The Davey-Stewartson model in quadratic media: a way to control pulses

Hervé LEBLOND, P.OM.A., U.M.R. C.N.R.S. 6136, Université d'Angers, France e-mail : herve.leblond@univ-angers.fr

The Davey-Stewartson model in quadratic media : a way to control pulses.

A. A generalization of the NonLinear Schrödinger equation.

A.1. One and two-dimensional Nonlinear Schrödinger equation.

The (1+1)-dimensional theory of envelope soliton propagations rests on the Nonlinear Schrödinger (NLS) equation, which reads :

iAu_z+Bu_xx+Cu|u|²=0 (1) u is the complex wave amplitude, z the propagation variable, x the pulse shape variable, and A, B, and C are real coefficients. It must be noticed that the variable x can either describe the transverse or the longitudinal pulse shape. In the latter case it is commonly written in optics as a time variable t.

A well-known, but essential property of the NLS equation is to be completely integrable by means of the Inverse Scattering Transform (IST) method for any value of the coefficients A, B, and C. It admits soliton solutions if BC > 0, and dark-soliton solutions if BC

< 0. The matter of the talk is a generalization of the NLS equation to (2+1) dimensions.

The most simple (2+1) dimensional generalization of the NLS equation reads :

iAu_z+Bu_xx+Cu_yy+Du|u|²=0 (2)

The notations are the same as above, A, B, C, and D are real coefficients. Equation (2) is called the two-dimensional nonlinear Schrödinger equation (2D NLS). It gives a correct account e.g. for spatial 2-dimensional light propagation in an isotropic Kerr medium. But the 2D NLS equation is never integrable, and does not admit any stable localized solution : self- focusing leads rather to collapse.

A.2. The Davey-Stewarson system.

We present another generalization of the NLS equation, that is "better" than the previous one, in the sense that it conserves many of its properties : the Davey-Stewartson (DS) model. It reads:

iAu_z+Bu_xx+Cu_yy+Du|u|²+Euψ=0

αψ_xx+βψ_yy=γ|u|²_xx+δ|u|²_yy (3)

The function  is real-valued, and A, B, C, D, E, , , , and  are real coefficients depending on the particular physical situation. This model was first introduced to describe the two- dimensional evolution of water wave packets [1]. It involves an auxiliary field  , that can be called rectified field, because in optics, it can result from optical rectification of the wave.

This field describes a solitary wave, in the sense of a single isolated oscillation, and not of an envelope soliton (there is no carrier wave).

The DS system reduces obviously to the 2DNLS equation if :

|

^{αβ δγ}

|

^{=αδ−βγ=0 , but}

only in this case. It is thus more general than the 2D NLS equation. In fact, the Davey- Stewartson model is the correct model for most (2+1)-dimensional single polarization propagation problems (see below).

The DS system is integrable by means of the IST method for some values of coefficients.

Multidimensional systems having this property are very rare [2]. Rigorously speaking, system (3) is called DS only when it is integrable. We use this terminology below.

A.3. Soliton solutions of DS.

Whatever the value of the coefficients, the above DS-type system (2) can be reduced to the following one :

iu_z+ε₁u_xx+u_yy+ε₂u|u|²+ν ψu =0 ψ_xx+μψ_yy=|u|²_xx

(4) where  and  are real constants, and : ε₁, ε₂=±1 . System (4) is integrable if and only if the two following conditions are satisfied :

μ=−ε₁

ν=−2ε₂ (5)

There are thus 4 integrable cases, depending on the signs of ε₁ and ε₂ . Line solitons exist for all sign cases (fig. 1). Lumps, algebraically decaying in all directions, exist if :

ε₁ε₂=−1 [3]. The 1-lmp solution is shown on fig. 2.

Figure 1. Line-soliton of DS. Figure 2. Lump solution of DS.

Figure 3. Dromion solution of DS. Figure 4. The rectified field  for the dromion solution of DS

Dromions, exponentially decaying in all directions, exist if : ε₁=+ 1 [4] (fig. 3). In this case the DS system is called DS I. The dromions involve nontrivial values for the auxiliary or rectified field  at infinity (fig. 4). These values can describe the input of the solitary waves, that interact with the main pulse.

B. Davey-Stewartson-type models in nonlinear optics.

B.1. The nonlinear mechanisms.

