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Existence and stability of the fundamental and higher-order solitons, which exist in nonlinear media with asymmetric response and periodic linear refractive index modulation, are presented. It is found that the existence of solitons results in the balance between linear refractive index modulation (optical lattices) and nonlinear refractive index induced by incident optical field. In addition, Dynamical properties of fundamental mode solitons are also investigated in detail, and may be applied in the fields of soliton controlling and steering.
© 2014 Optical Society of America
1. Introduction
Light propagation in nonlinear media has been studied extensively due to its novel properties and potential applications in all-optical controlling and steering [1, 2]. In recent years, the evolution of light beams in media with nonlocal nonlinearity has been well investigated [3–10]. Nonlocality plays an important role in many fields of physics [11], such as Bose-Einstein condensates [12], plasma physics [13], and nonlinear optics [3]. In nonlinear optics, nonlocality was found in photorefractive crystals [14–16], thermal nonlinear media [17–20], atomic vapor [21], and liquid crystals [22–25], etc.
Essentially, nonlocality tends to spread out nonlinear refractive index distribution induced by localized excitations. Thus, the intriguing properties of nonlocal solitons are remarkably different from those in media with local nonlinearity. For example, nonlocality allows suppression of the modulation instabilities of plane waves [26], prevents the catastrophic collapse of self-focused beams [27], and suppresses azimuthal instabilities of vortex-ring [28]. In particular, it can drastically modify the interaction between solitons, e.g., presenting attraction between out of phase solitons [29] and dark solitons beams [30].
2. Theoretical model
We consider the propagation of laser beam in the slab waveguide of focusing nonlinearity, with an imprinted periodic modulation of linear refractive index. The dynamics of the light field q(x, z) is governed by the following dimensionless equation [26]
Here x and z denote the transverse and longitudinal coordinates scaled to the beam width and diffraction length, respectively; the parameter p characterizes the lattice depth, while the function R(x) = cos2 (Ωx) describes lattice profile, where Ω is the modulation frequency. The change of the refractive index, induced by the laser beam with intensity I(x, z) = |q(x, z)|2, can be expressed as The real, localized G(x) with is the response function of the nonlocal medium, whose characteristic length determines the degree of nonlocality. For a homogeneous nonlocal medium, G(x) is usually a symmetric function and has been studied extensively. Nevertheless, in inhomogeneous nonlocal medium, the response function may be asymmetric. In this situation, Eq. (2) can be reduced to Here, the nonlocality parameter μ represents the magnitude of the nonlocal component of nonlinear response, and in most realistic situations μ < 1. Thus, Eqs. (1) and (2) can be written as [31] which describes the propagation of laser beam in a slab waveguide with self-focusing drift and asymmetric nonlocal nonlinearity in the presence of an imprinted optical lattice. It should be pointed out that the total energy flow is conserved. In this paper, we take nonlicality parameter μ ∼ 0.2, which corresponds to light beams with a width of the order of 3μm at the wavelength 0.5μm, propagating in a photorefractive material biased with a static electric field ∼ 400V/cm2 and maintained at room temperature [32]. For a beam width 15μm and wavelength 532nm, Ω = 1 corresponds to a lattice period 15μm, while p = 1 corresponds to a refractive index variation δn ≈ 0.0002. Therefore, the modulation depth of the refractive index can be considered as a small quanlity and is of the order of the nonlinear correction to the refractive index due to the nonlinear effect [33]. And q = 1 corresponds to a peak intensity of the order of 100mW/cm2, a length z ∼ 1 corresponds to about 0.7mm [34].3. Existence and properties of the fundamental mode solitons
Several studies have shown that asymmetric nonlocality significantly affects the soliton properties and their dynamical behaviours [31, 35, 36]. Particularly, in Kerr media, the asymmetric nonlocal response can result in the tunable self-bending of soliton light beams [31], and can alter the existence domains and the soliton stabilities [35].
When μ = 0 in Eq. (3), the nonlocal nonlinearity of the media is reduced to local nonlinearity. In such medium, the light field with symmetrical profile results to the symmetrical nonlinear refractive index so that the center of the light field is in coincidence with that of nonlinear refractive index distribution, as shown in Fig. 1. It means that the light field may self-trap in the waveguide induced by itself. Therefore, the stationary mode can exist in such system, even in the absence of optical lattice (that is, p = 0).
