Abstract

Analytical solutions for the slowly varying phase as well as the field amplitude are obtained for invariant pulse propagation in resonant absorbing media consisting of interacting two-level atoms. Such media, having many atoms within a cubic wavelength, have been shown to exhibit large frequency chirps in pulse propagation owing to near-dipole–dipole interactions. Departures from hyperbolic-secant pulse envelopes and pulse areas other than 2π have been shown to occur in invariant pulse propagation. We show that self-phase modulation results from the near-dipole–dipole interactions and can cause strong modification of pulse shapes, including temporal narrowing and multipulse behavior.

© 1991 Optical Society of America

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References

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  1. Y. Ben-Aryeh, C. M. Bowden, and J. C. Englund, Phys. Rev. A 34, 3917 (1986); Y. Ben-Aryeh and C. M. Bowden, Opt. Commun. 59, 224 (1986).
    [Crossref] [PubMed]
  2. See J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975), Chap. 4.
  3. F. A. Hopf, C. M. Bowden, and W. H. Louisell, Phys. Rev. A 29, 2591 (1984).
    [Crossref]
  4. F. A. Hopf and C. M. Bowden, Phys. Rev. A 32, 268 (1985).
    [Crossref] [PubMed]
  5. Y. Ben-Aryeh, C. M. Bowden, and J. C. Englund, Opt. Commun. 61, 147 (1987).
    [Crossref]
  6. R. Inguva and C. M. Bowden, Phys. Rev. A 41, 1670 (1990).
    [Crossref] [PubMed]
  7. Y. Ben-Aryeh and C. M. Bowden, Opt. Commun. 72, 335 (1989).
    [Crossref]
  8. F. Friedberg, S. R. Hartmann, and J. T. Manasseh, Phys. Rev. A 39, 3444 (1989).
    [Crossref] [PubMed]
  9. F. Friedberg, S. R. Hartmann, and J. T. Manasseh, Phys. Rev. A 40, 2446 (1989).
    [Crossref] [PubMed]
  10. C. R. Stroud, C. M. Bowden, and L. Allen, Opt. Commun. 67, 387 (1988).
    [Crossref]
  11. S. L. McCall and C. S. Hahn, Phys. Rev. 183, 457 (1969).
    [Crossref]
  12. L. Matulic and J. H. Eberly, Phys. Rev. A 6, 822 (1972).
    [Crossref]
  13. M. S. Malcuit, J. J. Maki, D. J. Simkin, and R. W. Boyd, Phys. Rev. Lett. 59, 1189 (1987).
    [Crossref] [PubMed]
  14. J. J. Maki, M. S. Malcuit, M. S. Raymer, R. W. Boyd, and P. D. Drummond, Phys. Rev. A 40, 5135 (1989).
    [Crossref] [PubMed]

1990 (1)

R. Inguva and C. M. Bowden, Phys. Rev. A 41, 1670 (1990).
[Crossref] [PubMed]

1989 (4)

Y. Ben-Aryeh and C. M. Bowden, Opt. Commun. 72, 335 (1989).
[Crossref]

F. Friedberg, S. R. Hartmann, and J. T. Manasseh, Phys. Rev. A 39, 3444 (1989).
[Crossref] [PubMed]

F. Friedberg, S. R. Hartmann, and J. T. Manasseh, Phys. Rev. A 40, 2446 (1989).
[Crossref] [PubMed]

J. J. Maki, M. S. Malcuit, M. S. Raymer, R. W. Boyd, and P. D. Drummond, Phys. Rev. A 40, 5135 (1989).
[Crossref] [PubMed]

1988 (1)

C. R. Stroud, C. M. Bowden, and L. Allen, Opt. Commun. 67, 387 (1988).
[Crossref]

1987 (2)

Y. Ben-Aryeh, C. M. Bowden, and J. C. Englund, Opt. Commun. 61, 147 (1987).
[Crossref]

M. S. Malcuit, J. J. Maki, D. J. Simkin, and R. W. Boyd, Phys. Rev. Lett. 59, 1189 (1987).
[Crossref] [PubMed]

1986 (1)

Y. Ben-Aryeh, C. M. Bowden, and J. C. Englund, Phys. Rev. A 34, 3917 (1986); Y. Ben-Aryeh and C. M. Bowden, Opt. Commun. 59, 224 (1986).
[Crossref] [PubMed]

1985 (1)

F. A. Hopf and C. M. Bowden, Phys. Rev. A 32, 268 (1985).
[Crossref] [PubMed]

1984 (1)

F. A. Hopf, C. M. Bowden, and W. H. Louisell, Phys. Rev. A 29, 2591 (1984).
[Crossref]

1972 (1)

L. Matulic and J. H. Eberly, Phys. Rev. A 6, 822 (1972).
[Crossref]

1969 (1)

S. L. McCall and C. S. Hahn, Phys. Rev. 183, 457 (1969).
[Crossref]

Allen, L.

