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## Optics, intensity of transmitted wave, polarization

1. The problem statement, all variables and given/known data
A linearly polarized wave incidates over a surface of a material with a higher refractive index than the incident media one. See picture for clarification. The polarization is such that the E field isn't perpendicular nor parallel to the plane of incidence. Rather, it's making an angle of 45° with it so it has a component that lies inside the plane of incidence and another one that is perpendicular to it, both have the same modulus, namely $$E/ \sqrt 2$$ if I didn't misunderstood the picture we had in the exam. (I'm writing the problem out of my memory and drawing their sketch is simply too hard so I just explain what it consist of).
1)Determine the intensity of the reflected wave if the intensity of the incident wave is $$I_0$$.
2)For what value of $$\theta _i$$ is the intensity of the reflected wave minimum?

2. Relevant equations
Fresnel equations I believe.

3. The attempt at a solution
1)I look at Hecht's book on Optics, page 345 (third edition I think). He writes $$R_{\parallel } = \frac{\tan ^2 (\theta _i - \theta _t)}{\tan ^2 (\theta _i + \theta _t))}$$ while $$R_{\perp } = \frac{\sin ^2 (\theta _i - \theta _t)}{\sin ^2 (\theta _i + \theta _t))}$$ and that $$R=\frac{R_{\parallel}+R_{\perp}}{2}$$. Using the fact that $$R=\frac{I_r}{I_0}$$ and using Snell's law for $$\theta _t$$, I reach that $$I_r=\frac{I_0}{2} \left ( \frac{\tan ^2 (\theta _i - \arcsin \left ( \frac{n_1\sin \theta _i}{n_2} \right ) )}{\tan ^2 (\theta _i + \arcsin \left ( \frac{n_1\sin \theta _i}{n_2} \right ))} + \frac{\sin ^2 (\theta _i - \arcsin \left ( \frac{n_1\sin \theta _i}{n_2}}{\sin ^2 (\theta _i + \arcsin \left ( \frac{n_1\sin \theta _i}{n_2})} \right ) \right )$$.
2)I must find the value of $$\theta _i$$ that minimizes the expression I got in 1). In my exam I said $$I_r$$ vanishes for $$\theta _i=0$$ (and I wrote it did NOT convince me, I was expecting something similar to Brewster's angle) since the numerator is worth 0. However I got 0 in my exam and now I realize that the denominator also vanishes so it's not that easy to minimize.
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Since I got no point for this exercise, it means I did it all wrong. I'd like to know what I did wrong and how to solve the problem.
Thank you very much for your time and help.
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