# 48 High Frequency Limit, Plasma Frequency

*Michael Fowler*

*Jackson 7.5D. Our
presentation here differs somewhat from Jackson’s in that we find it convenient,
as in our previous lectures, to work and present results using the wave
impedance **$Z.$** *

## Plasma Frequency: Phase Velocity and Impedance

Well above the highest frequency of the molecular oscillators in the Lorentz type model, the permittivity $\epsilon \left(\omega \right)$ (from the previous lecture) tends to a simple form we’ll call ${\epsilon}_{\text{pl}}$:

$$\frac{{\epsilon}_{\text{pl}}\left(\omega \right)}{{\epsilon}_{0}}\cong 1-\frac{{\omega}_{\text{pl}}^{2}}{{\omega}^{2}},\text{\hspace{1em}}{\omega}_{\text{pl}}^{2}=\frac{{n}_{e}{e}^{2}}{{\epsilon}_{0}m}.$$

This
${\omega}_{\text{pl}}$ is called the *plasma frequency*, as explained below, and systems for which this
oscillation predominates are called *plasmas*.

Two important examples of plasmas are the ionosphere, and the electron gas in some conductors.

At these high frequencies, the phase velocity ${v}_{\phi}=1/\sqrt{{\mu}_{0}\epsilon}=c\sqrt{\epsilon /{\epsilon}_{0}},$

becomes

$${v}_{\phi \text{pl}}\left(\omega \right)=c\sqrt{\frac{{\epsilon}_{0}}{{\epsilon}_{\text{pl}}\left(\omega \right)}}=c\frac{\omega}{\sqrt{{\omega}^{2}-{\omega}_{\text{pl}}^{2}}},$$

and the impedance

$${Z}_{\text{pl}}\left(\omega \right)={Z}_{0}\sqrt{\frac{1}{{\epsilon}_{\text{pl}}\left(\omega \right)}}={Z}_{0}\frac{\omega}{\sqrt{{\omega}^{2}-{\omega}_{\text{pl}}^{2}}}.$$

*Notation*! We will denote the electron density by ${n}_{e}.$ Jackson uses $NZ,$ we’ll reserve $Z$ for impedance in these notes.

### Understanding the Plasma Frequency: Longitudinal Oscillations

In this frequency range, the electrons jiggle back and forth so fast the "springs" and damping in the molecular oscillators are irrelevant$\u2014$it's like a gas of free electrons. (We also assume here that collisions between electrons are at a lower frequency.)

Imagine a unit cube of material: suppose all the electrons
are displaced by positive $x.$ If the electron density is ${n}_{e},$ and the electron charge $-e,$ there is a "surface" charge density $-{n}_{e}ex$ on the positive $x$ side, minus that on the other side,
consequently an electric field $\sigma /{\epsilon}_{0}={n}_{e}ex/{\epsilon}_{0}.$ This will give the electrons an acceleration $\ddot{x}=-\left({n}_{e}{e}^{2}/{\epsilon}_{0}m\right)x=-{\omega}_{\text{pl}}^{2}x,$ and hence simple harmonic motion, at this *plasma frequency*.

Notice this is a *longitudinal*
oscillation, more like a sound wave than a light wave$\u2014$but with an
important difference. If we imagine the electrons to be in sheets perpendicular
to the direction of the wave, then the interaction between sheets doesn’t drop
off with distance (recall electric field from a plane of charge). This is in
contrast to the mechanical forces as a sound wave passes through a solid or a
liquid. It follows that the *longitudinal*
plasma wave frequency depends only weakly on the wavelength of a disturbance
(what dependence there is comes from compressibility, which has a quantum
mechanical origin).

Note also that the field has the form $\overrightarrow{E}=\left(0,0,{E}_{z}\left(z\right)\right),$ which has no curl, so there is no accompanying magnetic field. For obvious reasons, these waves are sometimes called “electrostatic waves”. (However, in some systems they can generate radio waves via nonlinear mechanisms$\u2014$we won’t pursue this further here.)

