Abstract

We present a theoretical and experimental study of the time evolution of the saturation-spectroscopy line shape when the saturating beam is suddenly switched on. We consider two metastable levels coupled by two Doppler-broadened transitions with a common excited level. In this special case, the fast transients from the upper level are rapidly damped, and the slow transients from populations and from Raman coherence in the metastable levels are easy to observe and are used for studying the collisional relaxation. For the sake of clarity, the theoretical analysis is developed according to a perturbation calculation. Nevertheless, saturation of the metastable levels must be taken into account in a more realistic theory (velocity-selective optical pumping): We give only a few results that are in excellent agreement with the experiments performed on neon metastable levels 3P0 and 3P2.

© 1986 Optical Society of America

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  1. For a complete bibliography see Ref. 4.
  2. T. Hänsch, R. Keil, A. Schabert, Ch. Schmelzer, P. Toschek, “Interaction of laser light waves by dynamic Stark splitting,” Z. Phys. 226, 293 (1969).
    [CrossRef]
  3. T. Hänsch, P. Toschek, “Theory of three-level gas laser amplifier,” Z. Phys. 236, 213 (1970).
    [CrossRef]
  4. M. Ducloy, J. R. Leite, M. S. Feld, “Laser saturation spectroscopy in the time-delayed mode. Theory of optical free induction decay in coupled Doppler-broadened systems,” Phys. Rev. A 17, 623 (1978).
    [CrossRef]
  5. M. Ducloy, M. S. Feld, “Laser-induced transients in coupled Doppler-broadened systems,” J. Phys. (Paris) Lett. 37, L173 (1976). In this paper preliminary results are given on switching on transients, but with simplified relaxation processes.
    [CrossRef]
  6. J. R. Leite, M. Ducloy, A. Sachez, D. Seligon, M. S. Feld, “Measurement of molecular alignment relaxation rate in NH3using non-Lorentzian laser-induced saturation resonances,” Phys. Rev. Lett. 39, 1465 (1977); “Laser saturation resonance in NH3observed in the time delayed mode,” Phys. Rev. Lett. 39, 1469 (1977).
    [CrossRef]
  7. See, for instance, Ref. 4. Compared with this reference, we have an additional source term from spontaneous emission in equations for σb and σc.
  8. J. R. Leite, R. L. Sheffield, M. Ducloy, R. D. Sharma, M. S. Feld, “Theory of coherent three-level beats,” Phys. Rev. A 14, 1151 (1976).
    [CrossRef]
  9. D. Hennecart, “Etude des transferts collisionnels de population et d’alignment à l’intérieur des configurations 2p53s et 2p53p du néon,” Thèse d’Etat (Université de Caen, Caen, France, 1982), and references therein.
  10. The evolution of the shape of the population signal has been studied with a more convenient three-level system in which the common level a was the metastable state 3P0and b and c were excited levels. The coherent transient was observed as a broadening of the resonance for short time delays [M. Gorlicki, Thèse d’Etat (UniversitéParis-Nord, Paris, 1985) (to be published)]. Indeed, time resolution was not good enough to permit the observation of oscillating wings. For long delays, the broadening of the line and the growth of a background (both due to velocity-changing collisions) were studied in great detail: M. Gorlicki, A. Peuriot, M. Dumont, J. Phys. (Paris) Lett.41, L275 (1980); M. Dumont, M. Gorlicki, F. Manzano, Ann., Phys. (Paris) 7, 381 (1982); M. Gorlicki, Ch. Lerminiaux, M. Dumont, Phys. Rev. Lett. 49, 1394 (1982); M. Dumont, M. Gorlicki, Ch. Lerminiaux, in Spectral Line Shapes, K. Burnett, ed. (Walter de Gruyter, Berlin, 1983), Vol. 2, p. 881.
    [CrossRef]
  11. γb and γc are mainly due to velocity-changing collisions and metastability-exchange collisions. For the short time delays considered here, this can be seen as a simple relaxation, and the arrival term from this type of collision may be ignored. Nevertheless, this recovery is at the origin of the slow broadening of population signal from the lower level and of a very-long-term slow decrease of the Raman-signal amplitude. See Ref. 10.
  12. H. G. Kuhn, E. L. Lewis, Proc. R. Soc London Ser. A 299, 423 (1967).
    [CrossRef]
  13. M. Dumont, “Velocity selective optical pumping in saturation spectroscopy: transients from populations and coherences,” J. Opt. (Paris) (to be published); “Du pompage optique à l’absorption saturée résolue en temps,” in Hommage à Alfred Kastler, F. Laloe, ed., Ann. Phys. (Paris)10(1985).

1978 (1)

M. Ducloy, J. R. Leite, M. S. Feld, “Laser saturation spectroscopy in the time-delayed mode. Theory of optical free induction decay in coupled Doppler-broadened systems,” Phys. Rev. A 17, 623 (1978).
[CrossRef]

1977 (1)

J. R. Leite, M. Ducloy, A. Sachez, D. Seligon, M. S. Feld, “Measurement of molecular alignment relaxation rate in NH3using non-Lorentzian laser-induced saturation resonances,” Phys. Rev. Lett. 39, 1465 (1977); “Laser saturation resonance in NH3observed in the time delayed mode,” Phys. Rev. Lett. 39, 1469 (1977).
[CrossRef]

1976 (2)

J. R. Leite, R. L. Sheffield, M. Ducloy, R. D. Sharma, M. S. Feld, “Theory of coherent three-level beats,” Phys. Rev. A 14, 1151 (1976).
[CrossRef]

M. Ducloy, M. S. Feld, “Laser-induced transients in coupled Doppler-broadened systems,” J. Phys. (Paris) Lett. 37, L173 (1976). In this paper preliminary results are given on switching on transients, but with simplified relaxation processes.
[CrossRef]

1970 (1)

T. Hänsch, P. Toschek, “Theory of three-level gas laser amplifier,” Z. Phys. 236, 213 (1970).
[CrossRef]

1969 (1)

T. Hänsch, R. Keil, A. Schabert, Ch. Schmelzer, P. Toschek, “Interaction of laser light waves by dynamic Stark splitting,” Z. Phys. 226, 293 (1969).
[CrossRef]

1967 (1)

H. G. Kuhn, E. L. Lewis, Proc. R. Soc London Ser. A 299, 423 (1967).
[CrossRef]

Ducloy, M.

M. Ducloy, J. R. Leite, M. S. Feld, “Laser saturation spectroscopy in the time-delayed mode. Theory of optical free induction decay in coupled Doppler-broadened systems,” Phys. Rev. A 17, 623 (1978).
[CrossRef]

J. R. Leite, M. Ducloy, A. Sachez, D. Seligon, M. S. Feld, “Measurement of molecular alignment relaxation rate in NH3using non-Lorentzian laser-induced saturation resonances,” Phys. Rev. Lett. 39, 1465 (1977); “Laser saturation resonance in NH3observed in the time delayed mode,” Phys. Rev. Lett. 39, 1469 (1977).
[CrossRef]

J. R. Leite, R. L. Sheffield, M. Ducloy, R. D. Sharma, M. S. Feld, “Theory of coherent three-level beats,” Phys. Rev. A 14, 1151 (1976).
[CrossRef]

M. Ducloy, M. S. Feld, “Laser-induced transients in coupled Doppler-broadened systems,” J. Phys. (Paris) Lett. 37, L173 (1976). In this paper preliminary results are given on switching on transients, but with simplified relaxation processes.
[CrossRef]

Dumont, M.

