Plasma Frequency

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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: