Abstract

We examine the evolution of the optical phase in a nanosecond-pumped dye amplifier and show that the time-dependent gain introduces optical phase distortions. For the case of uniform pumping and weak input-beam intensity an analytic expression for the evolution of the optical phase is derived. Even for a temporally symmetric pump pulse the amplified pulse has an asymmetric frequency spectrum caused by the contribution of the fluorescence rate to the dynamics of the amplification process. When the injected beam is pulsed, the evolution of the optical phase varies with the relative timing between the input and the pump pulses. The model is tested against experimental results for an amplifier that uses DCM dye dissolved in dimethyl sulfoxide. The rms deviation between the measured and the calculated behavior of the instantaneous frequency is less than 6 MHz.

© 1994 Optical Society of America

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References

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  1. E. Cromwell, T. Trickl, Y. T. Lee, and A. H. Kung, Rev. Sci. Instrum. 60, 2888 (1989).
    [CrossRef]
  2. M. G. Littman, Appl. Opt. 23, 4465 (1984).
    [CrossRef] [PubMed]
  3. T. D. Raymond and A. V. Smith, Opt. Lett. 16, 33 (1991).
    [CrossRef] [PubMed]
  4. C. E. Hamilton, Opt. Lett. 17, 728 (1992).
    [CrossRef] [PubMed]
  5. J. M. Gilligan and E. E. Eyler, Phys. Rev. A 46, 3676 (1992).
    [CrossRef] [PubMed]
  6. E. A. Hildum, U. Boesl, D. H. McIntyre, R. G. Beausoleil, and T. W. Hänsch, Phys. Rev. Lett. 56, 576 (1986).
    [CrossRef] [PubMed]
  7. K. Danzmann, M. S. Fee, and S. Chu, Phys. Rev. A 39, 6072 (1989).
    [CrossRef] [PubMed]
  8. S. Schiemann, A. Kuhn, S. Steuerwald, and K. Bergmann, Phys. Rev. Lett. 71, 3637 (1993).
    [CrossRef] [PubMed]
  9. M. D. Levenson and S. S. Kano, Introduction to Nonlinear Laser Spectroscopy (Academic, Boston, Mass., 1988).
  10. M. Gehrtz, G. C. Bjorklund, and E. A. Whittaker, J. Opt. Soc. Am. B 2, 1510 (1985).
    [CrossRef]
  11. T. F. Gallagher, R. Kachru, F. Gounand, G. C. Bjorklund, and W. Lenth, Opt. Lett. 7, 28 (1982).
    [CrossRef] [PubMed]
  12. N. H. Tran, R. Kachru, P. Pillet, H. B. van Lindenvan den Heuvell, T. F. Gallagher, and J. P. Watjen, Appl. Opt. 23, 1353 (1984).
    [CrossRef]
  13. A systematic investigation of pulsed-laser FM spectroscopy is being conducted by the present authors together with J. C. Block and R. W. Field, Department of Chemistry, Massachusetts Institute of Technology, Cambridge, Mass. 02139. The pulsed FM technique is particularly effective for sensitive measurements of far-UV absorption.
  14. S. Gangopadhyay, N. Melikechi, and E. E. Eyler, J. Opt. Soc. Am. B 11, 231 (1994).
    [CrossRef]
  15. M. S. Fee, K. Danzmann, and S. Chu, Phys. Rev. A 45, 4911 (1992).
    [CrossRef] [PubMed]
  16. G. P. Agrawal and N. A. Olsson, IEEE J. Quantum Electron. 25, 2297 (1989).
    [CrossRef]
  17. E. Yablonovitch, Phys. Rev. A 10, 1888 (1974).
    [CrossRef]
  18. E. Yablonovitch, Phys. Rev. Lett. 60, 795 (1988).
    [CrossRef] [PubMed]
  19. B. Comaskey, T. Daly, C. Haynam, J. Morris, J. Paisner, and R. Young, in Pulsed Single-Frequency Lasers: Technology and Applications, W. K. Bischell and L. A. Rahn, eds., Proc. Soc. Photo-Opt. Instrum. Eng.912, 73 (1988).
    [CrossRef]
  20. L. M. Frantz and J. S. Nodvik, J. Appl. Phys. 34, 2346 (1963).
    [CrossRef]
  21. A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986).
  22. R. Wallenstein, Pulsed Dye Lasers, Laser HandbookM. L. Stich, ed. (North-Holland, Amsterdam, 1985), Vol. 3, p. 289.
  23. R. A. Haas and M. D. Rotter, Phys. Rev. A 43, 1573 (1991).
    [CrossRef] [PubMed]
  24. P. R. Hammond, IEEE J. Quantum Electron. 15, 624 (1979).
    [CrossRef]
  25. P. R. Hammond, Opt. Commun. 29, 331 (1979).
    [CrossRef]
  26. C. H. Henry, IEEE J. Quantum Electron. 18, 259 (1982).
    [CrossRef]
  27. K. Vahala, L. C. Chiu, S. Margalit, and A. Yariv, Appl. Phys. Lett. 42, 631 (1983).
    [CrossRef]
  28. M. Osiñski and J. Buus, IEEE J. Quantum Electron. 23, 9 (1987).
    [CrossRef]
  29. D. R. Lide, ed., CRC Handbook of Physics and Chemistry (CRC, Boca Raton, Fla., 1992).
  30. J. B. Birks, Photophysics of Aromatic Molecules (Wiley-Interscience, London, 1970), Chap. 3.

