Abstract

The multistability in a single-mode distributed feedback semiconductor laser with delayed optoelectronic feedback is observed experimentally. For a given delay time, the observed dynamical state of the laser output is critically dependent on the process of varying the delay time and is limited by the range of variation. Various routes of delay time variation results in multistabilities characterized by states of different time series and power spectra.

© 2007 Optical Society of America

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  1. S. Sivaprakasam and K. A. Shore, “Demonstration of opticalsynchronization of chaotic external-cavity laser diodes,” Opt. Lett 24,466–468 (1999)
    [Crossref]
  2. S. Sivaprakasam and K. A. Shore, “Signal masking for chaoticoptical communication using external-cavity diode lasers,” Opt. Lett 24,1200–1202 (1999)
    [Crossref]
  3. Y. Takiguchi, H. Fujino, and J. Ohtsubo, “Experimental synchronizationof chaotic oscillations in externally injected semiconductorlasers in a low-frequency fluctuation regime,” Opt. Lett 24,1570–1572 (1999)
    [Crossref]
  4. 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 62,011801-1-4 (2000)
    [Crossref]
  5. K. Kusumoto and J. Ohtsubo, “1.5-GHz message transmission based on synchronization of chaos in semiconductor lasers,” Opt. Lett 27989–991(2002)
    [Crossref]
  6. S. C. Chan and J. M. Liu, “Microwave frequency division and multiplication using an optically injected semiconductor laser,” IEEE J. Quantum Electron 41,1142–1147 (2005)
    [Crossref]
  7. R. Vicente, S. Tang, J. Mulet, C. R. Mirasso, and J. M. Liu, “Synchronization properties of two self-oscillating semiconductor lasers subject to delayed optoelectronic mutual coupling,” Phys. Rev. E 73,047201-1-4 (2006)
    [Crossref]
  8. G. Q. Xia, Z. M. Wu, and X. H. Jia, “Theoretical investigation on commanding the bistability and self-pulsation of bistable semiconductor laser diode using delayed optoelectronic feedback,” J. Lightwave Technol 23,4296–4304 (2005)
    [Crossref]
  9. C. H. Lee and S. Y. Shin, ”Self-pulsing, spectral bistability, and chaos in a semiconductor laser diode with optoelectronic feedback,” Appl. Phys. Lett 62,922–924 (1993)
    [Crossref]
  10. M. Nizette, T. Erneux, A. Gavrielides, V. Kovanis, and Simpson, “Bistability of pulsating intensities for double-locked laser diodes,” Phys. Rev. E 65,056610-1-4 (2002)
    [Crossref]
  11. B. Farias, T.Passerat de Silans, M. Chevrollier, and M. Oria, “Frequency bistability of a semiconductor laser under a frequency-dependent feedback,” Phys. Rev. Lett 94,173902-1-4 (2005)
    [Crossref] [PubMed]
  12. S. K. Hwang and J. M. Liu, “Attractors and basins of the locking-unlocking bistability in a semiconductor laser subject to strong optical injection,” Opt. Commun 169,167–176 (1999)
    [Crossref]
  13. S. Rajesh and V. M. Nandakumaran, “Control of bistability in a directly modulated semiconductor laser using delayed optoelectronic feedback.” Physica D 213,113–120 (2006)
    [Crossref]
  14. S. Tang and J. M. Liu. “chaotic pulsing and quasi-periodic route to chaos in a semiconductor laser with delayed opto-electronic feedback,” IEEE J. Quantum Electron 37,329–336 (2001)
    [Crossref]
  15. S. I. Turovets, J. Dellunde, and K. A. Shore, “Nonlinear dynamics of a laser diode subjected to both optical and electronic feedback,” J. Opt. Soc. Am. B 14,200–208 (1997)
    [Crossref]
  16. F. Lin and J. Liu, “Nonlinear dynamics of a semiconductor laser with delayed negative optoelectronic feedback,” IEEE J. Quantum Electron 39,562–568 (2003)
    [Crossref]
  17. L. Larger, J. P. Goedgebuer, and F. Delorme, “Optical encryption system using hyperchaos generated by an optoelectronic wavelength oscillator,” Phys. Rev. E 57,6618–6624 (1998)
    [Crossref]
  18. L. Larger, J. P. Goedgebuer, and J. M. Merolla, “Chaotic oscillator in wavelength: a new setup for investigating differential difference equations describing nonlinear dynamics,” IEEE J. Quantum Electron 34,594–601 (1998)
    [Crossref]

