Brillouin Zone Spectroscopy of Nonlinear Photonic Lattices

We present a novel, real-time, experimental technique for linear and nonlinear Brillouin zone spectroscopy of photonic lattices. The method relies on excitation with random-phase waves and far-field visualization of the spatial spectrum of the light exiting the lattice. Our technique facilitates mapping the borders of the extended Brillouin zones and the areas of normal and anomalous dispersion within each zone. For photonic lattices with defects (e.g., photonic crystal fibers), our technique enables far-field visualization of the defect mode overlaid on the extended Brillouin zone structure of the lattice. The technique is general and can be used for photonic crystal fibers as well as for periodic structures in areas beyond optics.

Figure 1

Brillouin zone structure of a 2D square lattice and a schematic of our spectroscopic setup. (a) First and second Brillouin zones of a 2D square lattice with the high-symmetry points ($\Gamma$, $X$, and $M$) marked with white dots. (b) First two bands of the transmission spectrum of a 2D square lattice. (c) Dispersion curves between the symmetry points of the first two bands. Negative curvature in these curves corresponds to normal diffraction regions. (d) Diagram of our setup for Fourier-space spectroscopy of photonic lattice.

Figure 2

Experimental linear mapping of the edges of Brillouin zones of square and hexagonal lattices. (a) Interference pattern of the array waves forming the square lattice. (b) Fourier spectrum of the input probe (broad circle) and lattice-forming beams (four sharp peaks). (c) Calculated extended BZ scheme of a square lattice. (d) True experimental picture depicting the Fourier spectrum of the probe beam at the output of the square lattice. (e)–(h) Same as (a)–(d) with a hexagonal lattice.

Figure 3

Defect modes in a hexagonal lattice. Interference pattern of the waves forming the hexagonal lattice (a) with no defect, (b) with a positive defect, and (c) with a negative defect. Experimental Fourier spectrum of the probe beam at the lattice output with induced 2D hexagonal lattice (d) with no defect, (e) with a positive defect, and (f) with a negative defect.

Figure 4

Nonlinear dispersion mapping of a square lattice. (a) Fourier spectrum of the input probe beam (circle) and of the lattice-forming beams (four dots). Experimental Fourier spectrum of the probe beam at the output face of the square lattice under (b) self-focusing nonlinearity and (c) self-defocusing nonlinearity.