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.

  • $\begingroup$ Since $k$ has to be real for non-attenuated propagation, this dispersion relation imposes a cutoff frequency, which will presumably vary with density. $\endgroup$ Apr 27, 2018 at 20:53
  • $\begingroup$ What do you mean by "cutoff frequency"? $\endgroup$
    – user63248
    Apr 28, 2018 at 12:05
  • $\begingroup$ When $\omega<\omega_p$, we have that $k^2<0$,which is impossible, so no propagation occurs for frequencies below $\omega_p$. $\endgroup$ Apr 28, 2018 at 16:18

1 Answer 1


The dispersion relation you mention, is the dispersion relation for O-waves, those are electromagnetic waves in an unmagnetized cold plasma, or waves where the electric field vector is parallel to the background magnetic field.

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.

If you vary the plasma density, you change the plasma frequency.

Consider an electromagnetic wave emitted onto a plasma whose frequency is slightly below the plasma frequency. It will be reflected by the plasma. If you are able to periodically vary the plasma density and reduce it such that the resulting plasma frequency is below the wave frequency, your wave can penetrate the plasma. Since you do this periodically, you will get a pulsed transmission.

  • $\begingroup$ 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? $\endgroup$
    – user63248
    May 6, 2018 at 10:41
  • $\begingroup$ @daljit97 yes, your understanding is correct. $\endgroup$
    – Alf
    May 6, 2018 at 14:50

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