Abstract

We investigate the dynamics of two semiconductor lasers with separate optical feedback when they are driven by a common signal injected from a chaotic laser under the condition of non-identical drive and response. We experimentally and numerically show conditions under which the outputs of the two lasers can be highly correlated with each other even though the correlation with the drive signal is low. In particular, the effects of the phase of the feedback light on the correlation characteristics are described. The maximum correlation between the two response lasers is obtained when the phase of the feedback light is matched between the two response lasers, while the minimum correlation is observed when the difference in the optical phase is π. On the other hand, the correlation between the drive and response is not sensitive to the phase of the feedback light, unlike the previously studied case of identical drive and response. We numerically examine the difference between the maximum and minimum cross correlations over a wide range of parameters, and show that it is largest when there is a balance between the injection strength and the feedback strength.

© 2009 Optical Society of America

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  1. S. Sivaprakasam and K. A. Shore, "Demonstration of optical synchronization of chaotic external-cavity laser diodes," Opt. Lett. 24, 466-468 (1999).
    [CrossRef]
  2. I. Fischer, Y. Liu, and P. Davis, "Synchronization of chaotic semiconductor laser dynamics on subnanosecond time scales and its potential for chaos communication," Phys. Rev. A,  62, 011801(R)-1--011801(R)-4 (2000).
    [CrossRef]
  3. H. Fujino and J. Ohtsubo, "Experimental synchronization of chaotic oscillations in external-cavity semiconductor lasers," Opt. Lett. 25, 625-627 (2000).
    [CrossRef]
  4. A. Locquet, C. Masoller, and C. R. Mirasso, "Synchronization regimes of optical-feedback-induced chaos in unidirectionally coupled semiconductor lasers," Phys. Rev. E 65, 056205-1--056205-12 (2002).
    [CrossRef]
  5. Y. Liu, Y. Takiguchi, P. Davis, T. Aida, S. Saito, and J. M. Liu, "Experimental observation of complete chaos synchronization in semiconductor lasers," Appl. Phys. Lett. 80, 4306-4308 (2002).
    [CrossRef]
  6. S. Tang and J. M. Liu, "Experimental verification of anticipated and retarded synchronization in chaotic semiconductor lasers," Phys. Rev. Lett. 90,194101 (2003).
    [CrossRef] [PubMed]
  7. T. Yamamoto, I. Oowada, H. Yip, A. Uchida, S. Yoshimori, K. Yoshimura, J. Muramatsu, S. Goto, and P. Davis, "Common-chaotic-signal induced synchronization in semiconductor lasers, " Opt. Express 15, 3974-3980 (2007).
    [CrossRef] [PubMed]
  8. D. Y. Tang, R. Dykstra, M. W. Hamilton, and N. R. Heckenberg, "Observation of generalized synchronization of chaos in a driven chaotic system," Phys. Rev. E 57, 5247-5251 (1998).
    [CrossRef]
  9. A. Uchida, K. Higa, T. Shiba, S. Yoshimori, F. Kuwashima, and H. Iwasawa, "Generalized synchronization of chaos in He-Ne lasers," Phys. Rev. E 68, 016215-1—016215-7 (2003).
    [CrossRef]
  10. A. Uchida, R. McAllister, R. Meucci, and R. Roy, "Generalized synchronization of chaos in identical systems with hidden degrees of freedom," Phys. Rev. Lett. 91, 174101-1—174101-4 (2003).
    [CrossRef]
  11. R. Vicente, T. Perez, and C. R. Mirasso, "Open- versus closed-loop performance of synchronized chaotic external-cavity semiconductor lasers," IEEE J. Quantum Electron. 38, 1197-1204 (2002).
    [CrossRef]
  12. M. Peil, T. Heil, I. Fischer, and W. Elsäßer, "Synchronization of chaotic semiconductor laser systems: a vectorial coupling-dependent scenario," Phys. Rev. Lett. 88, 1741011-1—1741011-4 (2002).
    [CrossRef]
  13. T. Heil, J. Mulet, I. Fischer, C. R. Mirasso, M. Peil, P. Colet, and W. Elsäßer, "On/off phase shift keying for chaos-encrypted communication using external-cavity semiconductor lasers," IEEE J. Quantum Electron. 38, 1162-1170 (2002).
    [CrossRef]
  14. A. Bogris, P. Rizomiliotis, K. E. Chlouverakis, A. Argyris, and D. Syvridis, "Feedback phase in optically generated chaos: a secret key for cryptographic applications," IEEE J. Quantum Electron. 44, 119-124 (2008).
    [CrossRef]

