Abstract

The nonlinear modulation dynamic properties of nonreciprocal indirect interband photonic transitions have been theoretically investigated in this paper. It is found that the period of such complete interband photonic transitions can be modulated for the relatively weak nonlinear effect, while interband transitions can be partially or even completely suppressed for relatively strong nonlinear effect, i.e., the dynamic behaviors of this system can be completely decided by nonlinear effects and dynamic modulations. Phase plane analysis is used to explain these dynamic behaviors.

© 2009 Optical Society of America

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  1. Z. Yu and S. Fan, “Complete optical isolation created by indirect interband photonic transitions,” Nat. Photon. 3, 91-94 (2009).
    [CrossRef]
  2. F. D. M. Haldane and S. Raghu, “Possible realization of directional optical waveguides in photonic crystals with broken time-reversal symmetry,” Phys. Rev. Lett. 100, 013904 (2008).
    [CrossRef] [PubMed]
  3. Z. Wang, Y. D. Chong, J. D. Joannopoulos, and M. Soljacic, “Reflection-free one-way edge modes in a gyromagnetic photonic crystal,” Phys. Rev. Lett. 100, 013905 (2008).
    [CrossRef] [PubMed]
  4. Z. Yu, G. Veronis, Z. Wang, and S. Fan, “One-way electromagnetic waveguide formed at the interface between a plasmonic metal under a static magnetic field and a photonic crystal,” Phys. Rev. Lett. 100, 023902 (2008).
    [CrossRef] [PubMed]
  5. S. Raghu and F. D. Haldane, “Analogs of quantum-Hall-effect edge states in photonic crystals,” Phys. Rev. A 78, 033834 (2008).
    [CrossRef]
  6. B. I. Halperin, “Quantized Hall conductance, current-carrying edge states, and the existence of extended states in a two-dimensional disordered potential,” Phys. Rev. B 25, 2185-2190 (1982).
    [CrossRef]
  7. R. Rammal, G. Toulouse, M. T. Jackel, and B. I. Halperin, “Quantized Hall conductance and edge states: two-dimensional strips with a periodic potential,” Phys. Rev. B 27, 5142-5145 (1983).
    [CrossRef]
  8. X. G. Wen, “Gapless boundary excitations in the quantum Hall states and in the chiral spin states,” Phys. Rev. B 43, 11025-11036 (1991).
    [CrossRef]
  9. A. Haché and M. Bourgeois, “Ultrafast all-optical switching in a silicon-based photonic crystal,” Appl. Phys. Lett. 77, 4089-4091 (2000).
    [CrossRef]
  10. 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 (2000).
    [CrossRef]
  11. M. Soljacic, M. Ibanescu, S. G. Johnson, Y. Fink, and J. D. Joannopoulos, “Optimal bistable switching I: nonlinear photonic crystals,” Phys. Rev. E 66, 055601 (2002).
    [CrossRef]
  12. M. F. Yanik, S. Fan, and M. Soljacic, “High-contrast all-optical bistable switching in photonic crystal microcavities,” Appl. Phys. Lett. 83, 2739-2741 (2003).
    [CrossRef]
  13. A. D. Bristow, J. P. R. Wells, W. H. Fan, A. M. Fox, M. S. Skolnick, D. M. Whittaker, A. Tahraoui, T. F. Krauss, and J. S. Roberts, “Ultrafast nonlinear response of AlGaAs two-dimensional photonic crystal waveguides,” Appl. Phys. Lett. 83, 851-853 (2003).
    [CrossRef]
  14. M. Soljacic, C. Luo, J. D. Joannopoulos, and S. Fan, “Nonlinear photonic crystal microdevices for optical integration,” Opt. Lett. 28, 637-639 (2003).
    [CrossRef] [PubMed]
  15. H. W. Tan, H. M. van Driel, S. L. Schweizer, R. B. Wehrspohn, and U. Gosele, “Nonlinear optical tuning of a two-dimensional silicon photonic crystal,” Phys. Rev. B 70, 205110 (2004).
    [CrossRef]
  16. X. Jiang, C. Zhou, X. Yu, S. Fan, M. Soljacic, and J. D. Joannopoulos, “Interplay between supercollimation and nonlinearity: a novel method of controlling beams in photonic crystals,” Appl. Phys. Lett. 91, 031105 (2007).
    [CrossRef]