2006 (2)

R. Vicente, S. Tang, J. Mulet, C. R. Mirasso, and J. M. Liu, “Synchronization properties of two self-oscillating semiconductor lasers subject to delayed optoelectronic mutual coupling,” Phys. Rev. E 73,047201-1-4 (2006)
[Crossref]

S. Rajesh and V. M. Nandakumaran, “Control of bistability in a directly modulated semiconductor laser using delayed optoelectronic feedback.” Physica D 213,113–120 (2006)
[Crossref]

2005 (3)

B. Farias, T.Passerat de Silans, M. Chevrollier, and M. Oria, “Frequency bistability of a semiconductor laser under a frequency-dependent feedback,” Phys. Rev. Lett 94,173902-1-4 (2005)
[Crossref] [PubMed]

G. Q. Xia, Z. M. Wu, and X. H. Jia, “Theoretical investigation on commanding the bistability and self-pulsation of bistable semiconductor laser diode using delayed optoelectronic feedback,” J. Lightwave Technol 23,4296–4304 (2005)
[Crossref]

S. C. Chan and J. M. Liu, “Microwave frequency division and multiplication using an optically injected semiconductor laser,” IEEE J. Quantum Electron 41,1142–1147 (2005)
[Crossref]

2003 (1)

F. Lin and J. Liu, “Nonlinear dynamics of a semiconductor laser with delayed negative optoelectronic feedback,” IEEE J. Quantum Electron 39,562–568 (2003)
[Crossref]

2002 (2)

M. Nizette, T. Erneux, A. Gavrielides, V. Kovanis, and Simpson, “Bistability of pulsating intensities for double-locked laser diodes,” Phys. Rev. E 65,056610-1-4 (2002)
[Crossref]

K. Kusumoto and J. Ohtsubo, “1.5-GHz message transmission based on synchronization of chaos in semiconductor lasers,” Opt. Lett 27989–991(2002)
[Crossref]

2001 (1)

S. Tang and J. M. Liu. “chaotic pulsing and quasi-periodic route to chaos in a semiconductor laser with delayed opto-electronic feedback,” IEEE J. Quantum Electron 37,329–336 (2001)
[Crossref]

2000 (1)

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 62,011801-1-4 (2000)
[Crossref]

1999 (4)

S. Sivaprakasam and K. A. Shore, “Demonstration of opticalsynchronization of chaotic external-cavity laser diodes,” Opt. Lett 24,466–468 (1999)
[Crossref]

S. Sivaprakasam and K. A. Shore, “Signal masking for chaoticoptical communication using external-cavity diode lasers,” Opt. Lett 24,1200–1202 (1999)
[Crossref]

Y. Takiguchi, H. Fujino, and J. Ohtsubo, “Experimental synchronizationof chaotic oscillations in externally injected semiconductorlasers in a low-frequency fluctuation regime,” Opt. Lett 24,1570–1572 (1999)
[Crossref]

S. K. Hwang and J. M. Liu, “Attractors and basins of the locking-unlocking bistability in a semiconductor laser subject to strong optical injection,” Opt. Commun 169,167–176 (1999)
[Crossref]

1998 (2)

L. Larger, J. P. Goedgebuer, and F. Delorme, “Optical encryption system using hyperchaos generated by an optoelectronic wavelength oscillator,” Phys. Rev. E 57,6618–6624 (1998)
[Crossref]

L. Larger, J. P. Goedgebuer, and J. M. Merolla, “Chaotic oscillator in wavelength: a new setup for investigating differential difference equations describing nonlinear dynamics,” IEEE J. Quantum Electron 34,594–601 (1998)
[Crossref]

1997 (1)

1993 (1)

C. H. Lee and S. Y. Shin, ”Self-pulsing, spectral bistability, and chaos in a semiconductor laser diode with optoelectronic feedback,” Appl. Phys. Lett 62,922–924 (1993)
[Crossref]

Chan, S. C.