1994 (1)

1993 (1)

S. Schiemann, A. Kuhn, S. Steuerwald, and K. Bergmann, Phys. Rev. Lett. 71, 3637 (1993).
[CrossRef] [PubMed]

1992 (3)

C. E. Hamilton, Opt. Lett. 17, 728 (1992).
[CrossRef] [PubMed]

J. M. Gilligan and E. E. Eyler, Phys. Rev. A 46, 3676 (1992).
[CrossRef] [PubMed]

M. S. Fee, K. Danzmann, and S. Chu, Phys. Rev. A 45, 4911 (1992).
[CrossRef] [PubMed]

1991 (2)

1989 (3)

K. Danzmann, M. S. Fee, and S. Chu, Phys. Rev. A 39, 6072 (1989).
[CrossRef] [PubMed]

E. Cromwell, T. Trickl, Y. T. Lee, and A. H. Kung, Rev. Sci. Instrum. 60, 2888 (1989).
[CrossRef]

G. P. Agrawal and N. A. Olsson, IEEE J. Quantum Electron. 25, 2297 (1989).
[CrossRef]

1988 (1)

E. Yablonovitch, Phys. Rev. Lett. 60, 795 (1988).
[CrossRef] [PubMed]

1987 (1)

M. Osiñski and J. Buus, IEEE J. Quantum Electron. 23, 9 (1987).
[CrossRef]

1986 (1)

E. A. Hildum, U. Boesl, D. H. McIntyre, R. G. Beausoleil, and T. W. Hänsch, Phys. Rev. Lett. 56, 576 (1986).
[CrossRef] [PubMed]

1985 (1)

1984 (2)

1983 (1)

K. Vahala, L. C. Chiu, S. Margalit, and A. Yariv, Appl. Phys. Lett. 42, 631 (1983).
[CrossRef]

1982 (2)

1979 (2)

P. R. Hammond, IEEE J. Quantum Electron. 15, 624 (1979).
[CrossRef]

P. R. Hammond, Opt. Commun. 29, 331 (1979).
[CrossRef]

1974 (1)

E. Yablonovitch, Phys. Rev. A 10, 1888 (1974).
[CrossRef]

1963 (1)

L. M. Frantz and J. S. Nodvik, J. Appl. Phys. 34, 2346 (1963).
[CrossRef]

Agrawal, G. P.

G. P. Agrawal and N. A. Olsson, IEEE J. Quantum Electron. 25, 2297 (1989).
[CrossRef]

Beausoleil, R. G.

E. A. Hildum, U. Boesl, D. H. McIntyre, R. G. Beausoleil, and T. W. Hänsch, Phys. Rev. Lett. 56, 576 (1986).
[CrossRef] [PubMed]

Bergmann, K.

