1. Introduction

Spatial optical solitons are self-trapped optical beams of finite spatial cross section that travel without the divergence associated with freely diffracting beams [1, 2]. Due to their novel physics as well as potential applications, spatial solitons are under intensive studies and rich dynamics have been discovered in the last decade. Solitons with a variety of shapes such as fundamental, multipole, vortex and other higher order solitons have been found in various nonlinear media [1, 2, 3]. The introducing of optical lattices into nonlinear medium significantly promotes the researches on solitons in optics [4, 5] and Bose-Einstein condensates [6, 7]. This is because the optical lattice provides an effective way to guide and control the dynamics of solitons. Many properties of solitons that cannot be observed in bulk uniform nonlinear media have been discovered in lattice modulated media [8, 9, 10, 11].

An important concentric optical lattice, Bessel lattice, with the radial intensity distribution obeying the pattern of Bessel function, has been paid intensive attentions these days. Such lattices can be induced by nondiffracting Bessel beams which are created in experiments by a conical prism (axicon) [12] or computer-generated hologram [13]. Various types of solitons have been predicted theoretically and observed experimentally in (modulated) Bessel lattices [9, 14, 15]. The unique cylindrical symmetry of such lattice allows the existence of stable ring-profile, vortex [16], multi-dipole [17] and spatiotemporal solitons [18] etc., provided that the lattice is modulated deep enough. Moreover, rotary (dipole) solitons were predicted to reside on different rings of Bessel lattices in a focusing cubic nonlinear medium [19, 20] while lights guided by Bessel-like lattices with a low refractive-index core were observed [21].

Necklace soliton, a special class of self-trapped beams, looks like a necklace with its intensity and phase modulated periodically along the azimuthal direction. Soljačić, et al. predicted theoretically the quasi-stable necklace beams in a Kerr nonlinear medium [22] and such beams were observed very recently [23]. Necklace vector solitons also exhibit quasi-stable evolution in a saturable nonlinear medium though their expanding rate was slowered [24]. Dynamics of necklace beam carrying (integer and fractional) angular momentum are presented [25]. It was shown that the presence of topological charges can slower the expansion of such beams. Metastable necklace solitons were observed in lead glass with thermal nonlocal nonlinearity [26]. Robust spatiotemporal necklace-ring solitons in the cubic-quintic Ginzburg-Landu system are reported in [27]. Dark hollow beams have also attracted attentions due to their potential applications in turbulent atmosphere [28], atomic optics, optical manipulation [29] etc. They can be generated by diverse ways in linear systems [29].

However, almost all necklace solitons reported so far inevitably experience expansions during their propagations due to a net outward force exerting on each “pearl” by all the others forming the necklace [22, 24, 25], with the only exception reported in [30] where stable necklace-like soliton was demonstrated in 2D photorefractive crystal imprinted with a square lattice. Yet, the necklace in the latter work is of a fixed number of eight “pearls” and these “pearls” do not distribute evenly on a circle but rather locate on a octagon which is composed of 4×4 waveguides. Moreover, all models supporting the necklace solitons focus on the self-focusing media. On the other hand, the researches on hollow beams with relatively large dark core were restricted to the linear systems.

Noticing that stable ring-profile vortex solitons exist in the defocusing cubic medium modulated by a Bessel optical lattice [16] and necklace solitons can be regarded as the combination of two vortex solitons with opposite charges, we naturally conjecture whether the necklace beams and hollow dark beams could be captured stably by the optical potential induced by the centrosymmetric Bessel lattices. In this paper, we investigate the stability and propagation dynamics of these complex entities. The formation of necklace in our settings is due to the interplay between self-defocusing, lattice trapping and light diffraction. To be specific, along the radial direction, ring-lattice induced higher-index trapping of “pearls” counterbalances the net outward force and the diffraction; along the azimuthal direction, self-defocusing nonlinearity and diffraction are balanced by the repulsive forces between neighboring “pearls” which avoids the overlapping of neighbor “pearls”. That is, in our setting, due to the presence of ring lattice, the necklace soliton maintains a stationary structure which guarantees its stability of whole structure during propagation, in contrast to the necklace solitons in the lattice-free system where the shrinkage or expanding of the whole structure upon propagation inevitably exists. We mention that we lay our attention in self-defocusing nonlinearity in the view of the possible collapse of “pearls” in self-focusing systems.

