Abstract

Analytical expressions for the optical response of active semiconductor waveguides are derived, taking into account the effects of propagation, gain, and index dispersion as well as pulse chirp. The dynamics of the gain and the index that are due to carrier density changes, carrier heating, and spectral holeburning are derived from semiclassical density-matrix equations, and instantaneous two-photon and Kerr effects are included phenomenologically. It is shown that dispersion of the gain strongly affects short-pulse pump-and-probe measurements, since it induces a coupling between the phase and the intensity of the probe pulses. Good agreement with published experimental data is obtained.

© 1996 Optical Society of America

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  1. J. E. Bowers, “High speed semiconductor laser design and performance,” Solid State Electron. 30, 1–11 (1987).
    [CrossRef]
  2. G. P. Agrawal and N. K. Dutta, Long-Wavelength Semiconductor Lasers (Van Nostrand Reinhold, New York, 1986).
    [CrossRef]
  3. E. Iannone and R. Sabella, “Performance evaluation of an optical multi-carrier network using wavelength converters based on FWM in semiconductor optical amplifiers,” J. Lightwave Technol. 13, 312–324 (1955), and references therein.
    [CrossRef]
  4. D. A. O. Davies, M. A. Fisher, D. J. Elton, S. D. Perrin, M. J. Adams, G. T. Kennedy, R. S. Grant, P. D. Roberts, and W. Sibbett, “Nonlinear switching in InGaAsP laser amplifier directional couplers biased at transparency,” Electron. Lett. 29, 1710–1711 (1993).
    [CrossRef]
  5. S. Kawanishi, T. Morioka, O. Kamatani, H. Takara, J. M. Jacob, and M. Saruwatari, “100 Gbit/s all-optical demultiplexing using four-wave mixing in a travelling wave laser diode amplifier,” Electron. Lett. 30, 981–982 (1994).
    [CrossRef]
  6. A. D. Ellis, M. C. Tatham, D. A. O. Davies, D. Nessett, D. G. Moodie, and G. Sherlock, “40 Gbit/s transmission over 202 km of standard fibre using midspan spectral inversion,” Electron. Lett. 31, 299–301 (1995).
    [CrossRef]
  7. G. Sucha, S. R. Bolton, D. S. Chemla, D. L. Sivco, and A. Y. Cho, “Carrier relaxation in InGaAs heterostructures,” Appl. Phys. Lett. 65, 1486–1488 (1994), and references therein.
    [CrossRef]
  8. 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]
  9. J. Mark and J. Mørk, “Subpicosecond gain dynamics in InGaAsP optical amplifiers: experiment and theory,” Appl. Phys. Lett. 61, 2281–2283 (1992).
    [CrossRef]
  10. C. T. Hultgren, D. J. Dougherty, and E. P. Ippen, “Above- and below-band femtosecond nonlinearities in active AlGaAs waveguides,” Appl. Phys. Lett. 61, 2767–2769 (1992).
    [CrossRef]
  11. K. L. Hall, G. Lenz, A. M. Darwish, and E. P. Ippen, “Subpicosecond gain and index nonlinearities in InGaAsP diode lasers,” Opt. Commun. 111, 589–612 (1994).
    [CrossRef]
  12. G. D. Sanders, C.-K. Sun, J. G. Fujimoto, H. K. Choi, C. A. Wang, and C. J. Stanton, “Carrier gain dynamics in InGaAs/AlGaAs strained-layer single-quantum-well diode lasers: comparison of theory and experiment,” Phys. Rev. B 50, 8539–8558 (1994).
    [CrossRef]
  13. J. Mørk and A. Mecozzi, “Response function for gain and refractive index dynamics in active semiconductor waveguides,” Appl. Phys. Lett. 65, 1736–1738 (1994).
    [CrossRef]
  14. M. Y. Hong, Y. H. Chang, A. Dienes, J. P. Heritage, and P. J. Delfyett, “Subpicosecond pulse amplification in semiconductor laser amplifiers: theory and experiment,” IEEE J. Quantum Electron. 30, 1122–1131 (1994).
    [CrossRef]
  15. M. Asada and Y. Suematsu, “Density matrix theory of semiconductor lasers with relaxation broadening model—gain and gain-suppression in semiconductor lasers,” IEEE J. Quantum Electron. QE-21, 434–441 (1985).
    [CrossRef]
  16. N. Ogasawara and R. Ito, “Longitudinal mode competition and asymmetric gain saturation in semiconductor injection lasers. II. Theory,” Jpn. J. Appl. Phys. 27, 615–626 (1988).
    [CrossRef]
  17. D. R. Hjelme and A. R. Mickelson, “Gain nonlinearities due to carrier density dependent dispersion in semiconductor lasers,” IEEE J. Quantum Electron. 25, 1625–1631 (1989).
    [CrossRef]
  18. A. Uskov, J. Mørk, and J. Mark, “Wave mixing in semiconductor laser amplifiers due to carrier heating and spectral-hole burning,” IEEE J. Quantum Electron. 30, 1769–1781 (1994).
    [CrossRef]
  19. Y. Nambu and A. Tomita, “Spectral holeburning and carrier-heating effect on the transient optical nonlinearity of highly carrier-injected semiconductors,” IEEE J. Quantum Electron. 30, 1981–1994 (1994).
    [CrossRef]
  20. H. Haug and S. W. Koch, Quantum Theory of the Optical and Electronic Properties of Semiconductors (World Scientific, Singapore, 1990).
    [CrossRef]
  21. W. W. Chow, S. W. Koch, and M. Sargent, Semiconductor-Laser Physics (Springer-Verlag, Berlin, 1994).
    [CrossRef]
  22. H. Haug and S. W. Koch, “Semiconductor laser theory with many-body effects,” Phys. Rev. A 39, 1887–1898 (1989).
    [CrossRef] [PubMed]
  23. R. Binder, D. Scott, A. E. Paul, M. Lindberg, K. Henneberger, and S. W. Koch, “Carrier–carrier scattering and optical dephasing in highly excited semiconductors,” Phys. Rev. B 45, 1107–1115 (1992).
    [CrossRef]
  24. D. C. Hutchings and E. W. Van Stryland, “Nondegenerate two-photon absorption in zinc blende semiconductors,” J. Opt. Soc. Am. B 9, 2065–2074 (1992).
    [CrossRef]
  25. M. Sheik-Bahae and E. W. Van Stryland, “Ultrafast nonlinearities in semiconductor laser amplifiers,” Phys. Rev. B 50, 14171–14178 (1994).
    [CrossRef]
  26. J. Mørk, J. Mark, and C. P. Seltzer, “Carrier heating in InGaAsP laser amplifiers due to two-photon absorption,” Appl. Phys. Lett. 64, 2206–2208 (1994).
    [CrossRef]
  27. A. D’Ottavi, E. Iannone, A. Mecozzi, S. Scotti, P. Spano, R. Dall’Ara, G. Guekos, and J. Eckner, “4.3 terahertz four-wave mixing spectroscopy of InGaAsP semiconductor amplifiers,” Appl. Phys. Lett. 65, 2633–2635 (1994).
    [CrossRef]
  28. A. Mecozzi, S. Scotti, A. D’Ottavi, E. Iannone, and P. Spano, “Four-wave mixing in traveling-wave semiconductor amplifiers,” IEEE J. Quantum Electron. 31, 689–699 (1995).
    [CrossRef]
  29. 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, 606–609 (1991).
    [CrossRef]
  30. A. Knorr, R. Binder, E. M. Wright, and S. W. Koch, “Amplification, absorption, and lossless propagation of femtosecond pulses in semiconductor amplifiers,” Opt. Lett. 18, 1538–1540 (1993).
    [CrossRef] [PubMed]
  31. A. Mecozzi and J. Mørk, “Theory of heterodyne pump–probe experiments with femtosecond pulses,” J. Opt. Soc. Am. B (to be published).
  32. Z. Vardeny and J. Tauc, “Picosecond coherence coupling in the pump and probe technique,” Opt. Commun. 39, 396–400 (1981).
    [CrossRef]
  33. A. Girndt, A. Knorr, M. Hofmann, and S. W. Koch, “Theoretical analysis of ultrafast pump–probe experiments in semiconductor amplifiers,” Appl. Phys. Lett. 66, 550–552 (1995).
    [CrossRef]
  34. M. Willatzen, J. Mark, J. Mørk, and C. P. Seltzer, “Carrier temperature and spectral holeburning dynamics in InGaAsP quantum well laser amplifiers,” Appl. Phys. Lett. 64, 143–145 (1994).
    [CrossRef]
  35. J. Mørk, A. Mecozzi, and C. Hultgren, “Spectral effects in short pulse pump–probe measurements,” Appl. Phys. Lett. 68, 449–451 (1996).
    [CrossRef]
  36. K. L. Hall, G. Lenz, E. P. Ippen, and G. Raybon, “Heterodyne pump–probe technique for time-domain studies of optical nonlinearities in waveguides,” Opt. Lett. 17, 874–876 (1992).
    [CrossRef]

1996 (1)

J. Mørk, A. Mecozzi, and C. Hultgren, “Spectral effects in short pulse pump–probe measurements,” Appl. Phys. Lett. 68, 449–451 (1996).
[CrossRef]

1995 (3)

A. Girndt, A. Knorr, M. Hofmann, and S. W. Koch, “Theoretical analysis of ultrafast pump–probe experiments in semiconductor amplifiers,” Appl. Phys. Lett. 66, 550–552 (1995).
[CrossRef]

