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\markright{{\sf Lecture 16 (Ch 7 part 1) Plane waves}}
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\chapter{Ch. 7, part 1}

\bigskip
\bigskip

\textsc{Lecture 16 - Plane Waves }

\newpage 

We start from Maxwell's equations with constant, real, frequency-independent 
$\varepsilon $ and $\mu .$ We will then generalize to frequency-dependent,
complex $\varepsilon $ and $\mu ,$ and we will also discuss briefly the case
if inhomogeneous and anisotropic media.

\section{Plane waves in a non-conducting medium}

Each cartesian component of $\mathbf{E}$ and $\mathbf{B}$ (and also $\mathbf{%
D}$, $\mathbf{H}$, $\mathbf{A}$, $\Phi $) obeys the \textsl{scalar wave
equation} 
\begin{equation}
\nabla ^2u-\frac 1{v^2}\frac{\partial ^2u}{\partial t^2}=0  \label{1}
\end{equation}
with 
\begin{equation}
v=\frac c{\sqrt{\varepsilon \mu }}=\frac cn  \label{2}
\end{equation}
The general solution of eq. (\ref{2}) is a linear superposition of \textsl{%
fundamental solutions }of the form 
\begin{equation}
u(\mathbf{r},\,t)=u_0\exp \left[ i(\mathbf{k\cdot r}-\omega t)\right] 
\label{3}
\end{equation}
where the amplitude $u_0$ is in general complex and the wave vector $\mathbf{%
k}$ must be such that 
\begin{equation}
k^2=k_x^2+k_y^2+k_z^2=\frac{\omega ^2}{v^2}=\varepsilon \mu \,\frac{\omega ^2%
}{c^2}  \label{4}
\end{equation}
Solutions that are well behaved for $\left| t\right| \rightarrow \infty $
and $r\rightarrow \infty $ are constructed using real values of $\omega $
and $\mathbf{k}$ (i.e. real $k_x,\,k_y,\,k_z$). However, solutions for
complex values of $\omega $ and $\mathbf{k}$ do exist and are useful (see
subsection on Evanescent Waves).

It is customary to write out separately the positive-frequency ($\omega >0$)
and negative-frequency components of the fields, although both must be
present in the physical solution, which is real. Quantum-mechanically they
correspond to photon creation and annihilation operators, respectively.

Equation (\ref{3}) describes a plane wave: the wave fronts are planes
perpendicular to $\mathbf{k}.$ Letting $\xi $ be the coordinate parallel to $%
\mathbf{k}$, the fundamental plane wave is 
\begin{equation}
u(\xi ,\,t)=u_0\exp \left[ i(k_\xi \xi -\omega t)\right]  \label{5}
\end{equation}
with 
\begin{equation}
k^2=k_\xi ^2=\varepsilon \mu \,\frac{\omega ^2}{c^2}=n^2\frac{\omega ^2}{c^2}
\label{6}
\end{equation}
In the simplest case, when $\varepsilon $ and $\mu $ are real and positive,
the \textsl{index of refraction} $n$ is the positive square root of $%
\varepsilon \mu $, and by definition $k=n|\omega |/c$ is positive. There are
then four solutions of the form (\ref{5}): two for $\omega >0$ and $k_\xi
=\pm k,$ and two for $\omega <0$ and $k_\xi =\pm k.$ It is enough to
consider complex solutions with $\omega >0$ and pick up their real part
(which automatically has $\omega <0$). The general \textsl{real }plane wave
solution for a given frequency is then 
\begin{equation}
u(\xi ,\,t)=\func{Re}\left[ Ae^{ik\xi -i\omega t}+Be^{-ik\xi -i\omega
t}\right]  \label{7}
\end{equation}
with 
\begin{equation}
vk=\omega ,\quad \quad \quad \quad v=c/n.  \label{8}
\end{equation}
More explicitly, with $A=\left| A\right| e^{i\alpha }$ and $B=\left|
B\right| e^{i\beta }$, we have 
\begin{equation}
u(\xi ,\,t)=\left| A\right| \cos (k\xi -\omega t+\alpha )+\left| B\right|
\cos (k\xi +\omega t+\beta )  \label{9}
\end{equation}
In (\ref{7}) or (\ref{9}) the first term describes a wave propagating in the
direction of increasing $\xi ,$ the second term a wave propagating in the
direction of decreasing $\xi .$

When $n$ does not depend on $\omega ,$ the \textsl{general plane wave }%
solution is of the form 
\begin{equation}
u(\xi ,\,t)=f(\xi -vt)+g(\xi +vt)  \label{10}
\end{equation}
with $f$ and $g$ arbitrary functions. This is obtained by linear (Fourier)
superposition of the solutions (\ref{7}) or (\ref{9}) with different
frequencies, or by direct solution of the one-dimensional wave equation.

The most general solution of the three-dimensional wave equation (\ref{1})
is a superposition of plane waves running in different directions, or a sum (%
\ref{3}) over different $\mathbf{k}$ values.

\subsubsection{Dispersion}

If the refractive index $n=\sqrt{\varepsilon \mu }$ is a function of $\omega
,$ the general solution is still a sum of monochromatic plane waves (\ref{3}%
), but eq.(\ref{10}) is no longer a valid expression for the general plane
wave solution. Instead, the shape of a plane wave train will change wit
time, due to the $\omega $-dependence of the phase velocity $v=c/n.$ The
medium is said to be \textsl{dispersive.}

\subsubsection{Transverse polarization}

While each cartesian component of the electromagnetic fields must satisfy
the scalar wave equation, there are additional constraints imposed by
Maxwell's equations, in particular by $\mathbf{\nabla \cdot D}=0.$ As a
result, in our case $\mathbf{k\cdot E}=0$ and the fundamental solution for
the electric field can be written 
\begin{equation}
\mathbf{E}(\mathbf{r},\,t)=(\mathbf{\epsilon }_1\,E_1+\mathbf{\epsilon }%
_2\,E_2)\exp \left[ i\left( \mathbf{k\cdot r}-\omega t\right) \right]
\label{11}
\end{equation}
where $\mathbf{\epsilon }_1$and $\mathbf{\epsilon }_2$ are two unit vectors
such that $\mathbf{\epsilon }_1,\,\mathbf{\epsilon }_2,\,\mathbf{k}$ define
a right-handed set of orthogonal axes. The amplitudes $E_1$ and $E_2$ are in
general complex.

Any unit vector such that $\mathbf{k\cdot \epsilon }=0$ is a possible 
\textsl{polarization vector}. It would be sufficient to consider real unit
vectors $\mathbf{\epsilon }_1$ and $\,\mathbf{\epsilon }_2,$ but it is often
convenient to introduce complex polarization vectors (see the treatment of
circular polarization in section 2).

The Maxwell equation $\nabla \times \mathbf{E=-}\dfrac 1c\,\dfrac{\partial 
\mathbf{B}}{\partial t}$ gives 
\[
\frac \omega c\,\mathbf{B}=\mathbf{k\times E,}
\]
while $\nabla \times \mathbf{H=}\dfrac 1c\,\dfrac{\partial \mathbf{D}}{%
\partial t}$ gives 
\[
\varepsilon \mu \,\frac \omega c\,\mathbf{E}=-\,\mathbf{k\times B.}
\]
These equations are mutually compatible: both imply that $\mathbf{B}$ is
orthogonal to $\mathbf{E}$ and to $\mathbf{k},$ and that $B=nE.$ Note that,
for real $n,$ $\mathbf{B}$ is in phase with $\mathbf{E}.$

\subsubsection{Energy and the Poynting vector}

With $\mathbf{E}$ and $\mathbf{B}$\textbf{\ }defined as above, the
time-averaged energy density is 
\[
\left\langle U\right\rangle =\frac \varepsilon {8\pi }\left( \left|
E_1\right| ^2+\left| E_2\right| ^2\right) 
\]
The energy flow is in the direction of $\mathbf{k}$ and has magnitude given
by the Poynting vector. The time-averaged energy flow is 
\[
\left\langle S\right\rangle =v\left\langle U\right\rangle =\frac c{8\pi }%
\sqrt{\frac \varepsilon \mu }\,\left( \left| E_1\right| ^2+\left| E_2\right|
^2\right) 
\]
Both $\left\langle U\right\rangle $ and $\left\langle S\right\rangle $ are
independent of the orientation of the mutually orthogonal unit vectors $%
\mathbf{\epsilon }_1$ and $\mathbf{\epsilon }_2.$

