Thursday, 26 January 2017

optics - 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?




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