A. Mecozzi, S. Scotti, A. D’Ottavi, E. Iannone, and P. Spano, “Four-wave mixing in traveling-wave semiconductor amplifiers,” IEEE J. Quantum Electron. 31, 689–699 (1995).
[CrossRef]

A. D. Ellis, M. C. Tatham, D. A. O. Davies, D. Nessett, D. G. Moodie, and G. Sherlock, “40 Gbit/s transmission over 202 km of standard fibre using midspan spectral inversion,” Electron. Lett. 31, 299–301 (1995).
[CrossRef]

1994 (12)

G. Sucha, S. R. Bolton, D. S. Chemla, D. L. Sivco, and A. Y. Cho, “Carrier relaxation in InGaAs heterostructures,” Appl. Phys. Lett. 65, 1486–1488 (1994), and references therein.
[CrossRef]

S. Kawanishi, T. Morioka, O. Kamatani, H. Takara, J. M. Jacob, and M. Saruwatari, “100 Gbit/s all-optical demultiplexing using four-wave mixing in a travelling wave laser diode amplifier,” Electron. Lett. 30, 981–982 (1994).
[CrossRef]

K. L. Hall, G. Lenz, A. M. Darwish, and E. P. Ippen, “Subpicosecond gain and index nonlinearities in InGaAsP diode lasers,” Opt. Commun. 111, 589–612 (1994).
[CrossRef]

G. D. Sanders, C.-K. Sun, J. G. Fujimoto, H. K. Choi, C. A. Wang, and C. J. Stanton, “Carrier gain dynamics in InGaAs/AlGaAs strained-layer single-quantum-well diode lasers: comparison of theory and experiment,” Phys. Rev. B 50, 8539–8558 (1994).
[CrossRef]

J. Mørk and A. Mecozzi, “Response function for gain and refractive index dynamics in active semiconductor waveguides,” Appl. Phys. Lett. 65, 1736–1738 (1994).
[CrossRef]

M. Y. Hong, Y. H. Chang, A. Dienes, J. P. Heritage, and P. J. Delfyett, “Subpicosecond pulse amplification in semiconductor laser amplifiers: theory and experiment,” IEEE J. Quantum Electron. 30, 1122–1131 (1994).
[CrossRef]

A. Uskov, J. Mørk, and J. Mark, “Wave mixing in semiconductor laser amplifiers due to carrier heating and spectral-hole burning,” IEEE J. Quantum Electron. 30, 1769–1781 (1994).
[CrossRef]

Y. Nambu and A. Tomita, “Spectral holeburning and carrier-heating effect on the transient optical nonlinearity of highly carrier-injected semiconductors,” IEEE J. Quantum Electron. 30, 1981–1994 (1994).
[CrossRef]

M. Sheik-Bahae and E. W. Van Stryland, “Ultrafast nonlinearities in semiconductor laser amplifiers,” Phys. Rev. B 50, 14171–14178 (1994).
[CrossRef]

J. Mørk, J. Mark, and C. P. Seltzer, “Carrier heating in InGaAsP laser amplifiers due to two-photon absorption,” Appl. Phys. Lett. 64, 2206–2208 (1994).
[CrossRef]

A. D’Ottavi, E. Iannone, A. Mecozzi, S. Scotti, P. Spano, R. Dall’Ara, G. Guekos, and J. Eckner, “4.3 terahertz four-wave mixing spectroscopy of InGaAsP semiconductor amplifiers,” Appl. Phys. Lett. 65, 2633–2635 (1994).
[CrossRef]

M. Willatzen, J. Mark, J. Mørk, and C. P. Seltzer, “Carrier temperature and spectral holeburning dynamics in InGaAsP quantum well laser amplifiers,” Appl. Phys. Lett. 64, 143–145 (1994).
[CrossRef]

1993 (2)

A. Knorr, R. Binder, E. M. Wright, and S. W. Koch, “Amplification, absorption, and lossless propagation of femtosecond pulses in semiconductor amplifiers,” Opt. Lett. 18, 1538–1540 (1993).
[CrossRef] [PubMed]

D. A. O. Davies, M. A. Fisher, D. J. Elton, S. D. Perrin, M. J. Adams, G. T. Kennedy, R. S. Grant, P. D. Roberts, and W. Sibbett, “Nonlinear switching in InGaAsP laser amplifier directional couplers biased at transparency,” Electron. Lett. 29, 1710–1711 (1993).
[CrossRef]

1992 (5)

J. Mark and J. Mørk, “Subpicosecond gain dynamics in InGaAsP optical amplifiers: experiment and theory,” Appl. Phys. Lett. 61, 2281–2283 (1992).
[CrossRef]

C. T. Hultgren, D. J. Dougherty, and E. P. Ippen, “Above- and below-band femtosecond nonlinearities in active AlGaAs waveguides,” Appl. Phys. Lett. 61, 2767–2769 (1992).
[CrossRef]

R. Binder, D. Scott, A. E. Paul, M. Lindberg, K. Henneberger, and S. W. Koch, “Carrier–carrier scattering and optical dephasing in highly excited semiconductors,” Phys. Rev. B 45, 1107–1115 (1992).
[CrossRef]

D. C. Hutchings and E. W. Van Stryland, “Nondegenerate two-photon absorption in zinc blende semiconductors,” J. Opt. Soc. Am. B 9, 2065–2074 (1992).
[CrossRef]

K. L. Hall, G. Lenz, E. P. Ippen, and G. Raybon, “Heterodyne pump–probe technique for time-domain studies of optical nonlinearities in waveguides,” Opt. Lett. 17, 874–876 (1992).
[CrossRef]

1991 (1)

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, 606–609 (1991).
[CrossRef]

1990 (1)

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]

1989 (2)

H. Haug and S. W. Koch, “Semiconductor laser theory with many-body effects,” Phys. Rev. A 39, 1887–1898 (1989).
[CrossRef] [PubMed]

D. R. Hjelme and A. R. Mickelson, “Gain nonlinearities due to carrier density dependent dispersion in semiconductor lasers,” IEEE J. Quantum Electron. 25, 1625–1631 (1989).
[CrossRef]

1988 (1)

N. Ogasawara and R. Ito, “Longitudinal mode competition and asymmetric gain saturation in semiconductor injection lasers. II. Theory,” Jpn. J. Appl. Phys. 27, 615–626 (1988).
[CrossRef]

1987 (1)

J. E. Bowers, “High speed semiconductor laser design and performance,” Solid State Electron. 30, 1–11 (1987).
[CrossRef]

1985 (1)

M. Asada and Y. Suematsu, “Density matrix theory of semiconductor lasers with relaxation broadening model—gain and gain-suppression in semiconductor lasers,” IEEE J. Quantum Electron. QE-21, 434–441 (1985).
[CrossRef]

1981 (1)

Z. Vardeny and J. Tauc, “Picosecond coherence coupling in the pump and probe technique,” Opt. Commun. 39, 396–400 (1981).
[CrossRef]

1955 (1)

E. Iannone and R. Sabella, “Performance evaluation of an optical multi-carrier network using wavelength converters based on FWM in semiconductor optical amplifiers,” J. Lightwave Technol. 13, 312–324 (1955), and references therein.
[CrossRef]

Adams, M. J.

D. A. O. Davies, M. A. Fisher, D. J. Elton, S. D. Perrin, M. J. Adams, G. T. Kennedy, R. S. Grant, P. D. Roberts, and W. Sibbett, “Nonlinear switching in InGaAsP laser amplifier directional couplers biased at transparency,” Electron. Lett. 29, 1710–1711 (1993).
[CrossRef]

Agrawal, G. P.

G. P. Agrawal and N. K. Dutta, Long-Wavelength Semiconductor Lasers (Van Nostrand Reinhold, New York, 1986).
[CrossRef]

Asada, M.

M. Asada and Y. Suematsu, “Density matrix theory of semiconductor lasers with relaxation broadening model—gain and gain-suppression in semiconductor lasers,” IEEE J. Quantum Electron. QE-21, 434–441 (1985).
[CrossRef]

Binder, R.

A. Knorr, R. Binder, E. M. Wright, and S. W. Koch, “Amplification, absorption, and lossless propagation of femtosecond pulses in semiconductor amplifiers,” Opt. Lett. 18, 1538–1540 (1993).
[CrossRef] [PubMed]

R. Binder, D. Scott, A. E. Paul, M. Lindberg, K. Henneberger, and S. W. Koch, “Carrier–carrier scattering and optical dephasing in highly excited semiconductors,” Phys. Rev. B 45, 1107–1115 (1992).
[CrossRef]

Bolton, S. R.

G. Sucha, S. R. Bolton, D. S. Chemla, D. L. Sivco, and A. Y. Cho, “Carrier relaxation in InGaAs heterostructures,” Appl. Phys. Lett. 65, 1486–1488 (1994), and references therein.
[CrossRef]

Bowers, J. E.

J. E. Bowers, “High speed semiconductor laser design and performance,” Solid State Electron. 30, 1–11 (1987).
[CrossRef]

Chang, Y. H.

M. Y. Hong, Y. H. Chang, A. Dienes, J. P. Heritage, and P. J. Delfyett, “Subpicosecond pulse amplification in semiconductor laser amplifiers: theory and experiment,” IEEE J. Quantum Electron. 30, 1122–1131 (1994).
[CrossRef]

Chemla, D. S.

G. Sucha, S. R. Bolton, D. S. Chemla, D. L. Sivco, and A. Y. Cho, “Carrier relaxation in InGaAs heterostructures,” Appl. Phys. Lett. 65, 1486–1488 (1994), and references therein.
[CrossRef]

Cho, A. Y.

G. Sucha, S. R. Bolton, D. S. Chemla, D. L. Sivco, and A. Y. Cho, “Carrier relaxation in InGaAs heterostructures,” Appl. Phys. Lett. 65, 1486–1488 (1994), and references therein.
[CrossRef]

Choi, H. K.

G. D. Sanders, C.-K. Sun, J. G. Fujimoto, H. K. Choi, C. A. Wang, and C. J. Stanton, “Carrier gain dynamics in InGaAs/AlGaAs strained-layer single-quantum-well diode lasers: comparison of theory and experiment,” Phys. Rev. B 50, 8539–8558 (1994).
[CrossRef]

Chow, W. W.

W. W. Chow, S. W. Koch, and M. Sargent, Semiconductor-Laser Physics (Springer-Verlag, Berlin, 1994).
[CrossRef]

D’Ottavi, A.

A. Mecozzi, S. Scotti, A. D’Ottavi, E. Iannone, and P. Spano, “Four-wave mixing in traveling-wave semiconductor amplifiers,” IEEE J. Quantum Electron. 31, 689–699 (1995).
[CrossRef]

A. D’Ottavi, E. Iannone, A. Mecozzi, S. Scotti, P. Spano, R. Dall’Ara, G. Guekos, and J. Eckner, “4.3 terahertz four-wave mixing spectroscopy of InGaAsP semiconductor amplifiers,” Appl. Phys. Lett. 65, 2633–2635 (1994).
[CrossRef]

Dall’Ara, R.

A. D’Ottavi, E. Iannone, A. Mecozzi, S. Scotti, P. Spano, R. Dall’Ara, G. Guekos, and J. Eckner, “4.3 terahertz four-wave mixing spectroscopy of InGaAsP semiconductor amplifiers,” Appl. Phys. Lett. 65, 2633–2635 (1994).
[CrossRef]

Darwish, A. M.

K. L. Hall, G. Lenz, A. M. Darwish, and E. P. Ippen, “Subpicosecond gain and index nonlinearities in InGaAsP diode lasers,” Opt. Commun. 111, 589–612 (1994).
[CrossRef]

Davies, D. A. O.

A. D. Ellis, M. C. Tatham, D. A. O. Davies, D. Nessett, D. G. Moodie, and G. Sherlock, “40 Gbit/s transmission over 202 km of standard fibre using midspan spectral inversion,” Electron. Lett. 31, 299–301 (1995).
[CrossRef]

D. A. O. Davies, M. A. Fisher, D. J. Elton, S. D. Perrin, M. J. Adams, G. T. Kennedy, R. S. Grant, P. D. Roberts, and W. Sibbett, “Nonlinear switching in InGaAsP laser amplifier directional couplers biased at transparency,” Electron. Lett. 29, 1710–1711 (1993).
[CrossRef]

Delfyett, P. J.

M. Y. Hong, Y. H. Chang, A. Dienes, J. P. Heritage, and P. J. Delfyett, “Subpicosecond pulse amplification in semiconductor laser amplifiers: theory and experiment,” IEEE J. Quantum Electron. 30, 1122–1131 (1994).
[CrossRef]

Dienes, A.

M. Y. Hong, Y. H. Chang, A. Dienes, J. P. Heritage, and P. J. Delfyett, “Subpicosecond pulse amplification in semiconductor laser amplifiers: theory and experiment,” IEEE J. Quantum Electron. 30, 1122–1131 (1994).
[CrossRef]

Dougherty, D. J.

C. T. Hultgren, D. J. Dougherty, and E. P. Ippen, “Above- and below-band femtosecond nonlinearities in active AlGaAs waveguides,” Appl. Phys. Lett. 61, 2767–2769 (1992).
[CrossRef]

Dutta, N. K.

G. P. Agrawal and N. K. Dutta, Long-Wavelength Semiconductor Lasers (Van Nostrand Reinhold, New York, 1986).
[CrossRef]

Eckner, J.

A. D’Ottavi, E. Iannone, A. Mecozzi, S. Scotti, P. Spano, R. Dall’Ara, G. Guekos, and J. Eckner, “4.3 terahertz four-wave mixing spectroscopy of InGaAsP semiconductor amplifiers,” Appl. Phys. Lett. 65, 2633–2635 (1994).
[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]

Ellis, A. D.

A. D. Ellis, M. C. Tatham, D. A. O. Davies, D. Nessett, D. G. Moodie, and G. Sherlock, “40 Gbit/s transmission over 202 km of standard fibre using midspan spectral inversion,” Electron. Lett. 31, 299–301 (1995).
[CrossRef]

Elton, D. J.

D. A. O. Davies, M. A. Fisher, D. J. Elton, S. D. Perrin, M. J. Adams, G. T. Kennedy, R. S. Grant, P. D. Roberts, and W. Sibbett, “Nonlinear switching in InGaAsP laser amplifier directional couplers biased at transparency,” Electron. Lett. 29, 1710–1711 (1993).
[CrossRef]

Fisher, M. A.

D. A. O. Davies, M. A. Fisher, D. J. Elton, S. D. Perrin, M. J. Adams, G. T. Kennedy, R. S. Grant, P. D. Roberts, and W. Sibbett, “Nonlinear switching in InGaAsP laser amplifier directional couplers biased at transparency,” Electron. Lett. 29, 1710–1711 (1993).
[CrossRef]

Fujimoto, J. G.

G. D. Sanders, C.-K. Sun, J. G. Fujimoto, H. K. Choi, C. A. Wang, and C. J. Stanton, “Carrier gain dynamics in InGaAs/AlGaAs strained-layer single-quantum-well diode lasers: comparison of theory and experiment,” Phys. Rev. B 50, 8539–8558 (1994).
[CrossRef]

Girndt, A.

A. Girndt, A. Knorr, M. Hofmann, and S. W. Koch, “Theoretical analysis of ultrafast pump–probe experiments in semiconductor amplifiers,” Appl. Phys. Lett. 66, 550–552 (1995).
[CrossRef]

Grant, R. S.

D. A. O. Davies, M. A. Fisher, D. J. Elton, S. D. Perrin, M. J. Adams, G. T. Kennedy, R. S. Grant, P. D. Roberts, and W. Sibbett, “Nonlinear switching in InGaAsP laser amplifier directional couplers biased at transparency,” Electron. Lett. 29, 1710–1711 (1993).
[CrossRef]

Guekos, G.

A. D’Ottavi, E. Iannone, A. Mecozzi, S. Scotti, P. Spano, R. Dall’Ara, G. Guekos, and J. Eckner, “4.3 terahertz four-wave mixing spectroscopy of InGaAsP semiconductor amplifiers,” Appl. Phys. Lett. 65, 2633–2635 (1994).
[CrossRef]

Hall, K. L.

K. L. Hall, G. Lenz, A. M. Darwish, and E. P. Ippen, “Subpicosecond gain and index nonlinearities in InGaAsP diode lasers,” Opt. Commun. 111, 589–612 (1994).
[CrossRef]

K. L. Hall, G. Lenz, E. P. Ippen, and G. Raybon, “Heterodyne pump–probe technique for time-domain studies of optical nonlinearities in waveguides,” Opt. Lett. 17, 874–876 (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]

Haug, H.

H. Haug and S. W. Koch, “Semiconductor laser theory with many-body effects,” Phys. Rev. A 39, 1887–1898 (1989).
[CrossRef] [PubMed]

H. Haug and S. W. Koch, Quantum Theory of the Optical and Electronic Properties of Semiconductors (World Scientific, Singapore, 1990).
[CrossRef]

Henneberger, K.

R. Binder, D. Scott, A. E. Paul, M. Lindberg, K. Henneberger, and S. W. Koch, “Carrier–carrier scattering and optical dephasing in highly excited semiconductors,” Phys. Rev. B 45, 1107–1115 (1992).
[CrossRef]

Heritage, J. P.

M. Y. Hong, Y. H. Chang, A. Dienes, J. P. Heritage, and P. J. Delfyett, “Subpicosecond pulse amplification in semiconductor laser amplifiers: theory and experiment,” IEEE J. Quantum Electron. 30, 1122–1131 (1994).
[CrossRef]

Hjelme, D. R.

D. R. Hjelme and A. R. Mickelson, “Gain nonlinearities due to carrier density dependent dispersion in semiconductor lasers,” IEEE J. Quantum Electron. 25, 1625–1631 (1989).
[CrossRef]

Hofmann, M.

A. Girndt, A. Knorr, M. Hofmann, and S. W. Koch, “Theoretical analysis of ultrafast pump–probe experiments in semiconductor amplifiers,” Appl. Phys. Lett. 66, 550–552 (1995).
[CrossRef]

Hong, M. Y.

M. Y. Hong, Y. H. Chang, A. Dienes, J. P. Heritage, and P. J. Delfyett, “Subpicosecond pulse amplification in semiconductor laser amplifiers: theory and experiment,” IEEE J. Quantum Electron. 30, 1122–1131 (1994).
[CrossRef]

Hultgren, C.

J. Mørk, A. Mecozzi, and C. Hultgren, “Spectral effects in short pulse pump–probe measurements,” Appl. Phys. Lett. 68, 449–451 (1996).
[CrossRef]

Hultgren, C. T.

C. T. Hultgren, D. J. Dougherty, and E. P. Ippen, “Above- and below-band femtosecond nonlinearities in active AlGaAs waveguides,” Appl. Phys. Lett. 61, 2767–2769 (1992).
[CrossRef]

Hutchings, D. C.

Iannone, E.

A. Mecozzi, S. Scotti, A. D’Ottavi, E. Iannone, and P. Spano, “Four-wave mixing in traveling-wave semiconductor amplifiers,” IEEE J. Quantum Electron. 31, 689–699 (1995).
[CrossRef]

A. D’Ottavi, E. Iannone, A. Mecozzi, S. Scotti, P. Spano, R. Dall’Ara, G. Guekos, and J. Eckner, “4.3 terahertz four-wave mixing spectroscopy of InGaAsP semiconductor amplifiers,” Appl. Phys. Lett. 65, 2633–2635 (1994).
[CrossRef]

E. Iannone and R. Sabella, “Performance evaluation of an optical multi-carrier network using wavelength converters based on FWM in semiconductor optical amplifiers,” J. Lightwave Technol. 13, 312–324 (1955), and references therein.
[CrossRef]

Ippen, E. P.

K. L. Hall, G. Lenz, A. M. Darwish, and E. P. Ippen, “Subpicosecond gain and index nonlinearities in InGaAsP diode lasers,” Opt. Commun. 111, 589–612 (1994).
[CrossRef]

C. T. Hultgren, D. J. Dougherty, and E. P. Ippen, “Above- and below-band femtosecond nonlinearities in active AlGaAs waveguides,” Appl. Phys. Lett. 61, 2767–2769 (1992).
[CrossRef]

K. L. Hall, G. Lenz, E. P. Ippen, and G. Raybon, “Heterodyne pump–probe technique for time-domain studies of optical nonlinearities in waveguides,” Opt. Lett. 17, 874–876 (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]

Ito, R.

N. Ogasawara and R. Ito, “Longitudinal mode competition and asymmetric gain saturation in semiconductor injection lasers. II. Theory,” Jpn. J. Appl. Phys. 27, 615–626 (1988).
[CrossRef]

Jacob, J. M.

S. Kawanishi, T. Morioka, O. Kamatani, H. Takara, J. M. Jacob, and M. Saruwatari, “100 Gbit/s all-optical demultiplexing using four-wave mixing in a travelling wave laser diode amplifier,” Electron. Lett. 30, 981–982 (1994).
[CrossRef]

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, 606–609 (1991).
[CrossRef]

Kamatani, O.

S. Kawanishi, T. Morioka, O. Kamatani, H. Takara, J. M. Jacob, and M. Saruwatari, “100 Gbit/s all-optical demultiplexing using four-wave mixing in a travelling wave laser diode amplifier,” Electron. Lett. 30, 981–982 (1994).
[CrossRef]

Kawanishi, S.

S. Kawanishi, T. Morioka, O. Kamatani, H. Takara, J. M. Jacob, and M. Saruwatari, “100 Gbit/s all-optical demultiplexing using four-wave mixing in a travelling wave laser diode amplifier,” Electron. Lett. 30, 981–982 (1994).
[CrossRef]

Kennedy, G. T.

D. A. O. Davies, M. A. Fisher, D. J. Elton, S. D. Perrin, M. J. Adams, G. T. Kennedy, R. S. Grant, P. D. Roberts, and W. Sibbett, “Nonlinear switching in InGaAsP laser amplifier directional couplers biased at transparency,” Electron. Lett. 29, 1710–1711 (1993).
[CrossRef]

Knorr, A.

A. Girndt, A. Knorr, M. Hofmann, and S. W. Koch, “Theoretical analysis of ultrafast pump–probe experiments in semiconductor amplifiers,” Appl. Phys. Lett. 66, 550–552 (1995).
[CrossRef]

A. Knorr, R. Binder, E. M. Wright, and S. W. Koch, “Amplification, absorption, and lossless propagation of femtosecond pulses in semiconductor amplifiers,” Opt. Lett. 18, 1538–1540 (1993).
[CrossRef] [PubMed]

Koch, S. W.

A. Girndt, A. Knorr, M. Hofmann, and S. W. Koch, “Theoretical analysis of ultrafast pump–probe experiments in semiconductor amplifiers,” Appl. Phys. Lett. 66, 550–552 (1995).
[CrossRef]

A. Knorr, R. Binder, E. M. Wright, and S. W. Koch, “Amplification, absorption, and lossless propagation of femtosecond pulses in semiconductor amplifiers,” Opt. Lett. 18, 1538–1540 (1993).
[CrossRef] [PubMed]

R. Binder, D. Scott, A. E. Paul, M. Lindberg, K. Henneberger, and S. W. Koch, “Carrier–carrier scattering and optical dephasing in highly excited semiconductors,” Phys. Rev. B 45, 1107–1115 (1992).
[CrossRef]

H. Haug and S. W. Koch, “Semiconductor laser theory with many-body effects,” Phys. Rev. A 39, 1887–1898 (1989).
[CrossRef] [PubMed]

W. W. Chow, S. W. Koch, and M. Sargent, Semiconductor-Laser Physics (Springer-Verlag, Berlin, 1994).
[CrossRef]

H. Haug and S. W. Koch, Quantum Theory of the Optical and Electronic Properties of Semiconductors (World Scientific, Singapore, 1990).
[CrossRef]

Lenz, G.

K. L. Hall, G. Lenz, A. M. Darwish, and E. P. Ippen, “Subpicosecond gain and index nonlinearities in InGaAsP diode lasers,” Opt. Commun. 111, 589–612 (1994).
[CrossRef]

K. L. Hall, G. Lenz, E. P. Ippen, and G. Raybon, “Heterodyne pump–probe technique for time-domain studies of optical nonlinearities in waveguides,” Opt. Lett. 17, 874–876 (1992).
[CrossRef]

Lindberg, M.

R. Binder, D. Scott, A. E. Paul, M. Lindberg, K. Henneberger, and S. W. Koch, “Carrier–carrier scattering and optical dephasing in highly excited semiconductors,” Phys. Rev. B 45, 1107–1115 (1992).
[CrossRef]

Mark, J.

J. Mørk, J. Mark, and C. P. Seltzer, “Carrier heating in InGaAsP laser amplifiers due to two-photon absorption,” Appl. Phys. Lett. 64, 2206–2208 (1994).
[CrossRef]

M. Willatzen, J. Mark, J. Mørk, and C. P. Seltzer, “Carrier temperature and spectral holeburning dynamics in InGaAsP quantum well laser amplifiers,” Appl. Phys. Lett. 64, 143–145 (1994).
[CrossRef]

A. Uskov, J. Mørk, and J. Mark, “Wave mixing in semiconductor laser amplifiers due to carrier heating and spectral-hole burning,” IEEE J. Quantum Electron. 30, 1769–1781 (1994).
[CrossRef]

J. Mark and J. Mørk, “Subpicosecond gain dynamics in InGaAsP optical amplifiers: experiment and theory,” Appl. Phys. Lett. 61, 2281–2283 (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]

Mecozzi, A.

J. Mørk, A. Mecozzi, and C. Hultgren, “Spectral effects in short pulse pump–probe measurements,” Appl. Phys. Lett. 68, 449–451 (1996).
[CrossRef]

A. Mecozzi, S. Scotti, A. D’Ottavi, E. Iannone, and P. Spano, “Four-wave mixing in traveling-wave semiconductor amplifiers,” IEEE J. Quantum Electron. 31, 689–699 (1995).
[CrossRef]

A. D’Ottavi, E. Iannone, A. Mecozzi, S. Scotti, P. Spano, R. Dall’Ara, G. Guekos, and J. Eckner, “4.3 terahertz four-wave mixing spectroscopy of InGaAsP semiconductor amplifiers,” Appl. Phys. Lett. 65, 2633–2635 (1994).
[CrossRef]

J. Mørk and A. Mecozzi, “Response function for gain and refractive index dynamics in active semiconductor waveguides,” Appl. Phys. Lett. 65, 1736–1738 (1994).
[CrossRef]

A. Mecozzi and J. Mørk, “Theory of heterodyne pump–probe experiments with femtosecond pulses,” J. Opt. Soc. Am. B (to be published).

Mickelson, A. R.

D. R. Hjelme and A. R. Mickelson, “Gain nonlinearities due to carrier density dependent dispersion in semiconductor lasers,” IEEE J. Quantum Electron. 25, 1625–1631 (1989).
[CrossRef]

Moodie, D. G.

A. D. Ellis, M. C. Tatham, D. A. O. Davies, D. Nessett, D. G. Moodie, and G. Sherlock, “40 Gbit/s transmission over 202 km of standard fibre using midspan spectral inversion,” Electron. Lett. 31, 299–301 (1995).
[CrossRef]

Morioka, T.

S. Kawanishi, T. Morioka, O. Kamatani, H. Takara, J. M. Jacob, and M. Saruwatari, “100 Gbit/s all-optical demultiplexing using four-wave mixing in a travelling wave laser diode amplifier,” Electron. Lett. 30, 981–982 (1994).
[CrossRef]

Mørk, J.

J. Mørk, A. Mecozzi, and C. Hultgren, “Spectral effects in short pulse pump–probe measurements,” Appl. Phys. Lett. 68, 449–451 (1996).
[CrossRef]

M. Willatzen, J. Mark, J. Mørk, and C. P. Seltzer, “Carrier temperature and spectral holeburning dynamics in InGaAsP quantum well laser amplifiers,” Appl. Phys. Lett. 64, 143–145 (1994).
[CrossRef]

J. Mørk, J. Mark, and C. P. Seltzer, “Carrier heating in InGaAsP laser amplifiers due to two-photon absorption,” Appl. Phys. Lett. 64, 2206–2208 (1994).
[CrossRef]

A. Uskov, J. Mørk, and J. Mark, “Wave mixing in semiconductor laser amplifiers due to carrier heating and spectral-hole burning,” IEEE J. Quantum Electron. 30, 1769–1781 (1994).
[CrossRef]

J. Mørk and A. Mecozzi, “Response function for gain and refractive index dynamics in active semiconductor waveguides,” Appl. Phys. Lett. 65, 1736–1738 (1994).
[CrossRef]

J. Mark and J. Mørk, “Subpicosecond gain dynamics in InGaAsP optical amplifiers: experiment and theory,” Appl. Phys. Lett. 61, 2281–2283 (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, 606–609 (1991).
[CrossRef]

A. Mecozzi and J. Mørk, “Theory of heterodyne pump–probe experiments with femtosecond pulses,” J. Opt. Soc. Am. B (to be published).

Nambu, Y.

Y. Nambu and A. Tomita, “Spectral holeburning and carrier-heating effect on the transient optical nonlinearity of highly carrier-injected semiconductors,” IEEE J. Quantum Electron. 30, 1981–1994 (1994).
[CrossRef]

Nessett, D.

A. D. Ellis, M. C. Tatham, D. A. O. Davies, D. Nessett, D. G. Moodie, and G. Sherlock, “40 Gbit/s transmission over 202 km of standard fibre using midspan spectral inversion,” Electron. Lett. 31, 299–301 (1995).
[CrossRef]

Ogasawara, N.

N. Ogasawara and R. Ito, “Longitudinal mode competition and asymmetric gain saturation in semiconductor injection lasers. II. Theory,” Jpn. J. Appl. Phys. 27, 615–626 (1988).
[CrossRef]

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, 606–609 (1991).
[CrossRef]

Paul, A. E.

R. Binder, D. Scott, A. E. Paul, M. Lindberg, K. Henneberger, and S. W. Koch, “Carrier–carrier scattering and optical dephasing in highly excited semiconductors,” Phys. Rev. B 45, 1107–1115 (1992).
[CrossRef]

Perrin, S. D.

D. A. O. Davies, M. A. Fisher, D. J. Elton, S. D. Perrin, M. J. Adams, G. T. Kennedy, R. S. Grant, P. D. Roberts, and W. Sibbett, “Nonlinear switching in InGaAsP laser amplifier directional couplers biased at transparency,” Electron. Lett. 29, 1710–1711 (1993).
[CrossRef]

Raybon, G.

Roberts, P. D.

D. A. O. Davies, M. A. Fisher, D. J. Elton, S. D. Perrin, M. J. Adams, G. T. Kennedy, R. S. Grant, P. D. Roberts, and W. Sibbett, “Nonlinear switching in InGaAsP laser amplifier directional couplers biased at transparency,” Electron. Lett. 29, 1710–1711 (1993).
[CrossRef]

Sabella, R.

E. Iannone and R. Sabella, “Performance evaluation of an optical multi-carrier network using wavelength converters based on FWM in semiconductor optical amplifiers,” J. Lightwave Technol. 13, 312–324 (1955), and references therein.
[CrossRef]

Sanders, G. D.

G. D. Sanders, C.-K. Sun, J. G. Fujimoto, H. K. Choi, C. A. Wang, and C. J. Stanton, “Carrier gain dynamics in InGaAs/AlGaAs strained-layer single-quantum-well diode lasers: comparison of theory and experiment,” Phys. Rev. B 50, 8539–8558 (1994).
[CrossRef]

Sargent, M.

W. W. Chow, S. W. Koch, and M. Sargent, Semiconductor-Laser Physics (Springer-Verlag, Berlin, 1994).
[CrossRef]

Saruwatari, M.

S. Kawanishi, T. Morioka, O. Kamatani, H. Takara, J. M. Jacob, and M. Saruwatari, “100 Gbit/s all-optical demultiplexing using four-wave mixing in a travelling wave laser diode amplifier,” Electron. Lett. 30, 981–982 (1994).
[CrossRef]

Scott, D.

R. Binder, D. Scott, A. E. Paul, M. Lindberg, K. Henneberger, and S. W. Koch, “Carrier–carrier scattering and optical dephasing in highly excited semiconductors,” Phys. Rev. B 45, 1107–1115 (1992).
[CrossRef]

Scotti, S.

A. Mecozzi, S. Scotti, A. D’Ottavi, E. Iannone, and P. Spano, “Four-wave mixing in traveling-wave semiconductor amplifiers,” IEEE J. Quantum Electron. 31, 689–699 (1995).
[CrossRef]

A. D’Ottavi, E. Iannone, A. Mecozzi, S. Scotti, P. Spano, R. Dall’Ara, G. Guekos, and J. Eckner, “4.3 terahertz four-wave mixing spectroscopy of InGaAsP semiconductor amplifiers,” Appl. Phys. Lett. 65, 2633–2635 (1994).
[CrossRef]

Seltzer, C. P.

J. Mørk, J. Mark, and C. P. Seltzer, “Carrier heating in InGaAsP laser amplifiers due to two-photon absorption,” Appl. Phys. Lett. 64, 2206–2208 (1994).
[CrossRef]

M. Willatzen, J. Mark, J. Mørk, and C. P. Seltzer, “Carrier temperature and spectral holeburning dynamics in InGaAsP quantum well laser amplifiers,” Appl. Phys. Lett. 64, 143–145 (1994).
[CrossRef]

Sheik-Bahae, M.

M. Sheik-Bahae and E. W. Van Stryland, “Ultrafast nonlinearities in semiconductor laser amplifiers,” Phys. Rev. B 50, 14171–14178 (1994).
[CrossRef]

Sherlock, G.

A. D. Ellis, M. C. Tatham, D. A. O. Davies, D. Nessett, D. G. Moodie, and G. Sherlock, “40 Gbit/s transmission over 202 km of standard fibre using midspan spectral inversion,” Electron. Lett. 31, 299–301 (1995).
[CrossRef]

Sibbett, W.

D. A. O. Davies, M. A. Fisher, D. J. Elton, S. D. Perrin, M. J. Adams, G. T. Kennedy, R. S. Grant, P. D. Roberts, and W. Sibbett, “Nonlinear switching in InGaAsP laser amplifier directional couplers biased at transparency,” Electron. Lett. 29, 1710–1711 (1993).
[CrossRef]

Sivco, D. L.

G. Sucha, S. R. Bolton, D. S. Chemla, D. L. Sivco, and A. Y. Cho, “Carrier relaxation in InGaAs heterostructures,” Appl. Phys. Lett. 65, 1486–1488 (1994), and references therein.
[CrossRef]

Spano, P.

A. Mecozzi, S. Scotti, A. D’Ottavi, E. Iannone, and P. Spano, “Four-wave mixing in traveling-wave semiconductor amplifiers,” IEEE J. Quantum Electron. 31, 689–699 (1995).
[CrossRef]

A. D’Ottavi, E. Iannone, A. Mecozzi, S. Scotti, P. Spano, R. Dall’Ara, G. Guekos, and J. Eckner, “4.3 terahertz four-wave mixing spectroscopy of InGaAsP semiconductor amplifiers,” Appl. Phys. Lett. 65, 2633–2635 (1994).
[CrossRef]

Stanton, C. J.

G. D. Sanders, C.-K. Sun, J. G. Fujimoto, H. K. Choi, C. A. Wang, and C. J. Stanton, “Carrier gain dynamics in InGaAs/AlGaAs strained-layer single-quantum-well diode lasers: comparison of theory and experiment,” Phys. Rev. B 50, 8539–8558 (1994).
[CrossRef]

Sucha, G.

G. Sucha, S. R. Bolton, D. S. Chemla, D. L. Sivco, and A. Y. Cho, “Carrier relaxation in InGaAs heterostructures,” Appl. Phys. Lett. 65, 1486–1488 (1994), and references therein.
[CrossRef]

Suematsu, Y.

M. Asada and Y. Suematsu, “Density matrix theory of semiconductor lasers with relaxation broadening model—gain and gain-suppression in semiconductor lasers,” IEEE J. Quantum Electron. QE-21, 434–441 (1985).
[CrossRef]

Sun, C.-K.

G. D. Sanders, C.-K. Sun, J. G. Fujimoto, H. K. Choi, C. A. Wang, and C. J. Stanton, “Carrier gain dynamics in InGaAs/AlGaAs strained-layer single-quantum-well diode lasers: comparison of theory and experiment,” Phys. Rev. B 50, 8539–8558 (1994).
[CrossRef]

Takara, H.

S. Kawanishi, T. Morioka, O. Kamatani, H. Takara, J. M. Jacob, and M. Saruwatari, “100 Gbit/s all-optical demultiplexing using four-wave mixing in a travelling wave laser diode amplifier,” Electron. Lett. 30, 981–982 (1994).
[CrossRef]

Tatham, M. C.

A. D. Ellis, M. C. Tatham, D. A. O. Davies, D. Nessett, D. G. Moodie, and G. Sherlock, “40 Gbit/s transmission over 202 km of standard fibre using midspan spectral inversion,” Electron. Lett. 31, 299–301 (1995).
[CrossRef]

Tauc, J.

Z. Vardeny and J. Tauc, “Picosecond coherence coupling in the pump and probe technique,” Opt. Commun. 39, 396–400 (1981).
[CrossRef]

Tomita, A.

Y. Nambu and A. Tomita, “Spectral holeburning and carrier-heating effect on the transient optical nonlinearity of highly carrier-injected semiconductors,” IEEE J. Quantum Electron. 30, 1981–1994 (1994).
[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, 606–609 (1991).
[CrossRef]

Uskov, A.

A. Uskov, J. Mørk, and J. Mark, “Wave mixing in semiconductor laser amplifiers due to carrier heating and spectral-hole burning,” IEEE J. Quantum Electron. 30, 1769–1781 (1994).
[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, 606–609 (1991).
[CrossRef]

Van Stryland, E. W.

M. Sheik-Bahae and E. W. Van Stryland, “Ultrafast nonlinearities in semiconductor laser amplifiers,” Phys. Rev. B 50, 14171–14178 (1994).
[CrossRef]

D. C. Hutchings and E. W. Van Stryland, “Nondegenerate two-photon absorption in zinc blende semiconductors,” J. Opt. Soc. Am. B 9, 2065–2074 (1992).
[CrossRef]

Vardeny, Z.

Z. Vardeny and J. Tauc, “Picosecond coherence coupling in the pump and probe technique,” Opt. Commun. 39, 396–400 (1981).
[CrossRef]

Wang, C. A.

G. D. Sanders, C.-K. Sun, J. G. Fujimoto, H. K. Choi, C. A. Wang, and C. J. Stanton, “Carrier gain dynamics in InGaAs/AlGaAs strained-layer single-quantum-well diode lasers: comparison of theory and experiment,” Phys. Rev. B 50, 8539–8558 (1994).
[CrossRef]

Willatzen, M.

M. Willatzen, J. Mark, J. Mørk, and C. P. Seltzer, “Carrier temperature and spectral holeburning dynamics in InGaAsP quantum well laser amplifiers,” Appl. Phys. Lett. 64, 143–145 (1994).
[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, 606–609 (1991).
[CrossRef]

Wright, E. M.

Appl. Phys. Lett. (10)

G. Sucha, S. R. Bolton, D. S. Chemla, D. L. Sivco, and A. Y. Cho, “Carrier relaxation in InGaAs heterostructures,” Appl. Phys. Lett. 65, 1486–1488 (1994), and references therein.
[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. Mark and J. Mørk, “Subpicosecond gain dynamics in InGaAsP optical amplifiers: experiment and theory,” Appl. Phys. Lett. 61, 2281–2283 (1992).
[CrossRef]

C. T. Hultgren, D. J. Dougherty, and E. P. Ippen, “Above- and below-band femtosecond nonlinearities in active AlGaAs waveguides,” Appl. Phys. Lett. 61, 2767–2769 (1992).
[CrossRef]

J. Mørk and A. Mecozzi, “Response function for gain and refractive index dynamics in active semiconductor waveguides,” Appl. Phys. Lett. 65, 1736–1738 (1994).
[CrossRef]

J. Mørk, J. Mark, and C. P. Seltzer, “Carrier heating in InGaAsP laser amplifiers due to two-photon absorption,” Appl. Phys. Lett. 64, 2206–2208 (1994).
[CrossRef]

A. D’Ottavi, E. Iannone, A. Mecozzi, S. Scotti, P. Spano, R. Dall’Ara, G. Guekos, and J. Eckner, “4.3 terahertz four-wave mixing spectroscopy of InGaAsP semiconductor amplifiers,” Appl. Phys. Lett. 65, 2633–2635 (1994).
[CrossRef]

A. Girndt, A. Knorr, M. Hofmann, and S. W. Koch, “Theoretical analysis of ultrafast pump–probe experiments in semiconductor amplifiers,” Appl. Phys. Lett. 66, 550–552 (1995).
[CrossRef]

M. Willatzen, J. Mark, J. Mørk, and C. P. Seltzer, “Carrier temperature and spectral holeburning dynamics in InGaAsP quantum well laser amplifiers,” Appl. Phys. Lett. 64, 143–145 (1994).
[CrossRef]

J. Mørk, A. Mecozzi, and C. Hultgren, “Spectral effects in short pulse pump–probe measurements,” Appl. Phys. Lett. 68, 449–451 (1996).
[CrossRef]

Electron. Lett. (3)

D. A. O. Davies, M. A. Fisher, D. J. Elton, S. D. Perrin, M. J. Adams, G. T. Kennedy, R. S. Grant, P. D. Roberts, and W. Sibbett, “Nonlinear switching in InGaAsP laser amplifier directional couplers biased at transparency,” Electron. Lett. 29, 1710–1711 (1993).
[CrossRef]

S. Kawanishi, T. Morioka, O. Kamatani, H. Takara, J. M. Jacob, and M. Saruwatari, “100 Gbit/s all-optical demultiplexing using four-wave mixing in a travelling wave laser diode amplifier,” Electron. Lett. 30, 981–982 (1994).
[CrossRef]

A. D. Ellis, M. C. Tatham, D. A. O. Davies, D. Nessett, D. G. Moodie, and G. Sherlock, “40 Gbit/s transmission over 202 km of standard fibre using midspan spectral inversion,” Electron. Lett. 31, 299–301 (1995).
[CrossRef]

IEEE J. Quantum Electron. (6)

M. Y. Hong, Y. H. Chang, A. Dienes, J. P. Heritage, and P. J. Delfyett, “Subpicosecond pulse amplification in semiconductor laser amplifiers: theory and experiment,” IEEE J. Quantum Electron. 30, 1122–1131 (1994).
[CrossRef]

M. Asada and Y. Suematsu, “Density matrix theory of semiconductor lasers with relaxation broadening model—gain and gain-suppression in semiconductor lasers,” IEEE J. Quantum Electron. QE-21, 434–441 (1985).
[CrossRef]

D. R. Hjelme and A. R. Mickelson, “Gain nonlinearities due to carrier density dependent dispersion in semiconductor lasers,” IEEE J. Quantum Electron. 25, 1625–1631 (1989).
[CrossRef]

A. Uskov, J. Mørk, and J. Mark, “Wave mixing in semiconductor laser amplifiers due to carrier heating and spectral-hole burning,” IEEE J. Quantum Electron. 30, 1769–1781 (1994).
[CrossRef]

Y. Nambu and A. Tomita, “Spectral holeburning and carrier-heating effect on the transient optical nonlinearity of highly carrier-injected semiconductors,” IEEE J. Quantum Electron. 30, 1981–1994 (1994).
[CrossRef]

A. Mecozzi, S. Scotti, A. D’Ottavi, E. Iannone, and P. Spano, “Four-wave mixing in traveling-wave semiconductor amplifiers,” IEEE J. Quantum Electron. 31, 689–699 (1995).
[CrossRef]

IEEE Photon. Technol. Lett. (1)

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, 606–609 (1991).
[CrossRef]

J. Lightwave Technol. (1)

E. Iannone and R. Sabella, “Performance evaluation of an optical multi-carrier network using wavelength converters based on FWM in semiconductor optical amplifiers,” J. Lightwave Technol. 13, 312–324 (1955), and references therein.
[CrossRef]

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

Jpn. J. Appl. Phys. (1)

N. Ogasawara and R. Ito, “Longitudinal mode competition and asymmetric gain saturation in semiconductor injection lasers. II. Theory,” Jpn. J. Appl. Phys. 27, 615–626 (1988).
[CrossRef]

Opt. Commun. (2)

K. L. Hall, G. Lenz, A. M. Darwish, and E. P. Ippen, “Subpicosecond gain and index nonlinearities in InGaAsP diode lasers,” Opt. Commun. 111, 589–612 (1994).
[CrossRef]

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[CrossRef]

Opt. Lett. (2)

Phys. Rev. A (1)

H. Haug and S. W. Koch, “Semiconductor laser theory with many-body effects,” Phys. Rev. A 39, 1887–1898 (1989).
[CrossRef] [PubMed]

Phys. Rev. B (3)

R. Binder, D. Scott, A. E. Paul, M. Lindberg, K. Henneberger, and S. W. Koch, “Carrier–carrier scattering and optical dephasing in highly excited semiconductors,” Phys. Rev. B 45, 1107–1115 (1992).
[CrossRef]

M. Sheik-Bahae and E. W. Van Stryland, “Ultrafast nonlinearities in semiconductor laser amplifiers,” Phys. Rev. B 50, 14171–14178 (1994).
[CrossRef]

G. D. Sanders, C.-K. Sun, J. G. Fujimoto, H. K. Choi, C. A. Wang, and C. J. Stanton, “Carrier gain dynamics in InGaAs/AlGaAs strained-layer single-quantum-well diode lasers: comparison of theory and experiment,” Phys. Rev. B 50, 8539–8558 (1994).
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G. P. Agrawal and N. K. Dutta, Long-Wavelength Semiconductor Lasers (Van Nostrand Reinhold, New York, 1986).
[CrossRef]

H. Haug and S. W. Koch, Quantum Theory of the Optical and Electronic Properties of Semiconductors (World Scientific, Singapore, 1990).
[CrossRef]

W. W. Chow, S. W. Koch, and M. Sargent, Semiconductor-Laser Physics (Springer-Verlag, Berlin, 1994).
[CrossRef]

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Figures (4)

Fig. 1
Fig. 1

Gain spectrum for a carrier density of N = 1.0 × 1024 m−3. The numbers indicate the different spectral positions for which the pump–probe dynamics are investigated in Figs. 24.

Fig. 2
Fig. 2

Different contributions to [(a) and (b)] the relative probe transmission change and [(c) and (d)] the probe phase change at spectral position 4 in Fig. 1. N, carrier density; CH, carrier heating; SHB, spectral hole burning; TPA, two-photon absorption; TOT, sum of contributions. The device length is L = 300 µm. The relative magnitude of the direct contributions and the spectral artifact can be seen from Figs. 3(a) (transmission) and 3(b) (phase), showing the total responses with and without the spectral artifact.

Fig. 3
Fig. 3

(a) Relative probe transmission change and (b) probe phase change at the different spectral positions marked in Fig. 1. Dashed curves, neglecting the spectral artifact; solid curves, including the spectral artifact. For curve 4 the different contributions to the total response are shown in Fig. 2.

