# Absorption given a complex refractive index

I'm learning about complex refractive index, and I know it has something to do with absorption, so a light wave which is propagating in $$z$$ gets absorbed by the following law: $$e^{- \frac{2 \pi}{\lambda_0} n_{\mathrm{Im}} z},$$ where $$n_{\mathrm{Im}}$$ is the imaginary part of the refractive index and $$\lambda_0$$ is the light wavelength in vacuum. How would I be able to compute the fraction of light absorbed by a media knowing its refractive index (for example, we can compute $$R$$ and $$T$$ being the reflection and transmission fractions) and how would the fact that $$n \in$$ complex numbers affect in the calculation of $$R$$ and $$T$$? Because I suppose that if a medium is somehow absorbent, then $$R$$ + $$T$$ shouldn't be equal to 1.

$$R + T + A = 1$$, where $$A$$ is the absorbed energy. This is just a statement of the conservation of energy. You can compute these values in 1D relatively easily with the Fresnel coefficients for any stack of materials, essentially just as you would for real-index materials. Just include the imaginary part in the calculation (there may occasionally be a couple of subtleties).
One thing that might help you (or might confuse you) is that once refractive index is complex, then $$r$$ and $$t$$, the field reflection and transmission coefficients (not power) are complex as well. What does this mean? It means that in addition to an amplitude change, there is also an arbitrary phase shift of the wave.