2008 (1)

A. Bogris, P. Rizomiliotis, K. E. Chlouverakis, A. Argyris, and D. Syvridis, "Feedback phase in optically generated chaos: a secret key for cryptographic applications," IEEE J. Quantum Electron. 44, 119-124 (2008).
[CrossRef]

2007 (1)

2003 (1)

S. Tang and J. M. Liu, "Experimental verification of anticipated and retarded synchronization in chaotic semiconductor lasers," Phys. Rev. Lett. 90,194101 (2003).
[CrossRef] [PubMed]

2002 (3)

Y. Liu, Y. Takiguchi, P. Davis, T. Aida, S. Saito, and J. M. Liu, "Experimental observation of complete chaos synchronization in semiconductor lasers," Appl. Phys. Lett. 80, 4306-4308 (2002).
[CrossRef]

R. Vicente, T. Perez, and C. R. Mirasso, "Open- versus closed-loop performance of synchronized chaotic external-cavity semiconductor lasers," IEEE J. Quantum Electron. 38, 1197-1204 (2002).
[CrossRef]

T. Heil, J. Mulet, I. Fischer, C. R. Mirasso, M. Peil, P. Colet, and W. Elsäßer, "On/off phase shift keying for chaos-encrypted communication using external-cavity semiconductor lasers," IEEE J. Quantum Electron. 38, 1162-1170 (2002).
[CrossRef]

2000 (1)

1999 (1)

1998 (1)

D. Y. Tang, R. Dykstra, M. W. Hamilton, and N. R. Heckenberg, "Observation of generalized synchronization of chaos in a driven chaotic system," Phys. Rev. E 57, 5247-5251 (1998).
[CrossRef]

Aida, T.

Y. Liu, Y. Takiguchi, P. Davis, T. Aida, S. Saito, and J. M. Liu, "Experimental observation of complete chaos synchronization in semiconductor lasers," Appl. Phys. Lett. 80, 4306-4308 (2002).
[CrossRef]

Argyris, A.

A. Bogris, P. Rizomiliotis, K. E. Chlouverakis, A. Argyris, and D. Syvridis, "Feedback phase in optically generated chaos: a secret key for cryptographic applications," IEEE J. Quantum Electron. 44, 119-124 (2008).
[CrossRef]

Bogris, A.

A. Bogris, P. Rizomiliotis, K. E. Chlouverakis, A. Argyris, and D. Syvridis, "Feedback phase in optically generated chaos: a secret key for cryptographic applications," IEEE J. Quantum Electron. 44, 119-124 (2008).
[CrossRef]

Chlouverakis, K. E.

A. Bogris, P. Rizomiliotis, K. E. Chlouverakis, A. Argyris, and D. Syvridis, "Feedback phase in optically generated chaos: a secret key for cryptographic applications," IEEE J. Quantum Electron. 44, 119-124 (2008).
[CrossRef]

Colet, P.

T. Heil, J. Mulet, I. Fischer, C. R. Mirasso, M. Peil, P. Colet, and W. Elsäßer, "On/off phase shift keying for chaos-encrypted communication using external-cavity semiconductor lasers," IEEE J. Quantum Electron. 38, 1162-1170 (2002).
[CrossRef]

Davis, P.

T. Yamamoto, I. Oowada, H. Yip, A. Uchida, S. Yoshimori, K. Yoshimura, J. Muramatsu, S. Goto, and P. Davis, "Common-chaotic-signal induced synchronization in semiconductor lasers, " Opt. Express 15, 3974-3980 (2007).
[CrossRef] [PubMed]

Y. Liu, Y. Takiguchi, P. Davis, T. Aida, S. Saito, and J. M. Liu, "Experimental observation of complete chaos synchronization in semiconductor lasers," Appl. Phys. Lett. 80, 4306-4308 (2002).
[CrossRef]

Dykstra, R.

D. Y. Tang, R. Dykstra, M. W. Hamilton, and N. R. Heckenberg, "Observation of generalized synchronization of chaos in a driven chaotic system," Phys. Rev. E 57, 5247-5251 (1998).
[CrossRef]

Elsäßer, W.

