Abstract

We perform a theoretical study of the nonlinear dynamics of nonlinear optical isolator devices based on coupled microcavities with gain and loss. This reveals a correspondence between the boundary of asymptotic stability in the nonlinear regime, where gain saturation is present, and the PT -breaking transition in the underlying linear system. For zero detuning and weak input intensity, the onset of optical isolation can be rigorously derived, and corresponds precisely to the transition into the PT -broken phase of the linear system. When the couplings to the external ports are unequal, the isolation ratio exhibits an abrupt jump at the transition point, whose magnitude is given by the ratio of the couplings. This phenomenon could be exploited to realize an actively controlled nonlinear optical isolator, in which strong optical isolation can be turned on and off by tiny variations in the inter-resonator separation.

© 2016 Optical Society of America

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References

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2015 (4)

Y. Shi, Z. Yu, and S. Fan, “Limitations of nonlinear optical isolators due to dynamic reciprocity,” Nature Phot. 9, 388–392 (2015).
[Crossref]

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

X.-Y. Lü, H. Jing, J.-Y. Ma, and Y. Wu, “PT-symmetry-breaking chaos in optomechanics,” Phys. Rev. Lett. 114, 253601 (2015).
[Crossref] [PubMed]

A. U. Hassan, H. Hodaei, M.-A. Miri, M. Khajavikhan, and D. N. Christodoulides, “Nonlinear reversal of the PT-symmetric phase transition in a system of coupled semiconductor microring resonators,” Phys. Rev. A 92, 063807 (2015).
[Crossref]

2014 (3)

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, 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, “Parity-time-symmetric whispering-gallery microcavities,” Nature Phys. 10, 394–398 (2014).
[Crossref]

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

2013 (2)

D. Jalas, A. Petrov, M. Eich, W. Freude, S. Fan, Z. Yu, and H. Renner, “What is and what is not an optical isolator,” Nature Phot. 7, 579–582 (2013).
[Crossref]

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

2012 (3)

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] [PubMed]

W. D. Heiss, “The physics of exceptional points,” J. Phys. A: Math. Theor. 45, 444016 (2012).
[Crossref]

H. Ramezani, T. Kottos, V. Kovanis., and D. N. Christodoulides, “Exceptional-point dynamics in photonic honeycomb lattices with PT symmetry,” Phys. Rev. A 85, 013818 (2012).
[Crossref]

2011 (1)

Y. D. Chong, L. Ge, and A. D. Stone, “PT-symmetry breaking and laser-absorber modes in optical scattering systems,” Phys. Rev. Lett. 196, 093902 (2011).
[Crossref]

2010 (5)

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

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

A. E. Miroshnichenko, E. Brasselet, and Y. S. Kivshar, “Reversible optical nonreciprocity in periodic structures with liquid crystals,” Appl. Phys. Lett. 96, 063302 (2010).
[Crossref]

C. G. Poulton, R. Pant, A. Byrnes, S. Fan, M. J. Steel, and B. J. Eggleton, “Design for broadband on-chip isolator using stimulated Brillouin scattering in dispersion-engineered chalcogenide waveguides,” Opt. Ex. 20, 21235–21246 (2010).
[Crossref]

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “PT-symmetric optical lattices,” Phys. Rev. A 81, 063807 (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] [PubMed]

2008 (5)

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam dynamics in PT-symmetric optical lattices,” Phys. Rev. Lett. 100, 103904 (2008).
[Crossref]

Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, “Optical solitons in PT periodic potentials,” Phys. Rev. Lett. 100, 030402 (2008)
[Crossref] [PubMed]

Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, “Analytical solutions to a class of nonlinear Schrdinger equations with PT-like potentials,” J. Phys. A 41, 244019 (2008).
[Crossref]

M. Krause, H. Renner, and E. Brinkmeyer, “Optical isolation in silicon waveguides based on nonreciprocal Raman amplification,” Electron. Lett. 44, 691–693 (2008).
[Crossref]

S. Zhang, D. A. Genov, Y. Wang, M. Liu, and X. Zhang, “Plasmon-induced transparency in metamaterials,” Phys. Rev. Lett. 101, 047401 (2008).
[Crossref] [PubMed]

2007 (2)

R. E. Hamam, A. Karalis, J. D. Joannopoulos, and M. Soljacić, “Coupled-mode theory for general free-space resonant scattering of waves,” Phys. Rev. A 75, 053801 (2007).
[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]

2005 (2)

2004 (2)

M. Soljacić and J. D. Joannopoulos, “Enhancement of nonlinear effects using photonic crystals,” Nature Mat. 3, 211–219 (2004).
[Crossref]

W. Suh, Z. Wang, and S. Fan, “Temporal coupled-mode theory and the presence of non-orthogonal modes in lossless multimode cavities,” IEEE J. Quantum Elect. 40, 1511–1518 (2004).
[Crossref]

2002 (1)

C. M. Bender, M. V. Berry, and A. Mandilara, “Generalized PT symmetry and real spectra,” J. Phys. A 35, L467–L471 (2002).
[Crossref]

2001 (1)

K. Gallo, G. Assanto, K. R. Parameswaran, and M. M. Fejer, “All optical diode in a periodically poled lithium niobate waveguide,” Appl. Phys. Lett. 79, 314–316 (2001).
[Crossref]

1998 (1)

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

Aimez, V.

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

Almeida, V. R.

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

Assanto, G.

K. Gallo, G. Assanto, K. R. Parameswaran, and M. M. Fejer, “All optical diode in a periodically poled lithium niobate waveguide,” Appl. Phys. Lett. 79, 314–316 (2001).
[Crossref]

Bahlmann, N.

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, 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, “Parity-time-symmetric whispering-gallery microcavities,” Nature Phys. 10, 394–398 (2014).
[Crossref]

C. M. Bender, M. V. Berry, and A. Mandilara, “Generalized PT symmetry and real spectra,” J. Phys. A 35, L467–L471 (2002).
[Crossref]

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

Berry, M. V.

C. M. Bender, M. V. Berry, and A. Mandilara, “Generalized PT symmetry and real spectra,” J. Phys. A 35, L467–L471 (2002).
[Crossref]

Bersch, C.

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

Boettcher, S.

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

Brasselet, E.

A. E. Miroshnichenko, E. Brasselet, and Y. S. Kivshar, “Reversible optical nonreciprocity in periodic structures with liquid crystals,” Appl. Phys. Lett. 96, 063302 (2010).
[Crossref]

Brinkmeyer, E.

M. Krause, H. Renner, and E. Brinkmeyer, “Optical isolation in silicon waveguides based on nonreciprocal Raman amplification,” Electron. Lett. 44, 691–693 (2008).
[Crossref]

Byrnes, A.

C. G. Poulton, R. Pant, A. Byrnes, S. Fan, M. J. Steel, and B. J. Eggleton, “Design for broadband on-chip isolator using stimulated Brillouin scattering in dispersion-engineered chalcogenide waveguides,” Opt. Ex. 20, 21235–21246 (2010).
[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,” Nature Phot. 8, 524–529 (2014).
[Crossref]

Chen, Y. F.

L. Feng, Y. L. Xu, W. S. Fegadolli, M. H. Lu, J. E. Oliveira, V. R. Almeida, Y. F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nature Mat. 12, 108–113 (2013).
[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. 196, 093902 (2011).
[Crossref]

Christodoulides, D. N.

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

A. U. Hassan, H. Hodaei, M.-A. Miri, M. Khajavikhan, and D. N. Christodoulides, “Nonlinear reversal of the PT-symmetric phase transition in a system of coupled semiconductor microring resonators,” Phys. Rev. A 92, 063807 (2015).
[Crossref]

H. Ramezani, T. Kottos, V. Kovanis., and D. N. Christodoulides, “Exceptional-point dynamics in photonic honeycomb lattices with PT symmetry,” Phys. Rev. A 85, 013818 (2012).
[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] [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,” Nature. Phys. 6, 192–195 (2010).
[Crossref]

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “PT-symmetric optical lattices,” Phys. Rev. A 81, 063807 (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] [PubMed]

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam dynamics in PT-symmetric optical lattices,” Phys. Rev. Lett. 100, 103904 (2008).
[Crossref]

Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, “Optical solitons in PT periodic potentials,” Phys. Rev. Lett. 100, 030402 (2008)
[Crossref] [PubMed]

Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, “Analytical solutions to a class of nonlinear Schrdinger equations with PT-like potentials,” J. Phys. A 41, 244019 (2008).
[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]

Dtsch, H.