2009 (1)

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

2008 (4)

F. D. M. Haldane and S. Raghu, “Possible realization of directional optical waveguides in photonic crystals with broken time-reversal symmetry,” Phys. Rev. Lett. 100, 013904 (2008).
[CrossRef] [PubMed]

Z. Wang, Y. D. Chong, J. D. Joannopoulos, and M. Soljacic, “Reflection-free one-way edge modes in a gyromagnetic photonic crystal,” Phys. Rev. Lett. 100, 013905 (2008).
[CrossRef] [PubMed]

Z. Yu, G. Veronis, Z. Wang, and S. Fan, “One-way electromagnetic waveguide formed at the interface between a plasmonic metal under a static magnetic field and a photonic crystal,” Phys. Rev. Lett. 100, 023902 (2008).
[CrossRef] [PubMed]

S. Raghu and F. D. Haldane, “Analogs of quantum-Hall-effect edge states in photonic crystals,” Phys. Rev. A 78, 033834 (2008).
[CrossRef]

2007 (1)

X. Jiang, C. Zhou, X. Yu, S. Fan, M. Soljacic, and J. D. Joannopoulos, “Interplay between supercollimation and nonlinearity: a novel method of controlling beams in photonic crystals,” Appl. Phys. Lett. 91, 031105 (2007).
[CrossRef]

2004 (1)

H. W. Tan, H. M. van Driel, S. L. Schweizer, R. B. Wehrspohn, and U. Gosele, “Nonlinear optical tuning of a two-dimensional silicon photonic crystal,” Phys. Rev. B 70, 205110 (2004).
[CrossRef]

2003 (3)

M. F. Yanik, S. Fan, and M. Soljacic, “High-contrast all-optical bistable switching in photonic crystal microcavities,” Appl. Phys. Lett. 83, 2739-2741 (2003).
[CrossRef]

A. D. Bristow, J. P. R. Wells, W. H. Fan, A. M. Fox, M. S. Skolnick, D. M. Whittaker, A. Tahraoui, T. F. Krauss, and J. S. Roberts, “Ultrafast nonlinear response of AlGaAs two-dimensional photonic crystal waveguides,” Appl. Phys. Lett. 83, 851-853 (2003).
[CrossRef]

M. Soljacic, C. Luo, J. D. Joannopoulos, and S. Fan, “Nonlinear photonic crystal microdevices for optical integration,” Opt. Lett. 28, 637-639 (2003).
[CrossRef] [PubMed]

2002 (1)

M. Soljacic, M. Ibanescu, S. G. Johnson, Y. Fink, and J. D. Joannopoulos, “Optimal bistable switching I: nonlinear photonic crystals,” Phys. Rev. E 66, 055601 (2002).
[CrossRef]

2000 (2)

A. Haché and M. Bourgeois, “Ultrafast all-optical switching in a silicon-based photonic crystal,” Appl. Phys. Lett. 77, 4089-4091 (2000).
[CrossRef]

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 (2000).
[CrossRef]

1991 (1)

X. G. Wen, “Gapless boundary excitations in the quantum Hall states and in the chiral spin states,” Phys. Rev. B 43, 11025-11036 (1991).
[CrossRef]

1983 (1)

R. Rammal, G. Toulouse, M. T. Jackel, and B. I. Halperin, “Quantized Hall conductance and edge states: two-dimensional strips with a periodic potential,” Phys. Rev. B 27, 5142-5145 (1983).
[CrossRef]

1982 (1)

B. I. Halperin, “Quantized Hall conductance, current-carrying edge states, and the existence of extended states in a two-dimensional disordered potential,” Phys. Rev. B 25, 2185-2190 (1982).
[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 (2000).
[CrossRef]

Bourgeois, M.

A. Haché and M. Bourgeois, “Ultrafast all-optical switching in a silicon-based photonic crystal,” Appl. Phys. Lett. 77, 4089-4091 (2000).
[CrossRef]

Bristow, A. D.

A. D. Bristow, J. P. R. Wells, W. H. Fan, A. M. Fox, M. S. Skolnick, D. M. Whittaker, A. Tahraoui, T. F. Krauss, and J. S. Roberts, “Ultrafast nonlinear response of AlGaAs two-dimensional photonic crystal waveguides,” Appl. Phys. Lett. 83, 851-853 (2003).
[CrossRef]

Chong, Y. D.