T. Heil, J. Mulet, I. Fischer, C. R. Mirasso, M. Peil, P. Colet, and W. Elsäßer, "On/off phase shift keying for chaos-encrypted communication using external-cavity semiconductor lasers," IEEE J. Quantum Electron. 38, 1162-1170 (2002).
[CrossRef]

Fischer, I.

T. Heil, J. Mulet, I. Fischer, C. R. Mirasso, M. Peil, P. Colet, and W. Elsäßer, "On/off phase shift keying for chaos-encrypted communication using external-cavity semiconductor lasers," IEEE J. Quantum Electron. 38, 1162-1170 (2002).
[CrossRef]

Fujino, H.

Goto, S.

Hamilton, M. W.

D. Y. Tang, R. Dykstra, M. W. Hamilton, and N. R. Heckenberg, "Observation of generalized synchronization of chaos in a driven chaotic system," Phys. Rev. E 57, 5247-5251 (1998).
[CrossRef]

Heckenberg, N. R.

D. Y. Tang, R. Dykstra, M. W. Hamilton, and N. R. Heckenberg, "Observation of generalized synchronization of chaos in a driven chaotic system," Phys. Rev. E 57, 5247-5251 (1998).
[CrossRef]

Heil, T.

T. Heil, J. Mulet, I. Fischer, C. R. Mirasso, M. Peil, P. Colet, and W. Elsäßer, "On/off phase shift keying for chaos-encrypted communication using external-cavity semiconductor lasers," IEEE J. Quantum Electron. 38, 1162-1170 (2002).
[CrossRef]

Liu, J. M.

S. Tang and J. M. Liu, "Experimental verification of anticipated and retarded synchronization in chaotic semiconductor lasers," Phys. Rev. Lett. 90,194101 (2003).
[CrossRef] [PubMed]

Y. Liu, Y. Takiguchi, P. Davis, T. Aida, S. Saito, and J. M. Liu, "Experimental observation of complete chaos synchronization in semiconductor lasers," Appl. Phys. Lett. 80, 4306-4308 (2002).
[CrossRef]

Liu, Y.

Y. Liu, Y. Takiguchi, P. Davis, T. Aida, S. Saito, and J. M. Liu, "Experimental observation of complete chaos synchronization in semiconductor lasers," Appl. Phys. Lett. 80, 4306-4308 (2002).
[CrossRef]

Mirasso, C. R.

T. Heil, J. Mulet, I. Fischer, C. R. Mirasso, M. Peil, P. Colet, and W. Elsäßer, "On/off phase shift keying for chaos-encrypted communication using external-cavity semiconductor lasers," IEEE J. Quantum Electron. 38, 1162-1170 (2002).
[CrossRef]

R. Vicente, T. Perez, and C. R. Mirasso, "Open- versus closed-loop performance of synchronized chaotic external-cavity semiconductor lasers," IEEE J. Quantum Electron. 38, 1197-1204 (2002).
[CrossRef]

Mulet, J.

T. Heil, J. Mulet, I. Fischer, C. R. Mirasso, M. Peil, P. Colet, and W. Elsäßer, "On/off phase shift keying for chaos-encrypted communication using external-cavity semiconductor lasers," IEEE J. Quantum Electron. 38, 1162-1170 (2002).
[CrossRef]

Muramatsu, J.

Ohtsubo, J.

Oowada, I.

Peil, M.

T. Heil, J. Mulet, I. Fischer, C. R. Mirasso, M. Peil, P. Colet, and W. Elsäßer, "On/off phase shift keying for chaos-encrypted communication using external-cavity semiconductor lasers," IEEE J. Quantum Electron. 38, 1162-1170 (2002).
[CrossRef]

Perez, T.

R. Vicente, T. Perez, and C. R. Mirasso, "Open- versus closed-loop performance of synchronized chaotic external-cavity semiconductor lasers," IEEE J. Quantum Electron. 38, 1197-1204 (2002).
[CrossRef]

Rizomiliotis, P.

A. Bogris, P. Rizomiliotis, K. E. Chlouverakis, A. Argyris, and D. Syvridis, "Feedback phase in optically generated chaos: a secret key for cryptographic applications," IEEE J. Quantum Electron. 44, 119-124 (2008).
[CrossRef]

Saito, S.

Y. Liu, Y. Takiguchi, P. Davis, T. Aida, S. Saito, and J. M. Liu, "Experimental observation of complete chaos synchronization in semiconductor lasers," Appl. Phys. Lett. 80, 4306-4308 (2002).
[CrossRef]

Shore, K. A.