Fig. 4
Fig. 4

Same as Fig. 3(a), but for a waveguide length of L = 600 µm.

Equations (113)

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ρ ˙ α , k ( t ) = γ 1 α [ ρ α , k ( t ) f α , k ( t ) ] γ h α [ ρ α , k ( t ) f α , k L ( t ) ] γ s [ ρ α , k ( t ) f α , k eq ] i ћ [ d k * ρ c v , k ( t ) d k ρ v c , k ( t ) ] × E ( z , t ) + Λ α , k ,     α = c , v ,
ρ ˙ c v , k ( t ) = ( i ω k γ 2 ) ρ c v , k ( t ) i ћ d k [ ρ c , k ( t ) + ρ v , k ( t ) 1 ] × E ( z , t ) ,
ρ c v , k = ρ c v , k * .
P ( z , t ) = 1 V k d k ( ρ c v , k + ρ v c , k ) ,
ћ ω k = E g + ћ 2 k 2 / 2 m r ,
E g = E g 0 + Δ E SX + Δ E CH ,
E ( z , t ) E ( z , t ) + 1 d k k k V | k k | ρ c v , k ,
ρ α , k ( t ) = ρ α , k ( 0 ) + Δ ρ α , k ( t ) ,
E ( z ,   t ) = E ( z , t ) exp ( i ω 0 t + i k 0 z ) + c.c . ,
ρ c v ( t ) = σ c v ( t ) exp ( i ω 0 t + i k 0 z ) ,
Δ ρ ˙ α , k ( t ) = γ 1 α [ Δ ρ α , k ( t ) Δ f α , k ( t ) ] γ h α [ Δ ρ α , k ( t ) Δ f α , k L ( t ) ] γ s Δ ρ α , k ( t ) i ћ [ d k * σ c v , k ( t ) E * ( z , t ) d k σ c v , k * ( t ) E ( z , t ) ] , α = c , v ,
σ ˙ c v , k ( t ) = [ i ( ω k ω 0 ) γ 2 ] σ c v , k ( t ) i ћ d k [ f c , k ( 0 ) + f v , k ( 0 ) 1 ] E ( z , t ) i ћ d k [ Δ ρ c , k ( t ) + Δ ρ v , k ( t ) 1 ] E ( z , t ) .
y ( Ω ) = y ( t ) exp ( i Ω t ) d t ,
y ( t ) = y ( Ω ) exp ( i Ω t ) d Ω 2 π ,
i Ω Δ ρ α , k ( Ω ) = γ 1 α [ Δ ρ α , k ( Ω ) Δ f α , k ( Ω ) ] γ h α [ Δ ρ α , k ( Ω ) Δ f α , k L ( Ω ) ] γ s Δ ρ α , k ( Ω ) = i ћ [ d k * σ c v , k ( Ω ) E * ( z , Ω ) d k σ c v , k * ( Ω ) E ( z , Ω ) ] ,
i Ω σ c v , k ( Ω ) = [ i ( ω k ω 0 ) γ 2 ] σ c v , k ( Ω ) i ћ d k [ f c , k ( 0 ) + f v , k ( 0 ) 1 ] E ( z , Ω ) i ћ d k [ Δ ρ c , k ( Ω ) + Δ ρ v , k ( Ω ) ] E ( z , Ω ) .
x ( Ω ) y ( Ω ) = x ( Ω Ω ) y ( Ω ) d Ω 2 π .
σ c v , k ( Ω ) = d k ћ L k ( ω 0 + Ω ) f c , k ( 0 ) + f v , k ( 0 ) 1 × E ( z , Ω ) + [ Δ ρ c , k ( Ω ) + Δ ρ v , k ( Ω ) ] E ( z , Ω ) ,
L k ( ω ) = 1 ω ω k + i γ 2 .
Δ ρ α , k ( Ω ) = 1 i Ω + γ 1 α γ 1 α Δ f α , k ( Ω ) i | d k | 2 ћ 2 f c , k ( 0 ) + f v , k ( 0 ) 1 × [ L k ( ω 0 + Ω ) E ( z , Ω ) E * ( z , Ω ) L k * ( ω 0 Ω ) E * ( z , Ω ) E ( z , Ω ) ] .
Δ ρ α , k ( Ω ) = 1 i Ω + γ 1 α γ 1 α Δ f α , k ( Ω ) 2 | d k | 2 σ S ћ 2 γ 2 | L k ( ω 0 ) | 2 × f c , k ( 0 ) + f v , k ( 0 ) 1 S ( z , Ω ) ,
S ( z , t ) = 2 0 n n g ћ ω 0 | E ( z , t ) | 2 = σ S | E ( z , t ) | 2 ,
S ( z , Ω ) = σ S E ( z , Ω ) E * ( z , Ω ) .
Δ f α , k = f α , k N Δ N + f α , k T α Δ T α ,
Δ f α , k L = f α , k N Δ N .
Δ N ( Ω ) = v g g 0 i Ω + γ s S ( z , Ω ) ,
Δ T α ( Ω ) = v g g 0 C α i Ω + γ h α S ( z , Ω ) .
C α = U α T α 1 U α N E α + σ α N g 0 ћ ω 0 ,     α = c , v ,
P ( z , t ) = P ( z , t ) exp ( i ω 0 t + i k 0 z ) + c .c .
P ( z , t ) = P ( 1 ) ( z , t ) + P ( 3 ) ( z , t ) .
P ( 1 ) ( z , Ω ) = 0 χ ( 1 ) ( ω 0 + Ω ) E ( z , Ω ) ,
χ ( 1 ) ( ω 0 + Ω ) = 1 V ћ 0 k L k ( ω 0 + Ω ) | d k | 2 f c , k ( 0 ) + f v , k ( 0 ) 1 ,
P ( 3 ) ( z , Ω ) = 0 χ ( 3 ) ( Ω ) E ( z , Ω ) ,
χ ( 3 ) ( Ω ) = χ N ( 3 ) ( Ω ) + χ T ( 3 ) ( Ω ) + χ SHB ( 3 ) ( Ω ) ,
χ N ( 3 ) ( Ω ) = 1 V ћ 0 k L k ( ω 0 ) | d k | 2 γ 1 i Ω + γ 1 × f c , k ( 0 ) N + f v , k ( 0 ) N Δ N ( Ω ) ,
χ T α ( 3 ) ( Ω ) = 1 V ћ 0 k L k ( ω 0 ) | d k | 2 γ 1 i Ω + γ 1 × f c , k ( 0 ) T c Δ T c ( Ω ) + f v , k ( 0 ) T v Δ T v ( Ω ) ,
χ SHB ( 3 ) ( Ω ) = 1 V ћ 0 k L k ( ω 0 ) 4 | d k | 4 γ 2 σ S ћ 2 × f c , k ( 0 ) + f v , k ( 0 ) 1 ( ω k ω 0 ) 2 + γ 2 2 1 i Ω + γ 1 S ( z , Ω ) .
2 E z 2 μ 0 σ 0 E t μ 0 0 2 E t 2 = μ 0 2 t 2 ( Γ P + P b ) .
2 i k 0 E ( z , Ω ) z = i μ 0 ( ω 0 + Ω ) σ 0 E ( z , Ω ) + k 0 2 ( ω 0 + Ω ) 2 c 2 { [ 1 + χ b ( ω 0 + Ω ) ] E ( z , Ω ) + Γ χ ( 1 ) ( ω 0 + Ω ) E ( z , Ω ) + Γ χ ( 3 ) ( Ω ) E ( z , Ω ) } .
P b ( z , Ω ) = 0 χ b ( ω 0 + Ω ) E ( z , Ω ) ,
n 2 ( ω ) = Re [ χ b ( ω ) ] + 1 ,
k 2 ( ω ) = ω 2 n 2 ( ω ) c 2 = ω 2 c 2 { 1 + Re [ χ b ( ω ) ] } .
E ( z , Ω ) z = ( ω 0 + Ω ) μ 0 σ ( ω 0 + Ω ) 2 k 0 E ( z , Ω ) + i 2 k 0 [ k 2 ( ω 0 + Ω ) k 0 2 ] E ( z , Ω ) + i 2 k 0 Γ ( ω 0 + Ω ) 2 c 2 [ χ ( 1 ) ( ω 0 + Ω ) E ( z , Ω ) + χ ( 3 ) ( Ω ) E ( z , Ω ] ,
σ ( ω ) = σ 0 + ω 0   Im [ χ b ( ω ) ] .
α int = μ 0 ω σ ( ω ) k 0 μ 0 ω 0 σ ( ω 0 ) k 0 ,
g ( ω ) = ω 2 k 0 c 2 Im [ χ ( 1 ) ( ω ) ] ,
κ ( ω ) = 1 2 k 0 [ k 2 ( ω ) k 2 ( ω 0 ) ] k ( ω ) k ( ω 0 ) ,
ξ ( ω ) = Γ g ( ω ) α int .
E ( z , Ω ) z = 1 2 ξ ( ω 0 + Ω ) + i κ ( ω 0 + Ω ) E ( z , Ω ) + i Γ ω 2 2 k 0 c 2 χ ( 3 ) ( Ω ) E ( z , Ω ) .
E ( z , t ) z = L ( t ) E ( z , t ) + R τ ( z , t ) ,
L = 1 2 ξ ω 0 + i t + i κ ω 0 + i t .
f ω 0 + i t = n = 0 f ( n ) ( ω 0 ) n ! i t n ,
