$q$ and $p$, are related to $\theta$ by $q=k\sin\theta$ and $p=k\cos\theta$. Once $\psi_A$ is obtained from the IIM, $\psi_B$ can be calculated using $$\begin{aligned}
\psi_B=-\frac{i}{k\epsilon}\frac{d\psi_A}{dx}+i\frac{\beta}{\epsilon}\psi_A.\end{aligned}$$

Starting from Eq. (\[eq:dirac\]), we are able to derive [*exact*]{} differential equations satisfied by $\mathit{r}$ and $\mathit{t}$ using the IIM developed in Ref. 45, which have the forms $$\begin{aligned}
&&\frac{1}{k}\frac{dr}{dl}=-e^{-i\theta}(\epsilon\tan{\theta}-\beta\sec{\theta})+2i(\epsilon\sec{\theta}-\beta\tan{\theta})r
\nonumber\\ && ~~~~~~~~~+e^{i\theta}(\epsilon\tan{\theta}-\beta\sec{\theta})r^2,\nonumber\\
&&\frac{1}{k}\frac{dt}{dl}=i(\epsilon\sec{\theta}-\beta\tan{\theta})t+e^{i\theta}(\epsilon\tan{\theta}-\beta\sec{\theta})rt.
\label{eq:rt}\end{aligned}$$ We can obtain $r$ and $t$ by integrating these equations numerically from $l=0$ to $l=L$ with the initial conditions $r(0)=0$ and $t(0)=1$.

If the potentials are nonzero in the transmitted region where $x<0$, we need to solve the same differential equations with the modified initial conditions obtained using the Fresnel formulas, which take the form $$\begin{aligned}
r(0)= \frac{e^{-i\theta}-Q}{e^{i\theta}+Q}, ~~
t(0)= \frac{2\cos\theta}{e^{i\theta}+Q},
\label{eq:refc}\end{aligned}$$ where $Q$ is defined by $$\begin{aligned}
&&Q=\frac{p_2}{k\epsilon_2}-i\frac{\beta_2}{\epsilon_2},\nonumber\\
&&\epsilon_2=1-u_2,~~\beta_2=\sin\theta+a_2.\end{aligned}$$ The parameters $u_2$, $a_2$, $\epsilon_2$ and $\beta_2$ are respectively the values of $u$, $a$, $\epsilon$ and $\beta$ in the region $x<0$ and $p_2$ is the negative $x$ component of the wave vector in the same region. Since $p_2$ satisfies ${p_2}^2+k^2{\beta_2}^2=k^2{\epsilon_2}^2$, we can calculate $p_2$ from $$\begin{aligned}
\frac{p_2}{k}=\left\{\begin{matrix} \mbox{sgn}(\epsilon_2)\sqrt{{\epsilon_2}^2-{\beta_2}^2} &\mbox{if }{\epsilon_2}^2\ge {\beta_2}^2 \\
i\sqrt{{\beta_2}^2-{\epsilon_2}^2} & \mbox{if }{\epsilon_2}^2< {\beta_2}^2 \end{matrix}.\right.\end{aligned}$$ After obtaining $r$ and $t$, we calculate the reflectance $R$ and the 