Nonlinear self-collimated sound beams in sonic crystals

We report the propagation of high-intensity sound beams in a sonic crystal, under self-collimation or reduced-divergence conditions. The medium is a fluid with elastic quadratic nonlinearity, where the dominating nonlinear effect is harmonic generation. The conditions for the efficient generation of narrow, nondiverging beam of second harmonic are discussed. Numerical simulations are in agreement with the analytical predictions made, based on the linear dispersion characteristics in modulated media and the nonlinear interaction in a quadratic medium under phase matching conditions.

Figure 1

Top: band structure (left) of a square lattice of rigid cylinders with $r=0.11a$, where $a$ is the lattice constant, immersed in water (right). Red and blue lines mark the bands (second and eighth) for which simultaneous self-collimation for both fundamental and second harmonic searched, along $\mathrm{\Gamma }-\mathrm{X}$ direction. Bottom: isofrequency contours for the second (left) and eighth bands (right). Points denote the wave vectors for both waves, lying on flat segments, respectively.

Figure 2

Dispersion curves involved in simultaneous self-collimation for the fundamental (red line) at the second band and second harmonic (blue line) at the eighth band. The “doubled” dispersion curve is represented (dashed red line) to identify phase matching of harmonics. The intersection denotes the frequency presenting phase matching. A closest view (inset) shows that for the self-collimated second harmonic, two solutions (modes A and B, with distinct $\mathrm{K}$) are found, phase and nonphase matched, respectively.

Figure 3

Spatial distribution of the pressure field for the Bloch modes of the fundamental (left) and the second harmonic (right) waves. The coupling coefficient is estimated to be $\kappa =0.85$.

Figure 4

Pressure distributions obtained by FDTD simulations. Normalized intensity cross section of fundamental (a) and second harmonic (b) at $x=80a$ propagating in the crystal (continuous line) and in a homogeneous (water) medium (dashed lines). Beam spatial distribution for simultaneously self-collimated harmonics in the sonic crystal (c),(d) and in homogeneous fluid (e),(f). Pressures are normalized to the maximum pressure.

Figure 5

Normalized field amplitude along the acoustical axis $y=0$, for the fundamental, second, and third harmonic beams. The dashed lines represent the analytical solutions for harmonic evolution on a plane wave propagating in a nondispersive medium.