On the monochromatic plane electromagnetic waves

Tian-Hao Zhang (张天昊) · 2026-02-15

Notes on the monochromatic plane wave and its polarization.
Ref: Xiong-Jun Liu (刘雄军)'s lecture notes at Peking University, 2023 Autumn.

Plane waves

Work in geometrized Gaussian unit ststem, Maxwell equations are

Eq. (1)

\begin{matrix} \nabla \cdot 𝑬 = 0, \nabla \times 𝑩 - \frac{𝜕 𝑬}{𝜕 𝑡} = 𝟎, \\ \nabla \cdot 𝑩 = 0, \nabla \times 𝑬 + \frac{𝜕 𝑩}{𝜕 𝑡} = 𝟎; \end{matrix}

wave equations are

Eq. (2)

\nabla^{2} 𝑬 - \frac{𝜕^{2} 𝑬}{𝜕 𝑡^{2}} = 𝟎, \nabla^{2} 𝑩 - \frac{𝜕^{2} 𝑩}{𝜕 𝑡^{2}} = 𝟎 .

Properties of plane waves

Def (plane wave) A plane wave is a solution to the wave equation that propagates along a fixed direction. A plane wave propagating along the $𝒏$ direction is of the form $𝑬 (𝒙, 𝑡) = 𝑬_{profile} (𝒏 \cdot 𝒙 - 𝑡) .$

From $𝜕^{} 𝑩 / 𝜕 𝑡^{} = - \nabla \times 𝑬$ we shall have

Eq. (3)

\begin{aligned} \frac{𝜕 𝐵_{𝑖}}{𝜕 𝑡} & = - 𝜀_{𝑖 𝑗 𝑘} \nabla^{𝑗} 𝐸_{profile}^{𝑘} \\ = - 𝜀_{𝑖 𝑗 𝑘} 𝑛^{𝑗} \frac{d 𝐸_{profile}^{𝑘}}{d 𝜑}, \end{aligned}

where $d^{} 𝐸_{profile}^{𝑘} / d 𝜑^{}$ is the derivative of $𝐸_{profile}^{𝑘}$ w.r.t. its argument $𝜑 = 𝒏 \cdot 𝒙 - 𝑡 .$ Now

Eq. (4)

\frac{𝜕 𝐸_{profile}^{𝑘}}{𝜕 𝑡} = - \frac{d 𝐸_{profile}^{𝑘}}{d 𝜑},

Eq. (5)

\begin{matrix} \frac{𝜕 𝐵_{𝑖}}{𝜕 𝑡} = 𝜀_{𝑖 𝑗 𝑘} 𝑛^{𝑗} \frac{𝜕 𝐸_{profile}^{𝑘}}{𝜕 𝑡} \\ \Rightarrow 𝐵_{𝑖} = 𝜀_{𝑖 𝑗 𝑘} 𝑛^{𝑗} 𝐸^{𝑘} + static magnetic filed . \end{matrix}

The static magnetic field is of no interest since we are considering time-dependent waves. Thus

Eq. (6)

𝑩 = 𝒏 \times 𝑬 .

Similarly, drop another static field, $\nabla \cdot 𝑬 = 0$ gives

Eq. (7)

\begin{matrix} 0 = \nabla_{𝑖} 𝐸_{profile}^{𝑖} = 𝑛_{𝑖} \frac{d 𝐸_{profile}^{𝑖}}{d 𝜑} = - \frac{𝜕}{𝜕 𝑡} (𝒏 \cdot 𝑬) \\ \Rightarrow 𝒏 \cdot 𝑬 = 0 . \end{matrix}

Prop (plane wave) $𝒏 \cdot 𝑬 = 0, 𝑩 = 𝒏 \times 𝑬 \Rightarrow 𝒏 \cdot 𝑩 = 0, 𝑬 \cdot 𝑩 = 0, | 𝑬 | = | 𝑩 | .$
Remark The propertity $𝒏 \cdot 𝑬 = 0$ will not hold in general in media where the charge density is not zero, whereas $𝑩 = 𝒏 \times 𝑬$ always holds.

Monochromatic plane waves

Def (monochromatic wave) A monochromatic wave is a solution to the wave equation that has a definite frequency, so that its time dependence is of sinusoidal form.

