Wave Propagation Through Inhomogeneous Plasmas

Let us start off by examining a very simple case. Consider a plane
electromagnetic wave,
of frequency , propagating along the -axis in an unmagnetized plasma
whose refractive index, , is a function of . We assume that
the wave normal is initially aligned along the -axis, and, furthermore, that
the wave starts off polarized in the -direction. It is
easily demonstrated that the wave normal subsequently remains aligned along
the -axis, and also that the polarization
state of the wave does not change.
Thus, the wave is fully described by

(553) |

(554) |

and

respectively. Here, is the wave-number in free space. Of course, the actual wave-number is .

The solution to Eq. (555) for the case of a homogeneous plasma, for which
is constant, is straightforward:

The solution (557) represents a wave of constant amplitude, , and phase, . According to Eq. (558), there are, in fact, two independent waves which can propagate through the plasma. The upper sign corresponds to a wave which propagates in the -direction, whereas the lower sign corresponds to a wave which propagates in the -direction. Both waves propagate with the constant phase velocity .

In general, if then the solution of Eq. (555) does not remotely resemble
the wave-like solution (557). However, in the limit in which is
a ``slowly varying'' function of (exactly how slowly varying is something which
will be established later on), we expect to recover wave-like solutions.
Let us suppose that is indeed a ``slowly varying'' function, and let us try
substituting the wave solution (557) into Eq. (555). We obtain

(560) |

It is clear from a comparison of Eqs. (559) and (561) that can be regarded as a ``slowly varying'' function of as long as its variation length-scale is far longer than the wave-length of the wave. In other words, provided that .

The second approximation to the solution is obtained by substituting Eq. (561) into
the right-hand side of Eq. (559):

(562) |

(563) |

Substitution of Eq. (564) into Eq. (557) yields the final result

It follows from Eq. (556) that

(566) |

Let us test to what extent the expression (565) is a good solution
of Eq. (555) by substituting this expression into the left-hand side
of the equation. The result is

(567) |

The solutions

to the non-uniform wave equations (555) and (556) are most commonly referred to as the

Recall, that when a propagating wave is normally incident on an *interface*,
where the
refractive index suddenly changes (for instance, when a light
wave propagating through
air is normally incident on a glass slab), there is generally
significant reflection of the wave. However, according to the WKB solutions,
(569)-(570), when a propagating wave is normally incident on a medium in which
the refractive index changes *slowly* along the direction of propagation of the
wave then the wave is not reflected at all. This is true
even if the refractive index
varies *very substantially* along the path of propagation of the wave,
as long as it varies *slowly*. The WKB
solutions imply that as the wave propagates through the medium its wave-length
gradually changes. In fact, the wave-length at position is approximately
. Equations (569)-(570) also imply that the amplitude
of the wave gradually changes as it propagates. In fact, the amplitude of the electric
field component is inversely proportional to , whereas the amplitude of the
magnetic field component is directly proportional to .
Note, however, that the energy
flux in the -direction, given by the the Poynting vector
, remains constant (assuming that is predominately
real).

Of course, the WKB solutions (569)-(570) are only *approximations*. In reality,
a wave propagating into a medium in which the refractive index is a slowly
varying function of position is subject to a small amount of reflection.
However, it is easily demonstrated that the ratio of the reflected amplitude
to the incident amplitude is of order
. Thus, as long as
the refractive index varies on a much longer length-scale than the wave-length
of the radiation, the reflected wave is negligibly small. This conclusion remains
valid as long as the inequality (568) is satisfied.
This inequality obviously
breaks down in the vicinity of a point where . We would, therefore,
expect strong reflection of the incident wave from such a point.
Furthermore, the WKB solutions also break down at a
point where
, since the amplitude of becomes
infinite.