linear vector potential corresponds to an external magnetic field of $\vert B_0\vert=Ea_m/(ev_FL)$ in the $-z$ direction, while the linear scalar potential corresponds to an external electric field of $\vert {\mathcal E}_0\vert=Eu_m/(eL)$ in the $-x$ direction. Since the ratio of $a$ and $u$ is a constant, we find $\theta_K=-60^\circ$, $-30^\circ$, $0^\circ$, $30^\circ$ and $60^\circ$ when $u_m$ and $a_m$ are given by $u_m=2$ and $a_m=1.732$, 1, 0, $-1$ and $-1.732$, which is clearly verified by the sharp transmission peaks shown in Fig. \[fig:a\]. These peaks become sharper as the thickness of the nonuniform region, $L$, gets larger. The angle $\theta_K$ can be tuned easily by tuning either the external magnetic field or the external electric field. We notice that our system in the present geometry can function as a very efficient directional filter.

Delocalization condition in the presence of random scalar and vector potentials {#sec:deloc}
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Now we return to the case where the potentials are random. We are interested in deriving the delocalization condition, which is in general distinct from the total transmission condition, Eq. (\[eq:ktc\]). The wave is delocalized and the localization length diverges only if the effective impedance $\eta$ is nonrandom, whereas the total transmission occurs when $\eta$ is 