Abstract

We predict that nonlinear waveguides which support frozen light associated with a degenerate photonic band edge, where the dispersion relation is locally quartic, exhibit a tunable, all-optical switching response. The thresholds for switching are orders-of-magnitude lower than at regular band edges. By adjusting the input condition, bistability can be eliminated, preventing switching hysteresis.

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References

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  1. H. M. Gibbs, Optical Bistability: Controlling Light with Light (Academic Press, Inc., Orlando, FL, 1985).
  2. G. Agrawal, Nonlinear Fiber Optics (Academic Press, 1989).
  3. J. Leuthold, C. Koos, and W. Freude, “Nonlinear silicon photonics,” Nat. Photonics4, 535–544 (2010).
    [CrossRef]
  4. M. Soljačić, M. Ibanescu, S. G. Johnson, Y. Fink, and J. D. Joannopoulos, “Optimal bistable switching in non-linear photonic crystals,” Phys. Rev. E66, 055601 (2002).
    [CrossRef]
  5. I. V. Kabakova, T. Walsh, C. M. de Sterke, and B. J. Eggleton, “Performance of field-enhanced optical switching in fiber bragg gratings,” J. Opt. Soc. Am. B27, 1343–1351 (2010).
    [CrossRef]
  6. B. E. A. Saleh, M. C. Teich, and B. E. Saleh, Fundamentals of Photonics, vol. 22 (Wiley Online Library, 1991).
    [CrossRef]
  7. L. L. Goddard, J. S. Kallman, and T. C. Bond, “Rapidly reconfigurable alloptical universal logic gates,” Proc. SPIE636863680H (2006).
    [CrossRef]
  8. W. Chen and D. L. Mills, “Gap solitons and the nonlinear optical response of superlattices,” Phys. Rev. Lett.58, 160–163 (1987).
    [CrossRef] [PubMed]
  9. C. M. de Sterke and J. E. Sipe, “Gap solitons,” Prog. Optics33, 203–260 (1994).
  10. B. J. Eggleton, C. M. de Sterke, A. B. Aceves, J. E. Sipe, T. A. Strasser, and R. E. Slusher, “Modulational instability and tunable multiple soliton generation in apodized fiber gratings,” Opt. Comm.149, 267–271 (1998).
    [CrossRef]
  11. N. G. R. Broderick, “Bistable switching in nonlinear bragg gratings,” Opt. Comm.148, 90–94 (1998).
    [CrossRef]
  12. F. Eilenberger, C. M. de Sterke, and B. J. Eggleton, “Soliton mediated optical quantization in the transmission of one-dimensional photonic crystals,” Opt. Express18, 12708–12718 (2010).
    [CrossRef] [PubMed]
  13. A. Figotin and I. Vitebskiy, “Slow wave phenomena in photonic crystals,” Laser Photon. Rev.5, 1863–8899 (2010).
  14. P. Colman, C. Husko, S. Combrié, I. Sagnes, C. W. Wong, and A. De Rossi, “Temporal solitons and pulse compression in photonic crystal waveguides,” Nat. Photonics4, 862–868 (2010).
    [CrossRef]
  15. A. Figotin and I. Vitebskiy, “Frozen light in photonic crystals with degenerate band edge,” Phys. Rev. E74, 066613 (2006).
    [CrossRef]
  16. N. Gutman, L. C. Botten, A. A. Sukhorukov, and C. M. de Sterke, “Degenerate band edges in optical fiber with multiple grating: efficient coupling to slow light,” Opt. Lett.36, 3257–3259 (2011).
    [CrossRef] [PubMed]
  17. N. Gutman, C. M. de Sterke, A. A. Sukhorukov, and L. C. Botten, “Slow and frozen light in optical waveguides with multiple gratings: Degenerate band edges and stationary inflection points,” Phys. Rev. A85, 033804 (2012).
    [CrossRef]
  18. A. A. Sukhorukov, C. J. Handmer, C. M. de Sterke, and M. J. Steel, “Slow light with flat or offset band edges in few-mode fiber with two gratings,” Opt. Express15, 17954 (2007).
    [CrossRef] [PubMed]
  19. A. A. Sukhorukov, A. V. Lavrinenko, D. N. Chigrin, D. E. Pelinovsky, and Y. S. Kivshar, “Slow-light dispersion in coupled periodic waveguides,” J. Opt. Soc. Am. B25, C65–C74 (2008).
    [CrossRef]
  20. P. Blown, C. Fisher, F. Lawrence, N. Gutman, and C. M. de Sterke, “Semi-analytic method for slow light photonic crystal waveguide design,” Phot. Nano. Fund. Appl. In Press (2012).
    [CrossRef]

2012 (2)

N. Gutman, C. M. de Sterke, A. A. Sukhorukov, and L. C. Botten, “Slow and frozen light in optical waveguides with multiple gratings: Degenerate band edges and stationary inflection points,” Phys. Rev. A85, 033804 (2012).
[CrossRef]

P. Blown, C. Fisher, F. Lawrence, N. Gutman, and C. M. de Sterke, “Semi-analytic method for slow light photonic crystal waveguide design,” Phot. Nano. Fund. Appl. In Press (2012).
[CrossRef]

2011 (1)

2010 (5)

J. Leuthold, C. Koos, and W. Freude, “Nonlinear silicon photonics,” Nat. Photonics4, 535–544 (2010).
[CrossRef]

I. V. Kabakova, T. Walsh, C. M. de Sterke, and B. J. Eggleton, “Performance of field-enhanced optical switching in fiber bragg gratings,” J. Opt. Soc. Am. B27, 1343–1351 (2010).
[CrossRef]

F. Eilenberger, C. M. de Sterke, and B. J. Eggleton, “Soliton mediated optical quantization in the transmission of one-dimensional photonic crystals,” Opt. Express18, 12708–12718 (2010).
[CrossRef] [PubMed]

A. Figotin and I. Vitebskiy, “Slow wave phenomena in photonic crystals,” Laser Photon. Rev.5, 1863–8899 (2010).

P. Colman, C. Husko, S. Combrié, I. Sagnes, C. W. Wong, and A. De Rossi, “Temporal solitons and pulse compression in photonic crystal waveguides,” Nat. Photonics4, 862–868 (2010).
[CrossRef]

2008 (1)

2007 (1)

2006 (2)

A. Figotin and I. Vitebskiy, “Frozen light in photonic crystals with degenerate band edge,” Phys. Rev. E74, 066613 (2006).
[CrossRef]

L. L. Goddard, J. S. Kallman, and T. C. Bond, “Rapidly reconfigurable alloptical universal logic gates,” Proc. SPIE636863680H (2006).
[CrossRef]

2002 (1)

M. Soljačić, M. Ibanescu, S. G. Johnson, Y. Fink, and J. D. Joannopoulos, “Optimal bistable switching in non-linear photonic crystals,” Phys. Rev. E66, 055601 (2002).
[CrossRef]

1998 (2)

B. J. Eggleton, C. M. de Sterke, A. B. Aceves, J. E. Sipe, T. A. Strasser, and R. E. Slusher, “Modulational instability and tunable multiple soliton generation in apodized fiber gratings,” Opt. Comm.149, 267–271 (1998).
[CrossRef]

N. G. R. Broderick, “Bistable switching in nonlinear bragg gratings,” Opt. Comm.148, 90–94 (1998).
[CrossRef]

1994 (1)

C. M. de Sterke and J. E. Sipe, “Gap solitons,” Prog. Optics33, 203–260 (1994).

1987 (1)

W. Chen and D. L. Mills, “Gap solitons and the nonlinear optical response of superlattices,” Phys. Rev. Lett.58, 160–163 (1987).
[CrossRef] [PubMed]

Aceves, A. B.

B. J. Eggleton, C. M. de Sterke, A. B. Aceves, J. E. Sipe, T. A. Strasser, and R. E. Slusher, “Modulational instability and tunable multiple soliton generation in apodized fiber gratings,” Opt. Comm.149, 267–271 (1998).
[CrossRef]

Agrawal, G.

G. Agrawal, Nonlinear Fiber Optics (Academic Press, 1989).

Blown, P.

P. Blown, C. Fisher, F. Lawrence, N. Gutman, and C. M. de Sterke, “Semi-analytic method for slow light photonic crystal waveguide design,” Phot. Nano. Fund. Appl. In Press (2012).
[CrossRef]

Bond, T. C.

