Loss of intensity in a prism due to transmissions and reflections Let's suppose that a beam of light polarized perpendicularly to the plane of incidence falls upon a prism of refractive index $n$ with intensity $I_0$, as shown in the image.

If we want to calculate the intensity that emerges through the point $3$, we will have to make the following calculations:


*

*Calculate the intensity $I^{(1)}_{t3}$ that comes out of $3$ after the path $1 \rightarrow 2 \rightarrow 3$: $I^{(1)}_{t3}=I^{(1)}_{i1}-I^{(1)}_{r1}-I^{(1)}_{t2}-I^{(1)}_{r3}$

*Calculate the intensity $I^{(2)}_{t3}$ that comes out of $3$ after the path $1 \rightarrow 2 \rightarrow 3 \rightarrow 2 \rightarrow 1 \rightarrow 2 \rightarrow 3$

*...


And then, you have to add them all up, so $I=\sum_{i=1}^{\infty}I^{(i)}_{t3}$. However, the problem becomes extremely tedious to solve as the number of faces increases. Do these phenomena have a special name or is there a general theory that describes them?
 A: Disrergarding interference (which would require you to compare the length of the paths to the wavelength of the light), you can simplify the description by only considering the intensity of the light on each segment of the path in each direction, never  caring about how many times it had been reflected inside the prism. Knoiwing the reflection and transition coefficients at each facet you can write a set of several equations that will link these intensities.
For example, if you have total internal reflection at point 2, you have 5 variables:
$$I_{in} = I_{1R}: \text{the incoming light; to the left side of point 1, going right} $$
$$I_{1L}: \text{the reflected light backtracking original ray; to the left side of point 1, going left} $$
$$I_{2R}: \text{the light in the prism, going right} $$
$$I_{2L}: \text{the light in the prism, going left} $$
$$I_{out} = I_{3R}: \text{the outgoing light; to the right side of point 3, going right} $$
Notably, there is no ingong light from the right side.
We will also use symbols $R_{ab}$ and $T_{ab}$, $a,b\in\{1,2,3\}$, denoting the reflection and transition coefficient affecting the light when it wants to pass from region $a$ to region $b$. Assuming no absorption and dispersion, we have $R_{ab}+T_{ab}=1$.
We have
$$I_{1L} = R_{12}I_{1R} + T_{21}I_{2L}$$
$$I_{2R} = T_{12}I_{1R} + R_{21}I_{2L}$$
$$I_{2L} = R_{23}I_{2R} $$
$$I_{3R} = T_{23}I_{2R} $$
By solving this set of equations we get, for example
$$I_{3R} = \frac{T_{12}T_{23}}{1-R_{21}R_{23}} I_{1R}$$
that is 
$$I_{out} = \frac{T_{12}T_{23}}{1-R_{21}R_{23}} I_{in}$$
If you don't assume the total full reflection at facet 2, the situation is a bit more complicated, but it can also be solved in a similar way.
A: The answer from Adam is fantastic. It made me realise the following which might also be of interest to you.
Your intuition about summing up all the contributions is correct. The final result is the same as an infinite sum of a geometric series,
$$I_{out} =I_{in} \frac{T_{12}T_{23}}{1-R_{21}R_{23}}= \frac{a}{1-r}$$
where $$a= I_{in}T_{12}T_{23}\\ r= R_{21}R_{23}$$
Here $a$ is the first term in the series, it is the intensity of which exits from the first pass, and $r$ is the common ratio which is a total reduction in intensity from entrance to exit for each full pass.
Knowing this you can generalise to $N$ surfaces,
$$a= I_{in}\prod_{i=1}^N T_{i,i+1} \\ r= \prod_{i=1}^NR_{i+1,i}$$
I looked at a related problem once in which light was attenuated by factor, $$e^{-\alpha d}$$ when travelling distance $d$ in the medium. Here $\alpha$ is the attenuation coefficient per unit length. This also has an analytical solution but the starting point was differential equations rather than series.
