Figure 1

(Color online) Band diagram of the complex potential defined by Eq. (2) for parameter values $\lambda =633\text{nm}$, $a=8\mu \text{m}$, $n_s=1.42$, $V_0=0.002$, and $\alpha =0.3$. The dotted curve shows, for comparison, the parabolic dispersion relation $E\left(\kappa \right)={\left(ƛ\kappa \right)}^2/\left(2n_s\right)$, folded inside the first Brillouin zone, corresponding to the critical case $\alpha ={\alpha }_c=1$. The picture on the right-hand side is an enlargement of the lowest band (solid curve), fitted by a sinusoidal curve (dotted curve), and almost overlapped with the solid one.Reuse & Permissions

Figure 2

(Color online) Wave packet broadening (discrete diffraction) and dynamic localization in the lattice of Fig. 1 for broad beam excitation at normal incidence (left panels) and for single site excitation at normal incidence (right panels). In (a) and (b), the external force is absent $\left(F_0=0\right)$. In (c) and (d) a sinusoidal force with period $\Lambda =1\text{cm}$ and amplitude $F_0=13.32\mu {\text{m}}^{-1}$, corresponding to $\Gamma =1.684$, is applied. In (e) and (f), the forcing amplitude is increased to $F_0=19.03\mu {\text{m}}^{-1}$, corresponding to the DL condition $\Gamma =2.405$.Reuse & Permissions

Figure 3

(Color online) Propagation of a broad Gaussian wave packet (input beam size $w=80\mu \text{m}$) in the complex lattice $V\left(x\right)=V_0\text{exp}\left(2\pi ix/a\right)$ for $V_0=0.0002$, $a=6\mu \text{m}$, $\lambda =633\text{nm}$, and $n_s=1.42$, subjected to a sinusoidal ac-like force $F\left(z\right)=F_0\text{cos}\left(2\pi z/\Lambda \right)$ with period $\Lambda =1\text{cm}$ and with increasing values of force amplitude $F_0$: (a) $F_0/F_c=0.5$, (b) $F_0/F_c=1$, (c) $F_0/F_c=1.5$, and (d) $F_0/F_c=3$, where the critical forcing $F_c$ is defined by Eq. (15). Left panels show snapshots of $\left|\psi \left(x,z\right)\right|$; the central panels depict the behavior of $k\left(z\right)$, normalized to the Bragg wave number $k_B$, as given by Eq. (8); the right panels show the evolution of normalized beam power $P\left(z\right)$. The horizontal dotted lines indicate the crossing points $z=z_0$ where new wave packets are generated by Bragg scattering. For the sake of clarity, in (d) only the first two crossing points are indicated. At the crossing points a rather abrupt increase of beam power $P\left(z\right)$ is observed. The crossing is linear in (c) and (d) and parabolic in (b).Reuse & Permissions