2. Model

We assume that an optical beam propagates along the z-axis in a defocusing cubic nonlinear medium with an imprinted Bessel lattice. The generic equation describing the evolution of optical wave packets in this setting is the nonlinear Schrödinger equation:

where transverse x,y and longitudinal z coordinates are scaled to the width and the diffraction length of the input beam respectively. A(x, y, z) is the normalized complex field amplitude of the beam. The function $R(x,y)=J_n^2\left[\sqrt{2\beta }r\right]$ represents the transverse refractive-index modulation profile, p describes the depth of the refractive-index modulation, β is the radial scale of the n-order Bessel function. As pointed out in [17], the nonlinearity in Eq. (1) can be practically realized in semiconductor photorefractive crystals, such as GaAs:Cr and CdTe:In, which exhibit strong photorefractivity (e.g. n ³ r ₄₁=152pm/V in CdTe:In). When the crystals are biased by a strong static electric field E ₀~10⁵ V/m, the propagation distance z=1 corresponds to about 1mm and the dimensionless amplitude A~1 corresponds to peak intensities ~50mW/cm ² for laser beams with the width ~10µm. Eq. (1) conserves several quantities, including the power, P=∫^∞ _-∞∫^∞ _-∞|A(x,y)|² dxdy. We search for stationary solutions in the form of A(x,y,z)= q(x,y)exp(ibz), where b is the real propagation constant, q(x,y) is the real function which must vanish at (x ²+y ²)^1/2→∞ and obey the equation:

The soliton profiles were found numerically by a two-dimensional relaxation algorithm. In numerical calculations, we vary the depth parameter p, the propagation constant b and fix transverse lattice scale β≡0.125. As done in [22], the initial iterative guess solution of q(x,y) was selected in the form of $\alpha \mathrm{sech}\left[\right(\frac{\sqrt{\left(x^2+y^2\right)}-L}{W}\left)\right]\cos \left(\Omega \theta \right)$ , where α is the amplitude, L stands for the radius of one of the Bessel lattice rings, W is the diameter of a “pearl” of the necklace soliton or the thickness of the ring soliton, Ω characterizes the periodical modulation along the azimuthal direction which relates to the number of “pearls” in necklace solitons. The initial iterative solution degenerates to the ring soliton guess if Ω=0. The stability of the soliton can be analyzed by considering the perturbed stationary solution form as A(x,y,z)=q(x,y)exp(ibz)+u(x,y)exp[i(b+λ)z]+ν*(x,y)exp[i(b-λ)z], where the perturbation components u,ν could grow with a complex rate λ during propagation. The soliton is stable if the imaginary parts of perturbation eigenvalues equal zero. Substituting the perturbed solution into Eq. (1), we obtain the linear eigen-equations:

 

Fig. 1. (a) Power and instability growth of 8-“pearl” necklace solitons residing on the first ring of the 1-order Bessel lattice vs b at p=30. (b, c) Profiles of solitons with p=30,b=0.6 in (b) and p=30,b=7.5 in (c). (d) Superposition contour of lattice and necklace soliton shown in (c). (e,f) Propagation simulation of soliton shown in (c) at z=0 and z=256 respectively. White noise with variance |σ_noise|=0.1 was added into the initial input.

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which can be solved numerically.

3. Discussions

First, we address properties of the necklace solitons supported by the 1-order Bessel lattice. The power of 8-“pearl” necklace soliton decreases with the growth of b. It vanishes when b approaches a certain value (b_cuttoff) [Fig. 1(a)]. The deeper the medium is modulated, the higher the power will be. An instance of instability growth with p=30 is also shown in the same figure. Such solitons will be stable if the propagation constant exceeds a certain value for fixed lattice depth. The “pearls” in necklace soliton shrink with the increment of b [Fig. 1(b) and 1(c)]. The distribution of necklace with respect to the position of lattice where it resides is shown in Fig. 1(d). Note that the ratio between soliton field and lattice has been adjusted for the visual effect. The soliton resides exactly on the first ring of the lattice. Figure 1(e) and 1(f) display one example of stable propagations of the perturbed necklace solitons. The stability and existence domains are similar to that of 44-“pearl” solitons that will be discussed later.

Some typical examples of necklace soliton profiles with different number of “pearls” are laid out in Fig. 2. These solutions are proved to be stable by numerical propagation simulation. We make three comments here. First, the necklace solitons can reside on any ring of the lattice theoretically provided that the lattice is modulated deep enough. The radii of solitons depend on the radii of lattice rings. Second, the maximum number of “pearls” is restricted by the radius of the ring where they reside. Assuming the “pearls” are round, the maximum number can be estimated roughly by n_max≈2πr/w, where r is the radius of lattice ring, w is the thickness of the ring. Third, Bessel lattices support the multistable states of necklace solitons [see e.g. Fig. 2(b) and 2(c)]. We even find the stable necklace solitons composed of different size “pearls”, e.g., an 8-“pearl” soliton with four “pearls” larger than the other “pearls” at the same parameters as in Fig. 1. Although the 14-“pearl” solutions with the same parameters as that of Fig. 2(b) will converge to uneven “pearl” distribution with b>1.7, they propagate stably if b∈ [1.46,1.9].

Now, we study the existence and stability domains of necklace solitons by selecting the 44-“pearl” soliton [Fig. 2(c)] as an example. Soliton solutions cannot be found above the b_upp curve which determines the existence domain of necklace solitons. A comprehensive linear stability analysis has revealed that the necklace solitons are completely stable if the propagation constant exceeds a certain value, provided that the lattice is deep enough (p≥25) [Fig. 3(a) and 3(b)]. The stability domain broadens with the growth of p. The instability region is also verified by directly simulating (by beam propagation method) the original Eq. (1) with the solutions derived from Eq. (2). The unstable solitons will decay into either radiation or ring solitons presented later. Fig. 3(c) and 3(d) display the stable evolution of necklace soliton shown in Fig. 2(c) under a strong random white noise perturbation.