The evolution of the shape of the population signal has been studied with a more convenient three-level system in which the common level a was the metastable state 3P0and b and c were excited levels. The coherent transient was observed as a broadening of the resonance for short time delays [M. Gorlicki, Thèse d’Etat (UniversitéParis-Nord, Paris, 1985) (to be published)]. Indeed, time resolution was not good enough to permit the observation of oscillating wings. For long delays, the broadening of the line and the growth of a background (both due to velocity-changing collisions) were studied in great detail: M. Gorlicki, A. Peuriot, M. Dumont, J. Phys. (Paris) Lett.41, L275 (1980); M. Dumont, M. Gorlicki, F. Manzano, Ann., Phys. (Paris) 7, 381 (1982); M. Gorlicki, Ch. Lerminiaux, M. Dumont, Phys. Rev. Lett. 49, 1394 (1982); M. Dumont, M. Gorlicki, Ch. Lerminiaux, in Spectral Line Shapes, K. Burnett, ed. (Walter de Gruyter, Berlin, 1983), Vol. 2, p. 881.
[CrossRef]

M. Dumont, “Velocity selective optical pumping in saturation spectroscopy: transients from populations and coherences,” J. Opt. (Paris) (to be published); “Du pompage optique à l’absorption saturée résolue en temps,” in Hommage à Alfred Kastler, F. Laloe, ed., Ann. Phys. (Paris)10(1985).

Feld, M. S.

M. Ducloy, J. R. Leite, M. S. Feld, “Laser saturation spectroscopy in the time-delayed mode. Theory of optical free induction decay in coupled Doppler-broadened systems,” Phys. Rev. A 17, 623 (1978).
[CrossRef]

J. R. Leite, M. Ducloy, A. Sachez, D. Seligon, M. S. Feld, “Measurement of molecular alignment relaxation rate in NH3using non-Lorentzian laser-induced saturation resonances,” Phys. Rev. Lett. 39, 1465 (1977); “Laser saturation resonance in NH3observed in the time delayed mode,” Phys. Rev. Lett. 39, 1469 (1977).
[CrossRef]

M. Ducloy, M. S. Feld, “Laser-induced transients in coupled Doppler-broadened systems,” J. Phys. (Paris) Lett. 37, L173 (1976). In this paper preliminary results are given on switching on transients, but with simplified relaxation processes.
[CrossRef]

J. R. Leite, R. L. Sheffield, M. Ducloy, R. D. Sharma, M. S. Feld, “Theory of coherent three-level beats,” Phys. Rev. A 14, 1151 (1976).
[CrossRef]

Gorlicki, M.

The evolution of the shape of the population signal has been studied with a more convenient three-level system in which the common level a was the metastable state 3P0and b and c were excited levels. The coherent transient was observed as a broadening of the resonance for short time delays [M. Gorlicki, Thèse d’Etat (UniversitéParis-Nord, Paris, 1985) (to be published)]. Indeed, time resolution was not good enough to permit the observation of oscillating wings. For long delays, the broadening of the line and the growth of a background (both due to velocity-changing collisions) were studied in great detail: M. Gorlicki, A. Peuriot, M. Dumont, J. Phys. (Paris) Lett.41, L275 (1980); M. Dumont, M. Gorlicki, F. Manzano, Ann., Phys. (Paris) 7, 381 (1982); M. Gorlicki, Ch. Lerminiaux, M. Dumont, Phys. Rev. Lett. 49, 1394 (1982); M. Dumont, M. Gorlicki, Ch. Lerminiaux, in Spectral Line Shapes, K. Burnett, ed. (Walter de Gruyter, Berlin, 1983), Vol. 2, p. 881.
[CrossRef]

