Abstract

We introduce new simultaneous, multiple-frequency, solitary-wave solutions to the traveling-wave parametric amplifier. Both degenerate and nondegenerate systems are treated including dispersion. These parametric amplifier simultons are shown to exhibit phase-dependent collisions. Spatial solitary waves are also found in the case of cw fields parametrically coupled in a planar waveguide.

© 1993 Optical Society of America

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References

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  1. G. A. Swartlander and C. T. Law, Phys. Rev. Lett. 69, 2503 (1992).
    [Crossref]
  2. G. R. Allan, S. R. Skinner, D. R. Anderson, and A. L. Smirl, Opt. Lett. 16, 156 (1991).
    [PubMed]
  3. V. E. Zakharov and A. B. Shabat, Sov. Phys. JETP 34, 62 (1972).
  4. D. J. Kaup, A. Rieman, and A. Bers, Rev. Mod. Phys. 51, 275 (1979).
    [Crossref]
  5. V. E. Zakharov and S. V. Manakov, Sov. Phys. JETP Lett. 18, 243 (1973).
  6. J. A. Armstrong, S. S. Jha, and N. S. Shiren, IEEE J. Quantum Electron. QE-6, 123 (1970).
    [Crossref]
  7. Y. N. Karamzin and A. P. Sukhorukov, Sov. Phys. JETP 41, 414 (1976); Y. N. Karamzin, A. P. Sukhorukov, and T. S. Filipchuk, Moscow Univ. Phys. Bull. 19, 73 (1978).
  8. M. G. Raymer, P. D. Drummond, and S. Carter, Opt. Lett. 16, 1189 (1991).
    [Crossref] [PubMed]
  9. M. J. Konopnicki, P. D. Drummond, and J. H. Eberly, Opt. Commun. 36, 313 (1981).
    [Crossref]
  10. M. J. Konopnicki and J. H. Eberly, Phys. Rev. A 49, 2567 (1981).
    [Crossref]
  11. G. Valiulis and K. Staliunas, Lith. Phys. J. 31, 38 (1991).

1992 (1)

G. A. Swartlander and C. T. Law, Phys. Rev. Lett. 69, 2503 (1992).
[Crossref]

1991 (3)

1981 (2)

M. J. Konopnicki, P. D. Drummond, and J. H. Eberly, Opt. Commun. 36, 313 (1981).
[Crossref]

M. J. Konopnicki and J. H. Eberly, Phys. Rev. A 49, 2567 (1981).
[Crossref]

1979 (1)

D. J. Kaup, A. Rieman, and A. Bers, Rev. Mod. Phys. 51, 275 (1979).
[Crossref]

1976 (1)

Y. N. Karamzin and A. P. Sukhorukov, Sov. Phys. JETP 41, 414 (1976); Y. N. Karamzin, A. P. Sukhorukov, and T. S. Filipchuk, Moscow Univ. Phys. Bull. 19, 73 (1978).

1973 (1)

V. E. Zakharov and S. V. Manakov, Sov. Phys. JETP Lett. 18, 243 (1973).

1972 (1)

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

1970 (1)

J. A. Armstrong, S. S. Jha, and N. S. Shiren, IEEE J. Quantum Electron. QE-6, 123 (1970).
[Crossref]

Allan, G. R.

Anderson, D. R.

Armstrong, J. A.

J. A. Armstrong, S. S. Jha, and N. S. Shiren, IEEE J. Quantum Electron. QE-6, 123 (1970).
[Crossref]

Bers, A.

D. J. Kaup, A. Rieman, and A. Bers, Rev. Mod. Phys. 51, 275 (1979).
[Crossref]

Carter, S.

Drummond, P. D.

M. G. Raymer, P. D. Drummond, and S. Carter, Opt. Lett. 16, 1189 (1991).
[Crossref] [PubMed]

M. J. Konopnicki, P. D. Drummond, and J. H. Eberly, Opt. Commun. 36, 313 (1981).
[Crossref]

Eberly, J. H.

M. J. Konopnicki, P. D. Drummond, and J. H. Eberly, Opt. Commun. 36, 313 (1981).
[Crossref]

M. J. Konopnicki and J. H. Eberly, Phys. Rev. A 49, 2567 (1981).
[Crossref]

Jha, S. S.

J. A. Armstrong, S. S. Jha, and N. S. Shiren, IEEE J. Quantum Electron. QE-6, 123 (1970).
[Crossref]

Karamzin, Y. N.

Y. N. Karamzin and A. P. Sukhorukov, Sov. Phys. JETP 41, 414 (1976); Y. N. Karamzin, A. P. Sukhorukov, and T. S. Filipchuk, Moscow Univ. Phys. Bull. 19, 73 (1978).

Kaup, D. J.

D. J. Kaup, A. Rieman, and A. Bers, Rev. Mod. Phys. 51, 275 (1979).
[Crossref]

Konopnicki, M. J.

