Plane Waves in Homogeneous Isotropic Media

Every calculation in the optics of continuous media begins by writing the light as a plane wave, so this chapter opens by pinning down exactly what that object is and what a medium does to it. We take the vacuum plane wave — its wavefront geometry, its transversality $\mathbf{k}\cdot\mathbf{E}=0$ , and its energy flow — from aq-em-monochromatic-plane-waves-vacuum, and we take the source-free optical wave equation $\nabla^2\mathbf{E}=\mu\varepsilon\,\partial_t^2\mathbf{E}$ from aq-optics-maxwell-equations-dielectric-media. What this article owns is the in-medium extension: the single scalar constraint that survives, the dispersion relation $k=n\omega/c$ , and the way the refractive index rescales every field ratio the vacuum wave carried. A monochromatic plane wave is a disturbance whose phase is constant over every plane perpendicular to a fixed direction $\hat{\mathbf{k}}$ , sliding forward rigidly at one speed. In a homogeneous medium the parameters $\varepsilon,\mu$ do not depend on position, so the wave keeps its shape; in an isotropic medium they are scalars, so $\mathbf{D}\parallel\mathbf{E}$ and there is no preferred direction. Under these two assumptions the plane-wave substitutions $\nabla\to i\mathbf{k}$ , $\partial_t\to-i\omega$ turn Maxwell's coupled differential equations into algebra, and the whole of propagation collapses onto $k=n\omega/c$ with $n=\sqrt{\varepsilon_r\mu_r}$ . Three payoffs organize the rest of the chapter. First, the medium enters only through $n$ : the crests slow to $v_p=c/n$ and the wave impedance drops to $\eta=\eta_0/n$ , so the same field amplitude carries a different intensity in glass than in vacuum. Second, when the index is complex, $\tilde n=n+i\kappa$ , the identical plane-wave solution describes an attenuated wave whose amplitude decays as $e^{-\kappa\omega z/c}$ — the seed of absorption. Third, because $n=n(\omega)$ in every real material, plane waves of different colours travel at different speeds, which is the origin of dispersion, group velocity, and pulse spreading. The plane wave is the vehicle that converts microscopic material response into observable propagation.

Key takeaways

A monochromatic plane wave $\mathbf{E}=\mathbf{E}_0\,e^{i(\mathbf{k}\cdot\mathbf{r}-\omega t)}$ is the fundamental solution of Maxwell's equations in a linear, homogeneous, isotropic medium; every field is a Fourier superposition of these.
Substituting it into the optical wave equation collapses everything to one scalar constraint, the dispersion relation $k=n\omega/c$ with $n=\sqrt{\varepsilon_r\mu_r}\approx\sqrt{\varepsilon_r}$ optically.
Transversality is not an extra postulate: $\nabla\!\cdot\!\mathbf{D}=0$ forces $\mathbf{k}\cdot\mathbf{E}=0$ , and $\hat{\mathbf{k}},\mathbf{E},\mathbf{H}$ form a right-handed orthogonal triad.
The medium rescales the vacuum ratios by $n$ : phase velocity $v_p=c/n$ , wavelength $\lambda=\lambda_0/n$ , and wave impedance $\eta=\sqrt{\mu/\varepsilon}=\eta_0/n$ with $\eta_0=376.730\ \Omega$ .
Intensity is $I=\langle S\rangle=\tfrac12 c\varepsilon_0 n\,|\mathbf{E}_0|^2=v_p\langle u\rangle$ : the same amplitude carries $n$ times more power in a denser medium, and electric and magnetic energies are equal.
A complex index $\tilde n=n+i\kappa$ turns the same solution into a decaying wave, $|\mathbf{E}|^2\propto e^{-\alpha z}$ with $\alpha=4\pi\kappa/\lambda_0$ — the plane-wave root of Beer–Lambert.
Key idea: in a linear homogeneous isotropic medium the substitutions $\nabla\to i\mathbf{k},\ \partial_t\to-i\omega$ reduce Maxwell's equations to algebra — transversality plus the single dispersion relation $k=n\omega/c$ — with the refractive index $n$ encoding the material entirely.

The core insight

In a linear homogeneous isotropic medium the plane-wave substitutions $\nabla\to i\mathbf{k}$ and $\partial_t\to-i\omega$ reduce Maxwell's equations to algebra: transversality $\mathbf{k}\cdot\mathbf{E}=0$ plus the scalar dispersion relation $k=n\omega/c$ , $n=\sqrt{\varepsilon_r\mu_r}$ . Everything a medium does to a wave — slow it, attenuate it, disperse it — is carried by that one index.

