Abstract

We show the advantages of controlling the unstable dynamics of a semiconductor laser subject to conventional optical feedback by means of a second filtered feedback branch. We give an overview of the analytical solutions of the double cavity feedback and show numerically that the region of stabilization is much larger when using a second branch with filtered feedback than when using a conventional feedback one.

© 2009 OSA

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References

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  1. A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. García-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, “Chaos-based communications at high bit rates using commercial fibre-optic links,” Nature 437(7066), 343–346 (2005).
  2. E. Ott, C. Grebogi, and J. A. Yorke, “Controlling chaos,” Phys. Rev. Lett. 64(11), 1196–1199 (1990).
    [PubMed]
  3. K. Pyragas, “Continuous control of chaos by self-controlling feedback,” Phys. Lett. A 170(6), 421–428 (1992).
  4. E. Schöll and K. Pyragas, “Tunable semiconductor oscillator based on self-control of chaos in the dynamic hall effect,” Europhys. Lett. 24(3), 159–164 (1993).
  5. C. Lourenço and A. Babloyantz, “Control of chaos in networks with delay: A model for synchronization of cortical tissue,” Neural Comput. 6(6), 1141–1154 (1994).
  6. F. R. Ruiz-Oliveras and A. N. Pisarchik, “Phase-locking phenomenon in a semiconductor laser with external cavities,” Opt. Express 14(26), 12859–12867 (2006).
    [PubMed]
  7. H. Erzgräber, D. Lenstra, B. Krauskopf, A. P. Fischer, and G. Vemuri, “Feedback phase sensitivity of a semiconductor laser subject to filtered optical feedback: experiment and theory,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 76(2), 026212 (2007).
    [PubMed]
  8. M. Yousefi, D. Lenstra, G. Vemuri, and A. Fischer, “Control of nonlinear dynamics of a semiconductor laser with filtered optical feedback,” IEE Proc., Optoelectron. 148(5-6), 233–237 (2001).
  9. R. Lang and K. Kobayashi, “External optical feedback effects on semiconductor injection laser properties,” IEEE J. Quantum Electron. 16(3), 347–355 (1980).
  10. B. Tromborg, J. H. Osmundsen, and H. Olesen, “Stability analysis for a semiconductor laser in an external cavity,” IEEE J. Quantum Electron. 20(9), 1023–1032 (1984).
  11. M. Wolfrum and D. Turaev, “Instabilities of lasers with moderately delayed optical feedback,” Opt. Commun. 212(1-3), 127–138 (2002).
  12. V. Z. Tronciu, H.-J. Wünsche, M. Wolfrum, and M. Radziunas, “Semiconductor laser under resonant feedback from a Fabry-Perot resonator: Stability of continuous-wave operation,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 73(4), 046205 (2006).
    [PubMed]
  13. V. Z. Tronciu, I. V. Ermakov, P. Colet, and C. R. Mirasso, “Chaotic dynamics of a semiconductor laser with double cavity feedback: Applications to phase shift keying modulation,” Opt. Commun. 281(18), 4747–4752 (2008).

2008 (1)

V. Z. Tronciu, I. V. Ermakov, P. Colet, and C. R. Mirasso, “Chaotic dynamics of a semiconductor laser with double cavity feedback: Applications to phase shift keying modulation,” Opt. Commun. 281(18), 4747–4752 (2008).

2007 (1)

H. Erzgräber, D. Lenstra, B. Krauskopf, A. P. Fischer, and G. Vemuri, “Feedback phase sensitivity of a semiconductor laser subject to filtered optical feedback: experiment and theory,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 76(2), 026212 (2007).
[PubMed]

2006 (2)

F. R. Ruiz-Oliveras and A. N. Pisarchik, “Phase-locking phenomenon in a semiconductor laser with external cavities,” Opt. Express 14(26), 12859–12867 (2006).
[PubMed]

V. Z. Tronciu, H.-J. Wünsche, M. Wolfrum, and M. Radziunas, “Semiconductor laser under resonant feedback from a Fabry-Perot resonator: Stability of continuous-wave operation,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 73(4), 046205 (2006).
[PubMed]

2005 (1)

A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. García-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, “Chaos-based communications at high bit rates using commercial fibre-optic links,” Nature 437(7066), 343–346 (2005).

2002 (1)

M. Wolfrum and D. Turaev, “Instabilities of lasers with moderately delayed optical feedback,” Opt. Commun. 212(1-3), 127–138 (2002).

2001 (1)

M. Yousefi, D. Lenstra, G. Vemuri, and A. Fischer, “Control of nonlinear dynamics of a semiconductor laser with filtered optical feedback,” IEE Proc., Optoelectron. 148(5-6), 233–237 (2001).

1994 (1)

C. Lourenço and A. Babloyantz, “Control of chaos in networks with delay: A model for synchronization of cortical tissue,” Neural Comput. 6(6), 1141–1154 (1994).

1993 (1)

E. Schöll and K. Pyragas, “Tunable semiconductor oscillator based on self-control of chaos in the dynamic hall effect,” Europhys. Lett. 24(3), 159–164 (1993).

1992 (1)

K. Pyragas, “Continuous control of chaos by self-controlling feedback,” Phys. Lett. A 170(6), 421–428 (1992).

1990 (1)

E. Ott, C. Grebogi, and J. A. Yorke, “Controlling chaos,” Phys. Rev. Lett. 64(11), 1196–1199 (1990).
[PubMed]

1984 (1)

B. Tromborg, J. H. Osmundsen, and H. Olesen, “Stability analysis for a semiconductor laser in an external cavity,” IEEE J. Quantum Electron. 20(9), 1023–1032 (1984).

1980 (1)

R. Lang and K. Kobayashi, “External optical feedback effects on semiconductor injection laser properties,” IEEE J. Quantum Electron. 16(3), 347–355 (1980).

Annovazzi-Lodi, V.

A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. García-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, “Chaos-based communications at high bit rates using commercial fibre-optic links,” Nature 437(7066), 343–346 (2005).

Argyris, A.

A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. García-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, “Chaos-based communications at high bit rates using commercial fibre-optic links,” Nature 437(7066), 343–346 (2005).

Babloyantz, A.

C. Lourenço and A. Babloyantz, “Control of chaos in networks with delay: A model for synchronization of cortical tissue,” Neural Comput. 6(6), 1141–1154 (1994).

Colet, P.

V. Z. Tronciu, I. V. Ermakov, P. Colet, and C. R. Mirasso, “Chaotic dynamics of a semiconductor laser with double cavity feedback: Applications to phase shift keying modulation,” Opt. Commun. 281(18), 4747–4752 (2008).

A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. García-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, “Chaos-based communications at high bit rates using commercial fibre-optic links,” Nature 437(7066), 343–346 (2005).

Ermakov, I. V.

V. Z. Tronciu, I. V. Ermakov, P. Colet, and C. R. Mirasso, “Chaotic dynamics of a semiconductor laser with double cavity feedback: Applications to phase shift keying modulation,” Opt. Commun. 281(18), 4747–4752 (2008).

Erzgräber, H.

H. Erzgräber, D. Lenstra, B. Krauskopf, A. P. Fischer, and G. Vemuri, “Feedback phase sensitivity of a semiconductor laser subject to filtered optical feedback: experiment and theory,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 76(2), 026212 (2007).
[PubMed]

Fischer, A.

M. Yousefi, D. Lenstra, G. Vemuri, and A. Fischer, “Control of nonlinear dynamics of a semiconductor laser with filtered optical feedback,” IEE Proc., Optoelectron. 148(5-6), 233–237 (2001).

Fischer, A. P.

H. Erzgräber, D. Lenstra, B. Krauskopf, A. P. Fischer, and G. Vemuri, “Feedback phase sensitivity of a semiconductor laser subject to filtered optical feedback: experiment and theory,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 76(2), 026212 (2007).
[PubMed]

Fischer, I.

A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. García-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, “Chaos-based communications at high bit rates using commercial fibre-optic links,” Nature 437(7066), 343–346 (2005).

García-Ojalvo, J.

A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. García-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, “Chaos-based communications at high bit rates using commercial fibre-optic links,” Nature 437(7066), 343–346 (2005).

Grebogi, C.

E. Ott, C. Grebogi, and J. A. Yorke, “Controlling chaos,” Phys. Rev. Lett. 64(11), 1196–1199 (1990).
[PubMed]

Kobayashi, K.

R. Lang and K. Kobayashi, “External optical feedback effects on semiconductor injection laser properties,” IEEE J. Quantum Electron. 16(3), 347–355 (1980).

Krauskopf, B.

H. Erzgräber, D. Lenstra, B. Krauskopf, A. P. Fischer, and G. Vemuri, “Feedback phase sensitivity of a semiconductor laser subject to filtered optical feedback: experiment and theory,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 76(2), 026212 (2007).
[PubMed]

Lang, R.