The DS system (3) or (4) is a good generalization of the NLS equation (1) for two reasons : 1) Like the NLS equation, the DS system belongs to the few set of partial differential equations that are completely integrable by means of the IST method. 2) In the same way as, in (1+1) dimensions, multiscale expansions for envelope propagation almost always lead to the NLS equation, the model obtained through the same procedure in higher dimensional cases is most generally of DS type. Convergence of the asymptotic method from the mathematical point of view has been recently proved in a general frame [5].

A rigorous derivation of the model equations for soliton propagation in ⁽²⁾ materials of the crystal classes 4 2m , 4 3m , 3m , 6mm has been performed [6]. The main results are listed below. The description of soliton propagation in ⁽²⁾ materials involves three nonlinear mechanisms, listed in the following table:

Process Name Nonlinear coefficient

ω+ω−ω→ ω Third-order Kerr effect χ⁽³⁾(ω,ω,−ω,ω) ω+ω→ 2ω

2ω−ω→ ω

Cascading χ⁽²⁾(ω,ω,2ω)×χ⁽²⁾(2ω,−ω,2ω) ω−ω → 0

0+ω→ ω

Optical rectification

& electro-optic effect χ⁽²⁾(ω,−ω,0)×χ⁽²⁾(0,ω,ω)

Far from phase-matching, the coefficients corresponding to the three mechanisms have the same order of magnitude. This is a priori true from the theoretical point of view of the multiscale expansion, and has been checked using published experimental values. To get convinced of this feature, notice that, far from phase matching, the second harmonic remains very small, therefore the cascaded term is not larger than the third order one. Notice also that

⁽³⁾ compares to the square of ⁽²⁾. For optical rectification there is also no resonance, thus the rectified field is very small, not directly observable, but cumulative effects on long-distance propagation of the main pulse are observable.

B.2. A model taking polarizations into account.

There are a priori 2 polarizations for the fundamental wave, and also 2 polarizations for the rectified field, which describes also a transverse electromagnetic wave. Taking into account the vectorial and tensorialstructure of the fields and susceptibilities, we derive a system of model equations, that is always of DS- type [6]. Due to the symmetry, for the 4 2m class of crystals, only 1 rectified field is involved, and the complete model consists in 3 equations. For the 3m class, the 4 fields intervene, and the 4 equations are integro- differential.

In the (1+1)-dimensional reduction of these models, the equation governing the rectified field integrates trivially, and an explicit expression of the rectified field is found. This reduces the system to 2 coupled NLS equations. Notice that, due to anisotropy, the energy exchange between the polarizations is allowed. The reduction of the latter system to a single polarization yields the NLSequation, but is possible for particular polarizations only.

At (2+1) or (3+1) dimensions, the equation relating the fundamental wave amplitude and the rectified field is nontrivial, except if some particular symmetry arises. In [6], this happens for the spatial case, the wave propagating along the optical axis of the uniaxial medium. The reduction of the (2+1) dimensional model to only one polarization for the fundamental, and one for the rectified field yields the DS system (3) [7]. Like in the (1+1) dimensional case, it is not always possible, but requires some particular polarization, some particular choice of the variables, and even, eventually, some condition to be satisfied by the

⁽²⁾ tensor .

B.3. Integrability conditions.

The two conditions under which these reduced DS type systems are completely integrable have been written down [7]. The first one expresses in a very simple way for the 6mm class:

1

n_e²(0)

(

^{χ^}xzx (2)

(ω,−ω)

)

^{2+ 1}

n_e²(2ω)

(

^{χ^}xzx

(2)(ω,ω)

)

²⁼³₂ ^{^χ}(3)xxxx(ω,ω,−ω) (6) It expresses an equilibrium between the coefficients of the third order Kerr effect and the sum of that of the quadratic processes.

The second integrability condition relies the dispersion coefficient kk'' to the difference between the group velocity vg of the wave and the velocity c/n(0) of the rectified field wave.

The signs of the coefficients are so that lump exist (except in a special case), and that dromions would exist for anomalous dispersion.

C. Solutions to the DS system : beam control.

C.1. Classification of the DS type systems.

The DS-type system (3) has several possible behaviours depending on its coefficients value. A classification is given by the following table :

type 1 

elliptic-elliptic + +

elliptic-hyperbolic + 

hyperbolic-elliptic  +

hyperbolic-hyperbolic  

Further, if 1>0 the system is said focusing if 2 > 0 and defocusing if 2 < 0 , according to the analogous property of the 2D NLS equation obtained by dropping the term u.

