Abstract

If an incident pulse is chirped, the critical parameters for self-induced transparency to occur in coherent pulse propagation can no longer be obtained from the well-known McCall-Hahn area theorem. We have been able to obtain these parameters by solving the Zakharov-Shabat eigenvalue equation for the bound-state eigenvalues. We find that critical (threshold) areas will be increased for a chirped incident pulse in almost all cases, except for a box profile or for a pulse that is approximately box-like in shape. In these latter cases, the chirped critical areas will instead decrease for the second and all higher branches. The first branch’s critical area is always increased due to chirping.

© 1979 Optical Society of America

Full Article  |  PDF Article

Corrections

William G. McKinley, L. V. Hmurcik, and D. J. Kaup, "Errata," J. Opt. Soc. Am. 69, 1633-1633 (1979)
https://www.osapublishing.org/josa/abstract.cfm?uri=josa-69-11-1633

References

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  1. S. L. McCall and E. L. Hahn, Phys. Rev. Lett. 18, 908 (1967).
    [CrossRef]
  2. S. L. McCall and E. L. Hahn, Phys. Rev. 183, 457 (1967).
    [CrossRef]
  3. J. C. Diels and E. L. Hahn, Phys. Rev. 8, 1084 (1973).
    [CrossRef]
  4. G. L. Lamb, Phys. Rev. Lett. 31, 196 (1973).
    [CrossRef]
  5. D. J. Kaup, Phys. Rev. A 16, 704 (1977).
    [CrossRef]
  6. M. J. Ablowitz, D. J. Kaup, and A. C. Newell, J. of Math Phys. 1852 (1974).
  7. R. E. Slusher and H. M. Gibbs, Phys. Rev. A 5, 1634 (1972).
    [CrossRef]
  8. Gardner, Green, Kruskal, and Miura, Phys. Rev. Lett. 19, 1095 (1967).
    [CrossRef]
  9. G. L. Lamb, Physica 66, 298 (1973).
    [CrossRef]
  10. D. W. McLaughlin and J. Corones, Phys. Rev. A 10, 2051 (1974).
    [CrossRef]
  11. V. E. Zakharov and A. B. Shabat, Soc. Phys. JETP 34, 62 (1972).
  12. M. J. Ablowitz, D. J. Kaup, A. C. Newell, and H. Segur, Stud. in App. Math. 53, 249 (1974).
  13. R. T. Deck and G. L. Lamb, Phys. Rev. A 12, 1503 (1975).
    [CrossRef]
  14. D. F. Dubois, D. W. Forslund, and E. A. Williams, Phys. Rev. Lett. 33, 1013 (1974).
    [CrossRef]
  15. D. D. Schnack and G. L. Lamb, in Coherence and Quantum Optics edited by L. Mandel and E. Wolf. (Plenum, New York, 1973), p. 23–33.
    [CrossRef]

1977 (1)

D. J. Kaup, Phys. Rev. A 16, 704 (1977).
[CrossRef]

1975 (1)

R. T. Deck and G. L. Lamb, Phys. Rev. A 12, 1503 (1975).
[CrossRef]

1974 (4)

D. F. Dubois, D. W. Forslund, and E. A. Williams, Phys. Rev. Lett. 33, 1013 (1974).
[CrossRef]

D. W. McLaughlin and J. Corones, Phys. Rev. A 10, 2051 (1974).
[CrossRef]

M. J. Ablowitz, D. J. Kaup, A. C. Newell, and H. Segur, Stud. in App. Math. 53, 249 (1974).

M. J. Ablowitz, D. J. Kaup, and A. C. Newell, J. of Math Phys. 1852 (1974).

1973 (3)

J. C. Diels and E. L. Hahn, Phys. Rev. 8, 1084 (1973).
[CrossRef]

G. L. Lamb, Phys. Rev. Lett. 31, 196 (1973).
[CrossRef]

G. L. Lamb, Physica 66, 298 (1973).
[CrossRef]

1972 (2)

V. E. Zakharov and A. B. Shabat, Soc. Phys. JETP 34, 62 (1972).

R. E. Slusher and H. M. Gibbs, Phys. Rev. A 5, 1634 (1972).
[CrossRef]

1967 (3)

Gardner, Green, Kruskal, and Miura, Phys. Rev. Lett. 19, 1095 (1967).
[CrossRef]

S. L. McCall and E. L. Hahn, Phys. Rev. Lett. 18, 908 (1967).
[CrossRef]

S. L. McCall and E. L. Hahn, Phys. Rev. 183, 457 (1967).
[CrossRef]

Ablowitz, M. J.

