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1 Plasma Frequency
1.1 Damped Simple Harmonic Oscillator
We can model and their atoms as a lattice of springs connecting to fixed points. Say that we send a plane wave at these springs that looks like:
We’ll say that the displacement of . The equation of motion for a single electron is then:
where is our dampening factor. Note that is absent here; we’re neglecting it because it is small for reasonable energies. It turns out this equation has the steady-state solution:
The limiting cases of this equation explain many phenomena. For example: (loss-less propagation).
In this case, our solution looks like:
From this we can use the dispersion relation to relate (frequency) and (phase).
The Dispersion Relation
Let’s solve Maxwell’s equations. Recall that is the current density:
where , and n is the # density of electrons. So on to Maxwell’s equations:
Combining these two equations we get:
This is our dispersion relation. The plasma frequency is defined as:
and the index of refraction is:
Rewritten in these terms, our dispersion relation in plasma is: