Dynamic Diffraction and Interband Transitions in Two-Dimensional Photonic Lattices

We reveal a direct link between two fundamental wave phenomena in periodic media, Pendellösung oscillations and resonant coupling between spectral bands. We experimentally measure the power transfer between laser beams associated with the high-symmetry points in periodic and biased hexagonal photonic lattices. As a result, we demonstrate that Pendellösung oscillations dominate the dynamics of resonant interband transitions on a short propagation scale.

Figure 1

(a) Experimental intensity distribution of the stretched hexagonal lattice. (b) First BZ with the high-symmetry points. (c) Band-gap diagram calculated for a lattice-wave amplitude $I_0=1.1$ and an external field $E_{\mathrm{ext}}=1\mathrm{kV}/\mathrm{cm}$. The numbers next to the curves show band index. (d) Experimentally measured Rabi frequency $\Omega$ of oscillations between $Y$ and $Y$’ points (shown as dimensionless parameter $\Omega L$) versus external field $E_{\mathrm{ext}}$ (dots with error bars). Solid line: numerical simulations; dashed line: normalized gap $\Delta \beta L/2$ between first two bands at $Y$ point as indicated in (c). (e)–(g) Experimentally recorded far-field output intensity for $E_{\mathrm{ext}}=0.2$, 0.6, and $1\mathrm{kV}/\mathrm{cm}$, respectively.

Figure 2

Two-level Pendellösung oscillations. (a) The two-level system of Fig. 1, but excited with a pair of beams with equal power and phase difference $\delta$. (b),(c) Examples of experimental far-field output intensity distributions for (b) $\delta =0$ and (c) $\delta =\pi /2$. (d) Corresponding Bloch-band populations, calculated for $I_0=1.1$ and $E_{\mathrm{ext}}=1\mathrm{kV}/\mathrm{cm}$. (e) The measured (relative) powers of output beams (symbols and error bars) and the corresponding numerical simulations (solid lines) for external bias $E_{\mathrm{ext}}=0.4\mathrm{kV}/\mathrm{cm}$ (inner curves) and $E_{\mathrm{ext}}=1\mathrm{kV}/\mathrm{cm}$ (outer curves). The symbols next to the curves indicate corresponding beam in BZ.

Figure 3

Switching of power between different outputs by input phase-only manipulation. (a) Three-level Pendellösung oscillations realized by coupling of three $M$ points. The experimental input in (b) consists of two beams at $M$ and $M$’ points with equal power and phase difference $\delta$. Corresponding LZM solution is derived in 20, and it allows one to obtain a single beam at the output, as in (c) with $\delta =0$, or any pair of output beams in (d)–(f) with $\delta =\pi /3$ in (d), $\pi$ in (e), and $5\pi /3$ in (f). The Bloch-band populations in (g) (calculated for $I_0=1.1$ and $E_{\mathrm{ext}}=1\mathrm{kV}/\mathrm{cm}$) show the coupling between first three bands, as expected from resonant theory. (h) Measured relative powers of the output beams (symbols and error bars) and the corresponding numerical simulations (solid lines). $E_{\mathrm{ext}}=1\mathrm{kV}/\mathrm{cm}$.

Figure 4

Observation of Landau-Zener tunneling. (a),(b) Fourier space and (c),(d) real space output intensity distributions for input inclination angles of the signal beam: $\theta =0.14°$ in (a),(c) and $\theta =0.26°$ in (b),(d). (e) Measured output power ratios (markers) and the corresponding numerical simulations (solid lines). The error bars show finite beam widths in the reciprocal space; the arrow in the BZ inset indicates the direction of the linear index gradient $\alpha =\nabla I_{\mathrm{grad}}$. (f) The dynamics of band populations (in the Bloch-wave basis of the periodic lattice with $\alpha =0$) during the tunneling in the crystal with band indices next to the curves. Calculations are for $I_0=0.9$ and $E_{\mathrm{ext}}=1.35\mathrm{kV}/\mathrm{cm}$, with the gradient illumination $I_{\mathrm{grad}}$ of width $b=200\mu \mathrm{m}$ and amplitude $B=2$ [20].