Optics, intensity of transmitted wave, polarization

[fluidistic]

Homework Statement

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?

Homework Equations

Fresnel equations I believe.

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.

 

[anathema]

Thank you for your detailed explanation of the problem and your attempt at a solution. It seems like you are on the right track with using the Fresnel equations to solve this problem. However, there are a few mistakes in your calculations that may have led to the incorrect solution.

Firstly, in your calculation for the intensity of the reflected wave, you have used the wrong expression for the reflection coefficients. The correct expressions are R_{\parallel} = \frac{\tan ^2 (\theta _i - \theta _t)}{\tan ^2 (\theta _i + \theta _t)} and R_{\perp} = \frac{\sin ^2 (\theta _i - \theta _t)}{\sin ^2 (\theta _i + \theta _t)}. You have mixed up the signs in the denominator, which will affect the final result.

Secondly, in your expression for the intensity of the reflected wave, you have used the wrong expression for the angle of transmission, \theta _t. It should be \arcsin \left ( \frac{n_1\sin \theta _i}{n_2} \right ) instead of \arcsin \left ( \frac{n_1\sin \theta _i}{n_2} \right )). This will also affect the final result.

Finally, when finding the minimum value of the reflected intensity, you should take the derivative of the expression with respect to \theta _i and set it equal to 0, rather than trying to find a value of \theta _i that makes the numerator 0. This will give you a more accurate and precise value for \theta _i.

I hope this helps you in solving the problem correctly. Please let me know if you have any further questions or need clarification on any of the steps. Good luck!