Abstract

The properties of a single-mode semiconductor laser with weak optical feedback from a distant reflector are modeled by Langevin equations for the plasma density and for a classical electric field. A novel comprehensive small-signal analysis of the deterministic model equations is performed near their stationary solution. The modifications of the relaxation oscillations between carrier and photon densities that are caused by the optical feedback are studied. Spectral properties are calculated by adding Langevin fluctuation terms of the dissipative processes. New analytic approximations for the dominant contributions to the time-averaged field autocorrelation spectrum are derived and compared with the results for a free-running laser.

© 1993 Optical Society of America

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  1. R. Lang and K. Kobayashi, “External optical feedback effects on semiconductor injection laser properties,” IEEE J. Quantum Electron. QE-16, 347–355 (1980).
    [CrossRef]
  2. R. W. Tkach and A. R. Charplyvy, “Regimes of feedback effects in 1.5 μ m distributed feedback lasers,” IEEE J. Lightwave Technol. LT-4, 1655–1661 (1986).
    [CrossRef]
  3. W. Schunk and K. Petermann, “Numerical analysis of the feedback regimes for a single-mode semiconductor laser with external feedback,” IEEE J. Quantum Electron. 24, 1242–1247 (1988).
    [CrossRef]
  4. M. Tamburrini, P. Spano, and S. Piazzolla, “Influence of an external cavity on semiconductor laser phase noise,” Appl. Phys. Lett. 43, 410–412 (1983).
    [CrossRef]
  5. P. Spano, S. Piazzolla, and M. Tamburrini, “Theory of noise in semiconductor lasers in the presence of optical feedback,” IEEE J. Quantum Electron. QE-20, 350–357 (1984).
    [CrossRef]
  6. G. P. Agrawal, “Line narrowing in a single-mode semiconductor laser due to optical feedback,” IEEE J. Quantum Electron. QE-20, 468–471 (1984).
    [CrossRef]
  7. J. Mørk, P. L. Christiansen, and B. Tromborg, “Limits of stable operation of AR-coated semiconductor lasers with strong optical feedback,” Electron. Lett. 24, 1065–1066 (1988).
    [CrossRef]
  8. D. Lenstra, B. H. Verbeek, and A. J. den Boef, “Coherence collapse in single-mode semiconductor lasers due to optical feedback,” IEEE J. Quantum Electron. QE-21, 674–679 (1985).
    [CrossRef]
  9. J. S. Cohen and D. Lenstra, “Spectral properties of the coherence collapsed state of a semiconductor laser with delayed optical feedback,” IEEE J. Quantum Electron. 25, 1143–1151 (1989).
    [CrossRef]
  10. J. S. Cohen, F. Wittgrefe, M. D. Hogerland, and J. P. Woerdman, “Optical spectra of a semiconductor laser with incoherent optical feedback,” IEEE J. Quantum Electron. 26, 982–990 (1990).
    [CrossRef]
  11. C. H. Henry and R. F. Kazarinov, “Instability of semiconductor lasers due to optical feedback from distant reflectors,” IEEE J. Quantum Electron. QE-22, 294–301 (1986).
    [CrossRef]
  12. J. Mørk, B. Tromborg, and P. L. Christiansen, “Bistability and low-frequency fluctuations in semiconductor lasers with optical feedback: a theoretical analysis,” IEEE J. Quantum Electron. 24, 123–133 (1988).
    [CrossRef]
  13. J. Sacher, W. Elsässer, and E. O. Goebel, “Intermittency in the coherence collapse of a semiconductor laser with external feedback,” Phys. Rev. Lett. 63, 2224–2227 (1989).
    [CrossRef] [PubMed]
  14. J. Sacher, W. Elsässer, and E. O. Goebel, “Nonlinear dynamics of semiconductor laser emission under variable feedback conditions,” IEEE J. Quantum Electron. 27, 373–391 (1991).
    [CrossRef]
  15. T. Mukai and K. Otsuka, “New route to optical chaos: successive-subharmonic-oscillation cascade in a semiconductor laser coupled to an external cavity,” Phys. Rev. Lett. 55, 1711–1714 (1985).
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  19. B. Tromborg and J. Mørk, “Stability analysis and the route to chaos for laser diodes with optical feedback,” IEEE Photon. Technol. Lett. 2, 549–552 (1990).
    [CrossRef]
  20. J. Mørk, J. Mark, and B. Tromborg, “Route to chaos and competition between relaxation oscillations for a semiconductor laser with optical feedback,” Phys. Rev. Lett. 65, 1999–2002 (1990).
    [CrossRef] [PubMed]
  21. J. Mørk, B. Tromborg, and J. Mark, “Chaos in semiconductor lasers with optical feedback: theory and experiment,” IEEE J. Quantum Electron 28, 93–108 (1992).
    [CrossRef]
  22. H. Haug, “Quantum mechanical rate equations for semiconductor lasers,” Phys. Rev. 184, 338–348 (1969).
    [CrossRef]
  23. K. Vahala and A. Yariv, “Semiclassical theory of noise in semiconductor lasers—Part II,” IEEE J. Quantum Electron. QE-19, 1103–1109 (1983).
  24. A. Ritter and H. Haug, “Theory of laser diodes with weak optical feedback. II. Limit-cycle behavior, quasi-periodicity, frequency locking, and route to chaos,” J. Opt. Soc. Am. B 10, 145–154 (1993).
    [CrossRef]
  25. K. Vahala, Ch. Harder, and A. Yariv, “Observation of relaxation resonance effects in the field spectrum of semiconductor lasers,” Appl. Phys. Lett. 42, 211–213 (1983).
    [CrossRef]
  26. K. Henneberger and H. Haug, “Nonlinear optics and transport in laser excited semiconductors,” Phys. Rev. B 38, 9759–9770 (1988).
    [CrossRef]
  27. H. Haug and K. Henneberger, “The kinetics of hole-burning in semiconductor lasers,” Z. Phys. B 83, 447–451 (1991).
    [CrossRef]
  28. K. L. Hall, J. Mark, E. P. Ippen, and G. Eisenstein, “Femtosecond gain dynamics in InGaAsP optical amplifiers,” Appl. Phys. Lett. 56, 1740–1742 (1990).
    [CrossRef]
  29. M. Willatzen, A. Uskov, J. Mørk, H. Olesen, and B. Tromborg, “Nonlinear gain supression in semiconductor lasers due to carrier heating,” IEEE Photon. Technol. Lett. 3, 606–609 (1991).
    [CrossRef]
  30. H. Haug and H. Haken, “Theory of noise in semiconductor laser emission,” Z. Phys. 204, 262–275 (1967).
    [CrossRef]
  31. B. Tromborg, J. H. Osmundsen, and H. Olesen, “Stability analysis for a semiconductor laser in an external cavity,” IEEE J. Quantum Electron. QE-20, 1023–1032 (1984).
    [CrossRef]
  32. J. Mørk and B. Tromborg, “The mechanism of mode selection for an external cavity laser,” IEEE Photon. Technol. Lett. 2, 21–23 (1990).
    [CrossRef]
  33. A. Mecozzi, S. Piazzolla, A. Sapia, and P. Spano, “Non-Gaussian statistics of frequency fluctuations in line-narrowed semiconductor lasers,” IEEE J. Quantum Electron. 24, 1985–1988 (1988).
    [CrossRef]
  34. J. Helms and K. Petermann, “A simple analytic expression for the stable operation range for laser diodes with optical feedback,” IEEE J. Quantum Electron. 26, 833–836 (1990).
    [CrossRef]
  35. J. S. Cohen, R. R. Drenten, and B. H. Verbeek, “The effect of optical feedback on the relaxation oscillation in semiconductor lasers,” IEEE J. Quantum Electron. 24, 1989–1995 (1988).
    [CrossRef]
  36. L. Goldberg, H. F. Taylor, A. Dandridge, J. F. Weller, and R. O. Miles, “Spectral characteristics of semiconductor lasers with optical feedback,” IEEE J. Quantum Electron. QE-18, 555–563 (1982).
    [CrossRef]
  37. J. O. Binder and G. D. Cormack, “Mode selection and stability of a semiconductor laser with weak optical feedback,” IEEE J. Quantum Electron. 25, 2255–2259 (1989).
    [CrossRef]
  38. H. Haken, Laser Theory, Vol. XXV/2c of Handbuch der Physik (Springer-Verlag, Berlin, 1970).
  39. M. Lax, in Bradeis University Summer Institute of Theoretical Physics, M. Chretin, E. P. Gross, and S. Deser, eds. (Gordon & Breach, New York, 1968).
  40. H. Haug, “Quantum mechanical theory of fluctuations in semiconductor lasers,” Z. Phys. 200, 57–68 (1967).
    [CrossRef]
  41. H. Haug and S. Schmitt-Rink, “Electron theory of the optical properties of laser excited semiconductors,” Prog. Quantum Electron. 9(1), 13, 14 (1984).
    [CrossRef]
  42. C. H. Henry, “Theory of the linewidth of semiconductor lasers,” IEEE J. Quantum Electron. QE-18, 259–264 (1982).
    [CrossRef]
  43. Y. Yamamoto and S. Machida, “High-impedance suppression of pump fluctuations and amplitude squeezing in semiconductor lasers,” Phys. Rev. A 35, 5114–5130 (1987).
    [CrossRef] [PubMed]
  44. M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions (Dover, New York, 1970), p. 375, ¶ 9.6.10; p. 376, ¶ 9.6.34.
  45. D. R. Hjelme, A. R. Mickelson, and R. G. Beausoleil, “Semiconductor laser stabilization by external optical feedback,” IEEE J. Quantum Electron. QE-27, 352–372 (1991).
    [CrossRef]

