Abstract

We investigate the propagation of periodic patterns in one and two dimensions for weak Kerr-type nonlinearity. Nonlinear amplitudes are introduced, which are related to the Fourier harmonics of a wave by polynomials of third and fifth degree. These amplitudes evolve in a particularly simple way and permit easy reconstruction of waveform after propagation. For the one-dimensional case, solutions are quasiperiodic, and solitonlike structures can be identified. For the two-dimensional case, recurrent and chaotic regimes exist depending on lattice type.

© 2006 Optical Society of America

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References

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  1. Y. S. Kivshar and G. P. Agraval, Optical Solitons: from Fibers to Photonic Crystals (Academic, 2003).
  2. V. E. Zakharov and A. B. Shabat, "Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media," Sov. Phys. JETP 61, 62-69 (1972).
  3. L. A. Takhtajan and L. D. Faddeev, Hamiltonian Methods in the Theory of Solitons (Springer-Verlag, 1987).
  4. A. Thyagaraja, "Recurrent motions in certain continuum dynamical systems," Phys. Fluids 22, 2093-2096 (1979).
    [CrossRef]
  5. N. Korneev and E. M. Rodríguez, "Spectral limits for periodic pattern propagation in Kerr media," Opt. Express 12, 3297-3306 (2004).
    [CrossRef] [PubMed]
  6. Y. C. Ma and M. J. Ablowitz, "The periodic cubic Schrödinger equation," Stud. Appl. Math. 65, 113-158 (1981).
  7. V. I. Arnold, Mathematical Methods of Classical Mechanics (Springer-Verlag, 1989).

2004

1981

Y. C. Ma and M. J. Ablowitz, "The periodic cubic Schrödinger equation," Stud. Appl. Math. 65, 113-158 (1981).

1979

A. Thyagaraja, "Recurrent motions in certain continuum dynamical systems," Phys. Fluids 22, 2093-2096 (1979).
[CrossRef]

1972

V. E. Zakharov and A. B. Shabat, "Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media," Sov. Phys. JETP 61, 62-69 (1972).

Ablowitz, M. J.

Y. C. Ma and M. J. Ablowitz, "The periodic cubic Schrödinger equation," Stud. Appl. Math. 65, 113-158 (1981).

Agraval, G. P.

Y. S. Kivshar and G. P. Agraval, Optical Solitons: from Fibers to Photonic Crystals (Academic, 2003).

Arnold, V. I.

V. I. Arnold, Mathematical Methods of Classical Mechanics (Springer-Verlag, 1989).

Faddeev, L. D.

L. A. Takhtajan and L. D. Faddeev, Hamiltonian Methods in the Theory of Solitons (Springer-Verlag, 1987).

Kivshar, Y. S.

Y. S. Kivshar and G. P. Agraval, Optical Solitons: from Fibers to Photonic Crystals (Academic, 2003).

Korneev, N.

Ma, Y. C.

Y. C. Ma and M. J. Ablowitz, "The periodic cubic Schrödinger equation," Stud. Appl. Math. 65, 113-158 (1981).

Rodríguez, E. M.

Shabat, A. B.

V. E. Zakharov and A. B. Shabat, "Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media," Sov. Phys. JETP 61, 62-69 (1972).

Takhtajan, L. A.

L. A. Takhtajan and L. D. Faddeev, Hamiltonian Methods in the Theory of Solitons (Springer-Verlag, 1987).

Thyagaraja, A.

A. Thyagaraja, "Recurrent motions in certain continuum dynamical systems," Phys. Fluids 22, 2093-2096 (1979).
[CrossRef]

Zakharov, V. E.

V. E. Zakharov and A. B. Shabat, "Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media," Sov. Phys. JETP 61, 62-69 (1972).

Opt. Express

Phys. Fluids

A. Thyagaraja, "Recurrent motions in certain continuum dynamical systems," Phys. Fluids 22, 2093-2096 (1979).
[CrossRef]

Sov. Phys. JETP

V. E. Zakharov and A. B. Shabat, "Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media," Sov. Phys. JETP 61, 62-69 (1972).

Stud. Appl. Math.

Y. C. Ma and M. J. Ablowitz, "The periodic cubic Schrödinger equation," Stud. Appl. Math. 65, 113-158 (1981).

Other

V. I. Arnold, Mathematical Methods of Classical Mechanics (Springer-Verlag, 1989).

L. A. Takhtajan and L. D. Faddeev, Hamiltonian Methods in the Theory of Solitons (Springer-Verlag, 1987).

Y. S. Kivshar and G. P. Agraval, Optical Solitons: from Fibers to Photonic Crystals (Academic, 2003).

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Figures (4)

Fig. 1
Fig. 1

Real parts of exact numerical solutions for complex amplitudes ψ k (solid curves) and the difference between exact solutions and analytic approximations with Eqs. (9, 10) (dotted curves close to zero). For all figures ψ 0 ( 0 ) = 1 , ψ 1 ( 0 ) = 0.8 , K = 1 , and other amplitudes are initially zero. (a) Solution for ψ 2 and ϰ = 0.05 , (b) solution for ψ 2 and ϰ = 0.005 , (c) solution for ψ 3 and ϰ = 0.005 . The relative error of reconstruction diminishes proportionally to ϰ.

