1. Introduction

Interfaces separating different physical media can support a special class of transversally localized waves known as surface waves or surface modes. They exhibit novel physics as well as potential applications in optical sensing, switching and for exploration of intrinsic and extrinsic surface characteristics etc. Linear surface waves have been investigated extensively in many branches of physics[1]. Nonlinear surface waves are paid special attentions in the context of fluids[2], solids[3], plasmas[1] and nonlinear optics[4, 5, 6, 7, 8].

Recently, a new implementation of surface solitary waves was proposed [5] and experimentally created[6] in nonlinear optics. Meanwhile, surface gap solitons were predicted[7] and created in an experiment[8] at an edge of a waveguiding array built into a self-defocusing continuous medium. Siviloglou et al. reported the experimental creation of discrete surface solitons supported by the quadratic nonlinearity[9]. Multipole-mode surface solitons were shown to exist at the interface between two distinct periodic lattices imprinted in Kerr-type nonlinear media[10]. Polychromatic interface solitons in nonlinear photonic lattices were predicted[11]. Moreover, two-dimensional vortex soliton can be captured stably by a surface between two optical lattice with different lattice parameters[12].

Recent studies show that solitons in Bessel optical lattice exhibit some unique features that cannot be found on other schemes. Non-diffracting Bessel beams of different orders can be created experimentally by illuminating through a narrow annular split placed in the focal plane of a lens or axicon[13] or by holographic techniques[14]. Bessel optical lattices have been shown to support stable vortex soliton[15], spiraling solitons[16] etc. The experimental observation of self-trapping and nonlinear localization of light in modulated Bessel lattices in the form of ring-shaped and single-site states was reported by Fischer et al.[17]. Discrete solitons and soliton rotation in Bessel-like photonic lattices were also obsevered[18]. Furthermore, vortex solitons with topological charge larger than unit remain robust in higher-order Bessel lattice under appropriate conditions[19] and spatiotemporal solitons were predicted to exist stably in Bessel optical lattices[20].

However, all studies on Bessel lattice soliton in nonlinear optics deal with the two-dimensional problems. So far, surface solitons are only observed in waveguide arrays[6, 8]. Thus, exploring other schemes for implementing surface solitons and investigating the dynamics of such solitons are of important significance. In the present paper, we put forward another way (in an optical modulation slab waveguide) for the realization of surface solitons. An interface separated by 1D Bessel potential with varying potential modulation depth beside the interface is presented. It can be realized easily in experiment by illustrating a Bessel beam with its polarization orthogonal to that of guiding beams into a slab waveguide. The depth of refractive-index modulation beside the center of waveguide can be adjusted by introducing an attenuating plate before the input Bessel beam. Dynamics of solitons supported by this model is discussed in detailed.

2. Model

Following Refs. [7, 15], we assume that an optical beam propagates along the z-axis in a slab waveguide with defocusing cubic nonlinearity and imprinted transverse refractive index modulation. The generic equation describing the evolution of optical wave packets in this setting is the nonlinear Schrödinger equation:

where x = X/X ₀, z = Z/L_d are the normalized transverse coordinate and propagation distance respectively, X ₀ is an arbitrary scale, L_d = k ₀ X ² ₀ is the diffraction length, corresponding to X ₀, k ₀ is the wave number, A(x,z) is the dimensionless amplitude of the beam. The function $R(x)=p_r{J}_n^2[\sqrt{2\beta }x]$ at x > 0 and $R(x)=p_l{J}_n^2[\sqrt{2\beta }(-x)]$ at x < 0 represents the transverse refractive-index profile, where p_l(p_r) describes the depth of the refractive-index modulation at the left (right) hand of the interface located at x = 0, β is the radial scale of the n-order Bessel function. Eq. (1) conserves several quantities, including the power P and Hamiltonian H:

We search for stationary solitons, A(x,z) = q(x)exp(ibz),where b is a real propagation constant, q(x) is the real function which must vanish at |x| → ∞ and obeys the equation:

In numerical calculations, we fix the radial scale, β ≡ 2, and vary p_l, p_r and the propagation constant b. The stability of the soliton can be analyzed by considering the perturbed stationary solution form as A(x,z) = q(x)exp(ibz) + u(x)exp [i(b+λ)z] λ v(x)exp [i(b-λ)z], where the perturbation components u,v could grow with a complex rate λ during propagation. The soliton is stable if the imaginary parts of eigenvalues (λ) equal zero. Substitute the perturbed solution into Eq. (1), we obtained the linear eigen-equations,

 

Fig. 1. Properties of (surface) soliton with a 0-order Bessel potential. (a) Power of surface soliton with p_l = 15 and different p_r versus b. (b) Profiles of ground-state soliton with p_l = p_r = 15. (c) Surface soliton profiles with p_l = 25 and p_r = 10. (d) Typical b_cutoff curve versus p_l for different p_r.

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Fig. 2. Evolution of the solitons with (a) p_l = p_r = 15,b = 0.2. (b)p_l = -23, p_r = 16,b = 1.1. (c) p_l = 8, p_r = 47,b = 2.6. The white noise with variance σ² _noise = 0.01 was added into the initial input.

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which can be solved numerically.