1993 (1)

1992 (1)

J. Mørk, B. Tromborg, and J. Mark, “Chaos in semiconductor lasers with optical feedback: theory and experiment,” IEEE J. Quantum Electron 28, 93–108 (1992).
[CrossRef]

1991 (4)

H. Haug and K. Henneberger, “The kinetics of hole-burning in semiconductor lasers,” Z. Phys. B 83, 447–451 (1991).
[CrossRef]

M. Willatzen, A. Uskov, J. Mørk, H. Olesen, and B. Tromborg, “Nonlinear gain supression in semiconductor lasers due to carrier heating,” IEEE Photon. Technol. Lett. 3, 606–609 (1991).
[CrossRef]

J. Sacher, W. Elsässer, and E. O. Goebel, “Nonlinear dynamics of semiconductor laser emission under variable feedback conditions,” IEEE J. Quantum Electron. 27, 373–391 (1991).
[CrossRef]

D. R. Hjelme, A. R. Mickelson, and R. G. Beausoleil, “Semiconductor laser stabilization by external optical feedback,” IEEE J. Quantum Electron. QE-27, 352–372 (1991).
[CrossRef]

1990 (6)

J. S. Cohen, F. Wittgrefe, M. D. Hogerland, and J. P. Woerdman, “Optical spectra of a semiconductor laser with incoherent optical feedback,” IEEE J. Quantum Electron. 26, 982–990 (1990).
[CrossRef]

K. L. Hall, J. Mark, E. P. Ippen, and G. Eisenstein, “Femtosecond gain dynamics in InGaAsP optical amplifiers,” Appl. Phys. Lett. 56, 1740–1742 (1990).
[CrossRef]

J. Mørk and B. Tromborg, “The mechanism of mode selection for an external cavity laser,” IEEE Photon. Technol. Lett. 2, 21–23 (1990).
[CrossRef]

J. Helms and K. Petermann, “A simple analytic expression for the stable operation range for laser diodes with optical feedback,” IEEE J. Quantum Electron. 26, 833–836 (1990).
[CrossRef]

B. Tromborg and J. Mørk, “Stability analysis and the route to chaos for laser diodes with optical feedback,” IEEE Photon. Technol. Lett. 2, 549–552 (1990).
[CrossRef]

J. Mørk, J. Mark, and B. Tromborg, “Route to chaos and competition between relaxation oscillations for a semiconductor laser with optical feedback,” Phys. Rev. Lett. 65, 1999–2002 (1990).
[CrossRef] [PubMed]

1989 (3)

J. S. Cohen and D. Lenstra, “Spectral properties of the coherence collapsed state of a semiconductor laser with delayed optical feedback,” IEEE J. Quantum Electron. 25, 1143–1151 (1989).
[CrossRef]

J. Sacher, W. Elsässer, and E. O. Goebel, “Intermittency in the coherence collapse of a semiconductor laser with external feedback,” Phys. Rev. Lett. 63, 2224–2227 (1989).
[CrossRef] [PubMed]

J. O. Binder and G. D. Cormack, “Mode selection and stability of a semiconductor laser with weak optical feedback,” IEEE J. Quantum Electron. 25, 2255–2259 (1989).
[CrossRef]

1988 (7)

G. C. Dente, P. S. Durkin, K. A. Wilson, and C. E. Moeller, “Chaos in the coherence collapse of semiconductor lasers,” IEEE J. Quantum Electron. 24, 2441–2447 (1988).
[CrossRef]

J. Mørk, B. Tromborg, and P. L. Christiansen, “Bistability and low-frequency fluctuations in semiconductor lasers with optical feedback: a theoretical analysis,” IEEE J. Quantum Electron. 24, 123–133 (1988).
[CrossRef]

W. Schunk and K. Petermann, “Numerical analysis of the feedback regimes for a single-mode semiconductor laser with external feedback,” IEEE J. Quantum Electron. 24, 1242–1247 (1988).
[CrossRef]

J. Mørk, P. L. Christiansen, and B. Tromborg, “Limits of stable operation of AR-coated semiconductor lasers with strong optical feedback,” Electron. Lett. 24, 1065–1066 (1988).
[CrossRef]

K. Henneberger and H. Haug, “Nonlinear optics and transport in laser excited semiconductors,” Phys. Rev. B 38, 9759–9770 (1988).
[CrossRef]

J. S. Cohen, R. R. Drenten, and B. H. Verbeek, “The effect of optical feedback on the relaxation oscillation in semiconductor lasers,” IEEE J. Quantum Electron. 24, 1989–1995 (1988).
[CrossRef]

A. Mecozzi, S. Piazzolla, A. Sapia, and P. Spano, “Non-Gaussian statistics of frequency fluctuations in line-narrowed semiconductor lasers,” IEEE J. Quantum Electron. 24, 1985–1988 (1988).
[CrossRef]

1987 (1)

Y. Yamamoto and S. Machida, “High-impedance suppression of pump fluctuations and amplitude squeezing in semiconductor lasers,” Phys. Rev. A 35, 5114–5130 (1987).
[CrossRef] [PubMed]

1986 (2)

R. W. Tkach and A. R. Charplyvy, “Regimes of feedback effects in 1.5 μ m distributed feedback lasers,” IEEE J. Lightwave Technol. LT-4, 1655–1661 (1986).
[CrossRef]

C. H. Henry and R. F. Kazarinov, “Instability of semiconductor lasers due to optical feedback from distant reflectors,” IEEE J. Quantum Electron. QE-22, 294–301 (1986).
[CrossRef]

1985 (2)

T. Mukai and K. Otsuka, “New route to optical chaos: successive-subharmonic-oscillation cascade in a semiconductor laser coupled to an external cavity,” Phys. Rev. Lett. 55, 1711–1714 (1985).
[CrossRef] [PubMed]

D. Lenstra, B. H. Verbeek, and A. J. den Boef, “Coherence collapse in single-mode semiconductor lasers due to optical feedback,” IEEE J. Quantum Electron. QE-21, 674–679 (1985).
[CrossRef]

1984 (4)

P. Spano, S. Piazzolla, and M. Tamburrini, “Theory of noise in semiconductor lasers in the presence of optical feedback,” IEEE J. Quantum Electron. QE-20, 350–357 (1984).
[CrossRef]

G. P. Agrawal, “Line narrowing in a single-mode semiconductor laser due to optical feedback,” IEEE J. Quantum Electron. QE-20, 468–471 (1984).
[CrossRef]

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

H. Haug and S. Schmitt-Rink, “Electron theory of the optical properties of laser excited semiconductors,” Prog. Quantum Electron. 9(1), 13, 14 (1984).
[CrossRef]

1983 (3)

M. Tamburrini, P. Spano, and S. Piazzolla, “Influence of an external cavity on semiconductor laser phase noise,” Appl. Phys. Lett. 43, 410–412 (1983).
[CrossRef]

K. Vahala and A. Yariv, “Semiclassical theory of noise in semiconductor lasers—Part II,” IEEE J. Quantum Electron. QE-19, 1103–1109 (1983).

K. Vahala, Ch. Harder, and A. Yariv, “Observation of relaxation resonance effects in the field spectrum of semiconductor lasers,” Appl. Phys. Lett. 42, 211–213 (1983).
[CrossRef]

1982 (2)

L. Goldberg, H. F. Taylor, A. Dandridge, J. F. Weller, and R. O. Miles, “Spectral characteristics of semiconductor lasers with optical feedback,” IEEE J. Quantum Electron. QE-18, 555–563 (1982).
[CrossRef]

C. H. Henry, “Theory of the linewidth of semiconductor lasers,” IEEE J. Quantum Electron. QE-18, 259–264 (1982).
[CrossRef]

1980 (1)

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

1969 (1)

H. Haug, “Quantum mechanical rate equations for semiconductor lasers,” Phys. Rev. 184, 338–348 (1969).
[CrossRef]

1967 (2)

H. Haug and H. Haken, “Theory of noise in semiconductor laser emission,” Z. Phys. 204, 262–275 (1967).
[CrossRef]

H. Haug, “Quantum mechanical theory of fluctuations in semiconductor lasers,” Z. Phys. 200, 57–68 (1967).
[CrossRef]

Agrawal, G. P.

G. P. Agrawal, “Line narrowing in a single-mode semiconductor laser due to optical feedback,” IEEE J. Quantum Electron. QE-20, 468–471 (1984).
[CrossRef]

Beausoleil, R. G.

D. R. Hjelme, A. R. Mickelson, and R. G. Beausoleil, “Semiconductor laser stabilization by external optical feedback,” IEEE J. Quantum Electron. QE-27, 352–372 (1991).
[CrossRef]

Bergé, P.

P. Bergé, Y. Pomeau, and C. Vidal, Order within Chaos (Wiley, New York, 1984).

Binder, J. O.

J. O. Binder and G. D. Cormack, “Mode selection and stability of a semiconductor laser with weak optical feedback,” IEEE J. Quantum Electron. 25, 2255–2259 (1989).
[CrossRef]

Charplyvy, A. R.

