Slow-Light Optical Bullets in Arrays of Nonlinear Bragg-Grating Waveguides

We demonstrate that propagation direction and velocity of optical pulses can be controlled independently in the structures with multiscale modulation of the refractive index in transverse and longitudinal directions. We reveal that, in arrays of waveguides with phase-shifted Bragg gratings, the refraction angle does not depend on the speed of light, allowing for efficient spatial steering of slow light. In this system, both spatial diffraction and temporal dispersion can be designed independently, and we identify the possibility for self-collimation of slow light when spatial diffraction is suppressed for certain propagation directions. We also show that broadening of pulses in space and time can be eliminated in nonlinear media, supporting the formation of slow-light optical bullets that remain localized irrespective of propagation direction.

Figure 1

(a),(b) Schematic of a waveguide array with (a) in-phase ($\phi =0$) and (b) phase-shifted ($\phi =\pi$) Bragg gratings; (c),(d) corresponding isofrequency contours for different detuning from the gap edge: $\omega =4$ (solid line), $\omega =2$ (dashed line), and $\omega =1.1$ (dashed-dotted line). (e)–(h) Beam refraction with incident angle corresponding to $K_b=\pi /2$ for (e),(f) $\omega =2$ and (g),(h) $\omega =1.1$. For all the plots, $\rho =1$ and $C=1$.

Figure 2

Family of the stationary (zero-velocity) light bullets characterized by (a) energy and (b) width along the transverse (solid curve) and longitudinal (dashed curve) directions vs the frequency tuning inside the spectral gap. (c)–(d) Intensity profiles of slow-light optical bullets for (c) $\omega =0.8$ and (d) $\omega =0.995$; brighter shading marks higher intensity.

Figure 3

Moving slow-light optical bullets with the frequency detuning $\omega =0.5$. Notations are the same as in Fig. 2, and the velocity is normalized to the speed of light away from the Bragg resonance. Profiles are shown for the velocities (c) $V=0.1$ and (d) $V=0.2$.

Figure 4

Snapshots of field intensities for an optical pulse propagating in a waveguide array structure show in Fig. 1b: (a)–(d) Linear broadening due to spatial diffraction and temporal dispersion; (e)–(h) nonlinear self-trapping in space and time and the formation of an optical bullet. The input pulse has a Gaussian profile with FWHM $\Delta t\simeq 13$.