M. J. Konopnicki and J. H. Eberly, Phys. Rev. A 49, 2567 (1981).
[Crossref]

M. J. Konopnicki, P. D. Drummond, and J. H. Eberly, Opt. Commun. 36, 313 (1981).
[Crossref]

Law, C. T.

G. A. Swartlander and C. T. Law, Phys. Rev. Lett. 69, 2503 (1992).
[Crossref]

Manakov, S. V.

V. E. Zakharov and S. V. Manakov, Sov. Phys. JETP Lett. 18, 243 (1973).

Raymer, M. G.

Rieman, A.

D. J. Kaup, A. Rieman, and A. Bers, Rev. Mod. Phys. 51, 275 (1979).
[Crossref]

Shabat, A. B.

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

Shiren, N. S.

J. A. Armstrong, S. S. Jha, and N. S. Shiren, IEEE J. Quantum Electron. QE-6, 123 (1970).
[Crossref]

Skinner, S. R.

Smirl, A. L.

Staliunas, K.

G. Valiulis and K. Staliunas, Lith. Phys. J. 31, 38 (1991).

Sukhorukov, A. P.

Y. N. Karamzin and A. P. Sukhorukov, Sov. Phys. JETP 41, 414 (1976); Y. N. Karamzin, A. P. Sukhorukov, and T. S. Filipchuk, Moscow Univ. Phys. Bull. 19, 73 (1978).

Swartlander, G. A.

G. A. Swartlander and C. T. Law, Phys. Rev. Lett. 69, 2503 (1992).
[Crossref]

Valiulis, G.

G. Valiulis and K. Staliunas, Lith. Phys. J. 31, 38 (1991).

Zakharov, V. E.

V. E. Zakharov and S. V. Manakov, Sov. Phys. JETP Lett. 18, 243 (1973).

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

IEEE J. Quantum Electron. (1)

J. A. Armstrong, S. S. Jha, and N. S. Shiren, IEEE J. Quantum Electron. QE-6, 123 (1970).
[Crossref]

Lith. Phys. J. (1)

G. Valiulis and K. Staliunas, Lith. Phys. J. 31, 38 (1991).

Opt. Commun. (1)

M. J. Konopnicki, P. D. Drummond, and J. H. Eberly, Opt. Commun. 36, 313 (1981).
[Crossref]

Opt. Lett. (2)

Phys. Rev. A (1)

M. J. Konopnicki and J. H. Eberly, Phys. Rev. A 49, 2567 (1981).
[Crossref]

Phys. Rev. Lett. (1)

G. A. Swartlander and C. T. Law, Phys. Rev. Lett. 69, 2503 (1992).
[Crossref]

Rev. Mod. Phys. (1)

D. J. Kaup, A. Rieman, and A. Bers, Rev. Mod. Phys. 51, 275 (1979).
[Crossref]

Sov. Phys. JETP (2)

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

Y. N. Karamzin and A. P. Sukhorukov, Sov. Phys. JETP 41, 414 (1976); Y. N. Karamzin, A. P. Sukhorukov, and T. S. Filipchuk, Moscow Univ. Phys. Bull. 19, 73 (1978).

Sov. Phys. JETP Lett. (1)

V. E. Zakharov and S. V. Manakov, Sov. Phys. JETP Lett. 18, 243 (1973).

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Figures (3)

Fig. 1
Fig. 1

Plot of |ϕ(ξ, τ)|2 with an ensemble of 100 input pulses. Added across the initial sech2 profile is a complex noise source whose standard deviation for both the real and the imaginary noise components is 10% of the peak amplitude required for solitary-wave propagation. The parameters used were k2/k1 = 1.8 and κ = 1.

Fig. 2
Fig. 2

Plot of |ϕ(ξ, τ)|2 for colliding simultons. The parameters used were k2/k1= 0.5, κ = 0.5.

Fig. 3
Fig. 3

Plot of |ϕ(ξ, τ)|2 for colliding simultons. The parameters are the same as in Fig. 2 except for the phase shift of π/2 in the initial phase of ϕ(0, τ) for one of the simultons and the shift of the center of the initial pulses from τ = 0 in Fig. 2 to τ = 15.

