random and the scalar potential is constant, it is more convenient to use the alternative expression Eq. (\[eq:imp3\]), from which a necessary condition for the nonrandomness of $\tilde\eta$ is seen to be $\epsilon=0$, or equivalently, $E=U$. In this case, however, the value of ${\tilde\eta}^2$ is $-1/\cos^2\theta$, and therefore the effective impedance $\tilde\eta$ is imaginary. Then the wave becomes evanescent and decays exponentially in a manner similar to localized waves. An exception to this occurs when $\beta_0$ ($=\sin\theta+a_0$) is also zero simultaneously. In later sections, we will show analytically and numerically that the localization length in this case indeed diverges.

Finally, when both $u$ and $a$ are random, a necessary condition for $\eta$ to be nonrandom is that $\delta a(x)$ and $\delta u(x)$ have to be proportional to each other, such that $\delta a(x)=f\delta u(x)$, as in our Model II. Then the condition that $\beta/\epsilon$ is uniform gives $$\begin{aligned}
\sin\theta_S=-a_0-f(1-u_0),
\label{eq:angle2}\end{aligned}$$ where the constant in the right-hand side should satisfy $\vert a_0+f(1-u_0)\vert<1$. In this case, we define $\theta_S$ as the delocalization angle. We also need an additional constraint $\vert f\vert <1$ to have a propagating wave. The effective impedance at $\theta_S$ given by $$\begin{aligned}
\eta^2=\frac{1-f^2}{\cos^2\theta_S}\end{aligned}$$ is not equal to 1. If 