Optics, intensity of transmitted wave, polarization

fluidistic

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 [tex]E/ \sqrt 2[/tex] 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 [tex]I_0[/tex].
2)For what value of [tex]\theta _i[/tex] 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 [tex]R_{\parallel } = \frac{\tan ^2 (\theta _i - \theta _t)}{\tan ^2 (\theta _i + \theta _t))}[/tex] while [tex]R_{\perp } = \frac{\sin ^2 (\theta _i - \theta _t)}{\sin ^2 (\theta _i + \theta _t))}[/tex] and that [tex]R=\frac{R_{\parallel}+R_{\perp}}{2}[/tex]. Using the fact that [tex]R=\frac{I_r}{I_0}[/tex] and using Snell's law for [tex]\theta _t[/tex], I reach that [tex]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 ) [/tex].
2)I must find the value of [tex]\theta _i[/tex] that minimizes the expression I got in 1). In my exam I said [tex]I_r[/tex] vanishes for [tex]\theta _i=0[/tex] (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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