1 Introduction.

The interaction of two plane waves in a Kerr medium (two wave mixing) is important in dynamic holography and nonlinear beam propagation studies. If the angle between beams is big, they do not produce higher diffraction orders, and their evolution is described using the Kogelnik diffraction theory [1]. If the angle is small enough, higher diffraction orders appear [2]. Three orders must be taken into account in the first approximation for the interaction of a strong beam with a weak one, when the additional order is formed symmetrically with respect to the strong beam. At least four orders are needed when the two initial beams have equal intensities. Higher diffraction orders are easily observable, in particular, in photorefractive crystals [3], and recently it was shown that the non-Bragg diffraction can produce in these materials big gains [4].

Three and four-orders interactions, as far as we know, were studied only numerically or with the undepleted pump approximation [3],[5]. Here we present analytical solutions. Their existence is related to the fact that the cases we consider prove to be completely integrable Hamiltonian tasks [6].

2 The Hamiltonian for interacting diffraction orders.

In a normalized form the nonlinear Schrödinger equation (NLSE) describing the propagation of wave in a Kerr medium is [7]:

Here ψ is the amplitude of wave depending on one transversal coordinate x and the propagation coordinate z; κ is the nonlinear coefficient, which is positive for self-focusing media and negative for self-defocusing ones. We are interested in the initial value problem when ψ(x, z) is known for z=z ₀, and we need to determine it for z>z ₀. The N-order interaction corresponds to the amplitude written as:

where S_k are the amplitudes of orders and the sum is taken over appropriate wavevectors k. For the odd number of orders we take for k values …-2K,-K, 0,K, 2K…. If the number of orders is even, they are … - 3/2K,-1/2K, 1/2K, 3/2K…. Here K=2π/Λ, where Λ is the interference fringes period. By substituting Eq.(2) into NLSE and combining the coefficients for Fourier harmonics, the infinite system of coupled nonlinear differential equations follows. We can, nevertheless, truncate this infinite system by assuming that higher diffraction orders are small and forcing the corresponding amplitudes to be zero. The coupled differential equations for the general case are written down in [5]. For three orders we have :

It is well known that NLSE is formally the equation of motion for the Hamiltonian system with infinite number of degrees of freedom. The corresponding Hamiltonian function is [7] :

The NLSE is given by the equation with Fréchet derivative i∂_zψ=δH/δψ*.

We substitute Eq.(2) in Eq.(6), and obtain a Hamiltonian which depends on S_k and S*_k. For soliton problems, the integration in Eq.(6) is done over the whole x axis. In our case we integrate over one period of interference fringes and normalize the result.

where every a can take any possible value of k. The Hamiltonian is real and symmetric with respect to S and its complexconj ugate.

The Hamiltonian equations for amplitudes are:

To obtain them in more familiar canonical form of mechanics, it is necessary to introduce real coordinates qk and momenta pk with S_k =1/√2(qk+ip_k ). For the optical task, the complexno tation is more natural.

The truncated Hamiltonian Eq.(7) gives Eqs.(3–5), this is verified by direct substitution. It is also not difficult to show that for N orders, Eqs.(8,7) give the same result as Eqs.(2,1).

As a completely integrable task, NLSE has an infinite number of integrals of motion which are called C_n in the original work of Zakharov and Shabat [7]. Of them, apart from the Hamiltonian itself, we will utilize the intensity : I=∑_k S_kS* _k , and the momentum: M=∑ _kkS_kS*_k. Their conservation for our case does not follow automatically from the argument of [7]. It is necessary to prove that truncated sums are integrals of motion for the truncated Hamiltonian. For any function U (S, S*) the condition for its conservation is the equality to zero of the Poisson bracket :

The conservation of Hamiltonian itself is evident from Eq.(9). The conservation of momentum and intensity can be proved for any number of orders by manipulations with Eq.(7), Eq.(9) and the quantity in question. It is also seen that I and M are in involution, i.e. {I,M}=0.