S. Schiemann, A. Kuhn, S. Steuerwald, and K. Bergmann, Phys. Rev. Lett. 71, 3637 (1993).
[CrossRef] [PubMed]

Birks, J. B.

J. B. Birks, Photophysics of Aromatic Molecules (Wiley-Interscience, London, 1970), Chap. 3.

Bjorklund, G. C.

Block, J. C.

A systematic investigation of pulsed-laser FM spectroscopy is being conducted by the present authors together with J. C. Block and R. W. Field, Department of Chemistry, Massachusetts Institute of Technology, Cambridge, Mass. 02139. The pulsed FM technique is particularly effective for sensitive measurements of far-UV absorption.

Boesl, U.

E. A. Hildum, U. Boesl, D. H. McIntyre, R. G. Beausoleil, and T. W. Hänsch, Phys. Rev. Lett. 56, 576 (1986).
[CrossRef] [PubMed]

Buus, J.

M. Osiñski and J. Buus, IEEE J. Quantum Electron. 23, 9 (1987).
[CrossRef]

Chiu, L. C.

K. Vahala, L. C. Chiu, S. Margalit, and A. Yariv, Appl. Phys. Lett. 42, 631 (1983).
[CrossRef]

Chu, S.

M. S. Fee, K. Danzmann, and S. Chu, Phys. Rev. A 45, 4911 (1992).
[CrossRef] [PubMed]

K. Danzmann, M. S. Fee, and S. Chu, Phys. Rev. A 39, 6072 (1989).
[CrossRef] [PubMed]

Comaskey, B.

B. Comaskey, T. Daly, C. Haynam, J. Morris, J. Paisner, and R. Young, in Pulsed Single-Frequency Lasers: Technology and Applications, W. K. Bischell and L. A. Rahn, eds., Proc. Soc. Photo-Opt. Instrum. Eng.912, 73 (1988).
[CrossRef]

Cromwell, E.

E. Cromwell, T. Trickl, Y. T. Lee, and A. H. Kung, Rev. Sci. Instrum. 60, 2888 (1989).
[CrossRef]

Daly, T.

B. Comaskey, T. Daly, C. Haynam, J. Morris, J. Paisner, and R. Young, in Pulsed Single-Frequency Lasers: Technology and Applications, W. K. Bischell and L. A. Rahn, eds., Proc. Soc. Photo-Opt. Instrum. Eng.912, 73 (1988).
[CrossRef]

Danzmann, K.

M. S. Fee, K. Danzmann, and S. Chu, Phys. Rev. A 45, 4911 (1992).
[CrossRef] [PubMed]

K. Danzmann, M. S. Fee, and S. Chu, Phys. Rev. A 39, 6072 (1989).
[CrossRef] [PubMed]

Eyler, E. E.

Fee, M. S.

M. S. Fee, K. Danzmann, and S. Chu, Phys. Rev. A 45, 4911 (1992).
[CrossRef] [PubMed]

K. Danzmann, M. S. Fee, and S. Chu, Phys. Rev. A 39, 6072 (1989).
[CrossRef] [PubMed]

Field, R. W.

A systematic investigation of pulsed-laser FM spectroscopy is being conducted by the present authors together with J. C. Block and R. W. Field, Department of Chemistry, Massachusetts Institute of Technology, Cambridge, Mass. 02139. The pulsed FM technique is particularly effective for sensitive measurements of far-UV absorption.

Frantz, L. M.

L. M. Frantz and J. S. Nodvik, J. Appl. Phys. 34, 2346 (1963).
[CrossRef]

Gallagher, T. F.

Gangopadhyay, S.

Gehrtz, M.

Gilligan, J. M.

J. M. Gilligan and E. E. Eyler, Phys. Rev. A 46, 3676 (1992).
[CrossRef] [PubMed]

Gounand, F.

Haas, R. A.

R. A. Haas and M. D. Rotter, Phys. Rev. A 43, 1573 (1991).
[CrossRef] [PubMed]

Hamilton, C. E.

