Derivation of absorption coefficient α.

Starting from Maxwell equations:
∇D=ρ
∇B=0
∇×E=∂B/∂t
∇×H=∂D/∂t (1)

To get a solution of EM wave, one need to use

B=μμ₀H and D=εε₀E and plug into the last two equations of (1).

Then we have:

∇²E+μμ₀εε₀∂²E/∂t²=0, the solution is

E=E₀e^-i(ωt+qr) (2)

where ω is the frequency and q is the wave vector and

q=ω√(μμ₀εε₀).

Note that normally μ=1 for non-magnetic material (or even for magnetic material at high frequency) and μ₀ε₀=1/c².

Therefore:

q=ω/c√ε (3).

Plug (3) into (2) and using the definition √ε=n+ik for imaginary dielectric constant (whee n is the refractive index and k is the extinction coefficient), one gets:

E=E₀e^{-iω(t+√εr/c)}=E₀e^{-iω[t-(n+ik)r/c]}=E₀e^{-iω[t-(n+ik)r/c]}e^-ωkr/c (4).

One can see that the electric field decays if there is an imaginary part of the dielectric constant.

By the definition of α:

E²=E₀²e^-αr, we get

α=2ωk/c.