Abstract

The macroscopic behavior of a semiconductor laser medium is described by use of modified rate equations. The model, valid on time scales greater than 10-13 s, explicitly treats carrier temperature as a dynamic variable and includes the nonlinear dependence of the gain function on carrier density and temperature. Gain suppression that is due to carrier heating is a natural consequence of the model and gives a qualitative explanation of subpicosecond gain dynamics experiments without introducing gain nonlinearity phenomenologically. We demonstrate the temperature behavior of the laser during transient dynamics near and well above threshold. By including carrier temperature as a dynamic variable we show that the laser response to an external perturbation exhibits a noticeable change in the damped oscillations of the photon density compared with that in models without temperature dynamics. Variation in the evolution of the gain function for different external pulse energies is also demonstrated.

© 1998 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. M. P. Kesler and E. P. Ippen, “Subpicosecond gain dynamics in GaAlAs laser diodes,” Appl. Phys. Lett. 51, 1765 (1987).
    [CrossRef]
  2. J. Shah, R. F. Leheny, and R. E. Nahory, “Hot-carrier effects in 1.3-μ In1−xGaxAsyP1−y light emitting diodes,” Appl. Phys. Lett. 39, 618 (1981).
    [CrossRef]
  3. T. L. Koch, L. C. Chiu, C. Harder, and A. Yariv, “Picosecond carrier dynamics and laser action in optically pumped buried heterostructure lasers,” Appl. Phys. Lett. 41, 6 (1982).
    [CrossRef]
  4. L. A. Rivlin, “Febrile reaction of electrons in semiconductor laser to an ultrashort light pulse,” Sov. J. Quantum Electron. 15, 453 (1985).
    [CrossRef]
  5. J. Shah and G. J. Iafrate, eds., Hot Carriers in Semiconductors, proceedings of the Fifth International Conference (Pergamon, London, 1988), and references therein.
  6. G. H. B. Thompson, Physics of Semiconductor Laser Devices (Wiley, New York, 1980).
  7. G. P. Agrawal and N. K. Dutta, Semiconductor Lasers (Van Nostrand Reinhold, New York, 1993).
  8. L. A. Coldren and S. W. Corzine, Diode Lasers and Photonic Integrated Circuits (Wiley, New York, 1995).
  9. V. M. Galitskii and V. F. Elesin, “Electron kinetics and stationary generation in semiconductor lasers,” Sov. Phys. JETP 37, 351 (1973).
  10. V. M. Galitskii and V. F. Elesin, “Kinetic theory of generation of a strong field in semiconductor lasers,” Sov. Phys. JETP 41, 104 (1975).
  11. I. A. Poluektov, “Theory of powerful light pulse propagation through the media under conditions of coherent resonant interaction,” Ph.D. dissertation (Lebedev Physics Institute, Moscow, 1981) (in Russian).
  12. C. M. Bowden and G. P. Agrawal, “Generalized Bloch–Maxwell formulation for semiconductor lasers,” Opt. Commun. 100, 147 (1993).
    [CrossRef]
  13. H. Haug and S. W. Koch, Quantum Theory of the Optical and Electronic Properties of Semiconductors (World Scientific, Singapore, 1993).
  14. F. Jahnke and S. W. Koch, “Many-body theory for semiconductor microcavity lasers,” Phys. Rev. A 52, 1712 (1995).
    [CrossRef] [PubMed]
  15. F. Jahnke and S. W. Koch, “Ultrafast intensity switching and nonthermal carrier effects in semiconductor microcavity lasers,” Appl. Phys. Lett. 67, 2278 (1995).
    [CrossRef]
  16. C. M. Bowden and G. P. Agrawal, “Maxwell–Bloch formulation for semiconductors: effects of coherent Coulomb exchange,” Phys. Rev. A 51, 4132 (1995).
    [CrossRef] [PubMed]
  17. O. N. Krokhin and Yu. M. Popov, “Slowing-down times of non-equilibrium current in semiconductors,” in Proceedings of International Conference on Semiconductor Physics (Czechoslovak Academy of Sciences, Prague, 1960), p. 126.
  18. L. A. Rivlin, “Nonthermal dynamics of amplification in two-component semiconductor laser,” Sov. J. Quantum Electron. 19, 1345 (1989).
    [CrossRef]
  19. M. Willatzen, A. Uskov, J. Mørk, H. Olesen, B. Tromborg, and A.-P. Jauho, “Nonlinear gain suppression in semiconductor lasers due to carrier heating,” IEEE Photon. Technol. Lett. 3, 606 (1991).
    [CrossRef]
  20. A. N. Oraevsky, M. M. Clark, and D. K. Bandy, “Many-temperature model of laser with dynamics,” Opt. Commun. 85, 360 (1991).
    [CrossRef]
  21. C. Z. Ning, R. A. Indik, and J. V. Moloney, “Self-consistent approach to thermal effects in vertical-cavity surface-emitting lasers,” J. Opt. Soc. Am. B 12, 1993 (1995).
    [CrossRef]
  22. A. V. Uskov, J. R. Karin, R. Nagarajan, and J. E. Bowers, “Dynamics of carrier heating and sweepout in waveguide saturable absorbers,” IEEE J. Sel. Top. Quantum Electron. 1, 552 (1995).
    [CrossRef]
  23. V. I. Tolstikhin and M. Willander, “Carrier heating effects in dynamic-single-frequency GaInAsP-InP laser diodes,” IEEE J. Quantum Electron. 31, 814 (1995).
    [CrossRef]
  24. C.-Y. Tsai, R. M. Spencer, Y.-H. Lo, and L. F. Eastman, “Nonlinear gain coefficients in semiconductor lasers: effects of carrier heating,” IEEE J. Quantum Electron. 32, 201 (1996).
    [CrossRef]
  25. K. Böer, Survey of Semiconductor Physics (Van Nostrand Reinhold, New York, 1992).
  26. V. Sa-yakanit, “Electron density of states in a Gaussian random potential: path-integral approach,” Phys. Rev. B 19, 2266 (1979).
    [CrossRef]
  27. W. Sritrakool, V. Sa-yakanit, and H. R. Glyde, “Band tails in disordered systems,” Phys. Rev. B 33, 1199 (1986).
    [CrossRef]
  28. L. A. Rivlin, A. T. Semenov, and S. D. Yakubovich, Dynamics and Spectra of Semiconductor Lasers (Radio I Svyaz, Moscow, 1983) (in Russian); L. A. Rivlin, “Dynamics of semiconductor lasers,” J. Sov. Laser Res. 7, 11 (1986).
  29. M. J. Adams, “Theoretical effects of exponential band tails on the properties of the injection laser,” Solid State Electron. 12, 661 (1969).
    [CrossRef]
  30. P. G. Eliseev, I. Ismailov, A. I. Krasil’nikov, and M. A. Man’ko, “Spectral characteristics of injection lasers,” Sov. Phys. Semicond. 1, 797 (1967).
  31. J. Mørk and A. Mecozzi, “Theory of the ultrafast optical response of active semiconductor waveguides,” J. Opt. Soc. Am. B 13, 1803 (1996).
    [CrossRef]
  32. B. N. Gomatam and A. P. DeFonzo, “Theory of hot carrier effects on non-linear gain in GaAs-GaAlAs lasers and amplifiers,” IEEE J. Quantum Electron. 26, 1689 (1990).
    [CrossRef]
  33. M. Willatzen, T. Takahashi, and Y. Arakawa, “Nonlinear gain effects due to carrier heating and spectral hole burning in strained-quantum well lasers,” IEEE Photon. Technol. Lett. 4, 682 (1992).
    [CrossRef]
  34. A. N. Oraevsky, T. Sarkisyan, and D. K. Bandy, “Dynamics of the temperature of a recombining ensemble of fermions,” JETP Lett. 62, 674 (1995).
  35. D. Bimberg and J. Mycielski, “Recombination-induced heating of free carriers in a semiconductor,” Phys. Rev. B 31, 5490 (1985).
    [CrossRef]
  36. K. L. Hall, J. Mark, E. P. Ippen, and G. Eisenstein, “Femtosecond gain dynamics in InGaAsP optical amplifiers,” Appl. Phys. Lett. 56, 1740 (1990).
    [CrossRef]
  37. J. Mark and J. Mørk, “Subpicosecond gain dynamics in InGaAsP optical amplifiers: experiment and theory,” Appl. Phys. Lett. 61, 2281 (1992).
    [CrossRef]
  38. R. A. Indik, R. Binder, M. Mlejnek, J. V. Moloney, S. Hughes, A. Knorr, and S. W. Koch, “Role of plasma cooling, heating, and memory effects in subpicosecond pulse propagation in semiconductor amplifiers,” Phys. Rev. A 53, 3614 (1996).
    [CrossRef] [PubMed]
  39. M. Asada, A. Kameyama, and Y. Suematsu, “Gain and intervalence band absorption in quantum-well lasers,” IEEE J. Quantum Electron. QE-20, 745 (1984).
    [CrossRef]
  40. G. E. Pikus, Fundamentals of the Theory of Semiconductor Devices (Nauka, Moscow, 1965).
  41. P. T. Landsberg, Recombination in Semiconductors (Cambridge U. Press, Cambridge, 1991).
  42. B. R. Nag, Theory of Electrical Transport in Semiconductors (Pergamon, London, 1972).
  43. R. Kubo, Statistical Mechanics (North-Holland, Amsterdam, 1971), Chap. 4.

1996 (3)

C.-Y. Tsai, R. M. Spencer, Y.-H. Lo, and L. F. Eastman, “Nonlinear gain coefficients in semiconductor lasers: effects of carrier heating,” IEEE J. Quantum Electron. 32, 201 (1996).
[CrossRef]

R. A. Indik, R. Binder, M. Mlejnek, J. V. Moloney, S. Hughes, A. Knorr, and S. W. Koch, “Role of plasma cooling, heating, and memory effects in subpicosecond pulse propagation in semiconductor amplifiers,” Phys. Rev. A 53, 3614 (1996).
[CrossRef] [PubMed]

J. Mørk and A. Mecozzi, “Theory of the ultrafast optical response of active semiconductor waveguides,” J. Opt. Soc. Am. B 13, 1803 (1996).
[CrossRef]

1995 (7)

C. Z. Ning, R. A. Indik, and J. V. Moloney, “Self-consistent approach to thermal effects in vertical-cavity surface-emitting lasers,” J. Opt. Soc. Am. B 12, 1993 (1995).
[CrossRef]

A. N. Oraevsky, T. Sarkisyan, and D. K. Bandy, “Dynamics of the temperature of a recombining ensemble of fermions,” JETP Lett. 62, 674 (1995).

A. V. Uskov, J. R. Karin, R. Nagarajan, and J. E. Bowers, “Dynamics of carrier heating and sweepout in waveguide saturable absorbers,” IEEE J. Sel. Top. Quantum Electron. 1, 552 (1995).
[CrossRef]

V. I. Tolstikhin and M. Willander, “Carrier heating effects in dynamic-single-frequency GaInAsP-InP laser diodes,” IEEE J. Quantum Electron. 31, 814 (1995).
[CrossRef]

F. Jahnke and S. W. Koch, “Many-body theory for semiconductor microcavity lasers,” Phys. Rev. A 52, 1712 (1995).
[CrossRef] [PubMed]

F. Jahnke and S. W. Koch, “Ultrafast intensity switching and nonthermal carrier effects in semiconductor microcavity lasers,” Appl. Phys. Lett. 67, 2278 (1995).
[CrossRef]

C. M. Bowden and G. P. Agrawal, “Maxwell–Bloch formulation for semiconductors: effects of coherent Coulomb exchange,” Phys. Rev. A 51, 4132 (1995).
[CrossRef] [PubMed]

1993 (1)

C. M. Bowden and G. P. Agrawal, “Generalized Bloch–Maxwell formulation for semiconductor lasers,” Opt. Commun. 100, 147 (1993).
[CrossRef]

1992 (2)

M. Willatzen, T. Takahashi, and Y. Arakawa, “Nonlinear gain effects due to carrier heating and spectral hole burning in strained-quantum well lasers,” IEEE Photon. Technol. Lett. 4, 682 (1992).
[CrossRef]

J. Mark and J. Mørk, “Subpicosecond gain dynamics in InGaAsP optical amplifiers: experiment and theory,” Appl. Phys. Lett. 61, 2281 (1992).
[CrossRef]

1991 (2)

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

A. N. Oraevsky, M. M. Clark, and D. K. Bandy, “Many-temperature model of laser with dynamics,” Opt. Commun. 85, 360 (1991).
[CrossRef]

1990 (2)

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

B. N. Gomatam and A. P. DeFonzo, “Theory of hot carrier effects on non-linear gain in GaAs-GaAlAs lasers and amplifiers,” IEEE J. Quantum Electron. 26, 1689 (1990).
[CrossRef]

1989 (1)

L. A. Rivlin, “Nonthermal dynamics of amplification in two-component semiconductor laser,” Sov. J. Quantum Electron. 19, 1345 (1989).
[CrossRef]

1987 (1)

M. P. Kesler and E. P. Ippen, “Subpicosecond gain dynamics in GaAlAs laser diodes,” Appl. Phys. Lett. 51, 1765 (1987).
[CrossRef]

1986 (1)

W. Sritrakool, V. Sa-yakanit, and H. R. Glyde, “Band tails in disordered systems,” Phys. Rev. B 33, 1199 (1986).
[CrossRef]

1985 (2)

D. Bimberg and J. Mycielski, “Recombination-induced heating of free carriers in a semiconductor,” Phys. Rev. B 31, 5490 (1985).
[CrossRef]

L. A. Rivlin, “Febrile reaction of electrons in semiconductor laser to an ultrashort light pulse,” Sov. J. Quantum Electron. 15, 453 (1985).
[CrossRef]

1984 (1)

M. Asada, A. Kameyama, and Y. Suematsu, “Gain and intervalence band absorption in quantum-well lasers,” IEEE J. Quantum Electron. QE-20, 745 (1984).
[CrossRef]

1982 (1)

T. L. Koch, L. C. Chiu, C. Harder, and A. Yariv, “Picosecond carrier dynamics and laser action in optically pumped buried heterostructure lasers,” Appl. Phys. Lett. 41, 6 (1982).
[CrossRef]

1981 (1)

J. Shah, R. F. Leheny, and R. E. Nahory, “Hot-carrier effects in 1.3-μ In1−xGaxAsyP1−y light emitting diodes,” Appl. Phys. Lett. 39, 618 (1981).
[CrossRef]

1979 (1)

V. Sa-yakanit, “Electron density of states in a Gaussian random potential: path-integral approach,” Phys. Rev. B 19, 2266 (1979).
[CrossRef]

1975 (1)

V. M. Galitskii and V. F. Elesin, “Kinetic theory of generation of a strong field in semiconductor lasers,” Sov. Phys. JETP 41, 104 (1975).

1973 (1)

V. M. Galitskii and V. F. Elesin, “Electron kinetics and stationary generation in semiconductor lasers,” Sov. Phys. JETP 37, 351 (1973).

1969 (1)

M. J. Adams, “Theoretical effects of exponential band tails on the properties of the injection laser,” Solid State Electron. 12, 661 (1969).
[CrossRef]

1967 (1)

P. G. Eliseev, I. Ismailov, A. I. Krasil’nikov, and M. A. Man’ko, “Spectral characteristics of injection lasers,” Sov. Phys. Semicond. 1, 797 (1967).

Adams, M. J.

M. J. Adams, “Theoretical effects of exponential band tails on the properties of the injection laser,” Solid State Electron. 12, 661 (1969).
[CrossRef]

Agrawal, G. P.

C. M. Bowden and G. P. Agrawal, “Maxwell–Bloch formulation for semiconductors: effects of coherent Coulomb exchange,” Phys. Rev. A 51, 4132 (1995).
[CrossRef] [PubMed]

C. M. Bowden and G. P. Agrawal, “Generalized Bloch–Maxwell formulation for semiconductor lasers,” Opt. Commun. 100, 147 (1993).
[CrossRef]

Arakawa, Y.

M. Willatzen, T. Takahashi, and Y. Arakawa, “Nonlinear gain effects due to carrier heating and spectral hole burning in strained-quantum well lasers,” IEEE Photon. Technol. Lett. 4, 682 (1992).
[CrossRef]

Asada, M.

M. Asada, A. Kameyama, and Y. Suematsu, “Gain and intervalence band absorption in quantum-well lasers,” IEEE J. Quantum Electron. QE-20, 745 (1984).
[CrossRef]

Bandy, D. K.

A. N. Oraevsky, T. Sarkisyan, and D. K. Bandy, “Dynamics of the temperature of a recombining ensemble of fermions,” JETP Lett. 62, 674 (1995).

A. N. Oraevsky, M. M. Clark, and D. K. Bandy, “Many-temperature model of laser with dynamics,” Opt. Commun. 85, 360 (1991).
[CrossRef]

Bimberg, D.

D. Bimberg and J. Mycielski, “Recombination-induced heating of free carriers in a semiconductor,” Phys. Rev. B 31, 5490 (1985).
[CrossRef]

Binder, R.

R. A. Indik, R. Binder, M. Mlejnek, J. V. Moloney, S. Hughes, A. Knorr, and S. W. Koch, “Role of plasma cooling, heating, and memory effects in subpicosecond pulse propagation in semiconductor amplifiers,” Phys. Rev. A 53, 3614 (1996).
[CrossRef] [PubMed]

Bowden, C. M.

C. M. Bowden and G. P. Agrawal, “Maxwell–Bloch formulation for semiconductors: effects of coherent Coulomb exchange,” Phys. Rev. A 51, 4132 (1995).
[CrossRef] [PubMed]

C. M. Bowden and G. P. Agrawal, “Generalized Bloch–Maxwell formulation for semiconductor lasers,” Opt. Commun. 100, 147 (1993).
[CrossRef]

Bowers, J. E.

A. V. Uskov, J. R. Karin, R. Nagarajan, and J. E. Bowers, “Dynamics of carrier heating and sweepout in waveguide saturable absorbers,” IEEE J. Sel. Top. Quantum Electron. 1, 552 (1995).
[CrossRef]

Chiu, L. C.

T. L. Koch, L. C. Chiu, C. Harder, and A. Yariv, “Picosecond carrier dynamics and laser action in optically pumped buried heterostructure lasers,” Appl. Phys. Lett. 41, 6 (1982).
[CrossRef]

Clark, M. M.

A. N. Oraevsky, M. M. Clark, and D. K. Bandy, “Many-temperature model of laser with dynamics,” Opt. Commun. 85, 360 (1991).
[CrossRef]

DeFonzo, A. P.

B. N. Gomatam and A. P. DeFonzo, “Theory of hot carrier effects on non-linear gain in GaAs-GaAlAs lasers and amplifiers,” IEEE J. Quantum Electron. 26, 1689 (1990).
[CrossRef]

Eastman, L. F.

C.-Y. Tsai, R. M. Spencer, Y.-H. Lo, and L. F. Eastman, “Nonlinear gain coefficients in semiconductor lasers: effects of carrier heating,” IEEE J. Quantum Electron. 32, 201 (1996).
[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 (1990).
[CrossRef]

Elesin, V. F.

V. M. Galitskii and V. F. Elesin, “Kinetic theory of generation of a strong field in semiconductor lasers,” Sov. Phys. JETP 41, 104 (1975).

V. M. Galitskii and V. F. Elesin, “Electron kinetics and stationary generation in semiconductor lasers,” Sov. Phys. JETP 37, 351 (1973).

Eliseev, P. G.

P. G. Eliseev, I. Ismailov, A. I. Krasil’nikov, and M. A. Man’ko, “Spectral characteristics of injection lasers,” Sov. Phys. Semicond. 1, 797 (1967).

Galitskii, V. M.

V. M. Galitskii and V. F. Elesin, “Kinetic theory of generation of a strong field in semiconductor lasers,” Sov. Phys. JETP 41, 104 (1975).

V. M. Galitskii and V. F. Elesin, “Electron kinetics and stationary generation in semiconductor lasers,” Sov. Phys. JETP 37, 351 (1973).

Glyde, H. R.

W. Sritrakool, V. Sa-yakanit, and H. R. Glyde, “Band tails in disordered systems,” Phys. Rev. B 33, 1199 (1986).
[CrossRef]

Gomatam, B. N.

B. N. Gomatam and A. P. DeFonzo, “Theory of hot carrier effects on non-linear gain in GaAs-GaAlAs lasers and amplifiers,” IEEE J. Quantum Electron. 26, 1689 (1990).
[CrossRef]

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 (1990).
[CrossRef]

Harder, C.

T. L. Koch, L. C. Chiu, C. Harder, and A. Yariv, “Picosecond carrier dynamics and laser action in optically pumped buried heterostructure lasers,” Appl. Phys. Lett. 41, 6 (1982).
[CrossRef]

Hughes, S.

R. A. Indik, R. Binder, M. Mlejnek, J. V. Moloney, S. Hughes, A. Knorr, and S. W. Koch, “Role of plasma cooling, heating, and memory effects in subpicosecond pulse propagation in semiconductor amplifiers,” Phys. Rev. A 53, 3614 (1996).
[CrossRef] [PubMed]

Indik, R. A.

R. A. Indik, R. Binder, M. Mlejnek, J. V. Moloney, S. Hughes, A. Knorr, and S. W. Koch, “Role of plasma cooling, heating, and memory effects in subpicosecond pulse propagation in semiconductor amplifiers,” Phys. Rev. A 53, 3614 (1996).
[CrossRef] [PubMed]

C. Z. Ning, R. A. Indik, and J. V. Moloney, “Self-consistent approach to thermal effects in vertical-cavity surface-emitting lasers,” J. Opt. Soc. Am. B 12, 1993 (1995).
[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 (1990).
[CrossRef]

M. P. Kesler and E. P. Ippen, “Subpicosecond gain dynamics in GaAlAs laser diodes,” Appl. Phys. Lett. 51, 1765 (1987).
[CrossRef]

Ismailov, I.

P. G. Eliseev, I. Ismailov, A. I. Krasil’nikov, and M. A. Man’ko, “Spectral characteristics of injection lasers,” Sov. Phys. Semicond. 1, 797 (1967).

Jahnke, F.

F. Jahnke and S. W. Koch, “Ultrafast intensity switching and nonthermal carrier effects in semiconductor microcavity lasers,” Appl. Phys. Lett. 67, 2278 (1995).
[CrossRef]

F. Jahnke and S. W. Koch, “Many-body theory for semiconductor microcavity lasers,” Phys. Rev. A 52, 1712 (1995).
[CrossRef] [PubMed]

Jauho, A.-P.

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

Kameyama, A.

M. Asada, A. Kameyama, and Y. Suematsu, “Gain and intervalence band absorption in quantum-well lasers,” IEEE J. Quantum Electron. QE-20, 745 (1984).
[CrossRef]

Karin, J. R.

A. V. Uskov, J. R. Karin, R. Nagarajan, and J. E. Bowers, “Dynamics of carrier heating and sweepout in waveguide saturable absorbers,” IEEE J. Sel. Top. Quantum Electron. 1, 552 (1995).
[CrossRef]

Kesler, M. P.

M. P. Kesler and E. P. Ippen, “Subpicosecond gain dynamics in GaAlAs laser diodes,” Appl. Phys. Lett. 51, 1765 (1987).
[CrossRef]

Knorr, A.

R. A. Indik, R. Binder, M. Mlejnek, J. V. Moloney, S. Hughes, A. Knorr, and S. W. Koch, “Role of plasma cooling, heating, and memory effects in subpicosecond pulse propagation in semiconductor amplifiers,” Phys. Rev. A 53, 3614 (1996).
[CrossRef] [PubMed]

Koch, S. W.

R. A. Indik, R. Binder, M. Mlejnek, J. V. Moloney, S. Hughes, A. Knorr, and S. W. Koch, “Role of plasma cooling, heating, and memory effects in subpicosecond pulse propagation in semiconductor amplifiers,” Phys. Rev. A 53, 3614 (1996).
[CrossRef] [PubMed]

F. Jahnke and S. W. Koch, “Ultrafast intensity switching and nonthermal carrier effects in semiconductor microcavity lasers,” Appl. Phys. Lett. 67, 2278 (1995).
[CrossRef]

F. Jahnke and S. W. Koch, “Many-body theory for semiconductor microcavity lasers,” Phys. Rev. A 52, 1712 (1995).
[CrossRef] [PubMed]

Koch, T. L.

T. L. Koch, L. C. Chiu, C. Harder, and A. Yariv, “Picosecond carrier dynamics and laser action in optically pumped buried heterostructure lasers,” Appl. Phys. Lett. 41, 6 (1982).
[CrossRef]

Krasil’nikov, A. I.

P. G. Eliseev, I. Ismailov, A. I. Krasil’nikov, and M. A. Man’ko, “Spectral characteristics of injection lasers,” Sov. Phys. Semicond. 1, 797 (1967).

Leheny, R. F.

J. Shah, R. F. Leheny, and R. E. Nahory, “Hot-carrier effects in 1.3-μ In1−xGaxAsyP1−y light emitting diodes,” Appl. Phys. Lett. 39, 618 (1981).
[CrossRef]

Lo, Y.-H.

C.-Y. Tsai, R. M. Spencer, Y.-H. Lo, and L. F. Eastman, “Nonlinear gain coefficients in semiconductor lasers: effects of carrier heating,” IEEE J. Quantum Electron. 32, 201 (1996).
[CrossRef]

Man’ko, M. A.

P. G. Eliseev, I. Ismailov, A. I. Krasil’nikov, and M. A. Man’ko, “Spectral characteristics of injection lasers,” Sov. Phys. Semicond. 1, 797 (1967).

Mark, J.

J. Mark and J. Mørk, “Subpicosecond gain dynamics in InGaAsP optical amplifiers: experiment and theory,” Appl. Phys. Lett. 61, 2281 (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 (1990).
[CrossRef]

Mecozzi, A.

Mlejnek, M.

R. A. Indik, R. Binder, M. Mlejnek, J. V. Moloney, S. Hughes, A. Knorr, and S. W. Koch, “Role of plasma cooling, heating, and memory effects in subpicosecond pulse propagation in semiconductor amplifiers,” Phys. Rev. A 53, 3614 (1996).
[CrossRef] [PubMed]

Moloney, J. V.

R. A. Indik, R. Binder, M. Mlejnek, J. V. Moloney, S. Hughes, A. Knorr, and S. W. Koch, “Role of plasma cooling, heating, and memory effects in subpicosecond pulse propagation in semiconductor amplifiers,” Phys. Rev. A 53, 3614 (1996).
[CrossRef] [PubMed]

C. Z. Ning, R. A. Indik, and J. V. Moloney, “Self-consistent approach to thermal effects in vertical-cavity surface-emitting lasers,” J. Opt. Soc. Am. B 12, 1993 (1995).
[CrossRef]

Mørk, J.

J. Mørk and A. Mecozzi, “Theory of the ultrafast optical response of active semiconductor waveguides,” J. Opt. Soc. Am. B 13, 1803 (1996).
[CrossRef]

J. Mark and J. Mørk, “Subpicosecond gain dynamics in InGaAsP optical amplifiers: experiment and theory,” Appl. Phys. Lett. 61, 2281 (1992).
[CrossRef]

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

Mycielski, J.

D. Bimberg and J. Mycielski, “Recombination-induced heating of free carriers in a semiconductor,” Phys. Rev. B 31, 5490 (1985).
[CrossRef]

Nagarajan, R.

A. V. Uskov, J. R. Karin, R. Nagarajan, and J. E. Bowers, “Dynamics of carrier heating and sweepout in waveguide saturable absorbers,” IEEE J. Sel. Top. Quantum Electron. 1, 552 (1995).
[CrossRef]

Nahory, R. E.

J. Shah, R. F. Leheny, and R. E. Nahory, “Hot-carrier effects in 1.3-μ In1−xGaxAsyP1−y light emitting diodes,” Appl. Phys. Lett. 39, 618 (1981).
[CrossRef]

Ning, C. Z.

Olesen, H.

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

Oraevsky, A. N.

A. N. Oraevsky, T. Sarkisyan, and D. K. Bandy, “Dynamics of the temperature of a recombining ensemble of fermions,” JETP Lett. 62, 674 (1995).

A. N. Oraevsky, M. M. Clark, and D. K. Bandy, “Many-temperature model of laser with dynamics,” Opt. Commun. 85, 360 (1991).
[CrossRef]

Rivlin, L. A.

L. A. Rivlin, “Nonthermal dynamics of amplification in two-component semiconductor laser,” Sov. J. Quantum Electron. 19, 1345 (1989).
[CrossRef]

L. A. Rivlin, “Febrile reaction of electrons in semiconductor laser to an ultrashort light pulse,” Sov. J. Quantum Electron. 15, 453 (1985).
[CrossRef]

Sarkisyan, T.

A. N. Oraevsky, T. Sarkisyan, and D. K. Bandy, “Dynamics of the temperature of a recombining ensemble of fermions,” JETP Lett. 62, 674 (1995).

Sa-yakanit, V.

W. Sritrakool, V. Sa-yakanit, and H. R. Glyde, “Band tails in disordered systems,” Phys. Rev. B 33, 1199 (1986).
[CrossRef]

V. Sa-yakanit, “Electron density of states in a Gaussian random potential: path-integral approach,” Phys. Rev. B 19, 2266 (1979).
[CrossRef]

Shah, J.

J. Shah, R. F. Leheny, and R. E. Nahory, “Hot-carrier effects in 1.3-μ In1−xGaxAsyP1−y light emitting diodes,” Appl. Phys. Lett. 39, 618 (1981).
[CrossRef]

Spencer, R. M.

C.-Y. Tsai, R. M. Spencer, Y.-H. Lo, and L. F. Eastman, “Nonlinear gain coefficients in semiconductor lasers: effects of carrier heating,” IEEE J. Quantum Electron. 32, 201 (1996).
[CrossRef]

Sritrakool, W.

W. Sritrakool, V. Sa-yakanit, and H. R. Glyde, “Band tails in disordered systems,” Phys. Rev. B 33, 1199 (1986).
[CrossRef]

Suematsu, Y.

M. Asada, A. Kameyama, and Y. Suematsu, “Gain and intervalence band absorption in quantum-well lasers,” IEEE J. Quantum Electron. QE-20, 745 (1984).
[CrossRef]

Takahashi, T.

M. Willatzen, T. Takahashi, and Y. Arakawa, “Nonlinear gain effects due to carrier heating and spectral hole burning in strained-quantum well lasers,” IEEE Photon. Technol. Lett. 4, 682 (1992).
[CrossRef]

Tolstikhin, V. I.

V. I. Tolstikhin and M. Willander, “Carrier heating effects in dynamic-single-frequency GaInAsP-InP laser diodes,” IEEE J. Quantum Electron. 31, 814 (1995).
[CrossRef]

Tromborg, B.

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

Tsai, C.-Y.

C.-Y. Tsai, R. M. Spencer, Y.-H. Lo, and L. F. Eastman, “Nonlinear gain coefficients in semiconductor lasers: effects of carrier heating,” IEEE J. Quantum Electron. 32, 201 (1996).
[CrossRef]

Uskov, A.

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

Uskov, A. V.

A. V. Uskov, J. R. Karin, R. Nagarajan, and J. E. Bowers, “Dynamics of carrier heating and sweepout in waveguide saturable absorbers,” IEEE J. Sel. Top. Quantum Electron. 1, 552 (1995).
[CrossRef]

Willander, M.

V. I. Tolstikhin and M. Willander, “Carrier heating effects in dynamic-single-frequency GaInAsP-InP laser diodes,” IEEE J. Quantum Electron. 31, 814 (1995).
[CrossRef]

Willatzen, M.

M. Willatzen, T. Takahashi, and Y. Arakawa, “Nonlinear gain effects due to carrier heating and spectral hole burning in strained-quantum well lasers,” IEEE Photon. Technol. Lett. 4, 682 (1992).
[CrossRef]

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

Yariv, A.

T. L. Koch, L. C. Chiu, C. Harder, and A. Yariv, “Picosecond carrier dynamics and laser action in optically pumped buried heterostructure lasers,” Appl. Phys. Lett. 41, 6 (1982).
[CrossRef]

Appl. Phys. Lett. (6)

M. P. Kesler and E. P. Ippen, “Subpicosecond gain dynamics in GaAlAs laser diodes,” Appl. Phys. Lett. 51, 1765 (1987).
[CrossRef]

J. Shah, R. F. Leheny, and R. E. Nahory, “Hot-carrier effects in 1.3-μ In1−xGaxAsyP1−y light emitting diodes,” Appl. Phys. Lett. 39, 618 (1981).
[CrossRef]

T. L. Koch, L. C. Chiu, C. Harder, and A. Yariv, “Picosecond carrier dynamics and laser action in optically pumped buried heterostructure lasers,” Appl. Phys. Lett. 41, 6 (1982).
[CrossRef]

F. Jahnke and S. W. Koch, “Ultrafast intensity switching and nonthermal carrier effects in semiconductor microcavity lasers,” Appl. Phys. Lett. 67, 2278 (1995).
[CrossRef]

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

J. Mark and J. Mørk, “Subpicosecond gain dynamics in InGaAsP optical amplifiers: experiment and theory,” Appl. Phys. Lett. 61, 2281 (1992).
[CrossRef]

IEEE J. Quantum Electron. (4)

V. I. Tolstikhin and M. Willander, “Carrier heating effects in dynamic-single-frequency GaInAsP-InP laser diodes,” IEEE J. Quantum Electron. 31, 814 (1995).
[CrossRef]

C.-Y. Tsai, R. M. Spencer, Y.-H. Lo, and L. F. Eastman, “Nonlinear gain coefficients in semiconductor lasers: effects of carrier heating,” IEEE J. Quantum Electron. 32, 201 (1996).
[CrossRef]

B. N. Gomatam and A. P. DeFonzo, “Theory of hot carrier effects on non-linear gain in GaAs-GaAlAs lasers and amplifiers,” IEEE J. Quantum Electron. 26, 1689 (1990).
[CrossRef]

M. Asada, A. Kameyama, and Y. Suematsu, “Gain and intervalence band absorption in quantum-well lasers,” IEEE J. Quantum Electron. QE-20, 745 (1984).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (1)

A. V. Uskov, J. R. Karin, R. Nagarajan, and J. E. Bowers, “Dynamics of carrier heating and sweepout in waveguide saturable absorbers,” IEEE J. Sel. Top. Quantum Electron. 1, 552 (1995).
[CrossRef]

IEEE Photon. Technol. Lett. (2)

M. Willatzen, T. Takahashi, and Y. Arakawa, “Nonlinear gain effects due to carrier heating and spectral hole burning in strained-quantum well lasers,” IEEE Photon. Technol. Lett. 4, 682 (1992).
[CrossRef]

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

J. Opt. Soc. Am. B (2)

JETP Lett. (1)

A. N. Oraevsky, T. Sarkisyan, and D. K. Bandy, “Dynamics of the temperature of a recombining ensemble of fermions,” JETP Lett. 62, 674 (1995).

