equations,[@kly; @kim1; @kim2; @kim3; @kim4; @kim5] we also derive the invariant imbedding equations for the reflection and transmission coefficients. By applying a stochastic averaging technique to them and solving the resulting equations numerically, we calculate the localization length exactly for an arbitrary strength of disorder. In addition, by applying the perturbation expansion method to the invariant imbedding equations, we derive concise analytical expressions for the localization length, which are extremely accurate in the weak and strong disorder regimes.

One of the main advantages of the IIM for the study of localization is that it is possible to perform the disorder averaging analytically in an exact manner and convert the random problem to an equivalent deterministic one. In other studies, disorder was introduced in the superlattice model by assigning the values of the potential and the layer widths randomly. The disorder averages of various quantities were obtained by repeating a large number of calculations for many random configurations and averaging over the results. In order to obtain reliable disorder averages using this type of method, one usually needs to do the calculations for a very large number of configurations.

Using our approaches, we calculate the dependencies of the localization length on 