Abstract

The generalized form of a nonlinear equation modeling the evolution of femtosecond pulses in a wide range of laser media is derived. Together with auxiliary equations, it describes both amplitude and phase changes owing to a variety of linear and nonlinear effects. These effects include gain saturation, more than one kind of self-phase modulation, and both gain and phase dispersions. The model is applied to previous experiments of ultrashort-pulse amplification in a dye-laser amplifier and is found to describe quantitatively the observed strong spectral changes. We show that the spectral changes are due both to gain dispersion and to the interplay of various dispersive and nonlinear effects.

© 1996 Optical Society of America

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  1. J. T. Verdeyen, Laser Electronics, 3d ed. (Prentice-HallEnglewood Cliffs, N.J., 1995), Chap. 10.
  2. C. V. Shank, “Physics of dye lasers,” Rev. Mod. Phys. 47, 649–657 (1975).
    [Crossref]
  3. A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986), Chap. 10.1.
  4. G. P. Agrawal and N. A. Olsson, “Self-phase modulation and spectral broadening of optical pulses in semiconductor laser amplifiers,” IEEE J. Quantum Electron. 25, 2297–2306 (1989).
    [Crossref]
  5. O. E. Martinez, R. L. Fork, and J. P. Gordon, “Theory of passively mode-locked lasers for the case of nonlinear complex propagation coefficients,” J. Opt. Soc. Am. B 2, 753–760 (1985).
    [Crossref]
  6. A. Dienes, L. W. Carr, and M. Y. Hong, “Compressible frequency chirping due to slow and fast self-phase modulation of ultrashort pulses in high-gain dye laser amplifiers,” IEEE J. Quantum Electron. 27, 1214–1220 (1991).
    [Crossref]
  7. R. Miranda, G. R. Jacobovitz, C. H. Brito-Cruz, and M. A. F. Scarparo, “Positive and negative chirping of laser pulses in a saturable absorber,” Opt. Lett. 11, 224–226 (1986).
    [Crossref]
  8. A. Dienes, J. P. Heritage, M. Y. Hong, and Y. H. Chang, “Time and spectral domain evolution of subpicosecond pulses in semiconductor laser amplifiers,” Opt. Lett. 17, 1602–1604 (1992).
    [Crossref] [PubMed]
  9. 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]
  10. P. J. Delfyett, A. Dienes, J. P. Heritage, M. Y. Hong, and Y. H. Chang, “Femtosecond hybrid mode-locked semiconductor lasers and amplifier dynamics,” Appl. Phys. B 58, 183–195 (1994).
    [Crossref]
  11. D. Nickel, D. Kühlke, and D. von der Linde, “Multipass dyecell amplifier for high-repetition-rate femtosecond optical pulses,” Opt. Lett. 14, 36–38 (1989).
    [Crossref] [PubMed]
  12. W. H. Knox, M. C. Downer, R. L. Fork, and C. V. Shank, “Amplified femtosecond optical pulses and continuum generation at a 5-kHz repetition rate,” Opt. Lett. 9, 552–555 (1984).
    [Crossref] [PubMed]
  13. G. P. Agrawal, Nonlinear Fiber Optics (Academic, San Diego, Calif., 1989), Chap. 2, pp. 44–48.
  14. Self-focusing owing to the nonlinear index n2 is responsible for Kerr-lens mode locking in solid-state laser oscillators; see, for example, H. A. House, J. G. Fujimoto, and E. P. Ippen, “Analytic theory of additive pulse and Kerr lens mode-locking,” IEEE J. Quantum Electron. 28, 2086–2096 (1992). However, these effects are small, typically only a few-percent change in beam area, unless pulse durations approach 10 fs.
    [Crossref]
  15. P. K. A. Wai, C. R. Menyuk, Y. C. Lee, and H. H. Chen, “Nonlinear pulse propagation in the neighborhood of the zero dispersion wavelength of monomode optical fibers,” Opt. Lett. 11, 464–467 (1986).
    [Crossref] [PubMed]
  16. K. Vahala, L. C. Chiu, S. Margalit, and A. Yariv, “On the linewidth enhancement factor α in semiconductor lasers,” Appl. Phys. Lett. 42, 631–633 (1983).
    [Crossref]
  17. M. Osinski and J. Burrus, “Linewidth broadening factor in semiconductor lasers—an overview,” IEEE J. Quantum Electron. QE-23, 9–29 (1987).
    [Crossref]
  18. M. S. Stix, M. P. Kesler, and E. P. Ippen, “Observations of subpicosecond dynamics in GaAlAs laser diodes,” Appl. Phys. Lett. 48, 1722–1724 (1986).
    [Crossref]
  19. P. J. Delfyett, Y. Silberberg, and G. A. Alphonse, “Hot-carrier thermalization induced self-phase modulation in semiconductor traveling wave amplifiers,” Appl. Phys. Lett. 59, 10–12 (1991).
    [Crossref]
  20. G. P. Agrawal and C. M. Bowden, “Concept of linewidth enhancement factor in semiconductor lasers: its usefulness and limitations,” IEEE Photon. Technol. Lett. 5, 640–643 (1993).
    [Crossref]
  21. I. P. Christov, M. M. Murnane, H. C. Kapteyn, J. Zhou, and C. P. Huang, “Fourth-order dispersion-limited solitary pulses,” Opt. Lett. 18, 1465–1467 (1994).
    [Crossref]
  22. J. Mørk, J. Mark, and C. P. Seltzer, “Subpicosecond gain dynamics in InGaAsP optical amplifiers: experiment and theory,” Appl. Phys. Lett. 64, 2206–2209 (1994).
  23. G. Boyer, M. Franco, J. P. Chambaret, A. Migus, A. Antonetti, P. Georges, F. Salin, and A. Brun, “Generation of 0.6 mJ pulses of 16 fs duration through high-repetition rate amplification of self-phase modulated pulses,” Appl. Phys. Letters,  53, 823–825 (1988); R. W. Schoenline, Lawrence Berkely National Laboratory, Berkely, Calif. 94720 (personal communication).
    [Crossref]
  24. S. H. Jiang and L. W. Casperson, “Ultrashort pulse propagation in dye laser amplifiers,” J. Appl. Phys. 73, 530–540 (1993).
    [Crossref]
  25. W. H. Knox, “Femtosecond optical amplification,” IEEE J. Quantum Electron. 24, 358–397 (1988).
    [Crossref]
  26. O. Teschke, A. J. Dienes, and J. R. Whinnery, “Theory and operation of high-power cw and long-pulse dye lasers,” IEEE J. Quantum Electron. QE-12, 383–395 (1976).
    [Crossref]
  27. A. Dienes and L. W. Carr, “Effect of pulse shape on self-phase-modulation-induced chirping of optical pulses in diode laser amplifiers,” J. Appl. Phys. 69, 1766–1768 (1991).
    [Crossref]
  28. Gain medium; F. Salin, G. Le Saux, P. Georges, A. Brun, C. Bagnall, and J. Zarzycki, “Efficient tunable solid-state laser near 630 nm using sulforhodamine 640-doped silica gel,” Opt. Lett. 14, 785–788 (1989). Rhodamine 101 is very similar to Rhodamine 640 with fluorescence peak shifted to 606 nm. Saturable-absorber absorption curve from Fisher Scientific Corporation.
    [Crossref] [PubMed]
  29. J. A. Valdmanis and R. L. Fork, “Design considerations for femtosecond pulse laser balancing self-phase modulation, group velocity dispersion, saturable absorption and saturable gain,” IEEE J. Quantum Electron. QE-22, 112–118 (1986).
    [Crossref]
  30. C. Chu, J. P. Heritage, R. S. Grant, K. X. Liu, A. Dienes, W. E. White, and A. Sullivan, “Direct measurement of the spectral phase of femtosecond pulses,” Opt. Lett. 20, 904–906 (1995).
    [Crossref] [PubMed]
  31. R. S. Grant and W. Sibbett, “Observations of ultrafast nonlinear refraction in an InGaAsP optical amplifier,” Appl. Phys. Lett. 58, 1119–1121 (1991).
    [Crossref]
  32. K. Wynne, G. D. Reid, and R. M. Hochstrasser, “Regenerative amplification of 30-fs pulses in Ti:sapphire at 5 kHz,” Opt. Lett. 19, 895–897 (1994).
    [Crossref] [PubMed]
  33. J. Zhou, C. P. Huang, M. M. Murnane, and H. C. Kapteyn, “Amplification of 26-fs, 2-TW pulses near the gain narrowing limit in Ti:sapphire,” Opt. Lett. 20, 64–66 (1995).
    [Crossref] [PubMed]
  34. I. P. Christov, H. C. Kapteyn, M. M. Murnane, C. P. Huang, and J. Zhou, “Space–time focusing of femtosecond pulses in a Ti:sapphire laser,” Opt. Lett. 20, 309–311 (1995).
    [Crossref] [PubMed]
  35. M. Y. Hong, Y. H. Chang, A. Dienes, J. P. Heritage, P. J. Defyette, Sol Dijaili, and F. Patterson, “Femtosecond self-and cross-phase modulation in semiconductor laser amplifiers,” to be submitted to IEEE .J. Quantum. Electron.

