The dynamics of localized nonlinear waves is an attractive subject in a wide range of real physical systems.^[1–3] In nonlinear optics, the theory and experiments on all-optical control have been developing at a fast pace in recent decades,^[4–10] and the dynamical properties of localized nonlinear optical waves (optical solitons) in nonlinear media open a new door to all-optical control.^[11–14] In general, the medium approximately presents local nonlinearity because the beamwidth is often considerably larger than the characteristic width, which is also width of nonlinear response function of medium.^[15–18] However, the nonlinearity of the media may also be remarkable nonlocal under the right conditions.^[19–21] In recent years, the nonlocality has become one of the hot topics in nonlinear optics due to its unique properties different from the locality and potential applications in all-optical controlling field.^[22–30]

In the general case, nonlocal nonlinear medium is homogeneous, and its response function is a symmetric function such as Gauss function, exponential function, and so on.^[25,31] Nevertheless, in some cases, nonlocal nonlinear medium can also be inhomogeneous, correspondingly, its response function may be asymmetric.^[32] Several studies have shown that asymmetric nonlocality significantly affects the soliton properties and their dynamical behaviors.^[33–36] Particularly, in Kerr media, the asymmetric nonlocal response can result in the tunable self-bending of soliton light beams,^[33] and can alter the existence domains and the stabilities of asymmetric nonlocal solitons.^[34]

In this paper, dynamical properties of a spatial soliton with phase-front curvature, which propagates in asymmetric nonlocal medium, are investigated in details. The results suggest that the phase-front curvature plays an important role in propagation and evolution of the asymmetric nonlocal soliton. More specifically, controllable soliton propagation can be achieved in asymmetrical nonlocal media by controlling the phase-front curvature of the asymmetric nonlocal soliton. These properties are of value to practical applications in all-optical controlling and steering fields.

We consider the propagation of a laser beam along the z axis of a slab waveguide with symmetric local and asymmetric nonlocal components of the focusing nonlinear response in the presence of an imprinted optical lattice. The dynamics of propagation is described by the modified nonlinear Schrödinger equation for the dimensionless complex light field amplitude q(z,x)^[32,33]

Equation (1) has stationary soliton solutions in the form of q₀(x,z) = w(x)exp(ibz), where w(x) is a real function and b is the propagation constant. Then, plugging it into Eq. (1), one can obtain the following equation

When other parameters keep unchanged, there is a critical value μ_cr of the nonlocality parameter. Stationary solitons can exist in the nonlinear system described by Eq. (2) only for μ < μ_cr.^[32] We can search for these stationary soliton solutions of Eq. (1) by solving Eq. (2) numerically with standard relaxation method,^[37] imaginary time method^[38,39] or shooting method.^[40] The properties of these stationary soliton solutions have been studied in Refs. [32] and [33] in great detail. Therefore, in the following section, we consider the dynamical properties of spatial soliton with initial phase-front curvature during its propagation in an asymmetric nonlocal nonlinear medium. To do so, the spatial soliton with initial phase-front curvature must first be obtained. By reshaping and phase imprinting technologies, the soliton, which can stably exist in the system described by Eq. (1), can have an initial phase-front curvature imposed on it.^[41,42]

Thus, in order to study the dynamical properties of the asymmetric nonlocal spatial soliton with an initial phase-front curvature during its propagation in the system (1), one should first obtain the stationary soliton q₀(x,z) = w(x)exp(ibz) which exists in the system (1); and then impose an initial phase-front curvature C on it at z = 0 by reshaping and phase imprinting technologies; finally, investigate its dynamical properties during the propagation by taking q(x,0) = q₀(x,0)exp(−iCx²/2) as an initial input.

