# Fresnel equations at normal incidence

1. ### brianpile

5
Hello everybody, I have a question about the Fresnel equations at normal incidence. My textbook gives the reflection coefficients as:

(1) perpendicular polarization

rs=( n1*cos(theta1)-n2*cos(theta2) ) / ( n1*cos(theta1)+n2*cos(theta2) )

(2) parallel polarization

rp=( n2*cos(theta1)-n1*cos(theta2) ) / ( n1*cos(theta1)+n2*cos(theta2) )

Now, to me, it seems that for normal incidence on a dielectric interface, the polarization loses its meaning since neither the E- or H-fields have components normal to the surface. Put another way, the plane of incidence has lost its meaning. In this case, shouldn't the 2 Fresnel equations for reflection give the same result? If you put 0 degrees in to those two equations they have opposite signs. What gives?

I'm probably spending too much time think about this, but any help making sense of this would be greatly appreciated!!

Thanks,
Brian

2. ### DiracRules

111
Wow, I didn't consider this aspect!!! Thank you for pointing this out! I'm studying these things right now.

Well, I think that you are right, the plane of incidence loses its meaning.
THOUGH, the thing here maybe another one - I say maybe because I've been thinking of it for 10 minutes, but as it makes sense to me I'll write it here.
The information from the reflection coefficients is not only about the amplitude of the reflected wave, but also about its phase.
Now, despite the lacking of a plane of incidence, at normal incidence the coefficients still have to tell you that the electric component has a 180° phase shift, while the magnetic one don't. This is why, I think, you get that the coefficients are opposite: as you say, there is no difference as regards the plane of incidence (the amplitudes of the reflected wave are the same), but there is still a difference in the phase of the reflected wave.

I think this is the explanation. I'm not sure, but it makes sense.Hope it is clear.

3. ### Redbelly98

12,037
Staff Emeritus
Welcome to PF.

It's a matter of the sign conventions used for the two polarizations.

For parallel polarization, rp is positive when E has an upward component for both the incident and reflected beams:

If you imagine the angle of incidence approaching zero (normal incidence case), this means that the incident and reflected electric fields are actually pointing in opposite directions.

For perpendicular polarization, rs is positive when E is in the same direction for both the incident and reflected beams. Changing the angle of incidence does not change the direction of E.

4. ### brianpile

5
Ok I see, thanks RedBelly. So to be consistent between the two polarizations, I should define the p-polarization as positive when the magnetic field receives a pi phase shift (and therefore the E-field does not).

5. ### Redbelly98

12,037
Staff Emeritus
I don't think that's right. In the figure I posted earlier, B would point in the same direction -- up, out of the page -- for both the incident and reflected waves.

6. ### brianpile

5
Right, I wasn't clear. If we defined the case when the H-field does change direction upon reflection (n1>n2) then the situation looks like this.

http://dl.dropbox.com/u/1505234/Fresnel Eqs - p-polariozation.bmp

and the boundary condition for the magnetic field would be Bi-Br=Bt. I think this change allows the signs of the reflection coefficients agree at normal incidence....like you said, just a convention.