 

Fig. 2. Profiles of necklace solitons. (a) 20-“pearls”, p=30,b=2.0 residing on the second ring of lattice. (b) 14-“pearls”, p=40,b=1.5 residing on the third ring of lattice. (c) 44-“pearls”, p=40,b=1.5 residing on the third ring of lattice. (d) 32-“pearls”, p=55,b=1.1 residing on the fourth ring of lattice.

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Fig. 3. (a) Existence and instability (shaded) domains of 44-“pearl” necklace solitons. (b) The imaginary part of the perturbation growth rate vs propagation constant at p=40. (c, d) Stable propagation of the necklace soliton shown in Fig. 2(c). z=0 in (c) and 256 in (d). White noise with variance |σ_noise|=0.1 was added into the initial input.

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To examine the effect of the global linked angular momentum on the dynamics of the necklace solitons, we perform propagation simulations of the necklace solitons by adding a phase twist (in the form of exp(imθ)) into the input beam. Figures 4(a) shows the phase structure of the 32-“pearl” soliton [Fig. 2(d)]. Note the phase of a single “pearl” now has a gradient distribution and the inner part of phase plot is induced by the numerical noise. The soliton rotates clockwise indeed without any tangible loss. The azimuthally twist phase and the strong white noise perturbation together do not destroy the stability of necklace soliton.

Next, we discuss the existence and stability properties of ring solitons in the defocusing cubic nonlinear medium modulated by 1-order Bessel lattice. [Figure 5(a)–5(c)] show some typical examples of ring solitons intensity distribution. The energy of such solitons resides mainly on one of the lattice rings for larger propagation constant. They become broader and cover other rings with the decrement of parameter b. This extension will destroy the stability sustained by Bessel lattice due to the radially natural diffraction together with attractive force between the beam components locating on different rings. It should be pointed out that ring solitons can exist in other rings of (different order) Bessel lattice provided the depth of lattice is deep enough [e.g. Fig. 5(c)]. The existence domain of ring solitons is determined by the lower and upper borders in terms of propagation constant b [Fig. 5(d)]. The power diverges when b approaches the lower cutoff (b_lower≡0) and vanish when b approaches the upper cutoff. We apply the linear stability analysis on such solutions and one example of the stability regions is displayed in Fig. 5(d). The existence and stability windows broaden with the increment of lattice depth. The curve separating the stable and unstable stationary solutions is derived by solving Eqs. (3). It also corresponds to the boundary between solitons locating on a single ring and multi-rings of lattice.

 

Fig. 4. (a) Phase structure of 32-“pearl” soliton [Fig. 2(d)] carrying global angular momentum in the form of exp(4iθ). (b–d) Stable clockwise rotation of the 32-“pearl” soliton. z=0 in (b), 128 in (c) and 256 in (d). White noise with variance |σ_noise|=0.1 was added into the initial input.

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Fig. 5. (a–c) Contour plots of ring solitons supported by 1-order Bessel lattice. The ring solitons reside on the second ring of lattice in (a),(b) and third ring in (c). (a) b=1.2, p=20. (b) b=0.79, p=20. (c) b=1.7, p=35. (d) Existence and instability (shaded) domains in the p,b plane for the ring solitons residing on the second ring of lattice.

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Finally, we would like to mention that similar complex entities, the so-called “soliton clusters” composed of several interacting components residing also on a ring with staircaselike phase distribution [31], can not propagate stably in this setting due to the phase difference between neighboring components is limited by 0≤θ≤π/2 which lead to the lack of repulsive forces along the azimuthal direction. Robust “soliton clusters” can form in media with competing optical nonlinearities [32, 33, 34]. We stress that although all aforementioned results are obtained in the case of 1-order lattices, we have verified that these findings can be generalized to the other order or other ring of lattices except for the central bright spot of 0-order Bessel lattice where the necklace solitons and ring solitons do not exist.

4. Conclusions

To summary, we have investigated the existence, stability and propagation dynamics of necklace solitons and ring solitons supported by Bessel lattice in a defocusing medium. We reveal that this scheme supports the completely stable necklace solitons and ring solitons and they can locate on any ring of the lattice under appropriate conditions. The maximum number of “pearls” in a necklace soliton is determined by the radius of lattice ring where it resides. Very interesting phenomena, multi-stable necklace solitons, are found to exist in Bessel lattices. The existence of such complex patterns can also be generalized to the case of Bose-Einstein condensates with repulsive interatomic interactions.

The work is supported by the National Natural Science Foundation of China (Grant No. 10704067) and the Department of Education of Zhejiang Province (Grant No. 20060493).