R. Lang and K. Kobayashi, “External optical feedback effects on semiconductor injection laser properties,” IEEE J. Quantum Electron. 16(3), 347–355 (1980).

Larger, L.

A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. García-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, “Chaos-based communications at high bit rates using commercial fibre-optic links,” Nature 437(7066), 343–346 (2005).

Lenstra, D.

H. Erzgräber, D. Lenstra, B. Krauskopf, A. P. Fischer, and G. Vemuri, “Feedback phase sensitivity of a semiconductor laser subject to filtered optical feedback: experiment and theory,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 76(2), 026212 (2007).
[PubMed]

M. Yousefi, D. Lenstra, G. Vemuri, and A. Fischer, “Control of nonlinear dynamics of a semiconductor laser with filtered optical feedback,” IEE Proc., Optoelectron. 148(5-6), 233–237 (2001).

Lourenço, C.

C. Lourenço and A. Babloyantz, “Control of chaos in networks with delay: A model for synchronization of cortical tissue,” Neural Comput. 6(6), 1141–1154 (1994).

Mirasso, C. R.

V. Z. Tronciu, I. V. Ermakov, P. Colet, and C. R. Mirasso, “Chaotic dynamics of a semiconductor laser with double cavity feedback: Applications to phase shift keying modulation,” Opt. Commun. 281(18), 4747–4752 (2008).

A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. García-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, “Chaos-based communications at high bit rates using commercial fibre-optic links,” Nature 437(7066), 343–346 (2005).

Olesen, H.

B. Tromborg, J. H. Osmundsen, and H. Olesen, “Stability analysis for a semiconductor laser in an external cavity,” IEEE J. Quantum Electron. 20(9), 1023–1032 (1984).

Osmundsen, J. H.

B. Tromborg, J. H. Osmundsen, and H. Olesen, “Stability analysis for a semiconductor laser in an external cavity,” IEEE J. Quantum Electron. 20(9), 1023–1032 (1984).

Ott, E.

E. Ott, C. Grebogi, and J. A. Yorke, “Controlling chaos,” Phys. Rev. Lett. 64(11), 1196–1199 (1990).
[PubMed]

Pesquera, L.

A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. García-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, “Chaos-based communications at high bit rates using commercial fibre-optic links,” Nature 437(7066), 343–346 (2005).

Pisarchik, A. N.

Pyragas, K.

E. Schöll and K. Pyragas, “Tunable semiconductor oscillator based on self-control of chaos in the dynamic hall effect,” Europhys. Lett. 24(3), 159–164 (1993).

K. Pyragas, “Continuous control of chaos by self-controlling feedback,” Phys. Lett. A 170(6), 421–428 (1992).

Radziunas, M.

V. Z. Tronciu, H.-J. Wünsche, M. Wolfrum, and M. Radziunas, “Semiconductor laser under resonant feedback from a Fabry-Perot resonator: Stability of continuous-wave operation,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 73(4), 046205 (2006).
[PubMed]

Ruiz-Oliveras, F. R.

Schöll, E.

E. Schöll and K. Pyragas, “Tunable semiconductor oscillator based on self-control of chaos in the dynamic hall effect,” Europhys. Lett. 24(3), 159–164 (1993).

Shore, K. A.

A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. García-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, “Chaos-based communications at high bit rates using commercial fibre-optic links,” Nature 437(7066), 343–346 (2005).

Syvridis, D.

A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. García-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, “Chaos-based communications at high bit rates using commercial fibre-optic links,” Nature 437(7066), 343–346 (2005).

Tromborg, B.

B. Tromborg, J. H. Osmundsen, and H. Olesen, “Stability analysis for a semiconductor laser in an external cavity,” IEEE J. Quantum Electron. 20(9), 1023–1032 (1984).

Tronciu, V. Z.

V. Z. Tronciu, I. V. Ermakov, P. Colet, and C. R. Mirasso, “Chaotic dynamics of a semiconductor laser with double cavity feedback: Applications to phase shift keying modulation,” Opt. Commun. 281(18), 4747–4752 (2008).

V. Z. Tronciu, H.-J. Wünsche, M. Wolfrum, and M. Radziunas, “Semiconductor laser under resonant feedback from a Fabry-Perot resonator: Stability of continuous-wave operation,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 73(4), 046205 (2006).
[PubMed]

Turaev, D.

M. Wolfrum and D. Turaev, “Instabilities of lasers with moderately delayed optical feedback,” Opt. Commun. 212(1-3), 127–138 (2002).

