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I posted a similar question a while back but never got a decent answer, most probably due to having too many details that obscured the real problem. This time around I'll try it more simply.

Fresnel's equation for reflectivity for s- and p-polarization are, according to Wikipedia, given by:

[tex]R_s = \left( \frac{n_1 \cos \theta - n_2 \sqrt{1 - \left( \frac{n_1}{n_2} \sin \theta \right)^2}}{n_1 \cos \theta + n_2 \sqrt{1 - \left( \frac{n_1}{n_2} \sin \theta \right)^2}} \right)^2[/tex]

[tex]R_p = \left( \frac{ n_1 \sqrt{1 - \left( \frac{n_1}{n_2}\sin\theta \right)^2} - n_2 \cos\theta }{ n_1 \sqrt{1 - \left( \frac{n_1}{n_2}\sin\theta \right)^2} + n_2 \cos\theta } \right)^2[/tex]

Now suppose we have a vacuum to metal interface so that the light travelling through the vacuum strikes and reflects off a metal surface. I think in this case the vacuum should have an index of refraction equal to unity while the metal gets a complex index of refraction: [itex]n_1 = 1[/itex] and [itex]n_2 = n(1 + i \kappa) = n + ik[/itex] where [itex]k[/itex] is the extinction coefficient (or is it [itex]\kappa[/itex]?).

Apparently, by substituting these values andmaking one assumption, that [itex]n^2+k^2 \gg 1[/itex], one should be able to arrive at the following much simpler expressions:

[tex]R_s = \frac{\left( n - \cos \theta \right)^2 + k^2}{\left( n + \cos \theta \right)^2 + k^2}[/tex]

[tex]R_p = \frac{\left( n - \sec \theta \right)^2 + k^2}{\left( n + \sec \theta \right)^2 + k^2}[/tex]

Graphing these formulas tells me that [itex]R_p[/itex] has a minimum reflectivity at some high angle, similar to non-metals (where this minimum actually goes to 0 for the Brewster angle) while [itex]R_s[/itex] is a smoothly increasing function.

The problem is as follows... I have been looking everywhere for a derivation, or even a simple statement of these formulas but I haven't found a decent reference anywhere. The only times I can find solid references to the reflectivity of a metal is for normal incidence, for [itex]\theta = 0[/itex] so that both [itex]\cos \theta = \sec \theta = 1[/itex], so that doesn't help.

I did find a few other references, but they either contradict themselves (describe in detail that Rp has the minimum in reflectivity, then show a graph and write Rs in it ...?!) or contradict other papers (some seem to mention Rp has the minimum, some mention it is Rs).

Some papers / books also don't mention s- or p-polarization at all but merely TE or TM modes. The same problem here: they contradict each other, some papers say Rs = TE, others seem to say Rs = TM...

The questions are simple (but maybe not to answer):

1. What's the correspondance between TE/TM mode and s-/p-polarization? Or is there no one definition and does it depend on something else..? I would think TM = p polarization and TE = s polarization, or vice versa. Which is it?

Same goes for perpendicular and parallel polarization. Perpendicular = TE or TM, s or p?

My view is this:

s (senkrecht) = perpendicular = TE

p (parallel) = parallel = TM.

Correct, or not?

2. Are the final formulas I've given for Rs and Rp correct, or did I switch them around (formula for Rs labeled Rp and vice versa)?

3. Finally: how does one get to that result.....? I don't see how the assumption that n^2+k^2 is large is any help, I've been trying for days to derive it but I am getting nowhere.

I did find one other 'derivation' (not in detail, which is what I want to see) which arrived at

[tex]R_p = \frac{(n^2+k^2) \cos^2 \theta - 2n\cos\theta + 1}{(n^2+k^2) \cos^2 \theta + 2n\cos\theta + 1}[/tex]

after dividing out [itex]\cos^2 \theta[/itex] and completing the square that does yield [itex]R_p[/itex]. The problem is that another statement (without derivation) I've got shows this to be [itex]R_s[/itex] instead?!

Any help is MUCH appreciated! Thanks!!

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# Fresnel's equation with complex n

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