Resonant Layers
Consider the situation, studied in the
previous section, in which a plane wave, polarized in the
-direction,
is launched along the
-axis, from an antenna located at large positive
,
and absorbed at a resonance located at
. In the vicinity
of the resonant point, the electric component of the wave satisfies
![$\displaystyle \frac{d^2 E_y}{dz^2} + \frac{k_0^2\,b}{z+{\rm i}\,\epsilon} E_y = 0,$](img2036.png) |
(6.55) |
where
and
.
The time-averaged Poynting flux in the
-direction is written
![$\displaystyle P_z = - \frac{(E_y\,B_x^{\,\ast} + E_y^{\,\ast}\,B_x)}{4 \,\mu_0}.$](img2037.png) |
(6.56) |
Now, the Faraday-Maxwell equation yields
![$\displaystyle {\rm i}\,\omega\,B_x = -\frac{d E_y}{dz}.$](img2038.png) |
(6.57) |
Thus, we have
![$\displaystyle P_z = -\frac{{\rm i}}{4\,\mu_0\,\omega} \left(\frac{d E_y}{dz}\, E_y^{\,\ast}
- \frac{d E_y^{\,\ast}}{dz} \,E_y\right).$](img2039.png) |
(6.58) |
Let us ascribe any variation of
with
to the wave energy emitted by the
plasma. We then obtain
![$\displaystyle \frac{d P_z}{dz} = W,$](img2041.png) |
(6.59) |
where
is the power emitted by the plasma per unit volume.
It follows that
![$\displaystyle W = -\frac{{\rm i}}{4\,\mu_0\,\omega}\left(\frac{d^2 E_y}{dz^{2}}\,E_y^{\,\ast}
- \frac{d^2 E_y^{\,\ast}}{dz^{2}}\,E_y\right).$](img2043.png) |
(6.60) |
Equations (6.55) and (6.60) yield
![$\displaystyle W = \left(\frac{k_0^{2}\,b}{2\,\mu_0\,\omega}\right)\left( \frac{\epsilon}{z^{2} + \epsilon^{2}}\right)\vert E_y\vert^{2}.$](img2044.png) |
(6.61) |
Note that
, because
, so wave energy is absorbed by the
plasma. It is clear, from the previous formula, that the absorption takes
place in a narrow layer, of thickness
, centered on the
resonance point,
.