Numerical values can be obtained from published experimental data, for KDP, the velocities arev_g()1.9510⁸ms^1 , and 0.65 10⁸ms ¹

) 0 (

 

n 

c ; for the lithium niobate, we get :

1

108

2 . 1 )

(   ms^

v_g  and 0.5 10⁸ms ¹

) 0 (

 

n 

c

.

The second equation of the DS model is thus hyperbolic. A solitary long wave, supported by the rectified field , is emitted by the main pulse, as shown of fig. 5.

C.2. Wave interaction.

The first equation of the DS system is

hyperbolic if the long wave  travels slower than the main pulse u : )

) ( 0

( _g 

n

c v (7)

If this condition is satisfied, the DS system describes an interaction between a localized envelope pulse and two solitary long waves, of line-soliton shape, that is resonant in some special sense [8].

The fig. 6 gives shows the principle of the interaction : u is the incident localized envelope pulse, travelling at speed V⃗ . 1

and 2 represent two incident solitary

"plane" waves, or rather line waves, propagating at speed V⃗₁ and V⃗₂ respectively. Obviously, the propagation direction is perpendicular to the wave line.

Mathematically, 1 and 2 are the limiting values at infinity of the rectified field . If u travels faster than the other waves, all will meet in the indicated interaction area.

Efficiency of the interaction necessitates a velocity matching, shown on figure 7. The intersection point of the two line of the waves travels

at some speed

V⃗

. The waves are matched if this velocity exactly coincides with that of the main pulse u. Then, seen from the main pulse, the two other waves apparently do not move, due to their self similarity along their line. The interaction lasts thus during all the propagation.

C.3. Beam control.

We present now some simulations, or solutions of the DS system (3) obtained by numerical computation. Such behaviour can arise in any physical situation for which the DS model is relevant. These results use physical values of the parameters valid for electromagnetic wave propagation in ferromagnets [8].

If the velocity matching is realized, through the interaction, it is possible to cut the pulse into two parts. This is shown on fig 8.

Figure 7. Velocity macthing.

Figure 6. Geometry of the wave interaction.

Figure 5. The solitary wave emitted by a two dimensional localized pulse.

The pulse may also been cut into four parts, as shown on fig. 9.

In a physical situation where the self-phase modulation is defocusing, in the absence of incident matched solitary wave, the pulse will be spread out. The interaction is able to prevent this effect, as shown on fig. 10. When self-phase modulation is focusing, interaction can prevent collapse and stabilize the pulse: see fig. 11. :

Figure 10. The interaction prevents the pulse from spreading out.

Figure 11. The interaction prevents collapse and stabilizes the pulse.

References :

[1] A. Davey and K. Stewartson, On three-dimensional packets of surface waves, Proc. R. Soc. Lond. A 338, 101-110 (1974).

[2] A.S. Fokas and M.J. Ablowitz, Phys. Rev. Lett. 51 7-10 (1983).

[3] J. Satsuma and M.J. Ablowitz, Two-dimensional lumps in nonlinear dispersive systems, J. Math. Phys. 20 (7), 1496 (1979).

[4] M. Boiti, J. J-P. Leon, L. Martina, and F. Pempinelli, Scattering of localized solitons in the plane, Phys. Lett. A 132 (8-9), 432-439 (1988).

[5] T. Colin, Rigorous derivation of the nonlinear Schrödinger equation and Davey- Stewartson systems from quadratic hyperbolic systems, preprint (1999).

Figure 9. The initial pulse has been cut into 4 parts through the interaction.

Figure 8. The pulse is cut into 2 parts through the interaction.

[6] H. Leblond, Propagation of optical localized pulses in ⁽²⁾ crystals :a (3+1)- dimensional model and its reduction to the NLS equation, J. Phys. A: Math.

Gen. 31, 3041-3066 (1998).

[7] H. Leblond, Bidimensional solitons in a quadratic medium, J. Phys. A: Math. Gen.

31, 5129-5143 (1998).

[8] H. Leblond, Electromagnetic waves in ferromagnets: a Davey-Stewartson type model, J. Phys. A: Math. Gen. 32, 7907-7932 (1999).