Fig. 1 Distribution of nonlinear refractive index n induced by light field (red curves) whose amplitude is |q| = e−x2 at different nonlocality parameter μ. Blue solid, dotted and dashed curves correspond to (a) μ = 0, −0.2, and 0.2; and (b) μ = 0, 0.1 and 0.2.
When μ ≠ 0, however, the media are characterized by asymmetrical nonlocal nonlinearity. Thus, a symmetrical beam can induce an asymmetrical distribution of the nonlinear refractive index so that the position of the maximum value of the refractive index does no longer coincide with that of the maximum value of the amplitude of light field. In the case of traveling in such medium, symmetrical 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 no longer coincide with that of the maximum value of the amplitudes of localized light field. Indeed, when μ < 0, the position of the maximum value of the nonlinear refractive index is shifted towards the negative x axis, which results in that the localized light field can be dragged towards the negative x axis by self-induced waveguide, as shown in Fig. 1(a). When μ > 0, the reverse occurs [see Fig. 1]. Also, from Fig. 1(b) one can see that, with the increasing of the absolute value of the nonlocality parameter μ, the nonlinear refractive index gets stronger and the asymmetry becomes more significant, which means that the larger the nonlocality parameter μ, the stronger the dragged force. Obviously, the larger the nonlocality parameter μ, the stronger the dragged force. Thus, one can infer that it can be inferred that the asymmetrical nonlocal nonlinearity will lead to the inexistence of symmetrical mode at p = 0.
Some studies have shown that, the optical lattices have bound effect on localized optical field. Judging from the above analysis, stationary state optical mode might possibly exist in the medium with asymmetrical nonlocal nonlinearity if the dragged effect can be balanced by the bound effect originated from optical lattices. In this paper, the properties of fundamental mode solitons, which exist in nonlinear media with asymmetric response in the presence of an imprinted optical lattice, are presented. Results show that the existence of these fundamental mode solitons depends on the competition between linear refractive index modulation (optical lattices) and nonlinear refractive index induced by incident optical field.
One can search for stationary soliton solutions of Eq. (4) numerically in the form of q(x, z) = w(x)exp(ibz) by using standard relaxation method [37, 38], imaginary time method [39, 40] or shooting method [41], where w(x) is a real function and b is the propagation constant. To elucidate the linear stability of soliton families, one can search for perturbed solutions in the form q(x;z) = w(x)exp(ibz) + u(x)exp[i(b + λ)z] + v* (x)exp[i(b − λ*)z] [42], where the perturbation components u and v could grow with a complex rate λ during propagation. Linearization of Eq. (4) around stationary solution w yields the eigenvalue problem
Clearly, stationary soliton is stable when the imaginary part of the complex rate λ is equal to zero. We have solved the system of Eqs. (5) and (6) numerically to investigate the linear stability of the stationary soliton. Due to the structural stability and convenience of application, it is important to understand the properties and dynamical behaviors of fundamental mode solitons in the system described by Eq. (4).The numerical results show that lattice depth p has an important impact on the properties of the fundamental mode solitons. To illustrate this, we can define the deviation of the position located by the peak amplitude of fundamental mode soliton from the central of the channel in which fundamental mode stays as its center deviation. By comparing Figs. 2(a) and 2(b) with Figs. 2(c) and 2(d), it was found that lattice depth p can significantly affect the amplitude profile of fundamental mode when other parameters are constant. More specifically, with an increase of lattice depth p, peak amplitude of fundamental mode increases, whereas its center deviation decreases synchronously, as shown in Figs. 2(e) and 2(f). The reason for this is that the bound effect on fundamental mode becomes stronger when lattice depth p increases. On the contrary, when lattice depth p decreases, the bound effect weakens. Hence, it can be imagined that the bound effect can no longer resist the dragged effect when the lattice depth p decreases to a certain extent, and this will lead to the inexistence of the fundamental mode. Obviously, there is a critical value pcr of the lattice depth in the case when other parameters are constant. When the value of lattice depth p is beyond critical lattice depth pcr, the fundamental mode can exist in the system (4). Nevertheless, for p < pcr, there is no any fundamental mode in it. Figures 3(a) and 3(b) present the dependence of critical lattice depth pcr on relevant parameter such as energy flow U or nonlocality parameter μ. It is evident that critical lattice depth pcr increases with an increase of nonlocality parameter μ or energy flow U while keeping other parameters unchanged.