1995 (3)

1994 (5)

K. Wynne, G. D. Reid, and R. M. Hochstrasser, “Regenerative amplification of 30-fs pulses in Ti:sapphire at 5 kHz,” Opt. Lett. 19, 895–897 (1994).
[Crossref] [PubMed]

I. P. Christov, M. M. Murnane, H. C. Kapteyn, J. Zhou, and C. P. Huang, “Fourth-order dispersion-limited solitary pulses,” Opt. Lett. 18, 1465–1467 (1994).
[Crossref]

J. Mørk, J. Mark, and C. P. Seltzer, “Subpicosecond gain dynamics in InGaAsP optical amplifiers: experiment and theory,” Appl. Phys. Lett. 64, 2206–2209 (1994).

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]

P. J. Delfyett, A. Dienes, J. P. Heritage, M. Y. Hong, and Y. H. Chang, “Femtosecond hybrid mode-locked semiconductor lasers and amplifier dynamics,” Appl. Phys. B 58, 183–195 (1994).
[Crossref]

1993 (2)

G. P. Agrawal and C. M. Bowden, “Concept of linewidth enhancement factor in semiconductor lasers: its usefulness and limitations,” IEEE Photon. Technol. Lett. 5, 640–643 (1993).
[Crossref]

S. H. Jiang and L. W. Casperson, “Ultrashort pulse propagation in dye laser amplifiers,” J. Appl. Phys. 73, 530–540 (1993).
[Crossref]

1992 (2)

A. Dienes, J. P. Heritage, M. Y. Hong, and Y. H. Chang, “Time and spectral domain evolution of subpicosecond pulses in semiconductor laser amplifiers,” Opt. Lett. 17, 1602–1604 (1992).
[Crossref] [PubMed]

Self-focusing owing to the nonlinear index n2 is responsible for Kerr-lens mode locking in solid-state laser oscillators; see, for example, H. A. House, J. G. Fujimoto, and E. P. Ippen, “Analytic theory of additive pulse and Kerr lens mode-locking,” IEEE J. Quantum Electron. 28, 2086–2096 (1992). However, these effects are small, typically only a few-percent change in beam area, unless pulse durations approach 10 fs.
[Crossref]

1991 (4)

P. J. Delfyett, Y. Silberberg, and G. A. Alphonse, “Hot-carrier thermalization induced self-phase modulation in semiconductor traveling wave amplifiers,” Appl. Phys. Lett. 59, 10–12 (1991).
[Crossref]

A. Dienes, L. W. Carr, and M. Y. Hong, “Compressible frequency chirping due to slow and fast self-phase modulation of ultrashort pulses in high-gain dye laser amplifiers,” IEEE J. Quantum Electron. 27, 1214–1220 (1991).
[Crossref]

R. S. Grant and W. Sibbett, “Observations of ultrafast nonlinear refraction in an InGaAsP optical amplifier,” Appl. Phys. Lett. 58, 1119–1121 (1991).
[Crossref]

A. Dienes and L. W. Carr, “Effect of pulse shape on self-phase-modulation-induced chirping of optical pulses in diode laser amplifiers,” J. Appl. Phys. 69, 1766–1768 (1991).
[Crossref]

1989 (3)

1988 (2)

W. H. Knox, “Femtosecond optical amplification,” IEEE J. Quantum Electron. 24, 358–397 (1988).
[Crossref]

G. Boyer, M. Franco, J. P. Chambaret, A. Migus, A. Antonetti, P. Georges, F. Salin, and A. Brun, “Generation of 0.6 mJ pulses of 16 fs duration through high-repetition rate amplification of self-phase modulated pulses,” Appl. Phys. Letters,  53, 823–825 (1988); R. W. Schoenline, Lawrence Berkely National Laboratory, Berkely, Calif. 94720 (personal communication).
[Crossref]

1987 (1)

M. Osinski and J. Burrus, “Linewidth broadening factor in semiconductor lasers—an overview,” IEEE J. Quantum Electron. QE-23, 9–29 (1987).
[Crossref]

1986 (4)

M. S. Stix, M. P. Kesler, and E. P. Ippen, “Observations of subpicosecond dynamics in GaAlAs laser diodes,” Appl. Phys. Lett. 48, 1722–1724 (1986).
[Crossref]

P. K. A. Wai, C. R. Menyuk, Y. C. Lee, and H. H. Chen, “Nonlinear pulse propagation in the neighborhood of the zero dispersion wavelength of monomode optical fibers,” Opt. Lett. 11, 464–467 (1986).
[Crossref] [PubMed]

R. Miranda, G. R. Jacobovitz, C. H. Brito-Cruz, and M. A. F. Scarparo, “Positive and negative chirping of laser pulses in a saturable absorber,” Opt. Lett. 11, 224–226 (1986).
[Crossref]

J. A. Valdmanis and R. L. Fork, “Design considerations for femtosecond pulse laser balancing self-phase modulation, group velocity dispersion, saturable absorption and saturable gain,” IEEE J. Quantum Electron. QE-22, 112–118 (1986).
[Crossref]

1985 (1)

1984 (1)

1983 (1)

K. Vahala, L. C. Chiu, S. Margalit, and A. Yariv, “On the linewidth enhancement factor α in semiconductor lasers,” Appl. Phys. Lett. 42, 631–633 (1983).
[Crossref]

1976 (1)

O. Teschke, A. J. Dienes, and J. R. Whinnery, “Theory and operation of high-power cw and long-pulse dye lasers,” IEEE J. Quantum Electron. QE-12, 383–395 (1976).
[Crossref]

1975 (1)

C. V. Shank, “Physics of dye lasers,” Rev. Mod. Phys. 47, 649–657 (1975).
[Crossref]

Agrawal, G. P.

G. P. Agrawal and C. M. Bowden, “Concept of linewidth enhancement factor in semiconductor lasers: its usefulness and limitations,” IEEE Photon. Technol. Lett. 5, 640–643 (1993).
[Crossref]

G. P. Agrawal and N. A. Olsson, “Self-phase modulation and spectral broadening of optical pulses in semiconductor laser amplifiers,” IEEE J. Quantum Electron. 25, 2297–2306 (1989).
[Crossref]

G. P. Agrawal, Nonlinear Fiber Optics (Academic, San Diego, Calif., 1989), Chap. 2, pp. 44–48.

Alphonse, G. A.

P. J. Delfyett, Y. Silberberg, and G. A. Alphonse, “Hot-carrier thermalization induced self-phase modulation in semiconductor traveling wave amplifiers,” Appl. Phys. Lett. 59, 10–12 (1991).
[Crossref]

Antonetti, A.

G. Boyer, M. Franco, J. P. Chambaret, A. Migus, A. Antonetti, P. Georges, F. Salin, and A. Brun, “Generation of 0.6 mJ pulses of 16 fs duration through high-repetition rate amplification of self-phase modulated pulses,” Appl. Phys. Letters,  53, 823–825 (1988); R. W. Schoenline, Lawrence Berkely National Laboratory, Berkely, Calif. 94720 (personal communication).
[Crossref]

Bagnall, C.

Bowden, C. M.

G. P. Agrawal and C. M. Bowden, “Concept of linewidth enhancement factor in semiconductor lasers: its usefulness and limitations,” IEEE Photon. Technol. Lett. 5, 640–643 (1993).
[Crossref]

Boyer, G.

G. Boyer, M. Franco, J. P. Chambaret, A. Migus, A. Antonetti, P. Georges, F. Salin, and A. Brun, “Generation of 0.6 mJ pulses of 16 fs duration through high-repetition rate amplification of self-phase modulated pulses,” Appl. Phys. Letters,  53, 823–825 (1988); R. W. Schoenline, Lawrence Berkely National Laboratory, Berkely, Calif. 94720 (personal communication).
[Crossref]

Brito-Cruz, C. H.

Brun, A.

Gain medium; F. Salin, G. Le Saux, P. Georges, A. Brun, C. Bagnall, and J. Zarzycki, “Efficient tunable solid-state laser near 630 nm using sulforhodamine 640-doped silica gel,” Opt. Lett. 14, 785–788 (1989). Rhodamine 101 is very similar to Rhodamine 640 with fluorescence peak shifted to 606 nm. Saturable-absorber absorption curve from Fisher Scientific Corporation.
[Crossref] [PubMed]

G. Boyer, M. Franco, J. P. Chambaret, A. Migus, A. Antonetti, P. Georges, F. Salin, and A. Brun, “Generation of 0.6 mJ pulses of 16 fs duration through high-repetition rate amplification of self-phase modulated pulses,” Appl. Phys. Letters,  53, 823–825 (1988); R. W. Schoenline, Lawrence Berkely National Laboratory, Berkely, Calif. 94720 (personal communication).
[Crossref]

Burrus, J.

M. Osinski and J. Burrus, “Linewidth broadening factor in semiconductor lasers—an overview,” IEEE J. Quantum Electron. QE-23, 9–29 (1987).
[Crossref]

Carr, L. W.

A. Dienes, L. W. Carr, and M. Y. Hong, “Compressible frequency chirping due to slow and fast self-phase modulation of ultrashort pulses in high-gain dye laser amplifiers,” IEEE J. Quantum Electron. 27, 1214–1220 (1991).
[Crossref]

A. Dienes and L. W. Carr, “Effect of pulse shape on self-phase-modulation-induced chirping of optical pulses in diode laser amplifiers,” J. Appl. Phys. 69, 1766–1768 (1991).
[Crossref]

Casperson, L. W.

S. H. Jiang and L. W. Casperson, “Ultrashort pulse propagation in dye laser amplifiers,” J. Appl. Phys. 73, 530–540 (1993).
[Crossref]

Chambaret, J. P.

