Abstract

Off-resonance light-pulse propagation in a homogeneously broadened medium composed of two-level atoms is numerically simulated. The dependence of the stabilization effect of the pulse area on the detuning and perpendicular relaxation rate is investigated. The dependence of carrier frequency on propagation distance is analyzed.

© 1995 Optical Society of America

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References

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  1. S. L. McCall and E. L. Hahn, “Self-induced transparency by pulsed coherent light,” Phys. Rev. Lett. 18, 908–911 (1967).
    [Crossref]
  2. S. L. McCall and E. L. Hahn, “Self-induced transparency,” Phys. Rev. 183, 457–485 (1969).
    [Crossref]
  3. M. J. Ablowitz, D. J. Kaup, and A. C. Newell, “Coherent pulse propagation, a dispersive, irreversible phenomenon,” J. Math. Phys. 15, 1852–1858 (1974).
    [Crossref]
  4. D. J. Kaup, “Coherent pulse propagation: a comparison of the complete solution with the McCall–Hahn theory and others,” Phys. Rev. A 16, 704–719 (1977).
    [Crossref]
  5. G. L. Lamb, “Analytical description of ultrashort optical pulse propagation in a resonant medium,” Rev. Mod. Phys. 43, 99–124 (1971).
    [Crossref]
  6. A. I. Maimistov, A. M. Basharov, S. O. Elyutin, and Yu. M. Sklyarov, “Present state of self-induced transparency theory,” Phys. Rep. 191, 1–108 (1990).
    [Crossref]
  7. A. M. Alhasan, J. Fiutak, and W. Miklaszewski, “The influence of the atomic relaxation on the resonant propagation of short light pulses,” Z. Phys. B 88, 349–358 (1992).
    [Crossref]
  8. W. Miklaszewski and J. Fiutak, “The effect of the homogeneous broadening on the propagation of the light pulses,” Z. Phys. B 93, 491–499 (1994).
    [Crossref]
  9. R. T. Deck and G. L. Lamb, “Phase variation in coherent-pulse propagation,” Phys. Rev. A 12, 1503–1512 (1975).
    [Crossref]
  10. L. V. Hmurcik and D. J. Kaup, “Solitons created by chirped initial profiles in coherent pulse propagation,” J. Opt. Soc. Am. 69, 597–604 (1979).
    [Crossref]
  11. J. C. Diels and E. L. Hahn, “Carrier-frequency distance dependence of a pulse propagating in a two-level system,” Phys. Rev. A 8, 1084–1110 (1973).
    [Crossref]
  12. L. Matulic and J. H. Eberly, “Analytic study of pulse chirping in self-induced transparency,” Phys. Rev. A 6, 822–836 (1972).
    [Crossref]
  13. J. C. Diels, “Self-induced transparency in near resonant media,” Phys. Lett. A 31, 111–112 (1970).
    [Crossref]
  14. J. C. Diels and E. L. Hahn, “Phase modulation effects in ruby,” Phys. Rev. A 10, 2501–2509 (1974).
    [Crossref]
  15. R. E. Slusher and H. M. Gibbs, “Self-induced transparency in atomic rubidium,” Phys. Rev. A 5, 1634–1659 (1972).
    [Crossref]
  16. G. L. Lamb, “Phase variation in coherent-pulse propagation,” Phys. Rev. Lett. 31, 196–199 (1973).
    [Crossref]
  17. J. Czub, J. Fiutak, and W. Miklaszewski, “On the scattering of a light pulse on a single atom perturbed by collisions with inert gas atoms,” Z. Phys. D 9, 287–295 (1988).
    [Crossref]
  18. A. Icsevgi and W. E. Lamb, “Propagation of light pulses in a laser amplifier,” Phys. Rev. 185, 517–545 (1969).
    [Crossref]
  19. J. Fiutak, S. Kryszewski, and W. Miklaszewski, “On the scattering of a light pulse on a single atom perturbed by collisions with inert gas atoms II. The S0–P1transition,” Z. Phys. D 15, 93–104 (1990).
    [Crossref]
  20. W. Miklaszewski and F. Rebentrost, “Classical path study of excitation of a collision system by ultrashort laser pulse,” Z. Phys. D 23, 249–261 (1992).
    [Crossref]
  21. T. Sizer and M. G. Raymer, “Atomic collisions in the presence of intense, ultrashort laser pulses,” Phys. Rev. A 36, 2643–2658 (1987).
    [Crossref] [PubMed]