Fig. 2 Profiles (column 1 in rows 1 and 2) and corresponding propagation (column 2 in rows 1 and 2) of fundamental mode solitons at p = 0.5 ((a) and (b) in row 1) and p = 5 ((c) and (d) in row 2). Dependence of (e) peak amplitude and (f) center deviation of the fundamental mode soliton on lattice depth p. Other parameters are U = 2, Ω = 1 and μ = 0.2.
Fig. 3 Critical lattice depth pcr versus (a) nonlocality parameter μ at U = 2 and (b) energy flow U at μ = 0.2; critical nonlocality parameter μcr versus (c) energy flow U at p = 1 and (d) lattice depth p at U = 2; critical energy flow Ucr versus (e) nonlocality parameter μ at p = 1 and (f) lattice depth p at μ = 0.2. The other parameter is Ω = 1.
Actually, nonlocality parameter μ also has significant effect on the existence and the properties of the fundamental mode solitons. By making the comparative analysis for their profiles at different nonlocality parameter μ presented in Figs. 4(a) and 4(b) and Figs. 4(c) and 4(d), one can see that nonlocality parameter μ governs the shape of the fundamental mode solitons when other parameters keep unchanged. The numerical results show that, with an increase of nonlocality parameter μ, peak amplitude decreases and its center deviation largens synchronously, as shown in Figs. 4(e) and 4(f). Similarly, one can define μcr as the critical nonlocality parameter. In the case of other parameters being unchanged, it was found that the fundamental mode can exist in the system (4) for μ < μcr, and once the nonlocality parameter μ is beyond the critical value μcr, any fundamental mode cannot exist in the system (4). Furthermore, one can note from Figs. 3(c) and 3(d) that the value of the critical nonlocality parameter μcr has been affected by lattice depth p or energy flow U with other parameters being constant. More specifically, the critical nonlocality parameter μcr decreases when the energy flow U increases, as illustrated in Fig. 3(c). However, with an increase of lattice depth p, the reverse occurs, that is, the critical nonlocality parameter μcr increases synchronously [see Fig. 3(d)].
Fig. 4 Profiles (column 1 in rows 1 and 2) and corresponding propagation (column 2 in rows 1 and 2) of fundamental mode solitons at μ = 0.1 ((a) and (b) in row 1) and μ = 0.3 ((c) and (d) in row 2). Dependence of (e) peak amplitude and (f) center deviation of the fundamental mode soliton on lattice depth μ. Other parameters are U = 2, Ω = 1 and p = 2.
Similarly, the properties of the fundamental mode solitons also depend heavily on its energy flow U. Figures 5(a)–5(d) present profiles and corresponding propagation of fundamental mode solitons at different energy flows. When the energy flow U increases, both the peak amplitude of the fundamental mode soliton and its center deviation show the increase trend, as shown in Figs. 5(e) and 5(f). Clearly, the larger the energy flow U, the stronger the asymmetric nonlinear refractive index, that is, the stronger the dragged force. To investigate the impact of the energy flow U on the existence of fundamental mode soliton, one can define Ucr as the critical energy flow. When the energy flow U is greater than its critical value Ucr, fundamental mode cannot exist in the system (4). Only for U < Ucr, fundamental mode would probably exist in such system. Numerical simulations have demonstrated that the critical energy flow Ucr decreases with the increasing of the nonlocality parameter μ [see Fig. 3(e)]. However, when the lattice depth p increases, the critical energy flow Ucr decreases, as shown in Fig. 3(f).
Fig. 5 Profiles (column 1 in rows 1 and 2) and corresponding propagation (column 2 in rows 1 and 2) of fundamental mode solitons at U = 1 ((a) and (b) in row 1) and U = 3 ((c) and (d) in row 2). Dependence of (e) peak amplitude and (f) center deviation of the fundamental mode soliton on energy flow U. Other parameters are μ = 0.2, Ω = 1 and p = 2.
