# Reflection coefficient as a function of frequency

I am trying to relate the equation for reflection coefficient in oblique mediums to the frequency but can't figure out how the frequency affects the reflection of light.

$n_1$= intrinsic impedance of medium 1

$n_2$= intrinsic impedance of medium 2

$\theta_1$ = incident angle

$\theta_2$= transmited angle

$$\Gamma= \frac{n_2 \cos(\theta_1) -n_1 \cos(\theta_2)}{n_2 \cos(\theta_1) + n_1 \cos(\theta_2)}$$

Can somebody explain to me how we can make this equation a function of frequency?

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• What do you mean by an "oblique medium"? Searching online it seems to refer to a type of fountain pen nib. – The Photon Nov 30 '14 at 5:54
• by "oblique mediums", I believe OP just means permeable or semi-permeable materials. This would be a better question for the Physics.SE, as this SE is more concerned with electronics design rather than the theory. – Funkyguy Nov 30 '14 at 6:31
• Theres a proper derivation ( long, and in most textbooks). You need to consider refractive index for this instead of impedance, because refractive index is directly dependent on wavelength (and hence frequency). – Plutonium smuggler Nov 30 '14 at 7:03
• The Fresnel equations are not frequency dependent, other than through the dependence on frequency of the two refractive indices. – Rob Jeffries Nov 30 '14 at 22:09

The reflexion co-efficient equation you have given is part of the Fresnel Equations. There are two ways wherein this equation varies with frequency: ultimately both down to the variation of the refractive indices $n_1$ and $n_2$ with frequency. Clearly the equation you have written can vary with frequency if $n_1$ and $n_2$ do so: also $\theta_2$ does because this is governed by Snell's law $n_2\,\sin\theta_2 = n_1\,\sin\theta_1$. So you're full variation is:
$$\Gamma(f,\,\theta_1) = \frac{n_2(f) \cos(\theta_1) -n_1(f) \cos\left(\arcsin\left(\frac{n_1(f)}{n_2(f)}\sin\theta_1\right)\right)}{n_2(f) \cos(\theta_1) + n_1(f) \cos\left(\arcsin\left(\frac{n_1(f)}{n_2(f)}\sin\theta_1\right)\right)}=\frac{n_2(f)^2 \cos(\theta_1) -n_1(f) \sqrt{n_2(f)^2-n_1(f)^2\sin\theta_1^2}}{n_2(f)^2 \cos(\theta_1) +n_1(f) \sqrt{n_2(f)^2-n_1(f)^2\sin\theta_1^2}}$$
Note that $\Gamma$ is a function of both $f$ and $\theta_1$: if there is only one interface, then of course $\theta_1$ will be fixed with frequency (it's set by how you aim the beam at the interface. But if there are many interfaces, the incidence angle on each will be set by the refracted transmission angle from the former interface in the stack, so it will then also vary with frequency.
First, for oblique incidence one has to distinguish the transverse electrical mode (TE) and the transverse magnetical (TM) polarisation. You give the Fresnel equation for the TE reflectivity. However you denote the impedamdanve by $$n$$ in place of, for example, $$Z$$. $$n$$ commonly denotes the refractive index and this is confusing. If $$n$$ would denote the refractive index then your equation would give the reflectivity if the TM mode instead.