Where this lives today

The plane-wave decomposition is the computational backbone of modern optics and photonics. Every thin-film coating design, waveguide-mode solver, and metasurface simulation expands the field in plane waves (the angular spectrum) and propagates each with its own $e^{ik_z z}$ ; a single line, $k=n\omega/c$ , is evaluated billions of times inside these codes. The wave impedance $\eta=\eta_0/n$ is the quantity that anti-reflection stacks and fibre-connector index-matching gels are engineered to match, and impedance mismatch is why a bare glass surface reflects 4% per face. In laser metrology the intensity relation $I=\tfrac12 c\varepsilon_0 n|E_0|^2$ converts a measured power and spot size into the peak field a sample sees — the number that decides whether an experiment is in the linear or the nonlinear regime. And the complex-index promotion $\tilde n=n+i\kappa$ is the entry point to everything that follows: absorption spectroscopy, ellipsometry (which reads $n$ and $\kappa$ off the polarization of a reflected plane wave), and the dispersion engineering that governs ultrafast pulses and long-haul fibre.

The physical picture

Stand in a field of parallel wavefronts sweeping past you. Each front is an infinite flat sheet on which the phase is the same everywhere; the whole comb of sheets slides rigidly along $\hat{\mathbf{k}}$ at speed $c/n$ , no sheet changing its value as it moves. The medium's only job is to slow that glide by the factor $n$ and, if it is lossy, to bleed a little amplitude out of each successive sheet. Because the medium is isotropic, the wave feels no preferred axis: pick any $\hat{\mathbf{k}}$ and the same dispersion relation holds — which is why a block of glass looks identical whichever way you turn it, unlike a crystal whose index depends on direction. The algebraic magic is that a plane wave diagonalises the derivative operators. Maxwell's coupled partial differential equations become a linear algebra problem in $\mathbf{k}$ and $\omega$ ; demanding a non-trivial solution collapses to one scalar equation, $k^2=n^2\omega^2/c^2$ . Everything about propagation in a simple medium hides inside that relation. Transversality then falls out for free: Gauss's law $\nabla\cdot\mathbf{D}=0$ becomes $\mathbf{k}\cdot\mathbf{E}=0$ , so $\mathbf{E}$ must lie in the wavefront plane, and Faraday's law completes a right-handed triad with the Poynting vector pointing along $\hat{\mathbf{k}}$ .

Figure

The transverse-EM triad in a medium: E ⟂ H ⟂ k with the fields in phase, wavefronts (planes of constant phase) stacked ⟂ k and spaced by λ = λ₀/n, the whole comb gliding at v_p = c/n

The second figure is the quantitative payoff a photonics engineer actually reaches for: what changes when the same field amplitude crosses from vacuum into a medium. The frequency is fixed by the source, but the crest speed, the wavelength, the impedance, and the delivered intensity all rescale by the single number $n$ .

Figure

Vacuum → glass (n = 1.50): the same field amplitude E₀ = 1 kV/m gives v_p = c/n = 1.999×10⁸ m/s, impedance η = η₀/n = 251.15 Ω, and intensity I = ½cε₀n|E₀|² = 1.99 kW/m² — the n-factor the vacuum wave does not carry

The core equations

A monochromatic plane wave is written with a complex exponential; physical fields are the real part. The convention $e^{i(\mathbf{k}\cdot\mathbf{r}-\omega t)}$ (a wave moving toward $+\hat{\mathbf{k}}$ as $t$ grows) is fixed once and used throughout — it sets the sign of $\kappa$ and of every susceptibility in the chapter.

