Tailoring phonon band structures with broken symmetry by shaping spatiotemporal modulations of stiffness in a one-dimensional elastic waveguide

Spatiotemporal modulations of the elastic properties of materials can be used to break time and parity symmetry of elastic waves. We show that the form of the elastic band structure depends not only on the spatial and temporal periodicity of a spatiotemporal modulation but also on its shape through its Fourier components. We demonstrate that hybridization gaps open from interactions between the Bloch modes of the periodic medium in absence of the temporal variation of the modulation and the combined sinusoidal components of the Fourier decomposition of the periodic modulation.

Figure 1

Schematic representation of a one-dimensional mass spring system with periodically varying stiffness (spring constant) moving at a velocity $V$.

Figure 2

Illustration of the three superlattices constructed by modulating the spring constant ${\beta }_n$ over position $n$.

Figure 3

(a) Band structures resulting from a moving periodic Gaussian modulations ${\beta }_n^{\left(1\right)}$. $A$ and $A$′ are band gaps arising from hybridization between zeroth-order modes $n=0$ [static modulation and see Fig. 4] and first-order modes $\left(n=1\right)$ resembling Stokes and anti-Stokes modes associated with Brillouin scattering due to the moving modulation. $B$ and $B$′ are hybridization gaps between second-order Stokes and anti-Stokes modes $\left(n=2\right)$ and zeroth-order modes. (b) Same band structure as (a) but with black, gray, and white solid lines serving as guides to the eyes and highlighting zeroth-order, first–order, and second-order modes, respectively. To enhance the contrast, the contour plots represent the SED intensity at the power 1/400. The frequency is in kHz and the wave number is in units of $L$.

Figure 4

Elastic band structures induced as a result of scattering by (a) static periodic Gaussian modulation, (b) moving modulation composed of a single sinusoidal function ${\beta }_n^{\left(2\right)}$ with the same spatial period as ${\beta }_n^{\left(1\right)}$, and (c) moving modulation ${\beta }_n^{\left(3\right)}$ composed of the first three sinusoidal Fourier components of ${\beta }_n^{\left(1\right)}$. To enhance the contrast, the contour plots represent the SED intensity at the power 1/400. Frequency is in kHz and wave number is in units of $L$.

Figure 5

Schematic illustration of the band structure of the static superlattice and of the Brillouin harmonic bands involved in the formation of the hybridization gaps $A$ and $A$′, and $B$ and $B$′. The gap $A$ is formed at the intersection of ${\omega }_0\left(k\right)$ and ${\omega }_0\left(k-K_1\right)-\mathrm{\Omega }$ (i.e., when $g=0$). The gap $A$′ forms at the intersection of ${\omega }_0\left(k-\frac{2\pi }{L}\right)$ and ${\omega }_0\left(k-\frac{2\pi }{L}+K_1\right)+\mathrm{\Omega }={\omega }_0\left(k\right)+\mathrm{\Omega }$ (i.e., when $g=-\frac{2\pi }{L}$). The gaps $B$ and $B$′ are occurring where the conditions ${\omega }_0^2\left(k+g-2K_1\right)-{[{\omega }_0\left(k+g\right)-2\mathrm{\Omega }]}^2$ and ${\omega }_0^2\left(k+g+2K_1\right)-{[{\omega }_0\left(k+g\right)+2\mathrm{\Omega }]}^2$ are satisfied. The $B$ point is the intersection between ${\omega }_0\left(k-\frac{2\pi }{L}+2K_1\right)={\omega }_0\left(k+\frac{2\pi }{L}\right)$ and ${\omega }_0\left(k-\frac{2\pi }{L}\right)-2\mathrm{\Omega }$. The $B$′ point is the intersection between ${\omega }_0\left(k+\frac{2\pi }{L}-2K_1\right)={\omega }_0\left(k-\frac{2\pi }{L}\right)$ and ${\omega }_0\left(k+\frac{2\pi }{L}\right)+2\mathrm{\Omega }$.