1994 (1)

W. Miklaszewski and J. Fiutak, “The effect of the homogeneous broadening on the propagation of the light pulses,” Z. Phys. B 93, 491–499 (1994).
[Crossref]

1992 (2)

A. M. Alhasan, J. Fiutak, and W. Miklaszewski, “The influence of the atomic relaxation on the resonant propagation of short light pulses,” Z. Phys. B 88, 349–358 (1992).
[Crossref]

W. Miklaszewski and F. Rebentrost, “Classical path study of excitation of a collision system by ultrashort laser pulse,” Z. Phys. D 23, 249–261 (1992).
[Crossref]

1990 (2)

J. Fiutak, S. Kryszewski, and W. Miklaszewski, “On the scattering of a light pulse on a single atom perturbed by collisions with inert gas atoms II. The S0–P1transition,” Z. Phys. D 15, 93–104 (1990).
[Crossref]

A. I. Maimistov, A. M. Basharov, S. O. Elyutin, and Yu. M. Sklyarov, “Present state of self-induced transparency theory,” Phys. Rep. 191, 1–108 (1990).
[Crossref]

1988 (1)

J. Czub, J. Fiutak, and W. Miklaszewski, “On the scattering of a light pulse on a single atom perturbed by collisions with inert gas atoms,” Z. Phys. D 9, 287–295 (1988).
[Crossref]

1987 (1)

T. Sizer and M. G. Raymer, “Atomic collisions in the presence of intense, ultrashort laser pulses,” Phys. Rev. A 36, 2643–2658 (1987).
[Crossref] [PubMed]

1979 (1)

1977 (1)

D. J. Kaup, “Coherent pulse propagation: a comparison of the complete solution with the McCall–Hahn theory and others,” Phys. Rev. A 16, 704–719 (1977).
[Crossref]

1975 (1)

R. T. Deck and G. L. Lamb, “Phase variation in coherent-pulse propagation,” Phys. Rev. A 12, 1503–1512 (1975).
[Crossref]

1974 (2)

M. J. Ablowitz, D. J. Kaup, and A. C. Newell, “Coherent pulse propagation, a dispersive, irreversible phenomenon,” J. Math. Phys. 15, 1852–1858 (1974).
[Crossref]

J. C. Diels and E. L. Hahn, “Phase modulation effects in ruby,” Phys. Rev. A 10, 2501–2509 (1974).
[Crossref]

1973 (2)

G. L. Lamb, “Phase variation in coherent-pulse propagation,” Phys. Rev. Lett. 31, 196–199 (1973).
[Crossref]

J. C. Diels and E. L. Hahn, “Carrier-frequency distance dependence of a pulse propagating in a two-level system,” Phys. Rev. A 8, 1084–1110 (1973).
[Crossref]

1972 (2)

L. Matulic and J. H. Eberly, “Analytic study of pulse chirping in self-induced transparency,” Phys. Rev. A 6, 822–836 (1972).
[Crossref]

R. E. Slusher and H. M. Gibbs, “Self-induced transparency in atomic rubidium,” Phys. Rev. A 5, 1634–1659 (1972).
[Crossref]

1971 (1)

G. L. Lamb, “Analytical description of ultrashort optical pulse propagation in a resonant medium,” Rev. Mod. Phys. 43, 99–124 (1971).
[Crossref]

1970 (1)

J. C. Diels, “Self-induced transparency in near resonant media,” Phys. Lett. A 31, 111–112 (1970).
[Crossref]

1969 (2)

S. L. McCall and E. L. Hahn, “Self-induced transparency,” Phys. Rev. 183, 457–485 (1969).
[Crossref]

A. Icsevgi and W. E. Lamb, “Propagation of light pulses in a laser amplifier,” Phys. Rev. 185, 517–545 (1969).
[Crossref]

1967 (1)

S. L. McCall and E. L. Hahn, “Self-induced transparency by pulsed coherent light,” Phys. Rev. Lett. 18, 908–911 (1967).
[Crossref]

Ablowitz, M. J.

M. J. Ablowitz, D. J. Kaup, and A. C. Newell, “Coherent pulse propagation, a dispersive, irreversible phenomenon,” J. Math. Phys. 15, 1852–1858 (1974).
[Crossref]

Alhasan, A. M.

A. M. Alhasan, J. Fiutak, and W. Miklaszewski, “The influence of the atomic relaxation on the resonant propagation of short light pulses,” Z. Phys. B 88, 349–358 (1992).
[Crossref]

Basharov, A. M.