\mathbf{E}(\mathbf{r},t)=\mathbf{E}_0\,e^{i(\mathbf{k}\cdot\mathbf{r}-\omega t)} \qquad (1)

where: $\mathbf{E}_0$ — constant complex amplitude vector (V/m); $\mathbf{k}$ — wavevector, magnitude $k$ , along the propagation direction (rad/m); $\omega$ — angular frequency (rad/s); $\mathbf{r},t$ — position and time. Surfaces of constant phase $\mathbf{k}\cdot\mathbf{r}-\omega t=\text{const}$ are planes $\perp\hat{\mathbf{k}}$ . Inserting the ansatz into the optical wave equation leaves exactly one scalar constraint linking $k$ to $\omega$ — the dispersion relation. All the material physics sits in the refractive index on its right-hand side.

k=n\,\frac{\omega}{c},\qquad n=\sqrt{\varepsilon_r\mu_r}\;\xrightarrow{\ \mu_r\approx1\ }\;\sqrt{\varepsilon_r} \qquad (2)

where: $n$ — refractive index; $\varepsilon_r$ — relative permittivity at frequency $\omega$ ; $\mu_r\approx1$ optically; $c=1/\sqrt{\varepsilon_0\mu_0}$ . Because $\varepsilon_r=\varepsilon_r(\omega)$ , equation (2) already contains dispersion: $k(\omega)$ is not simply proportional to $\omega$ . Transversality is a consequence of the divergence equations, not an extra assumption, and it is where isotropy earns its keep.

\mathbf{k}\cdot\mathbf{E}=0,\qquad \mathbf{k}\cdot\mathbf{H}=0 \qquad (3)

where: $\mathbf{H}$ — magnetic field (A/m). $\nabla\cdot\mathbf{D}=0$ gives $\varepsilon\,\mathbf{k}\cdot\mathbf{E}=0$ ; a scalar $\varepsilon$ (isotropy) then forces $\mathbf{E}\perp\mathbf{k}$ . In an anisotropic crystal only $\mathbf{D}\perp\mathbf{k}$ survives. Faraday's law ties $\mathbf{H}$ to $\mathbf{E}$ and fixes their ratio, the wave impedance, which is the vacuum value scaled down by $n$ .

\mathbf{H}=\frac{n}{\eta_0}\,\hat{\mathbf{k}}\times\mathbf{E},\qquad \eta=\frac{|\mathbf{E}|}{|\mathbf{H}|}=\sqrt{\frac{\mu}{\varepsilon}}=\frac{\eta_0}{n} \qquad (3\text{b})

where: $\hat{\mathbf{k}}$ — unit propagation vector; $\eta_0=\sqrt{\mu_0/\varepsilon_0}=376.730\ \Omega$ — vacuum wave impedance; $\eta$ — impedance of the medium. $(\hat{\mathbf{k}},\mathbf{E},\mathbf{H})$ is right-handed. The phase velocity and wavelength follow directly from (2); the frequency is set at the source and is continuous across any interface.

v_p=\frac{\omega}{k}=\frac{c}{n},\qquad \lambda=\frac{2\pi}{k}=\frac{\lambda_0}{n} \qquad (4)

where: $v_p$ — speed of surfaces of constant phase (m/s); $\lambda_0$ — vacuum wavelength. Only the spatial scale and speed rescale by $n$ ; $\nu=\omega/2\pi$ does not. The time-averaged Poynting vector is the wave's intensity; carrying the $n$ from the impedance gives the working formula used throughout photonics.

I=\langle\mathbf{S}\rangle\cdot\hat{\mathbf{k}}=\tfrac12\,\mathrm{Re}(\mathbf{E}\times\mathbf{H}^*)\cdot\hat{\mathbf{k}}=\frac{|\mathbf{E}_0|^2}{2\eta}=\tfrac12\,c\varepsilon_0 n\,|\mathbf{E}_0|^2 \qquad (5)

where: $I$ — intensity / irradiance (W/m²). The three forms are identical because $1/\eta=n/\eta_0=c\varepsilon_0 n$ . Intensity scales $\propto n$ for a fixed field amplitude. Finally, promoting the index to complex form turns the very same plane wave into an attenuated one — the single line that seeds the whole absorption story of this chapter.

\tilde n=n+i\kappa\;\Rightarrow\; \mathbf{E}\propto e^{-\kappa\omega z/c}\,e^{i(n\omega z/c-\omega t)},\qquad \alpha=\frac{2\kappa\omega}{c}=\frac{4\pi\kappa}{\lambda_0} \qquad (6)

where: $\kappa$ — extinction coefficient; $\alpha$ — intensity absorption coefficient (Beer–Lambert). The real part of $\tilde n$ advances the phase; the imaginary part decays the amplitude. Both $\tilde n$ (article 1487) and $\alpha$ (article 1488) are developed next.