L. L. Goddard, J. S. Kallman, and T. C. Bond, “Rapidly reconfigurable alloptical universal logic gates,” Proc. SPIE636863680H (2006).
[CrossRef]

Botten, L. C.

N. Gutman, C. M. de Sterke, A. A. Sukhorukov, and L. C. Botten, “Slow and frozen light in optical waveguides with multiple gratings: Degenerate band edges and stationary inflection points,” Phys. Rev. A85, 033804 (2012).
[CrossRef]

N. Gutman, L. C. Botten, A. A. Sukhorukov, and C. M. de Sterke, “Degenerate band edges in optical fiber with multiple grating: efficient coupling to slow light,” Opt. Lett.36, 3257–3259 (2011).
[CrossRef] [PubMed]

Broderick, N. G. R.

N. G. R. Broderick, “Bistable switching in nonlinear bragg gratings,” Opt. Comm.148, 90–94 (1998).
[CrossRef]

Chen, W.

W. Chen and D. L. Mills, “Gap solitons and the nonlinear optical response of superlattices,” Phys. Rev. Lett.58, 160–163 (1987).
[CrossRef] [PubMed]

Chigrin, D. N.

Colman, P.

P. Colman, C. Husko, S. Combrié, I. Sagnes, C. W. Wong, and A. De Rossi, “Temporal solitons and pulse compression in photonic crystal waveguides,” Nat. Photonics4, 862–868 (2010).
[CrossRef]

Combrié, S.

P. Colman, C. Husko, S. Combrié, I. Sagnes, C. W. Wong, and A. De Rossi, “Temporal solitons and pulse compression in photonic crystal waveguides,” Nat. Photonics4, 862–868 (2010).
[CrossRef]

De Rossi, A.

P. Colman, C. Husko, S. Combrié, I. Sagnes, C. W. Wong, and A. De Rossi, “Temporal solitons and pulse compression in photonic crystal waveguides,” Nat. Photonics4, 862–868 (2010).
[CrossRef]

de Sterke, C. M.

N. Gutman, C. M. de Sterke, A. A. Sukhorukov, and L. C. Botten, “Slow and frozen light in optical waveguides with multiple gratings: Degenerate band edges and stationary inflection points,” Phys. Rev. A85, 033804 (2012).
[CrossRef]

P. Blown, C. Fisher, F. Lawrence, N. Gutman, and C. M. de Sterke, “Semi-analytic method for slow light photonic crystal waveguide design,” Phot. Nano. Fund. Appl. In Press (2012).
[CrossRef]

N. Gutman, L. C. Botten, A. A. Sukhorukov, and C. M. de Sterke, “Degenerate band edges in optical fiber with multiple grating: efficient coupling to slow light,” Opt. Lett.36, 3257–3259 (2011).
[CrossRef] [PubMed]

F. Eilenberger, C. M. de Sterke, and B. J. Eggleton, “Soliton mediated optical quantization in the transmission of one-dimensional photonic crystals,” Opt. Express18, 12708–12718 (2010).
[CrossRef] [PubMed]

I. V. Kabakova, T. Walsh, C. M. de Sterke, and B. J. Eggleton, “Performance of field-enhanced optical switching in fiber bragg gratings,” J. Opt. Soc. Am. B27, 1343–1351 (2010).
[CrossRef]

A. A. Sukhorukov, C. J. Handmer, C. M. de Sterke, and M. J. Steel, “Slow light with flat or offset band edges in few-mode fiber with two gratings,” Opt. Express15, 17954 (2007).
[CrossRef] [PubMed]

B. J. Eggleton, C. M. de Sterke, A. B. Aceves, J. E. Sipe, T. A. Strasser, and R. E. Slusher, “Modulational instability and tunable multiple soliton generation in apodized fiber gratings,” Opt. Comm.149, 267–271 (1998).
[CrossRef]

C. M. de Sterke and J. E. Sipe, “Gap solitons,” Prog. Optics33, 203–260 (1994).

Eggleton, B. J.

Eilenberger, F.

Figotin, A.

A. Figotin and I. Vitebskiy, “Slow wave phenomena in photonic crystals,” Laser Photon. Rev.5, 1863–8899 (2010).

A. Figotin and I. Vitebskiy, “Frozen light in photonic crystals with degenerate band edge,” Phys. Rev. E74, 066613 (2006).
[CrossRef]

Fink, Y.