Vemuri, G.

H. Erzgräber, D. Lenstra, B. Krauskopf, A. P. Fischer, and G. Vemuri, “Feedback phase sensitivity of a semiconductor laser subject to filtered optical feedback: experiment and theory,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 76(2), 026212 (2007).
[PubMed]

M. Yousefi, D. Lenstra, G. Vemuri, and A. Fischer, “Control of nonlinear dynamics of a semiconductor laser with filtered optical feedback,” IEE Proc., Optoelectron. 148(5-6), 233–237 (2001).

Wolfrum, M.

V. Z. Tronciu, H.-J. Wünsche, M. Wolfrum, and M. Radziunas, “Semiconductor laser under resonant feedback from a Fabry-Perot resonator: Stability of continuous-wave operation,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 73(4), 046205 (2006).
[PubMed]

M. Wolfrum and D. Turaev, “Instabilities of lasers with moderately delayed optical feedback,” Opt. Commun. 212(1-3), 127–138 (2002).

Wünsche, H.-J.

V. Z. Tronciu, H.-J. Wünsche, M. Wolfrum, and M. Radziunas, “Semiconductor laser under resonant feedback from a Fabry-Perot resonator: Stability of continuous-wave operation,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 73(4), 046205 (2006).
[PubMed]

Yorke, J. A.

E. Ott, C. Grebogi, and J. A. Yorke, “Controlling chaos,” Phys. Rev. Lett. 64(11), 1196–1199 (1990).
[PubMed]

Yousefi, M.

M. Yousefi, D. Lenstra, G. Vemuri, and A. Fischer, “Control of nonlinear dynamics of a semiconductor laser with filtered optical feedback,” IEE Proc., Optoelectron. 148(5-6), 233–237 (2001).

Europhys. Lett. (1)

E. Schöll and K. Pyragas, “Tunable semiconductor oscillator based on self-control of chaos in the dynamic hall effect,” Europhys. Lett. 24(3), 159–164 (1993).

IEE Proc., Optoelectron. (1)

M. Yousefi, D. Lenstra, G. Vemuri, and A. Fischer, “Control of nonlinear dynamics of a semiconductor laser with filtered optical feedback,” IEE Proc., Optoelectron. 148(5-6), 233–237 (2001).

IEEE J. Quantum Electron. (2)

R. Lang and K. Kobayashi, “External optical feedback effects on semiconductor injection laser properties,” IEEE J. Quantum Electron. 16(3), 347–355 (1980).

B. Tromborg, J. H. Osmundsen, and H. Olesen, “Stability analysis for a semiconductor laser in an external cavity,” IEEE J. Quantum Electron. 20(9), 1023–1032 (1984).

Nature (1)

A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. García-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, “Chaos-based communications at high bit rates using commercial fibre-optic links,” Nature 437(7066), 343–346 (2005).

Neural Comput. (1)

C. Lourenço and A. Babloyantz, “Control of chaos in networks with delay: A model for synchronization of cortical tissue,” Neural Comput. 6(6), 1141–1154 (1994).

Opt. Commun. (2)

M. Wolfrum and D. Turaev, “Instabilities of lasers with moderately delayed optical feedback,” Opt. Commun. 212(1-3), 127–138 (2002).

V. Z. Tronciu, I. V. Ermakov, P. Colet, and C. R. Mirasso, “Chaotic dynamics of a semiconductor laser with double cavity feedback: Applications to phase shift keying modulation,” Opt. Commun. 281(18), 4747–4752 (2008).

Opt. Express (1)

Phys. Lett. A (1)

K. Pyragas, “Continuous control of chaos by self-controlling feedback,” Phys. Lett. A 170(6), 421–428 (1992).

Phys. Rev. E Stat. Nonlin. Soft Matter Phys. (2)

V. Z. Tronciu, H.-J. Wünsche, M. Wolfrum, and M. Radziunas, “Semiconductor laser under resonant feedback from a Fabry-Perot resonator: Stability of continuous-wave operation,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 73(4), 046205 (2006).
[PubMed]

H. Erzgräber, D. Lenstra, B. Krauskopf, A. P. Fischer, and G. Vemuri, “Feedback phase sensitivity of a semiconductor laser subject to filtered optical feedback: experiment and theory,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 76(2), 026212 (2007).
[PubMed]

Phys. Rev. Lett. (1)

E. Ott, C. Grebogi, and J. A. Yorke, “Controlling chaos,” Phys. Rev. Lett. 64(11), 1196–1199 (1990).
[PubMed]

Supplementary Material (21)

