Bloch Oscillations and Zener Tunneling in Two-Dimensional Photonic Lattices

We report on the first experimental observation of photonic Bloch oscillations and Zener tunneling in two-dimensional periodic systems. We study the propagation of an optical beam in a square lattice superimposed on a refractive index ramp. We observe oscillations of the beam inside the first Brilloin zone and tunneling of light from the first to the higher-order bands of the lattice band gap spectrum.

Figure 1

(a) Scheme of an optically induced square lattice. (b) Experimentally measured profile of the background illumination $I_m$ (dashed-dotted line) and the corresponding change in the refractive index (solid line). The dashed line indicates the initial position of the probe beam. (c) Calculated band structure of the lattice for $I_m=0$. The inset shows symmetry points in the first Brillouin zone. (d) Band structure of the lattice vs $I_m$. The horizontal line is the adiabatic value of the propagation constant; the dashed segment indicates Zener tunneling through the gap.

Figure 2

(Top) Numerically calculated position of the beam’s center of mass; solid line—Bloch oscillations in the lattice; dashed line—beam deflection without the lattice. The vertical line indicates the position of the exit facet of the crystal. The light intensity distributions corresponding to the positions (a), (b), and (c) are depicted in the three snapshots.

Figure 3

Light intensity profiles (top row) and corresponding Fourier spectra (bottom) measured at the crystal output for different initial inclinations of the input beam. The inclination is (a) $-0.095°$, (b) $-0.035°$, (c) 0°, (d) 0.035°, and (e) 0.12°. The arrow in (a) indicates the direction of the index gradient. White diamonds indicate the first Brillouin zone.

Figure 4

Light intensity profiles calculated (a)–(c) and measured (d)–(f) at the crystal output for a single site excitation. (a),(d) Discrete diffraction with no transverse gradient, (b),(e) weak and (c),(f) strong index gradients.