M. J. Ablowitz, D. J. Kaup, and A. C. Newell, J. of Math Phys. 1852 (1974).

M. J. Ablowitz, D. J. Kaup, A. C. Newell, and H. Segur, Stud. in App. Math. 53, 249 (1974).

Corones, J.

D. W. McLaughlin and J. Corones, Phys. Rev. A 10, 2051 (1974).
[CrossRef]

Deck, R. T.

R. T. Deck and G. L. Lamb, Phys. Rev. A 12, 1503 (1975).
[CrossRef]

Diels, J. C.

J. C. Diels and E. L. Hahn, Phys. Rev. 8, 1084 (1973).
[CrossRef]

Dubois, D. F.

D. F. Dubois, D. W. Forslund, and E. A. Williams, Phys. Rev. Lett. 33, 1013 (1974).
[CrossRef]

Forslund, D. W.

D. F. Dubois, D. W. Forslund, and E. A. Williams, Phys. Rev. Lett. 33, 1013 (1974).
[CrossRef]

Gardner,

Gardner, Green, Kruskal, and Miura, Phys. Rev. Lett. 19, 1095 (1967).
[CrossRef]

Gibbs, H. M.

R. E. Slusher and H. M. Gibbs, Phys. Rev. A 5, 1634 (1972).
[CrossRef]

Green,

Gardner, Green, Kruskal, and Miura, Phys. Rev. Lett. 19, 1095 (1967).
[CrossRef]

Hahn, E. L.

J. C. Diels and E. L. Hahn, Phys. Rev. 8, 1084 (1973).
[CrossRef]

S. L. McCall and E. L. Hahn, Phys. Rev. Lett. 18, 908 (1967).
[CrossRef]

S. L. McCall and E. L. Hahn, Phys. Rev. 183, 457 (1967).
[CrossRef]

Kaup, D. J.

D. J. Kaup, Phys. Rev. A 16, 704 (1977).
[CrossRef]

M. J. Ablowitz, D. J. Kaup, and A. C. Newell, J. of Math Phys. 1852 (1974).

M. J. Ablowitz, D. J. Kaup, A. C. Newell, and H. Segur, Stud. in App. Math. 53, 249 (1974).

Kruskal,

Gardner, Green, Kruskal, and Miura, Phys. Rev. Lett. 19, 1095 (1967).
[CrossRef]

Lamb, G. L.

R. T. Deck and G. L. Lamb, Phys. Rev. A 12, 1503 (1975).
[CrossRef]

G. L. Lamb, Physica 66, 298 (1973).
[CrossRef]

G. L. Lamb, Phys. Rev. Lett. 31, 196 (1973).
[CrossRef]

D. D. Schnack and G. L. Lamb, in Coherence and Quantum Optics edited by L. Mandel and E. Wolf. (Plenum, New York, 1973), p. 23–33.
[CrossRef]

McCall, S. L.

S. L. McCall and E. L. Hahn, Phys. Rev. Lett. 18, 908 (1967).
[CrossRef]

S. L. McCall and E. L. Hahn, Phys. Rev. 183, 457 (1967).
[CrossRef]

McLaughlin, D. W.

D. W. McLaughlin and J. Corones, Phys. Rev. A 10, 2051 (1974).
[CrossRef]

Miura,

Gardner, Green, Kruskal, and Miura, Phys. Rev. Lett. 19, 1095 (1967).
[CrossRef]

Newell, A. C.

M. J. Ablowitz, D. J. Kaup, and A. C. Newell, J. of Math Phys. 1852 (1974).

M. J. Ablowitz, D. J. Kaup, A. C. Newell, and H. Segur, Stud. in App. Math. 53, 249 (1974).

Schnack, D. D.

D. D. Schnack and G. L. Lamb, in Coherence and Quantum Optics edited by L. Mandel and E. Wolf. (Plenum, New York, 1973), p. 23–33.
[CrossRef]

Segur, H.

M. J. Ablowitz, D. J. Kaup, A. C. Newell, and H. Segur, Stud. in App. Math. 53, 249 (1974).

Shabat, A. B.

V. E. Zakharov and A. B. Shabat, Soc. Phys. JETP 34, 62 (1972).

Slusher, R. E.

R. E. Slusher and H. M. Gibbs, Phys. Rev. A 5, 1634 (1972).
[CrossRef]

Williams, E. A.

D. F. Dubois, D. W. Forslund, and E. A. Williams, Phys. Rev. Lett. 33, 1013 (1974).
[CrossRef]

Zakharov, V. E.