A. I. Maimistov, A. M. Basharov, S. O. Elyutin, and Yu. M. Sklyarov, “Present state of self-induced transparency theory,” Phys. Rep. 191, 1–108 (1990).
[Crossref]

Czub, J.

J. Czub, J. Fiutak, and W. Miklaszewski, “On the scattering of a light pulse on a single atom perturbed by collisions with inert gas atoms,” Z. Phys. D 9, 287–295 (1988).
[Crossref]

Deck, R. T.

R. T. Deck and G. L. Lamb, “Phase variation in coherent-pulse propagation,” Phys. Rev. A 12, 1503–1512 (1975).
[Crossref]

Diels, J. C.

J. C. Diels and E. L. Hahn, “Phase modulation effects in ruby,” Phys. Rev. A 10, 2501–2509 (1974).
[Crossref]

J. C. Diels and E. L. Hahn, “Carrier-frequency distance dependence of a pulse propagating in a two-level system,” Phys. Rev. A 8, 1084–1110 (1973).
[Crossref]

J. C. Diels, “Self-induced transparency in near resonant media,” Phys. Lett. A 31, 111–112 (1970).
[Crossref]

Eberly, J. H.

L. Matulic and J. H. Eberly, “Analytic study of pulse chirping in self-induced transparency,” Phys. Rev. A 6, 822–836 (1972).
[Crossref]

Elyutin, S. O.

A. I. Maimistov, A. M. Basharov, S. O. Elyutin, and Yu. M. Sklyarov, “Present state of self-induced transparency theory,” Phys. Rep. 191, 1–108 (1990).
[Crossref]

Fiutak, J.

W. Miklaszewski and J. Fiutak, “The effect of the homogeneous broadening on the propagation of the light pulses,” Z. Phys. B 93, 491–499 (1994).
[Crossref]

A. M. Alhasan, J. Fiutak, and W. Miklaszewski, “The influence of the atomic relaxation on the resonant propagation of short light pulses,” Z. Phys. B 88, 349–358 (1992).
[Crossref]

J. Fiutak, S. Kryszewski, and W. Miklaszewski, “On the scattering of a light pulse on a single atom perturbed by collisions with inert gas atoms II. The S0–P1transition,” Z. Phys. D 15, 93–104 (1990).
[Crossref]

J. Czub, J. Fiutak, and W. Miklaszewski, “On the scattering of a light pulse on a single atom perturbed by collisions with inert gas atoms,” Z. Phys. D 9, 287–295 (1988).
[Crossref]

Gibbs, H. M.

R. E. Slusher and H. M. Gibbs, “Self-induced transparency in atomic rubidium,” Phys. Rev. A 5, 1634–1659 (1972).
[Crossref]

Hahn, E. L.

J. C. Diels and E. L. Hahn, “Phase modulation effects in ruby,” Phys. Rev. A 10, 2501–2509 (1974).
[Crossref]

J. C. Diels and E. L. Hahn, “Carrier-frequency distance dependence of a pulse propagating in a two-level system,” Phys. Rev. A 8, 1084–1110 (1973).
[Crossref]

S. L. McCall and E. L. Hahn, “Self-induced transparency,” Phys. Rev. 183, 457–485 (1969).
[Crossref]

S. L. McCall and E. L. Hahn, “Self-induced transparency by pulsed coherent light,” Phys. Rev. Lett. 18, 908–911 (1967).
[Crossref]

Hmurcik, L. V.

Icsevgi, A.

A. Icsevgi and W. E. Lamb, “Propagation of light pulses in a laser amplifier,” Phys. Rev. 185, 517–545 (1969).
[Crossref]

Kaup, D. J.

L. V. Hmurcik and D. J. Kaup, “Solitons created by chirped initial profiles in coherent pulse propagation,” J. Opt. Soc. Am. 69, 597–604 (1979).
[Crossref]

D. J. Kaup, “Coherent pulse propagation: a comparison of the complete solution with the McCall–Hahn theory and others,” Phys. Rev. A 16, 704–719 (1977).
[Crossref]

M. J. Ablowitz, D. J. Kaup, and A. C. Newell, “Coherent pulse propagation, a dispersive, irreversible phenomenon,” J. Math. Phys. 15, 1852–1858 (1974).
[Crossref]

Kryszewski, S.