Derivation AFrom the wave equation to the dispersion relation and transversality

The goal is to show that in a source-free, linear, homogeneous, isotropic medium the plane-wave ansatz reduces the four Maxwell equations to two statements — a single scalar dispersion relation $k=n\omega/c$ and the transverse orthogonal triad — and that isotropy is exactly what makes $\mathbf{E}$ (not merely $\mathbf{D}$ ) transverse. We insert the ansatz, read off the algebraic replacements, and assemble the field geometry, recalling the vacuum plane-wave object and the wave equation by name from their owners.

Step 1. Recall from aq-optics-maxwell-equations-dielectric-media the source-free optical wave equation $\nabla^2\mathbf{E}=\mu\varepsilon\,\partial_t^2\mathbf{E}$ , obtained by taking the curl of Faraday's law and using $\nabla\cdot\mathbf{E}=0$ . Why: this is the object the plane wave must solve; the material enters only through the product $\mu\varepsilon$ .

Step 2. For the ansatz (1), $\nabla\to i\mathbf{k}$ and $\partial_t\to-i\omega$ , so the wave equation becomes $-k^2\mathbf{E}+\mu\varepsilon\,\omega^2\mathbf{E}=0$ , i.e. $k^2=\mu\varepsilon\,\omega^2=\varepsilon_r\mu_r\,\omega^2/c^2$ . Taking the positive root defines $n=\sqrt{\varepsilon_r\mu_r}$ and gives $k=n\omega/c$ . Why: a plane wave diagonalises the derivatives, so a differential equation becomes one algebraic constraint — the dispersion relation carries all of propagation.

Step 3. Gauss's law $\nabla\cdot\mathbf{D}=0$ becomes $i\varepsilon\,\mathbf{k}\cdot\mathbf{E}=0$ . Because $\varepsilon$ is a scalar (isotropy) and non-zero, $\mathbf{k}\cdot\mathbf{E}=0$ : the electric field is transverse. Likewise $\nabla\cdot\mathbf{B}=0$ gives $\mathbf{k}\cdot\mathbf{H}=0$ . Why: this is exactly the transversality recalled from aq-em-monochromatic-plane-waves-vacuum, but now the step $\mathbf{k}\cdot\mathbf{D}=0\Rightarrow\mathbf{k}\cdot\mathbf{E}=0$ needs isotropy — in a crystal it fails, and the wave acquires a longitudinal $\mathbf{E}$ component (birefringence).

Step 4. Faraday's law $\nabla\times\mathbf{E}=-\mu\,\partial_t\mathbf{H}$ becomes $i\mathbf{k}\times\mathbf{E}=i\omega\mu\,\mathbf{H}$ , so $\mathbf{H}=(\mathbf{k}\times\mathbf{E})/(\omega\mu)=(n/\eta_0)\,\hat{\mathbf{k}}\times\mathbf{E}$ using $k=n\omega/c$ and $\eta_0=c\mu_0$ . Hence $\mathbf{H}\perp\mathbf{E}$ , the triad $(\hat{\mathbf{k}},\mathbf{E},\mathbf{H})$ is right-handed, and $|\mathbf{E}|/|\mathbf{H}|=\eta_0/n=\eta$ . Why: this fixes the magnetic amplitude and the impedance, the quantity whose mismatch at a boundary sets the Fresnel reflection.

Step 5. Ampère's law $\nabla\times\mathbf{H}=\varepsilon\,\partial_t\mathbf{E}$ reproduces $k^2=\mu\varepsilon\,\omega^2$ , confirming the solution is complete and self-consistent — no independent constraint is left. Why: the fourth equation must not over-determine the wave; that it returns the dispersion relation is the consistency check. *Verified: $k=n\omega/c$ from $\mu\varepsilon=n^2/c^2$ ; $\mathbf{k}\cdot\mathbf{E}=0$ symbolically; $|\mathbf{E}|/|\mathbf{H}|=\eta_0/n$ ; $\eta_0=376.730\ \Omega$ in verify_aq-optics-plane-waves-isotropic-media.py.*

Go deeper · Derivation B · 1 minIntensity and the equal sharing of energy: I = ½cε₀n|E₀|² = v_p⟨u⟩

Derivation BIntensity and the equal sharing of energy: I = ½cε₀n|E₀|² = v_p⟨u⟩

The aim is to compute the time-averaged intensity of the plane wave from its fields, show that the electric and magnetic energy densities are equal, and prove that energy is transported along $\hat{\mathbf{k}}$ at exactly the phase velocity in a lossless non-dispersive medium. We form the mean Poynting vector, add the two energy densities, and take their ratio — the origin of the working intensity formula $I=\tfrac12 c\varepsilon_0 n|\mathbf{E}_0|^2$ .