The evolution of the shape of the population signal has been studied with a more convenient three-level system in which the common level a was the metastable state 3P0and b and c were excited levels. The coherent transient was observed as a broadening of the resonance for short time delays [M. Gorlicki, Thèse d’Etat (UniversitéParis-Nord, Paris, 1985) (to be published)]. Indeed, time resolution was not good enough to permit the observation of oscillating wings. For long delays, the broadening of the line and the growth of a background (both due to velocity-changing collisions) were studied in great detail: M. Gorlicki, A. Peuriot, M. Dumont, J. Phys. (Paris) Lett.41, L275 (1980); M. Dumont, M. Gorlicki, F. Manzano, Ann., Phys. (Paris) 7, 381 (1982); M. Gorlicki, Ch. Lerminiaux, M. Dumont, Phys. Rev. Lett. 49, 1394 (1982); M. Dumont, M. Gorlicki, Ch. Lerminiaux, in Spectral Line Shapes, K. Burnett, ed. (Walter de Gruyter, Berlin, 1983), Vol. 2, p. 881.
[CrossRef]

Hänsch, T.

T. Hänsch, P. Toschek, “Theory of three-level gas laser amplifier,” Z. Phys. 236, 213 (1970).
[CrossRef]

T. Hänsch, R. Keil, A. Schabert, Ch. Schmelzer, P. Toschek, “Interaction of laser light waves by dynamic Stark splitting,” Z. Phys. 226, 293 (1969).
[CrossRef]

Hennecart, D.

D. Hennecart, “Etude des transferts collisionnels de population et d’alignment à l’intérieur des configurations 2p53s et 2p53p du néon,” Thèse d’Etat (Université de Caen, Caen, France, 1982), and references therein.

Keil, R.

T. Hänsch, R. Keil, A. Schabert, Ch. Schmelzer, P. Toschek, “Interaction of laser light waves by dynamic Stark splitting,” Z. Phys. 226, 293 (1969).
[CrossRef]

Kuhn, H. G.

H. G. Kuhn, E. L. Lewis, Proc. R. Soc London Ser. A 299, 423 (1967).
[CrossRef]

Leite, J. R.

M. Ducloy, J. R. Leite, M. S. Feld, “Laser saturation spectroscopy in the time-delayed mode. Theory of optical free induction decay in coupled Doppler-broadened systems,” Phys. Rev. A 17, 623 (1978).
[CrossRef]

J. R. Leite, M. Ducloy, A. Sachez, D. Seligon, M. S. Feld, “Measurement of molecular alignment relaxation rate in NH3using non-Lorentzian laser-induced saturation resonances,” Phys. Rev. Lett. 39, 1465 (1977); “Laser saturation resonance in NH3observed in the time delayed mode,” Phys. Rev. Lett. 39, 1469 (1977).
[CrossRef]

J. R. Leite, R. L. Sheffield, M. Ducloy, R. D. Sharma, M. S. Feld, “Theory of coherent three-level beats,” Phys. Rev. A 14, 1151 (1976).
[CrossRef]

Lewis, E. L.

H. G. Kuhn, E. L. Lewis, Proc. R. Soc London Ser. A 299, 423 (1967).
[CrossRef]

Peuriot, A.

The evolution of the shape of the population signal has been studied with a more convenient three-level system in which the common level a was the metastable state 3P0and b and c were excited levels. The coherent transient was observed as a broadening of the resonance for short time delays [M. Gorlicki, Thèse d’Etat (UniversitéParis-Nord, Paris, 1985) (to be published)]. Indeed, time resolution was not good enough to permit the observation of oscillating wings. For long delays, the broadening of the line and the growth of a background (both due to velocity-changing collisions) were studied in great detail: M. Gorlicki, A. Peuriot, M. Dumont, J. Phys. (Paris) Lett.41, L275 (1980); M. Dumont, M. Gorlicki, F. Manzano, Ann., Phys. (Paris) 7, 381 (1982); M. Gorlicki, Ch. Lerminiaux, M. Dumont, Phys. Rev. Lett. 49, 1394 (1982); M. Dumont, M. Gorlicki, Ch. Lerminiaux, in Spectral Line Shapes, K. Burnett, ed. (Walter de Gruyter, Berlin, 1983), Vol. 2, p. 881.
[CrossRef]

Sachez, A.

