Complex dispersion relation of Rayleigh-Bloch waves trapped by slow inclusions

TL;DR

The paper investigates Rayleigh-Bloch waves guided along a periodic line of inclusions in an open medium and develops a framework of guided quasi-normal modes to derive a complex dispersion relation that includes radiation loss. By analyzing velocity and impedance contrasts (via $\\bar{\\rho}$, $\\bar{B}$, $\\bar{v}$, and $\\bar{Z}$) and by tracking QNMs, it shows that slow inclusions create additional, less-leaky bands that penetrate the sound cone and give rise to bound states in the continuum (BICs) due to symmetry and periodicity. The study reveals a rich spectrum with multiple leaky RB bands and BICs that depend on azimuthal symmetry $m$ and lattice geometry, including higher-order resonances up to $m=3$ and band folding near high-symmetry points. The results demonstrate that guided QNMs provide a practical route to characterize and design robust waveguiding with controlled radiation, with potential applicability to SAW and vector elastodynamics.

Abstract

Rayleigh-Bloch waves are guided acoustic waves propagating along a periodic line of inclusions placed inside an open, infinite medium. Below the sound cone, they are transversely evanescent on both sides of the line of inclusions. Guidance is then achieved without any cladding surrounding the segmented core. Inclusions usually impose definite boundary conditions, resulting in a single guided band. We consider instead the case of permeable, slow inclusions inside a fast medium. Introducing the concept of guided quasi-normal modes, we obtain the complex dispersion relation taking into account radiation at infinity. We thus show that multiple bands of leaky Rayleigh-Bloch waves appear and that guided bound states in the continuum arise as a result of the combination of symmetry and periodicity.

Figures (6)

• Figure 1: Waveguide composed of a line of periodic inclusions in an open, infinite, host medium. The central portion shows the primitive unit-cell with lattice constant $a$, terminated at top and bottom by perfectly matched layers (PML). The diameter of cylindrical inclusions is $d=0.6a$.
• Figure 2: Maps of the dispersion relation of Rayleigh-Bloch waves, computed using the resolvent band structure method laudePRB2018. The color scale is for the local density of states (LDOS) estimated from the response to a stochastic source term. Panels are for (a) hollow inclusions satisfying a Neumann boundary condition along their edge, (b) inclusions with $\bar{\rho}=\bar{B}=100$ ($\bar{v}=1$ and $\bar{Z}=100$), (c) inclusions with $\bar{\rho}=\bar{B}=2$ ($\bar{v}=1$ and $\bar{Z}=2$), (d) inclusions with $\bar{\rho}=1$ and $\bar{B}=1/4$ ($\bar{v}=1/2$ and $\bar{Z}=1/2$), (e) inclusions with $\bar{\rho}=2$ and $\bar{B}=1/2$ ($\bar{v}=1/2$ and $\bar{Z}=1$), (f) inclusions with $\bar{\rho}=4$ and $\bar{B}=1$ ($\bar{v}=1/2$ and $\bar{Z}=2$).
• Figure 3: Complex dispersion relation for leaky Rayleigh-Bloch waves treated as guided quasi-normal modes. The color scale is for the inverse of the quality factor. (a) For hollow inclusions (same conditions as in Fig. \ref{['fig2']}a) the first band is guided but the second band, extending inside the sound cone, is always strongly leaky. (b) For filled inclusions with $\bar{\rho}=1$ and $\bar{B}=1/4$ (same conditions as in Fig. \ref{['fig2']}d) the dispersion of the second band is lossless at both the X and the $\Gamma$ points. The latter solution is a bound state in the continuum (BIC) whereas the former is a guided wave. The real part of the pressure field of guided QNM solutions is shown within the primitive unit-cell at high-symmetry points.
• Figure 4: Dispersion relation of Rayleigh-Bloch waves for inclusions with $\bar{\rho}=1$ and $\bar{B}=1/9$ ($\bar{v}=1/3$ and $\bar{Z}=1/3$). (a) Map of the dispersion relation computed using the resolvent band structure method laudePRB2018. The color scale is for the local density of states (LDOS) estimated from the response to a stochastic source term. (b) Complex dispersion relation for leaky Rayleigh-Bloch waves treated as guided quasi-normal modes. The color scale is for the inverse of the quality factor. The real part of the pressure field of guided QNM solutions is shown within the primitive unit-cell at high-symmetry points. Examples of bound states in the continuum (BIC) are shown for bands 3 and 5 at the $\Gamma$ point. (c) The Rayleigh-Bloch wave of band $4$ at $ka/(2\pi)\approx0.236$ inside the first Brillouin zone is also a BIC.
• Figure 5: Dispersion relation of Rayleigh-Bloch waves for inclusions with $\bar{\rho}=1$ and $\bar{B}=1/16$ ($\bar{v}=1/4$ and $\bar{Z}=1/4$). (a) Map of the dispersion relation computed using the resolvent band structure method laudePRB2018. The color scale is for the local density of states (LDOS) estimated from the response to a stochastic source term. (b) Complex dispersion relation for leaky Rayleigh-Bloch waves treated as guided quasi-normal modes. The color scale is for the inverse of the quality factor. The real part of the pressure field of guided QNM solutions is shown within the primitive unit-cell at high-symmetry points. Examples of bound states in the continuum (BIC) are shown for bands 3 and 5 at the $\Gamma$ point, and band 8 at the X point.