Abstract

We show that Fano resonances created by two 𝒫𝒯 -symmetric nonlinear micro-resonators coupled to a waveguide, have line-shape and resonance position that depends on the direction of the incident light. We utilize these features in order to induce asymmetric transport, up to 47 dBs, in the optical C-window. Our theoretical proposal requires low input power and does not compromise the power or frequency characteristics of the output signal.

© 2014 Optical Society of America

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  1. Z. Yu, S. Fan, “Complete optical isolation created by indirect interband photonic transitions,” Nat. Photonics 3, 91–94 (2009).
    [CrossRef]
  2. M.S. Kang, A. Butsch, P.S.J. Russell, “Reconfigurable light-driven opto-acoustic isolators in photonic crystal fibre,” Nat. Photonics 5, 549–553 (2011).
    [CrossRef]
  3. K. Gallo, G. Assanto, “All-optical diode based on second-harmonic generation in an asymmetric waveguide,” JOSA B 16, 267–269 (1999).
    [CrossRef]
  4. M. Scalora, J.P. Dowling, C.M. Bowden, M.J. Bloemer, “The photonic band edge optical diode,” J. Appl. Phys. 76, 2023–2026 (1994).
    [CrossRef]
  5. S. Lepri, G. Casati, “Asymmetric wave propagation in nonlinear systems,” Phys. Rev. Lett. 106, 164101 (2011).
    [CrossRef] [PubMed]
  6. Y. Xu, A. E. Miroshnichenko, “Reconfigurable nonreciprocity with nonlinear Fano diode,” eprint arXiv:1311.2533 (2013).
  7. L. Fan, J. Wang, L.T. Varghese, H. Shen, B. Niu, Y. Xuan, A. M. Weiner, M. Qi, “An all-silicon passive optical diode,” Science 335, 447–450 (2012).
    [CrossRef]
  8. B. Peng, S. K. Ozdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, L. Yang, “Nonreciprocal light transmission in parity-time-symmetric whispering-gallery microcavities,” arXiv:1308.4564 (2013).
  9. K. J. Vahala, “Optical microcavities,” Nature (London) 424, 839–846 (2003).
    [CrossRef]
  10. N. Bender, S. Factor, J.D. Bodyfelt, H. Ramezani, D. N. Christodoulides, F. Ellis, T. Kottos, “Observation of asymmetric transport in structures with active nonlinearities,” Phys. Rev. Lett. 110, 234101 (2013).
    [CrossRef]
  11. A. E. Miroshnichenko, S. Flach, Y.S. Kivshar, “Fano resonances in nanoscale structures,” Rev. Mod. Phys. 82, 2257–2298 (2010).
    [CrossRef]
  12. K. Srinivasan, M. Borselli, O. Painter, A. Stintz, S. Krishna, “Cavity Q, mode volume, and lasing threshold in small diameter AlGaAs microdisks with embedded quantum dots,” Opt. Express 14, 1094–1105 (2006).
    [CrossRef] [PubMed]
  13. C. M. Bender, “Making sense of non-hermitian hamiltonians,” Rep. Prog. Phys. 70, 947–1018 (2007).
    [CrossRef]
  14. The 𝒫𝒯-symmetric phase transition can be also achieved by manipulating the coupling strength between the gain an loss elements while keeping the balanced gain and loss parameter constant. Specifically it can be shown that decreasing the coupling strength is equivalent to increasing the gain and loss parameter.
  15. K. G. Makris, R. El-Ganainy, D.N. Christodoulides, “Beam dynamics in PT symmetric optical lattices,” Phys. Rev. Lett. 100, 103904 (2008).
    [CrossRef] [PubMed]
  16. C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, D. Kip, “Observation of parity-time symmetry in optics,” Nat. Phys. 6, 192–195 (2010).
    [CrossRef]
  17. A. Regensburger, C. Bersch, M.A. Miri, G. Onishchukov, D. N. Christodoulides, U. Peschel, “Parity - time synthetic photonic lattices,” Nature 488, 167–171 (2012).
    [CrossRef] [PubMed]
  18. S. Longhi, “PT-symmetric laser absorber,” Phys. Rev. A 82, 031801 (2010).
    [CrossRef]
  19. Y.D. Chong, L. Ge, A.D. Stone, “PT-symmetry breaking and laser-absorber modes in optical scattering systems,” Phys. Rev. Lett. 106, 093902 (2011).
    [CrossRef] [PubMed]
  20. H. Ramezani, T. Kottos, R. El-Ganainy, D. N. Christodoulides, “Unidirectional nonlinear PT-symmetric optical structures,” Phys. Rev. A 82, 043803 (2010).
    [CrossRef]
  21. J. Schindler, Z. Lin, J. M. Lee, H. Ramezani, F. M. Ellis, T. Kottos, “PT-symmetric electronics,” J. Phys. A -Math and Theor. 45, 444029 (2012).
    [CrossRef]
  22. H. Ramezani, J. Schindler, F.M. Ellis, U. Günther, T. Kottos, “Bypassing the bandwidth theorem with PT symmetry,” Phys. Rev. A 85, 062122 (2012).
    [CrossRef]
  23. Z. Lin, J. Schindler, F.M. Ellis, T. Kottos, “Experimental observation of the dual behavior of PT-symmetric scattering,” Phys. Rev. A 85, 050101 (2012).
    [CrossRef]
  24. Y-F Xiao, M. Li, Y-C Liu, Y. Li, X. Sun, Q. Gong, “Asymmetric Fano resonance analysis in indirectly coupled microresonators,” Phys. Rev. A 82, 065804 (2010).
    [CrossRef]
  25. B-B Li, Y-F Xiao, C-L Zou, X-F Jiang, Y-C Liu, F-W Sun, Y Li, Q Gong, “Experimental controlling of Fano resonance in indirectly coupled whispering-gallery microresonators”, Applied Phys. Lett. 100, 021108 (2012).
    [CrossRef]
  26. Q. Xu, S. Sandhu, M. L. Povinelli, J. Shakya, S. Fan, M. Lipson, “Experimental realization of an on-chip all-optical analogue to electromagnetically induced transparency,” Phys. Rev. Lett. 96, 123901 (2006).
    [CrossRef] [PubMed]
  27. K. Totsuka, N. Kobayashi, M. Tomita, “Slow light in coupled-resonator-induced transparency,” Phys. Rev. Lett. 98, 213904 (2007).
    [CrossRef] [PubMed]
  28. D. D. Smith, H. Chang, K. A. Fuller, A. T. Rosenberger, R. W. Boyd, “Coupled-resonator-induced transparency,” Phys. Rev. A 69, 063804 (2004).
    [CrossRef]
  29. A. E. Miroshnichenko, B.A. Malomed, Y.S. Kivshar, “Nonlinearly PT-symmetric systems: spontaneous symmetry breaking and transmission resonances,” Phys. Rev. A 84, 012123 (2011).
    [CrossRef]
  30. J. Ctyroky, I. Richter, V. Sinor, “Dual resonance in a waveguide-coupled ring microresonator,” Optic. Quantum Electron. 38, 781–797 (2006).
    [CrossRef]