J. Fiutak, S. Kryszewski, and W. Miklaszewski, “On the scattering of a light pulse on a single atom perturbed by collisions with inert gas atoms II. The S0–P1transition,” Z. Phys. D 15, 93–104 (1990).
[Crossref]

Lamb, G. L.

R. T. Deck and G. L. Lamb, “Phase variation in coherent-pulse propagation,” Phys. Rev. A 12, 1503–1512 (1975).
[Crossref]

G. L. Lamb, “Phase variation in coherent-pulse propagation,” Phys. Rev. Lett. 31, 196–199 (1973).
[Crossref]

G. L. Lamb, “Analytical description of ultrashort optical pulse propagation in a resonant medium,” Rev. Mod. Phys. 43, 99–124 (1971).
[Crossref]

Lamb, W. E.

A. Icsevgi and W. E. Lamb, “Propagation of light pulses in a laser amplifier,” Phys. Rev. 185, 517–545 (1969).
[Crossref]

Maimistov, A. I.

A. I. Maimistov, A. M. Basharov, S. O. Elyutin, and Yu. M. Sklyarov, “Present state of self-induced transparency theory,” Phys. Rep. 191, 1–108 (1990).
[Crossref]

Matulic, L.

L. Matulic and J. H. Eberly, “Analytic study of pulse chirping in self-induced transparency,” Phys. Rev. A 6, 822–836 (1972).
[Crossref]

McCall, S. L.

S. L. McCall and E. L. Hahn, “Self-induced transparency,” Phys. Rev. 183, 457–485 (1969).
[Crossref]

S. L. McCall and E. L. Hahn, “Self-induced transparency by pulsed coherent light,” Phys. Rev. Lett. 18, 908–911 (1967).
[Crossref]

Miklaszewski, W.

W. Miklaszewski and J. Fiutak, “The effect of the homogeneous broadening on the propagation of the light pulses,” Z. Phys. B 93, 491–499 (1994).
[Crossref]

A. M. Alhasan, J. Fiutak, and W. Miklaszewski, “The influence of the atomic relaxation on the resonant propagation of short light pulses,” Z. Phys. B 88, 349–358 (1992).
[Crossref]

W. Miklaszewski and F. Rebentrost, “Classical path study of excitation of a collision system by ultrashort laser pulse,” Z. Phys. D 23, 249–261 (1992).
[Crossref]

J. Fiutak, S. Kryszewski, and W. Miklaszewski, “On the scattering of a light pulse on a single atom perturbed by collisions with inert gas atoms II. The S0–P1transition,” Z. Phys. D 15, 93–104 (1990).
[Crossref]

J. Czub, J. Fiutak, and W. Miklaszewski, “On the scattering of a light pulse on a single atom perturbed by collisions with inert gas atoms,” Z. Phys. D 9, 287–295 (1988).
[Crossref]

Newell, A. C.

M. J. Ablowitz, D. J. Kaup, and A. C. Newell, “Coherent pulse propagation, a dispersive, irreversible phenomenon,” J. Math. Phys. 15, 1852–1858 (1974).
[Crossref]

Raymer, M. G.

T. Sizer and M. G. Raymer, “Atomic collisions in the presence of intense, ultrashort laser pulses,” Phys. Rev. A 36, 2643–2658 (1987).
[Crossref] [PubMed]

Rebentrost, F.

W. Miklaszewski and F. Rebentrost, “Classical path study of excitation of a collision system by ultrashort laser pulse,” Z. Phys. D 23, 249–261 (1992).
[Crossref]

Sizer, T.

T. Sizer and M. G. Raymer, “Atomic collisions in the presence of intense, ultrashort laser pulses,” Phys. Rev. A 36, 2643–2658 (1987).
[Crossref] [PubMed]

Sklyarov, Yu. M.

A. I. Maimistov, A. M. Basharov, S. O. Elyutin, and Yu. M. Sklyarov, “Present state of self-induced transparency theory,” Phys. Rep. 191, 1–108 (1990).
[Crossref]

Slusher, R. E.