\subsubsection{Evanescent waves}

As noted in connection with eq. (\ref{1}), the fundamental solution still
holds if some of the numbers $k_x,\,k_y,\,k_z,\,\omega $ are complex,
provided that $k_x^2+k_y^2+\,k_z^2=\varepsilon \mu \,\omega ^2/c^2.$

In particular, solutions with real $k_x,\,k_y,\,\omega $ and purely
imaginary $\,k_z$ occur naturally in the problem of wave propagation across
a plane interface between two media at $z=0.$ Usually a solution of this
type is of interest for $z>0$ (for instance) and has $\,k_z=i\alpha $ with $%
\alpha >0.$ It is then of the form 
\[
u(\mathbf{r},\,t)=u_0\exp \left[ i(k_xx+\,k_yy-\omega t)\right] e^{-\alpha
z} 
\]
It vanishes for $z\rightarrow +\infty $ and is usually called an \textsl{%
evanescent wave}.

More generally, complex $\mathbf{k}$ solutions occur for wave propagation in
inhomogeneous media and are called \textsl{inhomogeneous plane waves} by
Jackson. If $\mathbf{k}=\mathbf{k}_R+i\mathbf{k}_I,$ we find that $%
k^2=\varepsilon \mu \omega ^2/c^2$ gives 
\[
k_R^2-k_I^2=\varepsilon \mu \omega ^2/c^2,\quad \quad \quad \mathbf{k}%
_R\cdot \mathbf{k}_I=0
\]
In the example above, $\mathbf{k}_R=k_x\mathbf{\hat{x}+}k_y\mathbf{\hat{y}}$
and $\mathbf{k}_I=\alpha \mathbf{\hat{z}}.$

\section{Polarization}

The tip of the vector 
\[
\mathbf{E}(\mathbf{r},\,t)=\func{Re}\left( (\mathbf{\epsilon }_1\,E_1+%
\mathbf{\epsilon }_2\,E_2)\exp \left[ i\left( \mathbf{k\cdot r}-\omega
t\right) \right] \right) 
\]
describes an ellipse (for fixed $\mathbf{\ k\cdot r}$ as a function of $t$,
and for fixed $t$ as a function of $\mathbf{k\cdot r}).$

If $E_1$ and $E_2$ have the same phase (so that $E_2/E_1$ is real), the
ellipse degenerates into a line segment. The radiation is then \textsl{%
linearly polarized} at an angle $\arctan (E_2/E_1)$ to the direction of $%
\mathbf{\epsilon }_1.$

If $E_1$ and $E_2$ are 90$^{\circ }$ out of phase (so that $E_2/E_1$ is pure
imaginary) the axes of the ellipse are in the directions of $\mathbf{%
\epsilon }_1$ and $\mathbf{\epsilon }_2,$ and the lengths of the semiaxes
are $\left| E_1\right| $ and $\left| E_2\right| .$ Without loss of
generality (up to an irrelevant phase factor) we can choose $E_1$ to be real
and $\mathbf{\epsilon }_1$ to point along the major axis. Then $E_1=a$ and $%
E_2=\pm ib,$ where $a$ and $b$ are the semiaxes.

If $E_2=iE_1$ (or $b=a$ and $E_2=+ib),$ the tip of $\mathbf{E}$ describes a
circle counterclockwise as a function of $t$ (the tip $\mathbf{E}$ describes
a circle clockwise as a function of $\mathbf{k\cdot r}$ for fixed $t).$ In
optics, light of this kind is said to be \textsl{left} circularly polarized.
If $E_2=iE_1,$ light is said to be \textsl{right} circularly polarized (in
the optics literature; in particle physics the opposite convention is used).
In general, for a given elliptical polarization one must still specify if
the motion of the tip of $\mathbf{E}$ is clockwise or counterclockwise.

\end{document}
