Subwavelength Su-Schrieffer-Heeger topological modes in acoustic waveguides

For the sake of clarity, we remind briefly the direct mapping approach proposed in Coutant et al. (2021a). We consider the propagation of acoustic waves at fixed frequency $\omega=kc_{0}$, with the harmonic convention $e^{-i\omega t}$. $k$ is the wavenumber and $c_{0}$ the speed of sound. The waves propagate inside a waveguide obtained by connecting segments of alternating cross sections $S_{A}$ and $S_{B}$, as illustrated in Fig. 1(a). All segments have the same length $\ell$. The waveguide is assumed to be sufficiently narrow so that propagation is one-dimensional (monomode propagation) along the $x$-axis. More precisely, we assume that the longitudinal length is much larger than the transverse ones. We describe the propagation using the transfer matrix formalism. By construction, the transfer matrix relates the acoustic pressure $p(x)$ and its derivative $p^{\prime}(x)$ from one side to the other of a segment:

where

Now, to relate the propagation inside one segment to nearby ones, we use the continuity of pressure and acoustic flow rate at each cross section change. This gives the junction conditions

The idea is to start at a given cross section change, and relate the pressure there to the left and to the right pressures using the transfer matrix. Considering the section change at $x_{n}^{A}$, this leads to the pair of equations

where $p^{\prime}_{\pm}(x)$ is short for the limit $\epsilon\to 0^{\pm}$ of $p^{\prime}(x+\epsilon)$. Similarly, starting from $x=x_{n}^{B}$, we obtain

Now, in both pairs of equations (4) and (5), we can eliminate the pressure derivative term by using the flow rate continuity. This leads to an eigenvalue problem for the discrete set of pressure amplitudes $A_{n}\equiv p(x_{n}^{A})$ and $B_{n}\equiv p(x_{n}^{B})$:

with

The eigenvalue problem obtained in equation 6 is exactly the SSH model Su et al. (1979); Asbóth et al. (2016). For finite waveguides, say made of an odd number $2N-1$ of segments and closed at both ends, the problem reduces to finding the eigenvalues of a $2N\times 2N$ matrix, analogous to the SSH Hamiltonian. At the closed ends the acoustic velocity vanishes, which imposes the condition $\cos(k\ell)A_{1}=B_{1}$, and $\cos(k\ell)B_{N}=A_{N}$ (this can be directly obtained from equations (4a) and (5b) ¹¹1We refer the reader to Coutant et al. (2021a) section III for an extended discussion on boundary conditions. In particular, it is explained how $H$ can be made hermitian by a rescaling of pressure values.). Hence, the eigenvalue problem of Eq. (6) has the form

with

and $X=\begin{pmatrix}A_{1},B_{1},\dots A_{N},B_{N}\end{pmatrix}^{T}$. The eigenvalue $E$ then gives the acoustic eigenfrequencies through the relation

The correspondance between waveguides of alternating cross sections was already established and studied in Coutant et al. (2021a), and also used to obtain two dimensional generalizations Zheng et al. (2019, 2020); Coutant et al. (2020); Coutant et al. (2021b). In such a setup, properties of the SSH model are directly realized in acoustics. In particular, the SSH model is known to host a topologically protected mode at $E=0$, which, from Eq. (10), induces topologically protected acoustic modes at $k\ell=\pm\pi/2+2n\pi$ with $n\in\mathbb{N}$. Unfortunately, this mapping limits the wavelength of theses modes to be at least of the order of a segment size. The novelty of the present work is to modify the energy-frequency relation of Eq. (10) in order to obtain SSH-like modes in different frequency ranges. With a proper configuration, we now show that $k\ell$ in Eq. (10) is replaced by a phase $\phi(k)$ that can be tailored to induce a topological mode with a lower frequency.

We now consider a broader class of waveguide generalizing section II.1, where each waveguide segment can host various scatterers, as shown in Fig. 1(b). Propagation in each segment is given in terms of a transfer matrix:

The key point of our generalization is to assume that the scattering induced is identical in each segment irrespectively of the cross section value, i.e.