The stability of the fundamental mode solitons is a very important issue for practical application. By solving the system of Eqs. (5) and (6) numerically, one can discover that the fundamental mode solitons are stable in the entire domain of their existence. However, perturbation has an important effect on their propagation. Figure 6 presents their evolvement against a white noise whose maximal value is 0.01 during propagation. Clearly, for the same perturbation, the larger the nonlocality parameter μ or the energy flow U, the poorer the stability of the fundamental mode solitons while keeping other parameters constant. Conversely, their stability becomes better with an increase of the lattice depth p.
Fig. 6 Propagation against white noise whose maximal value is 0.01 of fundamental solitons at (a) μ = 0.1 and (b) μ = 0.3 for p = 1 and U = 2; (c) p = 0.38 and (d) p = 5 for μ = 0.2 and U = 2; (e) U = 1 and (f) U = 2.5 for μ = 0.2 and p = 1. The other parameter is Ω = 1.
4. Existence and properties of the higher-order mode solitons
Existence and properties of the higher-order mode solitons in asymmetric nonlocal media are also investigated in detail. Figure 7 presents profiles of the two-pole mode solitons under different conditions of parameters. By comparing Fig. 7(a) with Fig. 7(b), one can see that the deviation of the maximum amplitude from the centers of the lattices becomes larger when the nonlocality parameter μ increases. The numerical results show that the two-pole mode soliton can no longer exist in the system (4) if the nonlocality parameter μ exceeds to a certain critical value μcr. On the other hand, the two-pole mode soliton can exist in the system (4) for μ < μcr. The critical nonlocality parameter μcr decreases when propagation constant b increases, but it increases and tends to saturation with the increasing of the lattice depth p, as shown in Figs. 8(c) and 8(d).
Fig. 7 Profiles of two-pole mode solitons (red lines) and optical lattices R(x) (blue lines) in the system (4) at (a) μ = 0.1, (b) μ = 0.35 for b = 1 and p = 1; (c) b = 0.7, (d) b = 2.1 for μ = 0.1 and p = 1; (e) p = 1, (f) p = 2.9 for μ = 0.1 and b = 2. The other parameter is Ω = 1.
Fig. 8 Critical lattice depth pcr for the dipole-mode soliton versus (a) propagation constant b at μ = 0.1 and (b) nonlocality parameter μ at b = 1; critical nonlocality parameter μcr for the dipole-mode soliton versus (c) propagation constant b at p = 1 and (d) lattice depth p at b = 1; critical propagation constant bcr for the dipole-mode soliton versus (e) lattice depth p at μ = 0.1 and (f) nonlocality parameter μ at p = 1. The other parameter is Ω = 1.
Similarly, with an increase of the propagation constant b, both the deviation and the magnitude of the maximum amplitude also become larger, as shown in Figs. 7(c) and 7(d). Clearly, there is a critical propagation constant bcr in the system (4). When propagation constant b is larger than bcr, the two-pole mode soliton may exist in the system (4). However, for b < bcr, the opposite occurs. From Figs. 8(e) and 8(f), one can discover that the critical propagation constant bcr increases when the lattice depth p decreases, whereas it decreases with an increase of the nonlocality parameter μ.
From Figs. 7(e) and 7(f), one can see that when the lattice depth p decreases, both the deviation and the magnitude of the maximum amplitude increase. Obviously, one can define pcr as critical lattice depth, the two-pole mode soliton may exist in the system (4) for p > pcr, but it don’t exist in such system when p is greater than pcr. The numerical results show that the critical lattice depth increases when the propagation constant b or the nonlocality parameter μ increases, as illustrated in Figs. 8(a) and 8(b).
Stability of the higher-order mode solitons is markedly different from that of the fundamental mode solitons. Figures 9(a), 9(c) and 9(e) address dependence of energy flow of the two-pole mode solitons on relevant parameters. Energy flow increases when the nonlocality parameter μ or the propagation constant b increases, and it decreases with an increase of the lattice depth p. By solving Eqs. (5) and (6) numerically, it is found that the two-pole mode solitons are not stable in all domain of their existence. From Figs. 9(b), 9(d) and 9(f), one can see that the two-pole mode solitons are usually unstable in regions near the edge of domain of their existence.
