Harari, A. U. Hassan, W. Chow, M. Soltani, D. Christodoulides, and M. Khajavikhan, “Ultrasensitive micro-scale parity-time-symmetric ring laser gyroscope,” Opt. Lett. 42, 1556–1559 (2017).
[Crossref]

H. Hodaei, M. A. Miri, A. U. Hassan, W. E. Hayenga, M. Heinrich, D. N. Christodoulides, and M. Khajavikhan, “Single mode lasing in transversely multi-moded PT-symmetric microring resonators,” Laser Photon. Rev. 10, 494–499 (2016).
[Crossref]

H. Hodaei, A. U. Hassan, W. E. Hayenga, M. A. Miri, D. N. Christodoulides, and M. Khajavikhan, “Dark-state lasers: mode management using exceptional points,” Opt. Lett. 41, 3049–3052 (2016).
[Crossref]

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

Kim, H. R.

K. H. Kim, M. S. Hwang, H. R. Kim, J. H. Choi, Y. S. No, and H. G. Park, “Direct observation of exceptional points in coupled photonic-crystal lasers with asymmetric optical gains,” Nat. Commun. 7, 13893 (2016).
[Crossref]

Kim, J.

Z. J. Wong, Y. L. Xu, J. Kim, K. O’Brien, Y. Wang, L. Feng, and X. Zhang, “Lasing and anti-lasing in a single cavity,” Nat. Photonics 10, 796–801 (2016).
[Crossref]

Kim, K. H.

K. H. Kim, M. S. Hwang, H. R. Kim, J. H. Choi, Y. S. No, and H. G. Park, “Direct observation of exceptional points in coupled photonic-crystal lasers with asymmetric optical gains,” Nat. Commun. 7, 13893 (2016).
[Crossref]

Kip, D.

C. E. Ruter, 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]

Kivshar, Y. S.

S. V. Suchkov, A. A. Sukhorukov, J. H. Huang, S. V. Dmitriev, C. Lee, and Y. S. Kivshar, “Nonlinear switching and solitons in PT-symmetric photonic systems,” Laser Photon. Rev. 10, 177–213 (2016).
[Crossref]

Klang, P.

M. Brandstetter, M. Liertzer, C. Deutsch, P. Klang, J. Schoberl, H. E. Tureci, G. Strasser, K. Unterrainer, and S. Rotter, “Reversing the pump dependence of a laser at an exceptional point,” Nat. Commun. 5, 4034 (2014).
[Crossref]

Konotop, V. V.

V. V. Konotop, J. K. Yang, and D. A. Zezyulin, “Nonlinear waves in PT-symmetric systems,” Rev. Mod. Phys. 88, 035002 (2016).
[Crossref]

Lee, C.

S. V. Suchkov, A. A. Sukhorukov, J. H. Huang, S. V. Dmitriev, C. Lee, and Y. S. Kivshar, “Nonlinear switching and solitons in PT-symmetric photonic systems,” Laser Photon. Rev. 10, 177–213 (2016).
[Crossref]

Li, M.

W. L. Liu, M. Li, R. S. Guzzon, E. J. Norberg, J. S. Parker, M. Z. Lu, L. A. Coldren, and J. P. Yao, “An integrated parity-time symmetric wavelength-tunable single-mode microring laser,” Nat. Commun. 8, 15389 (2017).
[Crossref]

Liertzer, M.

M. Brandstetter, M. Liertzer, C. Deutsch, P. Klang, J. Schoberl, H. E. Tureci, G. Strasser, K. Unterrainer, and S. Rotter, “Reversing the pump dependence of a laser at an exceptional point,” Nat. Commun. 5, 4034 (2014).
[Crossref]

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, 1063–1082 (2014).
[Crossref]

Liu, W. L.

W. L. Liu, M. Li, R. S. Guzzon, E. J. Norberg, J. S. Parker, M. Z. Lu, L. A. Coldren, and J. P. Yao, “An integrated parity-time symmetric wavelength-tunable single-mode microring laser,” Nat. Commun. 8, 15389 (2017).
[Crossref]

Longhi, S.

