High Frequency Limit

Consider the behavior of the dispersion relation (786) in the high frequency limit $\omega\gg \omega_i$ (for all $i$ ). In this case, the relation simplifies considerably to give

$$n^{\,2}(\omega) = 1 - \frac{\omega_p^{\,2}}{\omega^{\,2}},$$ (810)

where the quantity

$$\omega_p = \sqrt{\frac{N\,Z\,e^{\,2}}{\epsilon_0\, m}}$$ (811)

is called the plasma frequency. The wavenumber in the high frequency limit is given by

$$k = n\,\frac{\omega}{c} = \frac{(\omega^{\,2} - \omega_p^{\,2})^{1/2}}{c}.$$ (812)

This expression is only valid in dielectrics when $\omega\gg \omega_p$ . Thus, the refractive index is real, and slightly less than unity, giving waves that propagate without attenuation at a phase velocity slightly larger than the velocity of light in vacuum. However, in certain ionized media (in particular, in tenuous plasmas, such as occur in the ionosphere) the electrons are free, and the damping is negligible. In this case, Equations (811) and (813) are valid even when $\omega < \omega_p$ . It follows that a wave can only propagate through a tenuous plasma when its frequency exceeds the plasma frequency (in which case, it has a real wavenumber). If wave frequency is less than the plasma frequency then, according to Equation (813), the wavenumber is purely imaginary, and the wave is unable to propagate. This phenomenon accounts for the fact that long-wave and medium-wave (terrestrial) radio signals can be received even when the transmitter lies over the horizon. The frequency of these waves is less than the plasma frequency of the ionosphere, which reflects them (see Chapter 8), so they become trapped between the ionosphere and the surface of the Earth (which is also a good reflector of radio waves), and can, in certain cases, travel many times around the Earth before being attenuated. Unfortunately, this scheme does not work very well for medium-wave signals at night. The problem is that the plasma frequency of the ionosphere is proportional to the square root of the number density of free ionospheric electrons. These free electrons are generated through the ionization of neutral molecules by ultraviolet radiation from the Sun. Of course, there is no radiation from the Sun at night, so the density of free electrons starts to drop as the electrons gradually recombine with ions in the ionosphere. Eventually, the plasma frequency of the ionosphere falls below the frequency of medium-wave radio signals, causing them to be transmitted through the ionosphere into outer space. The ionosphere appears almost completely transparent to high frequency signals such as TV and FM radio signals. Thus, this type of signal is not reflected by the ionosphere. Consequently, to receive such signals it is necessary to be in the line of sight of the relevant transmitter.