Observation of Three-Dimensional Discrete-Continuous $X$ Waves in Photonic Lattices

We report on the generation of three-dimensional discrete $X$ waves in a femtosecond laser-written waveguide array. Our measurements constitute the first experimental observation of temporally localized three-dimensional discrete-continuous entities. The $X$ waves spontaneously emerge from a single-site excitation due to the discrete diffraction of the lattice and the normally dispersive as well as cubically nonlinear properties of the fused silica used as host material.

Figure 1

Dispersion relation of a two-dimensional waveguide array with square geometry and 25% diagonal coupling [Eq. (2) for $D=1$]: (a) cross section along the $\nu =0$ plane, (b) isosurface for $\beta =2(c_1+c_2+c_3+c_4)$.

Figure 2

(a) Experimental setup. (b) Denomination scheme of the individual channels. Measured (c) linear and (d) nonlinear discrete diffraction patterns at the samples end facet. (e) Refractive index of the bulk material according to the Sellmeier model. Simulated (f) linear and (g) nonlinear discrete diffraction patterns.

Figure 3

(a) Measured autocorrelation traces of each individual channel for excitation of channel $44$ (solid red curves) and respective simulations (dashed blue curves). Spatio-temporal intensity distribution of the simulated $X$ wave: (b) 3D plot, (c) intensity distribution in the outermost (channels $11–17$) and central (channels $41–47$) layer, respectively. The cone angle $2\vartheta$ of the $X$ wave is approximately 163°, as determined from the central layer.

Figure 4

Simulated evolution of the wave packet in a $15\times 15$ array (central layer, linear scale) for propagation distances (a) $z=0\mathrm{mm}$ [initial condition], (b) 105 mm [corresponding to experimental observations], (c) 150 mm. (d)–(f) Spectral distribution for the respective cases (logarithmic intensity scale).