Z. Wang, Y. D. Chong, J. D. Joannopoulos, and M. Soljacic, “Reflection-free one-way edge modes in a gyromagnetic photonic crystal,” Phys. Rev. Lett. 100, 013905 (2008).
[CrossRef] [PubMed]

Fan, S.

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

Z. Yu, G. Veronis, Z. Wang, and S. Fan, “One-way electromagnetic waveguide formed at the interface between a plasmonic metal under a static magnetic field and a photonic crystal,” Phys. Rev. Lett. 100, 023902 (2008).
[CrossRef] [PubMed]

X. Jiang, C. Zhou, X. Yu, S. Fan, M. Soljacic, and J. D. Joannopoulos, “Interplay between supercollimation and nonlinearity: a novel method of controlling beams in photonic crystals,” Appl. Phys. Lett. 91, 031105 (2007).
[CrossRef]

M. Soljacic, C. Luo, J. D. Joannopoulos, and S. Fan, “Nonlinear photonic crystal microdevices for optical integration,” Opt. Lett. 28, 637-639 (2003).
[CrossRef] [PubMed]

M. F. Yanik, S. Fan, and M. Soljacic, “High-contrast all-optical bistable switching in photonic crystal microcavities,” Appl. Phys. Lett. 83, 2739-2741 (2003).
[CrossRef]

Fan, W. H.

A. D. Bristow, J. P. R. Wells, W. H. Fan, A. M. Fox, M. S. Skolnick, D. M. Whittaker, A. Tahraoui, T. F. Krauss, and J. S. Roberts, “Ultrafast nonlinear response of AlGaAs two-dimensional photonic crystal waveguides,” Appl. Phys. Lett. 83, 851-853 (2003).
[CrossRef]

Fejer, M. M.

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 (2000).
[CrossRef]

Fink, Y.

M. Soljacic, M. Ibanescu, S. G. Johnson, Y. Fink, and J. D. Joannopoulos, “Optimal bistable switching I: nonlinear photonic crystals,” Phys. Rev. E 66, 055601 (2002).
[CrossRef]

Fox, A. M.

A. D. Bristow, J. P. R. Wells, W. H. Fan, A. M. Fox, M. S. Skolnick, D. M. Whittaker, A. Tahraoui, T. F. Krauss, and J. S. Roberts, “Ultrafast nonlinear response of AlGaAs two-dimensional photonic crystal waveguides,” Appl. Phys. Lett. 83, 851-853 (2003).
[CrossRef]

Gallo, K.

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 (2000).
[CrossRef]

Gosele, U.

H. W. Tan, H. M. van Driel, S. L. Schweizer, R. B. Wehrspohn, and U. Gosele, “Nonlinear optical tuning of a two-dimensional silicon photonic crystal,” Phys. Rev. B 70, 205110 (2004).
[CrossRef]

Haché, A.

A. Haché and M. Bourgeois, “Ultrafast all-optical switching in a silicon-based photonic crystal,” Appl. Phys. Lett. 77, 4089-4091 (2000).
[CrossRef]

Haldane, F. D.

S. Raghu and F. D. Haldane, “Analogs of quantum-Hall-effect edge states in photonic crystals,” Phys. Rev. A 78, 033834 (2008).
[CrossRef]

Haldane, F. D. M.

F. D. M. Haldane and S. Raghu, “Possible realization of directional optical waveguides in photonic crystals with broken time-reversal symmetry,” Phys. Rev. Lett. 100, 013904 (2008).
[CrossRef] [PubMed]

Halperin, B. I.

R. Rammal, G. Toulouse, M. T. Jackel, and B. I. Halperin, “Quantized Hall conductance and edge states: two-dimensional strips with a periodic potential,” Phys. Rev. B 27, 5142-5145 (1983).
[CrossRef]

B. I. Halperin, “Quantized Hall conductance, current-carrying edge states, and the existence of extended states in a two-dimensional disordered potential,” Phys. Rev. B 25, 2185-2190 (1982).
[CrossRef]

Ibanescu, M.

M. Soljacic, M. Ibanescu, S. G. Johnson, Y. Fink, and J. D. Joannopoulos, “Optimal bistable switching I: nonlinear photonic crystals,” Phys. Rev. E 66, 055601 (2002).
[CrossRef]

Jackel, M. T.

