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# Light Bullets in Nonlinear Periodically Curved Waveguide Arrays 

Michal Matuszewski $^{1}$, Ivan L. Garanovich$^{1}$, and Andrey A. Sukhorukov1 , 2 

$^{1}$Nonlinear Physics Centre and $^{2}$Centre for Ultra-high bandwidth Devices for Optical Systems (CUDOS), Research School of Physics and Engineering, Australian National University, Canberra, ACT 0200, Australia 

We predict that stable mobile spatio-temporal solitons can exist in arrays of periodically curved optical waveguides. We find two-dimensional light bullets in one-dimensional arrays with harmonic waveguide bending and three-dimensional bullets in square lattices with helical waveguide bending using variational formalism. Stability of the light bullet solutions is confirmed by the direct numerical simulations which show that the light bullets can freely move across the curved arrays. This mobility property is a distinguishing characteristic compared to previously considered discrete light bullets which were trapped to a specific lattice site. These results suggest new possibilities for flexible spatio-temporal manipulation of optical pulses in photonic lattices. 

Discrete solitons, the self-localized states which can exist in both one- and two-dimensional nonlinear pho- tonic lattices, have attracted a lot of attention over the recent years [1]. The interplay between the underly- ing periodicity and nonlinear self-action enables a whole lot of new opportunities for all-optical control, which are not possible in homogeneous media [1]. Of par- ticular interest are pulses which are localized both in space and in time as a result of balance between the self-focusing nonlinearity, diffraction, and anomalous dis- persion. Such spatio-temporal optical solitons are called light bullets [2]. Light bullets are inherently multi- dimensional, and consequently they are always unstable in Kerr media [2]. Discreteness in photonic lattices can provide an additional stabilization mechanism, and in- deed stable continuous-discrete solitons were found in nonlinear waveguide arrays [3, 4]. However, these dis- crete light bullets are strongly trapped in a few waveguide channels, and hence their mobility is practically reduced to the longitudinal dimension, whereas they cannot move across the photonic lattice. 

Recently, it was suggested that arrays of periodically curved optical waveguides offer unique opportunities for the control of both the strength and frequency disper- sion of the waveguide coupling [5–8]. Propagation of light beams in various types of the nonlinear lattices with mod- ified diffraction has been analyzed [9–14]. It was demon- strated that light beams can become localized in the peri- odically curved nonlinear waveguide arrays in the form of diffraction-managed solitons [15–17]. These diffraction- managed spatial solitons are reminiscent of dispersion- managed temporal solitons [18–20]. However, nonlinear propagation and localization of light pulses in diffraction managed photonic lattices have not been analyzed yet. 

In this work, we predict, for the first time to the best of our knowledge, that stable broad spatio-temporal solitons can exist in arrays of periodically curved optical waveg- uides, and such discrete light bullets are mobile and can move across the array. We find approximate light bullet 

PACS numbers: 42.65.Tg, 42.82.Et 

## I. INTRODUCTION 

solutions using the variational technique, where we as- sume a Gaussian shape of the spatial-temporal soliton as shown in Fig. 1(b). The results of the variational anal- ysis indicate that by optimizing the waveguide bending amplitude and period, one can achieve stable and mobile two- and three-dimensional light bullets. We emphasize that we find stable light bullets which spatial width is much larger than the lattice period. Hence, the light bullets would not be trapped in a single waveguide, re- taining mobility in the transverse dimension. This con- clusion is confirmed by direct numerical simulations. We also demonstrate that a similar approach can be used to achieve stabilization of multidimensional matter-wave solitons in Bose-Einstein condensates. 

## II. TWO-DIMENSIONAL LIGHT BULLETS 

## A. Discrete-continuous model equation 

First, we consider propagation of light pulses in a one- dimensional array of coupled optical waveguides, where waveguide axes are periodically curved in the longitudi- nal propagation direction z , see the sketch in Fig. 1(a). 

We model the light bullet dynamics with the modified discrete-continuous nonlinear Schrodinger propagation equation which takes into the periodic waveguide bend- ing [12, 21] and temporal dispersion [22–25]: 

Here n is the waveguide number, a$_{n}$ are the complex amplitudes of individual waveguide modes, z is the prop- agation distance, t is time, C is the coupling coefficient between the modes of straight neighboring waveguides, D is the second-order dispersion coefficient, γ is the non- linear coefficient, and p ( z ) is the phase coefficient related to the waveguide bending [21]. We consider all variables 

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