C. R. Stroud, C. M. Bowden, and L. Allen, Opt. Commun. 67, 387 (1988).
[Crossref]

Ben-Aryeh, Y.

Y. Ben-Aryeh and C. M. Bowden, Opt. Commun. 72, 335 (1989).
[Crossref]

Y. Ben-Aryeh, C. M. Bowden, and J. C. Englund, Opt. Commun. 61, 147 (1987).
[Crossref]

Y. Ben-Aryeh, C. M. Bowden, and J. C. Englund, Phys. Rev. A 34, 3917 (1986); Y. Ben-Aryeh and C. M. Bowden, Opt. Commun. 59, 224 (1986).
[Crossref] [PubMed]

Bowden, C. M.

R. Inguva and C. M. Bowden, Phys. Rev. A 41, 1670 (1990).
[Crossref] [PubMed]

Y. Ben-Aryeh and C. M. Bowden, Opt. Commun. 72, 335 (1989).
[Crossref]

C. R. Stroud, C. M. Bowden, and L. Allen, Opt. Commun. 67, 387 (1988).
[Crossref]

Y. Ben-Aryeh, C. M. Bowden, and J. C. Englund, Opt. Commun. 61, 147 (1987).
[Crossref]

Y. Ben-Aryeh, C. M. Bowden, and J. C. Englund, Phys. Rev. A 34, 3917 (1986); Y. Ben-Aryeh and C. M. Bowden, Opt. Commun. 59, 224 (1986).
[Crossref] [PubMed]

F. A. Hopf and C. M. Bowden, Phys. Rev. A 32, 268 (1985).
[Crossref] [PubMed]

F. A. Hopf, C. M. Bowden, and W. H. Louisell, Phys. Rev. A 29, 2591 (1984).
[Crossref]

Boyd, R. W.

J. J. Maki, M. S. Malcuit, M. S. Raymer, R. W. Boyd, and P. D. Drummond, Phys. Rev. A 40, 5135 (1989).
[Crossref] [PubMed]

M. S. Malcuit, J. J. Maki, D. J. Simkin, and R. W. Boyd, Phys. Rev. Lett. 59, 1189 (1987).
[Crossref] [PubMed]

Drummond, P. D.

J. J. Maki, M. S. Malcuit, M. S. Raymer, R. W. Boyd, and P. D. Drummond, Phys. Rev. A 40, 5135 (1989).
[Crossref] [PubMed]

Eberly, J. H.

L. Matulic and J. H. Eberly, Phys. Rev. A 6, 822 (1972).
[Crossref]

Englund, J. C.

Y. Ben-Aryeh, C. M. Bowden, and J. C. Englund, Opt. Commun. 61, 147 (1987).
[Crossref]

Y. Ben-Aryeh, C. M. Bowden, and J. C. Englund, Phys. Rev. A 34, 3917 (1986); Y. Ben-Aryeh and C. M. Bowden, Opt. Commun. 59, 224 (1986).
[Crossref] [PubMed]

Friedberg, F.

F. Friedberg, S. R. Hartmann, and J. T. Manasseh, Phys. Rev. A 39, 3444 (1989).
[Crossref] [PubMed]

F. Friedberg, S. R. Hartmann, and J. T. Manasseh, Phys. Rev. A 40, 2446 (1989).
[Crossref] [PubMed]

Hahn, C. S.

S. L. McCall and C. S. Hahn, Phys. Rev. 183, 457 (1969).
[Crossref]

Hartmann, S. R.

F. Friedberg, S. R. Hartmann, and J. T. Manasseh, Phys. Rev. A 40, 2446 (1989).
[Crossref] [PubMed]

F. Friedberg, S. R. Hartmann, and J. T. Manasseh, Phys. Rev. A 39, 3444 (1989).
[Crossref] [PubMed]

Hopf, F. A.

F. A. Hopf and C. M. Bowden, Phys. Rev. A 32, 268 (1985).
[Crossref] [PubMed]

F. A. Hopf, C. M. Bowden, and W. H. Louisell, Phys. Rev. A 29, 2591 (1984).
[Crossref]

Inguva, R.