J. R. Leite, M. Ducloy, A. Sachez, D. Seligon, M. S. Feld, “Measurement of molecular alignment relaxation rate in NH3using non-Lorentzian laser-induced saturation resonances,” Phys. Rev. Lett. 39, 1465 (1977); “Laser saturation resonance in NH3observed in the time delayed mode,” Phys. Rev. Lett. 39, 1469 (1977).
[CrossRef]

Schabert, A.

T. Hänsch, R. Keil, A. Schabert, Ch. Schmelzer, P. Toschek, “Interaction of laser light waves by dynamic Stark splitting,” Z. Phys. 226, 293 (1969).
[CrossRef]

Schmelzer, Ch.

T. Hänsch, R. Keil, A. Schabert, Ch. Schmelzer, P. Toschek, “Interaction of laser light waves by dynamic Stark splitting,” Z. Phys. 226, 293 (1969).
[CrossRef]

Seligon, D.

J. R. Leite, M. Ducloy, A. Sachez, D. Seligon, M. S. Feld, “Measurement of molecular alignment relaxation rate in NH3using non-Lorentzian laser-induced saturation resonances,” Phys. Rev. Lett. 39, 1465 (1977); “Laser saturation resonance in NH3observed in the time delayed mode,” Phys. Rev. Lett. 39, 1469 (1977).
[CrossRef]

Sharma, R. D.

J. R. Leite, R. L. Sheffield, M. Ducloy, R. D. Sharma, M. S. Feld, “Theory of coherent three-level beats,” Phys. Rev. A 14, 1151 (1976).
[CrossRef]

Sheffield, R. L.

J. R. Leite, R. L. Sheffield, M. Ducloy, R. D. Sharma, M. S. Feld, “Theory of coherent three-level beats,” Phys. Rev. A 14, 1151 (1976).
[CrossRef]

Toschek, P.

T. Hänsch, P. Toschek, “Theory of three-level gas laser amplifier,” Z. Phys. 236, 213 (1970).
[CrossRef]

T. Hänsch, R. Keil, A. Schabert, Ch. Schmelzer, P. Toschek, “Interaction of laser light waves by dynamic Stark splitting,” Z. Phys. 226, 293 (1969).
[CrossRef]

J. Phys. (Paris) Lett. (1)

M. Ducloy, M. S. Feld, “Laser-induced transients in coupled Doppler-broadened systems,” J. Phys. (Paris) Lett. 37, L173 (1976). In this paper preliminary results are given on switching on transients, but with simplified relaxation processes.
[CrossRef]

Phys. Rev. A (2)

M. Ducloy, J. R. Leite, M. S. Feld, “Laser saturation spectroscopy in the time-delayed mode. Theory of optical free induction decay in coupled Doppler-broadened systems,” Phys. Rev. A 17, 623 (1978).
[CrossRef]

J. R. Leite, R. L. Sheffield, M. Ducloy, R. D. Sharma, M. S. Feld, “Theory of coherent three-level beats,” Phys. Rev. A 14, 1151 (1976).
[CrossRef]

Phys. Rev. Lett. (1)

J. R. Leite, M. Ducloy, A. Sachez, D. Seligon, M. S. Feld, “Measurement of molecular alignment relaxation rate in NH3using non-Lorentzian laser-induced saturation resonances,” Phys. Rev. Lett. 39, 1465 (1977); “Laser saturation resonance in NH3observed in the time delayed mode,” Phys. Rev. Lett. 39, 1469 (1977).
[CrossRef]

Proc. R. Soc London Ser. A (1)

H. G. Kuhn, E. L. Lewis, Proc. R. Soc London Ser. A 299, 423 (1967).
[CrossRef]

Z. Phys. (2)

T. Hänsch, R. Keil, A. Schabert, Ch. Schmelzer, P. Toschek, “Interaction of laser light waves by dynamic Stark splitting,” Z. Phys. 226, 293 (1969).
[CrossRef]

T. Hänsch, P. Toschek, “Theory of three-level gas laser amplifier,” Z. Phys. 236, 213 (1970).
[CrossRef]

Other (6)

For a complete bibliography see Ref. 4.

