Abstract

An exact analysis is presented of the steady-state stability of a semiconductor laser subjected to feedback from a phase-conjugate mirror. Reduced stability occurs at low feedback whenever the effective external delay time is an integer multiple of the relaxation oscillation period. The role of a finite response time of the mirror is to enhance drastically the steady-state stability.

© 1998 Optical Society of America

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References

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  1. B. Tromborg, J. H. Osmundsen, and H. Olesen, IEEE J. Quantum Electron. QE-20, 1023 (1984).
    [CrossRef]
  2. A. Yariv, IEEE J. Quantum Electron. QE-14, 650 (1978).
    [CrossRef]
  3. G. H. M. van Tartwijk, H. J. C. van der Linden, and D. Lenstra, Opt. Lett. 17, 1590 (1992).
    [CrossRef] [PubMed]
  4. G. P. Agrawal and G. R. Gray, Phys. Rev. A 46, 5890 (1992).
    [CrossRef] [PubMed]
  5. A. Ritter and H. Haug, J. Opt. Soc. Am. B 10, 130 (1993).
    [CrossRef]
  6. J. S. Cohen, R. R. Drenten, and B. H. Verbeek, IEEE J. Quantum Electron. 24, 1989 (1988).
    [CrossRef]
  7. D. H. DeTienne, G. R. Gray, G. P. Agrawal, and D. Lenstra, IEEE J. Quantum Electron. 33, 838 (1997).
    [CrossRef]
  8. F. Mogensen, H. Olesen, and G. Jacobsen, IEEE J. Quantum Electron. QE-21, 784 (1985).
    [CrossRef]
  9. J. J. DiStefano, A. R. Stubberud, and I. J. Williams, Schaum's Outline Series on Feedback and Control Systems (McGraw-Hill, 1987).
  10. J. Mørk, B. Tromborg, and J. Mark, IEEE J. Quantum Electron. 28, 93 (1992).
    [CrossRef]
  11. A. Gavrielides, V. Kovanis, and T. Erneux, Opt. Commun. 136, 253 (1997).
    [CrossRef]

1997

D. H. DeTienne, G. R. Gray, G. P. Agrawal, and D. Lenstra, IEEE J. Quantum Electron. 33, 838 (1997).
[CrossRef]

A. Gavrielides, V. Kovanis, and T. Erneux, Opt. Commun. 136, 253 (1997).
[CrossRef]

1993

1992

G. H. M. van Tartwijk, H. J. C. van der Linden, and D. Lenstra, Opt. Lett. 17, 1590 (1992).
[CrossRef] [PubMed]

G. P. Agrawal and G. R. Gray, Phys. Rev. A 46, 5890 (1992).
[CrossRef] [PubMed]

J. Mørk, B. Tromborg, and J. Mark, IEEE J. Quantum Electron. 28, 93 (1992).
[CrossRef]

1988

J. S. Cohen, R. R. Drenten, and B. H. Verbeek, IEEE J. Quantum Electron. 24, 1989 (1988).
[CrossRef]

1985

F. Mogensen, H. Olesen, and G. Jacobsen, IEEE J. Quantum Electron. QE-21, 784 (1985).
[CrossRef]

1984

B. Tromborg, J. H. Osmundsen, and H. Olesen, IEEE J. Quantum Electron. QE-20, 1023 (1984).
[CrossRef]

1978

A. Yariv, IEEE J. Quantum Electron. QE-14, 650 (1978).
[CrossRef]

Agrawal, G. P.

D. H. DeTienne, G. R. Gray, G. P. Agrawal, and D. Lenstra, IEEE J. Quantum Electron. 33, 838 (1997).
[CrossRef]

G. P. Agrawal and G. R. Gray, Phys. Rev. A 46, 5890 (1992).
[CrossRef] [PubMed]

Cohen, J. S.

J. S. Cohen, R. R. Drenten, and B. H. Verbeek, IEEE J. Quantum Electron. 24, 1989 (1988).
[CrossRef]

DeTienne, D. H.

D. H. DeTienne, G. R. Gray, G. P. Agrawal, and D. Lenstra, IEEE J. Quantum Electron. 33, 838 (1997).
[CrossRef]

DiStefano, J. J.

J. J. DiStefano, A. R. Stubberud, and I. J. Williams, Schaum's Outline Series on Feedback and Control Systems (McGraw-Hill, 1987).

Drenten, R. R.

J. S. Cohen, R. R. Drenten, and B. H. Verbeek, IEEE J. Quantum Electron. 24, 1989 (1988).
[CrossRef]

Erneux, T.

A. Gavrielides, V. Kovanis, and T. Erneux, Opt. Commun. 136, 253 (1997).
[CrossRef]

Gavrielides, A.

A. Gavrielides, V. Kovanis, and T. Erneux, Opt. Commun. 136, 253 (1997).
[CrossRef]

Gray, G. R.

D. H. DeTienne, G. R. Gray, G. P. Agrawal, and D. Lenstra, IEEE J. Quantum Electron. 33, 838 (1997).
[CrossRef]

G. P. Agrawal and G. R. Gray, Phys. Rev. A 46, 5890 (1992).
[CrossRef] [PubMed]

Haug, H.

Jacobsen, G.

F. Mogensen, H. Olesen, and G. Jacobsen, IEEE J. Quantum Electron. QE-21, 784 (1985).
[CrossRef]

Kovanis, V.

A. Gavrielides, V. Kovanis, and T. Erneux, Opt. Commun. 136, 253 (1997).
[CrossRef]

Lenstra, D.