R. Rammal, G. Toulouse, M. T. Jackel, and B. I. Halperin, “Quantized Hall conductance and edge states: two-dimensional strips with a periodic potential,” Phys. Rev. B 27, 5142-5145 (1983).
[CrossRef]

Jiang, X.

X. Jiang, C. Zhou, X. Yu, S. Fan, M. Soljacic, and J. D. Joannopoulos, “Interplay between supercollimation and nonlinearity: a novel method of controlling beams in photonic crystals,” Appl. Phys. Lett. 91, 031105 (2007).
[CrossRef]

Joannopoulos, J. D.

Z. Wang, Y. D. Chong, J. D. Joannopoulos, and M. Soljacic, “Reflection-free one-way edge modes in a gyromagnetic photonic crystal,” Phys. Rev. Lett. 100, 013905 (2008).
[CrossRef] [PubMed]

X. Jiang, C. Zhou, X. Yu, S. Fan, M. Soljacic, and J. D. Joannopoulos, “Interplay between supercollimation and nonlinearity: a novel method of controlling beams in photonic crystals,” Appl. Phys. Lett. 91, 031105 (2007).
[CrossRef]

M. Soljacic, C. Luo, J. D. Joannopoulos, and S. Fan, “Nonlinear photonic crystal microdevices for optical integration,” Opt. Lett. 28, 637-639 (2003).
[CrossRef] [PubMed]

M. Soljacic, M. Ibanescu, S. G. Johnson, Y. Fink, and J. D. Joannopoulos, “Optimal bistable switching I: nonlinear photonic crystals,” Phys. Rev. E 66, 055601 (2002).
[CrossRef]

Johnson, S. G.

M. Soljacic, M. Ibanescu, S. G. Johnson, Y. Fink, and J. D. Joannopoulos, “Optimal bistable switching I: nonlinear photonic crystals,” Phys. Rev. E 66, 055601 (2002).
[CrossRef]

Krauss, T. F.

A. D. Bristow, J. P. R. Wells, W. H. Fan, A. M. Fox, M. S. Skolnick, D. M. Whittaker, A. Tahraoui, T. F. Krauss, and J. S. Roberts, “Ultrafast nonlinear response of AlGaAs two-dimensional photonic crystal waveguides,” Appl. Phys. Lett. 83, 851-853 (2003).
[CrossRef]

Luo, C.

Parameswaran, K. R.

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 (2000).
[CrossRef]

Raghu, S.

F. D. M. Haldane and S. Raghu, “Possible realization of directional optical waveguides in photonic crystals with broken time-reversal symmetry,” Phys. Rev. Lett. 100, 013904 (2008).
[CrossRef] [PubMed]

S. Raghu and F. D. Haldane, “Analogs of quantum-Hall-effect edge states in photonic crystals,” Phys. Rev. A 78, 033834 (2008).
[CrossRef]

Rammal, R.

R. Rammal, G. Toulouse, M. T. Jackel, and B. I. Halperin, “Quantized Hall conductance and edge states: two-dimensional strips with a periodic potential,” Phys. Rev. B 27, 5142-5145 (1983).
[CrossRef]

Roberts, J. S.

A. D. Bristow, J. P. R. Wells, W. H. Fan, A. M. Fox, M. S. Skolnick, D. M. Whittaker, A. Tahraoui, T. F. Krauss, and J. S. Roberts, “Ultrafast nonlinear response of AlGaAs two-dimensional photonic crystal waveguides,” Appl. Phys. Lett. 83, 851-853 (2003).
[CrossRef]

Schweizer, S. L.

H. W. Tan, H. M. van Driel, S. L. Schweizer, R. B. Wehrspohn, and U. Gosele, “Nonlinear optical tuning of a two-dimensional silicon photonic crystal,” Phys. Rev. B 70, 205110 (2004).
[CrossRef]

Skolnick, M. S.

A. D. Bristow, J. P. R. Wells, W. H. Fan, A. M. Fox, M. S. Skolnick, D. M. Whittaker, A. Tahraoui, T. F. Krauss, and J. S. Roberts, “Ultrafast nonlinear response of AlGaAs two-dimensional photonic crystal waveguides,” Appl. Phys. Lett. 83, 851-853 (2003).
[CrossRef]

Soljacic, M.

