$\vert f\vert<1$, $\eta$ is real and the disorder-averaged transmittance of a finite strip is smaller than 1. The total transmission can be obtained if $a_0$ is proportional to $u_0$ with the same proportionality constant $f$, such that $a(x)=f u(x)$. This case just corresponds to Eq. (\[eq:ktc\]) with both $u$ and $a$ as random functions. On the other hand, if $\vert f\vert\ge 1$, $\eta$ is either zero or imaginary and the wave decays exponentially.

Alternative derivation of the delocalization condition using the Fresnel formula
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Alternatively, we can derive the delocalization condition using the Fresnel formula for the reflection coefficient. We consider our stratified random medium as consisting of a large number of very thin strips. The reflection coefficient between two neighboring strips is written as $$\begin{aligned}
r=\frac{p/\epsilon-p^\prime/\epsilon^\prime-ik\left(\beta/\epsilon-\beta^\prime/\epsilon^\prime\right)}
{p/\epsilon+p^\prime/\epsilon^\prime+ik\left(\beta/\epsilon-\beta^\prime/\epsilon^\prime\right)},\end{aligned}$$ where $p$ ($p^\prime$) is the negative $x$ component of the wave vector in the first (second) strip with the parameters $\epsilon$ and $\beta$ ($\epsilon^\prime$ and $\beta^\prime$). Since $p$ satisfies $p^2+k^2\beta^2=k^2\epsilon^2$, we can introduce $\phi$ (and similarly $\phi^\prime$) such that $$\begin{aligned}
\frac{p}{\epsilon}=k\cos{\phi},~~\frac{\beta}{\epsilon}=\sin{\phi}.\end{aligned}$$ Then $r$ is written as $$\begin{aligned}
r=\frac{e^{-i\phi}-e^{-i\phi^\prime}}{e^{i\phi}+e^{-i\phi^\prime}},\end{aligned}$$ which vanishes when $\phi=\phi^\prime$, or equivalently, when $\beta/\epsilon=\beta^\prime/\epsilon^\prime$. If this condition is maintained throughout the system, we expect to have perfect transmission and delocalization.