Step 1. With $\mathbf{H}=(n/\eta_0)\hat{\mathbf{k}}\times\mathbf{E}$ and $\hat{\mathbf{k}}\times(\hat{\mathbf{k}}\times\mathbf{E})=-\mathbf{E}$ for transverse $\mathbf{E}$ , the mean flux is $\langle\mathbf{S}\rangle=\tfrac12\mathrm{Re}(\mathbf{E}\times\mathbf{H}^*)=\tfrac12(n/\eta_0)|\mathbf{E}_0|^2\,\hat{\mathbf{k}}$ . Since $n/\eta_0=c\varepsilon_0 n$ , the intensity is $I=\tfrac12 c\varepsilon_0 n|\mathbf{E}_0|^2$ . Why: this is the number a power meter reads; the factor $n$ is what the vacuum owner's $I=\tfrac12 c\varepsilon_0|\mathbf{E}_0|^2$ does not carry.

Step 2. The time-averaged energy density is $\langle u\rangle=\tfrac14\varepsilon|\mathbf{E}_0|^2+\tfrac14\mu|\mathbf{H}_0|^2$ . Using $|\mathbf{H}_0|=(n/\eta_0)|\mathbf{E}_0|$ and $\mu (n/\eta_0)^2=\mu\varepsilon_r\mu_r(\varepsilon_0/\mu_0)=\varepsilon$ (for $\mu_r\approx1$ ), the magnetic term equals the electric term. Why: equipartition of electric and magnetic energy is a defining feature of a freely propagating wave and makes the bookkeeping collapse to one term.

Step 3. Hence $\langle u\rangle=\tfrac12\varepsilon|\mathbf{E}_0|^2$ , and the energy-transport velocity is $\langle S\rangle/\langle u\rangle=(\tfrac12 c\varepsilon_0 n|\mathbf{E}_0|^2)/(\tfrac12\varepsilon_0\varepsilon_r|\mathbf{E}_0|^2)=cn/\varepsilon_r=c/n=v_p$ . Why: in a lossless non-dispersive medium the energy rides with the crests; only when $n$ disperses does the energy velocity depart from $v_p$ toward the group velocity (article 1491). *Verified: $I=\tfrac12 c\varepsilon_0 n|\mathbf{E}_0|^2$ (glass n=1.5, E₀=1 kV/m → 1.99 kW/m²), electric = magnetic energy, and $\langle S\rangle/\langle u\rangle=c/n$ in verify_aq-optics-plane-waves-isotropic-media.py.*

Limiting cases

Case	Result	Meaning
Vacuum	$n=1,\ k=\omega/c,\ v_p=c,\ \eta=376.73\ \Omega$	no medium to slow or load the wave; the vacuum impedance is universal
Transparent dielectric	$n=\sqrt{\varepsilon_r}\in\mathbb{R},\ v_p=c/n$	real index, no attenuation; $\mathbf{E}$ and $\mathbf{H}$ oscillate exactly in phase (glass, $n\approx1.5$ )
Non-magnetic optical medium	$\mu_r\approx1\Rightarrow n\approx\sqrt{\varepsilon_r},\ \eta=\eta_0/n$	magnetic response vanishes optically, so the index comes entirely from $\varepsilon_r$
Absorbing medium	$\tilde n=n+i\kappa,\ \|\mathbf{E}\|^2\propto e^{-\alpha z}$	complex index: phase advances with $n$ , amplitude decays with $\kappa$ , $\alpha=4\pi\kappa/\lambda_0$
Anisotropic crystal	$\mathbf{k}\cdot\mathbf{D}=0$ but $\mathbf{k}\cdot\mathbf{E}\neq0$	isotropy lost: $\mathbf{E}$ is no longer purely transverse; two indices per direction (birefringence)

Common trap“Any electromagnetic plane wave is automatically transverse, with E perpendicular to k.”

