Solving Light Refraction Boundary Conditions: Find T^-1

[drcrabs]

Can someone please help me with this question:

Light with frequency $$\omega$$ in media 1 ,with refractive index $$n_{1}$$ , is incident (normal) to an interface of media 2, with refractive index $$n_{2}$$, and then is incident on a second interface with refractive index $$n_{3}$$. Using boundary conditions show that the transmission coefficient is:

$$T^{-1} = \frac{1}{4n_1n_3} ((n_1+n_3)^2 + \frac{(n_1^2-n_2^2)(n_3^2-n_2^2)}{n_2^2} Sin(\frac{n_2d\omega}{c}))$$

So basically light starts in one media and passes though two different media and we get the above as the transmission coefficient

 

[Ahmes]

If I understand correctly, T is what comes out the n2 area, to n3 and infinity?
The point in this question is this:
1. find the genreal Fresnel Coefficients for reflection and transmition between two medias (you can find it around page 306 in Jackson 3re ed.).
2. Now look at your own wave - it is transmitted into n2, some passes into n3 but some reflects from the n2-n3 boundary back to n2, some passes back to n1 and some again is reflected to n3. The wave is reflected an infinite amount of times and there is an infinite sum involved of all the transmitted waves. Luckily, it's a geometric series.
P.S you must assume the media is non-magnetic to get this expression.
3. Don't forget the phase added to the wave while passing n2.

 

[Meir Achuz]

That problem is worked out on page 285 of Franklin "Classical Electromagnetism". The sin should be sin^2.