Fig. 9 Energy flow U and (b) imaginary part of the perturbation growth rate for the dipole-mode soliton versus nonlocality parameter μ at b = 1 and p = 1. (c) Energy flow U and (d) imaginary part of the perturbation growth rate for the dipole-mode soliton versus a propagation constant b at μ = 0.1 and p = 1. (e) Energy flow U and (f) imaginary part of the perturbation growth rate for the dipole-mode soliton versus lattice depth p at b = 2 and μ = 0.1. The other parameter is Ω = 1.
Other higher-order mode solitons, such as triple- and quadrupole-mode solitons, have similar properties with two-pole mode solitons. Specifically, the higher the order of solitons, the larger their unstable range.
5. Dynamic properties of the fundamental mode solitons
Properties of the fundamental mode in asymmetric nonlocal media can be applied to achieve effective control and steering of soliton in asymmetric nonlocal media. Actually, fundamental mode solitons are stable in the system (4). However, what may happen if optical lattices are suddenly removed when these fundamental mode solitons propagate in the system (4)? To address this problem, in the subsequent analysis the optical lattices are removed after propagating 20 diffraction lengths in the system (4). In this situation, propagation of light beam can be described by the following equation,
withNote that the fundamental mode soliton in the system (4) at p = p0 is used as the input beam in the following study. So it is stable during propagation in the first 20 diffraction lengths in the system described by Eqs. (7) and (8), as shown in Fig. 10. By comparing Fig. 10(a) with Fig. 10(b), Fig. 10(c) with Fig. 10(d), and Fig. 10(e) with Fig. 10(f), it is illustrated that the related parameters, such as the parameter p0, nonlocality parameter μ and energy flow U, play an important role in dynamical evolution of these fundamental modes during their propagation in the system described by Eqs. (7) and (8).
Fig. 10 Typical evolvement diagrams of the fundamental mode soliton in the system described by Eqs. (7) and (8) for different lattice depthes (a) p0 = 1 and (b) p0 = 2 at U = 2 and μ = 0.2; Typical evolvement diagrams of the fundamental mode soliton in the system described by Eqs. (7) and (8) for different nonlocality parameters (c) μ = 0.05 and (d) μ = 0.15 at U = 2 and p0 = 1; Typical evolvement diagrams of the fundamental mode soliton in the system described by Eqs. (7) and (8) for different energy flows (e) U = 1.8 and (f) U = 2.1 at μ = 0.2 and p0 = 1; Other parameter is Ω = 1. Dashed line, border between p = p0 and p = 0.
To investigate the dynamical properties of fundamental modes in such systems, one can define center shift Δ to be transverse shift of the soliton center at the output end (here, 50 diffraction lengths from the input terminal) from that at 20 diffraction lengths, and escape angle α is the deflection angle of the output beam. The dependence of center shift Δ and escape angle α on related parameters (i.e., the parameter p0, nonlocality parameter μ and energy flow U) is numerically studied in detail, and it is presented in Fig. 11. Research shows that center shift Δ and escape angle α decrease with an increase of the parameter p0, as shown in Figs. 11(a) and 11(b). On the contrary, Figs. 11(c)–11(f) show that center shift Δ and escape angle α increase synchronously when nonlocality parameter μ or energy flow U increases. These dynamical properties of the fundamental mode soliton may be widely applied in the fields of all-optical control, soliton steering, and so on.
Fig. 11 (a) Center shift Δ and (b) escape angle α versus lattice depth p at μ = 0.2 and U = 2; (c) center shift Δ and (d) escape angle α versus nonlocality parameter μ at p = 1 and U = 2; (e) center shift Δ and (f) escape angle α versus energy flow U at p = 1 and μ = 0.2. Other parameters is Ω = 1.
6. Conclusion
In summary, properties of the fundamental and higher-order mode solitons, which exist in nonlinear media with asymmetric response in the present of an imprinted optical lattice, are presented. Results show that the existence of these fundamental and higher-order mode solitons depends on the competition between linear refractive index modulation (optical lattices) and nonlinear refractive index induced by incident optical field.
Acknowledgments
This research was supported by the Scientific Research Plan of the Provincial Education Department in Hubei (Grant Nos. Q20101308 and D20121203), and the Technology Creative Project of Excellent Young Team of Hubei Province (Grant No. T201204). We are very grateful for the referee’s valuable comments and insightful discussion with Prof. Lu Li.
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