2013

N. Bender, S. Factor, J.D. Bodyfelt, H. Ramezani, D. N. Christodoulides, F. Ellis, T. Kottos, “Observation of asymmetric transport in structures with active nonlinearities,” Phys. Rev. Lett. 110, 234101 (2013).
[CrossRef]

2012

L. Fan, J. Wang, L.T. Varghese, H. Shen, B. Niu, Y. Xuan, A. M. Weiner, M. Qi, “An all-silicon passive optical diode,” Science 335, 447–450 (2012).
[CrossRef]

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

J. Schindler, Z. Lin, J. M. Lee, H. Ramezani, F. M. Ellis, T. Kottos, “PT-symmetric electronics,” J. Phys. A -Math and Theor. 45, 444029 (2012).
[CrossRef]

H. Ramezani, J. Schindler, F.M. Ellis, U. Günther, T. Kottos, “Bypassing the bandwidth theorem with PT symmetry,” Phys. Rev. A 85, 062122 (2012).
[CrossRef]

Z. Lin, J. Schindler, F.M. Ellis, T. Kottos, “Experimental observation of the dual behavior of PT-symmetric scattering,” Phys. Rev. A 85, 050101 (2012).
[CrossRef]

B-B Li, Y-F Xiao, C-L Zou, X-F Jiang, Y-C Liu, F-W Sun, Y Li, Q Gong, “Experimental controlling of Fano resonance in indirectly coupled whispering-gallery microresonators”, Applied Phys. Lett. 100, 021108 (2012).
[CrossRef]

2011

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

A. E. Miroshnichenko, B.A. Malomed, Y.S. Kivshar, “Nonlinearly PT-symmetric systems: spontaneous symmetry breaking and transmission resonances,” Phys. Rev. A 84, 012123 (2011).
[CrossRef]

M.S. Kang, A. Butsch, P.S.J. Russell, “Reconfigurable light-driven opto-acoustic isolators in photonic crystal fibre,” Nat. Photonics 5, 549–553 (2011).
[CrossRef]

S. Lepri, G. Casati, “Asymmetric wave propagation in nonlinear systems,” Phys. Rev. Lett. 106, 164101 (2011).
[CrossRef] [PubMed]

2010

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

A. E. Miroshnichenko, S. Flach, Y.S. Kivshar, “Fano resonances in nanoscale structures,” Rev. Mod. Phys. 82, 2257–2298 (2010).
[CrossRef]

H. Ramezani, T. Kottos, R. El-Ganainy, D. N. Christodoulides, “Unidirectional nonlinear PT-symmetric optical structures,” Phys. Rev. A 82, 043803 (2010).
[CrossRef]

Y-F Xiao, M. Li, Y-C Liu, Y. Li, X. Sun, Q. Gong, “Asymmetric Fano resonance analysis in indirectly coupled microresonators,” Phys. Rev. A 82, 065804 (2010).
[CrossRef]

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

2009

Z. Yu, S. Fan, “Complete optical isolation created by indirect interband photonic transitions,” Nat. Photonics 3, 91–94 (2009).
[CrossRef]

2008

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

2007

C. M. Bender, “Making sense of non-hermitian hamiltonians,” Rep. Prog. Phys. 70, 947–1018 (2007).
[CrossRef]

K. Totsuka, N. Kobayashi, M. Tomita, “Slow light in coupled-resonator-induced transparency,” Phys. Rev. Lett. 98, 213904 (2007).
[CrossRef] [PubMed]

2006

J. Ctyroky, I. Richter, V. Sinor, “Dual resonance in a waveguide-coupled ring microresonator,” Optic. Quantum Electron. 38, 781–797 (2006).
[CrossRef]

Q. Xu, S. Sandhu, M. L. Povinelli, J. Shakya, S. Fan, M. Lipson, “Experimental realization of an on-chip all-optical analogue to electromagnetically induced transparency,” Phys. Rev. Lett. 96, 123901 (2006).
[CrossRef] [PubMed]

K. Srinivasan, M. Borselli, O. Painter, A. Stintz, S. Krishna, “Cavity Q, mode volume, and lasing threshold in small diameter AlGaAs microdisks with embedded quantum dots,” Opt. Express 14, 1094–1105 (2006).
[CrossRef] [PubMed]

2004

D. D. Smith, H. Chang, K. A. Fuller, A. T. Rosenberger, R. W. Boyd, “Coupled-resonator-induced transparency,” Phys. Rev. A 69, 063804 (2004).
[CrossRef]

2003

K. J. Vahala, “Optical microcavities,” Nature (London) 424, 839–846 (2003).
[CrossRef]

1999

K. Gallo, G. Assanto, “All-optical diode based on second-harmonic generation in an asymmetric waveguide,” JOSA B 16, 267–269 (1999).
[CrossRef]

1994

M. Scalora, J.P. Dowling, C.M. Bowden, M.J. Bloemer, “The photonic band edge optical diode,” J. Appl. Phys. 76, 2023–2026 (1994).
[CrossRef]

Assanto, G.

