transmittance $T$ from $$\begin{aligned}
&&R=\vert r\vert^2,\nonumber\\
&&T=\left\{\begin{matrix} \frac{\mbox{sgn}(\epsilon_2)\sqrt{{\epsilon_2}^2-{\beta_2}^2}}
{\epsilon_2\cos\theta}\vert t\vert^2 &\mbox{if }{\epsilon_2}^2\ge {\beta_2}^2 \\
0 & \mbox{if }{\epsilon_2}^2< {\beta_2}^2 \end{matrix}.\right.\end{aligned}$$ With these definitions, $R$ and $T$ satisfy the law of energy conservation $R+T=1$.

One of the main advantages of using the IIM is that the disorder averaging can be performed analytically in an exact manner. We use Eq. (\[eq:rt\]) to calculate the disorder averages of various physical quantities consisting of $r$ and $t$ exactly. In this paper, we are mainly interested in the localization length $\xi$ defined by $$\begin{aligned}
\xi=-\lim_{L\rightarrow\infty}\left(\frac{L}{\langle \ln T\rangle}\right).\end{aligned}$$ The $\it nonrandom$ differential equation satisfied by $\langle \ln
T\rangle$ can be obtained using the second of Eq. (\[eq:rt\]) and Novikov’s formula[@nov] and, in the case of Model I, takes the form $$\begin{aligned}
-\frac{1}{k}\frac{{d\langle \ln T\rangle}}{dl}&=&C_1+{\rm Re}\left[2e^{i\theta}\left(C_0
- iC_2\right)Z_1\right.\nonumber\\&&\left. -e^{2i\theta}C_1Z_2\right],
\label{eq:iie1}\end{aligned}$$ where $Z_n$ ($n=1,2$) is equal to $\langle r^n\rangle$ and the parameters $C_0$, $C_1$ and $C_2$ are defined by $$\begin{aligned}
C_0&=&u_0\tan{\theta}+a_0\sec{\theta},~
C_1=g_u\tan^2{\theta}+g_a\sec^2{\theta},\nonumber\\
C_2&=&\left(g_u+g_a\right)\sec{\theta}\tan{\theta}.\end{aligned}$$ The $\it dimensionless$ disorder parameters $g_u$ and $g_a$, which can take any arbitrary nonnegative real values, are defined by $$g_u=\tilde g_u k,~g_a=\tilde g_a k.
\label{eq:ga2}$$ In general, our method can be applied to the case where $u_0$, $a_0$, $g_u$ and $g_a$ are arbitrary nonrandom functions 