Temporal dynamics of spatially localized waves in quadratic nonlinear waveguide arrays

We study experimentally and theoretically the temporal dynamics of laser pulses propagating under conditions of spatial self-focusing in quadratic nonlinear waveguide arrays made from periodically poled lithium niobate. We observed temporal pulse breakup and temporal pulse narrowing and studied the dynamics of these effects in different waveguides. We investigated the influence of the frequency dependence of the mode indices as a limiting factor for soliton formation. Our experimental results are in good agreement with the theoretical model developed from coupled-mode theory, providing a detailed understanding of pulse dynamics and beam distribution in waveguide arrays with quadratic nonlinearity.

Figure 1

(a) Sketch of waveguide array with period $d$. The red and yellow shading denotes the different crystal orientations, which form the quasi-phase-matching grating with a period ${\Lambda }_{\mathrm{QPM}}$. (b) FW and SH TM${}_{00}$ modes in an isolated array waveguide. (c) The diffraction relation of FW (solid blue line) and SH (red dash-dotted line) array supermodes. $\Delta \beta$ indicates the phase mismatch between the SH band and an isolated array waveguide mode (dotted blue line) at the FW. The black arrows emerging from the FW band denote the bifurcation points of unstaggered (upwards) solitons for focusing nonlinearity and staggered (downwards) solitons for the defocusing nonlinearity.

Figure 2

(a) Scheme of experimental setup showing the laser and beam shaping, the coupling to and from the sample with microscope objectives (MO), and the pulse-analysis setup. (b) Intensity (solid lines) and phase (dashed lines) of the input pulses at $\lambda =1552.5$ nm and $\lambda =1560.8$ nm.

Figure 3

(a)–(c) Pulse propagation in a homogeneous WGA without GVM and GVD, with Gaussian pulse excitation of 100 ps FWHM pulse length, and a phase mismatch $\Delta \beta L=51\pi$. Normalized power in (a) the central waveguide and (b) in the fourth waveguide from the center at the output of the WGA. (c) Normalized time-integrated power in all waveguides at the output of the WGA. (d), (e) Same as (a)–(c) in the WGA of our experiment with the experimental input pulse taking into account all experimental details, e.g., nonuniformities and dispersion.

Figure 4

Results for effective focusing nonlinearity. (a) Power dependence of time-averaged spatial FW output of WGA. (b) Comparison of spatial outputs of measurement (blue solid line), corresponding to a cut trough data plotted in (a), and simulation (red circles) for input peak power of 360 W. (c) Measured output spatiotemporal power distribution for FW pulse with 360 W input peak power. (d), (e) Comparison of the measured (blue dotted line) and simulated (red solid line) (d) temporal and (e) spectral powers in the different waveguides (waveguide number is indicated in each plot). The power of the input pulse is plotted in the center waveguide as a gray area for comparison.

Figure 5

Result for effective defocusing nonlinearity. (a) Power dependence of spatial FW output of WGA. (b) Comparison of spatial outputs of measurement (blue solid line), corresponding to cut trough the data plotted in panel (a), and simulation (red circles) for input peak power of 300 W. (c) Measured output spatiotemporal power distribution for FW pulse with 300 W input peak power. (d), (e) Comparison of the measured (blue dotted line) and simulated (red solid line) (d) temporal and (e) spectral powers in the different waveguides (waveguide number is indicated in each plot). The power of the input pulse is plotted in the center waveguide as a gray area for comparison.