The general electromagnetic plane wave
We consider an electromagnetic plane wave in a homogeneous, isotropic, source-free medium. Let \(\mathbf{s}\) be a unit vector in the direction of propagation. The wave depends on position and time through the single variable
\[ u=\mathbf{r}\cdot\mathbf{s}-vt. \]
The defining property of a plane wave is that the fields depend only on \(u\):
\[ \mathbf{E}=\mathbf{E}(u), \qquad \mathbf{H}=\mathbf{H}(u). \]
Relation between the electric and magnetic fields
Denote differentiation with respect to \(u\) by a prime. The chain rule gives
\[ \frac{\partial \mathbf{E}}{\partial t}=-v\mathbf{E}', \qquad \nabla\times\mathbf{E}=\mathbf{s}\times\mathbf{E}', \]
and the magnetic field satisfies the same relations. Substituting these expressions into Maxwell’s equations, with \(\mathbf{j}=0\), and using \(\mathbf{D}=\epsilon\mathbf{E}\) and \(\mathbf{B}=\mu\mathbf{H}\), gives
\[ \mathbf{s}\times\mathbf{H}'+\frac{\epsilon v}{c}\mathbf{E}'=0, \qquad \mathbf{s}\times\mathbf{E}'-\frac{\mu v}{c}\mathbf{H}'=0. \]
Using \(v/c=1/\sqrt{\epsilon\mu}\) and integrating with respect to \(u\), while neglecting constant fields, we obtain
\[ \mathbf{E}=-\sqrt{\frac{\mu}{\epsilon}}\,\mathbf{s}\times\mathbf{H}, \qquad \mathbf{H}=\sqrt{\frac{\epsilon}{\mu}}\,\mathbf{s}\times\mathbf{E}. \]
Taking the scalar product with \(\mathbf{s}\) gives
\[ \mathbf{E}\cdot\mathbf{s}=0, \qquad \mathbf{H}\cdot\mathbf{s}=0. \]
Thus, both fields are transverse to the direction of propagation. The vectors \(\mathbf{E}\), \(\mathbf{H}\), and \(\mathbf{s}\) are mutually perpendicular and form a right-handed triad. Taking the magnitudes of the field relations gives
\[ \sqrt{\mu}\,H=\sqrt{\epsilon}\,E, \qquad E=|\mathbf{E}|, \quad H=|\mathbf{H}|. \]
Energy density
The total energy density of the electromagnetic field is the sum of the electric and magnetic contributions:
\[ w=w_e+w_m =\frac{\epsilon E^2}{8\pi}+\frac{\mu H^2}{8\pi}. \]
The field-amplitude relation above gives
\[ \sqrt{\mu}\,H=\sqrt{\epsilon}\,E \quad\Longrightarrow\quad \mu H^2=\epsilon E^2. \]
Therefore, the electric and magnetic parts of the energy density are equal:
\[ \begin{aligned} w &=\frac{\epsilon E^2}{8\pi}+\frac{\mu H^2}{8\pi}\\ &=\frac{\epsilon E^2}{8\pi}+\frac{\epsilon E^2}{8\pi}\\ &=\frac{\epsilon}{4\pi}E^2 =\frac{\mu}{4\pi}H^2. \end{aligned} \tag{1}\]
The key step is to write the two contributions to the energy density and then use the field-amplitude relation to express one in terms of the other.
Energy flow
Since \(\mathbf{E}\times\mathbf{H}\) points in the direction of \(\mathbf{s}\), the Poynting vector is
\[ \mathbf{S}=\frac{c}{4\pi}\mathbf{E}\times\mathbf{H} =\frac{c}{4\pi}EH\,\mathbf{s}. \]
Using the field-amplitude relation and the energy density expression, together with \(v=c/\sqrt{\epsilon\mu}\), we find
\[ \mathbf{S}=\frac{c}{\sqrt{\epsilon\mu}}w\,\mathbf{s} =vw\,\mathbf{s}. \]
Thus, electromagnetic energy flows at the wave speed \(v\) in the direction of propagation \(\mathbf{s}\).
