By consideration of the formulae for the reflection and transmission of up to three films, it has been possible to form a generalization for an arbitrary number of films. This generalization is stated as a simple rule whereby the expressions for the reflected and transmitted amplitudes can be readily formed in terms of the Fresnel coefficients of the various boundaries. The generalization is proved by induction. It is also shown that the solutions of the problem by the consideration of the electromagnetic disturbance in each medium, and by the summation of multiple beams, lead to the same result.
A. W. CROOK, "The Reflection and Transmission of Light by Any System of Parallel Isotropic Films," J. Opt. Soc. Am. 38, 954-963 (1948)