Hammond, P. R.

P. R. Hammond, IEEE J. Quantum Electron. 15, 624 (1979).
[CrossRef]

P. R. Hammond, Opt. Commun. 29, 331 (1979).
[CrossRef]

Hänsch, T. W.

E. A. Hildum, U. Boesl, D. H. McIntyre, R. G. Beausoleil, and T. W. Hänsch, Phys. Rev. Lett. 56, 576 (1986).
[CrossRef] [PubMed]

Haynam, C.

B. Comaskey, T. Daly, C. Haynam, J. Morris, J. Paisner, and R. Young, in Pulsed Single-Frequency Lasers: Technology and Applications, W. K. Bischell and L. A. Rahn, eds., Proc. Soc. Photo-Opt. Instrum. Eng.912, 73 (1988).
[CrossRef]

Henry, C. H.

C. H. Henry, IEEE J. Quantum Electron. 18, 259 (1982).
[CrossRef]

Hildum, E. A.

E. A. Hildum, U. Boesl, D. H. McIntyre, R. G. Beausoleil, and T. W. Hänsch, Phys. Rev. Lett. 56, 576 (1986).
[CrossRef] [PubMed]

Kachru, R.

Kano, S. S.

M. D. Levenson and S. S. Kano, Introduction to Nonlinear Laser Spectroscopy (Academic, Boston, Mass., 1988).

Kuhn, A.

S. Schiemann, A. Kuhn, S. Steuerwald, and K. Bergmann, Phys. Rev. Lett. 71, 3637 (1993).
[CrossRef] [PubMed]

Kung, A. H.

E. Cromwell, T. Trickl, Y. T. Lee, and A. H. Kung, Rev. Sci. Instrum. 60, 2888 (1989).
[CrossRef]

Lee, Y. T.

E. Cromwell, T. Trickl, Y. T. Lee, and A. H. Kung, Rev. Sci. Instrum. 60, 2888 (1989).
[CrossRef]

Lenth, W.

Levenson, M. D.

M. D. Levenson and S. S. Kano, Introduction to Nonlinear Laser Spectroscopy (Academic, Boston, Mass., 1988).

Littman, M. G.

Margalit, S.

K. Vahala, L. C. Chiu, S. Margalit, and A. Yariv, Appl. Phys. Lett. 42, 631 (1983).
[CrossRef]

McIntyre, D. H.

E. A. Hildum, U. Boesl, D. H. McIntyre, R. G. Beausoleil, and T. W. Hänsch, Phys. Rev. Lett. 56, 576 (1986).
[CrossRef] [PubMed]

Melikechi, N.

Morris, J.

B. Comaskey, T. Daly, C. Haynam, J. Morris, J. Paisner, and R. Young, in Pulsed Single-Frequency Lasers: Technology and Applications, W. K. Bischell and L. A. Rahn, eds., Proc. Soc. Photo-Opt. Instrum. Eng.912, 73 (1988).
[CrossRef]

Nodvik, J. S.

L. M. Frantz and J. S. Nodvik, J. Appl. Phys. 34, 2346 (1963).
[CrossRef]

Olsson, N. A.

G. P. Agrawal and N. A. Olsson, IEEE J. Quantum Electron. 25, 2297 (1989).
[CrossRef]

Osiñski, M.

M. Osiñski and J. Buus, IEEE J. Quantum Electron. 23, 9 (1987).
[CrossRef]

Paisner, J.

B. Comaskey, T. Daly, C. Haynam, J. Morris, J. Paisner, and R. Young, in Pulsed Single-Frequency Lasers: Technology and Applications, W. K. Bischell and L. A. Rahn, eds., Proc. Soc. Photo-Opt. Instrum. Eng.912, 73 (1988).
[CrossRef]

Pillet, P.

Raymond, T. D.

Rotter, M. D.

R. A. Haas and M. D. Rotter, Phys. Rev. A 43, 1573 (1991).
[CrossRef] [PubMed]

Schiemann, S.

S. Schiemann, A. Kuhn, S. Steuerwald, and K. Bergmann, Phys. Rev. Lett. 71, 3637 (1993).
[CrossRef] [PubMed]

Siegman, A. E.

