Abstract

The nonlinear dynamics of doubly-degenerate vector four-wave mixing (FWM) are studied analytical and numerically, in phase space and in Stokes space. Depending on the initial conditions, vector FWM can evolve aperiodically or periodically, but not chaotically. The dynamics of vector FWM are similar to, but richer than, the dynamics of scalar FWM.

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References

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    [CrossRef]
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    [CrossRef]
  3. F. P. Bretherton, �??Resonant interactions between waves: The case of discrete oscillations,�?? J. Fluid Mech. 20, 457�??479 (1964).
    [CrossRef]
  4. C. J. McKinstrie and X. D. Cao, �??Nonlinear detuning of three-wave interactions,�?? J. Opt. Soc. Am. B 10, 898�??912 (1993).
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  5. G. Cappellini and S. Trillo, �??Third-order three-wave mixing in single-mode fibers: Exact solutions and spatial instability effects,�?? J. Opt. Soc. Am. B 8, 824�??838 (1991).
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  7. K. Inoue, �??Arrangement of orthogonal polarized signals for suppressing fiber four-wave mixing in optical multi-channel transmission systems,�?? IEEE Photon. Technol. Lett. 3, 560�??563 (1991).
    [CrossRef]
  8. J. Hansryd, P. A. Andrekson, M. Westlund, J. Li and P. O. Hedekvist, �??Fiber-based optical parametric amplifiers and their applications,�?? IEEE J. Sel. Top. Quantum Electron. 8, 506�??520 (2002).
    [CrossRef]
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    [CrossRef]
  10. K. Inoue, "Tunable and selective wavelength conversion using fiber four-wave mixing with two pump lights,�?? IEEE Photon. Technol. Lett. 6, 1451�??1453 (1994).
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  11. T. Tanemura and K. Kikuchi, "Highly efficient arbitrary wavelength conversion within entire C-band based on nondegenerate fiber four-wave mixing,�?? IEEE Photon. Technol. Lett. 16, 551�??553 (2004).
    [CrossRef]
  12. E. Ciaramella, F. Curti and S. Trillo, �??All-optical signal reshaping by means of four-wave mixing in optical fibers,�?? IEEE Photon. Technol. Lett. 13, 142�??144 (2001).
    [CrossRef]
  13. S. Radic, C. J. McKinstrie, R. M. Jopson, J. C. Centanni and A. R. Chraplyvy, �??All-optical regeneration in one- and two-pump parametric amplifers using highly nonlinear optical fiber,�?? IEEE Photon. Technol. Lett. 15, 957�??959 (2003).
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  14. C. J. McKinstrie, H. Kogelnik, R. M. Jopson, S. Radic and A. V. Kanaev, �??Four-wave mixing in fibers with random birefringence,�?? Opt. Express 12, 2033�??2055 (2004).
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    [CrossRef] [PubMed]
  16. Y. Silberberg and G. Stegeman, �??Bright spatial solitons in Kerr slab waveguides,�?? in Spatial Solitons, edited by S. Trillo and W. Torruellas (Springer-Verlag, 2001), pp. 37�??60.
  17. J. M. Manley and H. E. Rowe, �??Some general properties of nonlinear elements�??Part I. General energy relations,�?? Proc. IRE 44, 904�??913 (1956).
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  18. M. T. Weiss, �??Quantum derivation of energy relations analogous to those for nonlinear reactances,�?? Proc. IRE 45, 1012�??1013 (1957).
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  20. H. Kogelnik, R. M. Jopson and L. E. Nelson, �??Polarization-mode dispersion,�?? in Optical Fiber Telecommunications IVB, edited by I. Kaminow and T. Li (Academic Press, 2002), pp. 725�??861.