Equations (45)

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( z + i 2 k 1 2 t υ 2 ) Φ = χ * Ψ Φ * ,
[ z i ( k 0 ( 2 ) 2 k 0 ( 1 ) ) + i 2 k 2 2 t υ 2 ] Ψ = 1 2 χ Φ 2 ,
[ ξ + i 2 sgn ( k 1 ) 2 τ 2 ] ϕ = ψ ϕ * ,
[ ξ i z 0 ( k 0 ( 2 ) 2 k 0 ( 1 ) ) + i 2 k 2 | k 1 | 2 τ 2 ] ψ = 1 2 ϕ 2 ,
ϕ = Φ / Ψ 0 = ϕ 0 sech 2 ( κ τ ) exp ( i θ 1 ξ ) ,
ψ = Ψ / Ψ 0 = ψ 0 sech 2 ( κ τ ) exp ( i θ 2 ξ ) ,
θ 2 = 2 θ 1 .
θ 1 + 2 κ 2 sgn ( k 1 ) = 0 ,
θ 2 + 2 κ 2 k 2 | k 1 | z 0 [ k 0 ( 2 ) 2 k 0 ( 1 ) ] = 0 ,
z 0 [ k 0 ( 2 ) 2 k 0 ( 1 ) ] 2 κ 2 [ k 2 | k 1 | 2 sgn ( k 1 ) ] = 0 ,
3 i κ 2 sgn ( k 1 ) ϕ 0 + ϕ 0 * ψ 0 = 0 ,
3 i κ 2 k 2 | k 1 | ψ 0 1 2 ϕ 0 2 = 0.
ψ 0 = i | k 1 | ϕ 0 2 6 κ 2 k 2 ,
| ϕ 0 | 2 = 18 κ 4 k 2 k 1 .
ϕ i n ( 0 , τ ) = ϕ ( 0 , τ ) + ζ 1 ( τ ) ,
ψ i n ( 0 , τ ) = ψ ( 0 , τ ) + ζ 2 ( τ ) ,
ζ i ( τ ) ζ i * ( τ ) = 2 σ i 2 δ ( τ τ ) .
z 0 Δ k = 6 κ 2 k 2 / | k 1 | ,
ψ 0 = i ϕ 0 2 | k 1 | 6 κ 2 k 2 ,
| ϕ 0 | 2 = 18 κ 4 k 2 / k 1 .
ϕ = Φ / Ψ 0 = ϕ 0 sech ( κ τ ) tanh ( κ τ ) exp ( i θ 1 ξ ) ,
ψ = Ψ / Ψ 0 = ψ 0 sech 2 ( κ τ ) exp ( i θ 2 ξ )
ψ 0 = i ϕ 0 2 | k 1 | 6 κ 2 k 1 ,
| ϕ 0 | 2 = 18 κ 4 k 2 k 1
z 0 [ k 0 ( 2 ) 2 k 0 ( 2 ) ] + κ 2 [ k 2 | k 1 | + sgn ( k 1 ) ] = 0.
( z + i 2 k s 2 t υ 2 ) Φ s = 1 2 χ * Ψ Φ s * ,
[ z + ( 1 ω i 1 ω s ) t v + i 2 k i ( ω i ω s ) 2 2 t v 2 ] Φ i = 1 2 χ * Ψ Φ i * ,
[ z + ( 1 ω p 1 ω s ) t υ i Δ k + i 2 k p ( ω p ω s ) 2 t υ 2 ] Ψ = 1 2 χ Φ s Φ i ,
[ ξ + i 2 sgn ( k s ) | k s k i | 1 / 2 2 τ 2 ] ϕ s = ψ ϕ s * ,
[ ξ + i 2 sgn ( k i ) | k i k s | 1 / 2 2 τ 2 ] ϕ i = ψ ϕ i * ,
( ξ i z 0 Δ k + i 2 k p | k s k i | 1 / 2 2 τ 2 ) ψ = ϕ s ϕ i .
ϕ s = ϕ 0 s sech 2 ( κ τ ) exp ( i θ s ξ ) ,
ϕ i = ϕ 0 i sech 2 ( κ τ ) exp ( i θ i ξ ) ,
ψ = ψ 0 sech 2 ( κ τ ) exp ( i θ p ξ )
ψ 0 = i | k s k i | 1 / 2 ϕ 0 s ϕ 0 i 3 κ 2 k p ,
| ϕ 0 s | 2 = 3 κ 4 k p sgn ( k i ) | k s | ,
| ϕ 0 i | 2 = 3 κ 4 k p sgn ( k s ) | k i | ,
z 0 ( k 0 p k 0 s k 0 i ) 2 κ 2 | k s k i | 1 / 2 ( k p k s k i ) = 0.
z A 1 = i 2 k 1 2 x 2 A 1 + γ A 1 * A 2 ,
z A 2 = ( i Δ k + i 2 k 2 2 x 2 ) A 2 γ 2 A 1 2
z A 1 = ( i 2 k 1 2 x 2 ) A 1 + γ 2 A 2 * A 3 ,
z A 2 = ( i 2 k 2 2 x 2 ) A 2 + γ 2 A 1 * A 3 ,
z A 3 = ( i Δ k + i 2 k 3 2 x 2 ) A 3 γ 2 A 1 A 2
ξ ϕ = i 2 2 η 2 ϕ + ϕ * ψ ,
ξ ψ = ( i z 0 Δ k + i k 1 2 k 2 2 η 2 ) ψ 1 2 ϕ 2 ,

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