S. C. Chan and J. M. Liu, “Microwave frequency division and multiplication using an optically injected semiconductor laser,” IEEE J. Quantum Electron 41,1142–1147 (2005)
[Crossref]

Chevrollier, M.

B. Farias, T.Passerat de Silans, M. Chevrollier, and M. Oria, “Frequency bistability of a semiconductor laser under a frequency-dependent feedback,” Phys. Rev. Lett 94,173902-1-4 (2005)
[Crossref] [PubMed]

Davis, P.

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 62,011801-1-4 (2000)
[Crossref]

Dellunde, J.

Delorme, F.

L. Larger, J. P. Goedgebuer, and F. Delorme, “Optical encryption system using hyperchaos generated by an optoelectronic wavelength oscillator,” Phys. Rev. E 57,6618–6624 (1998)
[Crossref]

Erneux, T.

M. Nizette, T. Erneux, A. Gavrielides, V. Kovanis, and Simpson, “Bistability of pulsating intensities for double-locked laser diodes,” Phys. Rev. E 65,056610-1-4 (2002)
[Crossref]

Farias, B.

B. Farias, T.Passerat de Silans, M. Chevrollier, and M. Oria, “Frequency bistability of a semiconductor laser under a frequency-dependent feedback,” Phys. Rev. Lett 94,173902-1-4 (2005)
[Crossref] [PubMed]

Fischer, I.

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 62,011801-1-4 (2000)
[Crossref]

Fujino, H.

Y. Takiguchi, H. Fujino, and J. Ohtsubo, “Experimental synchronizationof chaotic oscillations in externally injected semiconductorlasers in a low-frequency fluctuation regime,” Opt. Lett 24,1570–1572 (1999)
[Crossref]

Gavrielides, A.

M. Nizette, T. Erneux, A. Gavrielides, V. Kovanis, and Simpson, “Bistability of pulsating intensities for double-locked laser diodes,” Phys. Rev. E 65,056610-1-4 (2002)
[Crossref]

Goedgebuer, J. P.

L. Larger, J. P. Goedgebuer, and J. M. Merolla, “Chaotic oscillator in wavelength: a new setup for investigating differential difference equations describing nonlinear dynamics,” IEEE J. Quantum Electron 34,594–601 (1998)
[Crossref]

L. Larger, J. P. Goedgebuer, and F. Delorme, “Optical encryption system using hyperchaos generated by an optoelectronic wavelength oscillator,” Phys. Rev. E 57,6618–6624 (1998)
[Crossref]

Hwang, S. K.

S. K. Hwang and J. M. Liu, “Attractors and basins of the locking-unlocking bistability in a semiconductor laser subject to strong optical injection,” Opt. Commun 169,167–176 (1999)
[Crossref]

Jia, X. H.

G. Q. Xia, Z. M. Wu, and X. H. Jia, “Theoretical investigation on commanding the bistability and self-pulsation of bistable semiconductor laser diode using delayed optoelectronic feedback,” J. Lightwave Technol 23,4296–4304 (2005)
[Crossref]

Kovanis, V.

M. Nizette, T. Erneux, A. Gavrielides, V. Kovanis, and Simpson, “Bistability of pulsating intensities for double-locked laser diodes,” Phys. Rev. E 65,056610-1-4 (2002)
[Crossref]

Kusumoto, K.

K. Kusumoto and J. Ohtsubo, “1.5-GHz message transmission based on synchronization of chaos in semiconductor lasers,” Opt. Lett 27989–991(2002)
[Crossref]

Larger, L.

L. Larger, J. P. Goedgebuer, and J. M. Merolla, “Chaotic oscillator in wavelength: a new setup for investigating differential difference equations describing nonlinear dynamics,” IEEE J. Quantum Electron 34,594–601 (1998)
[Crossref]

L. Larger, J. P. Goedgebuer, and F. Delorme, “Optical encryption system using hyperchaos generated by an optoelectronic wavelength oscillator,” Phys. Rev. E 57,6618–6624 (1998)
[Crossref]

Lee, C. H.