Opt. Commun. (2)

A. N. Oraevsky, M. M. Clark, and D. K. Bandy, “Many-temperature model of laser with dynamics,” Opt. Commun. 85, 360 (1991).
[CrossRef]

C. M. Bowden and G. P. Agrawal, “Generalized Bloch–Maxwell formulation for semiconductor lasers,” Opt. Commun. 100, 147 (1993).
[CrossRef]

Phys. Rev. A (3)

C. M. Bowden and G. P. Agrawal, “Maxwell–Bloch formulation for semiconductors: effects of coherent Coulomb exchange,” Phys. Rev. A 51, 4132 (1995).
[CrossRef] [PubMed]

F. Jahnke and S. W. Koch, “Many-body theory for semiconductor microcavity lasers,” Phys. Rev. A 52, 1712 (1995).
[CrossRef] [PubMed]

R. A. Indik, R. Binder, M. Mlejnek, J. V. Moloney, S. Hughes, A. Knorr, and S. W. Koch, “Role of plasma cooling, heating, and memory effects in subpicosecond pulse propagation in semiconductor amplifiers,” Phys. Rev. A 53, 3614 (1996).
[CrossRef] [PubMed]

Phys. Rev. B (3)

V. Sa-yakanit, “Electron density of states in a Gaussian random potential: path-integral approach,” Phys. Rev. B 19, 2266 (1979).
[CrossRef]

W. Sritrakool, V. Sa-yakanit, and H. R. Glyde, “Band tails in disordered systems,” Phys. Rev. B 33, 1199 (1986).
[CrossRef]

D. Bimberg and J. Mycielski, “Recombination-induced heating of free carriers in a semiconductor,” Phys. Rev. B 31, 5490 (1985).
[CrossRef]

Solid State Electron. (1)

M. J. Adams, “Theoretical effects of exponential band tails on the properties of the injection laser,” Solid State Electron. 12, 661 (1969).
[CrossRef]

Sov. J. Quantum Electron. (2)

L. A. Rivlin, “Nonthermal dynamics of amplification in two-component semiconductor laser,” Sov. J. Quantum Electron. 19, 1345 (1989).
[CrossRef]

L. A. Rivlin, “Febrile reaction of electrons in semiconductor laser to an ultrashort light pulse,” Sov. J. Quantum Electron. 15, 453 (1985).
[CrossRef]

Sov. Phys. JETP (2)

V. M. Galitskii and V. F. Elesin, “Electron kinetics and stationary generation in semiconductor lasers,” Sov. Phys. JETP 37, 351 (1973).

V. M. Galitskii and V. F. Elesin, “Kinetic theory of generation of a strong field in semiconductor lasers,” Sov. Phys. JETP 41, 104 (1975).

Sov. Phys. Semicond. (1)

P. G. Eliseev, I. Ismailov, A. I. Krasil’nikov, and M. A. Man’ko, “Spectral characteristics of injection lasers,” Sov. Phys. Semicond. 1, 797 (1967).

Other (13)

L. A. Rivlin, A. T. Semenov, and S. D. Yakubovich, Dynamics and Spectra of Semiconductor Lasers (Radio I Svyaz, Moscow, 1983) (in Russian); L. A. Rivlin, “Dynamics of semiconductor lasers,” J. Sov. Laser Res. 7, 11 (1986).

K. Böer, Survey of Semiconductor Physics (Van Nostrand Reinhold, New York, 1992).

G. E. Pikus, Fundamentals of the Theory of Semiconductor Devices (Nauka, Moscow, 1965).

P. T. Landsberg, Recombination in Semiconductors (Cambridge U. Press, Cambridge, 1991).

B. R. Nag, Theory of Electrical Transport in Semiconductors (Pergamon, London, 1972).

R. Kubo, Statistical Mechanics (North-Holland, Amsterdam, 1971), Chap. 4.

I. A. Poluektov, “Theory of powerful light pulse propagation through the media under conditions of coherent resonant interaction,” Ph.D. dissertation (Lebedev Physics Institute, Moscow, 1981) (in Russian).

H. Haug and S. W. Koch, Quantum Theory of the Optical and Electronic Properties of Semiconductors (World Scientific, Singapore, 1993).

O. N. Krokhin and Yu. M. Popov, “Slowing-down times of non-equilibrium current in semiconductors,” in Proceedings of International Conference on Semiconductor Physics (Czechoslovak Academy of Sciences, Prague, 1960), p. 126.

J. Shah and G. J. Iafrate, eds., Hot Carriers in Semiconductors, proceedings of the Fifth International Conference (Pergamon, London, 1988), and references therein.

G. H. B. Thompson, Physics of Semiconductor Laser Devices (Wiley, New York, 1980).

G. P. Agrawal and N. K. Dutta, Semiconductor Lasers (Van Nostrand Reinhold, New York, 1993).

L. A. Coldren and S. W. Corzine, Diode Lasers and Photonic Integrated Circuits (Wiley, New York, 1995).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (6)

Fig. 1
Fig. 1

Exact and approximate Fermi functions at 70 and 300 K.

Fig. 2
Fig. 2

Laser transition to cw behavior for pumping at twice the threshold rate: (a) photon density, (b) carrier density, (c) carrier temperature (the dotted line represents the lattice temperature).

Fig. 3
Fig. 3

Carrier temperature behavior (the dotted line represents the lattice temperature) for pumping: (a) slightly above threshold, (b) well above threshold.

Fig. 4
Fig. 4

Laser response (solid curves, with temperature dynamics; dotted curves, without temperature dynamics) to an external 0.1-pJ Gaussian pulse peaked at t=0: (a) 100-ps pulse width, (b) 10-ps pulse width.

Fig. 5
Fig. 5

Subpicosecond gain dynamics; gain saturation owing to carrier heating and recovery in the cases of (a) amplification, (b) transparency, (c) absorption. The 0.1-pJ external pulse is a Gaussian of width 0.4 ps peaked at t=0.

Fig. 6
Fig. 6

Subpicosecond gain dynamics in the case of strong absorption for four external pulse energies. In each case the external signal is a 0.4-ps Gaussian pulse peaked at t=0.