S. Longhi, “PT-symmetric mode-locking,” Opt. Lett. 41, 4518–4521 (2016).
[Crossref]

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

Lu, M. Z.

W. L. Liu, M. Li, R. S. Guzzon, E. J. Norberg, J. S. Parker, M. Z. Lu, L. A. Coldren, and J. P. Yao, “An integrated parity-time symmetric wavelength-tunable single-mode microring laser,” Nat. Commun. 8, 15389 (2017).
[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]

Makris, K. G.

C. E. Ruter, 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]

Miller, O. D.

Miri, M. A.

H. Hodaei, M. A. Miri, A. U. Hassan, W. E. Hayenga, M. Heinrich, D. N. Christodoulides, and M. Khajavikhan, “Single mode lasing in transversely multi-moded PT-symmetric microring resonators,” Laser Photon. Rev. 10, 494–499 (2016).
[Crossref]

H. Hodaei, A. U. Hassan, W. E. Hayenga, M. A. Miri, D. N. Christodoulides, and M. Khajavikhan, “Dark-state lasers: mode management using exceptional points,” Opt. Lett. 41, 3049–3052 (2016).
[Crossref]

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

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

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, 093902 (2009).
[Crossref]

No, Y. S.

K. H. Kim, M. S. Hwang, H. R. Kim, J. H. Choi, Y. S. No, and H. G. Park, “Direct observation of exceptional points in coupled photonic-crystal lasers with asymmetric optical gains,” Nat. Commun. 7, 13893 (2016).
[Crossref]

Norberg, E. J.

W. L. Liu, M. Li, R. S. Guzzon, E. J. Norberg, J. S. Parker, M. Z. Lu, L. A. Coldren, and J. P. Yao, “An integrated parity-time symmetric wavelength-tunable single-mode microring laser,” Nat. Commun. 8, 15389 (2017).
[Crossref]

O’Brien, K.

Z. J. Wong, Y. L. Xu, J. Kim, K. O’Brien, Y. Wang, L. Feng, and X. Zhang, “Lasing and anti-lasing in a single cavity,” Nat. Photonics 10, 796–801 (2016).
[Crossref]

Onishchukov, G.

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

Park, H. G.

K. H. Kim, M. S. Hwang, H. R. Kim, J. H. Choi, Y. S. No, and H. G. Park, “Direct observation of exceptional points in coupled photonic-crystal lasers with asymmetric optical gains,” Nat. Commun. 7, 13893 (2016).
[Crossref]

Parker, J. S.

W. L. Liu, M. Li, R. S. Guzzon, E. J. Norberg, J. S. Parker, M. Z. Lu, L. A. Coldren, and J. P. Yao, “An integrated parity-time symmetric wavelength-tunable single-mode microring laser,” Nat. Commun. 8, 15389 (2017).
[Crossref]

Peschel, U.

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

Pick, A.

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, 1063–1082 (2014).
[Crossref]

Regensburger, A.

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

Ren, J.

Rodriguez, A. W.

Rotter, S.

M. Brandstetter, M. Liertzer, C. Deutsch, P. Klang, J. Schoberl, H. E. Tureci, G. Strasser, K. Unterrainer, and S. Rotter, “Reversing the pump dependence of a laser at an exceptional point,” Nat. Commun. 5, 4034 (2014).
[Crossref]

Ruter, C. E.

C. E. Ruter, 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]

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, 093902 (2009).
[Crossref]

Schoberl, J.

M. Brandstetter, M. Liertzer, C. Deutsch, P. Klang, J. Schoberl, H. E. Tureci, G. Strasser, K. Unterrainer, and S. Rotter, “Reversing the pump dependence of a laser at an exceptional point,” Nat. Commun. 5, 4034 (2014).
[Crossref]

Segev, M.

C. E. Ruter, 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]

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, 093902 (2009).
[Crossref]

Soljacic, M.

Soltani, M.

Strasser, G.

M. Brandstetter, M. Liertzer, C. Deutsch, P. Klang, J. Schoberl, H. E. Tureci, G. Strasser, K. Unterrainer, and S. Rotter, “Reversing the pump dependence of a laser at an exceptional point,” Nat. Commun. 5, 4034 (2014).
[Crossref]

Suchkov, S. V.

S. V. Suchkov, A. A. Sukhorukov, J. H. Huang, S. V. Dmitriev, C. Lee, and Y. S. Kivshar, “Nonlinear switching and solitons in PT-symmetric photonic systems,” Laser Photon. Rev. 10, 177–213 (2016).
[Crossref]

Sukhorukov, A. A.

S. V. Suchkov, A. A. Sukhorukov, J. H. Huang, S. V. Dmitriev, C. Lee, and Y. S. Kivshar, “Nonlinear switching and solitons in PT-symmetric photonic systems,” Laser Photon. Rev. 10, 177–213 (2016).
[Crossref]

Tureci, H. E.

M. Brandstetter, M. Liertzer, C. Deutsch, P. Klang, J. Schoberl, H. E. Tureci, G. Strasser, K. Unterrainer, and S. Rotter, “Reversing the pump dependence of a laser at an exceptional point,” Nat. Commun. 5, 4034 (2014).
[Crossref]

Unterrainer, K.

M. Brandstetter, M. Liertzer, C. Deutsch, P. Klang, J. Schoberl, H. E. Tureci, G. Strasser, K. Unterrainer, and S. Rotter, “Reversing the pump dependence of a laser at an exceptional point,” Nat. Commun. 5, 4034 (2014).
[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, 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, 093902 (2009).
[Crossref]

Wang, Y.

Z. J. Wong, Y. L. Xu, J. Kim, K. O’Brien, Y. Wang, L. Feng, and X. Zhang, “Lasing and anti-lasing in a single cavity,” Nat. Photonics 10, 796–801 (2016).
[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]

Wittek, S.

H. Hodaei, A. U. Hassan, S. Wittek, H. Garcia-Gracia, R. El Ganainy, D. N. Christodoulides, and M. Khajavikhan, “Enhanced sensitivity at higher-order exceptional points,” Nature 548, 187–191 (2017).
[Crossref]

Wong, Z. J.

Z. J. Wong, Y. L. Xu, J. Kim, K. O’Brien, Y. Wang, L. Feng, and X. Zhang, “Lasing and anti-lasing in a single cavity,” Nat. Photonics 10, 796–801 (2016).
[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]

Xu, Y. L.

Z. J. Wong, Y. L. Xu, J. Kim, K. O’Brien, Y. Wang, L. Feng, and X. Zhang, “Lasing and anti-lasing in a single cavity,” Nat. Photonics 10, 796–801 (2016).
[Crossref]

Yang, J. K.

V. V. Konotop, J. K. Yang, and D. A. Zezyulin, “Nonlinear waves in PT-symmetric systems,” Rev. Mod. Phys. 88, 035002 (2016).
[Crossref]

Yao, J. P.

W. L. Liu, M. Li, R. S. Guzzon, E. J. Norberg, J. S. Parker, M. Z. Lu, L. A. Coldren, and J. P. Yao, “An integrated parity-time symmetric wavelength-tunable single-mode microring laser,” Nat. Commun. 8, 15389 (2017).
[Crossref]

Zezyulin, D. A.

V. V. Konotop, J. K. Yang, and D. A. Zezyulin, “Nonlinear waves in PT-symmetric systems,” Rev. Mod. Phys. 88, 035002 (2016).
[Crossref]

Zhang, X.

Z. J. Wong, Y. L. Xu, J. Kim, K. O’Brien, Y. Wang, L. Feng, and X. Zhang, “Lasing and anti-lasing in a single cavity,” Nat. Photonics 10, 796–801 (2016).
[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]

Zhen, B.

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, 1063–1082 (2014).
[Crossref]

Laser Photon. Rev. (2)

S. V. Suchkov, A. A. Sukhorukov, J. H. Huang, S. V. Dmitriev, C. Lee, and Y. S. Kivshar, “Nonlinear switching and solitons in PT-symmetric photonic systems,” Laser Photon. Rev. 10, 177–213 (2016).
[Crossref]

H. Hodaei, M. A. Miri, A. U. Hassan, W. E. Hayenga, M. Heinrich, D. N. Christodoulides, and M. Khajavikhan, “Single mode lasing in transversely multi-moded PT-symmetric microring resonators,” Laser Photon. Rev. 10, 494–499 (2016).
[Crossref]

Nat. Commun. (4)

W. L. Liu, M. Li, R. S. Guzzon, E. J. Norberg, J. S. Parker, M. Z. Lu, L. A. Coldren, and J. P. Yao, “An integrated parity-time symmetric wavelength-tunable single-mode microring laser,” Nat. Commun. 8, 15389 (2017).
[Crossref]

M. Brandstetter, M. Liertzer, C. Deutsch, P. Klang, J. Schoberl, H. E. Tureci, G. Strasser, K. Unterrainer, and S. Rotter, “Reversing the pump dependence of a laser at an exceptional point,” Nat. Commun. 5, 4034 (2014).
[Crossref]

K. H. Kim, M. S. Hwang, H. R. Kim, J. H. Choi, Y. S. No, and H. G. Park, “Direct observation of exceptional points in coupled photonic-crystal lasers with asymmetric optical gains,” Nat. Commun. 7, 13893 (2016).
[Crossref]

A. K. Jahromi, A. U. Hassan, D. Christodoulides, and A. Abouraddy, “Statistical parity-time-symmetric lasing in an optical fiber network,” Nat. Commun. 8, 1359 (2017).

Nat. Photonics (1)

Z. J. Wong, Y. L. Xu, J. Kim, K. O’Brien, Y. Wang, L. Feng, and X. Zhang, “Lasing and anti-lasing in a single cavity,” Nat. Photonics 10, 796–801 (2016).
[Crossref]

Nat. Phys. (1)

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

Nature (2)

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

H. Hodaei, A. U. Hassan, S. Wittek, H. Garcia-Gracia, R. El Ganainy, D. N. Christodoulides, and M. Khajavikhan, “Enhanced sensitivity at higher-order exceptional points,” Nature 548, 187–191 (2017).
[Crossref]

Opt. Express (1)

Opt. Lett. (3)

Phys. Rev. A (1)

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

Phys. Rev. Lett. (1)

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

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, 1063–1082 (2014).
[Crossref]

Rev. Mod. Phys. (1)

V. V. Konotop, J. K. Yang, and D. A. Zezyulin, “Nonlinear waves in PT-symmetric systems,” Rev. Mod. Phys. 88, 035002 (2016).
[Crossref]

Sci. Rep. (1)

L. Ge and R. El Ganainy, “Nonlinear modal interactions in parity-time (PT) symmetric lasers,” Sci. Rep. 6, 24889 (2016).
[Crossref]

Science (2)

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

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

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Figures (6)

