# Pseudo-Spin Polarized One-Way Elastic Wave Eigenstates in One-Dimensional Phononic Superlattices
## 2 Model and Method
### Reciprocity Condition  
Reciprocity implies that the signal received by a detector emitted by a vibration source remains the same upon the interchange of the source and the receiver. Since the considered system is linear, time-invariant, and bias-free, we expect reciprocity to be satisfied. In order to investigate reciprocity in the one-way propagation scenario, we focus on the continuum limit of the superlattice. We consider a finite continuous superlattice sandwiched between a source layer \"S\" and a detection layer \"D\" with impedances that may differ from those of materials 1 and 2. We calculate the transmission coefficient of the finite superlattice using transfer matrices, as is achieved in [19]. The transmission coefficient is given by:  
\[T=\frac{4\frac{Z_{D}}{Z_{5}}}{\left(\frac{Z_{D}}{Z_{1}}b-\frac{Z_{1}}{Z_{5}}c \right)^{2}+\left(d+\frac{Z_{D}}{Z_{5}}a\right)^{2}} \tag{29}\]  
where:  
\[a=\frac{\lambda-\mu}{2}\frac{\sin N\theta}{\sin\theta}+\cos N\theta\]\[b = \sigma\frac{\sin N\theta}{\sin\theta}\] \[c = \zeta\frac{\sin N\theta}{\sin\theta}\] \[d = -\frac{\lambda-\mu}{2}\frac{\sin N\theta}{\sin\theta}+\cos N\theta\]  
In these equations, we have:  
\[\lambda = \cos k_{1}d_{1}\cos k_{2}d_{2}-F\sin k_{1}d_{1}\sin k_{2}d_{2}\] \[\mu = \cos k_{1}d_{1}\cos k_{2}d_{2}-\frac{1}{F}\sin k_{1}d_{1}\sin k_{ 2}d_{2}\] \[\sigma = \sin k_{1}d_{1}\cos k_{2}d_{2}+F\cos k_{1}d_{1}\sin k_{2}d_{2}\] \[\zeta = -\sin k_{1}d_{1}\cos k_{2}d_{2}-\frac{1}{F}\cos k_{1}d_{1}\sin k _{2}d_{2}\]  
and \(\cos\theta=\frac{\lambda+\mu}{2}\). \(N\) is the number of unit cells in the superlattice. In Equation (29), \(Z_{1}\), \(Z_{D}\), and \(Z_{S}\) are the impedances of the type 1, detection, and source layers, which are chosen to be equal for illustrative purposes; however, varying \(Z_{D}\) and \(Z_{S}\) will change the results quantitively but not qualitatively. Here, instead of interchanging the source and detector, we swap media 1 and 2 by interchanging the indices 1 and 2 in Equation (29). In the long wavelength limit, we have \(k_{1}=\frac{\omega}{c_{1}}\) and \(k_{2}=\frac{\omega}{c_{2}^{2}}\), as well as \(c_{1}=a\sqrt{k_{1}}\) and \(c_{2}=a\sqrt{k_{2}}\). In Figure 2, we present the transmission coefficient as a function of the frequency for two superlattices. Figure 2: Transmission coefficient \(T\), corresponding to the first 6 bands of two finite superlattices, with \(N=20\) as a function of the frequency for (a) \(Z_{D}=Z_{S}=Z_{1}\). The two superlattices are related by an interchange of layer indices 1 and 2, which is equivalent to interchanging the source S and detector D. We have also chosen \(k_{2}d_{2}=1.2k_{1}d_{1}\). Each band supports 20 resonances, corresponding to the number of periods of the finite superlattice, \(N\). The two transmission coefficients are identical for all frequencies, confirming that reciprocity is achieved. In Figure 2, the gaps correspond to the usual Bragg scattering. These gaps arise when \(\left|\frac{\lambda+\mu}{2}\right|>1\), which corresponds to non-propagative modes. Each band is composed of \(N\) resonances. We also note that the lower envelopes of the transmission coefficient of the 2 nd and 4 th transmission bands are asymmetric, while the 1 st, 3 rd, and 5 th bands, which have conventional topologies, exhibit a more symmetrical behavior. We note in Figure 3a that the lower envelope of the transmission coefficient of the second band approaches one near \(\frac{\omega}{c_{1}}d_{1}=2.618\) (that is, \(k_{2}d_{2}=1.2\times 2.618=\pi\), which corresponds to the condition \(\sin k_{2}d_{2}=0\)). The amplitude of the backward-propagating wave equals zero under this condition. The same condition is satisfied for the fourth band \(\frac{\omega}{c_{1}}d_{1}=5.236\), where \(k_{2}d_{2}=2\pi\). This behavior arises for the even-numbered bands in the transmission spectrum. These bands are known to correspond to dispersion bands of the infinite superlattice with a non-conventional topology, for which the Berry or Zak phase is equal to \(\pi\).