1. Introduction

Optically induced photonic lattices attract a lot of interest due to intriguing waveguiding possibilities [1, 2]. Periodic or nonperiodic two-dimensional (2D) arrays of optically induced waveguides significantly modify the diffraction properties and strongly affect propagation of light [1, 3]. Until recently, only simple stationary structures have been described theoretically and generated experimentally in optically induced lattices [4–7]. The most important further step was the study of nonlinear modes with a nontrivial phase such as vortices [8–13].

Optical vortex solitons exist in nonlinear media with an infinite range of nonlocality [14]. Theoretical studies [15,16] suggest that the vortex-like structures can be supported by the lattice even in the self-focusing regime. In such a medium, strong stabilization and trapping of an optical vortex by the lattice was demonstrated from a broad range of initial conditions [9]. Optical lattice suppresses the rich instability-induced dynamics (rotating, diverging, and self-bending) of the filaments in a homogeneous nonlinear medium. The formation and interactions of spatial vortices have been studied mostly in the copropagation geometry, with a few exceptions in which the counterpropagating (CP) vortices, with or without lattice, were considered numerically in both space and time [17–19]; the formation and interactions of CP solitons can be found in a number of papers [20–24].

In this paper we investigate numerically properties of rotating CP mutually incoherent self-trapped vortex beams in different optically induced fixed photonic lattices, and determine the conditions to observe them. For hexagonal photonic lattice, regular rotation of vortex filaments is noticed by periodically increasing propagation distance. Owing to the presence of defect, we display some novel dynamical beam structures in photorefractive (PR) crystals that have no counterparts in the case without the lattice, particularly for the circular photonic lattices with negative defect. We found that vortex filaments rotate as a result of tunneling between lattice sites. When noise is added, the observed rotating structures remain stable.

2. The model

The behavior of CP vortices in photonic lattices is described by a time-dependent model for the formation of self-trapped CP optical beams [25], based on the theory of PR effect. The model consists of wave equations in the paraxial approximation for the propagation of CP beams and a relaxation equation for the generation of the space charge field in the PR crystal, in the isotropic approximation. The model equations in the computational space are of the form:

where F and B are the forward and the backward propagating beam envelopes, Δ is the transverse Laplacian, Γ is the dimensionless coupling constant, and E the homogenous part of the space charge field. The relaxation time of the crystal τ also depends on the total intensity, τ=τ₀/(1+I+ I_g). The quantity I=|F|²+|B|² is the laser light intensity, measured in units of the background intensity. When the propagation in photonic lattices is considered, Eq. (2) includes the transverse intensity distribution of the optically induced lattice array I_g formed by positioning Gaussian beams at the sites of the lattice, normalized by the dark irradiance I_d. A scaling x/x ₀→x, y/x ₀→y, z/L_D→z, is utilized in the writing of dimensionless propagation equations, where x ₀ is the typical FWHM beam waist and L_D is the diffraction length. The assumption, appropriate to the experimental conditions at hand, is that the mutually incoherent CP components interact only through the intensity-dependent space charge field.

It is well known that anisotropy could affect the predicted dynamic structures [7]. Effects of anisotropy in optically induced photonic lattices with a chessboard phase structure can be considerably suppressed if the orientation of the symmetry axes of the photonic lattice relative to the crystallographic c-axis is properly chosen [26]. This means that the isotropic model can be used as a useful approximation to the PR media.

The propagation equations are solved numerically, concurrently with the temporal equations, in the manner described in Ref. [17] and references cited therein. The dynamics is such that the space charge field builds up towards the steady state, which depends on the light distribution, which in turn is slaved to the change in the space charge field. As it will be seen, this simple type of dynamics does not preclude a more complicated dynamical behavior.

3. Counterpropagating rotating structures in hexagonal photonic lattice

First, we consider lattice array with a hexagonal/trigonal arrangement of beams and with the central beam absent. Such an arrangement is reminiscent of the holey fibers [27], except that there are no holes here, but laser beams that modulate the index of refraction [28]. The beams are assumed to be degenerate and incoherent with the forward and backward components.

