Extinction coefficient

The extinction coefficient for a particular substance is a measure of how well it absorbs electromagnetic radiation (EM waves), or the amount of "impedance" the material offers for the passage of electromagnetic radiation through it. If the EM wave can pass through very easily, the material has a low extinction coefficient. Conversely, if the radiation hardly penetrates the material, but rather quickly becomes "extinct" within it, the extinction coefficient is high.

A material can behave differently for different wavelengths of electromagnetic radiation. Glass is transparent to visible light, but many types of glass are opaque to ultra-violet wavelengths. In general, the extinction coefficient for any material is a function of the incident wavelength. The extinction coefficient is used widely in ultraviolet-visible spectroscopy.

Underlying Physics

The parameter used to describe the interaction of electromagnetic radiation with matter is the complex index of refraction, ñ, which is a combination of a real part and an imaginary part.

${\tilde {n}}=n-ik$

Here, n is also called the index of refraction, which sometimes leads to confusion. k on the other hand is called as the extinction coefficient and is responsible for the damping of an EM Wave inside the material as we shall proceed to illustrate.

Any EM wave, travelling in the material with velocity $v$ and angular frequency$\omega$ can be characterised by its time varying Electric Field as

$E(z,t)=E_{0}e^{i\omega (t-{\frac {z}{v}})}$

For simplicity, we restrict our analysis to propogation in one dimension denoted by $z$.

By definition of index of refraction,

${\tilde {n}}={\frac {c}{v}}$

Substituting for ${\tilde {n}}$ in the expression above, we get

${\frac {1}{v}}={\frac {n}{c}}-i{\frac {k}{c}}$.

Substituting this in the expression for the EM Wave above, we get,

$E(z,t)=E_{0}e^{i\omega (t-z({\frac {n}{c}}))}e^{-({\frac {k\omega }{c}})z}$

Looking at this expression we can see an oscillating Electric field with an exponetially damped amplitude due to the k term. It is this term causes the EM Wave to die out as it travels further into the material. Now, for any EM Wave, the intensity of the the wave, which corresponds to the energy it carries with it is simply the square of the magnitude of the accompanying Eletric field. Thus, using the expression for $E(z,t)$ derived above the intensity of the EM wave inside the material can be computed as

$I(z)=I_{0}e^{-{\frac {2\omega k}{c}}z}$

Now, there is a law by the name Beer-Lambert law which states that in any medium that is absorbing light, the decrease in intensity $I$ per unit length $z$ is proportional to the instantaneous value of $I$. In mathematical form this is

${\frac {dI\left(z\right)}{dz}}={-\alpha I\left(z\right)}$

which has the solution

${I\left(z\right)}={I_{0}e^{-\alpha z}}$ ,

where $I_{0}$ is the intensity of the electromagnetic radiation at the surface of the absorbing medium.

Comparing the expression for intensity we obtained above with the Beer-Lambert Law, we get

$\alpha ={\frac {2\omega k}{c}}$

Since c here denotes the speed of EM Wave in vacuum, we have

$c=\nu \lambda ={\frac {\omega }{2\pi }}\lambda$.

Substituting this in expression above and rearranging, we have our final expression for extinction coefficient as

$k={\frac {\lambda }{4\pi }}\alpha$ ,

where λ is the vacuum wavelength.