IEEE J. Sel. Top. Quantum Electron. (1)

J. Hansryd, P. A. Andrekson, M. Westlund, J. Li and P. O. Hedekvist, �??Fiber-based optical parametric amplifiers and their applications,�?? IEEE J. Sel. Top. Quantum Electron. 8, 506�??520 (2002).
[CrossRef]

IEEE Photon. Technol. Lett. (5)

K. Inoue, �??Arrangement of orthogonal polarized signals for suppressing fiber four-wave mixing in optical multi-channel transmission systems,�?? IEEE Photon. Technol. Lett. 3, 560�??563 (1991).
[CrossRef]

K. Inoue, "Tunable and selective wavelength conversion using fiber four-wave mixing with two pump lights,�?? IEEE Photon. Technol. Lett. 6, 1451�??1453 (1994).
[CrossRef]

T. Tanemura and K. Kikuchi, "Highly efficient arbitrary wavelength conversion within entire C-band based on nondegenerate fiber four-wave mixing,�?? IEEE Photon. Technol. Lett. 16, 551�??553 (2004).
[CrossRef]

E. Ciaramella, F. Curti and S. Trillo, �??All-optical signal reshaping by means of four-wave mixing in optical fibers,�?? IEEE Photon. Technol. Lett. 13, 142�??144 (2001).
[CrossRef]

S. Radic, C. J. McKinstrie, R. M. Jopson, J. C. Centanni and A. R. Chraplyvy, �??All-optical regeneration in one- and two-pump parametric amplifers using highly nonlinear optical fiber,�?? IEEE Photon. Technol. Lett. 15, 957�??959 (2003).
[CrossRef]

J. Fluid Mech. (1)

F. P. Bretherton, �??Resonant interactions between waves: The case of discrete oscillations,�?? J. Fluid Mech. 20, 457�??479 (1964).
[CrossRef]

J. Opt. Soc. Am. B (3)

Opt. Express (1)

Opt. Fiber Technol. (1)

S. Radic and C. J. McKinstrie, �??Two-pump parametric amplifiers,�?? Opt. Fiber Technol. 9, 7�??23 (2003).
[CrossRef]

Optical Fiber Telecommunications IVB (1)

H. Kogelnik, R. M. Jopson and L. E. Nelson, �??Polarization-mode dispersion,�?? in Optical Fiber Telecommunications IVB, edited by I. Kaminow and T. Li (Academic Press, 2002), pp. 725�??861.

Phys. Rev. (1)

J. A. Armstrong, N. Bloembergen, J. Ducuing and P. S. Pershan, �??Interactions between light waves in a nonlinear dielectric,�?? Phys. Rev. 127, 1918�??1939 (1962).
[CrossRef]

Proc. IEE (1)

A. Jurkus and P. N. Robson, �??Saturation effects in a traveling-wave parametric amplifier,�?? Proc. IEE Part B 107, 119�??122 (1960).
[CrossRef]

Proc. IRE (2)

J. M. Manley and H. E. Rowe, �??Some general properties of nonlinear elements�??Part I. General energy relations,�?? Proc. IRE 44, 904�??913 (1956).
[CrossRef]

M. T. Weiss, �??Quantum derivation of energy relations analogous to those for nonlinear reactances,�?? Proc. IRE 45, 1012�??1013 (1957).

Proc. Nat. Acad. Sci. (1)

J. P. Gordon and H. Kogelnik, �??PMD fundamentals: polarization mode dispersion in optical fibers,�?? Proc. Nat. Acad. Sci. 97, 4541�??4550 (2000).
[CrossRef] [PubMed]

Spatial Solitons (1)

Y. Silberberg and G. Stegeman, �??Bright spatial solitons in Kerr slab waveguides,�?? in Spatial Solitons, edited by S. Trillo and W. Torruellas (Springer-Verlag, 2001), pp. 37�??60.

Other (1)

P. F. Byrd and M. D. Friedman, Handbook of Elliptic Integrals for Enginers and Scientists, 2nd Edition (Springer-Verlag, 1971).

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Figures (14)

Fig. 1.
Fig. 1.

Phase sphere for scalar FWM. The solid curves denote periodic orbits, the dot-dashed curves denote aperiodic orbits (separatrices), the dashed curves denote the boundary of the phase sphere and the dots denote equilibrium points. (a) Side view. (b) Top view.

