# Pulsed transmission of electromagnetic waves through a plasma

A plasma has the following dispersion relation:

$$k^2 = \frac{\omega^2}{c^2}\left(1 - \frac{\omega_\mathrm{p}^2}{\omega^2}\right)$$

where $k$ is the magnitude of the wave-vector and $\omega$ is the angular frequency and $\omega_\mathrm{p}$ is the plasma angular frequency. Now my book mentions that it is possible that if electron density varies with time (e.g $n(t) = A\sin^2(10t)$), then the electromagnetic wave that is transmitted through the plasma can be pulsed. I don't understand why would that be the case and why wouldn't the wave simply be transmitted through it.

• Since $k$ has to be real for non-attenuated propagation, this dispersion relation imposes a cutoff frequency, which will presumably vary with density. – probably_someone Apr 27 '18 at 20:53
• What do you mean by "cutoff frequency"? – daljit97 Apr 28 '18 at 12:05
• When $\omega<\omega_p$, we have that $k^2<0$,which is impossible, so no propagation occurs for frequencies below $\omega_p$. – probably_someone Apr 28 '18 at 16:18

You can see that the dispersion relation has a characteristic frequency, $\omega_p$. That is the plasma frequency and electromagnetic waves with a frequency lower than the plasma frequency cannot penetrate the plasma, they are reflected (like rf-waves being reflected at the ionosphere). For this reason, the plasma frequency is also referred to as cut-off frequency. If the wave frequency is higher than the plasma frequency, it can penetrate the plasma (like microwaves in the GHz range which are used for satellite communication since they can easily pass the ionosphere).
The plasma frequency depends on the electron density of the plasma: $$\omega_p = \sqrt{\frac{n_e e^2}{m_e \epsilon_0}},$$ with $n_e$ the electron density, $e$ the elementary charge, $m_e$ the mass of the electron, and $\epsilon_0$ the free space permittivity.
• Ok so let's suppose that we have a time-varying plasma frequency $w_p$. Then sometimes then sometimes my wave penetrating will have a frequency lower than $w_p$ and sometimes higher. This will cause the wave to reflect and get transmitted periodically inside the plasma. Is my understanding correct? – daljit97 May 6 '18 at 10:41