G. Boyer, M. Franco, J. P. Chambaret, A. Migus, A. Antonetti, P. Georges, F. Salin, and A. Brun, “Generation of 0.6 mJ pulses of 16 fs duration through high-repetition rate amplification of self-phase modulated pulses,” Appl. Phys. Letters,  53, 823–825 (1988); R. W. Schoenline, Lawrence Berkely National Laboratory, Berkely, Calif. 94720 (personal communication).
[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]

P. J. Delfyett, A. Dienes, J. P. Heritage, M. Y. Hong, and Y. H. Chang, “Femtosecond hybrid mode-locked semiconductor lasers and amplifier dynamics,” Appl. Phys. B 58, 183–195 (1994).
[Crossref]

A. Dienes, J. P. Heritage, M. Y. Hong, and Y. H. Chang, “Time and spectral domain evolution of subpicosecond pulses in semiconductor laser amplifiers,” Opt. Lett. 17, 1602–1604 (1992).
[Crossref] [PubMed]

M. Y. Hong, Y. H. Chang, A. Dienes, J. P. Heritage, P. J. Defyette, Sol Dijaili, and F. Patterson, “Femtosecond self-and cross-phase modulation in semiconductor laser amplifiers,” to be submitted to IEEE .J. Quantum. Electron.

Chen, H. H.

Chiu, L. C.

K. Vahala, L. C. Chiu, S. Margalit, and A. Yariv, “On the linewidth enhancement factor α in semiconductor lasers,” Appl. Phys. Lett. 42, 631–633 (1983).
[Crossref]

Christov, I. P.

Chu, C.

Defyette, P. J.

M. Y. Hong, Y. H. Chang, A. Dienes, J. P. Heritage, P. J. Defyette, Sol Dijaili, and F. Patterson, “Femtosecond self-and cross-phase modulation in semiconductor laser amplifiers,” to be submitted to IEEE .J. Quantum. Electron.

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]

P. J. Delfyett, A. Dienes, J. P. Heritage, M. Y. Hong, and Y. H. Chang, “Femtosecond hybrid mode-locked semiconductor lasers and amplifier dynamics,” Appl. Phys. B 58, 183–195 (1994).
[Crossref]

P. J. Delfyett, Y. Silberberg, and G. A. Alphonse, “Hot-carrier thermalization induced self-phase modulation in semiconductor traveling wave amplifiers,” Appl. Phys. Lett. 59, 10–12 (1991).
[Crossref]

Dienes, A.

C. Chu, J. P. Heritage, R. S. Grant, K. X. Liu, A. Dienes, W. E. White, and A. Sullivan, “Direct measurement of the spectral phase of femtosecond pulses,” Opt. Lett. 20, 904–906 (1995).
[Crossref] [PubMed]

P. J. Delfyett, A. Dienes, J. P. Heritage, M. Y. Hong, and Y. H. Chang, “Femtosecond hybrid mode-locked semiconductor lasers and amplifier dynamics,” Appl. Phys. B 58, 183–195 (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. Dienes, J. P. Heritage, M. Y. Hong, and Y. H. Chang, “Time and spectral domain evolution of subpicosecond pulses in semiconductor laser amplifiers,” Opt. Lett. 17, 1602–1604 (1992).
[Crossref] [PubMed]

A. Dienes, L. W. Carr, and M. Y. Hong, “Compressible frequency chirping due to slow and fast self-phase modulation of ultrashort pulses in high-gain dye laser amplifiers,” IEEE J. Quantum Electron. 27, 1214–1220 (1991).
[Crossref]

A. Dienes and L. W. Carr, “Effect of pulse shape on self-phase-modulation-induced chirping of optical pulses in diode laser amplifiers,” J. Appl. Phys. 69, 1766–1768 (1991).
[Crossref]

M. Y. Hong, Y. H. Chang, A. Dienes, J. P. Heritage, P. J. Defyette, Sol Dijaili, and F. Patterson, “Femtosecond self-and cross-phase modulation in semiconductor laser amplifiers,” to be submitted to IEEE .J. Quantum. Electron.

Dienes, A. J.

O. Teschke, A. J. Dienes, and J. R. Whinnery, “Theory and operation of high-power cw and long-pulse dye lasers,” IEEE J. Quantum Electron. QE-12, 383–395 (1976).
[Crossref]

Dijaili, Sol

M. Y. Hong, Y. H. Chang, A. Dienes, J. P. Heritage, P. J. Defyette, Sol Dijaili, and F. Patterson, “Femtosecond self-and cross-phase modulation in semiconductor laser amplifiers,” to be submitted to IEEE .J. Quantum. Electron.

Downer, M. C.

Fork, R. L.

Franco, M.

G. Boyer, M. Franco, J. P. Chambaret, A. Migus, A. Antonetti, P. Georges, F. Salin, and A. Brun, “Generation of 0.6 mJ pulses of 16 fs duration through high-repetition rate amplification of self-phase modulated pulses,” Appl. Phys. Letters,  53, 823–825 (1988); R. W. Schoenline, Lawrence Berkely National Laboratory, Berkely, Calif. 94720 (personal communication).
[Crossref]

Fujimoto, J. G.

Self-focusing owing to the nonlinear index n2 is responsible for Kerr-lens mode locking in solid-state laser oscillators; see, for example, H. A. House, J. G. Fujimoto, and E. P. Ippen, “Analytic theory of additive pulse and Kerr lens mode-locking,” IEEE J. Quantum Electron. 28, 2086–2096 (1992). However, these effects are small, typically only a few-percent change in beam area, unless pulse durations approach 10 fs.
[Crossref]

Georges, P.

Gain medium; F. Salin, G. Le Saux, P. Georges, A. Brun, C. Bagnall, and J. Zarzycki, “Efficient tunable solid-state laser near 630 nm using sulforhodamine 640-doped silica gel,” Opt. Lett. 14, 785–788 (1989). Rhodamine 101 is very similar to Rhodamine 640 with fluorescence peak shifted to 606 nm. Saturable-absorber absorption curve from Fisher Scientific Corporation.
[Crossref] [PubMed]

G. Boyer, M. Franco, J. P. Chambaret, A. Migus, A. Antonetti, P. Georges, F. Salin, and A. Brun, “Generation of 0.6 mJ pulses of 16 fs duration through high-repetition rate amplification of self-phase modulated pulses,” Appl. Phys. Letters,  53, 823–825 (1988); R. W. Schoenline, Lawrence Berkely National Laboratory, Berkely, Calif. 94720 (personal communication).
[Crossref]

Gordon, J. P.

