Overview of General Fresnel Equations + Complex IORs

[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 http://iqst.ca/quantech/pubs/2013/fresnel-eoe.pdf (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

 

[Charles Link]

$Z=\sqrt{\frac{\mu}{\epsilon}}$. 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 $R=|\rho|^2$.

 

[Novgorod]

Sorry to hijack this thread, but my question directly relates to the topic. I want to calculate the electric field amplitude (not intensity) of the reflected beam. I know the polarization and complex refractive index ($n + ik$) for the particular wavelength, so I can calculate the reflectance coefficient $r$ for a particular (non-normal) incidence angle using the field form of the Fresnel equations (which can be found on the same Wikipedia page). This is directly related to the power/intensity reflectance $R$ via $R = \left|r\right|^2$, as mentioned before.

However, I want to know the reflected electric field amplitude rather than the intensity, so I have to use $r$ (not $R$) which is complex because the refractive index is complex ($k$ is non-zero, e.g. for metals at optical frequencies). Let's say the incoming beam is a simple plane wave with a (real) amplitude $E_i$, what's the correct way to calculate the reflected amplitude $E_r$? Is it $E_r = \mathrm{Re}[r]\cdot E_i$ or $E_r = \left|r\right|\cdot E_i$? The actual physical electric field amplitude must be real valued but what's the role of the imaginary part of $r$ (does it just add a phase?) and how to deal with it correctly?

 

[Charles Link]

I think your second way is correct. The imaginary part of $r$ will add both a phase and amplitude change. If you don't need any phase info, the second way will work.

 

[Novgorod]

Thanks for the quick reply! I just need to know the maximum electric field of the reflected plane wave, i.e. its amplitude. So I need the absolute value, not just the real part? I was a bit confused because we're used to taking the real part of the oscillatory phase term ($e^{i\omega t}$) to get the actual field value ;)...

 

[Charles Link]

Thanks for the quick reply! I just need to know the maximum electric field of the reflected plane wave, i.e. its amplitude. So I need the absolute value, not just the real part? I was a bit confused because we're used to taking the real part of the oscillatory phase term ($e^{i\omega t}$) to get the actual field value ;)...

Taking just the real part of $r$ would get you an incorrect result. In the event of a $\pi/2$ phase change, you would incorrectly compute zero reflection.

 

[Novgorod]

Makes sense, thanks!