M. Soljačić, M. Ibanescu, S. G. Johnson, Y. Fink, and J. D. Joannopoulos, “Optimal bistable switching in non-linear photonic crystals,” Phys. Rev. E66, 055601 (2002).
[CrossRef]

Fisher, C.

P. Blown, C. Fisher, F. Lawrence, N. Gutman, and C. M. de Sterke, “Semi-analytic method for slow light photonic crystal waveguide design,” Phot. Nano. Fund. Appl. In Press (2012).
[CrossRef]

Freude, W.

J. Leuthold, C. Koos, and W. Freude, “Nonlinear silicon photonics,” Nat. Photonics4, 535–544 (2010).
[CrossRef]

Gibbs, H. M.

H. M. Gibbs, Optical Bistability: Controlling Light with Light (Academic Press, Inc., Orlando, FL, 1985).

Goddard, L. L.

L. L. Goddard, J. S. Kallman, and T. C. Bond, “Rapidly reconfigurable alloptical universal logic gates,” Proc. SPIE636863680H (2006).
[CrossRef]

Gutman, N.

P. Blown, C. Fisher, F. Lawrence, N. Gutman, and C. M. de Sterke, “Semi-analytic method for slow light photonic crystal waveguide design,” Phot. Nano. Fund. Appl. In Press (2012).
[CrossRef]

N. Gutman, C. M. de Sterke, A. A. Sukhorukov, and L. C. Botten, “Slow and frozen light in optical waveguides with multiple gratings: Degenerate band edges and stationary inflection points,” Phys. Rev. A85, 033804 (2012).
[CrossRef]

N. Gutman, L. C. Botten, A. A. Sukhorukov, and C. M. de Sterke, “Degenerate band edges in optical fiber with multiple grating: efficient coupling to slow light,” Opt. Lett.36, 3257–3259 (2011).
[CrossRef] [PubMed]

Handmer, C. J.

Husko, C.

P. Colman, C. Husko, S. Combrié, I. Sagnes, C. W. Wong, and A. De Rossi, “Temporal solitons and pulse compression in photonic crystal waveguides,” Nat. Photonics4, 862–868 (2010).
[CrossRef]

Ibanescu, M.

M. Soljačić, M. Ibanescu, S. G. Johnson, Y. Fink, and J. D. Joannopoulos, “Optimal bistable switching in non-linear photonic crystals,” Phys. Rev. E66, 055601 (2002).
[CrossRef]

Joannopoulos, J. D.

M. Soljačić, M. Ibanescu, S. G. Johnson, Y. Fink, and J. D. Joannopoulos, “Optimal bistable switching in non-linear photonic crystals,” Phys. Rev. E66, 055601 (2002).
[CrossRef]

Johnson, S. G.

M. Soljačić, M. Ibanescu, S. G. Johnson, Y. Fink, and J. D. Joannopoulos, “Optimal bistable switching in non-linear photonic crystals,” Phys. Rev. E66, 055601 (2002).
[CrossRef]

Kabakova, I. V.

Kallman, J. S.

L. L. Goddard, J. S. Kallman, and T. C. Bond, “Rapidly reconfigurable alloptical universal logic gates,” Proc. SPIE636863680H (2006).
[CrossRef]

Kivshar, Y. S.

Koos, C.

J. Leuthold, C. Koos, and W. Freude, “Nonlinear silicon photonics,” Nat. Photonics4, 535–544 (2010).
[CrossRef]

Lavrinenko, A. V.

Lawrence, F.

P. Blown, C. Fisher, F. Lawrence, N. Gutman, and C. M. de Sterke, “Semi-analytic method for slow light photonic crystal waveguide design,” Phot. Nano. Fund. Appl. In Press (2012).
[CrossRef]

Leuthold, J.

J. Leuthold, C. Koos, and W. Freude, “Nonlinear silicon photonics,” Nat. Photonics4, 535–544 (2010).
[CrossRef]

Mills, D. L.

W. Chen and D. L. Mills, “Gap solitons and the nonlinear optical response of superlattices,” Phys. Rev. Lett.58, 160–163 (1987).
[CrossRef] [PubMed]

Pelinovsky, D. E.