Fig. 2.
Fig. 2.

Phase sphere for vector FWM. The solid curves denote periodic orbits, the dot-dashed curves denote aperiodic orbits (separatrices), the dashed curves denote the boundary of the phase sphere and the dots denote equilibrium points. (a) Side view. (b) Top view.

Fig. 3.
Fig. 3.

Phase sphere for vector FWM. The solid curves denote periodic orbits, the dot-dashed curves denote aperiodic orbits (separatrices), the dashed curves denote the boundary of the phase sphere and the dots denote equilibrium points. (a) Side view. (b) Top view.

Fig. 4.
Fig. 4.

Components of the solution vector plotted as functions of distance. (a) The solid and dot-dashed curves denote w and x, respectively. In this aperiodic solution y=x. (b) The solid and dashed curves denote the product and pump fluxes u and v, respectively.

Fig. 5.
Fig. 5.

Pseudo-particle energy plotted as a function of displacement. The solid curves denote the potential energies. (a) The dot-dashed line denotes the total energy associated with aperiodic motion and the dots denote equilibrium points. (b) The dashed line denotes the total energy associated with periodic motion and the circles denote turning points.

Fig. 6.
Fig. 6.

Components of the solution vector plotted as functions of distance. (a) The solid, dot-dashed and dashed curves denote w, x and y, respectively. (b) The solid and dashed curves denote the product and pump fluxes u and v, respectively.

Fig. 7.
Fig. 7.

Photon fluxes plotted as functions of distance. (a) The solid, dot-dashed medium-dashed and short-dashed curves denote F 1, G 1, F 2 and G 2, respectively. (b) The solid and dashed curves denote the product flux F 1+G 1 and the pump flux F 2+G 2, respectively.

Fig. 8.
Fig. 8.

Photon fluxes plotted as functions of distance. (a) The solid, dot-dashed mediumdashed and short-dashed curves denote F 1, G 1, F 2 and G 2, respectively. (b) The solid and dashed curves denote the product flux F 1+G 1 and the pump flux F 2+G 2, respectively.

Fig. 9.
Fig. 9.

Parametric plot of the Stokes vector of (a) wave 1 and (b) wave 2. Dots denote the initial Stokes vectors. The distance parameter varies from 0 to 6.

Fig. 10.
Fig. 10.

Parametric plot of the Stokes vector of (a) wave 1 and (b) wave 2. Dots denote the initial Stokes vectors. The distance parameter varies from 0 to 60.

Fig. 11.
Fig. 11.

Parametric plot of the Stokes vector of wave 1, viewed from different perspectives. Dots denote the initial Stokes vectors. The distance parameter varies from 0 to 60.

Fig. 12.
Fig. 12.

Parametric plot of the Stokes vector of wave 1, viewed from different perspectives. The distance parameter varies from 0 to 800.

Fig. 13.
Fig. 13.

Phase plane for scalar FWM. The solid curves denote periodic orbits, the dot-dashed curves denote aperiodic orbits and the dots denote equilibrium points.

Fig. 14.
Fig. 14.

Pseudo-particle energy plotted as a function of displacement. The solid curve denotes the potential energy, the dot-dashed and dashed lines denote the total energies E=0 and E=-1/8, respectively, the dots denote equilibrium points and the circles denote turning points.

Equations (93)

