# Pseudo-Spin Polarized One-Way Elastic Wave Eigenstates in One-Dimensional Phononic Superlattices
## 2 Model and Method
### Amplitudes of the Displacement Field  
To find the amplitudes, we use the transfer matrix method. Equation (8) can be rewritten as:  
\[ \begin{pmatrix}1&1\\ f\delta_{-}^{(1)}&f\delta_{+}^{(1)}\end{pmatrix}\begin{pmatrix}A_{-}^{N}\\ A_{-}^{N}\end{pmatrix} = \begin{pmatrix}\alpha_{2}&\beta_{2}\\ \alpha_{2}\delta_{-}^{(2)}&\beta_{2}\delta_{+}^{(2)}\end{pmatrix}\begin{pmatrix} B_{-}^{N-1}\\ B_{-}^{N-1}\end{pmatrix}  \tag{16a}\] \[ \begin{pmatrix}\alpha_{1}&\beta_{1}\\ f\alpha_{1}\delta_{-}^{(1)}&f\beta_{1}\delta_{+}^{(1)}\end{pmatrix}\begin{pmatrix} A_{-}^{N}\\ A_{-}^{N}\end{pmatrix} = \begin{pmatrix}1&1\\ \delta_{-}^{(2)}&\delta_{+}^{(2)}\end{pmatrix}\begin{pmatrix}B_{-}^{N}\\ B_{-}^{N}\end{pmatrix}  \tag{16b}\]  
Since the second relation, Equation (16b), applies to the unit cell \(N\)-1, as well as cell \(N\), it can be inverted to obtain a relation between \( \begin{pmatrix}B_{+}^{N-1}\\ B_{-}^{N-1}\end{pmatrix} \) and \( \begin{pmatrix}A_{+}^{N-1}\\ A_{-}^{N-1}\end{pmatrix} \). By inserting that relation into Equation (16a), we obtain the transfer matrix relating amplitudes in one unit cell to amplitudes in the neighboring cell:  
\[ \begin{pmatrix}A_{+}^{N}\\ A_{-}^{N}\end{pmatrix} = \begin{pmatrix}t_{11}&t_{12}\\ t_{21}&t_{22}\end{pmatrix}\begin{pmatrix}A_{+}^{N-1}\\ A_{-}^{N-1}\end{pmatrix}  \tag{17}\]  
The components of the transfer matrix are given by:  
\[t_{11} =\frac{1}{f\Big{(}\delta_{+}^{(2)}-\delta_{-}^{(2)}\Big{)}\Big{(} \delta_{+}^{(1)}-\delta_{-}^{(1)}\Big{)}}\alpha_{1}\Big{[}\Big{(}f\delta_{+}^ {(1)}-\delta_{-}^{(2)}\Big{)}\Big{(}\delta_{+}^{(2)}-f\delta_{-}^{(1)}\Big{)} \alpha_{2}+\Big{(}f\delta_{+}^{(1)}-\delta_{+}^{(2)}\Big{)}\Big{(}-\delta_{- }^{(2)}+f\delta_{-}^{(1)}\Big{)}\beta_{2}\Big{]}\] \[t_{12} =\frac{1}{f\Big{(}\delta_{+}^{(2)}-\delta_{-}^{(2)}\Big{)}\Big{(} \delta_{+}^{(1)}-\delta_{-}^{(1)}\Big{)}}\beta_{1}\Big{[}\Big{(}f\delta_{+}^ {(1)}-\delta_{-}^{(2)}\Big{)}\Big{(}\delta_{+}^{(2)}-f\delta_{+}^{(1)}\Big{)} \alpha_{2}+\Big{(}f\delta_{+}^{(1)}-\delta_{+}^{(2)}\Big{)}\Big{(}-\delta_{- }^{(2)}+f\delta_{+}^{(1)}\Big{)}\beta_{2}\Big{]}\] Here, \(t_{21}=t_{12}^{*}\) and \(t_{22}=t_{11}^{*}\). The star stands for a complex conjugation. Due to the periodicity of the system, solutions will take the form of Bloch waves: \[ \begin{pmatrix}A_{+}^{N}\\ A_{-}^{N}\end{pmatrix} =e^{iqL} \begin{pmatrix}A_{+}^{N-1}\\ A_{-}^{N-1}\end{pmatrix} \] (18) Combining Equations (17) and (18) leads to the eigenvalue problem [16]: \[\bigg{(} \begin{pmatrix}t_{11}&t_{12}\\ t_{21}&t_{22}\end{pmatrix} -e^{iqL} \begin{pmatrix}1&0\\ 0&1\end{pmatrix} \bigg{)} \begin{pmatrix}A_{+}\\ A_{-}\end{pmatrix} =0 \tag{19}\]  
If \(e^{iqL}\) is an eigenvalue of the transfer matrix, then the amplitudes \(A_{+}\) and \(A_{-}\) are given by:  
\[A_{+}=-t_{12} \tag{20a}\] \[A_{-}=t_{11}-e^{iqL} \tag{20b}\]The dispersion relation of Equation (9) can be rewritten as \(2\cos qL=t_{11}+t_{22}\), so Equation (20b) becomes:  
\[A_{-}=\frac{1}{2}(t_{11}-t_{22})-i\sin qL \tag{21}\]  
After a number of trigonometric and algebraic manipulations, we obtain:  
[4:04 PM] [PERSON]  
\[ \begin{array}{ll}A_{-}&=i\bigg{\{}\sin k_{1}d_{1}\cos k_{2}d_{2}\\ &+\frac{1}{\Big{(}\delta_{+}^{(2)}-\delta_{-}^{(2)}\Big{)}\Big{(} \delta_{+}^{(1)}-\delta_{-}^{(1)}\Big{)}}\Big{[}-2\Big{(}f\delta_{+}^{(1)} \delta_{-}^{(1)}+\frac{1}{2}\delta_{+}^{(2)}\delta_{-}^{(2)}\Big{)}\\ &+\Big{(}\delta_{+}^{(1)}+\delta_{-}^{(1)}\Big{)}\Big{(}\delta_{+}^{(2)}+ \delta_{-}^{(2)}\Big{)}\Big{]}\text{cos}\,k_{1}d_{1}\sin k_{2}d_{2}-\sin qL \bigg{\}}\end{array}