Normal incidence on 2 dielectric boundaries

[Robsta]

Homework Statement

I've not been able to do this question for years so I'd really appreciate some help.

Light is normaly incident from a medium 1 with impedance Z₁ through a layer of medium 2 of thickness L and impedance Z₂ into medium 3 of impedance Z₃. Obtain an expression for the total reflected intensity of light when the thickness corresponds to:
a) λ₂/4
b) λ₂/2

where λ₂ is the wavelength of the light in medium 2.

Homework Equations

I know that the tangential components of the E and B fields are continuous over the boundaries so at each interface:

E_1f + E_1b = E_2f +E_2b = E_3f
H_1f + H_1b = H_2f +H_2b = H_3f

where _f and _b signify forward and backward going waves

Also, E_1f/H_1f = Z₁
And E_1b/H_1b = -Z₁

I know how to do this for the one boundary problem where you get r = (Z₂ - Z₁)/(Z₁+Z₂) and t = 2Z₂/(Z₂+Z₁) But I can't apply it to the first boundary as now there's a wave coming the other way being reflected from the second boundary. It all seems intractably complicated but I know this is a fairly standard problem, so I'm finding it really frustrating.

3. The Attempt at a Solution

I don't really even know how to start. The wave in the middle material will bounce back and forwards seemingly infinitely and I don't know how to deal with that. I've just said in the middle there's the sum of a forward going and a backward going wave. I don't really know if the equations above are right for the middle section.

 

[ehild]

You find the answer in any book about thin film optics. Vasicek, Heavens fro example. Or Knittel https://archive.org/details/OpticsOfThinFilms.
If you have still problems, I write about it later.
All the multiple reflected waves add up to a wave traveling forward and an other, traveling backward. You can set up the relation between the waves in front of an interface and behind that interface, and also at the two boundaries of a medium. All these relationship can be written in matrix from.