We also assume that the scattering is reciprocal, and each segment is mirror symmetric. Under these assumptions, the transfer matrix has the general form:

with $\det(M)=1$. Note that the diagonal elements are equal due to mirror symmetry. As we shall see, the coefficient $\alpha$ plays a key role in our construction, and it is therefore useful to notice that it gives the dispersion relation of a medium obtained by connecting periodically segments described by $M$ (i.e. without cross section changes). Indeed, in this case solutions can be given in terms of Bloch waves satisfying $p(x_{n+1}^{A})=e^{i\phi}p(x_{n}^{A})$, with $\phi$ the Bloch phase. Since Bloch waves are also eigenvectors of the transfer matrix Markos and Soukoulis (2008), the two eigenvalues are $e^{\pm i\phi}$. Since their sum is the trace of $M$, the dispersion relation is simply given by $\cos(\phi)=\alpha$. Hence, the transfer matrix can be written under a form very similar to Eq. (2) with the help of the Bloch phase $\phi$:

where $Y$ is a complex number. Notice that by convention, we will now focus on $\phi>0$. We now follow exactly the same steps as in section II.1. We first write the propagation inside the segments

and

Again, using continuity of pressure and flow rate, one can get rid of pressure derivative in the preceding equations. Doing so, we obtain the exact same eigenvalue problem of Eq. (8) with the Hamiltonian $H$ of Eq. (9). However, the energy eigenvalue is now related to the acoustic frequencies through the Bloch phase $\phi(k)$, hence, Eq. (10) is replaced by

What we have shown is that for each eigenvalue $E$ of the SSH model (6) corresponds a set of eigenfrequencies of the waveguide obtained from Eq. (17) controlled by the Bloch phase $\phi(k)$.

As mentioned before, when the waveguide segments are made of straight tubes with no scatterer, as in Fig. 2(a), the Bloch phase is simply given by $\phi(k)=k\ell$. Hence, to obtain the spectrum of a finite waveguide with cross-section changes, we look for frequencies such that $k\ell=\phi_{n}=\mathrm{acos}(E_{n})$ with $E_{n}$ the eigenvalues of the SSH Hamiltonian of Eq. (9). This is illustrated in Fig. 2(b). In particular, since the SSH model has a gap around $E=0$, the cross-section changes induce the opening of a gap around $k\ell=\pi/2$.

We now consider the waveguide configuration where a Helmholtz resonator is put on the upper wall at the middle of each segment, as shown in Fig. 2(c). We use a low frequency model for the Helmholtz resonators Zheng et al. (2020), based on the following assumptions: (i) we only consider the lowest resonance frequency, (ii) we assume that the neck radius is much smaller than the typical wavelength (hence, $r\ll\ell$), (iii) we neglect dissipation. Under these assumptions, the resonators are characterized by two parameters: their resonance frequency $\omega_{0}=k_{0}c_{0}$ and their coupling to the waveguide $g$. In the limit of a small neck, these parameters can be approximated by

where $V_{c}$ is the volume of the cavity, $r$ the radius of the neck, $h$ its length, and $S_{0}$ the cross section of the waveguide. Since the neck radius is much smaller than the typical wavelength, the resonators induce a jump of the acoustic velocity from one side and the other of the hole, while the pressure $p$ is continuous. This jump translates into a jump for the pressure derivative $[p^{\prime}]=Y_{0}p$ with

Now, to apply the results of section II, the transfer matrices of each segment must be equal as required in Eq. (12). For this, we need to make sure that all resonators have the same resonance frequency $k_{0}$ and the same coupling constant $g$. Since the latter depends on the section of the segment, we need to adjust the geometries of the resonators accordingly. In a regime where Eq. (18) is valid, we might take $r_{A}^{2}/S_{A}=r_{B}^{2}/S_{B}$ and $r_{A}^{2}/V_{A}=r_{B}^{2}/V_{B}$.