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D A 1 = i H 1 A 1 + i γ A 1 A 2 A 2 ,
D A 2 = i H 2 A 2 + i γ A 2 A 1 A 1 ,
H j = β j + γ ( A j A j + 3 A k A k 2 ) + γ a k · σ 2
D A 1 = i γ A 1 A 2 A 2 ,
D A 2 = i γ A 2 A 1 A 1 ,
L = Σ j ( A j D A j D A j A j ) 2 i γ ( A 1 A 2 2 + A 2 A 1 2 ) 2 ,
D ( L | D A j ) = L | A j
H = Σ j ( A j | D A j D A j | A j ) 2 i L
H = γ ( A 1 | A 2 2 + A 2 | A 1 2 ) 2 .
D | A j = i H A j |
D A 1 | A 1 = i γ ( A 1 A 2 2 A 2 A 1 2 ) ,
D A 2 A 1 = i γ A 1 A 2 ( A 2 A 2 A 1 A 1 ) ,
D A 2 | A 2 = i γ ( A 2 A 1 2 A 1 A 2 2 ) .
D ( A 1 A 1 + A 2 A 2 ) = 0 .
D A 1 σ A 1 = i γ ( A 1 A 2 A 1 σ A 2 A 2 A 1 A 2 σ A 1 ) ,
D A 2 σ A 1 = i γ A 1 A 2 ( A 2 σ A 2 A 1 σ A 1 ) ,
D A 2 σ A 2 = i γ ( A 2 A 1 A 2 σ A 1 A 1 A 2 A 1 σ A 2 ) ,
D ( A 1 σ A 1 + A 2 σ A 2 ) = 0 .
A 1 A 2 2 = F 1 F 2 ( 1 + e 1 · e 2 ) 2 .
D = ( A 1 A 2 2 A 1 A 1 A 2 A 2 ) = 0 .
D [ ( A 1 | A 2 2 + A 1 | A 1 A 2 | A 2 ) 2 4 ] = 0 .
D ( A 1 σ A 2 2 A 1 σ A 1 · A 2 σ A 2 ) = 0 .
u = 4 x y ,
v = 4 x y ,
x = ( v u ) y ,
y = ( v u ) x ,
( u + v ) = 0 ,
w = 4 x y ,
x = 2 w y ,
y = 2 w x ,
( x 2 y 2 ) = 0 ,
( w 2 + x 2 + y 2 ) = 0 ,
w = ± 2 ( w c 2 w 2 ) ,
w ( z ) = ± w c tanh [ 2 w c ( z z 0 ) ] ,
x 2 ( z ) = x 0 2 + [ w 0 2 w 2 ( z ) ] 2 .
( w ) 2 = 4 ( w c 2 w 2 ) 2 .
( w ) 2 = 4 ( w 2 w 2 ) ( w + 2 w 2 ) ,
w ( z ) = w sn [ k ( z z 0 ) | m ] ,
x 2 ( z ) = x 0 2 + [ w 0 2 w 2 ( z ) ] 2 ,
y 2 ( z ) = y 0 2 + [ w 0 2 w 2 ( z ) ] 2 .
l = 4 K ( m ) k ,
H = F 1 F 2 cos ( 2 θ 12 ) + 2 ( F 1 F 2 G 1 G 2 ) 1 2 cos ( θ 12 + ϕ 12 ) + G 1 G 2 cos ( 2 ϕ 12 ) ,
D θ j = H F j ,
D F j = H θ j ,
D θ 1 = F 2 cos ( 2 θ 12 ) + ( F 2 G 1 G 2 F 1 ) 1 2 cos ( θ 12 + ϕ 12 ) ,
D F 1 = 2 F 1 F 2 sin ( 2 θ 12 ) + 2 ( F 1 F 2 G 1 G 2 ) 1 2 sin ( θ 12 + ϕ 12 ) ,
D θ 2 = F 1 cos ( 2 θ 12 ) + ( G 1 G 2 F 1 F 2 ) 1 2 cos ( θ 12 + ϕ 12 ) ,
D F 2 = 2 F 1 F 2 sin ( 2 θ 12 ) 2 ( F 1 F 2 G 1 G 2 ) 1 2 sin ( θ 12 + ϕ 12 ) .
D ( F 1 + F 2 ) = 0 ,
