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Absorption

In the last section we considered the changes that radiation undergoes at boundaries between dielectric materials. Those were purely geometric effects that can be fully understood in terms of ray optics. In the next sections we will leave behind the idea of rays and try to describe the interaction of electromagnetic waves with matter.

In particular we would like to understand the absorption of light as it passes through matter. This is phenomenologically described by the law of Lambert and Beer. They observed that the intensity of light is attenuated exponentially with the length d that the light propagates in the absorbing material:

I = I₀ exp{-αd}

The quantity α is called absorption coefficient.

Let us look a bit more closely at absorption phenomena. As always, we will try to squeeze in the harmonic oscillator, because it caused us so much trouble in the undergrad lab sessions.

Assume that the excitation is periodic. This is really straightforward, we want to understand interaction of electromagnetic waves with matter, and waves are indeed periodically changing electric fields.
We can further imagine that ions of a ionic crystal will respond to a periodic field by oscillations about their equilibrium positions. In most cases we can assume the potentials to be harmonic and a description with harmonic oscillators should be fairly accurate.
Within an individual atom we might be able to excite oscillations of electrons against the cores. The assumption of a harmonic potential may seem a bit far fetched here, we have to be careful. In any case the, the resonance frequencies should be much higher than in the ionic crystal because the electron has much lower mass.
In metals the free electron gas as a total can respond to an electromagnetic field. The free flow of current even for stationary fields (direct current) means that the electrons do not feel a restoring force but follow the fields instantly. Thus, we have to assume that the resonance frequency in the classical oscillator picture is zero.

See? The assumption of harmonic oscillators is always useful.

The equation of motion for our oscillating system with with mass m reads:

The damping is described by β, the resonance frequency of the system is denoted by ω₀. The periodic excitation force F is equal to eE₀exp{-i ωt}, E₀ being the amplitude of the electric field.

The oscillation of the system has an amplitude x₀ and as a whole forms a radiating dipole. If we sum up all dipole moments (Density n = N/V) we end up with a macroscopic polarization P. We can then calculate the dielectric constant ε via the following relation:

P = ε₀ (ε - 1)E.

We find that the dielectric constant ε is in fact no longer constant, but it depends on the excitaion frequency. This behaviour is called Dispersion.

A second important finding is that our dielectric function is a complex quantity. If we want to keep the useful relation between n and ε, we have to define a new quantity κ which we will call extinction coefficient.

A schematic dependence of the dielectric function and the refractive index on the frequency ω is given in the figure below.

Let us come back to our initial idea of wave propagation. For a correct description we have to plug our new refractive index into the equation.

E(x,t) = E₀ exp{i(n+iκ)kx - ωt}

Thus, our newly defined extiction coefficient multiplies out to an exponential damping of the wave in the material. We can relate κ to the absorption coeffient of the Lambert-Beer law:

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