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] [PubMed]

Eggleton, B. J.

C. G. Poulton, R. Pant, A. Byrnes, S. Fan, M. J. Steel, and B. J. Eggleton, “Design for broadband on-chip isolator using stimulated Brillouin scattering in dispersion-engineered chalcogenide waveguides,” Opt. Ex. 20, 21235–21246 (2010).
[Crossref]

Eich, M.

D. Jalas, A. Petrov, M. Eich, W. Freude, S. Fan, Z. Yu, and H. Renner, “What is and what is not an optical isolator,” Nature Phot. 7, 579–582 (2013).
[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,” Nature. Phys. 6, 192–195 (2010).
[Crossref]

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “PT-symmetric optical lattices,” Phys. Rev. A 81, 063807 (2010).
[Crossref]

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam dynamics in PT-symmetric optical lattices,” Phys. Rev. Lett. 100, 103904 (2008).
[Crossref]

Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, “Analytical solutions to a class of nonlinear Schrdinger equations with PT-like potentials,” J. Phys. A 41, 244019 (2008).
[Crossref]

Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, “Optical solitons in PT periodic potentials,” Phys. Rev. Lett. 100, 030402 (2008)
[Crossref] [PubMed]

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]

Fan, S.

Y. Shi, Z. Yu, and S. Fan, “Limitations of nonlinear optical isolators due to dynamic reciprocity,” Nature Phot. 9, 388–392 (2015).
[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, “Parity-time-symmetric whispering-gallery microcavities,” Nature Phys. 10, 394–398 (2014).
[Crossref]

D. Jalas, A. Petrov, M. Eich, W. Freude, S. Fan, Z. Yu, and H. Renner, “What is and what is not an optical isolator,” Nature Phot. 7, 579–582 (2013).
[Crossref]

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Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, “Optical solitons in PT periodic potentials,” Phys. Rev. Lett. 100, 030402 (2008)
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B. Peng, S. K. Özdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nature Phys. 10, 394–398 (2014).
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D. Jalas, A. Petrov, M. Eich, W. Freude, S. Fan, Z. Yu, and H. Renner, “What is and what is not an optical isolator,” Nature Phot. 7, 579–582 (2013).
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C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nature. Phys. 6, 192–195 (2010).
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[Crossref]

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C. G. Poulton, R. Pant, A. Byrnes, S. Fan, M. J. Steel, and B. J. Eggleton, “Design for broadband on-chip isolator using stimulated Brillouin scattering in dispersion-engineered chalcogenide waveguides,” Opt. Ex. 20, 21235–21246 (2010).
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S. Zhang, D. A. Genov, Y. Wang, M. Liu, and X. Zhang, “Plasmon-induced transparency in metamaterials,” Phys. Rev. Lett. 101, 047401 (2008).
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M. Wimmer, A. Regensburger, M. A. Miri, C. Bersch, D. N. Christodoulides, and U. Peschel, “Observation of optical solitons in PT-symmetric lattices,” Nature Comm. 6, 7782 (2015).
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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,” Nature Phot. 8, 524–529 (2014).
[Crossref]

Xu, Y. L.

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

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,” Nature Phot. 8, 524–529 (2014).
[Crossref]

Yang, 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, “Parity-time-symmetric whispering-gallery microcavities,” Nature Phys. 10, 394–398 (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, 328–332 (2014).
[Crossref] [PubMed]

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, 328–332 (2014).
[Crossref] [PubMed]

Yu, Z.

Y. Shi, Z. Yu, and S. Fan, “Limitations of nonlinear optical isolators due to dynamic reciprocity,” Nature Phot. 9, 388–392 (2015).
[Crossref]

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S. Zhang, D. A. Genov, Y. Wang, M. Liu, and X. Zhang, “Plasmon-induced transparency in metamaterials,” Phys. Rev. Lett. 101, 047401 (2008).
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S. Zhang, D. A. Genov, Y. Wang, M. Liu, and X. Zhang, “Plasmon-induced transparency in metamaterials,” Phys. Rev. Lett. 101, 047401 (2008).
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Appl. Phys. Lett. (2)

K. Gallo, G. Assanto, K. R. Parameswaran, and M. M. Fejer, “All optical diode in a periodically poled lithium niobate waveguide,” Appl. Phys. Lett. 79, 314–316 (2001).
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Electron. Lett. (1)

M. Krause, H. Renner, and E. Brinkmeyer, “Optical isolation in silicon waveguides based on nonreciprocal Raman amplification,” Electron. Lett. 44, 691–693 (2008).
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J. Opt. Soc. Am. B (2)

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Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, “Analytical solutions to a class of nonlinear Schrdinger equations with PT-like potentials,” J. Phys. A 41, 244019 (2008).
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C. M. Bender, M. V. Berry, and A. Mandilara, “Generalized PT symmetry and real spectra,” J. Phys. A 35, L467–L471 (2002).
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W. D. Heiss, “The physics of exceptional points,” J. Phys. A: Math. Theor. 45, 444016 (2012).
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Nature (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).
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Nature Comm. (1)

M. Wimmer, A. Regensburger, M. A. Miri, C. Bersch, D. N. Christodoulides, and U. Peschel, “Observation of optical solitons in PT-symmetric lattices,” Nature Comm. 6, 7782 (2015).
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Nature Mat. (2)

L. Feng, Y. L. Xu, W. S. Fegadolli, M. H. Lu, J. E. Oliveira, V. R. Almeida, Y. F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nature Mat. 12, 108–113 (2013).
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M. Soljacić and J. D. Joannopoulos, “Enhancement of nonlinear effects using photonic crystals,” Nature Mat. 3, 211–219 (2004).
[Crossref]

Nature Phot. (3)

D. Jalas, A. Petrov, M. Eich, W. Freude, S. Fan, Z. Yu, and H. Renner, “What is and what is not an optical isolator,” Nature Phot. 7, 579–582 (2013).
[Crossref]

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

Y. Shi, Z. Yu, and S. Fan, “Limitations of nonlinear optical isolators due to dynamic reciprocity,” Nature Phot. 9, 388–392 (2015).
[Crossref]

Nature Phys. (1)

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

Nature. Phys. (1)

C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nature. Phys. 6, 192–195 (2010).
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Phys. Rev. Lett. (7)

S. Zhang, D. A. Genov, Y. Wang, M. Liu, and X. Zhang, “Plasmon-induced transparency in metamaterials,” Phys. Rev. Lett. 101, 047401 (2008).
[Crossref] [PubMed]

X.-Y. Lü, H. Jing, J.-Y. Ma, and Y. Wu, “PT-symmetry-breaking chaos in optomechanics,” Phys. Rev. Lett. 114, 253601 (2015).
[Crossref] [PubMed]

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Science (1)

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, 328–332 (2014).
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Figures (8)

Fig. 1
Fig. 1

(a) Schematic of a resonator with saturable gain coupled to a lossy resonator, with both resonators coupled to optical fiber ports. Solid arrows indicate forward transmission (port 1 → 4), and dashed arrows indicate backward transmission (port 4 → 1). (b)–(d) Transmission characteristics in the linear (non-gain-saturated) regime, when the gain and loss are PT symmetric (g = γ = 0.4). Here, we plot the intensity in the active resonator (|a1|2) under forward transmission (solid lines), and in the passive resonator (|a2|2) under backward transmission (dashes), versus the frequency detuning. These resonator intensities are proportional to the forward and backward transmission intensities via Eqs. (3) and (6). In the PT -symmetric phase μ > γ, there are two transmission peaks; in the PT -broken phase μ < γ, these merge into a single peak.

Fig. 2
Fig. 2

(a)–(b) Domains in which the nonlinear coupled-mode equations have multiple steady-state solutions, for forward (a) and backward (b) transmission. Here, we show the parameter space defined by the frequency detuning Δω and inter-resonator coupling μ, with fixed γ = 0.4, sin = 0.5, and as = 3; symbols indicate the points in the parameter space corresponding to the curves in (c) and (d). Within the small-Δω region bounded by the red curves, the highest-intensity (or only) solution is asymptotically stable. (c)–(d) Plots showing the emergence of multiple solutions at several values of μ, fixing Δω = 0. The horizontal axis is the normalized intensity in the gain resonator, |a1/as|2; the vertical axis is the left-hand side of the cubic Eq. (14), and its counterpart for backward transmission; the steady-state coupled-mode equations are satisfied when the curves cross zero.

Fig. 3
Fig. 3

Isolation ratio versus μ/γ at zero frequency detuning (Δω = 0), for (a) weak inputs sin = 0.15 and as = 3, and (b) strong inputs regime sin = 9 and as = 3, using several choices of γ. In the weak-input regime, the isolation ratio is mainly determined by the PT -breaking parameter μ/γ. The system becomes reciprocal for μ/γ > 1, corresponding to the PT -symmetric phase of the linear system. (c) Close-up of the isolation ratio behavior in the weak-input regime, showing the kink in the dependence on μ/γ at the PT transition point μ/γ = 1. Circles show exact numerical solutions of the coupled-mode equations, and the solid curve shows the analytic approximations of Eqs. (22)(23).

Fig. 4
Fig. 4

Isolation ratio versus μ/γ for different microcavity-waveguide coupling rates. The system parameters are Δω = 0, sin = 0.03, as = 3, γ = 1, and g0 = 4γ. Thin solid lines show the analytic approximation in the weak-input limit ( s in γ a s), given by Eq. (27).

Fig. 5
Fig. 5

Transmittance Tb of a back-propagating wave, versus the normalized backward incident power |sb|2/γ|as|2. A forward-propagating wave with |sf|2/γ|as|2 = 10−4 is simultaneously present; the other parameters are as = 3, γ = κ1 = 1, and κ2 = 0.01γ. Note that Tb can exceed unity because of the presence of gain in the system; this feature can be suppressed if desired by adding loss to the waveguide leads.

Fig. 6
Fig. 6

(a) Lyapunov exponents for the highest-intensity steady-state solution under forward transmission, versus detuning Δω. Results are shown for μ ∈ {0.1, 0.2, 0.3, 0.4}. The other model parameters are fixed at γ = 0.4 and sin = 0.5. (b) Bounds of the asymptotic stability region under forward transmission, for several values of the amplitude sin, with fixed γ = 0.4 and as = 3. The bandwidth of the asymptotic stability region increases with μ, and diverges at μ = γ, which is the PT transition point of the linear system. (c) Bounds of the asymptotic stability region under backward transmission, with the same model parameters.

Fig. 7
Fig. 7

Time-dependent mode amplitudes under (a) forward transmission for Δω = 0 (three-solution domain), (b) forward transmission for Δω = 0.2 (one-solution domain), (c) backward transmission for Δω = 0 (three-solution domain), and (d) backward transmission for Δω = 0.2 (one-solution domain). The other model parameters are μ = 0.1, γ = 0.4, as = 3.0, and sin = 0.5. We start each simulation with initial conditions perturbed from a steady-state solution by δa1 = δa2 = 0.001. In the three-solution domain, perturbing the two lower-intensity solutions causes the system to evolve to the highest-intensity steady-state, which is asymptotically stable.

Fig. 8
Fig. 8

Beating amplitudes Δ|a1,2|2/|as|2, defined as the difference of the maximum and minimum values of |a1,2(t)|2/|as|2 over time t, versus the PT -breaking parameter μ/γ. The amplitudes a1,2(t) are solved numerically using the full time-dependent coupled-mode equations, using Δω = 0.5, γ = 0.4, sin = 0.5, and as = 3.