3 Solution for three orders.

If there are three integrals of motion for the Hamiltonian system with three degrees of freedom and they are in involution, it follows from the general theorem of mechanics that the trajectories of this system in the phase space lie on invariant 3-dimensional tori. The resulting movement is quasi-periodic, and the system is completely integrable [6],[8].

There is a general procedure for obtaining angle-action variables from known integrals, but we just use conserved quantities to obtain the equations for order intensities. Let us denote the order intensities as I _-1, I ₀, I ₁. Then I _-1+I ₀+I ₁=I, and I ₁-I _-1=M/K are conserved. It follows: $I_1=\frac{1}{2}(I+M⁄K-I_0)$, and $I_{-1}=\frac{1}{2}\left(I-M⁄K-I_0\right)$, thus all intensities can be linearly expressed in terms of the zeroth order one. Using the equations for order intensities, the integral given by the Hamiltonian is expressed as:

We rewrite Eq.(4) for the zeroth order intensity:

Comparing Eq.(11) and Eq.(10), and taking into account that

it is seen that ∂_zI0 is expressed as a square root of the fourth degree polynomial of I ₀, and it follows that the intensity is an elliptic function. So, all order intensities are periodic with a propagation length. It is not true for amplitudes, because phases generally do not return to their initial value after a period of intensity. The solutions for amplitudes can be obtained by writing down the equations for i∂_z ln(S_k )=i∂_zS_k/S_k . For example, for S ₀ we have

 

Fig. 1. The trajectory in the complex plane for S ₁. It is seen that the amplitude gains a constant phase shift after one period of intensity. The numerical solution of equations (3–5) for K=1.8, κ=0.4, and initial conditions S _-1=0.3, S ₀=1, S ₁=0.1+0.24i.

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Fig. 2. The maximal intensity transferred to the initially weak side order depending on the nonlinearity parameter q. The central beam intensity I ₀=1.

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In the right-hand side we have order intensities and the combination $S_0^{*2}$ S ₁ S _-1, and all of them can be expressed as functions of I ₀ using Eqs.(10–12). It follows that the phases are sums of periodic and linear functions of propagation length. The phase gains over the intensity period for 3 orders are not independent, because the combination $S_0^{*2}$ S ₁ S _-1 is periodic. In Figure 1 we present the computer calculation for a typical trajectory of the diffraction order. The formal solution in elliptic functions requiers the knowledge of the roots of the general fourth order polynomial, and thus it is quite cumbersome. Practically, the simplest way is the direct computer calculation. The most important property of three orders interaction is the quasi-periodical character of amplitudes and periodicity of intensities. So, it is possible to calculate amplitudes over one period of intensities, and to expand the solution for arbitrary propagation length.

The physically interesting case that cannot be reduced to the Kogelnik theory is the interaction of a strong central beam and a weak side beam with a formation of the third order. With the aid of Eqs.(10–12), it is possible to determine the maximum value of the intensity transfered to side beams by calculating the extrema of I ₀. The result is presented in Fig.2. The points of the derivative discontinuity in Fig.2. correspond to the transition from one branch of the solution to another. The maximal intensity, which can be transferred to side beams depends on the ratio q=2K ²/(κI), which we will call the nonlinearity parameter. When this ratio is bigger than 8 or smaller than 0 (this is the case of negative κ and self-defocusing), initially weak beams remain weak upon propagation. In these cases it is possible to neglect the formation of higher orders. The growth of solution to the finite value when 0<q<8 is related to the modulation instability. In this range of nonlinearity parameter, the initially weak beam grows exponentially in the undepleted pump approximation [9]. From the momentum conservation it follows that the intensity difference between two side orders remains constant with propagation. If the modulation instability is observed, the difference is of an order of the minimum side beam intensities, and it is much smaller than their maximal intensities.

