Nonadiabatic Landau-Zener Tunneling in Waveguide Arrays with a Step in the Refractive Index

Landau-Zener tunneling is discussed in connection with optical waveguide arrays. Light injected in a specific band of the Bloch spectrum in the propagation constant can be transmitted to another band, changing its physical properties. This can be achieved using two coupled waveguide arrays with different refractive indices. The step in the refractive index causes wave “acceleration” and thus induces strongly nonadiabatic Landau-Zener tunneling. Theoretically, the analysis is performed by considering a Schrödinger equation in a periodic potential with a step. The region of physical parameters where this phenomenon can occur is analytically determined and a realistic experimental setup is suggested. Its application could allow the realization of light filters.

Figure 1

Schematic picture of the combination of two waveguide arrays of the same spatial period but with different refractive indices (different gray levels in the main plot). The refractive index profile across the array with a step of size $A$ is shown in the upper inset.

Figure 2

Schematic band-gap structure and picture of the Landau-Zener tunneling process. $w$ is the gap between the bands (the height of the period potential), $d$ is the width of the upper band, and $\kappa$ is the initial detuning of the lower band mode wave number from zone boundary. $\beta$ is the dimensionless propagation constant [see formula (5)] and $\Delta \beta ={\beta }_{-}(0)-{\beta }_{-}(\kappa )$. $A$ is the height of the step. Initially, light is injected in the left array, populating the lower band mode (a). Going across the step, $\kappa$ decreases, reaching zero as the mode approaches the zone boundary causing Landau-Zener tunneling (b). After tunneling, most of the light intensity is transferred to the upper band mode, whose wave number decreases until the end of the step is reached. This is the light we observe in the right array (c). A smaller light intensity remains in the lower band and is observed in the left array (d).

Figure 3

Landau-Zener tunneling from the lower to the upper band. The step size $A=0.8$ is taken within the limits of Eq. (9). A step with a slope $\alpha =2.4$ is placed at $x=5$ and the lower band mode is injected into the first five periods. The inset shows the dependence of the transmission coefficient on the step slope $\alpha$. The solid line corresponds to the analytical formula (10) and the crosses are numerical simulations.

Figure 4

Upper graph: total reflection from the step. The step size, $A=1$, is greater than the upper bound of Eq. (9). Lower graph: penetration through the step without tunneling. The step size, $A=0.07$, fulfills the inequality (11). In both cases the step slope is infinite.