Grant, R. S.

C. Chu, J. P. Heritage, R. S. Grant, K. X. Liu, A. Dienes, W. E. White, and A. Sullivan, “Direct measurement of the spectral phase of femtosecond pulses,” Opt. Lett. 20, 904–906 (1995).
[Crossref] [PubMed]

R. S. Grant and W. Sibbett, “Observations of ultrafast nonlinear refraction in an InGaAsP optical amplifier,” Appl. Phys. Lett. 58, 1119–1121 (1991).
[Crossref]

Heritage, J. P.

C. Chu, J. P. Heritage, R. S. Grant, K. X. Liu, A. Dienes, W. E. White, and A. Sullivan, “Direct measurement of the spectral phase of femtosecond pulses,” Opt. Lett. 20, 904–906 (1995).
[Crossref] [PubMed]

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]

P. J. Delfyett, A. Dienes, J. P. Heritage, M. Y. Hong, and Y. H. Chang, “Femtosecond hybrid mode-locked semiconductor lasers and amplifier dynamics,” Appl. Phys. B 58, 183–195 (1994).
[Crossref]

A. Dienes, J. P. Heritage, M. Y. Hong, and Y. H. Chang, “Time and spectral domain evolution of subpicosecond pulses in semiconductor laser amplifiers,” Opt. Lett. 17, 1602–1604 (1992).
[Crossref] [PubMed]

M. Y. Hong, Y. H. Chang, A. Dienes, J. P. Heritage, P. J. Defyette, Sol Dijaili, and F. Patterson, “Femtosecond self-and cross-phase modulation in semiconductor laser amplifiers,” to be submitted to IEEE .J. Quantum. Electron.

Hochstrasser, R. M.

Hong, M. Y.

P. J. Delfyett, A. Dienes, J. P. Heritage, M. Y. Hong, and Y. H. Chang, “Femtosecond hybrid mode-locked semiconductor lasers and amplifier dynamics,” Appl. Phys. B 58, 183–195 (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. Dienes, J. P. Heritage, M. Y. Hong, and Y. H. Chang, “Time and spectral domain evolution of subpicosecond pulses in semiconductor laser amplifiers,” Opt. Lett. 17, 1602–1604 (1992).
[Crossref] [PubMed]

A. Dienes, L. W. Carr, and M. Y. Hong, “Compressible frequency chirping due to slow and fast self-phase modulation of ultrashort pulses in high-gain dye laser amplifiers,” IEEE J. Quantum Electron. 27, 1214–1220 (1991).
[Crossref]

M. Y. Hong, Y. H. Chang, A. Dienes, J. P. Heritage, P. J. Defyette, Sol Dijaili, and F. Patterson, “Femtosecond self-and cross-phase modulation in semiconductor laser amplifiers,” to be submitted to IEEE .J. Quantum. Electron.

House, H. A.

Self-focusing owing to the nonlinear index n2 is responsible for Kerr-lens mode locking in solid-state laser oscillators; see, for example, H. A. House, J. G. Fujimoto, and E. P. Ippen, “Analytic theory of additive pulse and Kerr lens mode-locking,” IEEE J. Quantum Electron. 28, 2086–2096 (1992). However, these effects are small, typically only a few-percent change in beam area, unless pulse durations approach 10 fs.
[Crossref]

Huang, C. P.

Ippen, E. P.

Self-focusing owing to the nonlinear index n2 is responsible for Kerr-lens mode locking in solid-state laser oscillators; see, for example, H. A. House, J. G. Fujimoto, and E. P. Ippen, “Analytic theory of additive pulse and Kerr lens mode-locking,” IEEE J. Quantum Electron. 28, 2086–2096 (1992). However, these effects are small, typically only a few-percent change in beam area, unless pulse durations approach 10 fs.
[Crossref]

M. S. Stix, M. P. Kesler, and E. P. Ippen, “Observations of subpicosecond dynamics in GaAlAs laser diodes,” Appl. Phys. Lett. 48, 1722–1724 (1986).
[Crossref]

Jacobovitz, G. R.

Jiang, S. H.

S. H. Jiang and L. W. Casperson, “Ultrashort pulse propagation in dye laser amplifiers,” J. Appl. Phys. 73, 530–540 (1993).
[Crossref]

Kapteyn, H. C.

Kesler, M. P.

M. S. Stix, M. P. Kesler, and E. P. Ippen, “Observations of subpicosecond dynamics in GaAlAs laser diodes,” Appl. Phys. Lett. 48, 1722–1724 (1986).
[Crossref]

Knox, W. H.

Kühlke, D.

Le Saux, G.

Lee, Y. C.

Liu, K. X.

Margalit, S.

K. Vahala, L. C. Chiu, S. Margalit, and A. Yariv, “On the linewidth enhancement factor α in semiconductor lasers,” Appl. Phys. Lett. 42, 631–633 (1983).
[Crossref]

Mark, J.

J. Mørk, J. Mark, and C. P. Seltzer, “Subpicosecond gain dynamics in InGaAsP optical amplifiers: experiment and theory,” Appl. Phys. Lett. 64, 2206–2209 (1994).

Martinez, O. E.

Menyuk, C. R.

Migus, A.

G. Boyer, M. Franco, J. P. Chambaret, A. Migus, A. Antonetti, P. Georges, F. Salin, and A. Brun, “Generation of 0.6 mJ pulses of 16 fs duration through high-repetition rate amplification of self-phase modulated pulses,” Appl. Phys. Letters,  53, 823–825 (1988); R. W. Schoenline, Lawrence Berkely National Laboratory, Berkely, Calif. 94720 (personal communication).
[Crossref]

Miranda, R.

Mørk, J.

J. Mørk, J. Mark, and C. P. Seltzer, “Subpicosecond gain dynamics in InGaAsP optical amplifiers: experiment and theory,” Appl. Phys. Lett. 64, 2206–2209 (1994).

Murnane, M. M.

Nickel, D.

Olsson, N. A.

G. P. Agrawal and N. A. Olsson, “Self-phase modulation and spectral broadening of optical pulses in semiconductor laser amplifiers,” IEEE J. Quantum Electron. 25, 2297–2306 (1989).
[Crossref]

Osinski, M.