» Media 1: MOV (2476 KB)     
» Media 2: MOV (2408 KB)     
» Media 3: MOV (3043 KB)     
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» Media 9: MOV (896 KB)     
» Media 10: MOV (663 KB)     
» Media 11: MOV (714 KB)     
» Media 12: MOV (888 KB)     
» Media 13: MOV (1259 KB)     
» Media 14: MOV (1082 KB)     
» Media 15: MOV (968 KB)     
» Media 16: MOV (927 KB)     
» Media 17: MOV (925 KB)     
» Media 18: MOV (888 KB)     
» Media 19: MOV (995 KB)     
» Media 20: MOV (990 KB)     
» Media 21: MOV (698 KB)     

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Figures (8)

Fig. 1
Fig. 1

Investigated setup. A laser with fiber-based external cavities. The cavities lengths are l 1 = 0.05 m and l 2 = 0.03 m. The refractive index of the optical fiber is n = 1.5.

Fig. 2
Fig. 2

(a) ECMs in the (ωs ,Ns )-plane (Media 1) for γ 1 = 20 ns−1, γ 2 = 15 ns−1, τ 1 = 0.5 ns, τ 2 = 0.3 ns, λ = 1.6 GHz and Δω = −8 GHz. (b) zoom of panel (a) around ωs = −13.9 GHz.

Fig. 3
Fig. 3

Lines of saddle-node bifurcation for φ = 0, (a) γ 1 = 12 ns−1, Δω = −6 GHz (Media 2) and (b) γ 1 = 20 ns−1, Δω = −8 GHz (Media 3). Other parameters are as in Fig. 2.

Fig. 4
Fig. 4

Optical spectra of a semiconductor laser with a CFB with (a) γ 1 = 12 ns−1 and (b) γ 1 = 20 ns−1. The other parameters are τ 1 = 0.5 ns, φ = 0. The red line shows the shape of the filter with λ = 1.6 GHz and (a) Δω = −6 GHz, (b) Δω = −8 GHz.

Fig. 5
Fig. 5

Control domain of the CFB with τ1 = 0.5 ns, γ 1 = 12 ns−1 and φ = 0 for (a), (d) γ 2 = 8 ns−1 (Media 4, Media 7); (b), (e) γ 2 = 10 ns−1 (Media 5, Media 8) and (c), (f) γ 2 = 15 ns−1 (Media 6, Media 9). The notations are: CW – continuous wave; P1, P2 – period one and two oscillations respectively; QP – quasi-periodic oscillation; LAC, HAC – low and high amplitude chaos respectively.

Fig. 6
Fig. 6

Effect of control of CFB. Parameters are (a) τ 2 = 0.3 ns, λ = 1.6 GHz (Media 10); (b) ψ = 2, λ = 1.6 GHz (Media 11); (c) ψ = 2, λ = ∞ (Media 12). Other parameters are as in Fig. 4(a).

Fig. 7
Fig. 7

Control domain of CFB with τ 1 = 0.5 ns, γ 1 = 20 ns−1 and φ = 0 for (a), (d) γ 2 = 15 ns−1 (Media 13, Media 16); (b), (e) γ 2 = 20 ns−1 (Media 14, Media 17) and (c), (f) γ 2 = 25 ns−1 (Media 15, Media 18).

Fig. 8
Fig. 8

Effect of control of CFB. Parameters are (a) τ 2 = 0.3 ns, λ = 1.6 GHz (Media 19); (b) ψ = 5, λ = 1.6 GHz (Media 20); (c) ψ = 5, λ = ∞ (Media 21). Other parameters are as in Fig. 4(b).

Equations (7)

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

r(ω)=λλ+i(Δωω).
E˙(t)=12(1+iα)[g(N(t)N0)/(1+s|E(t)|2)1/τph]E(t)+γ1E(tτ1)eiϕ+γ2F(t).
F˙(t)=λE(tτ2)eiψ+(iΔωλ)F(t) .
N˙(t)=I/eN(t)/τng(N(t)N0)|E(t)|2/(1+s|E(t)|2).
E(t)=Eseiωst,F(t)=Fseiωst+iΦs,N(t)=Ns,
ωs=D1sin(ϕωsτ1arctan(α))+D2effsin(ψωsτ2arctan(α)+δ),
Ns=[1/τph2γ1cos(ϕωsτ1)2γ2cos(δ)cos(ψωsτ2+δ)]/g+N0,

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