Sagnes, I.

P. Colman, C. Husko, S. Combrié, I. Sagnes, C. W. Wong, and A. De Rossi, “Temporal solitons and pulse compression in photonic crystal waveguides,” Nat. Photonics4, 862–868 (2010).
[CrossRef]

Saleh, B. E.

B. E. A. Saleh, M. C. Teich, and B. E. Saleh, Fundamentals of Photonics, vol. 22 (Wiley Online Library, 1991).
[CrossRef]

Saleh, B. E. A.

B. E. A. Saleh, M. C. Teich, and B. E. Saleh, Fundamentals of Photonics, vol. 22 (Wiley Online Library, 1991).
[CrossRef]

Sipe, J. E.

B. J. Eggleton, C. M. de Sterke, A. B. Aceves, J. E. Sipe, T. A. Strasser, and R. E. Slusher, “Modulational instability and tunable multiple soliton generation in apodized fiber gratings,” Opt. Comm.149, 267–271 (1998).
[CrossRef]

C. M. de Sterke and J. E. Sipe, “Gap solitons,” Prog. Optics33, 203–260 (1994).

Slusher, R. E.

B. J. Eggleton, C. M. de Sterke, A. B. Aceves, J. E. Sipe, T. A. Strasser, and R. E. Slusher, “Modulational instability and tunable multiple soliton generation in apodized fiber gratings,” Opt. Comm.149, 267–271 (1998).
[CrossRef]

Soljacic, M.

M. Soljačić, M. Ibanescu, S. G. Johnson, Y. Fink, and J. D. Joannopoulos, “Optimal bistable switching in non-linear photonic crystals,” Phys. Rev. E66, 055601 (2002).
[CrossRef]

Steel, M. J.

Strasser, T. A.

B. J. Eggleton, C. M. de Sterke, A. B. Aceves, J. E. Sipe, T. A. Strasser, and R. E. Slusher, “Modulational instability and tunable multiple soliton generation in apodized fiber gratings,” Opt. Comm.149, 267–271 (1998).
[CrossRef]

Sukhorukov, A. A.

Teich, M. C.

B. E. A. Saleh, M. C. Teich, and B. E. Saleh, Fundamentals of Photonics, vol. 22 (Wiley Online Library, 1991).
[CrossRef]

Vitebskiy, I.

A. Figotin and I. Vitebskiy, “Slow wave phenomena in photonic crystals,” Laser Photon. Rev.5, 1863–8899 (2010).

A. Figotin and I. Vitebskiy, “Frozen light in photonic crystals with degenerate band edge,” Phys. Rev. E74, 066613 (2006).
[CrossRef]

Walsh, T.

Wong, C. W.

P. Colman, C. Husko, S. Combrié, I. Sagnes, C. W. Wong, and A. De Rossi, “Temporal solitons and pulse compression in photonic crystal waveguides,” Nat. Photonics4, 862–868 (2010).
[CrossRef]

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

Laser Photon. Rev. (1)

A. Figotin and I. Vitebskiy, “Slow wave phenomena in photonic crystals,” Laser Photon. Rev.5, 1863–8899 (2010).

Nat. Photonics (2)

P. Colman, C. Husko, S. Combrié, I. Sagnes, C. W. Wong, and A. De Rossi, “Temporal solitons and pulse compression in photonic crystal waveguides,” Nat. Photonics4, 862–868 (2010).
[CrossRef]

J. Leuthold, C. Koos, and W. Freude, “Nonlinear silicon photonics,” Nat. Photonics4, 535–544 (2010).
[CrossRef]

Opt. Comm. (2)

B. J. Eggleton, C. M. de Sterke, A. B. Aceves, J. E. Sipe, T. A. Strasser, and R. E. Slusher, “Modulational instability and tunable multiple soliton generation in apodized fiber gratings,” Opt. Comm.149, 267–271 (1998).
[CrossRef]

N. G. R. Broderick, “Bistable switching in nonlinear bragg gratings,” Opt. Comm.148, 90–94 (1998).
[CrossRef]

Opt. Express (2)

Opt. Lett. (1)

Phot. Nano. Fund. Appl. (1)

