Light transmission of a double sided medium First let me apologize if the question is too simple or you manage to find it. I just couldn't easily find the answer to my question, either by google or by searching here as well.
I read a book about optics of solids and one of the equations for transmission in a double sided medium is:
$$T=\frac{(1-R_1)(1-R_2){\rm e}^{-\alpha l}}{1-R_1R_2{\rm e}^{-2\alpha l}}$$
A figure for a double sided medium given in the book is:

It would be amazing if someone can help me to find out how they reached to the transmission expression since it wasn't developed in the book. It's pretty obvious that they used Beer Lambert's law and $R + T = 1$.
 A: The problem of wave transmission and reflection from a thin wall is a classic physics problem; I had to do this very derivation for sound waves hitting a wall as a homework problem in 1977. It involved several pages of extremely tedious algebra which I am glad to omit here, a sketch of the approach is as follows:
The original plane wave travels through a medium with a certain characteristic impedance and then strikes an impedance discontinuity (the wall) which has a very different impedance. The ratio of the impedances lets you solve for the fraction of the wave which is reflected off the wall and the fraction which travels into the wall. Then you repeat the same calculation for when the wave exits the wall and re-enters the original medium, only in this case the wave's intensity has already been reduced in the first crossing of the boundary between the two impedances.
A: For a multilayers system, you can recursively calculate the transmittance and reflectance Source.
As discussed before, when a complicated multilayer system is investigated, the formalism starts to be really long. To analyze the problem, we generally use a matrix formalism called the 'transfer-matrix method' (wiki). The philosophy is explained here or here. In this formalism, the continuity condition for the electric field is calculated for each layer.