4 Symmetrical four orders interaction.

To solve for four-beam interaction in the general case it is necessary to find the fourth integral of motion. The procedure of deriving it from the fourth NLSE integral given in the paper of Zakharov and Shabat [7] fails: because of the trunctaion we do not obtain the conserved quantity here. We can, nevertheless, solve a task if initial conditions are such that S _-2=S ₂ and S _-1=S ₁. These relations are maintained with propagation. Resulting evolution equations include only two orders :

The Hamiltonian as an integral of motion is given by

Since the intensity I=2|S ₁|²+2|S ₂|² is also the conserved quantity, there are two integrals of motion. This permits us to state that the task is completely integrable. The equation for the second order intensity is:

and it is again possible with a help of H and I to express the right-hand side of this equation as a function of I ₂ only, reducing the solution to integration. The integral, nevertheless, is more complicated than the elliptic one. Again, the intensities are periodic with propagation distance and amplitudes are quasi-periodic.

5 Representation of solutions on the Poincaré sphere and their properties.

The Poincaré sphere representation is useful to figure out how beam intensities change with propagation. Instead of the two amplitudes S ₁, S ₂ we introduce Stokes parameters given by: A=I ₁-I ₂=S ₁ S*₁-S ₂ S*₂, B=-i(S ₁ S*₂-S*₁ S ₂), C=S ₁ S*₂+S*₁ S ₂. The intensity conservation leads to the identity:

thus the trajectories A(z),B(z), C(z) lie on a sphere. To determine their form one has to rewrite the Hamiltonian in terms of Stokes parameters. This gives the condition:

which means that projections of trajectories on A-C plane are concentric ellipses, and trajectories are intersections of elliptic cylinders parallel to B axis, and the sphere. In Fig.3, we show these ellipces built for illustration purposes with the direct numerical solution of motion equations. The ellipticity and orientation of axes of the ellipses do not depend on q parameter or the ellipse size. Their centers are given by $A_0=\frac{I}{6}\left(1-4q\right)$ and $C_0=\frac{I}{6}\left(q-1\right)$, thus with changes in q, the center position moves along the straight line. The A parameter gives the intensity of beams (I _1,2=I/4±A/2), thus the Figure 3 contains a wealth of information about behavior of order intensities in particular situations. For example, trajectories with one maximum per period or with two of them separated by one minimum are possible, and the maximal values can be obtained by solving algebraic ecuations. The self-similar solutions for which which order intensities do not change with propagation are also well seen. For the situation of the Fig.3, there are sixo f them, and two are unstable.

 

Fig. 3. Phase portraits of trajectories with different initial conditions on the Poincaré sphere for a four-beam interaction. The nonlinearity parameter is q=0.5.

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6 Discussion and conclusions.

The Hamiltonian method is a powerful tool for solving nonlinear propagation problems with a finite number of degrees of freedom. Once the problem in question is expressed in Hamiltonian form, the non-trivial integral of motion is obtained. If we are interested in the interaction of two nonlinear modes, and the intensity is conserved, we immediately have two integrals needed for a complete integrability with far-reaching consequences about the general behavior of solutions. An example is the symmetrical four-orders system discussed above. The three-order problem is solved by noting that the momentum is conserved as well.

For the interaction of bigger number of modes, additional integrals are needed. The direct substitution does not work for the fourth integral of Zakharov-Shabat [7]: a conserved quantity is not obtained. Even if the N-orders task is not completely integrable, truncated higher integrals of motion of Zakharov-Shabat can be used when orders beyond N are weak. It is exactly when truncated equations of motion are valid. Physically, the condition can be small nonlineartity (big q parameter), or limited propagation length when initially the energy was concentrated in a small number of central orders. It seems probable that the approximation to the analytical solution (action-angle variables) can still be obtained by using truncated integrals of motion, though some of these integrals are approximations.

In conclusion, we have shown that the propagation of three and symmetrical four diffraction orders in a Kerr medium can be reduced to a completely integrable Hamiltonian task. Integrals of motion can be obtained from known integrals of nonlinear Schödinger equation. In both cases that we have considered the order intesities are periodic functions of propagation length and the amplitudes are quasi-periodic.