Band-Gap Structure of Waveguide Arrays and Excitation of Floquet-Bloch Solitons

Band-gap structure of periodic waveguide arrays is investigated in the linear and the nonlinear regimes. Excitation of array modes belonging to high bands is demonstrated. Floquet-Bloch solitons are demonstrated experimentally and shown to be a generalization of discrete solitons.

Figure 1

Reduced band-gap diagram of a typical waveguide array (i.e., folded into the first Brillouin zone), where the propagation constant is plotted as a function of the Bloch wave number. The shaded regions represent the gaps. The four lowest lying bands (highest propagation constants) are shown.

Figure 2

Pure FB-wave excitation using the geometry shown in the inset. (a) Cross sections of the measured intensity distribution at the sample output as a function of the input-beam angle, at low (upper panel) and high (lower-panel) input powers. The high-power measurement was conducted at approximately 2 kW peak power. (b) Highly magnified images of the pure FB modes excited in each band at low powers. The high-power excitation reproduced the same modal shapes.

Figure 3

(a) Beam propagation simulation simulation results of the measurement described in Fig. 2, showing the beam at the array output, for low power (dotted line) and high power (solid line). The inset shows a magnified section of the plot, demonstrating the identical modal shapes of the low-power (diffracting) beam and the band No. 4 FB soliton formed at high power. (b) Photograph of the intensity distribution at the array output of a band No. 2 FB mode, at low, intermediate, and high output power (approximate peak powers are indicated). At low power, the input beam weakly excites band No. 3 (right side of the figure), and diffracts partly into the continuum (left side of the figure). At high power, a band No. 2 FB soliton is formed.

Figure 4

(a) Theoretically calculated diffraction of a typical array, plotted as a function of the propagation constant. The shaded areas represent the gaps. Bands 2 to 5 are seen, from right to left. Band 1, with the largest propagation constant, is not visible in this scale. (b) Measurement results of the intensity distribution at the array output, plotted as a function of the average output power, for the points indicated in the diffraction plot of part (a).

Figure 5

Measurement results for the case of a mixed excitation of two FB waves using the geometry shown in the inset. (a) Cross sections of the intensity distribution at the array output as a function of the input-beam angle. (b) Theoretically calculated band-gap diagram of the measured array, showing the two lowest-lying bands (highest $\beta$’s), and indicating the propagation direction of the FB modes. $k_x$ represents the $x$ component of the continuum $k$ vector. The shaded regions represent the gaps.