Reflectivity with complex refraction indices

So the general equation for the reflectivity at the interface between two materials is given by: $$R=\left(\frac{n_1-n_2}{n_1+n_2}\right)^2$$ in case of air/glass $n$ is real, but for, say, semiconductors or metals, where radiation is absorbed, $n$ is a complex number, with $\underline{n}=n_r-ik$. $k$ is described as the extinction coefficient and is related to the absorption coefficient with $\alpha=\frac{4\pi k}{\lambda}$, $\lambda$ being the wavelength.

I am looking to derive a formula for the reflectivity which only includes the real and imaginary parts of the complex refractive index. As far as I can tell, the equation above gives the reflectivity as long as the norm of the index is known, that is $$n_1=\sqrt{n_{r_1}^2+k_1^2} \\ n_2=\sqrt{n_{r_2}^2+k_2^2}$$ in the above formula for the reflectivity, I replaced the norms of the complex numbers and not the numbers themselves,obviously. So doing that, I get a fraction where square root terms appear. On the other hand Wikipedia writes(https://en.wikipedia.org/wiki/Refractive_index) $$R=\left|\frac{n_1-n_2}{n_1+n_2}\right|^2$$which also makes sense and leads to $$R=\frac{(n_{r_1}-n_{r_2})^2+(k_1-k_2)^2}{(n_{r_1}+n_{r_2})^2+(k_1+k_2)^2}$$ Which formula is right?

• As for me, the quantity $n = \sqrt{n_r^2 + k^2}$ does not make any sense. You don't want to mix the real and imaginary part of the refractive index as they describe different phenomena. – Ilya Mar 24 '17 at 16:45
• $n$ is the norm of $\underline{n}$, $\left|\underline{n}\right|=n=\sqrt{n_r^2+k^2}$ , $n_r$ being the real and $k$ being the imaginary part. If $k=0$, there is no point writing an index $\small r$, since it's $n=n_r$ – pitfermi Mar 24 '17 at 16:48
• Yes, I see that, but what is the physical meaning of this norm? The ratio of $n_r$ for two materials refers to the ratio of the wave speeds in them. $k$ is related to the decay rate of the amplitude. What about the norm? – Ilya Mar 24 '17 at 16:59
• If I knew i wouldn't have asked in the first place, but mathematically, the complex refraction index is automatically assigned a norm. why do you assume that it's false to build a norm out of 2 constants which describe a different phenomena? both are constants. there is no problem with units etc. – pitfermi Mar 24 '17 at 17:10

The first equation (Fresnel reflectivity) is derived assuming you have a lossless system, meaning $n$ is always real. When you introduce absorption you get the second formula as all the $k$ go to zero.