Sivaprakasam, S.

Syvridis, D.

A. Bogris, P. Rizomiliotis, K. E. Chlouverakis, A. Argyris, and D. Syvridis, "Feedback phase in optically generated chaos: a secret key for cryptographic applications," IEEE J. Quantum Electron. 44, 119-124 (2008).
[CrossRef]

Takiguchi, Y.

Y. Liu, Y. Takiguchi, P. Davis, T. Aida, S. Saito, and J. M. Liu, "Experimental observation of complete chaos synchronization in semiconductor lasers," Appl. Phys. Lett. 80, 4306-4308 (2002).
[CrossRef]

Tang, D. Y.

D. Y. Tang, R. Dykstra, M. W. Hamilton, and N. R. Heckenberg, "Observation of generalized synchronization of chaos in a driven chaotic system," Phys. Rev. E 57, 5247-5251 (1998).
[CrossRef]

Tang, S.

S. Tang and J. M. Liu, "Experimental verification of anticipated and retarded synchronization in chaotic semiconductor lasers," Phys. Rev. Lett. 90,194101 (2003).
[CrossRef] [PubMed]

Uchida, A.

Vicente, R.

R. Vicente, T. Perez, and C. R. Mirasso, "Open- versus closed-loop performance of synchronized chaotic external-cavity semiconductor lasers," IEEE J. Quantum Electron. 38, 1197-1204 (2002).
[CrossRef]

Yamamoto, T.

Yip, H.

Yoshimori, S.

Yoshimura, K.

Appl. Phys. Lett. (1)

Y. Liu, Y. Takiguchi, P. Davis, T. Aida, S. Saito, and J. M. Liu, "Experimental observation of complete chaos synchronization in semiconductor lasers," Appl. Phys. Lett. 80, 4306-4308 (2002).
[CrossRef]

IEEE J. Quantum Electron. (3)

R. Vicente, T. Perez, and C. R. Mirasso, "Open- versus closed-loop performance of synchronized chaotic external-cavity semiconductor lasers," IEEE J. Quantum Electron. 38, 1197-1204 (2002).
[CrossRef]

T. Heil, J. Mulet, I. Fischer, C. R. Mirasso, M. Peil, P. Colet, and W. Elsäßer, "On/off phase shift keying for chaos-encrypted communication using external-cavity semiconductor lasers," IEEE J. Quantum Electron. 38, 1162-1170 (2002).
[CrossRef]

A. Bogris, P. Rizomiliotis, K. E. Chlouverakis, A. Argyris, and D. Syvridis, "Feedback phase in optically generated chaos: a secret key for cryptographic applications," IEEE J. Quantum Electron. 44, 119-124 (2008).
[CrossRef]

Opt. Express (1)

Opt. Lett. (2)

Phys. Rev. E (1)

D. Y. Tang, R. Dykstra, M. W. Hamilton, and N. R. Heckenberg, "Observation of generalized synchronization of chaos in a driven chaotic system," Phys. Rev. E 57, 5247-5251 (1998).
[CrossRef]

Phys. Rev. Lett. (1)

S. Tang and J. M. Liu, "Experimental verification of anticipated and retarded synchronization in chaotic semiconductor lasers," Phys. Rev. Lett. 90,194101 (2003).
[CrossRef] [PubMed]

Other (5)

M. Peil, T. Heil, I. Fischer, and W. Elsäßer, "Synchronization of chaotic semiconductor laser systems: a vectorial coupling-dependent scenario," Phys. Rev. Lett. 88, 1741011-1—1741011-4 (2002).
[CrossRef]

A. Uchida, K. Higa, T. Shiba, S. Yoshimori, F. Kuwashima, and H. Iwasawa, "Generalized synchronization of chaos in He-Ne lasers," Phys. Rev. E 68, 016215-1—016215-7 (2003).
[CrossRef]

A. Uchida, R. McAllister, R. Meucci, and R. Roy, "Generalized synchronization of chaos in identical systems with hidden degrees of freedom," Phys. Rev. Lett. 91, 174101-1—174101-4 (2003).
[CrossRef]

A. Locquet, C. Masoller, and C. R. Mirasso, "Synchronization regimes of optical-feedback-induced chaos in unidirectionally coupled semiconductor lasers," Phys. Rev. E 65, 056205-1--056205-12 (2002).
[CrossRef]

I. Fischer, Y. Liu, and P. Davis, "Synchronization of chaotic semiconductor laser dynamics on subnanosecond time scales and its potential for chaos communication," Phys. Rev. A,  62, 011801(R)-1--011801(R)-4 (2000).
[CrossRef]

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Figures (7)

Fig. 1.
Fig. 1.