A monochromatic plane wave is of the form

Eq. (8)

𝑬 = Re (𝑬_{0} exp (i (𝒌 \cdot 𝒙 - 𝜔 𝑡)),

where $𝑬_{0} \in ℂ^{3}, 𝜔 > 0, 𝒌 = 𝒏 𝜔 .$ Since the Maxwell equations are linear, we could just work in the complex notation¹¹ Extra attention is needed when dealing with non-linear quantities, such as the energy density which is quadratic in the fields. with the understanding that we take the real part to get the physical quantity. Thus we simply write

Eq. (9)

𝑬 = 𝑬_{0} exp (i (𝒌 \cdot 𝒙 - 𝜔 𝑡)) .

Prop (monochromatic plane wave) $𝜔 = | 𝒌 |, 𝒌 \cdot 𝑬_{0} = 0, 𝑩_{0} = 𝒌 / 𝜔 \times 𝑬_{0} \Rightarrow 𝒌 \cdot 𝑩_{0} = 0, 𝑬_{0} \cdot 𝑩_{0} = 0, | 𝑬_{0} | = | 𝑩_{0} | .$

Polarization of a monochromatic plane EM wave

For a monochromatic plane EM wave, since $𝒏 \cdot 𝑬_{0} = 0,$ we could write

Eq. (10)

\begin{matrix} 𝑬_{0} = 𝐸_{0} {\hat{𝒆}}_{1} + {\tilde{𝐸}}_{0} {\hat{𝒆}}_{2}, \\ 𝑬 = (𝐸_{0} {\hat{𝒆}}_{1} + {\tilde{𝐸}}_{0} {\hat{𝒆}}_{2}) exp (i (𝒌 \cdot 𝒙 - 𝜔 𝑡)), \end{matrix}

where $𝐸_{0}, {\tilde{𝐸}}_{0} \in ℂ$ while ${\hat{𝒆}}_{1}, {\hat{𝒆}}_{2}$ are orthonormal and are both orthogonal to $𝒏 .$

WLOG set $𝒏 = {\hat{𝒆}}_{𝑧}, {\hat{𝒆}}_{1} = {\hat{𝒆}}_{𝑥}, {\hat{𝒆}}_{2} = {\hat{𝒆}}_{𝑦}, 𝐸_{0} \in ℝ_{\geq 0},$ then

Eq. (11)

𝑬 = [\begin{matrix} {\hat{𝒆}}_{𝑥} | {\hat{𝒆}}_{𝑦} | {\hat{𝒆}}_{𝑧} \end{matrix}] [\begin{matrix} 𝑎_{1} \\ 𝑎_{2} exp (i 𝛿) \\ 0 \end{matrix}] exp (i 𝜔 (𝑧 - 𝑡)),

where $𝑎_{1}, 𝑎_{2} \in ℝ_{\geq 0}$ and $𝛿 \in [- π, π)$ is the phase difference between ${\tilde{𝐸}}_{0}$ and $𝐸_{0} .$

If we fix a value of $𝑧$ (e.g. $𝑧 = 0$ ), then the end point of the vector

Eq. (12)

𝑬 = {\hat{𝒆}}_{𝑥} 𝑎_{1} cos (𝜔 𝑡) + {\hat{𝒆}}_{𝑦} 𝑎_{2} cos (𝜔 𝑡 - 𝛿)

will in general depict a ellipse in the $𝑥 𝑂 𝑦$ plane

Eq. (13)

\frac{𝑥^{2}}{𝑎_{1}^{2}} + \frac{𝑦^{2}}{𝑎_{2}^{2}} - \frac{2 𝑥 𝑦}{𝑎_{1} 𝑎_{2}} cos 𝛿 = {sin}^{2} 𝛿 .

It is in this sense that the monochromatic plane wave is elliptically polarized.

Linear polarization

Def (linear polarization) A monochromatic plane wave is said to be linearly polarized, if the direction of its electric field does not change, that is, if the ellipse in Eq. (13) degenerates into a straight line.

Using Eq. (11), a wave is linearly polarized if one of the following conditions holds:

$𝑎_{1} = 0, 𝑎_{2} \neq 0;$
$𝑎_{1} \neq 0, 𝑎_{2} = 0;$
$𝑎_{1} \neq 0, 𝑎_{2} \neq 0, 𝛿 = 0;$
$𝑎_{1} \neq 0, 𝑎_{2} \neq 0, 𝛿 = π .$

Note that the basis we used in Eq. (10) are linearly polarized.