The electron density in the ionosphere is 10^{18}$\u2014$10^{22}/m^{3}
so ${\omega}_{\text{pl}}$ is in the range 60 GHz to 6 THz (quoting
Jackson here). For metals, the density is of order 10^{29}/m^{3},
the plasma frequency in the ultraviolet (order 10^{16}.)

### Transverse EM Waves in a Plasma: $$ $\omega >{\omega}_{\text{pl}}$

What happens when a light (or other EM) wave enters a plasma?

In this lecture, we’re taking the permittivity to be

$${\epsilon}_{\text{pl}}\left(\omega \right)={\epsilon}_{0}\left({\omega}^{2}-{\omega}_{\text{pl}}^{2}\right)/{\omega}^{2},\text{\hspace{1em}}\text{with}{\omega}_{\text{pl}}^{2}={n}_{e}{e}^{2}/{\epsilon}_{0}m.$$

We also assume the plasma has no significant magnetic properties, that is, we take $\mu ={\mu}_{0}.$

This is a pretty good model for the electron gas in many conducting metals for the appropriate frequency range (including optical and higher), and an excellent model over a wide frequency range for the ionosphere.

As discussed earlier, the two important parameters in analyzing an electromagnetic wave entering or traversing a medium are the wave phase velocity (as in $\omega =vk$ ) and the impedance:

$${v}_{\phi \text{pl}}\left(\omega \right)=\frac{1}{\sqrt{{\epsilon}_{\text{pl}}\left(\omega \right){\mu}_{0}}},\text{\hspace{1em}}{Z}_{\text{pl}}\left(\omega \right)=\sqrt{\frac{{\mu}_{0}}{{\epsilon}_{\text{pl}}\left(\omega \right)}}.$$

Starting with the *kinetic*
properties, squaring $\omega ={v}_{\text{pl}}k$,
we find

$${k}^{2}={\omega}^{2}/{v}_{\text{pl}}^{2}={\omega}^{2}{\epsilon}_{\text{pl}}\left(\omega \right){\mu}_{0}={\epsilon}_{0}{\mu}_{0}\left({\omega}^{2}-{\omega}_{\text{pl}}^{2}\right),$$

and ${\epsilon}_{0}{\mu}_{0}=1/{c}^{2},$ so

$${\omega}^{2}={\omega}_{\text{pl}}^{2}+{c}^{2}{k}^{2}.$$

In contrast to the longitudinal plasma waves, these transverse waves have a frequency depending on wavelength.

For frequencies far above the plasma frequency, the waves
tend to ordinary light (or ultraviolet, etc.) with little contribution from the
plasma. However, for frequencies approaching ${\omega}_{\text{pl}}$ (from above) the group velocity $d\omega /dk$ tends to zero, and from $\overrightarrow{H}=\left(\widehat{\overrightarrow{k}}\times \overrightarrow{E}\right)/Z,$ and $Z\to \infty ,$ the magnetic field disappears, so we are
moving towards an oscillation more like the longitudinal one described above,
but now in the *transverse* direction.

Turning now to the *dynamic*
properties: suppose an ordinary electromagnetic wave, such as light, is
traveling through empty space and encounters a plasma. Will some be reflected?
In fact, we already have the necessary tools to find out: we know the plasma
impedance (assuming we’re in the right frequency range, and taking $\mu ={\mu}_{0}$ )

$${Z}_{\text{pl}}=\frac{1}{\sqrt{\epsilon \left(\omega \right)}}=\frac{1}{\sqrt{{\epsilon}_{0}}}\frac{\omega}{\sqrt{{\omega}^{2}-{\omega}_{\text{pl}}^{2}}}={Z}_{0}\frac{\omega}{\sqrt{{\omega}^{2}-{\omega}_{\text{pl}}^{2}}},$$

and, assuming normal incidence, we just substitute this in the equations we found in an earlier lecture,

$$\frac{{{E}^{\u2033}}_{0}}{{E}_{0}}=\frac{{Z}^{\prime}-Z}{{Z}^{\prime}+Z},\text{\hspace{1em}}\frac{{{E}^{\prime}}_{0}}{{E}_{0}}=\frac{2{Z}^{\prime}}{{Z}^{\prime}+Z},$$

to give the transmitted electric field

$$\frac{{{E}^{\prime}}_{0}}{{E}_{0}}=\frac{2{Z}_{\text{pl}}}{{Z}_{\text{pl}}+{Z}_{0}}=\frac{2\omega}{\omega +\sqrt{{\omega}^{2}-{\omega}_{\text{pl}}^{2}}}.$$

