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When light gets reflected or transmitted by another medium it's intensity splits for both the parallel and perpendicular component. Using Fresnel's equations calculating the reflected intensity which comes from the light staying in the same medium is not so difficult because the geometry and speed of the light doesn't change. (It would simply be the square of the reflection coefficient.)

But for the transmitted light this will change and I was wondering how this can be described mathematically.

With that I would also be able to prove that(as stated above) when light gets in contact with another medium the intensitiy from the incoming light equals the sum of the reflected and the transmitted intensities.

$I_0 = I_t + I_r$

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closed as off-topic by Buzz, ZeroTheHero, Jon Custer, Kyle Kanos, Gert Feb 5 at 19:42

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Check out this site

http://hyperphysics.phy-astr.gsu.edu/hbase/phyopt/freseq.html

And you will see the correct equation you are going for is

$r^2+t^2\frac{ (n2 \cos (\theta t))}{n1 \cos (\theta i)}=1$

For either perpendicular or parallel components. The second term gives the actual transmission coefficient.

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