C. H. Lee and S. Y. Shin, ”Self-pulsing, spectral bistability, and chaos in a semiconductor laser diode with optoelectronic feedback,” Appl. Phys. Lett 62,922–924 (1993)
[Crossref]

Lin, F.

F. Lin and J. Liu, “Nonlinear dynamics of a semiconductor laser with delayed negative optoelectronic feedback,” IEEE J. Quantum Electron 39,562–568 (2003)
[Crossref]

Liu, J.

F. Lin and J. Liu, “Nonlinear dynamics of a semiconductor laser with delayed negative optoelectronic feedback,” IEEE J. Quantum Electron 39,562–568 (2003)
[Crossref]

Liu, J. M.

R. Vicente, S. Tang, J. Mulet, C. R. Mirasso, and J. M. Liu, “Synchronization properties of two self-oscillating semiconductor lasers subject to delayed optoelectronic mutual coupling,” Phys. Rev. E 73,047201-1-4 (2006)
[Crossref]

S. C. Chan and J. M. Liu, “Microwave frequency division and multiplication using an optically injected semiconductor laser,” IEEE J. Quantum Electron 41,1142–1147 (2005)
[Crossref]

S. Tang and J. M. Liu. “chaotic pulsing and quasi-periodic route to chaos in a semiconductor laser with delayed opto-electronic feedback,” IEEE J. Quantum Electron 37,329–336 (2001)
[Crossref]

S. K. Hwang and J. M. Liu, “Attractors and basins of the locking-unlocking bistability in a semiconductor laser subject to strong optical injection,” Opt. Commun 169,167–176 (1999)
[Crossref]

Liu, Y.

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 62,011801-1-4 (2000)
[Crossref]

Merolla, J. M.

L. Larger, J. P. Goedgebuer, and J. M. Merolla, “Chaotic oscillator in wavelength: a new setup for investigating differential difference equations describing nonlinear dynamics,” IEEE J. Quantum Electron 34,594–601 (1998)
[Crossref]

Mirasso, C. R.

R. Vicente, S. Tang, J. Mulet, C. R. Mirasso, and J. M. Liu, “Synchronization properties of two self-oscillating semiconductor lasers subject to delayed optoelectronic mutual coupling,” Phys. Rev. E 73,047201-1-4 (2006)
[Crossref]

Mulet, J.

R. Vicente, S. Tang, J. Mulet, C. R. Mirasso, and J. M. Liu, “Synchronization properties of two self-oscillating semiconductor lasers subject to delayed optoelectronic mutual coupling,” Phys. Rev. E 73,047201-1-4 (2006)
[Crossref]

Nandakumaran, V. M.

S. Rajesh and V. M. Nandakumaran, “Control of bistability in a directly modulated semiconductor laser using delayed optoelectronic feedback.” Physica D 213,113–120 (2006)
[Crossref]

Nizette, M.

M. Nizette, T. Erneux, A. Gavrielides, V. Kovanis, and Simpson, “Bistability of pulsating intensities for double-locked laser diodes,” Phys. Rev. E 65,056610-1-4 (2002)
[Crossref]

Ohtsubo, J.

K. Kusumoto and J. Ohtsubo, “1.5-GHz message transmission based on synchronization of chaos in semiconductor lasers,” Opt. Lett 27989–991(2002)
[Crossref]

Y. Takiguchi, H. Fujino, and J. Ohtsubo, “Experimental synchronizationof chaotic oscillations in externally injected semiconductorlasers in a low-frequency fluctuation regime,” Opt. Lett 24,1570–1572 (1999)
[Crossref]

Oria, M.

B. Farias, T.Passerat de Silans, M. Chevrollier, and M. Oria, “Frequency bistability of a semiconductor laser under a frequency-dependent feedback,” Phys. Rev. Lett 94,173902-1-4 (2005)
[Crossref] [PubMed]

Rajesh, S.

S. Rajesh and V. M. Nandakumaran, “Control of bistability in a directly modulated semiconductor laser using delayed optoelectronic feedback.” Physica D 213,113–120 (2006)
[Crossref]

Shin, S. Y.