In particular, in the case when μ = 0, the system (1) is in essence a local one. The research results show that the propagation properties of localized optical field, which is stationary in local system, were remarkably influenced by the initial phase-front curvature C. Figure 1 presents typical evolution characteristics of fundamental mode solitons with different initial phase-front curvatures in local media. Obviously, one can see that initial phase-front curvature C does not change the trajectory of the light beam center by comparing Figs. 1(a), 1(b), and 1(c). However, it may make the beam width oscillatorily vary during propagation of the beam, as shown in Figs. 1(a) and 1(c). Furthermore, the effect of positive initial phase-front curvature on soliton propagation in local nonlinear system is slightly different from that of the negative one (see Figs. 1(a) and 1(c)). More specifically, one can see from Fig. 1(c) that the positive initial phase-front curvature makes the beam initially shrink; whereas the reverse occurs for the negative phase-front curvature, as shown in Fig. 1(a). Finally, it is worth pointing out that the beam may keep unchanged in waveform when C is equal to zero.

Fig. 1.

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	Fig. 1. Typical evolution plots of fundamental mode solitons with different initial phase-front curvatures (a) C = −0.5, (b) C = 0, (c) C = 0.5; (d) profile of fundamental mode soliton (red line), lattice profile (blue line), and nonlinear refractive index distribution (green line) in the system (1). Other parameters are p = 0.5, Ω = 0.5, μ = 0, and U = 2.

In general, the nonlocality parameter μ is usually nonzero, the corresponding system described by Eq. (1) is an asymmetrical nonlocal medium. In such a system, the effect of the initial phase-front curvature C on propagation of the stationary optical field is markedly different from that in local system. As illustrated in Fig. 2, the initial phase-front curvature C not only induces the oscillation of beam width, but it also can synchronously change the trajectory of the light beam during its propagation in an asymmetrical nonlocal medium. By comparing Fig. 2(b) with Fig. 2(c), one can easily discover that the oscillating amplitude of the light beam center increases in a transverse direction with an increase of the initial phase-front curvature C [see Fig. 3(a)]. One can define propagation distance of the light beam center along the z direction within a periodic transverse oscillation as its oscillating period, and results showed that the oscillating period increases when its initial phase-front curvature C increases, as shown in Fig. 3(b). Noticeably, the sign of initial phase-front curvature plays the same roles in width variation of light beam as the local case. However, two initial phase-front curvatures, which have the same absolute value and the opposite signs, make the light beam propagate along two distinct paths, as shown in Fig. 4.

Fig. 2.

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	Fig. 2. Typical evolution plots of fundamental mode solitons with different initial phase-front curvatures (a) C = 0, (b) C = 0.5, (c) C = 1; (d) profile of fundamental mode soliton (red line), lattice profile (blue line), and nonlinear refractive index distribution (green line) in the system (1). Other parameters are p = 0.5, Ω = 0.5, μ = 0.1, and U = 2.

Fig. 3.

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	Fig. 3. Dependence of (a) the oscillating amplitude and (b) the oscillating period for fundamental mode soliton on the initial phase-front curvature C during its propagation in the system (1). Other parameters are p = 0.5, Ω = 0.5, μ = 0.1 and U = 2.

Fig. 4.

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	Fig. 4. Typical evolution plots of fundamental mode solitons with different initial phase-front curvatures (a) C = −1 and (b) C = 1 in the system (1). Other parameters are p = 0.5, Ω = 0.5, μ = 0.1, and U = 2.

As can be seen from the above, in an asymmetrical nonlocal medium, the initial phase-front curvature can significantly affect the trajectory of the light beam during its propagation. The main reason for this phenomenon originates from asymmetrical nonlocal nonlinear response of material to optical field. In more specific terms, the soliton light beam can induce an asymmetrical distribution of nonlinear refractive index due to asymmetrical nonlocal nonlinearity, and the position of the maximum value of nonlinear refractive index does not coincide with that of the maximum value of the amplitudes of localized light field, as shown in Fig. 2(d). So, the localized light field can be dragged towards the transverse direction by the asymmetrical distribution of nonlinear refractive index. On the other hand, the optical lattices have bound effect on localized optical field.^[32] When C = 0, the initial phase-front curvature C cannot cause the oscillation of beam width, the distribution of nonlinear refractive index induced by the beam is invariable during its propagation. Then, the drag effect can always be balanced by the bound effect originated from optical lattices (the distribution of linear refractive index), which makes the light beam propagate in a straight line with constant beam width [see Fig. 2(a)]. However, for C ≠ 0, the beam width oscillates in a period when the light beam transmits ahead, thus it causes a periodically changing distribution of nonlinear refractive index, the drag force also varies periodically. But the optical lattices keeps constant, so that the bound force on localized optical field is invariable. Finally, the light beam oscillations in transverse direction during its forward propagation due to the combined effect of invariable bound force and periodically changing drag force, as shown in Figs. 2(b), 2(c), and Fig. 3. This property can be used to achieve effective control of the light beam, and to provide a new way to realize all-optical switches, optical routers, and optical networks.

