1. Introduction

In a number of cases the analysis of three-wave processes leads to finding various solitary waves in the form of coupled states, where waves of the same or even different nature become mutually trapped and propagate together [1–3]. In particular, such coupled states can be shaped via stationary co-directional collinear interaction of two optical modes with some non-optical third wave in a dispersive waveguide due to the balancing action of the square-law nonlinearity. The profiles of all the waves are steady at three different current frequencies, because the interaction exhibits itself as a mechanism of stabilizing self-action. Mismatching the wave numbers can be also included in the analysis, giving us an opportunity to follow the process more sequentially. The development of a quasi-stationary model for describing such a phenomenon and its verification are the subjects of this work. The presented approach offers a clear view of this phenomenon and predicts an opportunity of sculpturing multi-pulse coupled states. Moreover, it allows one to determine a certain topological charge, being intrinsic to each component of the coupled state, as well as to reveal the spontaneously broken symmetry. The analysis was examined experimentally using the Bragg acousto-optical interaction in a two-mode optical waveguide and all the predicted effects were observed.

2. Fundamental properties of stationary collinear three-wave coupled states

A three-wave co-directional collinear interaction with mismatched wave numbers is described by a set of three combined nonlinear partial differential equations [4]. In the particular case of the stationary regime that set can be reduced and written as

where x is the spatial coordinate, C_k (k=0, 1, 2) are the normalized complex amplitudes; 2q is the mismatch of wave numbers. Using the substitutions C_k=a_k exp(iφ_k); one can convert Eqs. (1) to the following equations for the real amplitudes a_k and the real phases φ_k

Here ζ_k, E_k, and λ_k are the constants determined by the boundary conditions. Equations (2) for the amplitudes a_k have the following solutions

The parameters α_k, β, η and the modulus κ=β/η of elliptic functions do not depend on the coordinate x. They are all determined by the boundary conditions and the mismatch q. The parameters α_k specify the backgrounds. When α_k=0, we yield ζ_k=0, λ₀=-η²β² (1-κ²), λ₁=η²β², λ₂=-η⁴(1-κ²) and linear dependences of the phases φ_k on the coordinate x. The terms b_k represent the oscillating portions of solutions, evaluating the extent of localization for the coupled state. For the functions b_k one can find

where n,m=0,1,2; k≠n≠m≠k; F_j (j=0,1,2) are the constants; F₀-F₁=F₂ and F₁>0. Equations (4) can be converted into three equations, independent of each other, with cubic-law nonlinearity, see Eqs. (5a). These equations can be considered as the motion equations d²b_i/dx²=-dU_i/db_i for some particles in the real-valued potentials U_i(b_i), see Eqs. (5b)

where q₀=-1, p₀=F₀, q₁=1, p₁=-F₁, q₂=-1, and p₂=-F₂; H_k=0 for the oscillating portions of solutions. For k=1 with b₁(x₀)=0, Eq.(5b) gives the potential that has a local minimum at b₁=0 and two absolute maxima at $b_1=\pm \sqrt{\frac{F_1}{2}}$ . The stationary localized solution connect these two maxima of U₁(b₁) and carry the topological charge Q=Δ [b₁(x→+∞)-b₁(x→-∞)], where Δ is a constant [2, 4, 5]. The topological charge Q reflects conservation of the boundary conditions of optical components inherent in the stationary coupled state. Substituting U₁(b₁) in Eq.(5a), we yield the solitary kink solution: $b_1\left(x\right)=\pm \sqrt{\frac{F_1}{2}}·\tanh \lfloor \sqrt{\left(\frac{F_1}{2}\right)}\left(x-x_0\right)\rfloor$ . The kink solution b₁(x) represents a shock wave of envelope or the dark optical component of the coupled state. The topological charge, associated with the wave b₁, can estimated as Q=±1 with Δ=(2F₁)^-1/2.

Let us look now at the waves b₀ and b₂. The potentials U_k(b_k) with k=(0, 2) are given by Eqs. (5b) with F₀>0 and F₂<0. Each of these potentials exhibits only one local maximum at b_k=0. The waves b₀ and b₂ shape the bright components $b_0\left(x\right)=\pm \sqrt{F_0}·\mathrm{sech}\left[\sqrt{F_0}\left(x-x_0\right)\right]$ and $b_2\left(x\right)=\pm \sqrt{-F_2}·\mathrm{sech}\left[\sqrt{-F_2}\left(x-x_0\right)\right]$ of the coupled state with the asymptotes b_k(x→-∞)=b_k(x→+∞)=0 and Q=0. For these waves the even potentials correspond to the even asymptotes b_k(x→-∞)=b_k(x→+∞)=0, but at any finite distance the symmetry in these waves turns to be broken, because the absolute minima of U_k(b_k) are degenerated, and they can be reached in two different points instead of one. The particles oscillate spontaneously only in one direction, towards the right or the left, of the local maxima of U_k(b_k). Since the symmetrical states with the least energy at b_k=0 are unstable, either of two signs can be realized in the relations b_k(x₀)=±|b_k(x₀)|. This phenomenon is known as the spontaneous breaking of symmetry [5] that is inherent in topologically uncharged bright components of the coupled state.