Circular polarization

Def (linear polarization) A monochromatic plane wave is said to be circularly polarized, if the ellipse in Eq. (13) degenerates into a circle.

Using Eq. (11), a wave is circularly polarized if $𝑎_{1} = 𝑎_{2} \neq 0$ and $𝛿 = \pm π / 2 .$
For $𝛿 = π / 2,$ the vector

Eq. (14)

𝑬 = 𝑎 ({\hat{𝒆}}_{𝑥} cos (𝜔 𝑡) + {\hat{𝒆}}_{𝑦} sin (𝜔 𝑡))

ratotes anticlockwise in the $𝑥 𝑂 𝑦$ plane; this wave is said to be right-hand circularly polarized (labelled by $𝜆 = +$ ) and have positive helicity²² Positive helicity means that the ratotion is parallel to the direction of propagation. Indeed, the helicity, which is defined as the component of spin angular momentum along the direction of propagation, of the mode characterized by wave vector $𝒌$ and polarization $𝜆 (= \pm),$ is given by $𝜎_{𝒌, 𝜆} = sgn (𝜆) {⟨ 𝐸 ⟩}_{𝒌, 𝜆} / 𝜔,$ where ${⟨ 𝐸 ⟩}_{𝒌, 𝜆}$ is the time-averaged total energy of the mode; this can be compared with the Planck formula in quantum mechanics $𝐸 = ℏ 𝜔 .$ In the transition from classical to quantum physics, the helicity of the EM field becomes the spin of the photon.. Likewise, the wave with $𝛿 = - π / 2$ is said to be left-hand circularly polarized (labelled by $𝜆 = -$ ) and have negative helicity.

It is sometimes convenient to expand $𝑬$ in terms of circularly-polarized waves, rather than linearly polarized waves as in Eq. (10). Define

Eq. (15)

{\hat{𝒆}}_{\pm} ≔ \frac{1}{\sqrt{2}} ({\hat{𝒆}}_{1} \pm 𝑖 {\hat{𝒆}}_{2});

that is

Eq. (16)

[\begin{matrix} {\hat{𝒆}}_{+} | {\hat{𝒆}}_{-} \end{matrix}] = [\begin{matrix} {\hat{𝒆}}_{1} | {\hat{𝒆}}_{2} \end{matrix}] [\begin{matrix} \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \\ \frac{𝑖}{\sqrt{2}} & - \frac{𝑖}{\sqrt{2}} \end{matrix}] .

Note that ${\hat{𝒆}}_{\pm}^{*} = {\hat{𝒆}}_{\mp} .$ It follows from the orthonormality of ${\hat{𝒆}}_{1}, {\hat{𝒆}}_{2}$ that

Eq. (17)

{\hat{𝒆}}_{+} \cdot {\hat{𝒆}}_{+} = 0, {\hat{𝒆}}_{-} \cdot {\hat{𝒆}}_{-} = 0, {\hat{𝒆}}_{+} \cdot {\hat{𝒆}}_{-} = 1 .

The inverse transform is

Eq. (18)

{\hat{𝒆}}_{1} = \frac{1}{\sqrt{2}} ({\hat{𝒆}}_{+} + {\hat{𝒆}}_{-}), {\hat{𝒆}}_{2} = - \frac{𝑖}{\sqrt{2}} ({\hat{𝒆}}_{+} - {\hat{𝒆}}_{-});

that is

Eq. (19)

[\begin{matrix} {\hat{𝒆}}_{1} | {\hat{𝒆}}_{2} \end{matrix}] = [\begin{matrix} {\hat{𝒆}}_{+} | {\hat{𝒆}}_{-} \end{matrix}] [\begin{matrix} \frac{1}{\sqrt{2}} & - \frac{𝑖}{\sqrt{2}} \\ \frac{1}{\sqrt{2}} & \frac{𝑖}{\sqrt{2}} \end{matrix}] .