We launch head-on CP vortices with opposite topological charges (±1) in the center of the lattice, parallel to the lattice beams. An optical waveguide array embedded in a PR crystal considerably changes the behavior of CP beams, as compared to the wave behavior in bulk media. CP vortices in the absence of lattice tend to form rotating dipoles, tripoles and quadrupoles, and quasi-stable and unstable structures [18], for wide region in Γ-L parameter plane and for very broad range of input FWHM of vortices. CP vortices in the presence of lattice tend to form only rotating tripole and unstable structures, but for very narrow region in Γ-L parameter plane and range of input FWHM of vortices. In the case of CP vortices one also observes strong pinning to the lattice and improved stability of the central basic vortex.

 

Fig. 1. Local and nonlocal rotation in triagonal lattice for various propagation distances: for propagation distances between these values chaotic behavior is observed (not shown). Movies of the intensity distribution of the forward field at its output face: (a) (1341 KB), (b) (1942 KB), (c) (1821 KB). Parameters: Γ=17, lattice spacing d=28 µm, FWHM of lattice beams 12.7 µm, input FWHM of vortices 26.2 µm, maximum lattice intensity I_g=5I_d, |F₀|²=|B_L|²=5I_d. All movies start after stable rotation is established (transient dynamics are not shown).

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The only clear rotation of vortex filaments in the presence of hexagonal photonic lattice with central defect is presented in Fig. 1 (transformation from vortex to rotating tripole is not shown in movies). Although it looks that vortex filaments rotate only in the center of the lattice (owing to the defect - as expected), they also rotate away from the center, but by tunneling between lattice sites. This phenomenon is demonstrated by the fact that, for all three cases presented in Fig. 1, the angular momenta of the vortex calculated on the whole lattice are considerably greater than the angular momenta of vortex calculated only for the central part of the lattice (for radii less than one lattice spacing). Vortex filaments rotate between lattice sites and along symmetry axes of the lattice (nonlocal rotation), with the same period as the tripole positioned at the place of central defect (local rotation). Physical origin of nonlocal rotation is Zener tunneling effect [29, 30].

Nonlocal rotation in periodic array, such as hexagonal photonic lattice, can exists only for some values of propagation distance L. It is well known that by increasing propagation distance, CP vortices produce chaotic behavior in the absence of photonic lattice [18], but in the presence of lattice pass through regular and irregular dynamical structures (and even nonpropagating modes occur). Lattice supports rotation only for some values of propagation distance, with “period” equal 1L_D (Fig. 1). For propagation distances between these values chaotic behavior is observed, as well as nonpropagating modes.

4. Counterpropagating rotating structures in circular photonic lattices

To investigate another CP rotating structures, we choose non-periodic but rotational symmetric lattice array arrangement of beams: circular photonic lattices (CPL) [31]. The rotational properties of the lattice can be improved by introducing negative defect in the manner described in Ref. [32], with defect function of the form f_D=-exp(-π(x ²+y ²)⁴/d ⁸). This is called a negative (repulsive) defect where light tends to escape to nearby lattice sites. The defect is restricted to a single lattice site in the lattice center, and the lattice intensity at the defect is zero. In the desire to find more interesting rotating structures, we choose only odd-fold symmetric CPL (Fig. 2).

 

Fig. 2. First circle for different odd-fold symmetric circular optically induced photonic lattices with negative defect.

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For CPL, the distance between lattice sites is constant in each concentric circle, but slightly differs for various circles. Ratio between distance of neighboring lattice sites in the first circle and FWHM of lattice beams is chosen to be 2.05 (Fig. 2). This value is very close to the value for standard periodic lattices, with ratio equal 2.

 

Fig. 3. Typical behavior in the parameter plane of CP vortices in sevenfold symmetric circular optically induced photonic lattice with negative defect. Input vortices have the opposite topological charge ±1. Insets list the possible outcomes from vortex collisions. For larger values of Γ and L there is a nonpropagating region. Parameters: lattice spacing 30 µm, FWHM of lattice beams 12.7 µm, input FWHM of vortices 16.4 µm, maximum lattice intensity I_g=5I_d, maximum input vortices intensity |F₀|²=|B_L|²=5I_d.