C. H. Lee and S. Y. Shin, ”Self-pulsing, spectral bistability, and chaos in a semiconductor laser diode with optoelectronic feedback,” Appl. Phys. Lett 62,922–924 (1993)
[Crossref]

Shore, K. A.

S. Sivaprakasam and K. A. Shore, “Demonstration of opticalsynchronization of chaotic external-cavity laser diodes,” Opt. Lett 24,466–468 (1999)
[Crossref]

S. Sivaprakasam and K. A. Shore, “Signal masking for chaoticoptical communication using external-cavity diode lasers,” Opt. Lett 24,1200–1202 (1999)
[Crossref]

S. I. Turovets, J. Dellunde, and K. A. Shore, “Nonlinear dynamics of a laser diode subjected to both optical and electronic feedback,” J. Opt. Soc. Am. B 14,200–208 (1997)
[Crossref]

Silans, T.Passerat de

B. Farias, T.Passerat de Silans, M. Chevrollier, and M. Oria, “Frequency bistability of a semiconductor laser under a frequency-dependent feedback,” Phys. Rev. Lett 94,173902-1-4 (2005)
[Crossref] [PubMed]

Simpson,

M. Nizette, T. Erneux, A. Gavrielides, V. Kovanis, and Simpson, “Bistability of pulsating intensities for double-locked laser diodes,” Phys. Rev. E 65,056610-1-4 (2002)
[Crossref]

Sivaprakasam, S.

S. Sivaprakasam and K. A. Shore, “Signal masking for chaoticoptical communication using external-cavity diode lasers,” Opt. Lett 24,1200–1202 (1999)
[Crossref]

S. Sivaprakasam and K. A. Shore, “Demonstration of opticalsynchronization of chaotic external-cavity laser diodes,” Opt. Lett 24,466–468 (1999)
[Crossref]

Takiguchi, Y.

Y. Takiguchi, H. Fujino, and J. Ohtsubo, “Experimental synchronizationof chaotic oscillations in externally injected semiconductorlasers in a low-frequency fluctuation regime,” Opt. Lett 24,1570–1572 (1999)
[Crossref]

Tang, S.

R. Vicente, S. Tang, J. Mulet, C. R. Mirasso, and J. M. Liu, “Synchronization properties of two self-oscillating semiconductor lasers subject to delayed optoelectronic mutual coupling,” Phys. Rev. E 73,047201-1-4 (2006)
[Crossref]

S. Tang and J. M. Liu. “chaotic pulsing and quasi-periodic route to chaos in a semiconductor laser with delayed opto-electronic feedback,” IEEE J. Quantum Electron 37,329–336 (2001)
[Crossref]

Turovets, S. I.

Vicente, R.

R. Vicente, S. Tang, J. Mulet, C. R. Mirasso, and J. M. Liu, “Synchronization properties of two self-oscillating semiconductor lasers subject to delayed optoelectronic mutual coupling,” Phys. Rev. E 73,047201-1-4 (2006)
[Crossref]

Wu, Z. M.

G. Q. Xia, Z. M. Wu, and X. H. Jia, “Theoretical investigation on commanding the bistability and self-pulsation of bistable semiconductor laser diode using delayed optoelectronic feedback,” J. Lightwave Technol 23,4296–4304 (2005)
[Crossref]

Xia, G. Q.

G. Q. Xia, Z. M. Wu, and X. H. Jia, “Theoretical investigation on commanding the bistability and self-pulsation of bistable semiconductor laser diode using delayed optoelectronic feedback,” J. Lightwave Technol 23,4296–4304 (2005)
[Crossref]

Appl. Phys. Lett (1)

C. H. Lee and S. Y. Shin, ”Self-pulsing, spectral bistability, and chaos in a semiconductor laser diode with optoelectronic feedback,” Appl. Phys. Lett 62,922–924 (1993)
[Crossref]

IEEE J. Quantum Electron (4)

S. Tang and J. M. Liu. “chaotic pulsing and quasi-periodic route to chaos in a semiconductor laser with delayed opto-electronic feedback,” IEEE J. Quantum Electron 37,329–336 (2001)
[Crossref]