In order to achieve controllable soliton propagation based on phase-front curvature in asymmetrical nonlocal media, one can truncate the optical lattice to a finite length along the longitudinal direction. In the following discussion, the function R(x,z) described lattice profile is expressed as

The effect of the initial phase-front curvature C on the propagation of the fundamental mode soliton is important for soliton steering and controlling. In this case, a phase-front curvature C is imposed on the fundamental mode soliton at z = 0 (i.e., z_i = 0). Numerical results show that the initial phase-front curvature C has a significant impact on dynamical behaviors of the fundamental mode soliton in the system described by Eqs. (1) and (3), as shown in Figs. 5(a) and 5(b). Specifically, figures 5(c) and 5(d) present the dependence of deviation distance Δ and escape angle α_end on the initial phase-front curvature C. As one can see from Figs. 5(a) and 5(b), with varying the initial phase-front curvature C, the fundamental mode soliton can observably change its trajectory, this causes deviation distance Δ and escape angle α_end to be simultaneously changed. It means that one can effectively control soliton propagation by adjusting the parameter C.

Fig. 5.

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	Fig. 5. Typical evolution plots of fundamental mode solitons with different initial phase-front curvatures (a) C = −0.5 and (b) C = 0.1 in the system described by Eqs. (1) and (3); dependence of (c) deviation distance Δ and (d) escape angle αend on initial phase-front curvature C for fundamental mode solitons. Other parameters are p = 0.5, Ω = 0.5, μ = 0.1, z0 = 20, and U = 2.

Similarly, in the system described by Eqs. (1) and (3), dynamical behavior of the fundamental mode soliton can also be effectively controlled by changing the position z_i at which the phase-front curvature C is imposed on it. Figures 6(a) and 6(b) present typical evolution plots of fundamental mode soliton for different positions z_i. Obviously, by changing the parameter z_i, one can make a fundamental mode soliton move along different trajectories during its transmission. Deviation distance Δ and escape angle α_end also change with the variation of the parameter z_i, as illustrated in Figs. 6(c) and 6(d). As is shown in Figs. 6(c) and 6(d), with an increase of the parameter z_i, the variation of the deviation distance Δ and the escape angle α_end is non-monotonous. Notably, the tendency of the variation of escape angle is similar to that of the deviation distance except several oscillations.

Fig. 6.

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	Fig. 6. Typical evolution plots of fundamental mode soliton when phase-front curvatures are imposed on it at different positions (a) zi = 10 and (b) zi = 18 in the system described by Eqs. (1) and (3); dependence of (c) deviation distance Δ and (d) escape angle αend on the position zi for fundamental mode solitons. Other parameters are p = 0.5, Ω = 0.5, μ = 0.1, z0 = 20, C = 0.5, and U = 2.

In summary, the phase-front curvature C has an important influence on the dynamical behavior of the fundamental mode soliton during its transmission in asymmetrical nonlocal media. The phase-front curvature C can be imposed on the fundamental mode soliton by reshaping or phase imprinting technologies. By changing the phase-front curvature C or its imposed position z_i, controllable soliton propagation in asymmetrical nonlocal media can be achieved. It is very useful for further understanding the properties of optical beams in asymmetrical nonlocal media and may be applied to all-optical controlling field.