*Exercise*:
Using $\overrightarrow{H}=\left(\widehat{\overrightarrow{k}}\times \overrightarrow{E}\right)/Z,$ find the transmitted energy using Poynting’s
theorem. Find what happens for $\omega \to \infty .$ Then take $\omega \to {\omega}_{\text{pl}}.$ In particular, look at the fields in this
limit.

### Transverse EM Waves in a Plasma: $\omega <{\omega}_{\text{pl}}$

Recall ${\epsilon}_{\text{pl}}\left(\omega \right)={\epsilon}_{0}\left({\omega}^{2}-{\omega}_{\text{pl}}^{2}\right)/{\omega}^{2},\text{\hspace{1em}}{v}_{\text{pl}}\left(\omega \right)=\frac{1}{\sqrt{{\epsilon}_{\text{pl}}\left(\omega \right){\mu}_{0}}}.$

From

$${k}^{2}={\omega}^{2}/{v}_{\text{pl}}^{2}={\omega}^{2}{\epsilon}_{\text{pl}}\left(\omega \right){\mu}_{0}={\epsilon}_{0}{\mu}_{0}\left({\omega}^{2}-{\omega}_{\text{pl}}^{2}\right),$$

for $\omega <{\omega}_{\text{pl}},\text{\hspace{0.33em}}{k}^{2}$ is negative,

$$ck=i\sqrt{{\omega}_{\text{pl}}^{2}-{\omega}^{2}},$$

and the wave decays exponentially after entering the plasma, ${E}^{\prime}\left(z\right)={E}^{\prime}\left(0\right){e}^{-z/\delta},\text{\hspace{0.33em}}\text{\hspace{0.33em}}\delta =c/\sqrt{{\omega}_{\text{pl}}^{2}-{\omega}^{2}}.$

The wave energy is *not*
being absorbed into the plasma, in our model there is no mechanism to do that, so
it has to be fully reflected, easy to confirm using the wave impedance

$${Z}_{\text{pl}}={Z}_{0}\frac{\omega}{\sqrt{{\omega}^{2}-{\omega}_{\text{pl}}^{2}}}=-i{Z}_{0}\frac{\omega}{\sqrt{{\omega}_{\text{pl}}^{2}-{\omega}^{2}}}.$$

So ${Z}_{\text{pl}}$ is pure imaginary, and the reflected amplitude

$$\frac{{{E}^{\u2033}}_{0}}{{E}_{0}}=\frac{{Z}_{\text{pl}}-{Z}_{0}}{{Z}_{\text{pl}}+{Z}_{0}}=\frac{\sqrt{{\omega}_{\text{pl}}^{2}-{\omega}^{2}}-i\omega}{\sqrt{{\omega}_{\text{pl}}^{2}-{\omega}^{2}}+i\omega},$$

has unit modulus, meaning all the energy *is* reflected, with phase shift $-2{\mathrm{tan}}^{-1}\left(\omega /\sqrt{{\omega}_{\text{pl}}^{2}-{\omega}^{2}}\right).$

In the limit of zero frequency, the field decays exponentially as stated above, and the field energy $\left(\propto {E}^{2}\right)$ has penetration length ${\scriptscriptstyle \frac{1}{2}}\delta =c/2{\omega}_{\text{pl}}.$

Notice, though, that on increasing the frequency of the
incoming EM wave, $\delta =c/\sqrt{{\omega}_{\text{pl}}^{2}-{\omega}^{2}}$ increases, and near the plasma frequency,
typically in the ultraviolet, some alkali metals become transparent, at least
in thin films, according to R. W. Wood in a 1933 paper, and Born and Wolf, and
Jackson calls it drastic (page 314). But gold foil is transparent if thin
enough, I can’t find evidence of a *thick *piece
of transparent alkali metal (say, 1 mm), so this search is left as an exercise
for the reader.