# Pseudo-Spin Polarized One-Way Elastic Wave Eigenstates in One-Dimensional Phononic Superlattices
## 2 Model and Method
### Reciprocity Condition
Comparing Equation (30a,b) and Equation (35a,b), we can establish constraints that would be imposed by translational invariance on the parameters \(A_{+}\), \(A_{-}\), \(B_{+}\), and \(B_{-}\), namely:  
\[ \begin{pmatrix}A_{+}\\ A_{-}\end{pmatrix} _{N}= \begin{pmatrix}e^{-ik_{1}NL}&0\\ 0&e^{ik_{1}NL}\end{pmatrix} e^{i\eta NL} \begin{pmatrix}A_{+}^{P}\\ A_{-}^{P}\end{pmatrix}  \tag{36a}\] \[ \begin{pmatrix}B_{+}\\ B_{-}\end{pmatrix} _{N}= \begin{pmatrix}e^{-ik_{1}(NL+d_{1})}&0\\ 0&e^{ik_{1}(NL+d_{1})}\end{pmatrix} e^{i\eta NL} \begin{pmatrix}B_{+}^{P}\\ B_{-}^{P}\end{pmatrix}  \tag{36b}\]  
These are actually unitary transformations, which rotate the representation amplitudes of the periodic system into the representation of the amplitudes in Equation (30a,b). By using Equation (32) and inserting Equation (36a,b) into Equation (33), i.e., \(N=1\) for the left hand side of the equal sign and \(N=0\) for the right hand side, one obtains the following relation:  
\[\left( \begin{pmatrix}t_{11}&t_{12}\\ t_{21}&t_{22}\end{pmatrix} -e^{i\eta L} \begin{pmatrix}1&0\\ 0&1\end{pmatrix} \right) \begin{pmatrix}A_{+}^{P}\\ A_{-}^{P}\end{pmatrix} =0. \tag{37}\]  
with:  
\[\mathbf{T}= \begin{pmatrix}t_{11}&t_{12}\\ t_{21}&t_{22}\end{pmatrix} =\mathbf{H}_{1}^{-1}(0)\mathbf{M}\mathbf{H}_{1}(0) \tag{38}\]After some algebraic manipulation, we find:  
\[t_{11}=\frac{\lambda+\mu}{2}+i\frac{\sigma-\zeta}{2},\;t_{12}=\frac{\lambda-\mu}{ 2}+i\frac{\sigma+\zeta}{2},\;\text{and}\;t_{21}=t_{12}^{*}\;\text{and}\;t_{22}=t _{11}^{*}.\]  
One can verify algebraically that Equation (37) is identical to Equation (19). We can also observe that the determinants of the unimodular matrices \(M\) and \(\mathbf{T}\) are identical and given by \(\lambda\mu-\sigma\zeta=1\). When solving the eigenvalues, \(\varepsilon\) of the matrix \(\mathbf{M}\) gives a characteristic equation of \(\varepsilon^{2}-(\lambda+\mu)+1=0\). The finite superlattice takes a propagative eigenmode when \(\left|\frac{\lambda+\mu}{2}\right|\leq 1\). By setting \(\cos\theta=\)\(\frac{\lambda+\mu}{2}\), the characteristic equation takes the form of the dispersion relation of the infinite superlattice (Equation (10)) in the limits of the long wavelength, provided one identifies \(\theta\) with \(qL\). Equations (37) and (19) achieve translational invariance and lead to one-way propagation (zero amplitudes for q < 0 and non-zero amplitudes for q > 0) when \(\sin k_{2}d_{2}=0\). When dealing with the finite superlattice, the unconstrained matrix \(\mathbf{M}\) is diagonalized with the eigenvalues \(\varepsilon_{+}=e^{i\theta}\) and \(\varepsilon_{-}=e^{-i\theta}\), which account for forward and backward propagation and are associated with the different pseudo-spins of the eigenvectors. The diagonalized matrix is then brought to the power \(N\) to calculate the transmission coefficient of Equation (29) for the finite lattice with \(N\) unit cells. The wave function of the finite superlattice will try to approach the Bloch wave solutions. Under the condition \(\sin k_{2}d_{2}=0\), the approximate Bloch wave will make the contribution of the backward-propagating wave (i.e., \(e^{-i\theta}\)) small compared to the forward wave, thereby leading to high values of the transmission coefficient. One-way propagation in the infinite superlattice arises from the combined effects of the constraint of the translational invariance of the wave function and of a Fabry-Perot resonance condition in the superlattice unit cell.