3. Discussion

First we consider the properties of surface solitons with 0–order Bessel modulation potential [Fig. 1]. The power of such solitons decreases monotonically with the increment of propagation constant [Fig. 1(a)] and vanishes at the upper cutoff b_co which determines the existence region of solitons. The soliton with a symmetrical profile corresponds to the ground-state soliton if p_l = p_r [Fig. 1(b)]. The solitons reside mainly in the center of 0–order Bessel potential if the value of b is large. They will extend and cover several nodes of Bessel function with the decreasing of propagation constant. On the other hand, the soliton profiles are asymmetric when p_l ≠ p_r [Fig. 1(c)]. It is the interface that supports such modes in the present scheme, that is, either component of surface soliton beside the interface cannot exist if the other component varnishes or is deformed seriously. We show in Fig. 1(d) those stationary solutions only exist on the upper side of b_cutoff curves. The existence region of surface solitons broadens with growth of the modulation parameter p_l(p_r). The b_cutoff curves with different p_r converges to a collectively asymptotic line when the value of p_l increases. Another very interesting result can be inferred from Fig. 1(d) is that there are families of stationary solutions which are proved to be stable when p_l × p_r < 0 [e.g. Fig. 2(b)].

 

Fig. 3. Properties of surfaces soliton with a 1–order Bessel potential. (a) Power of surface soliton with p_l = 15 and different p_l versus b. (b) Typical b_cutoff curves with p_l for different p_r.

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We applied the linear stability analysis [Eq. (4)] on such solutions and found that surface solitons are stable in the entire domain of their existence, i.e., all imaginary parts of eigenvalues (λ) equal zero. To confirm the instability analysis results, we perform extensive numerical simulations using Eq. (1) with the perturbed initial input A(x,z = 0) = q(x)[1 +ρ(x)], where ρ(x) is the white noise with variance σ² _noice. Some examples of evolution of the perturbed solitons are shown in Fig. 2.

For 1–order Bessel potential, the power of surface solitons also decreases monotonically with the growth of propagation constant [Fig. 3(a)]. The existence domain of soliton is smaller than that of 0–order Bessel potential with same parameters p_l,p_r [Fig. 3(b)]. Another important result is that the b_cutoff is determined by the value of max (p_l,p_r). The b_cutoff remains the same when p_l < p_r but increases almost linearly with b when p_r > p_l. Some typical soliton profiles are laid out in Fig. 4. One can see clearly the influences of the different potential beside the interface on soliton profiles. Similar to n = 0 cases, the profile of soliton is symmetrical with p_l = p_r but asymmetrical with different p_l and p_r. There is an intensity dip on the top of ground-state solitons since the Bessel potential now has a local minimum at x = 0 [Fig. 4(a). The dip is almost invisible if the ratio of p_l and p_r is relatively large. The left component of soliton profile steepens if p_l decreases which even holds for p_l < 0 [Fig. 4(b) and 4(c)]. The energy distribution of soliton beside the interface is proportional to the ratio between p_l and p_r. This can be explained in physics as that the deeper the refractive-index is modulated, the more energy is trapped. We performed again the instability analysis on the stationary solutions with 1–order Bessel potential and found similar results with the case of 0–order Bessel potential, that is, the solitons are stable in their existence domain. Numerical simulations of some examples are shown in Fig. 5. Note that the soliton is stable with different sign of p_l and p_r [Fig. 5(b) and 5(c)].

 

Fig. 4. Profiles of surface soliton with 1-order Bessel potential. (a) p_l = p_r = 15,b = 1.6, (b) p_l = -5, p_r = 25,b = 0.8, (c) p_l = -5, p_r = 2,b = 0.05, (d) p_l = 2, p_r = 35,b = 0.5

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Fig. 5. Evolution of the solitons shown in Fig. 4. The white noise with variance σ² _noise = 0.01 was added into the initial input.

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To explore the effect of Bessel potential on the stabilization of solitons further, we considered the higher-order Bessel potential (n ≥ 2) and found they also support the stable surface solitons. The properties of these solitons are similar to that of n = 0,1. The only difference is that their existence domain decreases when the n value increases for the same parameters p_l and p_r. Some examples of stable soltions and their propagation simulations are shown in Fig. 6. Note that for n = 5, surface soliton can still propagate stably. We perform instability analysis and propagation simulations for even higher-order Bessel potential(e.g. n = 15) and prove that the stationary solution is stable in its existence domain.

 

Fig. 6. Profiles and evolution of the solitons with higher-order Bessel potential. (a,d) p_l = p_r = 20,b = 0.2, (b,e) p_l = 10, p_r = 20,b = 0.5 with n = 2. (c,f) p_l = 100, p_r = 250,b = 1.8 with n = 5. The white noise with σ_{2 noise} = 0.01 was added into the initial input in (d,e,f).

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4. Conclusions

In conclusion, we present a new but simple model, an interface separated by different 1–D Bessel potential with a defocusing cubic nonlinearity. Numerical results show that this model supports stable surface solitons. Surface solitons with different-order Bessel potential are shown to be stable in the entire domain of their existence. Our theoretic results enrich the concept of surface soliton and may be generalized to the case of Bose-Einstein condensates trapped in 1D Bessel potential with repulsive interatomic interactions. The model could be realized in experiment based on experimental scheme presented in reference[15].