3. Weakly coupled states in the quasi-stationary case; the localization conditions

Here, we consider a regime of weak coupling, when two light modes are scattered from the pulse of a relatively slow wave, taken instead of the wave C₂ and being of non-optical nature. Because the number of interacting photons is several orders less than the number of scattering non-optical slow quanta in a medium, essentially effective Bragg scattering of light can be achieved without any observable influence of the scattering process on that non-optical wave. The velocities of light modes can be approximated by the same value c, because the length of a waveguide does not exceed 10 cm. In this regime, the above-mentioned set of three combined nonlinear partial differential equations [4], describing a three-wave co-directional collinear interaction with mismatched wave numbers, has to be transformed and falls into a homogeneous wave equation for a slow wave, which possesses the traveling-wave solution U(x-vt), v is its velocity, and the pair of combined equations

When the non-optical pulse U(x-vt)=u(x-vt)exp(iφ) has the constant phase φ, and C_0,1=a_0,1(x, t)exp(iΦ_0,1[x, t]), γ_0,1=∂Φ_0,1/∂x, Eqs. (6) can be converted into equations

It follows from Eqs. (8) that γ_0,1=±q (u/${\mathrm{a}}_{0,1}^2$) ∫u^-1(∂${\mathrm{a}}_{0,1}^2$/∂x) dx+Γ_0,1u/${\mathrm{a}}_{0,1}^2$, but here our consideration will be restricted by the simplest choice of Γ_0,1=0. Now, we focus on the process of localization in the case, when first, two facets of a waveguide at x=0 and x=L₀ bound the area of interaction and the spatial length l₀ of the non-optical pulse is much less than L₀; and second, the non-optical pulse u(x, t)=U₀(θ[z-vt]-θ[x-l₀-vt]) has a rectangular shape with the amplitude U₀. We analyze Eqs. (7) and (8) with the fixed magnitude of q and the natural boundary conditions a₀(x=0, t)=1, a₁(x=0, t)=0 and trace the dynamics of the phenomenon as far as the localizing pulse of the non-optical wave is incoming through the facet x=0, passing along a waveguide, and issuing through the facet x=L₀ with the constant velocity v. There are two possibilities. The first of them is connected with a quasi-stationary description of this effect with the assumption that v≪c, while the second one presupposes a weak inequality v<c. With a quasi-stationary approach, we may put ∂u/∂x≈0 in Eqs. (7), (8) everywhere, excluding the points x={0, l₀}, and yield γ_0,1=±q. Then, we follow three stages in the localization processes.

Stage 1: Localizing pulse is incoming through the facet x=0: Exploiting γ_0,1=±q, Eqs. (11) can be solved exactly. The intensities of light waves on x∈(0, l₀) are given by

To find the coefficients in Eqs. (9) we use the conservation law ${\mathrm{a}}_0^2$+${\mathrm{a}}_1^2$=1.

Stage 2: Localizing pulse is passing in a medium. The rectangular pulse as the whole is in a waveguide, so ∂u/∂x=0 and x=l₀ in Eqs. (9) for the region (l₀, L₀-l₀).

Stage3: Localizing pulse is issuing through the facet x=L₀. This stage is symmetrical to stage 1, so Eqs. (9), can be inverted and related to the region of x∈(L₀-l₀, L₀).

The first summand in |C₀|² in Eqs. (9) exhibits a background, whose level is determined by the mismatch q; the second one represents the oscillating portion of solution, i.e. the localized part of the incident light imposed on a background. The scattered light contains the only oscillating portion of field that gives the localization condition ${\mathrm{x}}_{\mathrm{C}}^2$ (${\mathrm{U}}_0^2$+q²)=π² N², where x_C is the spatial size of the localization area with v≪c and N=0, 1, 2, …

On the second possibility (v<c), it is reasonable to put u=αU₀x (α is to be found), when the localizing pulse is incoming through the facet x=0. In so doing, we have to take into account the fact that the solution to Eqs. (7) is known only if the last coefficients are proportional to u² [6], i.e. ${\mathrm{\gamma }}_{0.1}^2$∓2qγ_0,1=q²ζ²x² with ζ=const. That is why we are forced to exploit the smallness of mismatch, believing that q≪1, and to find approximate solutions to Eqs. (7), (8) at this stage. Resolving this algebraic equation relative to γ_0,1, we yield ${\gamma }_{\mathrm{0,1}}=\pm q\left(1\pm \sqrt{1+{\zeta }^2{x}^2}\right)$ . In terms of these values for γ_0,1, Eqs. (8) can be satisfied with an accuracy of q², while Eqs. (7) can be solved exactly. The intensities of light waves with α=ζ on the interval of x∈(0, l₀) are given by

To find the coefficients in Eqs. (10) we approximate γ_0,1 as γ₀≈-qxζ and γ₁=-q[(4/3)+xζ] on the interval of x∈(0, l₀) and then use the conservation law. Stage 2 with v<c is governed by Eqs. (9) as well, because again ∂u/∂x=0; finally, we can invert and apply Eqs. (10) to stage 3. The parameter α makes it possible to join Eqs. (9) and (10) at the point l₀, therefore the localization condition takes the form α²${\mathrm{x}}_{\mathrm{S}}^4$(${\mathrm{U}}_0^2$+q²)=4π²N², where x_S is the spatial size of localization area with v<c.

4. Computer simulation and experimental verification in the quasi-stationary case.

Shaping the optical components of solitary three-wave weakly coupled states was simulated using Eqs. (9). As an example, Fig.1 shows a set of plots for the scattered light intensity |C₁|², when both the amplitude U₀ and the mismatch q are fixed, while its width τ₀=l₀/v is increasing plot by plot in the temporal scale of τ_C=x_C/v. Figures 1(b) and (d) illustrate shaping the scattered optical components of one- and two-pulse weakly coupled states.

 

Fig. 1. Intensity of the scattered optical component (vertical axis) versus the time τ=x/v and the waveguide length L₀: a. τ₀<τ_C/2; b. τ₀=τ_C; c. 3τ_C/4<τ₀<2τ_C; d. τ₀=2τ_C.

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Verification of the analysis performed has been carried out due to our trial acousto-optical experiments with an optical wavelength of 633 nm in a two-mode crystalline waveguide based on calcium molybdate oriented along the x-axis (exact synchronism acoustic frequency 43.7 MHz, L₀=3 cm) and possessed the photoelastic constant p₄₅=0.06, making possible to couple two optical modes. The schematic arrangement of the experiments was similar to the scheme for acousto-optical filtering [7] and includes a continuous-wave polarized light beam, a crystalline waveguide, output analyzer, and photodetector. During the experiments rather effective (> 10%) Bragg scattering of the light was observed without any effect on the acoustic wave, when their powers were approximately equal to 100 mW each, so the regime of weak coupling had taken place. However, the emphasis was on the dynamics of shaping the coupled states and the quantitative estimation of their temporal characteristics and not on the amplitude parameters of this process. The intensity distributions in both incident and scattered optical components of coupled states as functions of the acoustic power density, the localizing pulse width τ₀, and the frequency mismatch Δf=q v/π has been measured. The oscilloscope traces in Fig. 2 illustrate the particular case, when only the localizing pulse width τ₀ is varied. One can see four sequential steps in shaping the optical components of the coupled states in a waveguide, which are in agreement with the analysis performed, see Eqs. (9) and Fig. 1.

 

Fig. 2 The incident (upper lines) and scattered (bottom lines) light intensities with Δf=0.4 MHz in a two-mode waveguide: a. τ₀=0.8 µs; b. τ₀=τ_C=2.5 µs; c. τ₀=4.2 µs; d. τ₀=2τ_C=5.0 µs.

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The amplitudes of one- and multi-pulse weakly coupled states are the same, when the acoustic power density P and the frequency mismatch Δf are fixed and only the acoustic pulse width τ₀ is varied, see Fig. 2 and Figs. 3(a) and (b). However, if the mismatch Δf increases, the amplitude of the coupled state decreases with fixed τ₀ and P, compare traces in Figs. 3(a) and (c).

 

Fig. 3. The incident (upper lines) and scattered (bottom lines) light intensities in a two-mode waveguide: a. τ₀=τ_C=2.5 µs; Δf=0.4 MHz; N=1, b. τ₀=3 τ_C=7.5 µs; Δf=0.4 MHz; N=3 c. τ₀=τ_C=2.5 µs; Δf=1.2 MHz; N=3 (the amplitude scale is enlarged nine times).

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5. Conclusion

We conclude from the analysis performed and the experimental data obtained that temporal Bragg solitary waves in the form of quasi-stationary three-wave weakly coupled states can be shaped due to collinear scattering of light by the non-optical wave in a two-mode waveguide. Experimentally observed stability of the coupled states gives us a hope that the adequate stability criterion [8] can be elaborated. The data in Fig. 3 show that the regime of multi-wave coupled states can be potentially applied to electronically controlled conversion of 1B/1B binary encoded electronic digital signals into 1B/NB binary encoded trains of optical pulses.