Experimental setup for synchronization by injection of common chaotic signal in semiconductor lasers with optical feedback. Amp, electronic amplifier; BS, beam splitter; FC, fiber collimator; ISO, optical isolator; L, lens; λ/2, half wave plate; M, mirror; NDF, neutral density filter; OSC, digital oscilloscope; PC, personal computer; PD, photodetector; PZT, piezo-transducer; SA, radio-frequency spectrum analyzer; SL, semiconductor laser.

Fig. 2.
Fig. 2.

Experimental result of temporal waveforms and their correlation plots for Response 1 and 2 at (a), (b) Δϕr1,r2 =0, and (c), (d) Δϕr1,r2 =π, where Δϕr1,r2 =ϕr1 -ϕr2 is the difference between optical feedback phases in Response 1 and 2. The cross correlation values are (b) 0.903 and (d) 0.015.

Fig. 3.
Fig. 3.

Experimental result of temporal waveforms and their correlation plots for Drive and Response 1 at (a), (b) Δϕd,r1 =0, and (c), (d) Δϕd,r1 =π, where Δϕd,r1 =ϕd -ϕr1 is the difference between optical feedback phases in Drive and Response 1. The cross correlation values are (b) 0.601 and (d) 0.599.

Fig. 4.
Fig. 4.

Experimental result of the cross correlation value between (a) Response 1 and 2, and between (b) Drive and Response 1, as a function of the position of the external mirror for Response 1 (optical phase difference) at the maximum feedback strength of the two Response lasers. The powers of the injection and feedback lights are 4.0 and 3.0 µW, respectively (measured in front of the Response 1 laser). (a) The cross correlation changes periodically as the optical feedback phase difference is changed. The maximum and minimum cross correlation values are 0.958 and 0.032, respectively. The period of the correlation curve is 1.5 µm, corresponding to the optical wavelength of the laser. (b) The cross correlation values stay between 0.45 and 0.65 and do not show periodical change.

Fig. 5.
Fig. 5.

Experimental result of the maximum and minimum cross correlation values (Cmax and Cmin ) between (a) Response 1 and 2, and between (b) Drive and Response 1, as a function of the feedback strength of the two Response lasers. The maximum value of cross correlation is obtained at zero phase difference (Δϕr1,r2 =0), whereas the minimum value is observed at Δϕr1,r2 =π. (a) The difference between maximum and minimum values of cross correlation (ΔC=Cmax -Cmin ) increases as the feedback strength is increased, and the maximum value of ΔC=0.880 is obtained at the maximum feedback strength which could be achieved in the experiment. (b) The values of maximum and minimum cross correlation gradually decrease as the feedback strength is increased. However, only relatively small change in ΔC is observed at different optical feedback strengths. The maximum value of ΔC is 0.105.

Fig. 6.
Fig. 6.

Numerical result of the contour plot of the cross correlation difference ΔC (=Cmax -Cmin ) between Response 1 and 2 as a function of the optical frequency detuning between Drive and Response (Δf) and the injection strength from Drive to Response (κinj ). Synchronization between Response 1 and 2 is achieved in the region above the dashed curve.

Fig. 7.
Fig. 7.

Numerical result of the contour plot of the cross correlation difference ΔC (=Cmax -Cmin ) between Response 1 and 2 as a function of the feedback strength of Response (κr ) and the injection strength from Drive to Response (κinj ). Synchronization between Response 1 and 2 is achieved in the region above the dashed curve.

Equations (5)

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C=(I1Iˉ1)(I2Iˉ2)σ1σ2
E˙(t)=12(1+iα)GN(N(t)Nth)E(t)+κDτinE(tτD),
N˙(t)=JD1τsN(t)GN(N(t)N0)E(t)2,
E˙j(t)=12(1+iα)GN(Nj(t)Nth)Ej(t)+κrτinEj(tτ)exp[iφj]+κinjτinE(t)exp[iΔωjt],
N˙j(t)=J1τsNj(t)GN(Nj(t)N0)Ej(t)2,

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