Fresnel coeff

chronite

1. The problem statement, all variables and given/known data

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





and I am trying to simplify to:







3. 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:

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

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 [tex]sin(2\theta)=2sin(\theta)*cos(\theta)[/tex], 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:

[tex] ((something)*cos(\theta_i) - (something else)*cos(\theta_t)) / ((something)*cos(\theta_i) + (something else)*cos(\theta_t)) [/tex]

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:
[tex] (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)) [/tex]

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.