D. Hennecart, “Etude des transferts collisionnels de population et d’alignment à l’intérieur des configurations 2p53s et 2p53p du néon,” Thèse d’Etat (Université de Caen, Caen, France, 1982), and references therein.

The evolution of the shape of the population signal has been studied with a more convenient three-level system in which the common level a was the metastable state 3P0and b and c were excited levels. The coherent transient was observed as a broadening of the resonance for short time delays [M. Gorlicki, Thèse d’Etat (UniversitéParis-Nord, Paris, 1985) (to be published)]. Indeed, time resolution was not good enough to permit the observation of oscillating wings. For long delays, the broadening of the line and the growth of a background (both due to velocity-changing collisions) were studied in great detail: M. Gorlicki, A. Peuriot, M. Dumont, J. Phys. (Paris) Lett.41, L275 (1980); M. Dumont, M. Gorlicki, F. Manzano, Ann., Phys. (Paris) 7, 381 (1982); M. Gorlicki, Ch. Lerminiaux, M. Dumont, Phys. Rev. Lett. 49, 1394 (1982); M. Dumont, M. Gorlicki, Ch. Lerminiaux, in Spectral Line Shapes, K. Burnett, ed. (Walter de Gruyter, Berlin, 1983), Vol. 2, p. 881.
[CrossRef]

γb and γc are mainly due to velocity-changing collisions and metastability-exchange collisions. For the short time delays considered here, this can be seen as a simple relaxation, and the arrival term from this type of collision may be ignored. Nevertheless, this recovery is at the origin of the slow broadening of population signal from the lower level and of a very-long-term slow decrease of the Raman-signal amplitude. See Ref. 10.

M. Dumont, “Velocity selective optical pumping in saturation spectroscopy: transients from populations and coherences,” J. Opt. (Paris) (to be published); “Du pompage optique à l’absorption saturée résolue en temps,” in Hommage à Alfred Kastler, F. Laloe, ed., Ann. Phys. (Paris)10(1985).

See, for instance, Ref. 4. Compared with this reference, we have an additional source term from spontaneous emission in equations for σb and σc.

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Figures (10)

Fig. 1
Fig. 1

Energy-level diagram.

Fig. 2
Fig. 2

Iteration method; scheme of the different paths leading to σbu(3). Level a is assumed to be empty at zero order (na = 0). Each arrow corresponds to the solution of a differential equation and introduces a new transient term 1 − exp(−Lit), where Li is really (γi) for a population and complex (Lij) for a coherence. Only the first Vp arrow from nb to σba(1) represents a stationary solution (Vp is constant).

Fig. 3
Fig. 3

Experimental setup.

Fig. 4
Fig. 4

Time diagram of pulses.

Fig. 5
Fig. 5

Saturated absorption signal obtained by scanning the probe frequency (time delay, 5 μsec; δs, 200 MHz). The right-hand-side resonance corresponds to counterpropagating beams (population signal). The left-hand-side resonance corresponds to copropagating beams (Raman + population signals).

Fig. 6
Fig. 6

Raman-signal line shape for t = 5 μsec. The experimental population signal is extracted first: One gets the curve A, which is the Raman signal alone. Then the Raman signal is decomposed into two Lorentzian curves, B and C. The narrow component B gives the Raman-coherence lifetime.

Fig. 7
Fig. 7

Raman-signal line shape from the perturbative calculation. For t = ∞ the signal has been decomposed into a narrow (N) and a broad (B) component. Values of the parameters are Γ = 5.5 MHz, Γbc = 1.5 MHz, and kp/ks = 1.048.

