Solving the Fresnel Equations with Polarization Parallel to Plane of Incidence

[chronite]

Homework Statement

The question involves the fresnel equations which I have derived. However, I seem to be missing something in the simplification. I arrive at these:

http://physics.tamuk.edu/~suson/html/4323/gifs/prop034.gif
http://physics.tamuk.edu/~suson/html/4323/gifs/prop035.gif


and I am trying to simplify to:
http://physics.tamuk.edu/~suson/html/4323/gifs/prop036.gif

The Attempt at a Solution

No matter how I use snell's law I can't seem to get them to simplify properly. Is there a trig identity that I'm missing? Currently I'm only interested in the coefficients if the field is polarized parallel to the plane of incidence.

Thanks for any input with the mathematics.

 

[2Tesla]

It's probably easier to start from the simplified forms and turn them into the unsimplified forms. Then you'll be able to see how to go the other way. The only trig identities you should need are:

$$sin(\theta1 - \theta2) = sin(\theta1)cos(\theta2) - sin(\theta2)cos(\theta1)$$
$$cos(\theta1 - \theta2) = cos(\theta1)cos(\theta2) + sin(\theta2)sin(\theta1)$$
(and the similar identities for addition)
$$sin^2(\theta) + cos^2(\theta) = 1$$
$$tan(\theta) = sin(\theta)/cos(\theta)$$

and Snell's law, of course. Good luck!

 

[chronite]

Thanks for the help!

I'm still having some difficulty. Not sure what I'm missing. For example, I keep ending up with:

r|| = (Sin[2*Theta1] - Sin[2*Theta2])/(Sin[2*Theta1] + Sin[2*Theta2])

I'm assuming Nair = 1.

 

[2Tesla]

But you're very close to the answer. Don't use the identity $$sin(2\theta)=2sin(\theta)*cos(\theta)$$, go back a step and write those terms out. Then, look at the equation you're trying to turn it into. It has a form like:

$$((something)*cos(\theta_i) - (something else)*cos(\theta_t)) / ((something)*cos(\theta_i) + (something else)*cos(\theta_t))$$

and your equation has this same form. Maybe you can find a way, by multiplying the numerator and denominator by the same thing and using Snell's law, to make them match?

 

[chronite]

Thanks for the help! I'm still not seeing something with this one:
$$(sin(\theta_i)+sin(\theta_i))*cos(\theta_i) - (sin(\theta_t)+sin(\theta_t))*cos(\theta_t)) / (sin(\theta_i)+sin(\theta_i))*cos(\theta_i) + (sin(\theta_t)+sin(\theta_t))*cos(\theta_t))$$

This is driving me nuts! I really appreciate your help.

 

[chronite]

Scratch that worked it out! Muchas Gracias!

 

[dibiz116]

thanks both of you!
i was just working on the same exact problem and having the same trouble.
this helped a lot.