Exercises

1. The electric polarization, ${\bf P}$, in a linear dielectric medium is related to the electric field-strength, ${\bf E}$, according to
   $${\bf P} =\epsilon_0\,(n^2-1)\,{\bf E},$$
   where $n$ is the refractive index. Any divergence of the polarization field is associated with a bound charge density
   $$\rho = -\nabla\cdot{\bf P},$$
   whereas any time variation generates a polarization current whose density is
   $${\bf j} = \frac{\partial{\bf P}}{\partial t}.$$
   Consider an electromagnetic wave propagating through a quasi-neutral, linear, dielectric medium. Assuming a common $\exp(-{\rm i}\,\omega\,t)$ time variation of the wave fields, demonstrate from Maxwell's equations that

   $$\nabla\times c\,{\bf B} = -{\rm i}\,k_0\,n^2\,{\bf E},$$

   $$\nabla\times {\bf E} ={\rm i}\,k_0\,c\,{\bf B},$$

   where $k_0=\omega/c$.

2. Consider an electromagnetic wave, polarized in the $y$-direction, that propagates in the $z$-direction through a medium of refractive index $n(z)$. Assuming that

   $${\bf E} = E_y(z)\,\exp(-{\rm i}\,\omega\,t)\,{\bf e}_y,$$

   $${\bf B} = B_x(z)\,\exp(-{\rm i}\,\omega\,t)\,{\bf e}_x,$$

   demonstrate that

   $$\frac{d^2 E_y}{dz^2} + k_0^{2}\,n^2\,E_y =0,$$

   $$\frac{d\,(c\,B_x)}{dz}+{\rm i}\,k_0\,n^2\,E_y =0,$$

   where $k_0=\omega/c$.

3. Consider an electromagnetic wave, polarized in the $y$-direction, that propagates in the $x$-$z$ plane through a medium of refractive index $n(z)$. Assuming that

   $${\bf E} = E_y(z)\,{\rm e}^{\,{\rm i}\,(k_x\,x-\omega\,t)}\,{\bf e}_y,$$

   $${\bf B} = B_x(z)\,{\rm e}^{\,{\rm i}\,(k_x\,x-\omega\,t)}\,{\bf e}_x + B_z(z)\,{\rm e}^{\,{\rm i}\,(k_x\,x-\omega\,t)}\,{\bf e}_z,$$

   demonstrate that

   $$\frac{d^2 E_y}{dz^2} + k_0^{2}\,q^2\,E_y =0,$$

   $$\frac{d\,(c\,B_x)}{dz}+ {\rm i}\,k_0\,q^2\,E_y =0,$$

   where
   $$q^2 = n^2-S^{2},$$
   and $S=k_x/k_0$, and $k_0=\omega/c$.

   Show that the WKB solutions take the form

   $$E_y(z) \simeq q^{-1/2}\,\exp\left(\pm {\rm i}\,k_0\int_0^z q\,dz'\right),$$

   $$c\,B_x(z) \simeq \mp q^{1/2}\,\exp\left(\pm {\rm i}\,k_0\int_0^z q\,dz'\right),$$

   and that the criterion for these solutions to be valid is
   $$\frac{1}{k_0^{2}}\left\vert\frac{3}{4}\left(\frac{1}{q^2}\,\frac{dq}{dz}\right)^2 - \frac{1}{2\,q^3}\,\frac{d^2q}{dz^2}\right\vert\ll 1.$$

4. Consider an electromagnetic wave, polarized in the $x$-$z$-plane, that propagates in the $x$-$z$ plane through a medium of refractive index $n(z)$. Assuming that

   $${\bf E} = E_x(z)\,{\rm e}^{\,{\rm i}\,(k_x\,x-\omega\,t)}\,{\bf e}_x+E_z(z)\,{\rm e}^{\,{\rm i}\,(k_x\,x-\omega\,t)}\,{\bf e}_z,$$

   $${\bf B} = B_y(z)\,{\rm e}^{\,{\rm i}\,(k_x\,x-\omega\,t)}\,{\bf e}_y,$$

   demonstrate that

   $$\frac{dE_x}{dz} -{\rm i}\,k_0\,\frac{q^2}{n^2}\,c\,B_y =0,$$

   $$\frac{d^2(c\,B_y)}{dz^2}-\frac{1}{n^2}\,\frac{dn^2}{dz}\,\frac{d\,(c\,B_y)}{dz}+k_0^{2}\,q^2\,c\,B_y =0,$$

   where
   $$q^2 = n^2-S^{2},$$
   and $S=k_x/k_0$, and $k_0=\omega/c$.

