# I Overview of General Fresnel Equations + Complex IORs

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1. Jul 10, 2017

### Geometrian

Hi,

My understanding is that when light (with some frequency and polarization) hits the interface between two media (each with some frequency-dependent material properties), the Fresnel equations apply. This tells us how much light reflects back versus refracts across the interface.

I'm looking for confirmation that this is accurate, plus the next level down of details. Specifically, I want to know that Fresnel equations for any wavelength, polarization, and type of material, giving me the reflection, transmissions, and polarizations of the reflected and transmitted rays. But, I may be getting ahead of myself; I can think of several things I'm unclear on:

• The formula given on Wikipedia:$$R_s = \left|\frac{ \sqrt{\frac{\mu_2}{\epsilon_2}} \cos \theta_i - \sqrt{\frac{\mu_1}{\epsilon_1}} \cos \theta_t }{ \sqrt{\frac{\mu_2}{\epsilon_2}} \cos \theta_i + \sqrt{\frac{\mu_1}{\epsilon_1}} \cos \theta_t }\right|^2\\ R_p = \left|\frac{ \sqrt{\frac{\mu_2}{\epsilon_2}} \cos \theta_t - \sqrt{\frac{\mu_1}{\epsilon_1}} \cos \theta_i }{ \sqrt{\frac{\mu_2}{\epsilon_2}} \cos \theta_t + \sqrt{\frac{\mu_1}{\epsilon_1}} \cos \theta_i }\right|^2$$
How general is it? In particular, the discussion sounds like it applies to all materials; is this the case? Also, can this formula be applied to compute for light of any polarization (not just "s" and "p")? If not to either, then what is the general form?

• I've learned that the quantity $Z=\frac{\mu}{\epsilon}$ is called the "wave impedance". I'm struggling to see how this relates to the complex-valued refractive index $\underline{n}=n+i\kappa$. I found a derivation (which I couldn't quite follow) that suggests that $Z_2=\underline{n_1}/\mu_1$ (and vice-versa). That works for the general case (i.e., having $\underline{n}$ complex-valued and dividing by $\mu$ works for all materials), right?

• I'm not familiar with representations of polarization (although I've encountered several that I plan to investigate more fully), but how does the polarization of the reflected and transmitted light relate to that of the incident light?

• What is the relationship between "non-magnetic" ($\mu\approx\mu_0$) and "dielectric"?

Thanks!

-G

2. Jul 11, 2017

$Z=\sqrt{\frac{\mu}{\epsilon}}$. The Fresnel relations also work with $n$ replaced by $\frac{1}{Z}$ because for most materials $\mu=\mu_o$, and index of refraction $n$ is proportional to $\sqrt{\epsilon}$. The complex impedance $Z$ is commonly used in r-f problems on transmission lines and coaxial cables, while the optics people prefer to work with index of refraction $n$. The good thing is you only need to learn the formulas once (for normal incidence it pays to memorize them), and you can replace $n$ by $\frac{1}{Z}$ in going from the optics case to the r-f case. $\\$ For normal incidence, polarization is not a factor, and reflection coefficient $\rho=\frac{E_r}{E_i}=\frac{n_1-n_2}{n_1+n_2}$, and transmission coefficient $\tau=\frac{E_t}{E_i}=\frac{2 n_1}{n_1+n_2}$. It also helps to know that intensity $I=n \, E^2$ other than some proportional constants, and energy reflection coefficient $R=|\rho|^2$.

Last edited: Jul 11, 2017