Equations (79)

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

dEdt=-12τc (1+iΔ)E+1/2(1+iα)gE,
dndt=J-dndtsp-gW,
n(μ, T)=ρ(ε)f(ε, μ, T)dε,
f(ε, μ, T)=1+expε-μkBT-1
U(μ, T)=ερ(ε)f(ε, μ, T)dε.
ddt U(μ, T)=-dUdtLat-dUdtsp+dUdtpump-ωgW+ωσfnW.
dUdtLat=1τL {U[μ(T), T]-U[μ(T0), T0]},
dUdtsp= 1τsε ερ(ε)f(ε, μ, T)dε,
ρ(ε)=ρ0 exp(ε/εd),
f(ε, μ, T)=1-12 expε-μkBTεμ12 expμ-εkBTε>μ.
n(μ, θ)=ρ0εd1-θ2 expμεd,
U(μ, θ)=n1-θ2 [μ(1-θ2)+εd(3θ2-1)],
n(μ, θ)=ρ0εd(1+θ2)exp(μ/εd),
U(μ, θ)=n(μ-εd+2εdθ2).
dUdtpumpQ.
Q=J(μ0-εd+2εdθ02),
Jτsp=ρ0εd(1+θ02)exp(μ0/εd).
g(n, T)=Γ(n-nt-nβΔT/T0).
g(T, μ, ω)=G(ω)tanhμ-ω2kBT.
G(ω)=c3τspω2η3 2meff32 ω-εg
G(ω)=π2c3τspω2η3 ξεd expω-εgεd
dEdt=-12τc (1+iΔ)E+1/2[1+iα]gE+κF,
dndt=J-dndtsp-gW,
dUdt=-1τL {U[μ(T), T]-U[μ(T0), T0]}-ωgW+ωσfnW+Q-dUdtsp,
εθ dθdt=-1τL {U(μ, T)-U[μ(T0), T0]}+(εn-ω)gW+ωσfnW-dUdtsp+Q+εndndtsp-Jεn.
εn-ω=μ+2εdθ02-ω.
tanhμ-ω2εdθ=1Gτc,
μ=ω+εdθ lnGτc+1Gτc-1.
μ=ω+2εdθ/Gτc.
g(θ, μ, ω)=G(ω) μ-ω2εdθ.
n(μ, θ)=ρ0εd(1+θ2)expωεdexpμ-ωεdnt1+μ-ωεd,
ntρ0εd(1+θ2)exp(ω/εd).
g(n, θ)=12 Gθ n-ntnt=γn(n-nt),
g(n, θ)=γn0[(n-nt0)(1-Δθ/θ0)-bnΔθ/θ0],
g=γn(n-nt)(1-SW).
g=γn0(n-nt0)[1-(b+1)Δθ/θ0]-γn0nt0bΔθ/θ0,
dWdt+1τc W-g(n)W+σfnW=KWext,
dndtsp=0 1τsε ρ(ε)f(ε, μ, T)dε,
1τsε=1τs0 εε0q.
εq=[εg+(ε-εg)]q=qεεgq-1-(q-1)εgq+ .
dndtsp=1τs0 q U(μ, T)εg-(q-1)n(μ, T),
dUdtsp=1τs0 [(q+1)U(μ, T)-qεgn(μ, T)].
1τsε=1τs0 με0q1+ε-μμq1τs0 με0q×1+q ε-μμ+q(q-1)2! ε-μμ2+.
dndtsp=nτs0 με0q1-q εdμ+2q εdμ θ2,
dUdtsp=nτs0 με0q[μ-(q+1)εd+2(q+1)εdθ2].
B(n, T)=4(2π)6 wcv(p, p)fec(p)fnv(p)dpdp,
B(n, T)=4C(2π)3 w(ε)fcmme εfvmmh εεdε,
ε=p22m,1m=1me+1mh,
C=4π2m3/2(2π)3,
fc,v(ε)=expε-μe,hkBT+1-1.
ne,h=2Cme,hm3/20expε-μe,hkBT+1-1εdε.
ne,h=Cπme,hm kBT3/2 expμe,hkBT,
B(n, T)=nenhB(T, ε)dε=nenhB(T),
B(T)=2π 1(2π)3 1C m2memh3/2 w0(kBT)3/2.
ne,h=43 me,hm μe,h3/2C,
B(n, T)=2 w0(2π)3 43 me,hm μe,h3/2,
C=2 w0(2π)3 n.
1τsp=2 w0(2π)3.
B(n, T)=B(T) n21+n/n0.
n0=2MkBT2π23/2,M=me+mh.
dEdt=-12τc (1+iΔ)E+1/2[1+iα]gE+κF,
F(t)=ΩFΩ(t)exp[-i(ΔΩt+ϕΩ)],
E=κ exp[-(1/2)D(t)]0t exp[(1/2)D(t)]F(t)dt,
D(t)=0t1τc-g+i(Δ-αg)dt.
EE*=κ2 exp[-Re D(t)]0tdt0tdt×exp[(1/2)D(t)+(1/2)D*(t)]F(t)F*(t).
F(t)F*(t)=Ω,ΩFΩ(t)FΩ*(t)×exp[-i(ΔΩt-ΔΩt+ϕΩ-ϕΩ)].
Ω,ΩFΩ(t)FΩ*(t)exp[-i(ΔΩt-ΔΩt+ϕΩ-ϕΩ)]
=ΩFΩ(t)FΩ*(t)exp[-iΔΩ(t-t)]+ΩΩFΩ(t)FΩ*(t)×exp[-i(ΔΩt-ΔΩt+ϕΩ-ϕΩ)].
ΩFΩ(t)FΩ*(t)exp[-iΔΩ(t-t)]
Fω(t)Fω*(t)Ω exp[-iΔΩ(t-t)],
ΩΩFΩ(t)FΩ*(t)
×exp[-i(ΔΩt-ΔΩt+ϕΩ-ϕΩ)]
Fω(t)Fω*(t)ΩΩ
×exp[-i(ΔΩt-ΔΩt+ϕΩ-ϕΩ)]
=0.
Fω(t)Fω*(t)Ω exp[-iΔΩ(t-t)]
Fω(t)Fω*(t)ΔΩ0  exp[-iΔΩ(t-t)]d(ΔΩ)2π Fω(t)Fω*(t)ΔΩ0 δ(t-t),
EE*=2π κ2ΔΩ0 exp[-Re D(t)]×0t exp[Re D(t)]|Fω(t)|2dt.
dWdt+1τc W-g(n)W+σfnW=KWext,

Metrics