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

Smith, A. V.

Steuerwald, S.

S. Schiemann, A. Kuhn, S. Steuerwald, and K. Bergmann, Phys. Rev. Lett. 71, 3637 (1993).
[CrossRef] [PubMed]

Tran, N. H.

Trickl, T.

E. Cromwell, T. Trickl, Y. T. Lee, and A. H. Kung, Rev. Sci. Instrum. 60, 2888 (1989).
[CrossRef]

Vahala, K.

K. Vahala, L. C. Chiu, S. Margalit, and A. Yariv, Appl. Phys. Lett. 42, 631 (1983).
[CrossRef]

van Lindenvan den Heuvell, H. B.

Wallenstein, R.

R. Wallenstein, Pulsed Dye Lasers, Laser HandbookM. L. Stich, ed. (North-Holland, Amsterdam, 1985), Vol. 3, p. 289.

Watjen, J. P.

Whittaker, E. A.

Yablonovitch, E.

E. Yablonovitch, Phys. Rev. Lett. 60, 795 (1988).
[CrossRef] [PubMed]

E. Yablonovitch, Phys. Rev. A 10, 1888 (1974).
[CrossRef]

Yariv, A.

K. Vahala, L. C. Chiu, S. Margalit, and A. Yariv, Appl. Phys. Lett. 42, 631 (1983).
[CrossRef]

Young, R.

B. Comaskey, T. Daly, C. Haynam, J. Morris, J. Paisner, and R. Young, in Pulsed Single-Frequency Lasers: Technology and Applications, W. K. Bischell and L. A. Rahn, eds., Proc. Soc. Photo-Opt. Instrum. Eng.912, 73 (1988).
[CrossRef]

Appl. Opt. (2)

Appl. Phys. Lett. (1)

K. Vahala, L. C. Chiu, S. Margalit, and A. Yariv, Appl. Phys. Lett. 42, 631 (1983).
[CrossRef]

IEEE J. Quantum Electron. (4)

M. Osiñski and J. Buus, IEEE J. Quantum Electron. 23, 9 (1987).
[CrossRef]

P. R. Hammond, IEEE J. Quantum Electron. 15, 624 (1979).
[CrossRef]

C. H. Henry, IEEE J. Quantum Electron. 18, 259 (1982).
[CrossRef]

G. P. Agrawal and N. A. Olsson, IEEE J. Quantum Electron. 25, 2297 (1989).
[CrossRef]

J. Appl. Phys. (1)

L. M. Frantz and J. S. Nodvik, J. Appl. Phys. 34, 2346 (1963).
[CrossRef]

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

Opt. Commun. (1)

P. R. Hammond, Opt. Commun. 29, 331 (1979).
[CrossRef]

Opt. Lett. (3)

Phys. Rev. A (5)

J. M. Gilligan and E. E. Eyler, Phys. Rev. A 46, 3676 (1992).
[CrossRef] [PubMed]

K. Danzmann, M. S. Fee, and S. Chu, Phys. Rev. A 39, 6072 (1989).
[CrossRef] [PubMed]

M. S. Fee, K. Danzmann, and S. Chu, Phys. Rev. A 45, 4911 (1992).
[CrossRef] [PubMed]

E. Yablonovitch, Phys. Rev. A 10, 1888 (1974).
[CrossRef]

R. A. Haas and M. D. Rotter, Phys. Rev. A 43, 1573 (1991).
[CrossRef] [PubMed]

Phys. Rev. Lett. (3)

E. Yablonovitch, Phys. Rev. Lett. 60, 795 (1988).
[CrossRef] [PubMed]

S. Schiemann, A. Kuhn, S. Steuerwald, and K. Bergmann, Phys. Rev. Lett. 71, 3637 (1993).
[CrossRef] [PubMed]

E. A. Hildum, U. Boesl, D. H. McIntyre, R. G. Beausoleil, and T. W. Hänsch, Phys. Rev. Lett. 56, 576 (1986).
[CrossRef] [PubMed]