Fig. 2
Fig. 2

Demonstration of conservation for nonlinear order intensity W 0 and linearity of phase for S 0 with propagation. S 0 ( t ) is calculated with a numerical solution for ψ using Eqs. (9, 10) with initial conditions ψ 0 ( 0 ) = 0.05 , ψ 1 ( 0 ) = 1 , ψ 2 ( 0 ) = 0.7 , ψ 3 ( 0 ) = 0.3 , K = 1 . (a) W 0 with second approximation (nearly horizontal line), first approximation (solid curve), and the value of diffraction-order intensity ψ ¯ 0 ψ 0 (dotted curve) for ϰ = 0.02 . (b) The difference between the exact phase of S 0 and phase values obtained with the initial condition for the first approximation, Eq. (6) (upper trace), and the second approximation, Eq. (10), ϰ = 0.005 .

Fig. 3
Fig. 3

Lattice configuration for 2D propagation.

Fig. 4
Fig. 4

Different long-distance propagation scenarios for the order ψ 1 , 1 (the geometry of Fig. 3). (a) For a nonrectangular lattice the propagation character is similar to the 1D case, and energy transfer to the order is small, a = 0.2 , ϰ = 0.01 , K = b = 1 , initially ψ 1 , 0 = ψ 0 , 1 = 1 , ψ 0 , 0 = 0.1 i , ψ 1 , 0 = 1 + 0.1 i . (b) A big energy transfer is possible for a nearly rectangular lattice [ a = 0.02 , ψ 1 , 0 = 0 , other parameters are the same as for the (a) case]. (c) Chaotic solution for the nearly rectangular lattice [the same parameters as for the (a) case, but a = 0.02 ].

Equations (17)

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i ψ t = Δ ψ + 2 ϰ ψ 2 ψ .
ψ ( x , t ) = k = exp ( i k K x ) ψ k ( t ) ,
i ψ k t = k 2 K 2 ψ k + 2 ϰ p , q ψ ¯ k + p + q ψ k + p ψ k + q .
S k = ψ k + ϰ K 2 p , q 0 ψ ¯ k + p + q ψ k + p ψ k + q p q .
W k = S k S ¯ k
v k = k 2 K 2 + 4 ϰ ( I W W k 2 ) + O ( ϰ 2 I 2 K 2 ) ,
S k ( t ) = S k ( 0 ) exp ( i v k t ) ,
ψ k = S k ϰ K 2 p , q 0 S ¯ k + p + q S k + p S k + q p q .
S k = ψ k + ϰ K 2 p , q ψ ¯ k + p + q ψ k + p ψ k + q p q + 2 ϰ 2 K 4 p , q , r , s ψ ¯ k + p + q ψ ¯ k + q + r + s ψ k + p ψ k + q + r ψ k + q + s p q ( p q + r s ) ϰ 2 K 4 p , q , r , s ψ ¯ k + p + q + r ψ ¯ k + p + q + s ψ k + p ψ k + q ψ k + p + q + r + s p q ( p q r s ) 2 ϰ 2 K 4 p , q , s 0 ψ ¯ k + s ψ ¯ k + p + q ψ k + p ψ k + s ψ k + q ( p q ) 2 ϰ 2 K 4 p , q ψ ¯ k ψ ¯ k + p + q ψ k + p ψ k + q ψ k ( p q ) 2 .
v k = k 2 K 2 + 4 ϰ ( I W W k 2 ) 2 ϰ 2 K 2 p 0 ( W k + p + 2 W k ) W k + p p 2 .
ψ 1 ( t ) S 1 ( t ) ,
ψ 2 ( t ) S 2 ( t ) ϰ K 2 S 0 ( t ) ¯ S 1 ( t ) 2
ψ 2 ( t ) ϰ K 2 ψ 0 ( t ) ¯ ψ 1 ( t ) 2 .
ψ ( 0 , 0 ) ( t ) S 0 , 0 ( t ) 2 ϰ K 2 S ( 1 , 1 ) ( t ) ¯ S ( 1 , 0 ) ( t ) S ( 0 , 1 ) ( t ) a .
J k = ( 1 ) k { 2 + 4 ϰ L 2 ψ k ψ ¯ k + 16 ϰ 2 L 4 [ p , q 0 ψ k ψ k + p + q ψ ¯ k + p ψ ¯ k + q + c.c. 4 π 2 p q + ψ k 2 ψ ¯ k 2 12 ( p 0 ψ ¯ k + p ψ k + p 2 π p ) 2 + p 0 ψ ¯ k + p ψ k + p ψ ¯ k ψ k 2 π 2 p 2 ] } + .
J k ( 1 ) k 2 = 4 ϰ L 2 W k + 16 ϰ 2 L 4 [ W k 2 12 ( p 0 W k + p 2 π p ) 2 + p 0 W k + p W k 2 π 2 p 2 ] ;
H = s s 2 K 2 W s + 2 ϰ s p W s W p ϰ s W s W s 2 ϰ 2 K 2 s p W s W p 2 ( s p ) 2 ,

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