Only in an isotropic medium. Transversality of $\mathbf{E}$ is not a property of Maxwell's equations alone — it is $\nabla\cdot\mathbf{D}=0$ , i.e. $\mathbf{k}\cdot\mathbf{D}=0$ , that is fundamental. In an isotropic medium $\mathbf{D}=\varepsilon\mathbf{E}$ with a scalar $\varepsilon$ , so $\mathbf{k}\cdot\mathbf{D}=0$ does imply $\mathbf{k}\cdot\mathbf{E}=0$ and $\mathbf{E}$ lies in the wavefront. In an anisotropic crystal $\varepsilon$ is a tensor, $\mathbf{D}=\overleftrightarrow{\varepsilon}\,\mathbf{E}$ , and $\mathbf{D}\perp\mathbf{k}$ no longer forces $\mathbf{E}\perp\mathbf{k}$ : the electric field acquires a longitudinal component, the Poynting vector tilts away from $\mathbf{k}$ (walk-off), and each direction carries two indices — the whole of birefringence. So "TEM everywhere" is a statement about isotropy, which this article assumes; drop it and the geometry changes. (A companion trap: the $e^{i(\mathbf{k}\cdot\mathbf{r}-\omega t)}$ sign convention is a choice — with the engineering $e^{i(\omega t-\mathbf{k}\cdot\mathbf{r})}$ a decaying wave needs $\mathrm{Im}\,\tilde n<0$ ; never mix the two mid-calculation.)

Worked examples

1 · Intensity carried by a plane wave in glass (confidence builder)

A monochromatic beam has electric-field amplitude $E_0=1.0\ \mathrm{kV/m}$ inside crown glass of index $n=1.50$ . Find the intensity it carries, and compare with the amplitude that would deliver the solar constant $1361\ \mathrm{W/m^2}$ in vacuum.

Step 1. Use $I=\tfrac12 c\varepsilon_0 n|E_0|^2$ . Insert $c=2.998\times10^8$ , $\varepsilon_0=8.854\times10^{-12}$ , $n=1.50$ , $E_0=10^3$ : $I=\tfrac12(2.998\times10^8)(8.854\times10^{-12})(1.50)(10^6)$ . Grouping, $\tfrac12 c\varepsilon_0=1.327\times10^{-3}$ , times $n|E_0|^2=1.50\times10^6$ gives $I=\mathbf{1.99\ kW/m^2}$ .

Step 2. The same field in vacuum ( $n=1$ ) would carry $I_0=1.99/1.50=1.33\ \mathrm{kW/m^2}$ — the intensity is larger in glass by the factor $n$ , because the medium loads the wave (lower impedance $\Rightarrow$ larger $H_0$ for the same $E_0$ ).

Step 3. For the solar constant in vacuum, invert $I=\tfrac12 c\varepsilon_0|E_0|^2$ : $E_0=\sqrt{2I/(c\varepsilon_0)}=\sqrt{2(1361)/(2.998\times10^8\cdot8.854\times10^{-12})}=\mathbf{1013\ V/m}$ . So bright sunlight is an optical field of only about a kilovolt per metre — a useful yardstick. I = 1.99 kW/m² in glass; sunlight ↔ E₀ = 1013 V/m.

2 · Impedance and the magnetic-field amplitude

For the same wave ( $E_0=1.0\ \mathrm{kV/m}$ , $n=1.50$ ), find the wave impedance of the glass and the magnetic-field amplitude $H_0$ , and confirm the intensity via $I=\tfrac12\eta H_0^2$ .

Step 1. Impedance: $\eta=\eta_0/n=376.730/1.50=\mathbf{251.15\ \Omega}$ . The medium presents a lower impedance than vacuum, exactly as a denser transmission line does.

Step 2. Magnetic amplitude: $H_0=E_0/\eta=10^3/251.15=\mathbf{3.982\ A/m}$ (equivalently $B_0=\mu_0 H_0=5.00\ \mu\mathrm{T}$ ). Because $\eta$ dropped, $H_0$ is larger than the vacuum value $E_0/\eta_0=2.654\ \mathrm{A/m}$ .

Step 3. Cross-check the intensity: $I=\tfrac12\eta H_0^2=\tfrac12(251.15)(3.982)^2=1.99\times10^3\ \mathrm{W/m^2}$ , matching Example 1. The three equivalent forms $I=|E_0|^2/2\eta=\tfrac12\eta H_0^2=\tfrac12 c\varepsilon_0 n|E_0|^2$ are one statement. η = 251.15 Ω, H₀ = 3.98 A/m, I = 1.99 kW/m² (consistent).