Stokes parameters

The magnitude and phase information is sometimes expressed in terms of the so called Stokes parameters $(𝑠_{0}, 𝑠_{1}, 𝑠_{2}, 𝑠_{2}),$ which are defined by (using the convensions and notations in Eq. (11))

Eq. (20)

\begin{aligned} 𝑠_{0} & ≔ 𝐸_{𝑥} 𝐸_{𝑥}^{*} + 𝐸_{𝑦} 𝐸_{𝑦}^{*} & = 𝑎_{1}^{2} + 𝑎_{2}^{2}, \\ 𝑠_{1} & ≔ 𝐸_{𝑥} 𝐸_{𝑥}^{*} - 𝐸_{𝑦} 𝐸_{𝑦}^{*} & = 𝑎_{1}^{2} - 𝑎_{2}^{2}, \\ 𝑠_{2} & ≔ 2 Re (𝐸_{𝑥}^{*} 𝐸_{𝑦}) & = 2 𝑎_{1} 𝑎_{2} cos 𝛿, \\ 𝑠_{3} & ≔ 2 Im (𝐸_{𝑥}^{*} 𝐸_{𝑦}) & = 2 𝑎_{1} 𝑎_{2} sin 𝛿 . \end{aligned}

Note that

Eq. (21)

𝑠_{0}^{2} = 𝑠_{1}^{2} + 𝑠_{2}^{2} + 𝑠_{3}^{2} .

The parameter $𝑠_{0}$ characterizes the intensity of the wave, while $𝑠_{1}$ characterizes the amount of $𝑥$ polarization versus $𝑦$ polatization; the third independent parameter, either $𝑠_{2}$ or $𝑠_{3}$ , characterizes the phase difference between the $𝑦$ and $𝑥$ polarized waves.

Prop (characterization of linear/circular polarization)
- Linear polarization corresponds to $𝑠_{3} = 0;$
- circular polarization corresponds to $𝑠_{1} = 𝑠_{2} = 0 .$

Doppler effect

The world lines of the observer and the illuminant (light source) are both time-like. Assume that at spacetime point $𝑝$ the illuminant whose 4-velocity is $𝑉^{𝑎}$ in abstract index notation, emits a monochromatic plane wave (at least in the geometric optics approximation sense) with 4-wavevector $𝐾^{𝑎},$ which is received at spacetime point $𝑞$ by the observer whose 4-velocity is $𝑈^{𝑎} .$

Note that ${𝐾^{𝑎} |}_{𝑝} = {𝐾^{𝑎} |}_{𝑞},$ the angular-frequency measured by the illuminant at emitting is

Eq. (22)

𝜔 = - 𝐾_{𝑎} 𝑉^{𝑎}, 𝑉^{𝑎} \in 𝑇_{𝑝} ℝ^{4}

and by the observer at receiving is

Eq. (23)

𝜔^{'} = - 𝐾_{𝑎} 𝑈^{𝑎}, 𝑈^{𝑎} \in 𝑇_{𝑞} ℝ^{4} .

Since the Minkowski space $(ℝ^{4}, 𝜂, \nabla = 𝜕)$ is flat, $𝑈^{𝑎} \in 𝑇_{𝑞} ℝ^{4}$ can be can be identified with the vector $𝑈^{𝑎} \in 𝑇_{𝑝} ℝ^{4}$ via the global parallelism; from now on it is understood that all the calculations are done at $𝑝 .$

Let $𝛾 = - 𝑉^{𝑎} 𝑈_{𝑎},$ then

Eq. (24)

𝑈^{𝑎} = 𝛾 (𝑉^{𝑎} + 𝑢^{𝑎}),

where $𝛾 𝑢^{𝑎}$ is the projection of $𝑈^{𝑎}$ on the “space plane” perpendicular to $𝑉^{𝑎} .$ Also

Eq. (25)

𝐾_{𝑎} = - 𝜔 𝑉_{𝑎} + 𝑘_{𝑎},

thus

Eq. (26)

\begin{aligned} 𝜔^{'} & = - 𝐾_{𝑎} 𝑈^{𝑎} \\ = - 𝛾 (- 𝜔 𝑉_{𝑎} + 𝑘_{𝑎}) (𝑉^{𝑎} + 𝑢^{𝑎}) \\ = 𝛾 (𝜔 - 𝑘_{𝑎} 𝑢^{𝑎}) \\ \overset{in 3d notation}{=} 𝛾 (𝜔 - 𝒌 \cdot 𝒖) . \end{aligned}

Let $𝜃$ be the angle between $𝒌$ and $𝒖$ and note that $| 𝒌 | = 𝜔,$ the result is written

Eq. (27)

𝜔^{'} = 𝛾 𝜔 (1 - 𝑢 cos 𝜃)

where $𝑢 = | 𝒖 |, 𝛾 = 1 / \sqrt{1 - 𝑢^{2}} .$ Note that $𝒖$ is just the 3-velocity of the observer relative to the illuminant.