K. Gallo, G. Assanto, “All-optical diode based on second-harmonic generation in an asymmetric waveguide,” JOSA B 16, 267–269 (1999).
[CrossRef]

Bender, C. M.

C. M. Bender, “Making sense of non-hermitian hamiltonians,” Rep. Prog. Phys. 70, 947–1018 (2007).
[CrossRef]

B. Peng, S. K. Ozdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, L. Yang, “Nonreciprocal light transmission in parity-time-symmetric whispering-gallery microcavities,” arXiv:1308.4564 (2013).

Bender, N.

N. Bender, S. Factor, J.D. Bodyfelt, H. Ramezani, D. N. Christodoulides, F. Ellis, T. Kottos, “Observation of asymmetric transport in structures with active nonlinearities,” Phys. Rev. Lett. 110, 234101 (2013).
[CrossRef]

Bersch, C.

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

Bloemer, M.J.

M. Scalora, J.P. Dowling, C.M. Bowden, M.J. Bloemer, “The photonic band edge optical diode,” J. Appl. Phys. 76, 2023–2026 (1994).
[CrossRef]

Bodyfelt, J.D.

N. Bender, S. Factor, J.D. Bodyfelt, H. Ramezani, D. N. Christodoulides, F. Ellis, T. Kottos, “Observation of asymmetric transport in structures with active nonlinearities,” Phys. Rev. Lett. 110, 234101 (2013).
[CrossRef]

Borselli, M.

Bowden, C.M.

M. Scalora, J.P. Dowling, C.M. Bowden, M.J. Bloemer, “The photonic band edge optical diode,” J. Appl. Phys. 76, 2023–2026 (1994).
[CrossRef]

Boyd, R. W.

D. D. Smith, H. Chang, K. A. Fuller, A. T. Rosenberger, R. W. Boyd, “Coupled-resonator-induced transparency,” Phys. Rev. A 69, 063804 (2004).
[CrossRef]

Butsch, A.

M.S. Kang, A. Butsch, P.S.J. Russell, “Reconfigurable light-driven opto-acoustic isolators in photonic crystal fibre,” Nat. Photonics 5, 549–553 (2011).
[CrossRef]

Casati, G.

S. Lepri, G. Casati, “Asymmetric wave propagation in nonlinear systems,” Phys. Rev. Lett. 106, 164101 (2011).
[CrossRef] [PubMed]

Chang, H.

D. D. Smith, H. Chang, K. A. Fuller, A. T. Rosenberger, R. W. Boyd, “Coupled-resonator-induced transparency,” Phys. Rev. A 69, 063804 (2004).
[CrossRef]

Chong, Y.D.

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

Christodoulides, D. N.

N. Bender, S. Factor, J.D. Bodyfelt, H. Ramezani, D. N. Christodoulides, F. Ellis, T. Kottos, “Observation of asymmetric transport in structures with active nonlinearities,” Phys. Rev. Lett. 110, 234101 (2013).
[CrossRef]

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

H. Ramezani, T. Kottos, R. El-Ganainy, D. N. Christodoulides, “Unidirectional nonlinear PT-symmetric optical structures,” Phys. Rev. A 82, 043803 (2010).
[CrossRef]

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

Christodoulides, D.N.

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

Ctyroky, J.

J. Ctyroky, I. Richter, V. Sinor, “Dual resonance in a waveguide-coupled ring microresonator,” Optic. Quantum Electron. 38, 781–797 (2006).
[CrossRef]

Dowling, J.P.

M. Scalora, J.P. Dowling, C.M. Bowden, M.J. Bloemer, “The photonic band edge optical diode,” J. Appl. Phys. 76, 2023–2026 (1994).
[CrossRef]

El-Ganainy, R.

H. Ramezani, T. Kottos, R. El-Ganainy, D. N. Christodoulides, “Unidirectional nonlinear PT-symmetric optical structures,” Phys. Rev. A 82, 043803 (2010).
[CrossRef]

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

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

Ellis, F.

N. Bender, S. Factor, J.D. Bodyfelt, H. Ramezani, D. N. Christodoulides, F. Ellis, T. Kottos, “Observation of asymmetric transport in structures with active nonlinearities,” Phys. Rev. Lett. 110, 234101 (2013).
[CrossRef]

Ellis, F. M.

J. Schindler, Z. Lin, J. M. Lee, H. Ramezani, F. M. Ellis, T. Kottos, “PT-symmetric electronics,” J. Phys. A -Math and Theor. 45, 444029 (2012).
[CrossRef]

Ellis, F.M.

Z. Lin, J. Schindler, F.M. Ellis, T. Kottos, “Experimental observation of the dual behavior of PT-symmetric scattering,” Phys. Rev. A 85, 050101 (2012).
[CrossRef]

H. Ramezani, J. Schindler, F.M. Ellis, U. Günther, T. Kottos, “Bypassing the bandwidth theorem with PT symmetry,” Phys. Rev. A 85, 062122 (2012).
[CrossRef]

Factor, S.

N. Bender, S. Factor, J.D. Bodyfelt, H. Ramezani, D. N. Christodoulides, F. Ellis, T. Kottos, “Observation of asymmetric transport in structures with active nonlinearities,” Phys. Rev. Lett. 110, 234101 (2013).
[CrossRef]

Fan, L.