Physics of the portrait

The plane wave is the atom of wave optics: not a real beam (which is finite and therefore a superposition of plane waves) but the exact eigenmode from which every real field is built by Fourier synthesis. Its power is that it turns partial differential equations into algebra, so that a medium's entire influence is compressed into one number, $n$ .

Origins · 1 minplane wave and how this was discovered

plane wave

the monochromatic eigenmode E₀e^{i(k·r−ωt)} — the building block of wave optics · electromagnetic optics of media

The plane wave is less a discovery than a language. Its roots lie in the wave equation that d'Alembert wrote for a vibrating string in 1747, whose solutions are rigidly translating profiles $f(x\mp vt)$ ; the sinusoidal, single-frequency member of that family became the workhorse once Fourier showed in 1822 that any disturbance is a superposition of sinusoids. When Maxwell identified light as an electromagnetic wave in 1865 and computed its speed from $\varepsilon_0,\mu_0$ , the transverse plane wave became the concrete object that carried the identification: $\mathbf{E}$ and $\mathbf{H}$ perpendicular to each other and to the propagation, in phase, with the ratio $\eta_0=\sqrt{\mu_0/\varepsilon_0}$ . John Henry Poynting (1884) and Oliver Heaviside gave the energy flux $\mathbf{S}=\mathbf{E}\times\mathbf{H}$ its meaning, so that the plane wave's intensity became a measurable transport of energy rather than a formal quantity. The idealization is deliberately unphysical — an infinite sheet of infinite extent and duration — yet it is controlled: a real beam of waist $w_0$ behaves as a plane wave over its Rayleigh range, and a laser's coherence time can exceed a second, so within those windows every formula here applies with negligible error. Everything in this chapter — refractive index, absorption, phase and group velocity, dispersion, Kramers–Kronig — is bookkeeping performed on this one object as it meets matter.

History

1747

— d'Alembert's wave equation for the vibrating string: rigidly translating solutions

1822

— Fourier establishes that any disturbance is a superposition of sinusoids (plane-wave basis)

1865

— Maxwell identifies light as an electromagnetic wave and computes its speed from ε₀, μ₀

1884

— Poynting formulates the energy-flux vector S = E × H

1885

— Heaviside independently arrives at the E × H energy flux, confirming the transport law

Going further

Texts. Hecht, Optics, 5th ed., §3.2–3.5 · Saleh & Teich, Fundamentals of Photonics, 3rd ed., Ch. 5 (electromagnetic optics) · Loudon, The Quantum Theory of Light, 3rd ed., Ch. 1 · Foot, Atomic Physics, §7.1 · Jackson, Classical Electrodynamics, 3rd ed., §7.1.

Provenance. The vacuum plane wave, its transversality $\mathbf{k}\cdot\mathbf{E}=0$ , and its energy flux are recalled by name from aq-em-monochromatic-plane-waves-vacuum; the source-free optical wave equation $\nabla^2\mathbf{E}=\mu\varepsilon\,\partial_t^2\mathbf{E}$ from aq-optics-maxwell-equations-dielectric-media. A verifier should confirm: (1) $k=n\omega/c$ from $\mu\varepsilon=n^2/c^2$ (Jackson §7.1, Hecht §3.2); (2) the field ratio $|E|/|H|=\eta=\eta_0/n$ with $\eta_0=376.730\ \Omega$ ; (3) the intensity $I=\tfrac12 c\varepsilon_0 n|E_0|^2$ and its equality with $v_p\langle u\rangle$ via electric–magnetic equipartition (Saleh & Teich Ch. 5); (4) $\alpha=4\pi\kappa/\lambda_0$ from the complex-index propagation factor (Loudon Ch. 1). The worked numbers (glass $n=1.5$ : $v_p=1.999\times10^8$ m/s, $\eta=251.15\ \Omega$ , $I=1.99$ kW/m²; sunlight $E_0=1013$ V/m) are recomputable from the formulas.

Prerequisites. Maxwell's Equations in Dielectric Media · Dielectric Susceptibility and Permittivity · Electric Polarization and the Displacement Field. Leads to. The Refractive Index · Phase Velocity · Complex Refractive Index and the Extinction Coefficient · Group Velocity and Group Index.

Is a plane wave a real thing, or just a mathematical convenience?

