1. Introduction

Generally the scattering of light by elastic waves in a photo-elastic medium represents a parametric process in a system with a square-law nonlinearity. This phenomenon makes possible shaping spatial-temporal multi-wave solitons, whose field components can differ in physical nature, but are coupled with each other [1–3]. Such a type of soliton, so called three-wave coupled states, have already been found and investigated in two-mode waveguides [4,5], when spatial-temporal coupled states could be shaped with collinear scattering of the light by coherent acoustic phonons. Here, we discuss a new phenomenon, namely, Bragg spatial solitons in the form of three- and four-wave coupled states with just the non-collinear regime of acousto-optical interaction in a tellurium dioxide crystal. The peculiarity of these solitons lies in the fact that one or two of their optical components can be completely locked by the other components. The development of a quasi-stationary model for describing multi-wave coupled states and the obtained experimental data are subjects of the work presented.

2. One- and two-phonon scattering of light in a uniaxial crystal

The strongly nonlinear behavior of optical components with Bragg acousto-optical interaction in anisotropic medium can be easily achieved in an experiment without any observable influence of the scattering process on the acoustic wave since the number of interacting photons is 10⁵ times less than the number of phonons injected into a medium, where the powers of the incident light and ultrasound are close to each other. In this case, the amplitude of the acoustic wave is governed by a homogeneous wave equation, and it is agreed that the regime of so-called weak coupling takes place. Let us assume that the area of propagation for the acoustic wave, traveling almost perpendicularly to the light beams, is bounded by two planes x = 0 and x = L in a uniaxial crystal, and take into account both angular and frequency mismatches in the wave vectors. Usually, the Bragg acousto-optical process includes three waves, incident and scattered light modes as well as an acoustic mode, and incorporates conserving both the energy and the momentum for each partial act of a one-phonon scattering. However, under certain conditions, i.e. at a set of the angles of light incidence on selected crystal cuts and at fixed frequency of acoustic wave, one can observe Bragg scattering of the light caused by participating two phonons. The conservation laws are given by (ω₁ = ω₀ + Ω , k→₁ = k→₀ + K→ , ω₂ = ω₀ + 2Ω , and k→₂ = k→₀ + 2K→ simultaneously (ω_m, k→_m and Ω, K→ are the frequencies and wave vectors of light and acoustic waves, m = 0,1,2). Such a four-wave process occurs at the frequency of the acoustic wave, peculiar to just a two-phonon scattering, which can be determined from Ω₀ = 2πλ^-1v ∣ ${\mathrm{n}}_0^2$ - ${\mathrm{n}}_1^2$ ∣^1/2, here n ≠ n₁ are the refractive indices of uniaxial crystal, v is the ultrasound velocity, λ is the incident light wavelength. The polarization of light in the zero-th and the second orders is orthogonal to the polarization in the first order, whereas the frequencies of light beams in the first and second orders are shifted by Ω₀ and 2Ω₀, respectively, with respect to the zero-th order.

3. Three-wave non-collinear acousto-optical weakly coupled states

In this case, the evolution of the complex amplitudes C₀(x) and C₁(x) of light waves, describing a one-phonon Bragg non-collinear acousto-optical interaction, is governed by the well-known set of combined equations [6,7]

We approximate the factors q_0,1 as q₀ ≈ q₁ ≈q = 2πΔn/(λcosθ). The amplitude Δn of varying the refractive indices due to the action of a continuous-wave ultrasound reflects the photo-elastic properties of the crystal and includes the amplitude of acoustic wave; θ is the angle of incidence for the wave C₀. The parameter η = k_0,x-k_1,x represents the joint angular-frequency mismatch. Using the boundary conditions ∣ C₀ (x = 0) ∣² = 1, C₁ (x = 0) = 0, and the conservation law ∣ C₀∣² + ∣C₁∣ = 1, resulting from Eqs. (1), we write the solutions to Eqs. (1) in terms of the light intensities

These solutions involve the contributions of two types. The first summand in ∣C₀∣² is independent of the coordinate x and exhibits the contribution of some background, while the second one, representing the oscillating portion of the solution, describes localizing the incident light field. The oscillating portion of the field is imposed on the background, whereby the existence of the joint angular-frequency mismatch η and its absolute value determine the level of that background. The intensity of the scattered light contains the only oscillating portion of a field. Applying the additional condition of localizing the scattered light within the spatial interval L of interaction, we find from Eqs. (2) that $L\sqrt{q^2+{\eta }^2}=\mathrm{\pi n}$, where n is a whole number. The intensity ∣C₁∣² will be nonzero only in the spatial interval occupied by elastic wave and, therefore, the envelope of scattered light wave will be localized; i.e. the distribution of ∣C₁∣² over the transverse extent of elastic wave has n partial peaks in its envelope, while the intensity ∣C₁∣⁰ has n holes. Here, we have assumed that the transverse distribution of acoustic power density is rather uniform in behavior. When all these phenomena take place, we may say that a n -pulse weakly coupled state is shaped with a three-wave non-collinear acousto-optical interaction. The efficiency ξ, of localization can be obtained from Eqs. (2): ξ, = q²(q² +η²)^-1. It depends on the acoustic power density P ~q² and the joint mismatch η. With η ≠ 0, one can shape the non-collinear weakly coupled acousto-optical states at a low level of the power density P, but the efficiency I of localization will be less than unity. The two-dimensional plots for the spatial-frequency distributions of the optical components in a two-pulse three-wave coupled state are presented in Fig. 1.

 

Fig. 1. Spatial-frequency distributions for optical components in a two-pulse three-wave weakly coupled state with q = 1, I = 1, n = 2, and L = 2π : (a) the zero-th order of scattering; (b) the first order of scattering. The distribution (b) is locked with η = 0 .

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4. Shaping four-wave non-collinear acousto-optical weakly coupled states via two-phonon light scattering

A set of equations for the amplitudes C_m (x) of light waves (m = 0,1,2), with stationary two-phonon light scattering in Bragg regime is given by [6,7]

The parameters η_m=k_m,x-k_m+1,x, explained in terms of x-components for the light wave vectors, represent the joint angular-frequency mismatches. The factor q describes both the material properties and the acoustic power density and it is set equal to a constant. We analyze Eqs. (3) with the simplest boundary conditions ∣ C₀(x = 0)∣² = I², C_l,2 (x = 0) = 0 and exploit the conservation law ∣C₀∣² +∣C₁∣² +∣C₂∣² =I², resulting from Eqs. (3), where I² is the intensity of continuous-wave incident light. The exact solutions to Eqs. (3) in this regime can be written as

Here the numbers a_m are real roots of the cubic algebraic equation a³ -(η₀ +η)a² - (2q² -ηη₀)a + q²η = 0 and η = η₀ +η₁. As follows from Eqs. (4–6), the intensities ∣C_m(x)∣² are periodic in x, so such values x_n exist that ∣C₀(x_n=0)∣² =I², C_l,2 (x_n = 0) = 0. Thus, the intensities of scattered waves are zero outside the area occupied by the acoustic wave, i.e. the effect of localization occurs for the scattered light. Inside this area, the spatial distributions of the scattered waves contain a number of peaks, and simultaneously the distribution of incident light has holes at the same positions. If η₀ = η₁ = 0, we find

The condition of localization for the scattered components in Eqs. (7) within the spatial interval (0, x_n) has the form of x_n q = πn√2 . Now we assume the precise angular alignment and expend η₀ and η₁ into a power series only in terms of the frequency detuning Δf = ∣f - f₀∣ for the current frequency f relative to the frequency f₀. In the second order approximation the diagram of wave vectors gives us η₀ ≈ πλ ${\mathrm{n}}_0^{-1}$v^-2(Δf)² and η₁ ≈ πλ, ${\mathrm{n}}_0^{-1}$ v^-2 (4f₀ (Δf) + 7 (Δf)²). Therefore, in the first order approximation, we may put η₀ ≈ 0 and η ≈ η₁ ∝ Δf. A graphic example of the spatial distributions for light intensities inside the rectangular acoustic pulse for this case is presented in Fig. 2. It is seen that well localized and uniform distribution of light with η₁ = 0 becomes to be broken due to the increasing mismatch η₁, i.e. the detuning Δf, or converted into other states of localization. These plots demonstrate various opportunities for shaping multi-pulse four-wave weakly coupled states via two-phonon light scattering and for observing their optical components.

 

Fig. 2. Distributions for the triplet of optical components in a multi-pulse four-wave weakly coupled state with q = 1, I = 1, and x = 2π : a five-pulse component of incident light in the zero-th order of scattering (a); a ten-pulse component in the first order (b); a five-pulse component in the second order (c). The distributions (b) and (c) are completely locked with η₁ = 0 .

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5. Experimental data

A schematic arrangement of the experiment is shown in Fig. 3. It is similar to the set-up for acousto-optical processing [8] and includes a continuous-wave wide-aperture light beam, a non-collinear crystalline cell, and the CCD linear array. The incident beam was precisely oriented at the Bragg angle relative to acoustic beam to minimize the effect of angular mismatches and thereby to provide electronic control over measuring the contribution of frequency detuning.

 

Fig. 3. Schematic arrangement of the experiment (a) and the simplest distribution of optical components inside the rectangular acoustic pulse for a non-collinear four-wave weakly coupled state (b), see Eqs. (7) with n = 1.

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Observation of optical components in stationary three- and four-wave coupled states has been carried out with experiments using a cell made of tellurium dioxide crystal oriented along [II0]-axis (f₀ =63.3 MHz, v=0.6 mm/μs) on a light wavelength of λ= 633 nm and λ = 488 nm, respectively. The Bragg scattering of circularly polarized light, which gives the maximum efficiency of interaction, was performed without any effect on the acoustic wave. The incident light power was about 1 W, while the acoustic pulse power was in excess of 3.3 W. Acoustic pulses had the rectangular shape, whose width was varied to control stability of the states formed. The intensities of the optical components in four-wave coupled states have been measured as functions of the product qx and the detuning Δf on the first interval of localization (n = 1). The corresponding oscilloscope traces are presented in Figs. 4 and 5. Because L = 1.3 cm was constant, the acoustic power was varied to control the product qx .

 

Fig. 4. Oscilloscope traces for the intensities of the optical components in three-wave coupled states (m = 1, 2) versus the product qx at various values of the frequency detuning Δf.

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Fig. 5. Oscilloscope traces for the intensities of the optical components in four-wave coupled states versus the product qx at various values of the frequency detuning Δf.

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6. Conclusion

We conclude from Figs. 4 and 5 that stationary three- and four-wave coupled states have been shaped and they are tolerant to small levels of detuning. Experimentally observed stability of optical components as well as preliminary estimations show that the stability criterion, by analogy with the result obtained for a system associated with the nonlinear Schroedinger equation [9], can be elaborated for this type of Bragg spatial soliton. The presence of frequency detuning decreases the efficiency of localization and leads to compressing the coupled states. Moreover, when the detuning grows, one can observe the conversion of a four-wave state into a three-wave state, because two-phonon scattering becomes forbidden. The first interval of localization for a three-wave coupled state turns out to be shorter than for a four-wave one (qx₁ decreases to 3.25 at Δf = 482 kHz) in agreement with Refs. [4–6].