Proving polarization of reflected wave in oblique incidence

[yungman]

As shown in the attachment, the book assumes $\hat E_{||}^r=\hat y_r=(\hat x \cos \theta_r +\hat z \sin\theta_r)$. Why? How do you proof this. I have another post here about the Normal Incidence and no luck.

I am not even convinced that the reflected E is even in the Plane of Incidence, how do you even proof this. I have 5 EM books only Griffiths even attempt to proof in a way I don't even agree for the Normal Incidence. Every book pretty much just give the polarity. If anyone have article or notes, please share with me.

Thanks

Alan

 

[sudu.ghonge]

Try solving the problem there. I faced the very same hurdle. The boundary conditions beautifully bring out the result

 

[yungman]

Try solving the problem there. I faced the very same hurdle. The boundary conditions beautifully bring out the result

Actually this is exactly what I was struggling with in the other post:
https://www.physicsforums.com/showthread.php?t=694386

I am going no where, I spent like two days thinking about that problem already, please take a look and give me some insight. I have the solution manual and I just don't agree with it. The hint make an assumption the the reflected wave is tangential as a starter and where is the proof that the reflected wave is even tangential to the boundary?

For me, the proof I can accept is if $\hat n_R=\hat x \cos \theta_R+\hat y \sin\theta_R+ \hat z f(\theta_R)$ and proof that y and z component are both zero. Or better yet, proof reflection of a TEM wave is also a TEM wave.

Thanks