the plane wave (and exponentially decaying/diverging) solutions given by Eq. (\[eq:bogouniform\]) with $ \varphi=\pm\frac{\delta}{2} $.\
The solution of the scattering problem is defined by the one that has the following asymptotic form[@Kovrizhin; @Kagan; @DanshitaYokoshiKurihara]: $$\begin{aligned}
        \begin{pmatrix}u \\ v \end{pmatrix} \rightarrow \begin{cases}  w_1\left(x,-\frac{\delta}{2}\right)+r\, w_2\left(x,-\frac{\delta}{2}\right) & (x\rightarrow-\infty) \\ t\, w_1\left(x,+\frac{\delta}{2}\right) & (x\rightarrow+\infty). \end{cases} \label{eq:uvasymptotic}
    \end{aligned}$$ Here the exponentially decaying waves, which are $ w_4(x,-\frac{\delta}{2}) $ in $ x\rightarrow -\infty $ and $ w_3(x,+\frac{\delta}{2}) $ in $ x\rightarrow+\infty $, can be also included. However, they are irrelevant in the definition of transmission and reflection coefficients. The calculation of $ W $ shows that $$\begin{aligned}
        \frac{W(+\infty)}{2\mathrm{i}}=&|t|^2\left[ (k_{1}+p)|\bar{u}_{1}|^2+(k_{1}-p)|\bar{v}_{1}|^2 \right]\!, \\
        \begin{split}
        \frac{W(-\infty)}{2\mathrm{i}}=&(k_{1}+p)|\bar{u}_{1}|^2+(k_{1}-p)|\bar{v}_{1}|^2 \\
        &\ +|r|^2\left[ (k_{2}+p)|\bar{u}_{2}|^2+(k_{2}-p)|\bar{v}_{2}|^2 \right].
        \end{split}
    \end{aligned}$$ Since $ W(+\infty)=W(-\infty) $, the transmission coefficient $ T $ and the reflection coefficient $ R $ are naturally defined as $$\begin{aligned}
        T&=|t|^2, \label{eq:tc} \\
        R&=\frac{(-k_{2}-p)|\bar{u}_{2}|^2+(-k_{2}+p)|\bar{v}_{2}|^2}{(k_{1}+p)|\bar{u}_{1}|^2+(k_{1}-p)|\bar{v}_{1}|^2}|r|^2. \label{eq:defofR}
    \end{aligned}$$ By this definition, $ T+R=1 $ always holds. If one chooses the normalization $ \bar{u}_i $ and $ \bar{v}_i $ as Eq. (\[eq:ucoeff\]) and (\[eq:vcoeff\]), one can show $$\begin{aligned}
        \frac{(-k_{2}-p)|\bar{u}_{2}|^2+(-k_{2}+p)|\bar{v}_{2}|^2}{(k_{1}+p)|\bar{u}_{1}|^2+(k_{1}-p)|\bar{v}_{1}|^2} = 1-\frac{pp_{\text{L}}\epsilon^2}{2(p^2-p_{\text{L}}^2)^3}+O(\epsilon^4).
    \end{aligned}$$ So, if one is only interested in the leading order, one can approximate $ R \simeq |r|^2 $.

Summary of Main Result and Numerical Verification 