R. W. Tkach and A. R. Charplyvy, “Regimes of feedback effects in 1.5 μ m distributed feedback lasers,” IEEE J. Lightwave Technol. LT-4, 1655–1661 (1986).
[CrossRef]

Christiansen, P. L.

J. Mørk, P. L. Christiansen, and B. Tromborg, “Limits of stable operation of AR-coated semiconductor lasers with strong optical feedback,” Electron. Lett. 24, 1065–1066 (1988).
[CrossRef]

J. Mørk, B. Tromborg, and P. L. Christiansen, “Bistability and low-frequency fluctuations in semiconductor lasers with optical feedback: a theoretical analysis,” IEEE J. Quantum Electron. 24, 123–133 (1988).
[CrossRef]

Cohen, J. S.

J. S. Cohen, F. Wittgrefe, M. D. Hogerland, and J. P. Woerdman, “Optical spectra of a semiconductor laser with incoherent optical feedback,” IEEE J. Quantum Electron. 26, 982–990 (1990).
[CrossRef]

J. S. Cohen and D. Lenstra, “Spectral properties of the coherence collapsed state of a semiconductor laser with delayed optical feedback,” IEEE J. Quantum Electron. 25, 1143–1151 (1989).
[CrossRef]

J. S. Cohen, R. R. Drenten, and B. H. Verbeek, “The effect of optical feedback on the relaxation oscillation in semiconductor lasers,” IEEE J. Quantum Electron. 24, 1989–1995 (1988).
[CrossRef]

Cormack, G. D.

J. O. Binder and G. D. Cormack, “Mode selection and stability of a semiconductor laser with weak optical feedback,” IEEE J. Quantum Electron. 25, 2255–2259 (1989).
[CrossRef]

Dandridge, A.

L. Goldberg, H. F. Taylor, A. Dandridge, J. F. Weller, and R. O. Miles, “Spectral characteristics of semiconductor lasers with optical feedback,” IEEE J. Quantum Electron. QE-18, 555–563 (1982).
[CrossRef]

den Boef, A. J.

D. Lenstra, B. H. Verbeek, and A. J. den Boef, “Coherence collapse in single-mode semiconductor lasers due to optical feedback,” IEEE J. Quantum Electron. QE-21, 674–679 (1985).
[CrossRef]

Dente, G. C.

G. C. Dente, P. S. Durkin, K. A. Wilson, and C. E. Moeller, “Chaos in the coherence collapse of semiconductor lasers,” IEEE J. Quantum Electron. 24, 2441–2447 (1988).
[CrossRef]

Drenten, R. R.

J. S. Cohen, R. R. Drenten, and B. H. Verbeek, “The effect of optical feedback on the relaxation oscillation in semiconductor lasers,” IEEE J. Quantum Electron. 24, 1989–1995 (1988).
[CrossRef]

Durkin, P. S.

G. C. Dente, P. S. Durkin, K. A. Wilson, and C. E. Moeller, “Chaos in the coherence collapse of semiconductor lasers,” IEEE J. Quantum Electron. 24, 2441–2447 (1988).
[CrossRef]

Eisenstein, G.

K. L. Hall, J. Mark, E. P. Ippen, and G. Eisenstein, “Femtosecond gain dynamics in InGaAsP optical amplifiers,” Appl. Phys. Lett. 56, 1740–1742 (1990).
[CrossRef]

Elsässer, W.

J. Sacher, W. Elsässer, and E. O. Goebel, “Nonlinear dynamics of semiconductor laser emission under variable feedback conditions,” IEEE J. Quantum Electron. 27, 373–391 (1991).
[CrossRef]

J. Sacher, W. Elsässer, and E. O. Goebel, “Intermittency in the coherence collapse of a semiconductor laser with external feedback,” Phys. Rev. Lett. 63, 2224–2227 (1989).
[CrossRef] [PubMed]

Goebel, E. O.

J. Sacher, W. Elsässer, and E. O. Goebel, “Nonlinear dynamics of semiconductor laser emission under variable feedback conditions,” IEEE J. Quantum Electron. 27, 373–391 (1991).
[CrossRef]

J. Sacher, W. Elsässer, and E. O. Goebel, “Intermittency in the coherence collapse of a semiconductor laser with external feedback,” Phys. Rev. Lett. 63, 2224–2227 (1989).
[CrossRef] [PubMed]

Goldberg, L.

L. Goldberg, H. F. Taylor, A. Dandridge, J. F. Weller, and R. O. Miles, “Spectral characteristics of semiconductor lasers with optical feedback,” IEEE J. Quantum Electron. QE-18, 555–563 (1982).
[CrossRef]

Haken, H.

H. Haug and H. Haken, “Theory of noise in semiconductor laser emission,” Z. Phys. 204, 262–275 (1967).
[CrossRef]

H. Haken, Laser Theory, Vol. XXV/2c of Handbuch der Physik (Springer-Verlag, Berlin, 1970).

Hall, K. L.

K. L. Hall, J. Mark, E. P. Ippen, and G. Eisenstein, “Femtosecond gain dynamics in InGaAsP optical amplifiers,” Appl. Phys. Lett. 56, 1740–1742 (1990).
[CrossRef]

Harder, Ch.

K. Vahala, Ch. Harder, and A. Yariv, “Observation of relaxation resonance effects in the field spectrum of semiconductor lasers,” Appl. Phys. Lett. 42, 211–213 (1983).
[CrossRef]

Haug, H.

A. Ritter and H. Haug, “Theory of laser diodes with weak optical feedback. II. Limit-cycle behavior, quasi-periodicity, frequency locking, and route to chaos,” J. Opt. Soc. Am. B 10, 145–154 (1993).
[CrossRef]

H. Haug and K. Henneberger, “The kinetics of hole-burning in semiconductor lasers,” Z. Phys. B 83, 447–451 (1991).
[CrossRef]

K. Henneberger and H. Haug, “Nonlinear optics and transport in laser excited semiconductors,” Phys. Rev. B 38, 9759–9770 (1988).
[CrossRef]

H. Haug and S. Schmitt-Rink, “Electron theory of the optical properties of laser excited semiconductors,” Prog. Quantum Electron. 9(1), 13, 14 (1984).
[CrossRef]

H. Haug, “Quantum mechanical rate equations for semiconductor lasers,” Phys. Rev. 184, 338–348 (1969).
[CrossRef]

H. Haug, “Quantum mechanical theory of fluctuations in semiconductor lasers,” Z. Phys. 200, 57–68 (1967).
[CrossRef]

H. Haug and H. Haken, “Theory of noise in semiconductor laser emission,” Z. Phys. 204, 262–275 (1967).
[CrossRef]

Helms, J.

J. Helms and K. Petermann, “A simple analytic expression for the stable operation range for laser diodes with optical feedback,” IEEE J. Quantum Electron. 26, 833–836 (1990).
[CrossRef]

Henneberger, K.

H. Haug and K. Henneberger, “The kinetics of hole-burning in semiconductor lasers,” Z. Phys. B 83, 447–451 (1991).
[CrossRef]

K. Henneberger and H. Haug, “Nonlinear optics and transport in laser excited semiconductors,” Phys. Rev. B 38, 9759–9770 (1988).
[CrossRef]

Henry, C. H.

C. H. Henry and R. F. Kazarinov, “Instability of semiconductor lasers due to optical feedback from distant reflectors,” IEEE J. Quantum Electron. QE-22, 294–301 (1986).
[CrossRef]

C. H. Henry, “Theory of the linewidth of semiconductor lasers,” IEEE J. Quantum Electron. QE-18, 259–264 (1982).
[CrossRef]

Hjelme, D. R.

D. R. Hjelme, A. R. Mickelson, and R. G. Beausoleil, “Semiconductor laser stabilization by external optical feedback,” IEEE J. Quantum Electron. QE-27, 352–372 (1991).
[CrossRef]

Hogerland, M. D.

J. S. Cohen, F. Wittgrefe, M. D. Hogerland, and J. P. Woerdman, “Optical spectra of a semiconductor laser with incoherent optical feedback,” IEEE J. Quantum Electron. 26, 982–990 (1990).
[CrossRef]

Ippen, E. P.

K. L. Hall, J. Mark, E. P. Ippen, and G. Eisenstein, “Femtosecond gain dynamics in InGaAsP optical amplifiers,” Appl. Phys. Lett. 56, 1740–1742 (1990).
[CrossRef]

Kazarinov, R. F.

C. H. Henry and R. F. Kazarinov, “Instability of semiconductor lasers due to optical feedback from distant reflectors,” IEEE J. Quantum Electron. QE-22, 294–301 (1986).
[CrossRef]

Kobayashi, K.

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

Lang, R.

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

Lax, M.

M. Lax, in Bradeis University Summer Institute of Theoretical Physics, M. Chretin, E. P. Gross, and S. Deser, eds. (Gordon & Breach, New York, 1968).

Lenstra, D.

J. S. Cohen and D. Lenstra, “Spectral properties of the coherence collapsed state of a semiconductor laser with delayed optical feedback,” IEEE J. Quantum Electron. 25, 1143–1151 (1989).
[CrossRef]

D. Lenstra, B. H. Verbeek, and A. J. den Boef, “Coherence collapse in single-mode semiconductor lasers due to optical feedback,” IEEE J. Quantum Electron. QE-21, 674–679 (1985).
[CrossRef]

Machida, S.

Y. Yamamoto and S. Machida, “High-impedance suppression of pump fluctuations and amplitude squeezing in semiconductor lasers,” Phys. Rev. A 35, 5114–5130 (1987).
[CrossRef] [PubMed]

Mark, J.

J. Mørk, B. Tromborg, and J. Mark, “Chaos in semiconductor lasers with optical feedback: theory and experiment,” IEEE J. Quantum Electron 28, 93–108 (1992).
[CrossRef]

K. L. Hall, J. Mark, E. P. Ippen, and G. Eisenstein, “Femtosecond gain dynamics in InGaAsP optical amplifiers,” Appl. Phys. Lett. 56, 1740–1742 (1990).
[CrossRef]

J. Mørk, J. Mark, and B. Tromborg, “Route to chaos and competition between relaxation oscillations for a semiconductor laser with optical feedback,” Phys. Rev. Lett. 65, 1999–2002 (1990).
[CrossRef] [PubMed]

Mecozzi, A.

A. Mecozzi, S. Piazzolla, A. Sapia, and P. Spano, “Non-Gaussian statistics of frequency fluctuations in line-narrowed semiconductor lasers,” IEEE J. Quantum Electron. 24, 1985–1988 (1988).
[CrossRef]

Mickelson, A. R.

D. R. Hjelme, A. R. Mickelson, and R. G. Beausoleil, “Semiconductor laser stabilization by external optical feedback,” IEEE J. Quantum Electron. QE-27, 352–372 (1991).
[CrossRef]

Miles, R. O.

L. Goldberg, H. F. Taylor, A. Dandridge, J. F. Weller, and R. O. Miles, “Spectral characteristics of semiconductor lasers with optical feedback,” IEEE J. Quantum Electron. QE-18, 555–563 (1982).
[CrossRef]

Moeller, C. E.

G. C. Dente, P. S. Durkin, K. A. Wilson, and C. E. Moeller, “Chaos in the coherence collapse of semiconductor lasers,” IEEE J. Quantum Electron. 24, 2441–2447 (1988).
[CrossRef]

Mørk, J.

J. Mørk, B. Tromborg, and J. Mark, “Chaos in semiconductor lasers with optical feedback: theory and experiment,” IEEE J. Quantum Electron 28, 93–108 (1992).
[CrossRef]

M. Willatzen, A. Uskov, J. Mørk, H. Olesen, and B. Tromborg, “Nonlinear gain supression in semiconductor lasers due to carrier heating,” IEEE Photon. Technol. Lett. 3, 606–609 (1991).
[CrossRef]

J. Mørk and B. Tromborg, “The mechanism of mode selection for an external cavity laser,” IEEE Photon. Technol. Lett. 2, 21–23 (1990).
[CrossRef]

B. Tromborg and J. Mørk, “Stability analysis and the route to chaos for laser diodes with optical feedback,” IEEE Photon. Technol. Lett. 2, 549–552 (1990).
[CrossRef]

J. Mørk, J. Mark, and B. Tromborg, “Route to chaos and competition between relaxation oscillations for a semiconductor laser with optical feedback,” Phys. Rev. Lett. 65, 1999–2002 (1990).
[CrossRef] [PubMed]

J. Mørk, B. Tromborg, and P. L. Christiansen, “Bistability and low-frequency fluctuations in semiconductor lasers with optical feedback: a theoretical analysis,” IEEE J. Quantum Electron. 24, 123–133 (1988).
[CrossRef]

J. Mørk, P. L. Christiansen, and B. Tromborg, “Limits of stable operation of AR-coated semiconductor lasers with strong optical feedback,” Electron. Lett. 24, 1065–1066 (1988).
[CrossRef]

Mukai, T.

T. Mukai and K. Otsuka, “New route to optical chaos: successive-subharmonic-oscillation cascade in a semiconductor laser coupled to an external cavity,” Phys. Rev. Lett. 55, 1711–1714 (1985).
[CrossRef] [PubMed]

Olesen, H.

M. Willatzen, A. Uskov, J. Mørk, H. Olesen, and B. Tromborg, “Nonlinear gain supression in semiconductor lasers due to carrier heating,” IEEE Photon. Technol. Lett. 3, 606–609 (1991).
[CrossRef]

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

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. QE-20, 1023–1032 (1984).
[CrossRef]

Otsuka, K.

T. Mukai and K. Otsuka, “New route to optical chaos: successive-subharmonic-oscillation cascade in a semiconductor laser coupled to an external cavity,” Phys. Rev. Lett. 55, 1711–1714 (1985).
[CrossRef] [PubMed]

Petermann, K.

J. Helms and K. Petermann, “A simple analytic expression for the stable operation range for laser diodes with optical feedback,” IEEE J. Quantum Electron. 26, 833–836 (1990).
[CrossRef]

W. Schunk and K. Petermann, “Numerical analysis of the feedback regimes for a single-mode semiconductor laser with external feedback,” IEEE J. Quantum Electron. 24, 1242–1247 (1988).
[CrossRef]

Piazzolla, S.

A. Mecozzi, S. Piazzolla, A. Sapia, and P. Spano, “Non-Gaussian statistics of frequency fluctuations in line-narrowed semiconductor lasers,” IEEE J. Quantum Electron. 24, 1985–1988 (1988).
[CrossRef]

P. Spano, S. Piazzolla, and M. Tamburrini, “Theory of noise in semiconductor lasers in the presence of optical feedback,” IEEE J. Quantum Electron. QE-20, 350–357 (1984).
[CrossRef]

M. Tamburrini, P. Spano, and S. Piazzolla, “Influence of an external cavity on semiconductor laser phase noise,” Appl. Phys. Lett. 43, 410–412 (1983).
[CrossRef]

Pomeau, Y.

P. Bergé, Y. Pomeau, and C. Vidal, Order within Chaos (Wiley, New York, 1984).

Ritter, A.

Sacher, J.

J. Sacher, W. Elsässer, and E. O. Goebel, “Nonlinear dynamics of semiconductor laser emission under variable feedback conditions,” IEEE J. Quantum Electron. 27, 373–391 (1991).
[CrossRef]

J. Sacher, W. Elsässer, and E. O. Goebel, “Intermittency in the coherence collapse of a semiconductor laser with external feedback,” Phys. Rev. Lett. 63, 2224–2227 (1989).
[CrossRef] [PubMed]

Sapia, A.

A. Mecozzi, S. Piazzolla, A. Sapia, and P. Spano, “Non-Gaussian statistics of frequency fluctuations in line-narrowed semiconductor lasers,” IEEE J. Quantum Electron. 24, 1985–1988 (1988).
[CrossRef]

Schmitt-Rink, S.

H. Haug and S. Schmitt-Rink, “Electron theory of the optical properties of laser excited semiconductors,” Prog. Quantum Electron. 9(1), 13, 14 (1984).
[CrossRef]

Schunk, W.

W. Schunk and K. Petermann, “Numerical analysis of the feedback regimes for a single-mode semiconductor laser with external feedback,” IEEE J. Quantum Electron. 24, 1242–1247 (1988).
[CrossRef]

Schuster, H. G.

H. G. Schuster, Deterministic Chaos, 2nd ed. (VCH, Weinheim, Germany, 1989).

Spano, P.

A. Mecozzi, S. Piazzolla, A. Sapia, and P. Spano, “Non-Gaussian statistics of frequency fluctuations in line-narrowed semiconductor lasers,” IEEE J. Quantum Electron. 24, 1985–1988 (1988).
[CrossRef]

P. Spano, S. Piazzolla, and M. Tamburrini, “Theory of noise in semiconductor lasers in the presence of optical feedback,” IEEE J. Quantum Electron. QE-20, 350–357 (1984).
[CrossRef]

M. Tamburrini, P. Spano, and S. Piazzolla, “Influence of an external cavity on semiconductor laser phase noise,” Appl. Phys. Lett. 43, 410–412 (1983).
[CrossRef]

Tamburrini, M.

P. Spano, S. Piazzolla, and M. Tamburrini, “Theory of noise in semiconductor lasers in the presence of optical feedback,” IEEE J. Quantum Electron. QE-20, 350–357 (1984).
[CrossRef]

M. Tamburrini, P. Spano, and S. Piazzolla, “Influence of an external cavity on semiconductor laser phase noise,” Appl. Phys. Lett. 43, 410–412 (1983).
[CrossRef]

Taylor, H. F.

L. Goldberg, H. F. Taylor, A. Dandridge, J. F. Weller, and R. O. Miles, “Spectral characteristics of semiconductor lasers with optical feedback,” IEEE J. Quantum Electron. QE-18, 555–563 (1982).
[CrossRef]

Tkach, R. W.

R. W. Tkach and A. R. Charplyvy, “Regimes of feedback effects in 1.5 μ m distributed feedback lasers,” IEEE J. Lightwave Technol. LT-4, 1655–1661 (1986).
[CrossRef]

Tromborg, B.

J. Mørk, B. Tromborg, and J. Mark, “Chaos in semiconductor lasers with optical feedback: theory and experiment,” IEEE J. Quantum Electron 28, 93–108 (1992).
[CrossRef]

M. Willatzen, A. Uskov, J. Mørk, H. Olesen, and B. Tromborg, “Nonlinear gain supression in semiconductor lasers due to carrier heating,” IEEE Photon. Technol. Lett. 3, 606–609 (1991).
[CrossRef]

J. Mørk and B. Tromborg, “The mechanism of mode selection for an external cavity laser,” IEEE Photon. Technol. Lett. 2, 21–23 (1990).
[CrossRef]

J. Mørk, J. Mark, and B. Tromborg, “Route to chaos and competition between relaxation oscillations for a semiconductor laser with optical feedback,” Phys. Rev. Lett. 65, 1999–2002 (1990).
[CrossRef] [PubMed]

B. Tromborg and J. Mørk, “Stability analysis and the route to chaos for laser diodes with optical feedback,” IEEE Photon. Technol. Lett. 2, 549–552 (1990).
[CrossRef]

J. Mørk, P. L. Christiansen, and B. Tromborg, “Limits of stable operation of AR-coated semiconductor lasers with strong optical feedback,” Electron. Lett. 24, 1065–1066 (1988).
[CrossRef]

J. Mørk, B. Tromborg, and P. L. Christiansen, “Bistability and low-frequency fluctuations in semiconductor lasers with optical feedback: a theoretical analysis,” IEEE J. Quantum Electron. 24, 123–133 (1988).
[CrossRef]

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

Uskov, A.

M. Willatzen, A. Uskov, J. Mørk, H. Olesen, and B. Tromborg, “Nonlinear gain supression in semiconductor lasers due to carrier heating,” IEEE Photon. Technol. Lett. 3, 606–609 (1991).
[CrossRef]

Vahala, K.

K. Vahala and A. Yariv, “Semiclassical theory of noise in semiconductor lasers—Part II,” IEEE J. Quantum Electron. QE-19, 1103–1109 (1983).

K. Vahala, Ch. Harder, and A. Yariv, “Observation of relaxation resonance effects in the field spectrum of semiconductor lasers,” Appl. Phys. Lett. 42, 211–213 (1983).
[CrossRef]

Verbeek, B. H.

J. S. Cohen, R. R. Drenten, and B. H. Verbeek, “The effect of optical feedback on the relaxation oscillation in semiconductor lasers,” IEEE J. Quantum Electron. 24, 1989–1995 (1988).
[CrossRef]

D. Lenstra, B. H. Verbeek, and A. J. den Boef, “Coherence collapse in single-mode semiconductor lasers due to optical feedback,” IEEE J. Quantum Electron. QE-21, 674–679 (1985).
[CrossRef]

Vidal, C.

P. Bergé, Y. Pomeau, and C. Vidal, Order within Chaos (Wiley, New York, 1984).

Weller, J. F.

L. Goldberg, H. F. Taylor, A. Dandridge, J. F. Weller, and R. O. Miles, “Spectral characteristics of semiconductor lasers with optical feedback,” IEEE J. Quantum Electron. QE-18, 555–563 (1982).
[CrossRef]

Willatzen, M.

M. Willatzen, A. Uskov, J. Mørk, H. Olesen, and B. Tromborg, “Nonlinear gain supression in semiconductor lasers due to carrier heating,” IEEE Photon. Technol. Lett. 3, 606–609 (1991).
[CrossRef]

Wilson, K. A.

G. C. Dente, P. S. Durkin, K. A. Wilson, and C. E. Moeller, “Chaos in the coherence collapse of semiconductor lasers,” IEEE J. Quantum Electron. 24, 2441–2447 (1988).
[CrossRef]

Wittgrefe, F.

J. S. Cohen, F. Wittgrefe, M. D. Hogerland, and J. P. Woerdman, “Optical spectra of a semiconductor laser with incoherent optical feedback,” IEEE J. Quantum Electron. 26, 982–990 (1990).
[CrossRef]

Woerdman, J. P.

J. S. Cohen, F. Wittgrefe, M. D. Hogerland, and J. P. Woerdman, “Optical spectra of a semiconductor laser with incoherent optical feedback,” IEEE J. Quantum Electron. 26, 982–990 (1990).
[CrossRef]

Yamamoto, Y.

Y. Yamamoto and S. Machida, “High-impedance suppression of pump fluctuations and amplitude squeezing in semiconductor lasers,” Phys. Rev. A 35, 5114–5130 (1987).
[CrossRef] [PubMed]

Yariv, A.

K. Vahala, Ch. Harder, and A. Yariv, “Observation of relaxation resonance effects in the field spectrum of semiconductor lasers,” Appl. Phys. Lett. 42, 211–213 (1983).
[CrossRef]

K. Vahala and A. Yariv, “Semiclassical theory of noise in semiconductor lasers—Part II,” IEEE J. Quantum Electron. QE-19, 1103–1109 (1983).

Appl. Phys. Lett. (3)

M. Tamburrini, P. Spano, and S. Piazzolla, “Influence of an external cavity on semiconductor laser phase noise,” Appl. Phys. Lett. 43, 410–412 (1983).
[CrossRef]

K. Vahala, Ch. Harder, and A. Yariv, “Observation of relaxation resonance effects in the field spectrum of semiconductor lasers,” Appl. Phys. Lett. 42, 211–213 (1983).
[CrossRef]

K. L. Hall, J. Mark, E. P. Ippen, and G. Eisenstein, “Femtosecond gain dynamics in InGaAsP optical amplifiers,” Appl. Phys. Lett. 56, 1740–1742 (1990).
[CrossRef]

Electron. Lett. (1)

J. Mørk, P. L. Christiansen, and B. Tromborg, “Limits of stable operation of AR-coated semiconductor lasers with strong optical feedback,” Electron. Lett. 24, 1065–1066 (1988).
[CrossRef]

IEEE J. Lightwave Technol. (1)

R. W. Tkach and A. R. Charplyvy, “Regimes of feedback effects in 1.5 μ m distributed feedback lasers,” IEEE J. Lightwave Technol. LT-4, 1655–1661 (1986).
[CrossRef]

IEEE J. Quantum Electron (1)

J. Mørk, B. Tromborg, and J. Mark, “Chaos in semiconductor lasers with optical feedback: theory and experiment,” IEEE J. Quantum Electron 28, 93–108 (1992).
[CrossRef]

IEEE J. Quantum Electron. (20)

G. C. Dente, P. S. Durkin, K. A. Wilson, and C. E. Moeller, “Chaos in the coherence collapse of semiconductor lasers,” IEEE J. Quantum Electron. 24, 2441–2447 (1988).
[CrossRef]

J. Sacher, W. Elsässer, and E. O. Goebel, “Nonlinear dynamics of semiconductor laser emission under variable feedback conditions,” IEEE J. Quantum Electron. 27, 373–391 (1991).
[CrossRef]

W. Schunk and K. Petermann, “Numerical analysis of the feedback regimes for a single-mode semiconductor laser with external feedback,” IEEE J. Quantum Electron. 24, 1242–1247 (1988).
[CrossRef]

P. Spano, S. Piazzolla, and M. Tamburrini, “Theory of noise in semiconductor lasers in the presence of optical feedback,” IEEE J. Quantum Electron. QE-20, 350–357 (1984).
[CrossRef]

G. P. Agrawal, “Line narrowing in a single-mode semiconductor laser due to optical feedback,” IEEE J. Quantum Electron. QE-20, 468–471 (1984).
[CrossRef]

D. Lenstra, B. H. Verbeek, and A. J. den Boef, “Coherence collapse in single-mode semiconductor lasers due to optical feedback,” IEEE J. Quantum Electron. QE-21, 674–679 (1985).
[CrossRef]

J. S. Cohen and D. Lenstra, “Spectral properties of the coherence collapsed state of a semiconductor laser with delayed optical feedback,” IEEE J. Quantum Electron. 25, 1143–1151 (1989).
[CrossRef]

J. S. Cohen, F. Wittgrefe, M. D. Hogerland, and J. P. Woerdman, “Optical spectra of a semiconductor laser with incoherent optical feedback,” IEEE J. Quantum Electron. 26, 982–990 (1990).
[CrossRef]

C. H. Henry and R. F. Kazarinov, “Instability of semiconductor lasers due to optical feedback from distant reflectors,” IEEE J. Quantum Electron. QE-22, 294–301 (1986).
[CrossRef]

J. Mørk, B. Tromborg, and P. L. Christiansen, “Bistability and low-frequency fluctuations in semiconductor lasers with optical feedback: a theoretical analysis,” IEEE J. Quantum Electron. 24, 123–133 (1988).
[CrossRef]

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

K. Vahala and A. Yariv, “Semiclassical theory of noise in semiconductor lasers—Part II,” IEEE J. Quantum Electron. QE-19, 1103–1109 (1983).

A. Mecozzi, S. Piazzolla, A. Sapia, and P. Spano, “Non-Gaussian statistics of frequency fluctuations in line-narrowed semiconductor lasers,” IEEE J. Quantum Electron. 24, 1985–1988 (1988).
[CrossRef]

J. Helms and K. Petermann, “A simple analytic expression for the stable operation range for laser diodes with optical feedback,” IEEE J. Quantum Electron. 26, 833–836 (1990).
[CrossRef]

J. S. Cohen, R. R. Drenten, and B. H. Verbeek, “The effect of optical feedback on the relaxation oscillation in semiconductor lasers,” IEEE J. Quantum Electron. 24, 1989–1995 (1988).
[CrossRef]

L. Goldberg, H. F. Taylor, A. Dandridge, J. F. Weller, and R. O. Miles, “Spectral characteristics of semiconductor lasers with optical feedback,” IEEE J. Quantum Electron. QE-18, 555–563 (1982).
[CrossRef]

J. O. Binder and G. D. Cormack, “Mode selection and stability of a semiconductor laser with weak optical feedback,” IEEE J. Quantum Electron. 25, 2255–2259 (1989).
[CrossRef]

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

C. H. Henry, “Theory of the linewidth of semiconductor lasers,” IEEE J. Quantum Electron. QE-18, 259–264 (1982).
[CrossRef]

D. R. Hjelme, A. R. Mickelson, and R. G. Beausoleil, “Semiconductor laser stabilization by external optical feedback,” IEEE J. Quantum Electron. QE-27, 352–372 (1991).
[CrossRef]

IEEE Photon. Technol. Lett. (3)

J. Mørk and B. Tromborg, “The mechanism of mode selection for an external cavity laser,” IEEE Photon. Technol. Lett. 2, 21–23 (1990).
[CrossRef]

M. Willatzen, A. Uskov, J. Mørk, H. Olesen, and B. Tromborg, “Nonlinear gain supression in semiconductor lasers due to carrier heating,” IEEE Photon. Technol. Lett. 3, 606–609 (1991).
[CrossRef]

B. Tromborg and J. Mørk, “Stability analysis and the route to chaos for laser diodes with optical feedback,” IEEE Photon. Technol. Lett. 2, 549–552 (1990).
[CrossRef]

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

Phys. Rev. (1)

H. Haug, “Quantum mechanical rate equations for semiconductor lasers,” Phys. Rev. 184, 338–348 (1969).
[CrossRef]

Phys. Rev. A (1)

Y. Yamamoto and S. Machida, “High-impedance suppression of pump fluctuations and amplitude squeezing in semiconductor lasers,” Phys. Rev. A 35, 5114–5130 (1987).
[CrossRef] [PubMed]

Phys. Rev. B (1)

K. Henneberger and H. Haug, “Nonlinear optics and transport in laser excited semiconductors,” Phys. Rev. B 38, 9759–9770 (1988).
[CrossRef]

Phys. Rev. Lett. (3)

J. Mørk, J. Mark, and B. Tromborg, “Route to chaos and competition between relaxation oscillations for a semiconductor laser with optical feedback,” Phys. Rev. Lett. 65, 1999–2002 (1990).
[CrossRef] [PubMed]

T. Mukai and K. Otsuka, “New route to optical chaos: successive-subharmonic-oscillation cascade in a semiconductor laser coupled to an external cavity,” Phys. Rev. Lett. 55, 1711–1714 (1985).
[CrossRef] [PubMed]

J. Sacher, W. Elsässer, and E. O. Goebel, “Intermittency in the coherence collapse of a semiconductor laser with external feedback,” Phys. Rev. Lett. 63, 2224–2227 (1989).
[CrossRef] [PubMed]

Prog. Quantum Electron. (1)

H. Haug and S. Schmitt-Rink, “Electron theory of the optical properties of laser excited semiconductors,” Prog. Quantum Electron. 9(1), 13, 14 (1984).
[CrossRef]

Z. Phys. (2)

H. Haug and H. Haken, “Theory of noise in semiconductor laser emission,” Z. Phys. 204, 262–275 (1967).
[CrossRef]

H. Haug, “Quantum mechanical theory of fluctuations in semiconductor lasers,” Z. Phys. 200, 57–68 (1967).
[CrossRef]

Z. Phys. B (1)

H. Haug and K. Henneberger, “The kinetics of hole-burning in semiconductor lasers,” Z. Phys. B 83, 447–451 (1991).
[CrossRef]

Other (5)

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions (Dover, New York, 1970), p. 375, ¶ 9.6.10; p. 376, ¶ 9.6.34.

H. Haken, Laser Theory, Vol. XXV/2c of Handbuch der Physik (Springer-Verlag, Berlin, 1970).

M. Lax, in Bradeis University Summer Institute of Theoretical Physics, M. Chretin, E. P. Gross, and S. Deser, eds. (Gordon & Breach, New York, 1968).

H. G. Schuster, Deterministic Chaos, 2nd ed. (VCH, Weinheim, Germany, 1989).

P. Bergé, Y. Pomeau, and C. Vidal, Order within Chaos (Wiley, New York, 1984).

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Figures (10)

Fig. 1
Fig. 1

Lower part: Stability range (hatched area) of the stationary solution of minimum linewidth (ρ = −arctan α). Dotted and dashed curves, analytic approximations for lower and upper critical feedback rates, respectively. Upper part: Critical RO frequency ωc[Ω(J)]. The Ωτ interval (τ = 1.25 ns) corresponds to a current range of 1.082 < J/Jth < 2.104.

Fig. 2
Fig. 2

Solid curves, real and imaginary parts of the complex eigenfrequencies of the linearized model (ρ = −arctan α, ητin = 0.18 × 10−3, τ = 2 ns). Dotted curves, approximations (2.17). Dashed–dotted curve, decay rate −Γ(Ω) of the solitary RO frequency ωR0 = (Ω2 − Γ2)1/2. Dashed horizontal lines, approximate asymptotic limit, Eq. (2.22a), of RO modes.

Fig. 3
Fig. 3

(a) Time-averaged frequency autocorrelation spectrum with (solid curve) and without (dotted curve) external feedback (ρ = arctan α, ητin = 3 × 10−3, J/Jth ≃ 1.52, τ = 0.5 ns). The amplitude is normalized to the linewidth of the solitary laser. The arrows mark the characteristic frequencies. (b) Complex eigenfrequencies of D(z); similar to Fig. 2. The vertical line marks the current for the noise spectrum given in (a).

Fig. 4
Fig. 4

As in Fig. 3, for ρ = −arctan α, ητin = 1.8 × 10−3, J/Jth ≃ 1.52, τ = 0.7 ns.

Fig. 5
Fig. 5

As in Fig. 4W but with higher feedback, ητin = 5.5 × 10−3.

Fig. 6
Fig. 6

Comparison of the asymptotic relation, Eqs. (2.22) (solid curves), with numerical results with D(z) = 0 for Ω(J)τ = 5.5π (dotted curves), 9π (dashed curves), 18π(dashed-dotted curves); τ = 2 ns and ρ = −arctan α.

Fig. 7
Fig. 7

Time-averaged field autocorrelation spectrum of a solitary laser diode at J = 1.2Jth.

Fig. 8
Fig. 8

As in Fig. 7, at J = 1.52Jth. Inset, linear plot of the RO sidebands.

Fig. 9
Fig. 9

Second-order expansion of the time-averaged field autocorrelation spectrum in the presence of weak optical feedback (parameters same as in Fig. 5).

Fig. 10
Fig. 10

As in Fig. 9, with parameters the same as in Fig. 3(a).

Tables (1)

Tables Icon

Table 1 Parameters

Equations (90)