R τ ( z , t ) = Γ i 2 k 0 c 2 ω 2 χ ( 3 ) ( Ω ) E ( z , Ω ) exp ( i Ω t ) d Ω i Γ ω 0 2 n 0 c 2 χ ( 3 ) ( t ) E ( z , t ) = h ( t t ) | E ( z , t ) | 2 d t   E ( z , t ) .
i Γ ω 0 2 n 0 c χ ( 3 ) ( Ω ) = 1 σ S h ( Ω ) S ( z , Ω ) = 1 σ S [ h N ( Ω ) + h T ( Ω ) + h SHB ( Ω ) ] S ( z , Ω ) ,
h N ( Ω ) = 1 2 Γ σ S v g g 0 g 0 N ( 1 i α N ) × γ 1 ( i Ω + γ 1 ) ( i Ω + γ s ) ,
h N ( t ) = 1 2 Γ σ S v g g 0 g 0 N ( 1 i α N ) × [ exp ( t / τ s ) exp ( t / τ 1 ) ] u ( t ) ,
α N = Re [ χ ( 1 ) ] / N Im [ χ ( 1 ) ] / N .
h T ( Ω ) = 1 2 Γ σ S T c γ h c g 0 ( 1 i α T c ) × γ 1 ( i Ω + γ h c ) ( i Ω + γ 1 ) ,
h T ( t ) = 1 2 Γ σ S T c γ h c g 0 ( 1 i α T c ) [ exp ( t / τ h c ) exp ( t / τ 1 ) ] u ( t ) .
T c = v g τ h c C c g 0 T .
h SHB ( Ω ) = 1 2 Γ σ S SHB g 0 γ 1 ( 1 i α SHB ) 1 i Ω + γ 1 ;
h SHB ( t ) = 1 2 Γ σ S SHB g 0 γ 1 ( 1 i α SHB ) exp ( t / τ 1 ) u ( t ) .
SHB = 2 τ 1 τ 2 ћ ω 0 | d k | 2 ћ 2 0 n n g ,
h TPA ( Ω ) = σ S v g ћ ω 0 β 2 ( 1 i α 2 ) ,
h TPA ( t ) = σ S v g ћ ω 0 β 2 ( 1 i α 2 ) δ ( t ) ,
α 2 = 4 π λ n 2 β 2 ,
E ( z , t ) = E 0 ( z , t + τ ) + E 1 ( z , t ) ,
E 1 ( z , t ) = E 1 ( z , t ) + Δ E 1 ( z , t ) ,
E 1 ( z , t ) z = L 1 E 1 ( z , t ) .
L j = 1 2 ξ j ω 0 + i t + i κ j ω 0 + i t ,     j = 0,1 .
Δ E 1 ( z , t ) z = L 1 Δ E 1 ( z , t ) + R τ ( z , t ) ,
R τ ( z , t ) = d t h ( t + τ t ) | E 0 ( z , t ) | 2 E 1 ( z , t ) .
E 0 ( z , t ) z = L 0 E 0 ( z , t ) .
E j ( z , t ) = exp ( z L j ) E j ( 0 , t ) .
Δ E 1 ( z , t ) = 0 z d z exp [ ( z z ) L 1 ] R τ ( z , t ) .
I ( τ ) = 0 z d t E 1 * ( L , t ) Δ E 1 ( L , t ) ,
I ( τ ) T = Δ T ( τ ) 2 T + i Δ ϕ 1 ¯ ( τ ) ,
T = | E 1 ( L , t ) | 2 d t ,
Δ ϕ 1 ¯ ( τ ) = Δ ϕ 1 ( L , t ) | E 1 ( L , t ) | 2 d t | E 1 ( L , t ) | 2 d t .
I ( τ ) = 0 L d z d t   E 1 * ( L , t ) { exp [ ( L z ) L 1 ] R τ ( z , t ) } .
I ( τ ) = 0 L d z d t exp [ ( L z ) L 1 ] E 1 ( L , t ) * R τ ( z , t ) ,
L 1 = 1 2 ξ 1 ω 0 + i t i κ 1 ω 0 + i t .
I ( τ ) = 0 L d z d t exp ( L z ) L 1 + L L 1 E 1 ( 0 , t ) * × [ exp ( z L 1 ) E 1 ( 0 , t ) ] d t h ( t + τ t ) × | exp ( z L 0 ) E 0 ( 0 , t ) | 2 .
exp [ ( L z ) L 1 ] exp ( L L 1 ) = exp [ ( L z ) L 1 + L L 1 ] .
ξ ˆ j = ξ j ω 0 + i t ,
κ ˆ j = κ j ω 0 + i t ,
I ( τ ) = 0 L d z d t exp ξ ˆ 1 2 ( 2 L z ) + i z κ ˆ 1 E 1 ( 0 , t ) * × exp z 2 ξ ˆ 1 + i z κ ˆ 1 E 1 ( 0 , t ) × d t h ( t + τ t ) × exp z 2 ξ ˆ 0 + i z κ ˆ 0 E 0 ( 0 , t ) 2 .
ξ ˆ j = ξ j + ξ j i t ,
κ ˆ j = κ j + κ j i t ,
f a ( t ) = exp a t f ( t ) ,
f a ( t ) = d Ω 2 π f ( Ω ) exp [ i Ω ( t + a ) ] = f ( t + a ) .
I ( τ ) = exp ( ξ 1 L ) 0 L d z   exp ( ξ 0 z ) d t × E 1 * 0 , t + i ξ 1 2 ( 2 L z ) κ 1 z × E 1 0 , t + i ξ 1 2 z κ 1 z d t h ( t + τ t ) × E 0 0 , t + i ξ 0 2 z κ 0 z 2 .
I ( τ ) = exp ( ξ 1 L ) 0 L d z exp ( ξ 0 z ) d t × E 1 * 0 , t + i ξ 1 2 ( 2 L z ) × E 1 0 , t + i ξ 1 2 z d t h ( t + τ t ) × E 0 0 , t + i ξ 0 2 z Δ κ z 2 ,
Δ κ = κ 0 κ 1
E j ( 0 , t ) = [ A j ( t ) ] 1 + i β ,
E j ( 0 , t ) = E j ( 0,0 ) exp ( 1 + i β ) t 2 2 τ 0 2 .
I ( τ ) = d t   H ( t ) G ( 2 ) ( τ t ) ,
H ( t ) = exp ( ξ 1 L ) 0 L d z   exp ( ξ 0 z ) 1 + i ξ 1 2 ( L z ) + β ξ 1 2 L β ξ 0 2 z Δ κ z t h ( t )
G ( 2 ) ( t ) = d t [ A 0 ( t ) ] 2 [ A 1 ( t t ) ] 2 .
H ( t ) = exp ( ξ 0 L ) ξ 0 exp ( ξ 0 L ) 1 + ( i + β ) ξ 0 2 ξ 0 × [ exp ( ξ 0 L ) 1 ξ 0 L ] t + Δ κ ξ 0 × [ ( 1 ξ 0 L ) exp ( ξ 0 L ) 1 ] t h ( t ) .
ρ T = α ξ 0 ' L 2 τ 0 exp ( ξ 0 L ) 1 ξ 0 L ξ 0 L [ exp ( ξ 0 L ) 1 ] α ξ 0 ' L 4 τ 0 1 1 6 ξ 0 L + O [ ( ξ 0 L ) 3 ] .
I ( τ ) = exp ( ξ 1 L ) 0 L d z   exp ( ξ 0 z ) d t   h ( t ) d t × E 1 * 0 , t + t τ + i ξ 1 2 ( 2 L z ) × E 1 0 , t + t τ + i ξ 1 2 z × E 0 0 , t + i ξ 0 2 z Δ κ z 2 .
E 1 0 , s + i ξ 1 2 ζ [ A 1 ( s ) ] 1 + i β + ( 1 + i β ) i ξ 1 ζ 2 [ A 1 ( s ) ] i β s A 1 ( s ) ,
E 0 0 , t + i ξ 0 2 z Δ κ z 2 1 β ξ 0 2 z + Δ κ z t [ A 0 ( t ) ] 2 ,
I ( τ ) = exp ( ξ 1 L ) 0 L d z   exp ( ξ 0 z ) d t   h ( t ) d t × 1 i ξ 1 2 ( L z ) + β ξ 1 2 L t [ A 1 ( t + t τ ) ] 2 × 1 β ξ 0 2 z + Δ κ z t [ A 0 ( t ) ] 2 .
I ( τ ) = exp ( ξ 1 L ) 0 L d z   exp ( ξ 0 z ) d t   h ( t ) d t [ A 0 ( t ) ] 2 × 1 i ξ 1 2 ( L z ) + β ξ 1 2 L β ξ 0 2 z Δ κ z t × [ A 1 ( t + t τ ) ] 2 .
I ( τ ) = exp ( ξ 1 L ) 0 L d z   exp ( ξ 0 z ) d t   H ( t ) d t × [ A 0 ( t ) ] 2 [ A 1 ( t + t τ ) ] 2 ,
H ( t ) = 1 + i ξ 1 2 ( L z ) + β ξ 1 2 L β ξ 0 2 z Δ κ z t h ( t ) .
I ( τ ) = exp ( ξ 0 L ) d z   h ( t ) 0 L d z   exp ( ξ 0 z ) × G ( 2 ) ( τ t Δ κ z ) .
I Δ κ = 0 = exp ( ξ 0 L ) exp ( ξ 0 L ) 1 ξ 0 d t   h ( t ) G ( 2 ) ( τ t ) ,
I ( τ ) = d t   I Δ κ ' = 0 ( t ) W ( τ t ) = I Δ κ = 0 W ( τ ) .
W ( t ) = N   exp ( ξ 0 t / Δ κ ) [ u ( t ) u ( τ WO t ) ] ,
W ( t ) d t = 1 .

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