R. Inguva and C. M. Bowden, Phys. Rev. A 41, 1670 (1990).
[Crossref] [PubMed]

Jackson, J. D.

See J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975), Chap. 4.

Louisell, W. H.

F. A. Hopf, C. M. Bowden, and W. H. Louisell, Phys. Rev. A 29, 2591 (1984).
[Crossref]

Maki, J. J.

J. J. Maki, M. S. Malcuit, M. S. Raymer, R. W. Boyd, and P. D. Drummond, Phys. Rev. A 40, 5135 (1989).
[Crossref] [PubMed]

M. S. Malcuit, J. J. Maki, D. J. Simkin, and R. W. Boyd, Phys. Rev. Lett. 59, 1189 (1987).
[Crossref] [PubMed]

Malcuit, M. S.

J. J. Maki, M. S. Malcuit, M. S. Raymer, R. W. Boyd, and P. D. Drummond, Phys. Rev. A 40, 5135 (1989).
[Crossref] [PubMed]

M. S. Malcuit, J. J. Maki, D. J. Simkin, and R. W. Boyd, Phys. Rev. Lett. 59, 1189 (1987).
[Crossref] [PubMed]

Manasseh, J. T.

F. Friedberg, S. R. Hartmann, and J. T. Manasseh, Phys. Rev. A 40, 2446 (1989).
[Crossref] [PubMed]

F. Friedberg, S. R. Hartmann, and J. T. Manasseh, Phys. Rev. A 39, 3444 (1989).
[Crossref] [PubMed]

Matulic, L.

L. Matulic and J. H. Eberly, Phys. Rev. A 6, 822 (1972).
[Crossref]

McCall, S. L.

S. L. McCall and C. S. Hahn, Phys. Rev. 183, 457 (1969).
[Crossref]

Raymer, M. S.

J. J. Maki, M. S. Malcuit, M. S. Raymer, R. W. Boyd, and P. D. Drummond, Phys. Rev. A 40, 5135 (1989).
[Crossref] [PubMed]

Simkin, D. J.

M. S. Malcuit, J. J. Maki, D. J. Simkin, and R. W. Boyd, Phys. Rev. Lett. 59, 1189 (1987).
[Crossref] [PubMed]

Stroud, C. R.

C. R. Stroud, C. M. Bowden, and L. Allen, Opt. Commun. 67, 387 (1988).
[Crossref]

Opt. Commun. (3)

Y. Ben-Aryeh, C. M. Bowden, and J. C. Englund, Opt. Commun. 61, 147 (1987).
[Crossref]

Y. Ben-Aryeh and C. M. Bowden, Opt. Commun. 72, 335 (1989).
[Crossref]

C. R. Stroud, C. M. Bowden, and L. Allen, Opt. Commun. 67, 387 (1988).
[Crossref]

Phys. Rev. (1)

S. L. McCall and C. S. Hahn, Phys. Rev. 183, 457 (1969).
[Crossref]

Phys. Rev. A (8)

L. Matulic and J. H. Eberly, Phys. Rev. A 6, 822 (1972).
[Crossref]

J. J. Maki, M. S. Malcuit, M. S. Raymer, R. W. Boyd, and P. D. Drummond, Phys. Rev. A 40, 5135 (1989).
[Crossref] [PubMed]

F. Friedberg, S. R. Hartmann, and J. T. Manasseh, Phys. Rev. A 39, 3444 (1989).
[Crossref] [PubMed]

F. Friedberg, S. R. Hartmann, and J. T. Manasseh, Phys. Rev. A 40, 2446 (1989).
[Crossref] [PubMed]

R. Inguva and C. M. Bowden, Phys. Rev. A 41, 1670 (1990).
[Crossref] [PubMed]

Y. Ben-Aryeh, C. M. Bowden, and J. C. Englund, Phys. Rev. A 34, 3917 (1986); Y. Ben-Aryeh and C. M. Bowden, Opt. Commun. 59, 224 (1986).
[Crossref] [PubMed]

F. A. Hopf, C. M. Bowden, and W. H. Louisell, Phys. Rev. A 29, 2591 (1984).
[Crossref]

F. A. Hopf and C. M. Bowden, Phys. Rev. A 32, 268 (1985).
[Crossref] [PubMed]

Phys. Rev. Lett. (1)

M. S. Malcuit, J. J. Maki, D. J. Simkin, and R. W. Boyd, Phys. Rev. Lett. 59, 1189 (1987).
[Crossref] [PubMed]

Other (1)

See J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975), Chap. 4.

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Figures (4)

Fig. 1
Fig. 1

Pulse envelope ɛ (dashed curve) and pulse quadrature phase, ɛ cos ϕ (solid curve) versus retarded time τ, for ɛ = 10 and γ ¯ = 0 in units of β.

Fig. 2
Fig. 2

Pulse envelope ɛ (dashed curve) and pulse quadrature phase, ɛ cos ϕ (solid curve) versus retarded time τ, for the conditions of Fig. 1 except that γ ¯ = −0.6.

Fig. 3
Fig. 3

Pulse envelope ɛ (dashed curve) and pulse quadrature phase ɛ cos ϕ (solid curve) versus retarded time τ, for the conditions of Fig. 1 except that γ ¯ = −0.95. The horizontal scale is same as that for Figs. 1 and 2.

Fig. 4
Fig. 4

Pulse quadrature phase ɛ cos ϕ versus retarded time τ, for the conditions of Fig. 1 except that γ ¯ = −0.99. The horizontal scale is same as that for Figs. 1 and 2.

Equations (26)

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u ˙ = - ( γ - ϕ ˙ + w ) v ,
v ˙ = ( γ - ϕ ˙ + w ) u + κ ɛ w ,
w ˙ = - κ ɛ v ,
ɛ ˙ = - ( κ / m 2 ) v ,
( ϕ ˙ + Δ k δ ) ɛ = κ m 2 u .
w = w 0 + ( m 2 / 2 ) ɛ 2 ,
ϕ ¨ ɛ 2 + ϕ ˙ ɛ ˙ 2 + 1 2 [ Δ k δ - γ - w 0 ] ɛ ˙ 2 - m 2 4 ɛ 2 ɛ ˙ 2 = 0.
Δ k δ - γ - w 0 = 0.
ϕ ˙ = c 1 ɛ 2 + m 2 8 ɛ 2 ,
u = m 2 κ ( c 1 ɛ + γ ¯ ɛ + m 2 8 ɛ 3 ) ,
u = m 2 κ ( c 1 ɛ + γ ¯ ɛ + m 2 8 ɛ 3 ) + c 3 ,
ɛ 2 ( ɛ ˙ ) 2 = ( κ 2 / 4 ) ( - D ɛ 8 + F ɛ 6 + B ɛ 4 - A ɛ 2 - G ) .
G = 4 c 1 2 κ 2 ,             A = 8 c 1 γ ¯ κ 2 ,             F = - γ ¯ m 2 κ 2 - 1 , D = 2 m 4 16 κ 2 ,             B = 4 m 2 ( 1 - γ ¯ 2 m 2 κ 2 - c 1 m 4 4 κ 2 ) .
S 0 S d S ( - D S 4 + F S 3 + B S 2 - A S - G ) 1 / 2 = κ τ ,
ɛ = 2 m sech ( κ m τ ) ,             = c 1 = γ = 0.
B = 4 m 2 ( 1 - γ ¯ 2 m 2 κ 2 ) ,             A = G = 0 ,             c 1 = 0 ,             0.
S 0 S d S S ( - D S 2 + F S + B ) 1 / 2 = κ τ .
P ( S ) S 2 ( - D S 2 + F S + B ) ,
S = 0 ,             S ± = F 2 D [ 1 ± ( 1 + 4 B D F 2 ) 1 / 2 ] .
- ( β - ) < γ < ( β + ) ,
ɛ = ɛ 0 cos ( ½ Φ ) [ cosh 2 ( λ τ ) - sin 2 ( ½ Φ ) ] 1 / 2 ,
ɛ 0 = λ 2 1 / 2 κ ( g + b ) 1 / 2 ,             λ 2 = κ 2 B / 4 , tan Φ = f / b ,             b = - F / 4 , f 2 = B D / 4 ,             g = ( f 2 + b 2 ) 1 / 2 .
ϕ ˙ = M cosh 2 λ τ - Q ,
M = m 2 ɛ 0 2 8 cos 2 ( 1 2 Φ ) ,             Q = sin 2 ( 1 2 Φ ) .
ϕ ( τ ) = ϕ 0 + m 2 ɛ 0 2 8 λ cot ( 1 2 Φ ) arctan [ tan ( 1 2 Φ ) tanh ( λ τ ) ] ,
8 m 2 - + ϕ ˙ d τ = 2 λ [ ɛ 0 2 ( Φ / 2 ) tan ( Φ / 2 ) ] .

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