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

E ( t ) = Re ( [ I ( t ) ] 1 / 2 exp { - i [ ω L 0 t + Φ ( t ) ] } ) .
d d t I ( t ) = { G [ Δ N ( t ) , I ( t ) ] - 1 τ p } I ( t ) + 2 η [ I ( t ) I ( t - τ ) ] 1 / 2 × cos [ ω L 0 τ + Φ ( t ) - Φ ( t - τ ) ] + F I ( t ) ,
d d t Φ ( t ) = α 2 A Δ N ( t ) - η [ I ( t - τ ) I ( t ) ] 1 / 2 × sin [ ω L 0 τ + Φ ( t ) - Φ ( t - τ ) ] + F Φ ( t ) ,
d d t Δ N ( t ) = Δ J - Δ N ( t ) T - G [ Δ N ( t ) , I ( t ) ] I ( t ) + F N ( t ) .
G [ Δ N ( t ) , I ( t ) ] = A Δ N ( t ) + ( 1 / τ p ) 1 + I ( t ) [ 1 - I ( t ) ] [ A Δ N ( t ) + 1 τ p ] .
I s = I ( t ) = I ( t - τ ) , Δ N s = Δ N ( t ) , G s = G ( Δ N s , I s ) = G ( t ) , Φ ( t ) = ( ω L - ω L 0 ) t .
Δ ω L = ω L - ω L 0 = α 2 A Δ N s - η sin ρ = α I s 2 τ p - η ( 1 + α 2 ) 1 / 2 sin ( ρ + arctan α ) ,
ρ = ω L τ mod 2 π ,             α = α / ( 1 - I s ) , G s - ( 1 / τ p ) = - 2 η cos ρ ,
I s = Δ J - ( Δ N s / T ) G s .
I ( t ) = I s + δ I ( t ) ,             Φ ( t ) = Δ ω L t + ϕ ( t ) , Δ N ( t ) = Δ N s + δ N ( t ) .
D ( z ) D Φ ( z ) = z + C [ 1 - exp ( - z τ ) ] , C η ( 1 + α 2 ) 1 / 2 cos ( ρ + arctan α )
σ 0 = 1 + C τ > 0.
d d t ϕ ( t ) - Δ ω L + α I s 2 τ p - η ( 1 + α 2 ) 1 / 2 × sin [ ω L τ + ( arctan α ) + ϕ ( t ) - ϕ ( t - τ ) ] + F ϕ ( t ) - α 2 I s F I ( t ) .
d d t ϕ ( t ) + η ( 1 + α 2 ) 1 / 2 cos ( ω L τ + arctan α ) × [ ϕ ( t ) - ϕ ( t - τ ) ] = F Φ ( t ) - α 2 I s F I ( t ) .
D Φ ( z ) ϕ ( z ) = F Φ ( z ) - α 2 I s F I ( z ) .
D ( z ) z ( z 2 + 2 { Γ + η ˜ ( z ) [ cos ρ - ( i Ω z ) 2 ( 1 + α 2 ) 1 / 2 2 × cos ( ρ + arctan α ) ] } z + Ω 2 ) .
D ( z ) z D R + ( z ) D R - ( z ) ,
D R ± ( z ) = z + Γ - b η ˜ ( z ) i Ω , b = - ½ ( 1 + α 2 ) 1 / 2 cos ( ρ - arctan α ) .
γ = Γ - b η [ 1 - exp ( γ τ ) cos ( ω τ ) ] ,
ω = Ω + b η exp ( γ τ ) sin ( ω τ ) .
Γ τ = 2 x ( 1 - x 2 ) 1 / 2 arcsin x ,             x = ( Γ 2 b η cu ) 1 / 2 < 1 ,             b > 0.
ω - Ω sin ( ω τ ) = b η exp ( γ τ ) .
γ = Γ - b η + ω - Ω tan ( ω τ ) .
( ω - Ω ) τ sin ( ω τ ) exp [ - ( ω - Ω ) τ tan ( ω τ ) ] = b η τ exp ( Γ τ - b η τ ) .
        Ω τ > ω n τ > { ( 2 n - 1 ) π for b > 0 2 n π for b < 0
Ω τ < ω n τ < { ( 2 n - 1 ) π for b > 0 2 n π for b < 0 .
γ ± = Γ - b η + ± δ / τ exp ( ± δ ) - 1 ,
δ ( exp δ ) - 1 exp [ - δ ( exp δ ) - 1 ] = b η τ exp ( Γ τ - b η τ ) < exp ( - 1 ) ,             0 < δ < .
γ = Γ - b η + Δ ω tan ( Δ ω τ ) ,
Δ ω τ sin ( Δ ω τ ) exp [ - Δ ω τ tan ( Δ ω τ ) ] = b η τ exp ( Γ τ - b η τ ) > exp ( - 1 ) ,             0 < Δ ω τ < π .
D ( z ) Ω 2 D Φ ( z ) ,             if             z / Ω 2 1.
ω n τ sin ( ω n τ ) exp [ ω n τ tan ( ω n τ ) ] = - C τ exp ( C τ ) ,
γ n = C + ω n tan ( ω n τ ) .
F μ ( t ) = 0 ,             μ = I , Φ , N ,
F μ ( t ) F ν ( t ) = 2 D μ ν δ ( t - t ) ,
2 D I I = 2 V R I s ,             2 D Φ Φ = R 2 V I s ,             2 D I Φ = 0 ,
2 D N I = - 2 R I s V ( 1 - 1 2 n sp ) ,             2 D N Φ = 0 ,
2 D N N = 1 V [ 2 R I s ( 1 - 1 2 n sp ) + J th ] .
R = n sp ( ω L ) G s ( ω L ) ,             n sp = 1 1 - exp [ β ( ω L - μ ) ] .
1 D ( z ) D ( - z ) = 1 Ω 2 σ 0 2 1 δ 2 - z 2 + n = 1 1 D ( z n ) D ( - z n ) ( 1 z - z n - 1 z + z n ) + n = 1 1 [ D ( z n ) D ( - z n ) ] * ( 1 z - z n * - 1 z + z n * ) ,
σ 0 = 1 Ω 2 lim D ( z ) z .
D ( z n ) D ( - z n ) = 8 γ n z n 4 σ n cos θ n exp ( i 3 θ n ) ,
θ n = arctan ( γ n / ω n ) .
S E ( ω ) = - + d t 1 I s E ( t + t ) E * ( t ) t exp [ i ( ω + ω L ) t ]
S E ( t ) = I I s E ( t + t ) E * ( t ) t [ 1 + δ I ( t + t ) + δ I ( t ) 2 I s ] × exp { - i [ ϕ ( t + t ) - ϕ ( t ) + ω L t ] } t .
S E ( t ) = [ 1 - i C I Φ ( t ) ] exp { - [ C Φ Φ ( t ) + i ω L t ] } ,
C Φ Φ ( t ) = ½ [ ϕ ( t + t ) - ϕ ( t ) ] 2 ,
C I Φ ( t ) = 1 2 I s [ δ I ( t + t ) + δ I ( t ) ] [ ϕ ( t + t ) - ϕ ( t ) ] .
C Φ Φ ( 0 ) ( t ) = 1 2 π - + d ω [ 1 - cos ( ω t ) ] [ c Φ Φ ( 0 ) ( ω ) + c Φ Φ ( 0 ) ( - ω ) ] ,
c Φ Φ ( 0 ) ( ω ) = R 4 I s V × [ 1 + α 2 ω 2 + δ 2 + α 2 2 Γ cos θ Γ cos ( 3 θ ) - ( ω - ω R 0 ) sin ( 3 θ ) ( ω - ω R 0 ) 2 + Γ 2 ] , θ = arctan ( Γ / ω R 0 ) ,             ω R 0 = ( Ω 2 - Γ 2 ) 1 / 2 .
C Φ Φ ( 0 ) ( t ) = R 4 I s V [ ( 1 + α 2 ) t + α 2 2 Γ cos ( 3 θ ) - exp ( - Γ t ) cos ( ω R 0 t - 3 θ ) cos θ ] .
C I Φ ( 0 ) ( t ) = 1 2 π - + d ω sin ( ω t ) [ c I Φ ( 0 ) ( ω ) - c I Φ ( 0 ) ( - ω ) ] ,
c I Φ ( 0 ) ( ω ) = - α R / Γ 4 I s V cos θ { r Ω T × [ sin ( 2 θ ) P ω + Γ cos ( 2 θ ) - ( ω - ω R 0 ) sin ( 2 θ ) ( ω - ω R 0 ) 2 + Γ 2 ] + q A I s Ω Γ ( ω - ω R 0 ) 2 + Γ 2 } - α R / Ω 2 I s V cos θ Γ N Γ [ ( tan Θ ) P ω + 1 cos Θ Γ ( cos Θ ) - ( ω - ω R 0 ) ( sin Θ ) ( ω - ω R 0 ) 2 + Γ 2 ] ,
Θ arctan ( r Γ Γ N cos θ Ω T ) < 2 θ ,             q = D N I / D I I , r = 1 + A I s T ( 1 - q ) .
C I Φ ( 0 ) ( t ) = α R / Ω 2 I s V cos θ Γ N Γ × sgn ( t ) ( sin Θ ) + exp ( - Γ t ) sin ( ω R 0 t - Θ ) cos Θ .
S E 0 ( ω ) = m = - + n = 0 + 2 ( m + n ) ! n ! ( g Φ Φ 2 ) m + 2 n × exp [ - g Φ Φ cos ( 3 θ ) ] ( ω - n ω R 0 ) 2 + λ m + 2 n 2 × ( { ( 1 + u 0 m ) cos ( 3 m θ ) + [ u 1 + ( m + 2 n ) u 2 ] sin ( 3 m θ ) } λ m + 2 n - { ( 1 + u 0 m ) sin ( 3 m θ ) - [ u 1 + ( m + 2 n ) u 2 ] cos ( 3 m θ ) } ( ω - m ω R 0 ) ) ,
g Φ Φ = α 2 R / Γ 8 I s V cos θ , λ m = m Γ + Γ 0 , Γ 0 = ( 1 + α 2 ) R 4 I s V , u 0 = - 4 α Γ N Ω cos ( 3 θ - Θ ) cos Θ , u 1 = - α 2 Ω T R / Ω I s V r , u 2 = u 0 tan ( 3 θ - Θ ) .