We can now compute the transfer matrix of a segment of length $\ell$ with a resonator at the middle point. Under the form of Eq. (13), the matrix coefficients are given by

As in the preceding section (Eq. (14)), the dispersion relation is directly obtained from the diagonal elements of $M$. This gives the Bloch phase as a function of the frequency $k$:

The acoustic eigenfrequencies of a finite waveguide can now be obtained by using the dispersion relation with the spectral values of the phase (see Fig. 2(c,d)).

The SSH model is classical for its topological properties. In particular, the bulk-edge correspondance guarantees that when it is in its topological phase, edge modes are present in the middle of the gap, more specifically at $E=0$. This can be seen either by computing the appropriate topological invariant, the winding number or the Zak phase Asbóth et al. (2016), or by explicitly computing the edge mode. In particular, in the limit of a semi-infinite waveguide ($N\to\infty$) where the right end is sent to infinity, a direct verification shows that the mode

is a solution of the eigenvalue problem (6) with $E=0$. Moreover, it is localized on the left edge if $s<t$, which corresponds to the topological phase. In Fig. 3(a), we show the spectrum of the SSH model for varying parameter $s$ and $t=1-s$. As we just argued, for $t<s$ we only have extended modes in the passing bands (trivial phase), while for $s<t$, we have two edge modes localized on each end of the chain (topological phase). From relation (17), for each of the eigenvalues $E_{n}$ corresponds a value of the Bloch phase $\phi_{n}>0$ and vice-versa. This is shown in Fig. 3(b). One can then obtain the frequency spectrum of the acoustic system from the dispersion relation (14). In particular, when $s<t$, we will obtain two topologically protected edge modes (one is symmetric and the other antisymmetric) for two frequencies corresponding to $\phi\simeq\pi/2$, as we see in Fig. 3(b).

Turning back to the configuration of Fig. 2(c), the zero energy mode of the SSH model (seen in equation (22)) induces a localized topological acoustic mode each time $\phi(k)=\pi/2$. This is illustrated in Fig. 4(a). The isolated eigenfrequencies we see near $\phi=\pi/2$ (when $s<t$) correspond to edge modes. We compare this configuration with a waveguide of alternating cross sections and without resonators, shown in Fig. 2(c). We see that the presence of Helmholtz resonators opens a gap near the resonance frequency $k_{0}$, which in turn, induces extra edge modes at low frequencies $k<k_{0}$. This means that, by taking $k_{0}$ very small, we can obtain topological acoustic modes at arbitrary low frequencies. This is our main result: by changing the scatterer inside each segment, the mapping to the SSH model is exact and realized for any desired frequency range through Eq. (17). In Fig. 4(c), we compare the pressure profile of the edge mode on the left for waveguides with (red curve) or without resonators (purple curve).

To confirm this subwavelength topological edge wave we have performed the numerical computation of eigenfrequencies and eigenmodes of a cavity closed by hard walls. It solves the 2D Helmholtz equation $\triangle p+k^{2}p$ with Neumann boundary conditions at the walls, using Finite Element Method. We follow the approach described previously, but take an even number of segments ($N=8$), which allows us to isolate one unique edge mode Coutant et al. (2021a) at the extremity with the smallest width (here $S_{A}$). Each segment is decorated by an Helmholtz resonator with neck width $r$, neck length $h$ and a cavity of width $L_{x}$ and height $L_{y}$. These geometrical parameters are finely tuned to get an equal Bloch phase following equation (21) in the range of frequencies $0\leq k\ell\leq\pi$, for both segments of cross section $S_{A}$ and $S_{B}$. The values of these tuned paramters are found to be $S_{A}=0.1\ell$, $r_{A}=0.0217\ell$, $h_{A}=0.2\ell$, $L_{xA}=0.2\ell$, $L_{yA}=0.3\ell$ and $S_{B}=0.2\ell$, $r_{B}=0.05\ell$, $h_{B}=0.2\ell$, $L_{xB}=0.3\ell$, $L_{yB}=0.4\ell$; they correspond to resonance frequency $k_{0}\ell=1.226$ and a coupling parameter $g\ell=0.914$. The corresponding edge mode is confirmed to appear (it is the fifth mode of the cavity) and is displayed in Fig. 5. It can be remarked that the amplitude is strongly concentrated in the first resonator at the left.