Rev. Sci. Instrum. (1)

E. Cromwell, T. Trickl, Y. T. Lee, and A. H. Kung, Rev. Sci. Instrum. 60, 2888 (1989).
[CrossRef]

Other (7)

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

R. Wallenstein, Pulsed Dye Lasers, Laser HandbookM. L. Stich, ed. (North-Holland, Amsterdam, 1985), Vol. 3, p. 289.

D. R. Lide, ed., CRC Handbook of Physics and Chemistry (CRC, Boca Raton, Fla., 1992).

J. B. Birks, Photophysics of Aromatic Molecules (Wiley-Interscience, London, 1970), Chap. 3.

M. D. Levenson and S. S. Kano, Introduction to Nonlinear Laser Spectroscopy (Academic, Boston, Mass., 1988).

B. Comaskey, T. Daly, C. Haynam, J. Morris, J. Paisner, and R. Young, in Pulsed Single-Frequency Lasers: Technology and Applications, W. K. Bischell and L. A. Rahn, eds., Proc. Soc. Photo-Opt. Instrum. Eng.912, 73 (1988).
[CrossRef]

A systematic investigation of pulsed-laser FM spectroscopy is being conducted by the present authors together with J. C. Block and R. W. Field, Department of Chemistry, Massachusetts Institute of Technology, Cambridge, Mass. 02139. The pulsed FM technique is particularly effective for sensitive measurements of far-UV absorption.

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Figures (11)

Fig. 1
Fig. 1

Top, Model calculations of finst for amplification at 650 nm with DCM dye in DMSO. The pump energy is 1 mJ for the solid curve and 10 mJ for the dashed curve. Bottom, Truncated Gaussian pump pulse used for this simulation.

Fig. 2
Fig. 2

Same as Fig. 1 but at 670 nm.

Fig. 3
Fig. 3

Top, Calculated finst for a Gaussian pump pulse for amplification at 670 nm with DCM in DMSO. The pump energy is 1 mJ for the solid curve and 10 mJ for the dashed curve. Bottom, Gaussian pump pulse with 5 ns FWHM used for this simulation.

Fig. 4
Fig. 4

Same as Fig. 3 but for a Gaussian pump pulse with a 20-ns FWHM.

Fig. 5
Fig. 5

Top, Calculated finst for a double-peaked pump pulse for DCM in DMSO at 670 nm. Bottom, Double Gaussian pump pulse with peaks are separated by 20 ns. The pump energy is 10 mJ.

Fig. 6
Fig. 6

Effects of relative timing between injected and pump pulses. (a) Top, Calculated finst when the peaks of the pulses are synchronous. Middle, Gaussian pump pulse with a 20-ns FWHM. Bottom, Gaussian injected pulse with a 10-ns FWHM. (b), (c) Same as (a) but for delays of +5 and −5 ns between the pulse peaks.

Fig. 7
Fig. 7

Effects of the irradiance of the injected beam on finst. Top, Calculated finst for irradiances of 250 W/cm2 (dashed curve) and 2.5 × 10 cm4 W/cm2 (solid curve). Bottom, Gaussian pump pulse, with a 20-ns FWHM.

Fig. 8
Fig. 8

Experimental arrangement for measuring finst.

Fig. 9
Fig. 9

Top, Measured finst for amplified pulses at 661.95 nm, for which no shifts are expected. The pump energy is 7.5 mJ. For the measured waveforms only the frequency behavior between the 10% intensity points is shown. The error bars are calculated as described in Ref. 14. The rms deviation between the measured waveform and the expected zero shift is 1.7 MHz, measured between the 10% intensity points. Bottom, Envelope of amplified pulse.

Fig. 10
Fig. 10

Top, Comparison of measured (dashed curve) and calculated (solid curve) finst as in Fig. 9 but at 636 nm, where αd′(ω) = 2.8 × 10−27 m3. The rms deviation between the measured and the calculated waveforms is 5 MHz. Bottom, Envelope of amplified pulse.