S E 0 ( ω ) = m = - + 2 Γ 0 I m ( g Φ Φ ) ( ω - m ω R 0 ) 2 + Γ 0 2 exp ( - g Φ Φ ) ,
C Φ Φ ( t ) = 1 2 π - + d ω [ 1 - cos ( ω t ) ] [ c Φ Φ ( ω ) + c Φ Φ ( - ω ) ] ,
c Φ Φ ( ω ) - c Φ Φ ( m ) ( ω ) = R 4 I s V { 1 σ 0 2 1 + α 2 ω 2 + δ 2 + p α 2 2 γ p z p / Ω 4 σ p cos θ p × [ γ p cos ( 3 θ p + μ p ) - ( ω - ω p ) sin ( 3 θ p + μ p ) ( ω - ω p ) 2 + γ p 2 - 2 z p / Ω η ( 1 + α 2 ) 1 / 2 Ω × γ p cos ( 2 θ p + μ p ) - ( ω - ω p ) sin ( 2 θ p + μ p ) ( ω - ω p ) 2 + γ p 2 × sin ( ω τ ) ] } ,
σ 0 = 1 + ( 1 + α 2 ) 1 / 2 η τ ,             μ p = arg ( σ p ) , c Φ Φ ( m ) ( ω ) = c Φ Φ ( m ) ( - ω ) .
C Φ Φ ( ω ) - c Φ Φ ( m ) ( ω ) R 4 I s V [ 1 σ 0 2 1 + α 2 ω 2 + δ 2 + p α 2 ( 1 - ζ p ) 2 γ p z p / Ω 4 σ p ( cos θ p ) ( cos ξ p ) × γ p ( cos ψ p Φ Φ ) - ( ω - ω p ) ( sin ψ p Φ Φ ) ( ω - ω p ) 2 + γ p 2 ] ,
ζ p = 2 ( 1 + α 2 ) 1 / 2 ω p Ω η Ω sin ( ω p τ ) , ξ p = arctan ( ζ p 1 - ζ p γ p ω p ) θ p , ψ p Φ Φ = 3 θ p + μ p + ξ p .
C I Φ ( t ) = 1 2 π - + d ω sin ( ω t ) [ c I Φ ( ω ) - c I Φ ( - ω ) ] ,
c I Φ ( ω ) - c I Φ ( m ) ( ω ) - α R 4 I s V { 2 Ω σ 0 2 [ r Ω T - q ( 1 + α 2 ) 1 / 2 η τ A I s Ω ] P ω + p 1 Ω T r + | z p Ω | 2 [ q A I s T + 4 sin 2 ( ω p τ / 2 ) ( 1 + α 2 ) 1 / 2 η T ] γ p z p / Ω 3 σ p ( cos θ p ) ( cos Θ p ) × γ p ( cos ψ p I Φ ) - ( ω - ω p ) ( sin ψ p I Φ ) ( ω - ω p ) 2 + γ p 2 } ,
c Φ Φ ( m ) ( ω ) = - c I Φ ( m ) ( - ω ) ,             ψ p I Φ = μ p + Θ p , Θ p arctan { r sin ( 2 θ p ) r + z p / Ω 2 [ q A I s T + 4 η T ( 1 + α 2 ) 1 / 2 sin 2 ( ω p τ / 2 ) ] } .
S E ( 2 ) ( ω ) = C 0 2 γ 0 ω 2 + γ 0 2 [ 1 + C Φ Φ ( ω ) + C I Φ ( ω ) + C Φ Φ ( ω ) C I Φ ( ω ) + 1 2 C Φ Φ ( ω ) C Φ Φ ( ω ) ] ,
C Φ Φ ( ω ) = c Φ Φ ( ω ) + c Φ Φ ( - ω ) - 2 γ 0 ω 2 + δ 2 ,             γ 0 = Γ 0 / σ 0 2 ,
C I Φ ( ω ) = c I Φ ( ω ) - c I Φ ( - ω ) .
f ( ω ) g ( ω ) = 1 2 π - + d ω f ( ω - ω ) g ( ω ) ,
C 0 = exp [ - 1 2 π - + d ω C Φ Φ ( ω ) ] .
S E ( 2 ) ( ω ) = S Φ Φ ( 2 ) ( ω ) + S I Φ ( 2 ) ( ω ) .
S Φ Φ ( 2 ) ( ω ) = s Φ Φ ( 2 ) ( ω ) + s Φ Φ ( 2 ) ( - ω ) , 1 C 0 s Φ Φ ( 2 ) ( ω ) γ 0 ω 2 + γ 0 2 + p , q a p Φ Φ a p Φ Φ 4 ( 2 γ + γ 0 ) cos ( ψ p Φ Φ - ψ q Φ Φ ) - ( ω - ω p + ω q ) sin ( ψ p Φ Φ - ψ q Φ Φ ) ( ω - ω p + ω q ) 2 + ( 2 γ + γ 0 ) 2 + p a p Φ Φ ( γ + γ 0 ) ( cos ψ p Φ Φ ) - ( ω - ω p ) ( sin ψ p Φ Φ ) ( ω - ω p ) 2 + ( γ + γ 0 ) 2 + p , q a p Φ Φ a q Φ Φ 4 ( 2 γ + γ 0 ) cos ( ψ p Φ Φ + ψ q Φ Φ ) - ( ω - ω p - ω q ) sin ( ψ p Φ Φ + ψ q Φ Φ ) ( ω - ω p - ω q ) 2 + ( 2 γ + γ 0 ) 2 + c Φ Φ ( m ) ( ω ) + p a p Φ Φ ( cos ψ p Φ Φ ) c Φ Φ ( m ) ( ω - ω p ) .
S I Φ ( 2 ) ( ω ) = s I Φ ( 2 ) ( ω ) - s I Φ ( 2 ) ( - ω ) , 1 C 0 s I Φ ( 2 ) ( ω ) a ^ I Φ ω ω 2 + γ 0 2 + p , q a p I Φ a q Φ Φ 2 ( 2 γ + γ 0 ) cos ( ψ p I Φ - ψ q Φ Φ ) - ( ω - ω p + ω q ) sin ( ψ p I Φ - ψ q Φ Φ ) ( ω - ω p + ω q ) 2 + ( 2 γ + γ 0 ) 2 + p a p I Φ ( γ + γ 0 ) ( cos ψ p I Φ ) - ( ω - ω p ) ( sin ψ p I Φ ) ( ω - ω p ) 2 + ( γ + γ 0 ) 2 + p , q a p I Φ a q Φ Φ 2 ( 2 γ + γ 0 ) cos ( ψ p I Φ + ψ q Φ Φ ) - ( ω - ω p - ω q ) sin ( ψ p I Φ + ψ q Φ Φ ) ( ω - ω p - ω q ) 2 + ( 2 γ + γ 0 ) 2 + c I Φ ( m ) ( ω ) + p [ a p Φ Φ ( cos ψ p Φ Φ ) c I Φ ( m ) ( ω - ω p ) + a p I Φ ( cos ψ p I Φ ) c Φ Φ ( m ) ( ω - ω p ) ] .
[ z + 2 Γ N G s - 2 Γ I 0 - A I s z + 2 Γ I + η ˜ ( z ) ( cos ρ ) 2 I s η ˜ ( z ) ( sin ρ ) - α 2 A - η ˜ ( z ) 2 I s ( sin ρ ) z + η ˜ ( z ) ( cos ρ ) ] [ δ N ( z ) δ I ( z ) ϕ ( z ) ] = [ F N ( z ) F I ( z ) F Φ ( z ) ] ,
η ˜ ( z ) = η [ 1 - exp ( - z τ ) ] ,             Γ N = 1 2 ( 1 T + A I s ) , Γ I = I s 2 G s ,
α = α 1 - I s ,             A = ( 1 - I s ) A ,             = 1 - I s .
D ( z ) = z 3 + 2 [ Γ + η ˜ ( z ) ( cos ρ ) ] z 2 + [ Ω 2 + 2 ( Γ + Γ N ) η ˜ ( z ) ( cos ρ ) + η ˜ 2 ( z ) ] z + 2 Γ N η ˜ 2 ( z ) + Ω 2 ( 1 + α ˜ 2 ) 1 / 2 η ˜ ( z ) cos ( ρ + arctan α ˜ ) ,
Ω 2 = A G s I s ( 1 + A T ) ,             α ˜ = α ( 1 - 4 Γ N Γ I Ω 2 ) .
G ( t ) = 1 τ p + A Δ N ( t ) - G s I ( t ) .
γ n = Γ + η { c [ 1 - cos ( ω n τ - θ n ) cos θ n exp ( γ n τ ) ] - a ^ [ 1 - cos ( ω n τ + θ n ) cos θ n exp ( γ n τ ) ] } ,
a ^ = a Ω / z n 2 , a = ½ ( 1 + α 2 ) 1 / 2 cos ( ρ + arctan α ) , c = cos ρ , θ n = arctan ( γ n / ω n ) .
D ( z n ) D ( - z n ) = 8 γ n z n 4 σ n ( cos θ n ) exp ( i 3 θ n ) ,
σ n = { 1 + η τ [ c - a ^ + i ( tan θ n ) ( c + a ^ ) ] exp ( γ n τ - i ω n τ ) + η z n [ i | z n ω n | 2 ( c + a ^ ) exp ( γ n τ ) sin ( ω n τ ) + 2 a ^ ( cos θ n ) - exp ( γ n τ ) cos ( ω n τ + θ n ) cos θ n ] } × ( 1 + η τ ( c - a ^ ) sinh ( γ n τ ) γ n τ exp ( i ω n τ ) + η z n { z n ω n [ c exp ( γ n τ ) + a ^ exp ( - γ n τ ) ] sin ( ω n τ ) - 2 a ^ [ 1 - cos ( ω n τ ) cosh ( γ n τ ) + ( tan θ n ) sin ( ω n τ ) sinh ( γ n τ ) ] } ) .
1 D ( i ω p ) D ( - i ω p ) 1 4 Ω 4 ( Γ - 2 b η ) 2 ,
σ p [ 1 + b η τ exp ( γ p τ ) ] [ 1 + b η τ sin ( γ p τ ) γ p τ ] ,
γ p Γ - 2 b η 1 + b η τ { [ exp ( γ p τ ) - 1 ] / ( γ p τ ) } .
1 D ( i ω p ) D ( - i ω p ) | res 1 4 Ω 4 σ p γ p 2 .
Ψ = D ( i ω p ) D ( - i ω p ) [ D ( i ω p ) D ( - i ω p ) ] res [ 1 + b η τ exp ( γ p τ ) - 1 γ p τ ] [ 1 + b η τ exp ( γ p τ ) ] [ 1 + b η τ sinh ( γ p τ ) γ p τ ] .
c Φ Φ ( ω ) R 4 I s V { 1 ω 2 + δ 2 D 0 ( i ω ) D 0 ( - i ω ) D ( i ω ) D ( - i ω ) + α 2 Ω 4 D ( i ω ) D ( - i ω ) × [ 1 - 2 η / Ω ( 1 + α 2 ) 1 / 2 ( 1 - ω 2 - Ω 2 α 2 Ω 2 ) ω Ω sin ( ω τ ) ] } .

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