D. H. DeTienne, G. R. Gray, G. P. Agrawal, and D. Lenstra, IEEE J. Quantum Electron. 33, 838 (1997).
[CrossRef]

G. H. M. van Tartwijk, H. J. C. van der Linden, and D. Lenstra, Opt. Lett. 17, 1590 (1992).
[CrossRef] [PubMed]

Mark, J.

J. Mørk, B. Tromborg, and J. Mark, IEEE J. Quantum Electron. 28, 93 (1992).
[CrossRef]

Mogensen, F.

F. Mogensen, H. Olesen, and G. Jacobsen, IEEE J. Quantum Electron. QE-21, 784 (1985).
[CrossRef]

Mørk, J.

J. Mørk, B. Tromborg, and J. Mark, IEEE J. Quantum Electron. 28, 93 (1992).
[CrossRef]

Olesen, H.

F. Mogensen, H. Olesen, and G. Jacobsen, IEEE J. Quantum Electron. QE-21, 784 (1985).
[CrossRef]

B. Tromborg, J. H. Osmundsen, and H. Olesen, IEEE J. Quantum Electron. QE-20, 1023 (1984).
[CrossRef]

Osmundsen, J. H.

B. Tromborg, J. H. Osmundsen, and H. Olesen, IEEE J. Quantum Electron. QE-20, 1023 (1984).
[CrossRef]

Ritter, A.

Stubberud, A. R.

J. J. DiStefano, A. R. Stubberud, and I. J. Williams, Schaum's Outline Series on Feedback and Control Systems (McGraw-Hill, 1987).

Tromborg, B.

J. Mørk, B. Tromborg, and J. Mark, IEEE J. Quantum Electron. 28, 93 (1992).
[CrossRef]

B. Tromborg, J. H. Osmundsen, and H. Olesen, IEEE J. Quantum Electron. QE-20, 1023 (1984).
[CrossRef]

van der Linden, H. J. C.

van Tartwijk, G. H. M.

Verbeek, B. H.

J. S. Cohen, R. R. Drenten, and B. H. Verbeek, IEEE J. Quantum Electron. 24, 1989 (1988).
[CrossRef]

Williams, I. J.

J. J. DiStefano, A. R. Stubberud, and I. J. Williams, Schaum's Outline Series on Feedback and Control Systems (McGraw-Hill, 1987).

Yariv, A.

A. Yariv, IEEE J. Quantum Electron. QE-14, 650 (1978).
[CrossRef]

IEEE J. Quantum Electron.

B. Tromborg, J. H. Osmundsen, and H. Olesen, IEEE J. Quantum Electron. QE-20, 1023 (1984).
[CrossRef]

A. Yariv, IEEE J. Quantum Electron. QE-14, 650 (1978).
[CrossRef]

J. S. Cohen, R. R. Drenten, and B. H. Verbeek, IEEE J. Quantum Electron. 24, 1989 (1988).
[CrossRef]

D. H. DeTienne, G. R. Gray, G. P. Agrawal, and D. Lenstra, IEEE J. Quantum Electron. 33, 838 (1997).
[CrossRef]

F. Mogensen, H. Olesen, and G. Jacobsen, IEEE J. Quantum Electron. QE-21, 784 (1985).
[CrossRef]

J. Mørk, B. Tromborg, and J. Mark, IEEE J. Quantum Electron. 28, 93 (1992).
[CrossRef]

J. Opt. Soc. Am. B

Opt. Commun.

A. Gavrielides, V. Kovanis, and T. Erneux, Opt. Commun. 136, 253 (1997).
[CrossRef]

Opt. Lett.

Phys. Rev. A

G. P. Agrawal and G. R. Gray, Phys. Rev. A 46, 5890 (1992).
[CrossRef] [PubMed]

Other

J. J. DiStefano, A. R. Stubberud, and I. J. Williams, Schaum's Outline Series on Feedback and Control Systems (McGraw-Hill, 1987).

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Figures (3)

Fig. 1
Fig. 1

Feedback rate γp at which the laser changes stability as a function of the cavity round-trip time τ, normalized to the RO period (which is constant and equal to 1.3  ns). The shaded regions indicate stable behavior with tm=100 ps. For low feedback the laser is stable for all mirror-response times. The critical value at which instability sets in depends on τ in an oscillatory way. For moderate amounts of feedback (the top set of curves) a sluggish mirror response gives rise to stable behavior over the whole range of τ investigated, whereas an instantaneous mirror response yields stable behavior in a small region for short cavities only.

Fig. 2
Fig. 2

Stability of a PCF laser as a function of γp and the pump detuning δ0. tm=100 ps, except at the dotted curve. The shaded region is the region of stable behavior for a cavity with ωRτ/2π=1. The dotted curve represents an instantaneous mirror response at ωRτ/2π=0.5; in this case the region of stability shrinks to a narrow stripe. When δ0>γp1+α21/2 steady-state solutions (fixed points) do not exist at all. The line representing δ0=γp1+α21/2 is displayed only where it separates stable from nonexistent fixed points.

Fig. 3
Fig. 3

Magnification of the region near the origin of Fig.  2. Notice the destabilizing influence of the RO when it matches an external round-trip resonance (compare the solid and the dashed curves). The dotted curve is an extension of Fig.  3 of Ref.  3, which was obtained for an instantaneously responding mirror.

Tables (1)

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Table 1 Values of the Parameters Used in the Calculations

Equations (3)

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E·=12ξΔN-ϵΓ0P1+ϵP+iαξΔNE+γptmexp2iδ0t-τ2-tE* θ-τ×exp-1tm+iδ0t-θdθ,
N·=J-NT1-Γ0+ξΔN1+ϵPP,
12πiCf sf sds=N-P,

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