Fig. 11
Fig. 11

Same as Fig. 10 but for an injected beam at 677.6 nm, where αd′(ω) = −2.8 × 10−27 m3. The rms deviation between the measured and the calculated waveforms is 5.4 MHz.

Tables (1)

Tables Icon

Table 1 Effect of Pump Intensity on the Chirp of the Amplified Pulsea

Equations (29)

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

( - 2 z 2 + 1 c 2 2 t 2 ) E ( z , t ) = - μ 0 2 t 2 P ( z , t ) .
E ( z , t ) = e ^ E ( z , t ) exp [ ( i ( k · z - ω t ) ] + c . c . ,
E ( z , t ) = A ( z , t ) exp [ i ϕ ( z , t ) ] .
P s ( z , t ) = 0 χ s ( ω ) E ( z , t ) ,
P d ( z , t ) = 0 χ d ( ω , z , t ) E ( z , t ) ,
N = N 0 ( z , t ) + N 1 ( z , t ) ,
χ d ( ω , z , t ) = α 0 ( ω ) N 0 ( z , t ) + α 1 ( ω ) N 1 ( z , t ) ,
[ z + n s ( ω ) c t ] E ( z , t ) = - ω 2 2 k c 2 [ χ d ( ω , z , t ) - i χ d ( ω , z , t ) ] E ( z , t ) ,
[ z + n s ( ω ) c t ] A ( z , t ) = { - ω 2 2 k c 2 [ α 1 ( ω ) N 1 ( z , t ) ] } A ( z , t ) ,
[ z + n s ( ω ) c t ] ϕ ( z , t ) = ω 2 2 k c 2 [ α 1 ( ω ) N 1 ( z , t ) + α 0 ( ω ) N 0 ( z , t ) ] ,
t τ = t - n s ( ω ) c z ,
z z = z .
A ( z , τ ) = A ( 0 , t ) exp [ - ω 2 2 k c 2 α 1 ( ω ) N 1 ( z , τ ) z ] ,
ϕ ( z , τ ) = ϕ ( 0 , τ ) + ω 2 2 k c 2 0 z [ α 1 ( ω ) N 1 ( z , τ ) + α 0 ( ω ) N 0 ( z , τ ) ] d z ,
f inst ( z , τ ) = - 1 2 π d d τ ϕ ( z , τ ) .
f inst ( z , τ ) = f inst ( 0 , τ ) - 1 2 λ n s ( ω ) × d d τ { 0 z [ α 1 ( ω ) N 1 ( z , τ ) + α 0 ( ω ) N 0 ( z , τ ) ] d z } ,
f inst ( z , τ ) = f inst ( 0 , τ ) - α d ( ω ) 2 λ n s ( ω ) d d τ [ 0 z N 1 ( z , τ ) d z ] ,
α d ( ω ) = α 1 ( ω ) - α 0 ( ω ) .
f inst ( L , τ ) = f inst ( 0 , τ ) - L 2 n s ( ω ) λ α d ( ω ) d d τ N 1 ( τ ) .
g ( ω , τ ) = σ ( ω ) N 1 ( τ ) - σ a ( ω ) N 0 ( τ ) ,
σ s ( ω ) = - ω c n s ( ω ) α 1 ( ω ) ,
σ a ( ω ) = ω c n s ( ω ) α 0 ( ω ) .
f inst ( L , τ ) = f inst ( 0 , τ ) + L 4 π ρ ( ω ) d d τ g ( ω , τ ) ,
ρ ( ω ) = α d ( ω ) α d ( w ) .
d d τ N 1 ( τ ) = γ p ( τ ) N - ( γ p ( τ ) + γ s ( τ ) + γ f ) N 1 ( τ ) ,
γ p ( τ ) = σ a ( ω p ) λ p I p ( τ ) h c ,
γ s ( τ ) = σ s ( ω ) λ I ( τ ) h c ,
α 1 ( ω ) = 2 P 0 [ ω α 1 ( ω ) π ( ω 2 - ω 2 ) ] d ω ,
α 1 ( ω ) = - 2 c n s ( ω ) P 0 [ σ s ( ω ) π ( ω 2 - ω 2 ) ] d ω .

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