Equations (48)

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d a 1 d t = ( i Δ ω 1 + g ) a 1 i μ a 2
d a 2 d t = ( i Δ ω 2 γ ) a 2 i μ a 1 + κ 2 s in
I F = κ 1 | a 1 | 2 .
d a 1 d t = ( i Δ ω 1 + g ) a 1 i μ a 2 + κ 1 s in
d a 2 d t = ( i Δ ω 2 γ ) a 2 i μ a 1
I B = κ 2 | a 2 | 2 ,
g = 1 2 ( g γ 1 κ 1 )
γ = 1 2 ( γ 2 + κ 2 ) ,
κ 1 = κ 2 = γ 1 = γ 2 ,
Δ ω 1 = Δ ω 2 Δ ω ,
g = g 0 1 + | a 1 / a s | 2 .
R T F T B = I F ( I in ) I B ( I in ) ,
Δ ω = i g γ 2 ± μ 2 γ g ( g γ 2 ) 2 .
| α | 2 x 3 + ( 2 | α 1 | 2 2 β ) x 2 + ( | α 2 | 2 2 β ) x β = 0 ,
α = ( i Δ ω + γ ) 2 + μ 2 γ ( i Δ ω + γ ) , β = μ 2 Δ ω 2 + γ 2 1 γ | s in a s | 2 , x = | a 1 a s | 2 .
β = 1 γ | s in a s | 2 .
α = 1 + ( μ γ ) 2
β = | s in / a s | 2 γ × { ( μ γ ) 2 , ( Forward ) 1 , ( Backward ) .
s in γ a s .
R = I F I B = ( μ / γ ) 2 x F x B ,
x ( x 1 ( μ / γ ) 2 1 + ( μ / γ ) 2 ) 2 0 .
R ( μ / γ ) 2 for μ / γ < 1 , s in γ a s .
R 1 for μ / γ > 1 , s in γ a s .
γ 1 + κ 1 = γ 2 + κ 2 = 2 γ .
g = 2 γ 1 + | a 1 / a s | 2 γ ,
β κ 2 γ β ( Forward ) β κ 1 γ β ( Backward ) .
R = ( κ 1 / κ 2 ) ( μ / γ ) 2 x F / x { ( κ 1 / κ 2 ) ( μ / γ ) 2 1 for for μ / γ < 1 μ / γ > 1.
d a 1 d t = g a 1 i μ a 2
d a 2 d t = γ a 2 i μ a 1 + κ 2 s f
d a 1 d t = g a 1 i μ a 2 + κ 1 s b
d a 2 d t = γ a 2 i μ a 1 .
g = 2 γ 1 + | a 1 / a s | 2 + | a 1 / a s | 2 γ .
T b = κ 2 | a 2 | 2 | s b | 2 .
d a 1 d t = ( i Δ ω γ + κ 1 2 + 2 γ 1 + | a 1 / a s | 2 ) a 1 ( t ) i μ a 2 ( t ) ,
d a 2 d t = ( i Δ ω γ + κ 2 2 ) a 2 ( t ) i μ a 1 ( t ) + κ 2 s in .
a 1 ( t ) = a ˜ 1 + ρ 1 ( t )
a 2 ( t ) = a ˜ 2 + ρ 2 ( t ) ,
2 γ 1 + | a s | 2 ( a ˜ 1 + ρ 1 ) ( a ˜ 1 * + ρ 1 * ) 2 γ 1 + | a s | 2 ( | a ˜ 1 | 2 + a ˜ 1 ρ 1 * + a ˜ 1 * ρ 1 )
2 γ 1 + | a ˜ 1 / a s | 2 [ 1 a ˜ 1 ρ 1 * ( t ) + a ˜ 1 * ρ 1 ( t ) | a s | 2 + | a ˜ 1 | 2 ] .
d ρ 1 d t = A ρ 1 ( t ) + B ρ 1 * ( t ) + C ρ 2 ( t )
d ρ 2 d t = C ρ 1 ( t ) + D ρ 2 ( t ) ,
A = i Δ ω γ + κ 1 2 + 2 γ 1 + | a ˜ 1 / a s | 2 2 γ | a ˜ 1 / a s | 2 ( 1 + | a ˜ 1 / a s | 2 ) 2
B = 2 γ a ˜ 1 2 / | a s | 2 ( 1 + | a ˜ 1 / a s | 2 ) 2
C = i μ
D = i Δ ω γ + κ 2 2
ρ 1 ( t ) = u 1 e λ t + v 1 * e λ * t
ρ 2 ( t ) = u 2 e λ t + v 2 * e λ * t .
[ A B C 0 B * A * 0 C * C 0 D 0 0 C * 0 D * ] [ u 1 v 1 u 2 v 2 ] = λ [ u 1 v 1 u 2 v 2 ] .

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