# Pseudo-Spin Polarized One-Way Elastic Wave Eigenstates in One-Dimensional Phononic Superlattices
## 2 Model and Method
### Reciprocity Condition
The odd-numbered bands are associated with conventional topologies and zero Berry or Zak phases. This type of asymmetry in the transmission of topologically non-conventional bands has been observed theoretically for electronic waves in finite semiconductor superlattices [20] and electromagnetic superlattices [18]. In Figure 3, we can clearly see that the asymmetry in the dispersion of the transmission of the even-numbered bands is associated with the Fabry-Perot resonance condition of medium 2 in a single unit cell, \(\sin k_{2}d_{2}=0\), where the transmission coefficient is equal to one. This is exactly the condition for which, in an infinite superlattice, spin-selective one-way propagation occurs, as imposed by the requirement that the waves take the form of Bloch waves, i.e., the waves satisfy translational invariance. Note that for a single unit cell, gaps exist that again correspond to the condition \(\left|\frac{\lambda+\mu}{2}\right|>1\). To shed more light on the relationship between the behaviors of the finite system and the infinite system, below we give the transfer matrix across one unit cell derived in [19]. The displacement field in the two media 1 and 2 of a unit cell is expressed as a general ansatz formed as a linear combination of transmitted and reflected waves:  
\[a=\frac{\lambda-\mu}{2}\frac{\sin N\theta}{\sin\theta}+\cos N\theta\]0 \[a=\frac{\lambda-\mu}{2}\frac{\sin N\theta}{\sin\theta}+\cos N\theta\]1  
Here, \(x\) is the coordinate along the direction perpendicular to the layers forming the superlattice. There is no specific constraint on the form of the amplitudes \(A_{+}\), \(A_{-}\), \(B_{+}\), and  
\[a=\frac{\lambda-\mu}{2}\frac{\sin N\theta}{\sin\theta}+\cos N\theta\]2 The same scheme as in \[a=\frac{\lambda-\mu}{2}\frac{\sin N\theta}{\sin\theta}+\cos N\theta\]3 The black solid line is for a single unit cell \(N=1\), sandwiched between the source and the detector. The arrows mark the location where \(\sin k_{2}d_{2}=0\). The stress is expressed as:  
\[a=\frac{\lambda-\mu}{2}\frac{\sin N\theta}{\sin\theta}+\cos N\theta\]4 \[a=\frac{\lambda-\mu}{2}\frac{\sin N\theta}{\sin\theta}+\cos N\theta\]5  
These definitions allow us to introduce the two-component vector for \(j=1\), 2:  
\[\mathbf{W}_{j}(x)= \begin{pmatrix}U_{j}(x)\\ S_{j}(x)\end{pmatrix} = \begin{pmatrix}e^{ik_{j}x}&e^{-ik_{j}x}\\ i\omega Z_{j}e^{ik_{j}x}&-i\omega Z_{j}e^{-ik_{j}x}\end{pmatrix}\begin{pmatrix}A _{+}\text{ \emph{or} }B_{+}\\ A_{-}\text{ \emph{or} }B_{-}\end{pmatrix}  \tag{32}\]  
Here, we define \(\mathbf{H}_{j}(x)= \begin{pmatrix}e^{ik_{j}x}&e^{-ik_{j}x}\\ i\omega Z_{j}e^{ik_{j}x}&-i\omega Z_{j}e^{-ik_{j}x}\end{pmatrix} \). In a stack of unit cells, using the conditions of the continuity of displacement and stress, one can relate \(W_{1}(x=d_{1}+d_{2})\) right after a unit cell to \(W_{1}(x=0)\) right before a unit cell using the following relation:  
\[\mathbf{W}_{1}(x=L=d_{1}+d_{2})=\mathbf{M}\mathbf{W}_{1}(x=0) \tag{33}\]  
where:  
\[\mathbf{M}= \begin{pmatrix}\lambda&\frac{1}{Z_{1}\omega}\sigma\\ Z_{1}\omega\zeta&\mu\end{pmatrix}  \tag{34}\]  
The symbol in Equation (34) was defined earlier. We remark that there is no constraint imposed on the ansatz of Equation (30a,b). For the periodic infinite superlattice and a unit cell labelled, \(N\), we must use a continuous version of the Bloch wave ansatz given by Equation (28a,b):  
\[U_{1}^{N}(x)=e^{i\eta NL}\Big{(}A_{+}^{P}e^{ik_{1}(x-NL)}+A_{-}^{P}e^{-ik_{1}( x-NL)}\Big{)}e^{i\omega t}\text{ \emph{for} }NL<x<NL+d_{1} \tag{35a}\] \[U_{2}^{N}(x)=e^{i\eta NL}\Big{(}B_{+}^{P}e^{ik_{1}(x-NL-d_{1})}+B_{-}^{P}e^{- ik_{1}(x-NL-d_{1})}\Big{)}e^{i\omega t}\text{ \emph{for} }NL+d_{1}<x<(N+1)L \tag{35b}\]  
This ansatz achieves translational periodicity, as it represents Bloch waves. In Equation (35a,b), the upper script \"p\" stands for periodic. This is necessary as the periodicity redefines the amplitude parameters.