M. Osinski and J. Burrus, “Linewidth broadening factor in semiconductor lasers—an overview,” IEEE J. Quantum Electron. QE-23, 9–29 (1987).
[Crossref]

Patterson, F.

M. Y. Hong, Y. H. Chang, A. Dienes, J. P. Heritage, P. J. Defyette, Sol Dijaili, and F. Patterson, “Femtosecond self-and cross-phase modulation in semiconductor laser amplifiers,” to be submitted to IEEE .J. Quantum. Electron.

Reid, G. D.

Salin, F.

Gain medium; F. Salin, G. Le Saux, P. Georges, A. Brun, C. Bagnall, and J. Zarzycki, “Efficient tunable solid-state laser near 630 nm using sulforhodamine 640-doped silica gel,” Opt. Lett. 14, 785–788 (1989). Rhodamine 101 is very similar to Rhodamine 640 with fluorescence peak shifted to 606 nm. Saturable-absorber absorption curve from Fisher Scientific Corporation.
[Crossref] [PubMed]

G. Boyer, M. Franco, J. P. Chambaret, A. Migus, A. Antonetti, P. Georges, F. Salin, and A. Brun, “Generation of 0.6 mJ pulses of 16 fs duration through high-repetition rate amplification of self-phase modulated pulses,” Appl. Phys. Letters,  53, 823–825 (1988); R. W. Schoenline, Lawrence Berkely National Laboratory, Berkely, Calif. 94720 (personal communication).
[Crossref]

Scarparo, M. A. F.

Seltzer, C. P.

J. Mørk, J. Mark, and C. P. Seltzer, “Subpicosecond gain dynamics in InGaAsP optical amplifiers: experiment and theory,” Appl. Phys. Lett. 64, 2206–2209 (1994).

Shank, C. V.

Sibbett, W.

R. S. Grant and W. Sibbett, “Observations of ultrafast nonlinear refraction in an InGaAsP optical amplifier,” Appl. Phys. Lett. 58, 1119–1121 (1991).
[Crossref]

Siegman, A. E.

A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986), Chap. 10.1.

Silberberg, Y.

P. J. Delfyett, Y. Silberberg, and G. A. Alphonse, “Hot-carrier thermalization induced self-phase modulation in semiconductor traveling wave amplifiers,” Appl. Phys. Lett. 59, 10–12 (1991).
[Crossref]

Stix, M. S.

M. S. Stix, M. P. Kesler, and E. P. Ippen, “Observations of subpicosecond dynamics in GaAlAs laser diodes,” Appl. Phys. Lett. 48, 1722–1724 (1986).
[Crossref]

Sullivan, A.

Teschke, O.

O. Teschke, A. J. Dienes, and J. R. Whinnery, “Theory and operation of high-power cw and long-pulse dye lasers,” IEEE J. Quantum Electron. QE-12, 383–395 (1976).
[Crossref]

Vahala, K.

K. Vahala, L. C. Chiu, S. Margalit, and A. Yariv, “On the linewidth enhancement factor α in semiconductor lasers,” Appl. Phys. Lett. 42, 631–633 (1983).
[Crossref]

Valdmanis, J. A.

J. A. Valdmanis and R. L. Fork, “Design considerations for femtosecond pulse laser balancing self-phase modulation, group velocity dispersion, saturable absorption and saturable gain,” IEEE J. Quantum Electron. QE-22, 112–118 (1986).
[Crossref]

Verdeyen, J. T.

J. T. Verdeyen, Laser Electronics, 3d ed. (Prentice-HallEnglewood Cliffs, N.J., 1995), Chap. 10.

von der Linde, D.

Wai, P. K. A.

Whinnery, J. R.

O. Teschke, A. J. Dienes, and J. R. Whinnery, “Theory and operation of high-power cw and long-pulse dye lasers,” IEEE J. Quantum Electron. QE-12, 383–395 (1976).
[Crossref]

White, W. E.

Wynne, K.

Yariv, A.

K. Vahala, L. C. Chiu, S. Margalit, and A. Yariv, “On the linewidth enhancement factor α in semiconductor lasers,” Appl. Phys. Lett. 42, 631–633 (1983).
[Crossref]

Zarzycki, J.

Zhou, J.

Appl. Phys. B (1)

P. J. Delfyett, A. Dienes, J. P. Heritage, M. Y. Hong, and Y. H. Chang, “Femtosecond hybrid mode-locked semiconductor lasers and amplifier dynamics,” Appl. Phys. B 58, 183–195 (1994).
[Crossref]

Appl. Phys. Lett. (5)

K. Vahala, L. C. Chiu, S. Margalit, and A. Yariv, “On the linewidth enhancement factor α in semiconductor lasers,” Appl. Phys. Lett. 42, 631–633 (1983).
[Crossref]

M. S. Stix, M. P. Kesler, and E. P. Ippen, “Observations of subpicosecond dynamics in GaAlAs laser diodes,” Appl. Phys. Lett. 48, 1722–1724 (1986).
[Crossref]

P. J. Delfyett, Y. Silberberg, and G. A. Alphonse, “Hot-carrier thermalization induced self-phase modulation in semiconductor traveling wave amplifiers,” Appl. Phys. Lett. 59, 10–12 (1991).
[Crossref]

J. Mørk, J. Mark, and C. P. Seltzer, “Subpicosecond gain dynamics in InGaAsP optical amplifiers: experiment and theory,” Appl. Phys. Lett. 64, 2206–2209 (1994).