The harmonic electromagnetic plane wave
For a time-harmonic plane wave, every Cartesian component of the electric and magnetic fields has the form
\[ a\cos(\tau+\delta)=\operatorname{Re}\{ae^{-i(\tau+\delta)}\}, \qquad a>0, \]
where
\[ \tau=\omega\left(t-\frac{\mathbf{r}\cdot\mathbf{s}}{v}\right) =\omega t-\mathbf{k}\cdot\mathbf{r}. \]
Because the fields are transverse, choose the \(z\)-axis along the propagation direction. At a fixed point in space, only the \(x\)- and \(y\)-components are nonzero:
\[ \begin{aligned} E_x&=a_1\cos(\tau+\delta_1),\\ E_y&=a_2\cos(\tau+\delta_2). \end{aligned} \]
The relative phase
\[ \delta=\delta_2-\delta_1 \]
determines the polarization state.
Complex representation and time averages
It is convenient to represent the real fields by complex vectors:
\[ \mathbf{E}(\mathbf{r},t)=\operatorname{Re}\{\mathbf{E}_0(\mathbf{r})e^{-i\omega t}\}, \qquad \mathbf{H}(\mathbf{r},t)=\operatorname{Re}\{\mathbf{H}_0(\mathbf{r})e^{-i\omega t}\}. \]
The real part is understood at the end of a linear calculation. The time-averaged energy densities and Poynting vector are
\[ \langle w_e\rangle=\frac{\epsilon}{16\pi}\mathbf{E}_0\cdot\mathbf{E}_0^*, \qquad \langle w_m\rangle=\frac{\mu}{16\pi}\mathbf{H}_0\cdot\mathbf{H}_0^*, \]
\[ \langle\mathbf{S}\rangle =\frac{c}{8\pi}\operatorname{Re}(\mathbf{E}_0\times\mathbf{H}_0^*). \]
For a nonconducting medium with no sources or absorbers,
\[ \nabla\cdot\langle\mathbf{S}\rangle=0. \]
a) Elliptic polarization
Eliminating \(\tau\) from the two field components gives
\[ \left(\frac{E_x}{a_1}\right)^2 +\left(\frac{E_y}{a_2}\right)^2 -2\frac{E_xE_y}{a_1a_2}\cos\delta =\sin^2\delta. \]
This is an ellipse in the plane perpendicular to the propagation direction, so the wave is generally elliptically polarized. The magnetic vector is also elliptically polarized.
Let \(a\) and \(b\) be the major and minor semiaxes, with \(a\geq b\), and let \(\psi\) be the angle between the major axis and the \(x\)-axis. If
\[ \tan\alpha=\frac{a_2}{a_1}, \]
then
\[ \begin{aligned} a^2+b^2&=a_1^2+a_2^2,\\ \tan 2\psi&=(\tan 2\alpha)\cos\delta. \end{aligned} \]
The ellipticity can be described by an angle \(\chi\):
\[ \sin 2\chi=(\sin 2\alpha)\sin\delta, \qquad \tan\chi=\pm\frac{b}{a}. \]
The sign of \(\chi\) distinguishes the two senses in which the electric-vector endpoint travels around the ellipse.
b) Linear and circular polarization
Two important special cases occur when the ellipse becomes a straight line or a circle.
Linear polarization
If
\[ \delta=m\pi, \qquad m=0,\pm1,\pm2,\ldots, \]
then
\[ \frac{E_y}{E_x}=(-1)^m\frac{a_2}{a_1}, \]
and the electric vector remains on one fixed line. The wave is linearly polarized.
Circular polarization
If
\[ a_1=a_2=a, \qquad \delta=\frac{m\pi}{2}, \qquad m=\pm1,\pm3,\ldots, \]
then
\[ E_x^2+E_y^2=a^2, \]
so the polarization ellipse is a circle. With the convention used here, \(\sin\delta>0\) corresponds to right-handed polarization and \(\sin\delta<0\) to left-handed polarization.
In complex notation,
\[ \frac{E_y}{E_x}=\frac{a_2}{a_1}e^{-i\delta}. \]
For linear polarization this ratio is real; for right- and left-handed circular polarization it is \(-i\) and \(+i\), respectively.
c) Stokes parameters and the Poincare sphere
The polarization state of a plane monochromatic wave can be described by the four Stokes parameters
\[ \begin{aligned} s_0&=a_1^2+a_2^2,\\ s_1&=a_1^2-a_2^2,\\ s_2&=2a_1a_2\cos\delta,\\ s_3&=2a_1a_2\sin\delta. \end{aligned} \]
Only three are independent because
\[ s_0^2=s_1^2+s_2^2+s_3^2. \]
If \(\psi\) is the orientation angle of the ellipse and \(\chi\) is its ellipticity angle, then
\[ \begin{aligned} s_1&=s_0\cos 2\chi\cos 2\psi,\\ s_2&=s_0\cos 2\chi\sin 2\psi,\\ s_3&=s_0\sin 2\chi. \end{aligned} \]
Thus \((s_1,s_2,s_3)\) can be regarded as the coordinates of a point on a sphere of radius \(s_0\). This is the Poincare sphere: linear polarization lies on its equator, while the two circular polarization states lie at opposite poles.
Harmonic vector waves of arbitrary form
The preceding results can be extended to a time-harmonic vector wave whose amplitude and phase vary with position. Write its Cartesian components as
\[ \begin{aligned} V_x(\mathbf{r},t)&=a_1(\mathbf{r})\cos[\omega t-g_1(\mathbf{r})],\\ V_y(\mathbf{r},t)&=a_2(\mathbf{r})\cos[\omega t-g_2(\mathbf{r})],\\ V_z(\mathbf{r},t)&=a_3(\mathbf{r})\cos[\omega t-g_3(\mathbf{r})]. \end{aligned} \]
Here \(a_s(\mathbf{r})\) and \(g_s(\mathbf{r})\) are real functions of position. For a plane harmonic wave, the amplitudes are constant and the phase functions have the common spatial dependence \(\mathbf{k}\cdot\mathbf{r}\).
Each component can be written as a combination of \(\cos\omega t\) and \(\sin\omega t\):
\[ V_x(\mathbf{r},t)=p_x(\mathbf{r})\cos\omega t+q_x(\mathbf{r})\sin\omega t, \]
with
\[ p_x=a_1\cos g_1, \qquad q_x=a_1\sin g_1, \]
and analogous expressions for the \(y\)- and \(z\)-components. Define two real vectors
\[ \mathbf{p}=(p_x,p_y,p_z), \qquad \mathbf{q}=(q_x,q_y,q_z). \]
The general real vector wave then has the compact form
\[ \mathbf{V}(\mathbf{r},t) =\mathbf{p}(\mathbf{r})\cos\omega t +\mathbf{q}(\mathbf{r})\sin\omega t. \]
By Fourier analysis, an arbitrary vector wave can be represented as a superposition of waves of this type.
Complex vector representation
As for scalar waves, introduce the complex vector amplitude
\[ \mathbf{U}(\mathbf{r})=\mathbf{p}(\mathbf{r})+i\mathbf{q}(\mathbf{r}). \]
Then the real field is recovered from
\[ \mathbf{V}(\mathbf{r},t) =\operatorname{Re}\{\mathbf{U}(\mathbf{r})e^{-i\omega t}\}. \]
For linear operations, it is enough to work with the complex field and take the real part at the end. The complex conjugate is
\[ \mathbf{U}^*=\mathbf{p}-i\mathbf{q}. \]
Useful products are
\[ \begin{aligned} \mathbf{U}\cdot\mathbf{U} &=\mathbf{p}^2-\mathbf{q}^2+2i\,\mathbf{p}\cdot\mathbf{q},\\ \mathbf{U}\cdot\mathbf{U}^* &=\mathbf{p}^2+\mathbf{q}^2. \end{aligned} \]
The second expression is real and non-negative, so it is useful when calculating time-averaged quantities. In particular, the electric and magnetic fields may be written as
\[ \begin{aligned} \mathbf{E}(\mathbf{r},t)&=\operatorname{Re}\{\mathbf{E}_0(\mathbf{r})e^{-i\omega t}\},\\ \mathbf{H}(\mathbf{r},t)&=\operatorname{Re}\{\mathbf{H}_0(\mathbf{r})e^{-i\omega t}\}. \end{aligned} \]
Their time-averaged energy densities and Poynting vector are therefore obtained directly from the complex amplitudes:
\[ \langle w_e\rangle=\frac{\epsilon}{16\pi}\mathbf{E}_0\cdot\mathbf{E}_0^*, \qquad \langle w_m\rangle=\frac{\mu}{16\pi}\mathbf{H}_0\cdot\mathbf{H}_0^*, \]
\[ \langle\mathbf{S}\rangle =\frac{c}{8\pi}\operatorname{Re}(\mathbf{E}_0\times\mathbf{H}_0^*). \]
Thus the complex-vector method separates the rapid oscillation in time from the spatially varying vector amplitude, while retaining the full polarization information.
Polarization ellipse at a fixed point
At a fixed point, suppress the position dependence and write
\[ \mathbf{V}(t)=\mathbf{p}\cos\omega t+\mathbf{q}\sin\omega t. \]
Introduce two new vectors
\[ \mathbf{a}=\mathbf{p}\cos\varepsilon+\mathbf{q}\sin\varepsilon, \qquad \mathbf{b}=-\mathbf{p}\sin\varepsilon+\mathbf{q}\cos\varepsilon. \]
Choose \(\varepsilon\) so that \(\mathbf{a}\cdot\mathbf{b}=0\) and \(a\geq b\). Then
\[ \tan 2\varepsilon =\frac{2\mathbf{p}\cdot\mathbf{q}}{p^2-q^2}. \]
Let
\[ A=2\mathbf{p}\cdot\mathbf{q}, \qquad B=p^2-q^2. \]
Since \(\tan 2\varepsilon=A/B\), the pair \((\cos 2\varepsilon,\sin 2\varepsilon)\) is proportional to \((B,A)\). Normalizing it with \(\sin^2 2\varepsilon+\cos^2 2\varepsilon=1\) gives
\[ \sin 2\varepsilon =\frac{2\mathbf{p}\cdot\mathbf{q}} {\sqrt{(p^2-q^2)^2+4(\mathbf{p}\cdot\mathbf{q})^2}}, \]
\[ \cos 2\varepsilon =\frac{p^2-q^2} {\sqrt{(p^2-q^2)^2+4(\mathbf{p}\cdot\mathbf{q})^2}}. \]
The common sign is chosen so that \(a\geq b\). With
\[ D=\sqrt{(p^2-q^2)^2+4(\mathbf{p}\cdot\mathbf{q})^2}, \]
the principal semiaxes are
\[ a^2=\frac12(p^2+q^2+D), \qquad b^2=\frac12(p^2+q^2-D). \]
The field can therefore be written as
\[ \mathbf{V}(t) =\mathbf{a}\cos(\omega t-\varepsilon) +\mathbf{b}\sin(\omega t-\varepsilon), \]
so its endpoint traces an ellipse whose principal axes are along \(\mathbf{a}\) and \(\mathbf{b}\).
Circular and linear polarization
For circular polarization, \(a=b\), so \(D=0\). Because \(D^2\) is a sum of two non-negative terms,
\[ p^2=q^2, \qquad \mathbf{p}\cdot\mathbf{q}=0. \]
Thus \(\mathbf{p}\) and \(\mathbf{q}\) are perpendicular and have equal lengths. The magnitude of \(\mathbf{V}\) remains constant while its direction rotates.
For linear polarization, the minor semiaxis vanishes: \(b=0\). The semiaxis formula then gives
\[ p^2q^2=(\mathbf{p}\cdot\mathbf{q})^2. \]
This is the equality case of the Cauchy–Schwarz inequality, so \(\mathbf{p}\) and \(\mathbf{q}\) are parallel or antiparallel. Hence \(\mathbf{V}(t)\) always lies along one fixed direction and moves back and forth on a straight line.
In short, equal perpendicular vectors \(\mathbf{p}\) and \(\mathbf{q}\) produce circular polarization, parallel vectors produce linear polarization, and the general case produces elliptical polarization. For a vector wave of arbitrary form, this classification is local and may vary from point to point.