R. E. Slusher and H. M. Gibbs, “Self-induced transparency in atomic rubidium,” Phys. Rev. A 5, 1634–1659 (1972).
[Crossref]

J. Math. Phys. (1)

M. J. Ablowitz, D. J. Kaup, and A. C. Newell, “Coherent pulse propagation, a dispersive, irreversible phenomenon,” J. Math. Phys. 15, 1852–1858 (1974).
[Crossref]

J. Opt. Soc. Am. (1)

Phys. Lett. A (1)

J. C. Diels, “Self-induced transparency in near resonant media,” Phys. Lett. A 31, 111–112 (1970).
[Crossref]

Phys. Rep. (1)

A. I. Maimistov, A. M. Basharov, S. O. Elyutin, and Yu. M. Sklyarov, “Present state of self-induced transparency theory,” Phys. Rep. 191, 1–108 (1990).
[Crossref]

Phys. Rev. (2)

S. L. McCall and E. L. Hahn, “Self-induced transparency,” Phys. Rev. 183, 457–485 (1969).
[Crossref]

A. Icsevgi and W. E. Lamb, “Propagation of light pulses in a laser amplifier,” Phys. Rev. 185, 517–545 (1969).
[Crossref]

Phys. Rev. A (7)

R. T. Deck and G. L. Lamb, “Phase variation in coherent-pulse propagation,” Phys. Rev. A 12, 1503–1512 (1975).
[Crossref]

T. Sizer and M. G. Raymer, “Atomic collisions in the presence of intense, ultrashort laser pulses,” Phys. Rev. A 36, 2643–2658 (1987).
[Crossref] [PubMed]

J. C. Diels and E. L. Hahn, “Phase modulation effects in ruby,” Phys. Rev. A 10, 2501–2509 (1974).
[Crossref]

R. E. Slusher and H. M. Gibbs, “Self-induced transparency in atomic rubidium,” Phys. Rev. A 5, 1634–1659 (1972).
[Crossref]

D. J. Kaup, “Coherent pulse propagation: a comparison of the complete solution with the McCall–Hahn theory and others,” Phys. Rev. A 16, 704–719 (1977).
[Crossref]

J. C. Diels and E. L. Hahn, “Carrier-frequency distance dependence of a pulse propagating in a two-level system,” Phys. Rev. A 8, 1084–1110 (1973).
[Crossref]

L. Matulic and J. H. Eberly, “Analytic study of pulse chirping in self-induced transparency,” Phys. Rev. A 6, 822–836 (1972).
[Crossref]

Phys. Rev. Lett. (2)

S. L. McCall and E. L. Hahn, “Self-induced transparency by pulsed coherent light,” Phys. Rev. Lett. 18, 908–911 (1967).
[Crossref]

G. L. Lamb, “Phase variation in coherent-pulse propagation,” Phys. Rev. Lett. 31, 196–199 (1973).
[Crossref]

Rev. Mod. Phys. (1)

G. L. Lamb, “Analytical description of ultrashort optical pulse propagation in a resonant medium,” Rev. Mod. Phys. 43, 99–124 (1971).
[Crossref]

Z. Phys. B (2)

A. M. Alhasan, J. Fiutak, and W. Miklaszewski, “The influence of the atomic relaxation on the resonant propagation of short light pulses,” Z. Phys. B 88, 349–358 (1992).
[Crossref]

W. Miklaszewski and J. Fiutak, “The effect of the homogeneous broadening on the propagation of the light pulses,” Z. Phys. B 93, 491–499 (1994).
[Crossref]

Z. Phys. D (3)

J. Czub, J. Fiutak, and W. Miklaszewski, “On the scattering of a light pulse on a single atom perturbed by collisions with inert gas atoms,” Z. Phys. D 9, 287–295 (1988).
[Crossref]

J. Fiutak, S. Kryszewski, and W. Miklaszewski, “On the scattering of a light pulse on a single atom perturbed by collisions with inert gas atoms II. The S0–P1transition,” Z. Phys. D 15, 93–104 (1990).
[Crossref]

W. Miklaszewski and F. Rebentrost, “Classical path study of excitation of a collision system by ultrashort laser pulse,” Z. Phys. D 23, 249–261 (1992).
[Crossref]

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Figures (12)

Fig. 1
Fig. 1

Generalized pulse area [Eq. (22)] versus input pulse area for different input pulse detunings. The T2G pulse has duration T = 0.01/γ, Γ = γ/2, and z = 100z0. The dotted curve corresponds to Δ = 10γ.

Fig. 2
Fig. 2

Integral I 2 = 0 ρ 2 ( z , τ ) d τ as a function of the input pulse area for different propagation distances. The T2G pulse has duration T = 0.01, Δ = 200γ, and Γ = γ/2.

Fig. 3
Fig. 3

Dependence of the generalized pulse area on the propagation distance for different input pulse detunings. The T2G pulse has duration T = 0.01, and Γ = γ/2.