It is an idealization — an infinite sheet of infinite duration that no source can produce — but a controlled one. Any real field is an exact superposition of plane waves (its angular spectrum in space, its frequency spectrum in time), so nothing is lost by working with one plane wave at a time and adding them up. A collimated laser beam of waist $w_0$ behaves as a single plane wave over its Rayleigh range $z_R=\pi w_0^2/\lambda$ — metres for a millimetric visible beam — and over the coherence length set by its linewidth. Within that window every formula in this article applies to better than a part in $10^3$ . The corrections (finite width $\to$ diffraction, finite duration $\to$ linewidth) are themselves organized as small departures from the plane wave, which is why it is the right starting point rather than a crude one.

The frequency doesn't change entering a medium but the wavelength does — why?

The frequency is imposed by the source and by the boundary: at an interface the fields on both sides must oscillate in step at every instant, which is only possible if they share the same $\omega$ . What the medium changes is the speed of the crests, $v_p=c/n$ . Since $v_p=\nu\lambda$ with $\nu$ fixed, the wavelength must shrink to $\lambda=\lambda_0/n$ — the crests pack closer together while ticking past at the same rate. This is exactly why colour (a frequency) is preserved when light enters water or glass, even though the wave physically slows and compresses.

How can the intensity be larger in glass for the same E₀ — where does the extra power come from?

No extra power is created; the comparison just holds a different quantity fixed. Intensity is $I=\tfrac12 c\varepsilon_0 n|E_0|^2$ , so for a fixed field amplitude it rises with $n$ . The physical reason is the impedance: $\eta=\eta_0/n$ is lower in glass, so the same $E_0$ is accompanied by a larger magnetic amplitude $H_0=E_0/\eta$ , and the energy flux $\tfrac12 E_0 H_0$ grows. If instead you held the power fixed (as when a beam refracts into glass), the field amplitude would adjust and there is no paradox. In practice, what is continuous across a normal interface is the tangential $E$ and $H$ , and matching them gives the Fresnel transmission — a bookkeeping this article's impedance sets up.

Why does isotropy matter so much for transversality?

Because the fundamental divergence law is about $\mathbf{D}$ , not $\mathbf{E}$ : $\nabla\cdot\mathbf{D}=0$ gives $\mathbf{k}\cdot\mathbf{D}=0$ always. Whether this makes $\mathbf{E}$ transverse depends on how $\mathbf{D}$ relates to $\mathbf{E}$ . In an isotropic medium $\mathbf{D}=\varepsilon\mathbf{E}$ with a scalar $\varepsilon$ , so $\mathbf{D}\parallel\mathbf{E}$ and $\mathbf{k}\cdot\mathbf{E}=0$ follows. In a crystal $\varepsilon$ is a tensor, $\mathbf{D}$ and $\mathbf{E}$ point in different directions, and $\mathbf{k}\cdot\mathbf{E}$ is generally non-zero: the wave has a longitudinal electric component, the energy flow (Poynting vector) walks off from $\mathbf{k}$ , and each propagation direction supports two waves with two indices. Birefringence, wave plates, and phase-matching in nonlinear crystals all live in that broken isotropy.

What is the wave impedance really, and why do engineers care about it?

It is the ratio of the electric to the magnetic field amplitude in the wave, $\eta=|E|/|H|=\sqrt{\mu/\varepsilon}=\eta_0/n$ , and it plays exactly the role that characteristic impedance plays on a transmission line. When a wave meets a boundary between two media, the reflection is governed by the mismatch of their impedances: equal impedances $\Rightarrow$ no reflection, large mismatch $\Rightarrow$ strong reflection. That is why a single-layer anti-reflection coating uses an index $n_f=\sqrt{n_0 n_s}$ (a geometric-mean impedance match) and why index-matching fluid on a fibre connector kills the back-reflection. The 4% reflection from each face of a bare glass plate is just the impedance step from $\eta_0$ to $\eta_0/1.5$ .

Does energy always travel at the phase velocity c/n?

In a lossless, non-dispersive medium, yes: the derivation above gives $\langle S\rangle/\langle u\rangle=v_p=c/n$ . But that relied on $n$ being constant. When $n=n(\omega)$ — which is always true in a real material — a pulse (a superposition of frequencies) has its envelope, and hence its energy and information, move at the group velocity $v_g=c/n_g$ , generally different from $v_p$ . Near an absorption line the two can differ dramatically, and $v_p$ can even exceed $c$ without anything physical exceeding $c$ . So "energy travels at $c/n$ " is the correct non-dispersive statement; the dispersive correction is the subject of the phase-velocity and group-velocity articles that follow.