Fig. 8
Fig. 8

Experimental curves for P = 50 mTorr. The time delays between the beginning of the saturating pulse and the beginning of the boxcar gate are A, 60 nsec; B, 160 nsec; C, 560 nsec. The population signal has not been extracted. It is small for these delays (positive for A and B and almost zero for C). Line broadening and wing oscillations are evident on curves A and B. The positions of the first minima and maxima (arrows) correspond to A, t = 75 nsec and B, 175 nsec. Taking into account the 30-nsec gate width, this is in excellent agreement with the measured time delays (estimated uncertainty 10 nsec).

Fig. 9
Fig. 9

A, Evolution of the Raman-signal width as a function of time (P = 20 mTorr). For t > 0.4 μsec, it is the width of the narrow Lorentzian component. For t < 0.4 μsec, there is no precise method for extracting the narrow component. Then the plotted points are approximate values. B, Raman-signal height at 20 mTorr (the population signal has been extracted). Its evolution is comparable to the theoretical calculation of Fig. 10 (Γ = 5.5 MHz; Γbc = 1.5 MHz; γb−1 = γc−1 = 2 μsec; s = 5). C, Raman height for P = 200 mTorr (not at the same scale); the slow decrease because of saturation does not exist.

Fig. 10
Fig. 10

(A) Time evolution of Raman-signal height from perturbation theory (s = 0) and from VSOP theory. Γ = 5.5 MHz, Γbc = 1.5 MHz, γb−1 = γc−1 = 2 μsec, kp/ks = 1.048, γab/γa = 0.068, γac/γa = 0.495. (B) Comparison of Raman line shapes for s = 0 (dashed curves) and s = 5 (solid curves). For t = 5 μsec the narrow component is strongly decreased and broadened by saturation (the measured value for Γbc becomes 2 MHz instead of 1.5 MHz). For short time delays, saturation has a much smaller effect.

Tables (1)

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Table 1 Glossary of Symbols

Equations (12)

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σ b ( 0 ) = n b ( v ) ,             σ c ( 0 ) = n c ( v ) ,             σ b a ( 1 ) = i 2 n b V p L a b - 1 .
Δ I p = l π 4 ω p V s 2 V p 2 ( k s + k p ) u exp ( - δ s k s u ) × Re ( N c P + N c R c + N b R b ) ,
P = 1 - exp ( - γ a t ) γ a ( L - γ a ) ( 1 - γ a b γ b - γ a ) + 1 - exp ( - L t ) L ( γ a - L ) × ( 1 - γ a b γ b - L ) - γ a b [ 1 - exp ( - γ b t ) ] γ b ( γ a - γ b ) ( L - γ b ) ,
R c = k m k M 1 L - L b c [ 1 - exp ( - L b c t ) L b c - 1 - exp ( - L t ) L ] 1 - 2 ,
R b = k p + k s k s L b c 2 { 1 - exp ( - L b c t ) - k p k s [ 1 - exp ( - k s k p L b c t ) ] } η ( 1 - ) 2 ,
L = Γ + i Δ = k s Γ a b + k p Γ a c k s + k p + i k s δ p - k p δ s k s + k p ,
L b c = k m k M ( Γ b c + k M - k m k m Γ m + i k s δ p - k p δ s k m ) , { = + 1 for copropagating waves = - 1 for counterpropagating waves , { η = 0 ; k m = k s ; k M = k p ; Γ m = Γ a c for k s < k p η = 1 ; k m = k p ; k M = k s ; Γ m = Γ a b for k s > k p .
Re ( P a ) ( t = ) = 1 γ a Γ Γ 2 + Δ 2 .
Re ( P b ) - γ a b γ a γ b 1 Γ 2 + Δ 2 [ 1 - exp ( - γ b t ) ] .
Re ( R c ) Γ b c Γ b c 2 + Δ p 2 - Γ ( β + 1 ) Γ 2 ( β + 1 ) 2 + Δ p 2 ,
Γ b c = Γ b c + ( β - 1 ) Γ = Γ b c + 0.048 Γ .
Γ b , γ c , Γ b c V s γ a , Γ a b , Γ a c .

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