D ( G 1 + G 2 ) = 0 .
a α = A A * A A * ,
a β = A A * + A * A ,
a γ = i ( A A * A * A ) .
s α = ( F 1 G 1 ) + ( F 2 G 2 ) ,
s β = 2 ( F 1 G 1 ) 1 2 cos ( θ 1 ϕ 1 ) + 2 ( F 2 G 2 ) 1 2 cos ( θ 2 ϕ 2 ) ,
s γ = 2 ( F 1 G 1 ) 1 2 sin ( θ 1 ϕ 1 ) 2 ( F 2 G 2 ) 1 2 sin ( θ 2 ϕ 2 ) .
w 2 = [ ( F 1 + G 1 ) ( F 2 + G 2 ) ] 2 4
x 2 + y 2 = F 1 F 2 + 2 ( F 1 F 2 G 1 G 2 ) 1 2 cos ( θ 12 ϕ 12 ) + G 1 G 2 .
J = ( s α 2 + s β 2 + s γ 2 ) 4 .
D A 1 = i γ A 2 2 A 1 * ,
D A 2 = i γ A 1 2 A 2 * ,
D θ 1 = γ F 2 cos ( 2 θ ) ,
D θ 2 = γ F 1 cos ( 2 θ ) ,
D F 1 = 2 γ F 1 F 2 sin ( 2 θ ) ,
D F 2 = 2 γ F 1 F 2 sin ( 2 θ ) .
H= γ F 1 F 2 cos ( 2 θ ) ,
D θ j = H F j ,
D F j = H θ j .
H = F ( 1 F ) cos ( 2 θ )
θ = H F ,
F = H θ .
( F ) 2 = 4 [ F 2 ( 1 F ) 2 H 0 2 ] ,
F ( z ) = F 0 exp ( 2 z ) ( 1 F 0 ) + F 0 exp ( 2 z ) ,
F ( z ) = [ 1 + tanh ( z z 0 ) ] 2 ,
a = [ 1 ( 1 + 4 H 0 ) 1 2 ] 2 ,
b = [ 1 ( 1 4 H 0 ) 1 2 ] 2 ,
c = [ 1 + ( 1 4 H 0 ) 1 2 ] 2 ,
d = [ 1 + ( 1 + 4 H 0 ) 1 2 ] 2 ,
F ( z ) = b ( c a ) a ( c b ) sn 2 [ k ( z + z 0 ) m ] ( c a ) ( c b ) sn 2 [ k ( z + z 0 ) m ] ,
z 0 = b F 0 d x 2 [ ( x a ) ( x b ) ( c x ) ( d x ) ] 1 2 .
( w ) 2 = 4 ( w 2 w 2 ) ( w + 2 w 2 ) ,
ρ = a , θ = cos 1 ( a α ρ ) , ϕ = tan 1 ( a γ a β ) .
a α = ρ cos θ , a β = ρ sin θ cos ϕ , a γ = ρ sin θ sin ϕ .
| A = ρ 1 2 exp ( i ψ ) [ cos ( θ 2 ) , sin ( θ 2 ) exp ( i ϕ ) ] .
L = j ( ρ j ψ j + ρ j s j 2 ϕ j ) ρ 1 ρ 2 [ c 1 2 c 2 2 cos ( 2 ψ 12 )
+ 2 c 1 c 2 s 1 s 2 cos ( 2 ψ 12 + ϕ 12 ) + s 1 2 s 2 2 cos ( 2 ψ 12 + 2 ϕ 12 ) ] ,
ρ = F + G , ψ = θ f , θ = 2 tan 1 = [ ( G F ) 1 2 ] , ϕ = ϕ f ϕ f ,
F = ρ c 2 , G = ρ s 2 , θ f = ψ , ϕ f = ϕ + ψ .
H = ( p 1 q 1 ) ( p 2 q 2 ) cos ( 2 ψ 12 )
+ 2 [ ( p 1 q 1 ) q 1 ( p 2 q 2 ) q 2 ] 1 2 cos ( 2 ψ 12 + ϕ 12 )
+ q 1 q 2 cos ( 2 ψ 12 + 2 ϕ 12 ) .
ψ = θ f , p = F + G , ϕ = ϕ f θ f , q = G .
θ f = ψ , F = p q , ϕ f = ϕ + ψ , G = q .

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