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

1992

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

1991

1981

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

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

1979

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

1976

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

1973

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

1972

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

1970

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

Bers, A.

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

Carter, S.

Drummond, P. D.

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

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

Eberly, J. H.

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

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

Jha, S. S.

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

Karamzin, Y. N.

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

Kaup, D. J.

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

Konopnicki, M. J.

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

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

Law, C. T.

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

Manakov, S. V.

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

Raymer, M. G.

Rieman, A.

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

Shabat, A. B.

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

Shiren, N. S.

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

Skinner, S. R.

Smirl, A. L.

Staliunas, K.

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

Sukhorukov, A. P.

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

Swartlander, G. A.

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

Valiulis, G.

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

Zakharov, V. E.

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

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

IEEE J. Quantum Electron.

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

Lith. Phys. J.

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

Opt. Commun.

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

Opt. Lett.

Phys. Rev. A

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

Phys. Rev. Lett.

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

Rev. Mod. Phys.

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

Sov. Phys. JETP

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

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

Sov. Phys. JETP Lett.

V. E. Zakharov, 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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