L. Fan, J. Wang, L.T. Varghese, H. Shen, B. Niu, Y. Xuan, A. M. Weiner, M. Qi, “An all-silicon passive optical diode,” Science 335, 447–450 (2012).
[CrossRef]

Fan, S.

Z. Yu, S. Fan, “Complete optical isolation created by indirect interband photonic transitions,” Nat. Photonics 3, 91–94 (2009).
[CrossRef]

Q. Xu, S. Sandhu, M. L. Povinelli, J. Shakya, S. Fan, M. Lipson, “Experimental realization of an on-chip all-optical analogue to electromagnetically induced transparency,” Phys. Rev. Lett. 96, 123901 (2006).
[CrossRef] [PubMed]

B. Peng, S. K. Ozdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, L. Yang, “Nonreciprocal light transmission in parity-time-symmetric whispering-gallery microcavities,” arXiv:1308.4564 (2013).

Flach, S.

A. E. Miroshnichenko, S. Flach, Y.S. Kivshar, “Fano resonances in nanoscale structures,” Rev. Mod. Phys. 82, 2257–2298 (2010).
[CrossRef]

Fuller, K. A.

D. D. Smith, H. Chang, K. A. Fuller, A. T. Rosenberger, R. W. Boyd, “Coupled-resonator-induced transparency,” Phys. Rev. A 69, 063804 (2004).
[CrossRef]

Gallo, K.

K. Gallo, G. Assanto, “All-optical diode based on second-harmonic generation in an asymmetric waveguide,” JOSA B 16, 267–269 (1999).
[CrossRef]

Ge, L.

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

Gianfreda, M.

B. Peng, S. K. Ozdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, L. Yang, “Nonreciprocal light transmission in parity-time-symmetric whispering-gallery microcavities,” arXiv:1308.4564 (2013).

Gong, Q

B-B Li, Y-F Xiao, C-L Zou, X-F Jiang, Y-C Liu, F-W Sun, Y Li, Q Gong, “Experimental controlling of Fano resonance in indirectly coupled whispering-gallery microresonators”, Applied Phys. Lett. 100, 021108 (2012).
[CrossRef]

Gong, Q.

Y-F Xiao, M. Li, Y-C Liu, Y. Li, X. Sun, Q. Gong, “Asymmetric Fano resonance analysis in indirectly coupled microresonators,” Phys. Rev. A 82, 065804 (2010).
[CrossRef]

Günther, U.

H. Ramezani, J. Schindler, F.M. Ellis, U. Günther, T. Kottos, “Bypassing the bandwidth theorem with PT symmetry,” Phys. Rev. A 85, 062122 (2012).
[CrossRef]

Jiang, X-F

B-B Li, Y-F Xiao, C-L Zou, X-F Jiang, Y-C Liu, F-W Sun, Y Li, Q Gong, “Experimental controlling of Fano resonance in indirectly coupled whispering-gallery microresonators”, Applied Phys. Lett. 100, 021108 (2012).
[CrossRef]

Kang, M.S.

M.S. Kang, A. Butsch, P.S.J. Russell, “Reconfigurable light-driven opto-acoustic isolators in photonic crystal fibre,” Nat. Photonics 5, 549–553 (2011).
[CrossRef]

Kip, D.

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

Kivshar, Y.S.

A. E. Miroshnichenko, B.A. Malomed, Y.S. Kivshar, “Nonlinearly PT-symmetric systems: spontaneous symmetry breaking and transmission resonances,” Phys. Rev. A 84, 012123 (2011).
[CrossRef]

A. E. Miroshnichenko, S. Flach, Y.S. Kivshar, “Fano resonances in nanoscale structures,” Rev. Mod. Phys. 82, 2257–2298 (2010).
[CrossRef]

Kobayashi, N.

K. Totsuka, N. Kobayashi, M. Tomita, “Slow light in coupled-resonator-induced transparency,” Phys. Rev. Lett. 98, 213904 (2007).
[CrossRef] [PubMed]

Kottos, T.

N. Bender, S. Factor, J.D. Bodyfelt, H. Ramezani, D. N. Christodoulides, F. Ellis, T. Kottos, “Observation of asymmetric transport in structures with active nonlinearities,” Phys. Rev. Lett. 110, 234101 (2013).
[CrossRef]

H. Ramezani, J. Schindler, F.M. Ellis, U. Günther, T. Kottos, “Bypassing the bandwidth theorem with PT symmetry,” Phys. Rev. A 85, 062122 (2012).
[CrossRef]

J. Schindler, Z. Lin, J. M. Lee, H. Ramezani, F. M. Ellis, T. Kottos, “PT-symmetric electronics,” J. Phys. A -Math and Theor. 45, 444029 (2012).
[CrossRef]

Z. Lin, J. Schindler, F.M. Ellis, T. Kottos, “Experimental observation of the dual behavior of PT-symmetric scattering,” Phys. Rev. A 85, 050101 (2012).
[CrossRef]

H. Ramezani, T. Kottos, R. El-Ganainy, D. N. Christodoulides, “Unidirectional nonlinear PT-symmetric optical structures,” Phys. Rev. A 82, 043803 (2010).
[CrossRef]

Krishna, S.

Lee, J. M.

J. Schindler, Z. Lin, J. M. Lee, H. Ramezani, F. M. Ellis, T. Kottos, “PT-symmetric electronics,” J. Phys. A -Math and Theor. 45, 444029 (2012).
[CrossRef]

Lei, F.

B. Peng, S. K. Ozdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, L. Yang, “Nonreciprocal light transmission in parity-time-symmetric whispering-gallery microcavities,” arXiv:1308.4564 (2013).

Lepri, S.

S. Lepri, G. Casati, “Asymmetric wave propagation in nonlinear systems,” Phys. Rev. Lett. 106, 164101 (2011).
[CrossRef] [PubMed]

Li, B-B

B-B Li, Y-F Xiao, C-L Zou, X-F Jiang, Y-C Liu, F-W Sun, Y Li, Q Gong, “Experimental controlling of Fano resonance in indirectly coupled whispering-gallery microresonators”, Applied Phys. Lett. 100, 021108 (2012).
[CrossRef]

Li, M.

Y-F Xiao, M. Li, Y-C Liu, Y. Li, X. Sun, Q. Gong, “Asymmetric Fano resonance analysis in indirectly coupled microresonators,” Phys. Rev. A 82, 065804 (2010).
[CrossRef]

Li, Y

B-B Li, Y-F Xiao, C-L Zou, X-F Jiang, Y-C Liu, F-W Sun, Y Li, Q Gong, “Experimental controlling of Fano resonance in indirectly coupled whispering-gallery microresonators”, Applied Phys. Lett. 100, 021108 (2012).
[CrossRef]

Li, Y.

Y-F Xiao, M. Li, Y-C Liu, Y. Li, X. Sun, Q. Gong, “Asymmetric Fano resonance analysis in indirectly coupled microresonators,” Phys. Rev. A 82, 065804 (2010).
[CrossRef]

Lin, Z.

J. Schindler, Z. Lin, J. M. Lee, H. Ramezani, F. M. Ellis, T. Kottos, “PT-symmetric electronics,” J. Phys. A -Math and Theor. 45, 444029 (2012).
[CrossRef]

Z. Lin, J. Schindler, F.M. Ellis, T. Kottos, “Experimental observation of the dual behavior of PT-symmetric scattering,” Phys. Rev. A 85, 050101 (2012).
[CrossRef]

Lipson, M.

Q. Xu, S. Sandhu, M. L. Povinelli, J. Shakya, S. Fan, M. Lipson, “Experimental realization of an on-chip all-optical analogue to electromagnetically induced transparency,” Phys. Rev. Lett. 96, 123901 (2006).
[CrossRef] [PubMed]

Liu, Y-C

B-B Li, Y-F Xiao, C-L Zou, X-F Jiang, Y-C Liu, F-W Sun, Y Li, Q Gong, “Experimental controlling of Fano resonance in indirectly coupled whispering-gallery microresonators”, Applied Phys. Lett. 100, 021108 (2012).
[CrossRef]

Y-F Xiao, M. Li, Y-C Liu, Y. Li, X. Sun, Q. Gong, “Asymmetric Fano resonance analysis in indirectly coupled microresonators,” Phys. Rev. A 82, 065804 (2010).
[CrossRef]

Long, G. L.

B. Peng, S. K. Ozdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, L. Yang, “Nonreciprocal light transmission in parity-time-symmetric whispering-gallery microcavities,” arXiv:1308.4564 (2013).

Longhi, S.

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

Makris, K. G.

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

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

Malomed, B.A.

A. E. Miroshnichenko, B.A. Malomed, Y.S. Kivshar, “Nonlinearly PT-symmetric systems: spontaneous symmetry breaking and transmission resonances,” Phys. Rev. A 84, 012123 (2011).
[CrossRef]

Miri, M.A.

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

Miroshnichenko, A. E.

A. E. Miroshnichenko, B.A. Malomed, Y.S. Kivshar, “Nonlinearly PT-symmetric systems: spontaneous symmetry breaking and transmission resonances,” Phys. Rev. A 84, 012123 (2011).
[CrossRef]

A. E. Miroshnichenko, S. Flach, Y.S. Kivshar, “Fano resonances in nanoscale structures,” Rev. Mod. Phys. 82, 2257–2298 (2010).
[CrossRef]

Y. Xu, A. E. Miroshnichenko, “Reconfigurable nonreciprocity with nonlinear Fano diode,” eprint arXiv:1311.2533 (2013).

Monifi, F.

B. Peng, S. K. Ozdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, L. Yang, “Nonreciprocal light transmission in parity-time-symmetric whispering-gallery microcavities,” arXiv:1308.4564 (2013).

Niu, B.

L. Fan, J. Wang, L.T. Varghese, H. Shen, B. Niu, Y. Xuan, A. M. Weiner, M. Qi, “An all-silicon passive optical diode,” Science 335, 447–450 (2012).
[CrossRef]

Nori, F.

B. Peng, S. K. Ozdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, L. Yang, “Nonreciprocal light transmission in parity-time-symmetric whispering-gallery microcavities,” arXiv:1308.4564 (2013).

Onishchukov, G.

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

Ozdemir, S. K.

B. Peng, S. K. Ozdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, L. Yang, “Nonreciprocal light transmission in parity-time-symmetric whispering-gallery microcavities,” arXiv:1308.4564 (2013).

Painter, O.

Peng, B.

B. Peng, S. K. Ozdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, L. Yang, “Nonreciprocal light transmission in parity-time-symmetric whispering-gallery microcavities,” arXiv:1308.4564 (2013).

Peschel, U.

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

Povinelli, M. L.

Q. Xu, S. Sandhu, M. L. Povinelli, J. Shakya, S. Fan, M. Lipson, “Experimental realization of an on-chip all-optical analogue to electromagnetically induced transparency,” Phys. Rev. Lett. 96, 123901 (2006).
[CrossRef] [PubMed]

Qi, M.

L. Fan, J. Wang, L.T. Varghese, H. Shen, B. Niu, Y. Xuan, A. M. Weiner, M. Qi, “An all-silicon passive optical diode,” Science 335, 447–450 (2012).
[CrossRef]

Ramezani, H.

