total transmission can arise only when the parameter $\beta/\epsilon$ is a constant independent of $x$. In that case, the expression for $\eta$ is simplified to $$\begin{aligned}
\eta^2=\frac{1}{\cos^2\theta}\left[1-\left(\frac{\beta}{\epsilon}\right)^2\right].
\label{eq:imp2}\end{aligned}$$

Sometimes, it is more convenient to rewrite Eq. (\[eq:dirac\]) as another alternative form $$\begin{aligned}
\frac{d}{dx}\left(\frac{1}{\beta}\frac{d\psi_A}{dx}\right)-\frac{1}{\epsilon}\frac{d}{dx}\left(\frac{\epsilon}{\beta}\right)
\frac{d\psi_A}{dx}+p^2\beta\tilde\eta^2\psi_A=0,\end{aligned}$$ where $\tilde\eta$ is given by $$\begin{aligned}
{\tilde\eta}^2=\frac{1}{\cos^2\theta}\left[\left(\frac{\epsilon}{\beta}\right)^2+\frac{1}{k\epsilon}\frac{d}{dx}
\left(\frac{\epsilon}{\beta}\right)-1\right].
\label{eq:imp4}\end{aligned}$$ When $\epsilon/\beta$ is independent of $x$, these equations are simplified to $$\begin{aligned}
&&\frac{d}{dx}\left(\frac{1}{\beta}\frac{d\psi_A}{dx}\right)+p^2\beta\tilde\eta^2\psi_A=0,\nonumber\\
&&{\tilde\eta}^2=\frac{1}{\cos^2\theta}\left[\left(\frac{\epsilon}{\beta}\right)^2-1\right].
\label{eq:imp3}\end{aligned}$$ Again, the uniformity of $\epsilon/\beta$ implies that of the effective wave impedance $\tilde\eta$.

Generalized Klein tunneling in inhomogeneous potentials {#sec:kl}
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The condition for total transmission is obtained by setting $\eta$ equal to 1 in Eq. (\[eq:imp2\]). We obtain two solutions, which are $a=-u\sin\theta$ and $a=(u-2)\sin\theta$, respectively. Only the first of these two is the real solution because, in the second case, $\beta/\epsilon$ changes discontinuously from $\sin\theta$ to $-\sin\theta$ as the wave enters obliquely from the incident region and the derivative term in Eq. (\[eq:imp1\]) is nonzero. Therefore the expression for the incident angle $\theta_K$ at which the transmittance is identically equal to 1 becomes $$\begin{aligned}
\sin\theta_K=-\frac{a}{u}=-\frac{ev_FA_y}{U}.
\label{eq:ktc}\end{aligned}$$

If there is no vector potential, then this condition is satisfied at $\theta_K=0$ for any arbitrary functional form of $U(x)$, which corresponds to the 