Fig. 4
Fig. 4

Modulus of the pulse’s Fourier transform for different relatively small propagation distances. The T2G pulse has duration T = 0.01, and Γ = γ/2.

Fig. 5
Fig. 5

Pulse spectrum for different relatively long propagation distances: (a) the whole spectrum, (b) the spectrum around the atomic frequency. The T2G pulse has duration T = 0.01, and Γ = γ/2.

Fig. 6
Fig. 6

Generalized pulse area function [Eq. (26)] versus retarded time for different propagation distances. The T2G pulse has duration T = 0.01, and Γ = γ/2.

Fig. 7
Fig. 7

Evolution of the pulse envelope in the course of propagation. The T2G pulse has duration T = 0.01, and Γ = γ/2.

Fig. 8
Fig. 8

Generalized pulse area versus input pulse area for different coherence damping rates. The T2G pulse has duration T = 0.01, and z = 100z0.

Fig. 9
Fig. 9

Generalized pulse area as a function of the propagation distance for different detunings. The T2G pulse has duration T = 0.01.

Fig. 10
Fig. 10

Mean carrier frequency versus input pulse area for different input pulse detunings. The T2G pulse has duration T = 0.01, and Γ = γ/2.

Fig. 11
Fig. 11

Average carrier frequency as a function of the input pulse area for different coherence relaxation rates. The T2G pulse has duration T = 0.01.

Fig. 12
Fig. 12

Mean carrier frequency versus propagation distance for different input pulse detunings. The T2G pulse has duration T = 0.01.

Equations (27)

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

z v ( z , τ ) = α ρ 3 ( z , τ ) ,
τ ρ 2 ( z , τ ) = γ ρ 2 ( z , τ ) - 2 Re [ v ( z , τ ) * ρ 3 ( z , τ ) ] ,
τ ρ 3 ( z , τ ) = ( - Γ + i Δ ) ρ 3 ( z , τ ) + v ( z , τ ) [ 2 ρ 2 ( z , τ ) - 1 2 ] ,
Γ = γ 2 + γ 12 coll .
Δ = ω 0 - ω 21 .
v ( 0 , τ ) = v in ( τ ) ,
θ in = 2 0 v in ( τ ) d τ ,
E in 0 v in ( τ ) 2 d τ .
v 0 = 0 v in ( τ ) 2 d τ / 0 v in ( τ ) d τ ,
T = [ 0 v in ( τ ) d τ ] 2 / 0 v in ( τ ) 2 d τ .
θ in = 2 v 0 T ,
E in v 0 2 T .
ρ 2 ( z , 0 ) = ρ 3 ( z , 0 ) = 0.
V ( z , ω ) = 1 2 π 0 v ( z , τ ) exp ( i ω τ ) d τ .
ω = - ω V ( z , ω ) 2 d ω - V ( z , ω ) 2 d ω .
ρ 3 = - 1 2 2 sin θ ( z , τ ) ,
θ ( z , τ ) = 2 0 τ v ( z , τ ) d τ .
2 z τ θ ( z , τ ) = - α 2 sin θ ( z , τ ) .
2 ρ 2 ( z , τ ) 1 ,
V ( z , ω ) = V ( 0 , ω ) exp [ - α Γ z / 2 Γ 2 + ( ω + Δ ) 2 ] × exp [ - i ω + Δ Γ 2 + ( ω + Δ ) 2 ] .
θ ( z ) = 2 0 v ( z , τ ) d τ = 2 π V ( z , 0 ) .
θ Δ ( z ) = 2 | 0 v ( z , τ ) exp ( - i Δ τ ) d τ | = 2 π V ( z , - Δ ) ,
θ Δ ( z ) = θ Δ ( 0 ) exp ( - α 2 Γ z ) .
v in ( τ ) = 64 27 2 π v 0 ( τ T ) 2 exp [ - 8 9 π ( τ T ) 2 ] .
d E ( z ) d z I 2 ( z ) = 0 ρ 2 ( z , τ ) d τ .
θ Δ ( z , τ ) = 2 | 0 τ v ( z , τ ) exp ( - i Δ τ ) d τ |
ω weak = - ω V ( 0 , ω ) 2 × exp [ - α Γ z Γ 2 + ( ω + Δ ) 2 ] d ω / - × V ( 0 , ω ) 2 exp [ - α Γ z Γ 2 + ( ω + Δ ) 2 ] d ω ,

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