N. Bender, S. Factor, J.D. Bodyfelt, H. Ramezani, D. N. Christodoulides, F. Ellis, T. Kottos, “Observation of asymmetric transport in structures with active nonlinearities,” Phys. Rev. Lett. 110, 234101 (2013).
[CrossRef]

H. Ramezani, J. Schindler, F.M. Ellis, U. Günther, T. Kottos, “Bypassing the bandwidth theorem with PT symmetry,” Phys. Rev. A 85, 062122 (2012).
[CrossRef]

J. Schindler, Z. Lin, J. M. Lee, H. Ramezani, F. M. Ellis, T. Kottos, “PT-symmetric electronics,” J. Phys. A -Math and Theor. 45, 444029 (2012).
[CrossRef]

H. Ramezani, T. Kottos, R. El-Ganainy, D. N. Christodoulides, “Unidirectional nonlinear PT-symmetric optical structures,” Phys. Rev. A 82, 043803 (2010).
[CrossRef]

Regensburger, A.

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

Richter, I.

J. Ctyroky, I. Richter, V. Sinor, “Dual resonance in a waveguide-coupled ring microresonator,” Optic. Quantum Electron. 38, 781–797 (2006).
[CrossRef]

Rosenberger, A. T.

D. D. Smith, H. Chang, K. A. Fuller, A. T. Rosenberger, R. W. Boyd, “Coupled-resonator-induced transparency,” Phys. Rev. A 69, 063804 (2004).
[CrossRef]

Russell, P.S.J.

M.S. Kang, A. Butsch, P.S.J. Russell, “Reconfigurable light-driven opto-acoustic isolators in photonic crystal fibre,” Nat. Photonics 5, 549–553 (2011).
[CrossRef]

Rüter, C. E.

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

Sandhu, S.

Q. Xu, S. Sandhu, M. L. Povinelli, J. Shakya, S. Fan, M. Lipson, “Experimental realization of an on-chip all-optical analogue to electromagnetically induced transparency,” Phys. Rev. Lett. 96, 123901 (2006).
[CrossRef] [PubMed]

Scalora, M.

M. Scalora, J.P. Dowling, C.M. Bowden, M.J. Bloemer, “The photonic band edge optical diode,” J. Appl. Phys. 76, 2023–2026 (1994).
[CrossRef]

Schindler, J.

H. Ramezani, J. Schindler, F.M. Ellis, U. Günther, T. Kottos, “Bypassing the bandwidth theorem with PT symmetry,” Phys. Rev. A 85, 062122 (2012).
[CrossRef]

J. Schindler, Z. Lin, J. M. Lee, H. Ramezani, F. M. Ellis, T. Kottos, “PT-symmetric electronics,” J. Phys. A -Math and Theor. 45, 444029 (2012).
[CrossRef]

Z. Lin, J. Schindler, F.M. Ellis, T. Kottos, “Experimental observation of the dual behavior of PT-symmetric scattering,” Phys. Rev. A 85, 050101 (2012).
[CrossRef]

Segev, M.

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

Shakya, J.

Q. Xu, S. Sandhu, M. L. Povinelli, J. Shakya, S. Fan, M. Lipson, “Experimental realization of an on-chip all-optical analogue to electromagnetically induced transparency,” Phys. Rev. Lett. 96, 123901 (2006).
[CrossRef] [PubMed]

Shen, H.

L. Fan, J. Wang, L.T. Varghese, H. Shen, B. Niu, Y. Xuan, A. M. Weiner, M. Qi, “An all-silicon passive optical diode,” Science 335, 447–450 (2012).
[CrossRef]

Sinor, V.

J. Ctyroky, I. Richter, V. Sinor, “Dual resonance in a waveguide-coupled ring microresonator,” Optic. Quantum Electron. 38, 781–797 (2006).
[CrossRef]

Smith, D. D.

D. D. Smith, H. Chang, K. A. Fuller, A. T. Rosenberger, R. W. Boyd, “Coupled-resonator-induced transparency,” Phys. Rev. A 69, 063804 (2004).
[CrossRef]

Srinivasan, K.

Stintz, A.

Stone, A.D.

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

Sun, F-W

B-B Li, Y-F Xiao, C-L Zou, X-F Jiang, Y-C Liu, F-W Sun, Y Li, Q Gong, “Experimental controlling of Fano resonance in indirectly coupled whispering-gallery microresonators”, Applied Phys. Lett. 100, 021108 (2012).
[CrossRef]

Sun, X.

Y-F Xiao, M. Li, Y-C Liu, Y. Li, X. Sun, Q. Gong, “Asymmetric Fano resonance analysis in indirectly coupled microresonators,” Phys. Rev. A 82, 065804 (2010).
[CrossRef]

Tomita, M.

K. Totsuka, N. Kobayashi, M. Tomita, “Slow light in coupled-resonator-induced transparency,” Phys. Rev. Lett. 98, 213904 (2007).
[CrossRef] [PubMed]

Totsuka, K.

K. Totsuka, N. Kobayashi, M. Tomita, “Slow light in coupled-resonator-induced transparency,” Phys. Rev. Lett. 98, 213904 (2007).
[CrossRef] [PubMed]

Vahala, K. J.

K. J. Vahala, “Optical microcavities,” Nature (London) 424, 839–846 (2003).
[CrossRef]

Varghese, L.T.

L. Fan, J. Wang, L.T. Varghese, H. Shen, B. Niu, Y. Xuan, A. M. Weiner, M. Qi, “An all-silicon passive optical diode,” Science 335, 447–450 (2012).
[CrossRef]

Wang, J.

L. Fan, J. Wang, L.T. Varghese, H. Shen, B. Niu, Y. Xuan, A. M. Weiner, M. Qi, “An all-silicon passive optical diode,” Science 335, 447–450 (2012).
[CrossRef]

Weiner, A. M.