Fig. 1.
Fig. 1. Schematic of the proposed PT-symmetric fiber-ring laser, composed of two coupled fiber-ring cavities with gain and losses. The coupling between cavities is controlled by means of phase shifts φ 1 and φ 2 . Arrows indicate the direction of propagation.
Fig. 2.
Fig. 2. (a) Ratio of two linear mode eigenvalues, | μ + / μ | and (b) the relative phase of the eigenmode amplitudes arg ( u 1 / u 2 ) μ + = Re ( ν + ) versus the difference of phases and gain/losses in two fiber cavities. White dotted lines indicate the PT-breaking boundary. (c), (d) The absolute eigenvalues shown with solid ( | μ + | ) and dashed ( | μ | ) lines versus the gain coefficient for fixed losses ( g 2 = 0.7 ) and different phases (c)  Δ φ = 1.5 and (d)  Δ φ = 1 . Horizontal dotted line marks the level | μ | = 1 corresponding to stationary modes with balanced gain and losses.
Fig. 3.
Fig. 3. (a) Stationary regimes of laser operation with nonlinear gain saturation: no lasing (white background), pair of PT-symmetric laser modes (grey shading), or one mode in PT-broken regime (yellow shading). (b), (c) Characteristic mode amplification versus power for points A and B marked in (a) corresponding to different lasing regimes. Solid circles mark stable and the open circle marks unstable regimes with balanced gain and loss (zero mode amplification). Background shading marks PT-symmetric and broken regimes. Saturable gain parameter g h = 0.23 (1 dB).
Fig. 4.
Fig. 4. Power dependencies in stationary lasing regimes. (a) A ratio of power generated in passive and active cavities, P 2 / P 1 , is unity in PT-symmetrical region and less than unity in PT-broken area. (b), (c) Dependence of the lasing power in two cavities on the gain in PT-symmetric and PT-broken regimes corresponding to different phase shift Δ φ = 1.5 and 0.8, respectively, both shown with dashed lines in plot (a).
Fig. 5.
Fig. 5. (a) Dependence of generated power P 1 on phase shift Δ φ and loss g 2 at fixed gain g 1 = 1.0 has non-trivial form resulting from PT transition. In the PT-symmetric area, the higher are the losses | g 2 | , the lower is the lasing power as it should be in a conventional laser, whereas in the case of PT-broken regime, the generation power increases with the increase of losses. Panels (b)–(d) are cross sections of a 3D surface indicated on panel (a) over dotted lines.
Fig. 6.
Fig. 6. Dynamical properties of a PT-symmetric fiber laser. (a) Two trapping regions shown with shading according to Eq. (18), shown in the plane of relative phases and powers in two fiber cavities. Laser dynamics is confined to one region according to the initial conditions. Solid and dashed lines indicate possible stationary lasing states: PT-symmetric—vertical lines at P 2 / P 1 = 1 , and PT-broken—horizontal lines at relative phases { 0 , ± π } . (b), (c) Dynamical evolution demonstrating bi-stability on the PT-symmetric regime. Shown are relative (b) phases and (c) powers, which converge to one of two stationary states marked with solid circles in (a). Parameters are g 1 = 2.3 , g 2 = 0.7 , and Δ φ = 2.95 .

Equations (18)

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L = ( e g 1 / 2 0 0 e g 2 / 2 ) ( 0 1 1 0 ) ( 1 / 2 i / 2 i / 2 1 / 2 ) ( e i φ 1 0 0 e i φ 2 ) ( 1 / 2 i / 2 i / 2 1 / 2 ) ( e g 1 / 2 0 0 e g 2 / 2 ) ,
L = i e g ˜ + i φ ˜ ( e Δ g / 2 cos ( Δ φ 2 ) sin ( Δ φ 2 ) sin ( Δ φ 2 ) e Δ g / 2 cos ( Δ φ 2 ) ) ,
PT e g ˜ i φ ˜ L = e g ˜ i φ ˜ LPT .
P = ( 0 1 1 0 ) ,
PL ( g 1 , g 2 , φ 1 , φ 2 ) P 1 = L ( g 2 , g 1 , φ 2 , φ 1 ) .
TL = ( L * ) 1 T .
μ ± = i e g ˜ + i φ ˜ μ ˜ ± , μ ˜ ± = 1 / μ ˜ = cosh ( Δ g / 2 ) cos ( Δ φ / 2 ) ± cosh 2 ( Δ g / 2 ) cos 2 ( Δ φ / 2 ) 1
( u 1 u 2 ) = ( exp ( i ν ± / 2 ) exp ( i ν ± / 2 ) ) ,
ν ± = i log [ μ ˜ ± e Δ g / 2 cos ( Δ φ 2 ) sin ( Δ φ 2 ) ] .
Δ g PT = 2 cosh 1 [ 1 cos ( Δ φ / 2 ) ] ,
ν + = ν ,
cos ( Δ φ / 2 ) < 1 / cosh ( | g 2 | ) ,
cos ( Δ φ / 2 ) > 1 / cosh ( | g 2 | )
g 1 = g 0 1 + P ˜ 1 g h ,
J = Im ( u 1 u 2 * ) = | u 1 u 2 | sin [ arg ( u 1 ) arg ( u 2 ) ] .
J e 2 g ˜ J .
sign ( J ) = sign { sin [ arg ( u 1 ) arg ( u 2 ) ] } = const .
0 < arg ( u 1 ) arg ( u 2 ) < π , π < arg ( u 1 ) arg ( u 2 ) < 0 .

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