V. E. Zakharov and A. B. Shabat, Soc. Phys. JETP 34, 62 (1972).

J. of Math Phys. 1852 (1)

M. J. Ablowitz, D. J. Kaup, and A. C. Newell, J. of Math Phys. 1852 (1974).

Phys. Rev. (2)

S. L. McCall and E. L. Hahn, Phys. Rev. 183, 457 (1967).
[CrossRef]

J. C. Diels and E. L. Hahn, Phys. Rev. 8, 1084 (1973).
[CrossRef]

Phys. Rev. A (4)

D. J. Kaup, Phys. Rev. A 16, 704 (1977).
[CrossRef]

R. E. Slusher and H. M. Gibbs, Phys. Rev. A 5, 1634 (1972).
[CrossRef]

D. W. McLaughlin and J. Corones, Phys. Rev. A 10, 2051 (1974).
[CrossRef]

R. T. Deck and G. L. Lamb, Phys. Rev. A 12, 1503 (1975).
[CrossRef]

Phys. Rev. Lett. (4)

D. F. Dubois, D. W. Forslund, and E. A. Williams, Phys. Rev. Lett. 33, 1013 (1974).
[CrossRef]

Gardner, Green, Kruskal, and Miura, Phys. Rev. Lett. 19, 1095 (1967).
[CrossRef]

S. L. McCall and E. L. Hahn, Phys. Rev. Lett. 18, 908 (1967).
[CrossRef]

G. L. Lamb, Phys. Rev. Lett. 31, 196 (1973).
[CrossRef]

Physica (1)

G. L. Lamb, Physica 66, 298 (1973).
[CrossRef]

Soc. Phys. JETP (1)

V. E. Zakharov and A. B. Shabat, Soc. Phys. JETP 34, 62 (1972).

Stud. in App. Math. (1)

M. J. Ablowitz, D. J. Kaup, A. C. Newell, and H. Segur, Stud. in App. Math. 53, 249 (1974).

Other (1)

D. D. Schnack and G. L. Lamb, in Coherence and Quantum Optics edited by L. Mandel and E. Wolf. (Plenum, New York, 1973), p. 23–33.
[CrossRef]

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Figures (10)

FIG. 1
FIG. 1

Graph of ητp vs θ ¯ 0 for box profile with chirp strengths Δ = 0, 2, 5, 9.

FIG. 2
FIG. 2

Graph of ητp vs θ ¯ 0 for the sech profile with Δ = 0, 0.5, 1.5 and with β = 12.

FIG. 3
FIG. 3

Graph of ητp vs θ ¯ 0 for the sech profile with Δ = 0, 2.5, 5 and with β = 12.

FIG. 4
FIG. 4

Graph of ητp vs θ ¯ 0 for the Gaussian profile with Δ = 0, 5, 7 and with β = 3.12.

FIG. 5
FIG. 5

Graph of ητp vs θ ¯ 0 for the sech profile with η small and computer precision increased. β = 12.

FIG. 6A
FIG. 6A

Graph of ητp vs θ ¯ 0 for the sech profile with Δ = 0, 0.05, 0.1, with η very small, and with only the first branch plotted.β = 12.

FIG. 6B
FIG. 6B

Graph of ητp vs θ ¯ 0 for the sech profile with Δ = 0, 0.05, 0.1, with η very small, and with only the second branch plotted. β = 12.

FIG. 6C
FIG. 6C

Graph of ητp vs θ ¯ 0 for the sech profile with Δ = 0, 0.05, 0.1, with η very small, and with only the third branch plotted. β = 12.

FIG. 7
FIG. 7

Graph of ητp vs θ ¯ 0 for the “clipped” Gaussian profile with Δ = 0, 1.5 and with Δ = 3.12.

FIG. 8
FIG. 8

The results of Deck and Lamb are graphed along with ours for Δ = 0, 0.1.

Equations (38)