L. Fan, J. Wang, L.T. Varghese, H. Shen, B. Niu, Y. Xuan, A. M. Weiner, M. Qi, “An all-silicon passive optical diode,” Science 335, 447–450 (2012).
[CrossRef]

Xiao, Y-F

B-B Li, Y-F Xiao, C-L Zou, X-F Jiang, Y-C Liu, F-W Sun, Y Li, Q Gong, “Experimental controlling of Fano resonance in indirectly coupled whispering-gallery microresonators”, Applied Phys. Lett. 100, 021108 (2012).
[CrossRef]

Y-F Xiao, M. Li, Y-C Liu, Y. Li, X. Sun, Q. Gong, “Asymmetric Fano resonance analysis in indirectly coupled microresonators,” Phys. Rev. A 82, 065804 (2010).
[CrossRef]

Xu, Q.

Q. Xu, S. Sandhu, M. L. Povinelli, J. Shakya, S. Fan, M. Lipson, “Experimental realization of an on-chip all-optical analogue to electromagnetically induced transparency,” Phys. Rev. Lett. 96, 123901 (2006).
[CrossRef] [PubMed]

Xu, Y.

Y. Xu, A. E. Miroshnichenko, “Reconfigurable nonreciprocity with nonlinear Fano diode,” eprint arXiv:1311.2533 (2013).

Xuan, Y.

L. Fan, J. Wang, L.T. Varghese, H. Shen, B. Niu, Y. Xuan, A. M. Weiner, M. Qi, “An all-silicon passive optical diode,” Science 335, 447–450 (2012).
[CrossRef]

Yang, L.

B. Peng, S. K. Ozdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, L. Yang, “Nonreciprocal light transmission in parity-time-symmetric whispering-gallery microcavities,” arXiv:1308.4564 (2013).

Yu, Z.

Z. Yu, S. Fan, “Complete optical isolation created by indirect interband photonic transitions,” Nat. Photonics 3, 91–94 (2009).
[CrossRef]

Zou, C-L

B-B Li, Y-F Xiao, C-L Zou, X-F Jiang, Y-C Liu, F-W Sun, Y Li, Q Gong, “Experimental controlling of Fano resonance in indirectly coupled whispering-gallery microresonators”, Applied Phys. Lett. 100, 021108 (2012).
[CrossRef]

Applied Phys. Lett.

B-B Li, Y-F Xiao, C-L Zou, X-F Jiang, Y-C Liu, F-W Sun, Y Li, Q Gong, “Experimental controlling of Fano resonance in indirectly coupled whispering-gallery microresonators”, Applied Phys. Lett. 100, 021108 (2012).
[CrossRef]

J. Appl. Phys.

M. Scalora, J.P. Dowling, C.M. Bowden, M.J. Bloemer, “The photonic band edge optical diode,” J. Appl. Phys. 76, 2023–2026 (1994).
[CrossRef]

J. Phys. A -Math and Theor.

J. Schindler, Z. Lin, J. M. Lee, H. Ramezani, F. M. Ellis, T. Kottos, “PT-symmetric electronics,” J. Phys. A -Math and Theor. 45, 444029 (2012).
[CrossRef]

JOSA B

K. Gallo, G. Assanto, “All-optical diode based on second-harmonic generation in an asymmetric waveguide,” JOSA B 16, 267–269 (1999).
[CrossRef]

Nat. Photonics

Z. Yu, S. Fan, “Complete optical isolation created by indirect interband photonic transitions,” Nat. Photonics 3, 91–94 (2009).
[CrossRef]

M.S. Kang, A. Butsch, P.S.J. Russell, “Reconfigurable light-driven opto-acoustic isolators in photonic crystal fibre,” Nat. Photonics 5, 549–553 (2011).
[CrossRef]

Nat. Phys.

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

Nature

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

Nature (London)

K. J. Vahala, “Optical microcavities,” Nature (London) 424, 839–846 (2003).
[CrossRef]

Opt. Express

Optic. Quantum Electron.

J. Ctyroky, I. Richter, V. Sinor, “Dual resonance in a waveguide-coupled ring microresonator,” Optic. Quantum Electron. 38, 781–797 (2006).
[CrossRef]

Phys. Rev. A

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

H. Ramezani, T. Kottos, R. El-Ganainy, D. N. Christodoulides, “Unidirectional nonlinear PT-symmetric optical structures,” Phys. Rev. A 82, 043803 (2010).
[CrossRef]

H. Ramezani, J. Schindler, F.M. Ellis, U. Günther, T. Kottos, “Bypassing the bandwidth theorem with PT symmetry,” Phys. Rev. A 85, 062122 (2012).
[CrossRef]

Z. Lin, J. Schindler, F.M. Ellis, T. Kottos, “Experimental observation of the dual behavior of PT-symmetric scattering,” Phys. Rev. A 85, 050101 (2012).
[CrossRef]

Y-F Xiao, M. Li, Y-C Liu, Y. Li, X. Sun, Q. Gong, “Asymmetric Fano resonance analysis in indirectly coupled microresonators,” Phys. Rev. A 82, 065804 (2010).
[CrossRef]

D. D. Smith, H. Chang, K. A. Fuller, A. T. Rosenberger, R. W. Boyd, “Coupled-resonator-induced transparency,” Phys. Rev. A 69, 063804 (2004).
[CrossRef]

A. E. Miroshnichenko, B.A. Malomed, Y.S. Kivshar, “Nonlinearly PT-symmetric systems: spontaneous symmetry breaking and transmission resonances,” Phys. Rev. A 84, 012123 (2011).
[CrossRef]

Phys. Rev. Lett.

Q. Xu, S. Sandhu, M. L. Povinelli, J. Shakya, S. Fan, M. Lipson, “Experimental realization of an on-chip all-optical analogue to electromagnetically induced transparency,” Phys. Rev. Lett. 96, 123901 (2006).
[CrossRef] [PubMed]

K. Totsuka, N. Kobayashi, M. Tomita, “Slow light in coupled-resonator-induced transparency,” Phys. Rev. Lett. 98, 213904 (2007).
[CrossRef] [PubMed]

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

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

S. Lepri, G. Casati, “Asymmetric wave propagation in nonlinear systems,” Phys. Rev. Lett. 106, 164101 (2011).
[CrossRef] [PubMed]

N. Bender, S. Factor, J.D. Bodyfelt, H. Ramezani, D. N. Christodoulides, F. Ellis, T. Kottos, “Observation of asymmetric transport in structures with active nonlinearities,” Phys. Rev. Lett. 110, 234101 (2013).
[CrossRef]

Rep. Prog. Phys.

C. M. Bender, “Making sense of non-hermitian hamiltonians,” Rep. Prog. Phys. 70, 947–1018 (2007).
[CrossRef]

Rev. Mod. Phys.

A. E. Miroshnichenko, S. Flach, Y.S. Kivshar, “Fano resonances in nanoscale structures,” Rev. Mod. Phys. 82, 2257–2298 (2010).
[CrossRef]

Science

L. Fan, J. Wang, L.T. Varghese, H. Shen, B. Niu, Y. Xuan, A. M. Weiner, M. Qi, “An all-silicon passive optical diode,” Science 335, 447–450 (2012).
[CrossRef]

Other

B. Peng, S. K. Ozdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, L. Yang, “Nonreciprocal light transmission in parity-time-symmetric whispering-gallery microcavities,” arXiv:1308.4564 (2013).

Y. Xu, A. E. Miroshnichenko, “Reconfigurable nonreciprocity with nonlinear Fano diode,” eprint arXiv:1311.2533 (2013).

The 𝒫𝒯-symmetric phase transition can be also achieved by manipulating the coupling strength between the gain an loss elements while keeping the balanced gain and loss parameter constant. Specifically it can be shown that decreasing the coupling strength is equivalent to increasing the gain and loss parameter.

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Figures (5)

Fig. 1
Fig. 1

(a) The transmittances T (λ) of the linear system for incident waves from the gain (filled black circles) and from the loss (black line) side are compared with the corresponding T (λ) in the case of microdisks with Kerr non-linearity. The resonance from the loss side (green line) experienced a small red-shift with respect to the linear structure. In contrast, the transmittance curve (both line-shape and resonance position) of an incident wave entering the structure from the gain side (red dotted-dashed line) is different. Insets: Schematic of the photonic circuit. The color coding indicates the intensity strength of the field inside the circuit. (b) The transmission phases (taken in the interval [−π, π]) are plotted as a function of wavelength λ. The colors and the line-type indicate the same scattering process as the one used in the upper panel. The blue arrows (and the blue dots) mark the wavelengths for which we have the Fano resonances.

Fig. 2
Fig. 2

The transmittances T (λ) for a left (red curve) and right (green curve) incident light for the non-linear system shown in Fig. 1(a). The frequency window contains two Fano resonances. Here the two microdisks are placed in distance 100nm so that the system is in the exact phase. In the inset we report the difference between left and right transmittances. The maximum asymmetry that has been found is as high as 5.5dBs. This has to be contrasted with the case of Fig. 1(a) where the system is in the broken phase and the maximum asymmetry reached values as high as 25dBs.

Fig. 3
Fig. 3

(a) Transmittances T (λ) for a single gain or loss microdisk coupled to a waveguide. The filled green circles (green line) correspond to a lossy disk in the absence (presence) of Kerr nonlinearity. The filled red circles (dotted-dashed line) correspond to a gain disk in the absence (presence) of Kerr nonlinearity. A red-shift of the resonance position and a strong modification of T (λ) is observed. (b) The transmission spectra of two coupled linear microdisks with various gain and loss configurations (see legends in the figure). (c) The same as in (b) but now the two microdisks have a Kerr non-linearity. In the ���� -symmetric configuration we report the transmission spectra for both left (gain side) and right (loss side) incident beams.

Fig. 4
Fig. 4

Transmittance for different input powers. The asymmetric transport is maintained for a broad range of input power levels. Dashed lines correspond to TL(λ) (gain side) while solid lines to TR(λ) (lossy side). The transport asymmetry is as high as 46.5 dB without compromising the outgoing optical intensity which is as high as −5 dBs.

Fig. 5
Fig. 5

(a) Left (gain) TL and right (loss) TR transmittances for the theoretical model of Eq. (1). Notice the red-shift associated with TL (pointed with a black arrow), in the neighborhood of the second Fano resonance. In this domain the asymmetry is most pronounced TLTR. The parameters used in this simulation are VG = VL = 0.5, N = 1, χ = 0.0125 and γ = 0.02. The red-shadowed area on the right of the graph around λ ≈ 4.5, corresponds to a bi-stability behavior which however is away from the Fano resonance regime and thus does not affect asymmetric transport. (b) A density plot of the transmission asymmetry 10|log10(TL) − log10(TR)| versus the gain and loss parameter γ and the non-linearity χ. The maximum asymmetry is observed for large values of χ and γ (upper right corner of the χγ parameter space).

Equations (5)

Equations on this page are rendered with MathJax. Learn more.

i ϕ ˙ n = { C ( ϕ n 1 + ϕ n + 1 ) + V G ϕ G δ n , 0 + V L ϕ L δ n , N } i ϕ ˙ G / L = { ( E i γ ) ϕ G / L + χ | ϕ G / L | 2 ϕ G / L + V G / L ϕ 0 / N }
ω A n = C ( A n 1 + A n + 1 ) + V G A G δ n , 0 + V L A L δ n , N ω A G / L = E A G / L i γ A G / L + χ | A G / L | 2 A G / L + V G / L A 0 / N
A n = { I e i q n + r e i q n n 0 α e i q n + β e i q n 0 n N t e i q n N n
r L = i V G A G + V L A L e i q N 2 C sin q ; t L ( I + i V G A G + V L A L e i q N 2 C sin q )
( E ω i γ ) A G + χ | A G | 2 A G + V G ( I + r L ) = 0 ; ( E ω + i γ ) A L + χ | A L | 2 A L + V L e i q N t L = 0

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