F. Lin and J. Liu, “Nonlinear dynamics of a semiconductor laser with delayed negative optoelectronic feedback,” IEEE J. Quantum Electron 39,562–568 (2003)
[Crossref]

S. C. Chan and J. M. Liu, “Microwave frequency division and multiplication using an optically injected semiconductor laser,” IEEE J. Quantum Electron 41,1142–1147 (2005)
[Crossref]

L. Larger, J. P. Goedgebuer, and J. M. Merolla, “Chaotic oscillator in wavelength: a new setup for investigating differential difference equations describing nonlinear dynamics,” IEEE J. Quantum Electron 34,594–601 (1998)
[Crossref]

J. Lightwave Technol (1)

G. Q. Xia, Z. M. Wu, and X. H. Jia, “Theoretical investigation on commanding the bistability and self-pulsation of bistable semiconductor laser diode using delayed optoelectronic feedback,” J. Lightwave Technol 23,4296–4304 (2005)
[Crossref]

J. Opt. Soc. Am. B (1)

Opt. Commun (1)

S. K. Hwang and J. M. Liu, “Attractors and basins of the locking-unlocking bistability in a semiconductor laser subject to strong optical injection,” Opt. Commun 169,167–176 (1999)
[Crossref]

Opt. Lett (4)

K. Kusumoto and J. Ohtsubo, “1.5-GHz message transmission based on synchronization of chaos in semiconductor lasers,” Opt. Lett 27989–991(2002)
[Crossref]

S. Sivaprakasam and K. A. Shore, “Demonstration of opticalsynchronization of chaotic external-cavity laser diodes,” Opt. Lett 24,466–468 (1999)
[Crossref]

S. Sivaprakasam and K. A. Shore, “Signal masking for chaoticoptical communication using external-cavity diode lasers,” Opt. Lett 24,1200–1202 (1999)
[Crossref]

Y. Takiguchi, H. Fujino, and J. Ohtsubo, “Experimental synchronizationof chaotic oscillations in externally injected semiconductorlasers in a low-frequency fluctuation regime,” Opt. Lett 24,1570–1572 (1999)
[Crossref]

Phys. Rev. A (1)

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 62,011801-1-4 (2000)
[Crossref]

Phys. Rev. E (3)

R. Vicente, S. Tang, J. Mulet, C. R. Mirasso, and J. M. Liu, “Synchronization properties of two self-oscillating semiconductor lasers subject to delayed optoelectronic mutual coupling,” Phys. Rev. E 73,047201-1-4 (2006)
[Crossref]

M. Nizette, T. Erneux, A. Gavrielides, V. Kovanis, and Simpson, “Bistability of pulsating intensities for double-locked laser diodes,” Phys. Rev. E 65,056610-1-4 (2002)
[Crossref]

L. Larger, J. P. Goedgebuer, and F. Delorme, “Optical encryption system using hyperchaos generated by an optoelectronic wavelength oscillator,” Phys. Rev. E 57,6618–6624 (1998)
[Crossref]

Phys. Rev. Lett (1)

B. Farias, T.Passerat de Silans, M. Chevrollier, and M. Oria, “Frequency bistability of a semiconductor laser under a frequency-dependent feedback,” Phys. Rev. Lett 94,173902-1-4 (2005)
[Crossref] [PubMed]

Physica D (1)

S. Rajesh and V. M. Nandakumaran, “Control of bistability in a directly modulated semiconductor laser using delayed optoelectronic feedback.” Physica D 213,113–120 (2006)
[Crossref]

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Figures (3)

Fig. 1.
Fig. 1.

Schematic diagram of the experimental setup. Dashed line: optical path; Solid line: microwave path.

Fig. 2.
Fig. 2.

Power spectra and time series experimentally observed. The delay time: (a) 13.13 ns, (b) 13.27 ns, and (c) 13.67 ns.

Fig. 3.
Fig. 3.

Dependence of the peak frequency of power spectrum on the delay time, where the arrows mark the routes of varying the delay time, and different symbols characterize different output states of SL observed experimentally.

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