Non-relativistic limit of the Doppler effect

In this subsection, SI units is used.

For the light waves,

Eq. (28)

𝜔^{'} = 𝜔 (1 - \frac{𝑢}{𝑐} cos 𝜃) .

For other non-relativistic waves, the Doppler shift is most easily derive from the Galilean transformation.

Since the wave equation $((1 / 𝑣_{p}^{2}) 𝜕^{2} / 𝜕 𝑡^{2} - \nabla^{2}) 𝑓 = 0$ is invariant under the Galilean transformation, the plane wave phase $𝒌 \cdot 𝒙 - 𝜔 𝑡$ is also invariant. Since in Newtonian mechanics, the concept of the length is absolute, the wave-length³³ The wave-length is defined as the distance between the two material points (连续介质力学中的物质点) whose phase difference is $2 π .$ and thus the wave-vector $𝒌$ is Galilean invariant. Thus, under the Galilean transformation $𝒙^{'} = 𝒙 - 𝒖 𝑡,$ the angular-frequency transforms according to

Eq. (29)

\begin{matrix} 𝜔^{'} 𝑡 - 𝒌 \cdot 𝒙^{'} = 𝜔 - 𝒌 \cdot 𝒙 \\ \Rightarrow 𝜔^{'} = 𝜔 - 𝒌 \cdot 𝒖, \end{matrix}

where $𝜔$ is the angular-frequency measured by the illuminant and $𝜔^{'}$ by the observer, $𝒖$ is the velocity of the observer relative to the illuminant.

In space data analysis, let $𝜔_{sc}$ and $𝑽$ be the angular-frequency of the wave and the bulk velocity of the plasma as observed by the spacecraft, respectively, then the angular-frequency in the plasma frame is $𝜔_{pl} = 𝜔_{sc} - 𝒌 \cdot 𝑽 .$

Electrostatic waves

The electrostatic field satisfies the equations

Eq. (30)

\nabla^{2} 𝜑 = - 4 π 𝜌, 𝑬 = - \nabla 𝜑,

from which we derive the Fourier transform

Eq. (31)

𝜑 (𝒌) = \frac{4 π 𝜌 (𝒌)}{𝑘^{2}}, 𝑬 (𝒌) = - 𝑖 \frac{4 π 𝜌 (𝒌)}{𝑘^{2}} 𝒌 .

In particular, the electrostatic field is longitudinal (instead of transverse) $𝑬 (𝒌) ∥ 𝒌 .$

When coupling to the matters (such as plasmas), it is possible for an electrostatic pertubation to propagate through the system; this is an electrostatic wave.
If the coupling is linear, then the mode characterized by wave vector $𝒌$ evolves as (let $𝜔 = 𝜔 (𝒌) = 𝜔_{r} (𝒌) + 𝑖 𝛾 (𝒌) \in ℂ$ be the dispersion relation which encodes the linear interaction of the electric field and the matter)

Eq. (32)

\begin{aligned} 𝑬_{𝒌} (𝒙, 𝑡) & = \hat{𝒌} 𝐸_{0} (𝒌) exp (i (𝒌 \cdot 𝒙 - 𝜔 (𝒌) 𝑡)) \\ = \hat{𝒌} 𝐸_{0} (𝒌) exp (𝛾 (𝒌) 𝑡) exp (i (𝒌 \cdot 𝒙 - 𝜔_{r} (𝒌) 𝑡)), \end{aligned}

where $𝐸_{0} (𝒌) \in ℂ$ is determined by solving the linear system (including boundary and initial conditions). This is sometimes referred to as the “plane wave ansatz.” Note that if the coupling is non-linear (as is generally the case), then the most general solution need not possess a monochromatic plane wave form (although we could always try to search for solutions of this form).⁴⁴ The essence of the “plane wave ansatz”, or the “superposition principle” in linear systems, is two fold. On the one hand, the physical quantity under consideration can be decomposed into Fourier modes — we can always do this with aid of the mathematical Fourier theory. On the other hand, each Fourier mode evolves independently, and there is no interaction between any two of the modes — this is a consequence of linearity. Note, however, that if there exist non-linear phenomena, then different Fourier modes will indeed couple and cannot be solved independently. In such systems, for example, a pertubation with angular-frequency $𝜔$ might excite frequency multiplications $2 𝜔, 3 𝜔, \dots .$