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Some typical examples of head-on counterpropagation of two centered vortices in sevenfold symmetric CPL, are shown in Fig. 3, which represents the phase diagram in the plane of control parameters Γ and L. We choose vortices with the opposite topological charges because they tend to form rotating structures. For lower values of Γ or L we see stable structures in the form of the well-preserved core vortex at the central defect with some filaments of vortices mostly focused onto the neighboring lattice sites. Above this region stable rotating tripoles and quadrupoles exist. For higher values of the parameters, we identify the following situations: dipole-like rotating structures, irregular rotating structures, rotating-breathing tripole and unstable structures (i.e. constantly changing structures of unrecognizable shape). Above this region a nonpropagating domain is observed. Analogous parameter planes can be found for any of odd-fold symmetric CPL, with similar structures (mentioned above).

 

Fig. 4. Characteristic examples shown in Fig. 3. Movies of the intensity distribution of the forward beam at its output face: (a) the rotating quadrupole (1173 KB), (b) the rotating-breathing tripole (1640 KB), (c) the dipole-like rotating structure (1147 KB), (d) the irregular rotating structure (1808 KB) and (e) unstable structure (2423 KB). Parameters Γ and L are given in the figures.

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The most illustrative cases from Fig. 3 are presented in Fig. 4, as movies in the transverse plane. Most of these structures can be observed in the case without lattice (of course, with no filaments of vortices focus onto the lattice sites). The only exceptions are rotating-breathing tripole (Fig. 4(b)) and irregular rotating structures (Fig. 4(d)). Both rotating structures can exist only in the presence of lattice and owing to the negative defect, and have no analogy with CP vortices behavior in bulk media. Region where these behaviors are observed is close to the chaotic region, and thus restricts this region. The generated discrete-vortex regular rotating structures are composed of several phase-correlated lobes with a total phase ramp of 2π, while relative phase between the neighboring lobes changes in equal steps. The observed phase structures closely resemble that of a soliton cluster [33].

All stable rotating structures (tripole and quadrupole) exist in all odd-fold symmetric optically induced CPL with negative defect. Rotating tripoles are presented in Fig. 5, with noise of 3% added to the input beam intensity and phase. Most of regular rotating structures are stable for noise of 10% or greater. The transient dynamics lasts relatively short because of the presence of noise. Again, one can notice that vortex filaments also rotate by tunneling between lattice sites, away from the center. The angular momentum of CP vortices is not conserved. The system experiences a considerable loss of angular momentum, owing to the presence of the fixed (massive) lattice.

 

Fig. 5. Rotating tripole, forward field, phase-intensity noise is 3%. Movies of the intensity distribution for: (a) threefold symmetric CPL, Γ=13.8, L=2L_D=8 mm, input FWHM of vortices 16.4 µm (1226 KB), (b) fivefold symmetric CPL, Γ=11, L=2.5L_D, input FWHM of vortices 19.6 µm (2290 KB) (c) sevenfold symmetric CPL, Γ=12.5, L=2L_D, input FWHM of vortices 16.4 µm (2184 KB), (d) ninefold symmetric CPL, Γ=11, L=2.5L_D, input FWHM of vortices 18 µm (2286 KB). Ratio between distance of two neighboring Gaussians in the first circle of lattices in different circular geometries and FWHM of that Gaussians is 2.05. Other parameters: I_g=5I_d, |F0|²=|BL|2=5I_d.

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Higher-order CP vortices (topological charges ±2 and ±3) in the presence of lattice tend to form the well-preserved core vortex at the central defect with some filaments mostly focused onto the lattice sites, without any dynamics or rotation.

5. Conclusions

In summary, we report on the various aspects of rotational properties of counterpropagating self-trapped vortices in isotropic local saturable photorefractive media, in the presence of an optically-induced photonic lattice. The rotation effects in these systems are determined by transport through tunneling between lattice sites. We display regular nonlocal rotation of vortex filaments in hexagonal/trigonal lattice with a central waveguiding defect, by periodically increasing propagation distance for one L_D. Including negative defect in CPL, we discover novel types of rotating beam structures that have no analogy in the case without the lattice. Generation of rotating structures requires a balance between the discrete diffraction and self-focusing experienced by the vortices.