R. S. Grant and W. Sibbett, “Observations of ultrafast nonlinear refraction in an InGaAsP optical amplifier,” Appl. Phys. Lett. 58, 1119–1121 (1991).
[Crossref]

Appl. Phys. Letters (1)

G. Boyer, M. Franco, J. P. Chambaret, A. Migus, A. Antonetti, P. Georges, F. Salin, and A. Brun, “Generation of 0.6 mJ pulses of 16 fs duration through high-repetition rate amplification of self-phase modulated pulses,” Appl. Phys. Letters,  53, 823–825 (1988); R. W. Schoenline, Lawrence Berkely National Laboratory, Berkely, Calif. 94720 (personal communication).
[Crossref]

IEEE J. Quantum Electron. (8)

J. A. Valdmanis and R. L. Fork, “Design considerations for femtosecond pulse laser balancing self-phase modulation, group velocity dispersion, saturable absorption and saturable gain,” IEEE J. Quantum Electron. QE-22, 112–118 (1986).
[Crossref]

W. H. Knox, “Femtosecond optical amplification,” IEEE J. Quantum Electron. 24, 358–397 (1988).
[Crossref]

O. Teschke, A. J. Dienes, and J. R. Whinnery, “Theory and operation of high-power cw and long-pulse dye lasers,” IEEE J. Quantum Electron. QE-12, 383–395 (1976).
[Crossref]

M. Osinski and J. Burrus, “Linewidth broadening factor in semiconductor lasers—an overview,” IEEE J. Quantum Electron. QE-23, 9–29 (1987).
[Crossref]

Self-focusing owing to the nonlinear index n2 is responsible for Kerr-lens mode locking in solid-state laser oscillators; see, for example, H. A. House, J. G. Fujimoto, and E. P. Ippen, “Analytic theory of additive pulse and Kerr lens mode-locking,” IEEE J. Quantum Electron. 28, 2086–2096 (1992). However, these effects are small, typically only a few-percent change in beam area, unless pulse durations approach 10 fs.
[Crossref]

G. P. Agrawal and N. A. Olsson, “Self-phase modulation and spectral broadening of optical pulses in semiconductor laser amplifiers,” IEEE J. Quantum Electron. 25, 2297–2306 (1989).
[Crossref]

A. Dienes, L. W. Carr, and M. Y. Hong, “Compressible frequency chirping due to slow and fast self-phase modulation of ultrashort pulses in high-gain dye laser amplifiers,” IEEE J. Quantum Electron. 27, 1214–1220 (1991).
[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]

IEEE Photon. Technol. Lett. (1)

G. P. Agrawal and C. M. Bowden, “Concept of linewidth enhancement factor in semiconductor lasers: its usefulness and limitations,” IEEE Photon. Technol. Lett. 5, 640–643 (1993).
[Crossref]

J. Appl. Phys. (2)

A. Dienes and L. W. Carr, “Effect of pulse shape on self-phase-modulation-induced chirping of optical pulses in diode laser amplifiers,” J. Appl. Phys. 69, 1766–1768 (1991).
[Crossref]

S. H. Jiang and L. W. Casperson, “Ultrashort pulse propagation in dye laser amplifiers,” J. Appl. Phys. 73, 530–540 (1993).
[Crossref]

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

Opt. Lett. (11)

R. Miranda, G. R. Jacobovitz, C. H. Brito-Cruz, and M. A. F. Scarparo, “Positive and negative chirping of laser pulses in a saturable absorber,” Opt. Lett. 11, 224–226 (1986).
[Crossref]

A. Dienes, J. P. Heritage, M. Y. Hong, and Y. H. Chang, “Time and spectral domain evolution of subpicosecond pulses in semiconductor laser amplifiers,” Opt. Lett. 17, 1602–1604 (1992).
[Crossref] [PubMed]

P. K. A. Wai, C. R. Menyuk, Y. C. Lee, and H. H. Chen, “Nonlinear pulse propagation in the neighborhood of the zero dispersion wavelength of monomode optical fibers,” Opt. Lett. 11, 464–467 (1986).
[Crossref] [PubMed]

D. Nickel, D. Kühlke, and D. von der Linde, “Multipass dyecell amplifier for high-repetition-rate femtosecond optical pulses,” Opt. Lett. 14, 36–38 (1989).
[Crossref] [PubMed]

W. H. Knox, M. C. Downer, R. L. Fork, and C. V. Shank, “Amplified femtosecond optical pulses and continuum generation at a 5-kHz repetition rate,” Opt. Lett. 9, 552–555 (1984).
[Crossref] [PubMed]

C. Chu, J. P. Heritage, R. S. Grant, K. X. Liu, A. Dienes, W. E. White, and A. Sullivan, “Direct measurement of the spectral phase of femtosecond pulses,” Opt. Lett. 20, 904–906 (1995).
[Crossref] [PubMed]

K. Wynne, G. D. Reid, and R. M. Hochstrasser, “Regenerative amplification of 30-fs pulses in Ti:sapphire at 5 kHz,” Opt. Lett. 19, 895–897 (1994).
[Crossref] [PubMed]

J. Zhou, C. P. Huang, M. M. Murnane, and H. C. Kapteyn, “Amplification of 26-fs, 2-TW pulses near the gain narrowing limit in Ti:sapphire,” Opt. Lett. 20, 64–66 (1995).
[Crossref] [PubMed]

I. P. Christov, H. C. Kapteyn, M. M. Murnane, C. P. Huang, and J. Zhou, “Space–time focusing of femtosecond pulses in a Ti:sapphire laser,” Opt. Lett. 20, 309–311 (1995).
[Crossref] [PubMed]

Gain medium; F. Salin, G. Le Saux, P. Georges, A. Brun, C. Bagnall, and J. Zarzycki, “Efficient tunable solid-state laser near 630 nm using sulforhodamine 640-doped silica gel,” Opt. Lett. 14, 785–788 (1989). Rhodamine 101 is very similar to Rhodamine 640 with fluorescence peak shifted to 606 nm. Saturable-absorber absorption curve from Fisher Scientific Corporation.
[Crossref] [PubMed]

I. P. Christov, M. M. Murnane, H. C. Kapteyn, J. Zhou, and C. P. Huang, “Fourth-order dispersion-limited solitary pulses,” Opt. Lett. 18, 1465–1467 (1994).
[Crossref]

Rev. Mod. Phys. (1)

C. V. Shank, “Physics of dye lasers,” Rev. Mod. Phys. 47, 649–657 (1975).
[Crossref]

Other (4)

A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986), Chap. 10.1.

J. T. Verdeyen, Laser Electronics, 3d ed. (Prentice-HallEnglewood Cliffs, N.J., 1995), Chap. 10.

G. P. Agrawal, Nonlinear Fiber Optics (Academic, San Diego, Calif., 1989), Chap. 2, pp. 44–48.

M. Y. Hong, Y. H. Chang, A. Dienes, J. P. Heritage, P. J. Defyette, Sol Dijaili, and F. Patterson, “Femtosecond self-and cross-phase modulation in semiconductor laser amplifiers,” to be submitted to IEEE .J. Quantum. Electron.

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Figures (7)

Fig. 1
Fig. 1

(a) Simplified geometry of a single pass in the DLA. Ip is a quasi-cw pump beam and Is is an ultrashort-pulse signal. (b) Level scheme of an amplifier dye. (c) Absorption and emission cross sections of an amplifier dye (Rhodamine 6G).

Fig. 2
Fig. 2

Kramers–Kronig calculations of the parameter α for the dye Rhodamine 6G. (a) Imχ (i.e., gain) versus wavelength for various fractions of dye molecules in the excited state ranging from 0.1 to 0.9, (b) Re χ versus wavelength, (c) α at various wavelengths versus fraction of molecules in the excited state from

Fig. 3
Fig. 3

Theoretical shapes for frequency dispersion Og(ω) = f(ω)/f(ω0) of the gain dye and of the saturable absorber [Oa(ω)] used in the experiment. Solid curve, gain dye; dashed curve, saturable absorber.

Fig. 4
Fig. 4

Comparison of the experimental results with the theoretical model. The parameters are listed in Tables 1 and 2.

Fig. 5
Fig. 5

Pulse shapes corresponding to the theoretical spectral curves of Fig. 4. The pulse durations are given in Table 4.

Fig. 6
Fig. 6

Ratio of the (normalized) output and input spectra F(ω) = |Vout(ω)|2/|Vin(ω)|2 for the cases b (dashed curve) and c (solid curve) of Fig. 4.

Fig. 7
Fig. 7

Calculated spectral output of the amplifier, with various terms in model equation (12) set deliberately to zero. For each case, the dashed curve is the correct match from Fig. 4. (a) case a with quadratic phase on the input spectrum set to zero (K = 0); (b) case b with all self-phase modulations set to zero: α = 0, n2 = 0; (c) case c with gain (and absorber) dispersions set to zero.

Tables (4)

Tables Icon

Table 2 Physical Parameters

Tables Icon

Table 3 Matching of Experimental and Theoretical Data for Case c

Tables Icon

Table 4 Comparison of Experimental and Theoretical Input–Output Pulses

Equations (22)

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2 E ¯ r c 2 2 E ¯ t 2 = 0 ,
E ¯ ( x , y , z , t ) = ½ x ˆ F ( x , y ) V ( t , z ) exp [ i ( ω 0 t β 0 z ) ] + c . c . ,
E ¯ ( x , y , z , ω ) = x ˆ F ( x , y ) V ( ω , z ) exp ( i β 0 z ) ,
r = n b 2 + χ ˜ ( ω ) .
V ( ω , z ) z = i { ω c [ 1 + χ m ( ω ) + Γ χ ˜ ( ω , N ) ] 1 / 2 β 0 } × V ( ω , z ) .
Γ = S | F ( x , y ) | 2 d S / + + | F ( x , y ) | 2 d x d y .
2 F x 2 + 2 F y 2 + ( n b 2 n ¯ 2 ) ω 0 2 c 2 F = 0 ,
[ z + i v g ( ω ω 0 ) + i β 2 2 ( ω ω 0 ) 2 + i β 3 6 ( ω ω 0 ) 3 + γ 2 ] × V ( ω , z ) = { i ω 0 Γ 2 c n ¯ [ χ R ( ω 0 ) + i χ I ( ω 0 ) ] + O g ( ω ) i O p ( ω ) } V ( ω , z ) .
O g ( ω ) = ω 0 Γ 2 c n ¯ [ ( ω ω 0 ) χ I ω | ω 0 + 1 2 ( ω ω 0 ) 2 2 χ I ω 2 | ω 0 + 1 6 ( ω ω 0 ) 3 3 χ I ω 3 | ω 0 + ] ,
O p ( ω ) = ω 0 Γ 2 c n ¯ [ ( ω ω 0 ) χ R ω | ω 0 + 1 2 ( ω ω 0 ) 2 2 χ R ω 2 | ω 0 + 1 6 ( ω ω 0 ) 3 3 χ R ω 3 | ω 0 + ] .
( z i 2 β 2 2 τ 2 β 3 6 3 τ 3 + γ 2 ) V ( τ , z ) = { ω 0 Γ 2 c n ¯ [ χ I ( ω 0 ) i χ R ( ω 0 ) ] + O ˆ g ( τ ) i O ˆ p ( τ ) } V ( τ ) ,
O ˆ p ( τ ) = ω 0 Γ 2 c n ¯ { i χ R ω τ | ω 0 1 2 2 χ R ω 2 2 τ 2 | ω 0 + i 1 6 3 χ R ω 3 3 τ 3 | ω 0 + } ,
O ˆ g ( τ ) = ω 0 Γ 2 c n ¯ { i χ I ω τ | ω 0 1 2 2 χ I ω 2 2 τ 2 | ω 0 + i 1 6 3 χ I ω 3 3 τ 3 | ω 0 + } .
[ z i 2 β 2 2 τ 2 1 6 β 3 3 τ 3 + γ 2 + 1 A ( γ 2 p + i ω 0 c n 2 ) × | V ( τ ) | 2 ] V ( τ , z ) = { ω 0 Γ 2 c n ¯ [ χ I ( ω 0 ) i χ R ( ω 0 ) ] + O ˆ g ( τ ) i O ˆ p ( τ ) } V ( τ ) ,
ω 0 Γ 2 c n ¯ [ χ I ( ω 0 ) i χ R ( ω 0 ) ] = g 2 i β ,
g = σ ( N N 0 ) ,
N τ = R N τ e Γ σ ( N N 0 ) | V ( τ ) | 2 A ћ ω ,
g ( τ ) = g 0 exp [ 1 W S τ | V ( τ ) | 2 d τ ] ,
[ z i 2 β 2 2 τ 2 1 6 β 3 3 τ 3 + γ 2 + 1 A ( γ 2 p + i ω 0 c n 2 ) × | V ( τ ) | 2 ] V ( τ , z ) = [ g ( t ) ( 1 + i α ) + Δ g ( τ ) ( 1 + i ζ ) + O ˆ g ( τ ) i O ˆ p ( τ ) ] V ( τ ) ,
g ( τ ) = N 1 ( τ ) σ e N 0 ( τ ) σ a .
O g ( ω ) = f ( ω ) / f ( ω 0 ) ,
[ 2 C 1 exp ( a Ω 1 ) + exp ( b Ω 1 ) + 2 C 2 exp ( c Ω 2 ) + exp ( b Ω 2 ) ] × exp ( j K Ω 1 2 ) ,

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