imaginary, which leads to a finite decay rate independent of the disorder strength $g_a$ in the $g_a\rightarrow\infty$ limit. The analytical expression of the localization length in the large disorder limit to be derived in the next section will show just such a behavior.

Finally, in the case where $\delta a(x)$ and $\delta u(x)$ are proportional to each other such that $\delta a(x)=f\delta u(x)$, we find from Eq. (\[eq:imp1\]) that in the strong disorder limit, the impedance approaches a constant given by $\eta^2\approx (1-f^2)/\cos^2\theta$. When $\vert f\vert$ is smaller than 1, this constant is real and we have a complete delocalization and a diverging localization length for all $\theta$ in the large disorder limit, whereas, when $\vert f\vert\ge 1$, the impedance is either zero or imaginary. In this latter case, the localization length approaches a constant as the disorder parameter increases to infinity. These behaviors will be also confirmed in the next sections.

Analytical expressions for the localization length in the weak and strong disorder regimes {#sec:form}
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Weak disorder regime
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Starting from the invariant imbedding equations, Eqs. (\[eq:iie1\]), (\[eq:iie2\]), (\[eq:iie3\]) and (\[eq:iie4\]), and applying the perturbation theory, it is possible to derive accurate analytical expressions for the localization length 