   Show that the WKB solutions take the form

   $$c\,B_y(z) \simeq n\,q^{-1/2}\,\exp\left(\pm {\rm i}\,k_0\int_0^z q\,dz'\right),$$

   $$E_x(z) \simeq \pm n^{-1}\,q^{1/2}\,\exp\left(\pm {\rm i}\,k_0\int_0^z q\,dz'\right),$$

   and that the criterion for these solutions to be valid is
   $$\frac{1}{k_0^{2}}\left\vert\frac{3}{4}\left(\frac{1}{q^2}\,\frac{... ...2n}{dz^2}-2\,\left(\frac{1}{n}\,\frac{dn}{dz}\right)^2\right]\right\vert\ll 1.$$

5. An electromagnetic wave pulse of frequency $\omega$ is launched vertically from ground level, travels upward into the ionosphere, is reflected, and returns to ground level. If $\tau(\omega)$ is the net travel time of the pulse then the so-called equivalent height of reflection is defined $h(\omega)=c\,\tau(\omega)/2$. It follows that $h$ is the altitude of the reflection layer calculated on the assumption that the pulse always travels at the velocity of light in vacuum. Let ${\mit\Pi}_e(z)$ be the ionospheric plasma frequency, where $z$ measures altitude above the ground. Neglect collisions and the Earth's magnetic field.
   1. Demonstrate that
      $$h(\omega) = \int_0^{z_0(\omega)}\frac{\omega}{\left[\omega^2- {\mit\Pi}_e^{2}(z)\right]^{1/2}}\,dz,$$
      where ${\mit\Pi}_e(z_0)= \omega$.
   2. Show that if $z_0(\omega)$ is a monotonically increasing function of $\omega$ then the previous integral can be inverted to give
      $$z_0(\omega) = \frac{2}{\pi}\int_0^{\pi/2}h(\omega\,\sin\alpha)\,d\alpha,$$
      or, equivalently,
      $$z({\mit\Pi}_e) = \frac{2}{\pi}\int_0^{\pi/2}h({\mit\Pi}_e\,\sin\alpha)\,d\alpha.$$
      (Hint: This is a form of Abel inversion. See Budden 1985.)

   3. Demonstrate that if
      $$h(\omega)= h_0 + \delta\left(\frac{\omega}{{\mit\Pi}_0}\right)^p,$$
      where $h_0$, $\delta$, and ${\mit\Pi}_0$ are positive constants, then ${\mit\Pi}_e(z)=0$ for $z<h_0$, and
      $${\mit\Pi}_e(z) =\left[\frac{\pi\,{\Gamma}(1+p)}{{\Gamma}(1/2+p/2)... .../2)}\right]^{1/p} \frac{{\mit\Pi}_0}{2}\left(\frac{z-h_0}{\delta}\right)^{1/p}$$
      for $z\geq h_0$. Here, ${\Gamma}(z)$ is a gamma function (Abramowitz and Stegun 1965).

6. Suppose that the refractive index, $n(z)$, of the ionosphere is given by $n^2=1-\alpha\,(z-h_0)$ for $z\geq h_0$, and $n^2=1$ for $z<h_0$, where $\alpha$ and $h_0$ are positive constants, and the Earth's magnetic field and curvature are both neglected. Here, $z$ measures altitude above the Earth's surface.
   1. A point transmitter sends up a wave packet at an angle $\theta$ to the vertical. Show that the packet returns to Earth a distance
      $$x_0 = 2\,h_0\,\tan\theta + \frac{2}{\alpha}\,\sin2\theta$$
      from the transmitter. Demonstrate that if $\alpha\,h_0<1/4$ then for some values of $x_0$ the previous equation is satisfied by three different values of $\theta$. In other words, wave packets can travel from the transmitter to the receiver via one of three different paths. Show that the critical case $\alpha\,h_0=1/4$ corresponds to $\theta=\pi/3$ and $x_0=6\sqrt{3}\,h_0$.
   2. A point radio transmitter emits a pulse of radio waves uniformly in all directions. Show that the pulse first returns to the Earth a distance $4\,h_0\,(2/\alpha\,h_0-1)^{1/2}$ from the transmitter, provided that $\alpha\,h_0<2$.