P. Blown, C. Fisher, F. Lawrence, N. Gutman, and C. M. de Sterke, “Semi-analytic method for slow light photonic crystal waveguide design,” Phot. Nano. Fund. Appl. In Press (2012).
[CrossRef]

Phys. Rev. A (1)

N. Gutman, C. M. de Sterke, A. A. Sukhorukov, and L. C. Botten, “Slow and frozen light in optical waveguides with multiple gratings: Degenerate band edges and stationary inflection points,” Phys. Rev. A85, 033804 (2012).
[CrossRef]

Phys. Rev. E (2)

A. Figotin and I. Vitebskiy, “Frozen light in photonic crystals with degenerate band edge,” Phys. Rev. E74, 066613 (2006).
[CrossRef]

M. Soljačić, M. Ibanescu, S. G. Johnson, Y. Fink, and J. D. Joannopoulos, “Optimal bistable switching in non-linear photonic crystals,” Phys. Rev. E66, 055601 (2002).
[CrossRef]

Phys. Rev. Lett. (1)

W. Chen and D. L. Mills, “Gap solitons and the nonlinear optical response of superlattices,” Phys. Rev. Lett.58, 160–163 (1987).
[CrossRef] [PubMed]

Proc. SPIE (1)

L. L. Goddard, J. S. Kallman, and T. C. Bond, “Rapidly reconfigurable alloptical universal logic gates,” Proc. SPIE636863680H (2006).
[CrossRef]

Prog. Optics (1)

C. M. de Sterke and J. E. Sipe, “Gap solitons,” Prog. Optics33, 203–260 (1994).

Other (3)

B. E. A. Saleh, M. C. Teich, and B. E. Saleh, Fundamentals of Photonics, vol. 22 (Wiley Online Library, 1991).
[CrossRef]

H. M. Gibbs, Optical Bistability: Controlling Light with Light (Academic Press, Inc., Orlando, FL, 1985).

G. Agrawal, Nonlinear Fiber Optics (Academic Press, 1989).

Supplementary Material (2)

» Media 1: AVI (703 KB)     
» Media 2: AVI (1021 KB)     

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Figures (3)

Fig. 1
Fig. 1

(a) Dispersion of propagating modes at a RBE (red) and DBE (blue). (c–d) Complex band structure of (c) RBE ( Media 1) and (d) DBE ( Media 2). Blue and red curves correspond to the propagating modes with real wavenumbers as shown in (a), black curves represent evanescent modes with complex wavenumbers. (b) Two-mode waveguide where DBE can be realized.

Fig. 2
Fig. 2

(a) Reflection versus the input parameters (α, β) at the DBE frequency for a wave-guide of length ρ1L = 12. (b) Field intensity inside the waveguide for the condition of minimum (blue) and maximum (red) reflection, corresponding to points A and B in (a), respectively. Dashed black line shows asymptotic for the blue line at small z. (c) Field enhancement inside the waveguide versus waveguide length (loglog scale) at input state A, showing that the enhancement is ∝ L2.

Fig. 3
Fig. 3

(a–b) Transmission vs. the input flux at the band edge frequency: (a) the DBE with input amplitudes i1 = 1 (red) and i2 = 1 (blue), corresponding to points C and D in Fig. 2(a), respectively; (b) the DBE with input amplitudes corresponding to point A in Fig. 2(a) (blue) and a RBE (black). Solid lines indicate stable while and plus (+) signs — unstable stationary solutions. (c) Threshold power for switching (100% transmission) versus the normalized waveguide length. (d) Field profile inside the waveguide at the nonlinear resonance.

Equations (1)

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i E 1 ± t ± i v E 1 ± z + ρ 1 E 1 + ρ 2 E 2 + [ Γ 11 ( | E 1 ± | 2 + 2 | E 1 | 2 ) + 2 Γ 12 ( | E 2 ± | 2 + | E 2 | 2 ) ] E 1 ± + 2 Γ 12 E 2 ± E 2 E 1 * = 0 , i E 2 ± t ± i v E 2 ± z ± δ E 2 ± + ρ 2 E 1 + [ Γ 22 ( | E 22 ± | 2 + 2 | E 2 | 2 ) + 2 Γ 12 ( | E 1 ± | 2 + | E 1 | 2 ) ] E 2 ± + 2 Γ 12 E 1 ± E 1 E 2 ± * = 0 ,

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