Cutoffs

We have seen that electromagnetic wave propagation (in one dimension) through an inhomogeneous plasma, in the physically relevant limit in which the variation length-scale of the plasma is much greater than the wave-length of the wave, is well described by the WKB solutions, (545)-(546). However, these solutions break down in the immediate vicinity of a cutoff, where $n^2=0$, or a resonance, where $n^2\rightarrow\infty$. Let us now examine what happens to electromagnetic waves propagating through a plasma when they encounter a cutoff or a resonance.

Suppose that a cutoff is located at $z=0$, so that

$$n^2 = a\,z+ O(z^2)$$ (547)

in the immediate vicinity of this point, where $a>0$. It is evident, from the WKB solutions, (545)-(546), that the cutoff point lies at the boundary between a region ($z>0$) in which electromagnetic waves propagate, and a region ($z<0$) in which the waves are evanescent. In a physically realistic solution, we would expect the wave amplitude to decay (as $z$ decreases) in the evanescent region $z<0$. Let us search for such a wave solution.

In the immediate vicinity of the cutoff point, $z=0$, Eqs. (531) and (547) yield

$$\frac{d^2 E_y}{d\hat{z}^2} + \hat{z}\,E_y = 0,$$ (548)

where

$$\hat{z} = (k_0^{~2}\,a)^{1/3}\,z.$$ (549)

Equation (548) is a standard equation, known as Airy's equation, and possesses two independent solutions, denoted ${\rm Ai}(-\hat{z})$ and ${\rm Bi}(-\hat{z})$.¹⁴ The second solution, ${\rm Bi}(-\hat{z})$, is unphysical, since it blows up as $\hat{z}\rightarrow-\infty$. The physical solution, ${\rm Ai}(-\hat{z})$, has the asymptotic behaviour

$${\rm Ai}(-\hat{z})\sim \frac{1}{2\,\sqrt{\pi}}\, \vert\hat{z... .../4}\,\exp\! \left(-\frac{2}{3}\,\vert\hat{z}\vert^{3/2}\right)$$ (550)

in the limit $\hat{z}\rightarrow-\infty$, and

$${\rm Ai}(-\hat{z})\sim \frac{1}{\sqrt{\pi}}\, \hat{z}^{-1/4}\,\sin\! \left(\frac{2}{3}\,\hat{z}^{3/2}+\frac{\pi}{4}\right)$$ (551)

in the limit $\hat{z}\rightarrow +\infty$.

Suppose that a unit amplitude plane electromagnetic wave, polarized in the $y$-direction, is launched from an antenna, located at large positive $z$, towards the cutoff point at $z=0$. It is assumed that $n=1$ at the launch point. In the non-evanescent region, $z>0$, the wave can be represented as a linear combination of propagating WKB solutions:

$$E_y(z) = n^{-1/2}\,\exp\left(- {\rm i}\, k_0 \!\int_0^z \!n\... ...,n^{-1/2}\,\exp\left(+{\rm i}\, k_0 \!\int_0^z \!n\,dz\right).$$ (552)

The first term on the right-hand side of the above equation represents the incident wave, whereas the second term represents the reflected wave. The complex constant $R$ is the coefficient of reflection. In the vicinity of the cutoff point (i.e., $z$ small and positive, or $\hat{z}$ large and positive) the above expression reduces to

$$E_y(\hat{z}) = (k_0/a)^{1/6}\,\left[ \hat{z}^{-1/4}\exp\!\le... ...xp\!\left(+{\rm i}\,\frac{2}{3}\,\hat{z}^{3/2}\right) \right].$$ (553)

However, we have another expression for the wave in this region. Namely,

$$E_y(\hat{z}) = C\,{\rm Ai}(-\hat{z}) \simeq \frac{C}{\sqrt{\... ...,\sin\! \left(\frac{2}{3}\,\hat{z}^{3/2}+\frac{\pi}{4}\right),$$ (554)

where $C$ is an arbitrary constant. The above equation can be written

$$E_y(\hat{z}) =\frac{C}{2}\sqrt{\frac{{\rm i}}{\pi}} \left[\h... ...xp\!\left(+{\rm i}\,\frac{2}{3}\,\hat{z}^{3/2}\right) \right].$$ (555)

A comparison of Eqs. (553) and (555) yields

$$R = -{\rm i}.$$ (556)

In other words, at a cutoff point there is total reflection, since $\vert R\vert=1$, with a $-\pi/2$ phase-shift.