Z. Wang, Y. D. Chong, J. D. Joannopoulos, and M. Soljacic, “Reflection-free one-way edge modes in a gyromagnetic photonic crystal,” Phys. Rev. Lett. 100, 013905 (2008).
[CrossRef] [PubMed]

X. Jiang, C. Zhou, X. Yu, S. Fan, M. Soljacic, and J. D. Joannopoulos, “Interplay between supercollimation and nonlinearity: a novel method of controlling beams in photonic crystals,” Appl. Phys. Lett. 91, 031105 (2007).
[CrossRef]

M. Soljacic, C. Luo, J. D. Joannopoulos, and S. Fan, “Nonlinear photonic crystal microdevices for optical integration,” Opt. Lett. 28, 637-639 (2003).
[CrossRef] [PubMed]

M. F. Yanik, S. Fan, and M. Soljacic, “High-contrast all-optical bistable switching in photonic crystal microcavities,” Appl. Phys. Lett. 83, 2739-2741 (2003).
[CrossRef]

M. Soljacic, M. Ibanescu, S. G. Johnson, Y. Fink, and J. D. Joannopoulos, “Optimal bistable switching I: nonlinear photonic crystals,” Phys. Rev. E 66, 055601 (2002).
[CrossRef]

Tahraoui, A.

A. D. Bristow, J. P. R. Wells, W. H. Fan, A. M. Fox, M. S. Skolnick, D. M. Whittaker, A. Tahraoui, T. F. Krauss, and J. S. Roberts, “Ultrafast nonlinear response of AlGaAs two-dimensional photonic crystal waveguides,” Appl. Phys. Lett. 83, 851-853 (2003).
[CrossRef]

Tan, H. W.

H. W. Tan, H. M. van Driel, S. L. Schweizer, R. B. Wehrspohn, and U. Gosele, “Nonlinear optical tuning of a two-dimensional silicon photonic crystal,” Phys. Rev. B 70, 205110 (2004).
[CrossRef]

Toulouse, G.

R. Rammal, G. Toulouse, M. T. Jackel, and B. I. Halperin, “Quantized Hall conductance and edge states: two-dimensional strips with a periodic potential,” Phys. Rev. B 27, 5142-5145 (1983).
[CrossRef]

van Driel, H. M.

H. W. Tan, H. M. van Driel, S. L. Schweizer, R. B. Wehrspohn, and U. Gosele, “Nonlinear optical tuning of a two-dimensional silicon photonic crystal,” Phys. Rev. B 70, 205110 (2004).
[CrossRef]

Veronis, G.

Z. Yu, G. Veronis, Z. Wang, and S. Fan, “One-way electromagnetic waveguide formed at the interface between a plasmonic metal under a static magnetic field and a photonic crystal,” Phys. Rev. Lett. 100, 023902 (2008).
[CrossRef] [PubMed]

Wang, Z.

Z. Yu, G. Veronis, Z. Wang, and S. Fan, “One-way electromagnetic waveguide formed at the interface between a plasmonic metal under a static magnetic field and a photonic crystal,” Phys. Rev. Lett. 100, 023902 (2008).
[CrossRef] [PubMed]

Z. Wang, Y. D. Chong, J. D. Joannopoulos, and M. Soljacic, “Reflection-free one-way edge modes in a gyromagnetic photonic crystal,” Phys. Rev. Lett. 100, 013905 (2008).
[CrossRef] [PubMed]

Wehrspohn, R. B.

H. W. Tan, H. M. van Driel, S. L. Schweizer, R. B. Wehrspohn, and U. Gosele, “Nonlinear optical tuning of a two-dimensional silicon photonic crystal,” Phys. Rev. B 70, 205110 (2004).
[CrossRef]

Wells, J. P. R.

A. D. Bristow, J. P. R. Wells, W. H. Fan, A. M. Fox, M. S. Skolnick, D. M. Whittaker, A. Tahraoui, T. F. Krauss, and J. S. Roberts, “Ultrafast nonlinear response of AlGaAs two-dimensional photonic crystal waveguides,” Appl. Phys. Lett. 83, 851-853 (2003).
[CrossRef]

Wen, X. G.

X. G. Wen, “Gapless boundary excitations in the quantum Hall states and in the chiral spin states,” Phys. Rev. B 43, 11025-11036 (1991).
[CrossRef]

Whittaker, D. M.

A. D. Bristow, J. P. R. Wells, W. H. Fan, A. M. Fox, M. S. Skolnick, D. M. Whittaker, A. Tahraoui, T. F. Krauss, and J. S. Roberts, “Ultrafast nonlinear response of AlGaAs two-dimensional photonic crystal waveguides,” Appl. Phys. Lett. 83, 851-853 (2003).
[CrossRef]

Yanik, M. F.

M. F. Yanik, S. Fan, and M. Soljacic, “High-contrast all-optical bistable switching in photonic crystal microcavities,” Appl. Phys. Lett. 83, 2739-2741 (2003).
[CrossRef]

Yu, X.

X. Jiang, C. Zhou, X. Yu, S. Fan, M. Soljacic, and J. D. Joannopoulos, “Interplay between supercollimation and nonlinearity: a novel method of controlling beams in photonic crystals,” Appl. Phys. Lett. 91, 031105 (2007).
[CrossRef]

Yu, Z.

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

Z. Yu, G. Veronis, Z. Wang, and S. Fan, “One-way electromagnetic waveguide formed at the interface between a plasmonic metal under a static magnetic field and a photonic crystal,” Phys. Rev. Lett. 100, 023902 (2008).
[CrossRef] [PubMed]

Zhou, C.

X. Jiang, C. Zhou, X. Yu, S. Fan, M. Soljacic, and J. D. Joannopoulos, “Interplay between supercollimation and nonlinearity: a novel method of controlling beams in photonic crystals,” Appl. Phys. Lett. 91, 031105 (2007).
[CrossRef]

Appl. Phys. Lett. (5)

M. F. Yanik, S. Fan, and M. Soljacic, “High-contrast all-optical bistable switching in photonic crystal microcavities,” Appl. Phys. Lett. 83, 2739-2741 (2003).
[CrossRef]

A. D. Bristow, J. P. R. Wells, W. H. Fan, A. M. Fox, M. S. Skolnick, D. M. Whittaker, A. Tahraoui, T. F. Krauss, and J. S. Roberts, “Ultrafast nonlinear response of AlGaAs two-dimensional photonic crystal waveguides,” Appl. Phys. Lett. 83, 851-853 (2003).
[CrossRef]

A. Haché and M. Bourgeois, “Ultrafast all-optical switching in a silicon-based photonic crystal,” Appl. Phys. Lett. 77, 4089-4091 (2000).
[CrossRef]

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 (2000).
[CrossRef]

X. Jiang, C. Zhou, X. Yu, S. Fan, M. Soljacic, and J. D. Joannopoulos, “Interplay between supercollimation and nonlinearity: a novel method of controlling beams in photonic crystals,” Appl. Phys. Lett. 91, 031105 (2007).
[CrossRef]

Nat. Photon. (1)

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

Opt. Lett. (1)

Phys. Rev. A (1)

S. Raghu and F. D. Haldane, “Analogs of quantum-Hall-effect edge states in photonic crystals,” Phys. Rev. A 78, 033834 (2008).
[CrossRef]

Phys. Rev. B (4)

B. I. Halperin, “Quantized Hall conductance, current-carrying edge states, and the existence of extended states in a two-dimensional disordered potential,” Phys. Rev. B 25, 2185-2190 (1982).
[CrossRef]

R. Rammal, G. Toulouse, M. T. Jackel, and B. I. Halperin, “Quantized Hall conductance and edge states: two-dimensional strips with a periodic potential,” Phys. Rev. B 27, 5142-5145 (1983).
[CrossRef]

X. G. Wen, “Gapless boundary excitations in the quantum Hall states and in the chiral spin states,” Phys. Rev. B 43, 11025-11036 (1991).
[CrossRef]

H. W. Tan, H. M. van Driel, S. L. Schweizer, R. B. Wehrspohn, and U. Gosele, “Nonlinear optical tuning of a two-dimensional silicon photonic crystal,” Phys. Rev. B 70, 205110 (2004).
[CrossRef]

Phys. Rev. E (1)

M. Soljacic, M. Ibanescu, S. G. Johnson, Y. Fink, and J. D. Joannopoulos, “Optimal bistable switching I: nonlinear photonic crystals,” Phys. Rev. E 66, 055601 (2002).
[CrossRef]

Phys. Rev. Lett. (3)

F. D. M. Haldane and S. Raghu, “Possible realization of directional optical waveguides in photonic crystals with broken time-reversal symmetry,” Phys. Rev. Lett. 100, 013904 (2008).
[CrossRef] [PubMed]

Z. Wang, Y. D. Chong, J. D. Joannopoulos, and M. Soljacic, “Reflection-free one-way edge modes in a gyromagnetic photonic crystal,” Phys. Rev. Lett. 100, 013905 (2008).
[CrossRef] [PubMed]

Z. Yu, G. Veronis, Z. Wang, and S. Fan, “One-way electromagnetic waveguide formed at the interface between a plasmonic metal under a static magnetic field and a photonic crystal,” Phys. Rev. Lett. 100, 023902 (2008).
[CrossRef] [PubMed]

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Figures (4)

Fig. 1
Fig. 1

Schematic of nonreciprocal interband photonic transitions. (a) Structure of a silicon waveguide and dynamical modulation is applied to the dark region. (b) Band structure of the silicon waveguide. Frequency and wave vector shifts as induced by dynamic modulation and marked with arrows. Gray dots (pink online) correspond to mode 1 (at frequency ω 1 ) and mode 2(at frequency ω 2 ), respectively.

Fig. 2
Fig. 2

The spatial evolution of the photonic flux N between two modes. Red ( N ( 0 ) = 1 ) and blue ( N ( 0 ) = 0 ) curves represent mode 1 and mode 2, respectively. The dynamic modulation and nonlinear parameters are (a) c = 0.155 , γ = 0 , ( a ) c = 0.155 , γ = 0.5 , (b) c = 0.12 , γ = 0 , ( b ) c = 0.12 , γ = 0.5 , (c) c = 0.03 , γ = 0 , ( c ) c = 0.03 , γ = 0.5 .

Fig. 3
Fig. 3

Evolution of the phase space motions (the first column) with system parameters (a) c = 0.03 , γ = 0.5 , (b) c = 0.12 , γ = 0.5 , (c) c = 0.155 , γ = 0.5 , (d) c = 0.155 , γ = 0 . The second column is the corresponding energy curve at ϕ = 0 (solid curve) and ϕ = ± π (dashed curve). The arrows indicate the shifting direction of the fixed points with the increase of ( c γ ) . Red curves (online) represent the evolution of our system with the initial condition: M ( 0 ) = 1 , R ( 0 ) = F ( 0 ) = 0 .

Fig. 4
Fig. 4

The spatial evolution of the photonic flux N between two modes with dissipative ( α = 0.01 ) . Curves are corresponding to mode 1 ( N ( 0 ) = 1 ) and mode 2 ( N ( 0 ) = 0 ) , respectively. System parameters are (a) c = 0.155 , γ = 0 , (b) c = 0.155 , γ = 0.5 , (c) c = 0.12 , γ = 0.5 , (d) c = 0.03 , γ = 0.5 .

Equations (12)

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ϵ ( x , y , t ) = f ( x ) cos ( Δ ω t ( Δ k ) y ) ,
E ( x , y , t ) = n = 1 2 a n ( y ) E n ( x ) e i ( k n y + ω n t ) ,
d d y ( a 1 a 2 ) = ( i γ ( | a 1 | 2 + 2 | a 2 | 2 ) i c e i Δ k y i c e i Δ k y i γ ( 2 | a 1 | 2 + | a 2 | 2 ) ) ( a 1 a 2 ) ,
a ̇ 1 = d d y a 1 = i γ ( | a 1 | 2 + 2 | a 2 | 2 ) a 1 + i c a 2 ,
a ̇ 2 = d d y a 2 = i γ ( 2 | a 1 | 2 + | a 2 | 2 ) a 2 + i c a 1 ,
M ̇ = 4 c F ,
R ̇ = γ M F ,
F ̇ = γ M R + c M .
M ̇ = 2 c 1 M 2 sin ϕ ,
ϕ ̇ = γ M + 2 c M 1 M 2 cos ϕ .
H = γ 2 M 2 2 c 1 M 2 cos ϕ .
d d y ( a 1 a 2 ) = ( α 1 i γ ( | a 1 | 2 + 2 | a 2 | 2 ) i c e i Δ k y i c e i Δ k y α 2 i γ ( 2 | a 1 | 2 + | a 2 | 2 ) ) ( a 1 a 2 ) ,

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