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E ( x , t ) = Ɛ ( x , t ) cos ( k x ω t + δ )
E = Re [ ( x , t ) e i ( k x ω t ) ] ,
υ 1 τ + i ζ υ 1 = ( 1 / 2 ) κ υ 2
υ 2 τ i ζ υ 2 = ( 1 / 2 ) κ * υ 1 ,
θ ( x ) = κ d t Ɛ ( x , t ) ,
τ = t n x / c ,
χ = x .
κ = 4 η exp [ i ( e + b χ 2 ξ τ ) ] / cosh ( f + d χ 2 η τ ) ,
ϕ [ 1 0 ] e i ζ τ , τ ,
ψ [ 0 1 ] e i ζ τ , τ + .
ϕ ( a ( ζ ) e i ζ τ b ( ζ ) e i ζ τ ) ,
a = w ( ϕ , ψ ) ϕ 1 ψ 2 ϕ 2 ψ 1 .
Arg [ a ( ξ ) ] | ξ = + ξ = = 2 π N ,
1 2 τ δ ( x , t ) .
δ = Δ β ( τ / τ p ) 2 / [ β + ( τ / τ p ) 2 ] .
γ n = Δ 2 [ ( 2 n 1 ) π 2 ( 2 ( 2 n 1 ) π ( 2 n 1 ) π 6 ) + 0 ( Δ 4 ) .
κ Ɛ = ( θ ¯ 0 / π τ p ) sech ( τ / τ p ) ,
κ Ɛ = ( θ ¯ 0 / π τ p ) exp [ ( τ / τ p ) 2 ] ,
exp i Δ ( τ / τ p ) 2
exp i β Δ ( τ / τ p ) 2 / [ β + ( τ / τ p ) 2 ]
γ 1 = 0.256 , γ 2 = 0.011 , γ 3 = 0.093 ,
( τ ) = ( τ ) ,
υ 1 τ ( ζ , τ ) + i ζ υ 1 ( ζ , τ ) = ( 1 / 2 ) κ ( τ ) υ 2 ( ζ , τ ) , υ 2 τ ( ζ , τ ) i ζ υ 2 ( ζ , τ ) = ( 1 / 2 ) κ * ( τ ) υ 1 ( ζ , τ ) .
υ 1 τ * ( ζ , τ ) i ζ * υ 1 * ( ζ , τ ) = κ * ( τ ) 2 υ 2 * ( ζ , τ ) , υ 2 * τ ( ζ , τ ) + i ζ * υ 2 * ( ζ , τ ) = κ ( τ ) 2 υ 1 * ( ζ , τ ) .
ϕ ( ζ , τ ) = ( 0 1 1 0 ) ψ * ( ζ * , τ ) .
a ( ζ ) = a * ( ζ * ) .
υ 1 = u 1 e i ζ τ , υ 2 = u 2 e + i ζ τ ,
a = u 1 | τ ,
u 1 = A ( τ ) cos Q / 2 + B ( τ ) sin Q / 2 , u 2 = B ( τ ) cos Q / 2 A ( τ ) sin Q / 2 ,
Q ( τ ) = κ τ Ɛ d τ .
( B / A ) | τ = [ cot Q / 2 ] | τ = cot ( θ ¯ 0 / 2 ) .
( B / A ) τ = κ / 2 { ( B / A ) 2 + 1 + ( 2 B / A ) i sin Q sin ( δ + 2 ζ τ ) [ ( B / A ) 2 + 1 ] cos ( δ + 2 ζ τ ) + i [ 1 ( B / A ) 2 ] sin ( δ + 2 ζ τ ) cos Q }
( B / A ) τ = 1 2 θ ¯ 0 { [ ( B / A ) 2 + 1 ] Δ 2 ( τ / τ p ) 4 2 + ( 2 B / A ) i sin [ θ ¯ 0 ( τ / τ p + 1 / 2 ) ] Δ τ 2 + [ 1 ( B / A ) 2 ] i cos [ θ ¯ 0 ( τ / τ p + 1 / 2 ) ] Δ τ 2 }
( B / A ) = θ ¯ 0 2 [ Δ 2 2 1 / 2 τ d τ τ 4 + i 1 / 2 τ d τ Δ τ 2 cos [ θ ¯ 0 ( τ + 1 / 2 ) ] ,
( B / A ) = Δ ² θ ¯ 0 2 [ sin θ ¯ 0 ( 1 2 θ ¯ 0 4 ) + cos θ ¯ 0 ( 1 θ ¯ 0 5 1 8 θ ¯ 0 3 ) + sin 2 θ ¯ 0 ( 49 128 θ ¯ 0 2 + 1 32 θ ¯ 0 6 ) + cos 2 θ ¯ 0 ( 3 16 θ ¯ 0 3 + 11 8 θ ¯ 0 5 ) + 1 12 θ ¯ 0 3 1 θ ¯ 0 5 ] .
γ n = Δ 2 ( 2 n 1 ) 2 π 2 ( 2 ( 2 n 1 ) π ( 2 n 1 ) π 6 ) .
γ n = 4 Δ 2 d τ 2 β τ [ β + τ 2 ] 2 sin [ ( 2 n 1 ) tan 1 sinh τ ) ] × τ d τ 2 β τ [ β + ( τ ) 2 ] 2 cos [ ( 2 n 1 ) ( tan 1 sinh τ ) ] .
γ 1 = 0.2664 γ 2 = 0.01136 γ 3 = 0.093.