Phase Velocity

Aim a laser through a block of glass and ask the simplest possible question: how fast does the light move? The answer comes in layers, and this article owns the first one. We take from aq-em-phase-group-velocity the two velocity definitions every wave carries — the phase velocity $v_p=\omega/k$ of a single monochromatic wave's crests and the group velocity $v_g=\mathrm{d}\omega/\mathrm{d}k$ of a modulation envelope — and from aq-optics-plane-waves-isotropic-media the in-medium dispersion relation $k=n\omega/c$ . Combining them is one line: $v_p=c/n(\omega)$ . The crest speed is the thing the refractive index literally measures, and everything the index does — Snell bending, the shrunken wavelength $\lambda_0/n$ , interference fringes, phase matching — is phase-velocity bookkeeping. The article's second job is to defuse the scandal hiding in that line. Every material has frequency bands where $n(\omega)<1$ — just above each resonance, and throughout the X-ray region — and there $v_p>c$ . Nothing is wrong. A monochromatic wave is infinite and featureless: a crest is a label on a phase, not a mark you can time, so no information rides at $v_p$ . Signals are modulations, and modulations travel at the group velocity, developed next in aq-optics-group-velocity-group-index. The free-electron plasma makes the audit exact: at $\omega=2\omega_p$ the index is $n=0.866$ and the crests do $v_p=1.1547c$ — while the plasma's dispersion relation pins $v_p v_g=c^2$ , so the envelope moves at $v_g=0.866c$ , safely below light speed. Faster-than- $c$ crests and slower-than- $c$ signals are two faces of the same dispersion curve.

Key takeaways

The phase velocity $v_p=\omega/k$ is the speed of the constant-phase surfaces of one monochromatic plane wave; with the in-medium dispersion relation $k=n\omega/c$ (recalled by name) it becomes $v_p=c/n(\omega)$ .
In crown glass $n=1.5$ : $v_p=c/1.5=1.9986\times10^8$ m/s, and the wavelength compresses to $\lambda=\lambda_0/n$ while the frequency stays fixed.
Because $n=n(\omega)$ , each colour has its own crest speed — the root of chromatic dispersion and of every prism spectrum.
Wherever $n<1$ (above every resonance, the whole X-ray band, a plasma above $\omega_p$ ) the crests outrun $c$ : $v_p>c$ is routine and harmless, because a crest carries no information.
Only the real part of the complex index $\tilde n=n+i\kappa$ sets $v_p$ ; the imaginary part $\kappa$ sets attenuation, not speed.
Signals and energy ride the group velocity: for the plasma, $v_p v_g=c^2$ exactly, so $n=0.866$ gives $v_p=1.1547c$ and $v_g=0.866c<c$ .
Key idea: set the phase $kx-\omega t$ constant and differentiate — the crest speed is $v_p=\omega/k=c/n$ , a property of one Fourier component, not of a signal; where $n<1$ it exceeds $c$ with no causal consequence.

The core insight

Set the phase $kx-\omega t=\text{const}$ and differentiate: the constant-phase surface moves at $v_p=\omega/k=c/n(\omega)$ . This is the speed of a crest of one Fourier component — the number the refractive index measures — not the speed of any signal, which is why $v_p>c$ in the $n<1$ bands above resonances is physically harmless.

Where this lives today

Phase velocity is the working variable wherever accumulated phase matters more than arrival time. Phase matching in nonlinear optics is the purest case: second-harmonic generation is efficient only if fundamental and harmonic crests stay locked, $n(2\omega)=n(\omega)$ , and birefringent or quasi-phase-matched crystals are engineered entirely around equalizing two phase velocities. Fabry–Pérot cavities and laser resonators are ruled by the round-trip phase $2knL=2n\omega L/c$ ; refractometry reads $n=c/v_p$ from Snell's law to five decimals; and every interferometer — from gravitational-wave detectors to on-chip Mach–Zehnders — converts a phase-velocity difference into a fringe shift. The $n<1$ side is equally industrial. X-ray telescopes and synchrotron beamlines steer photons with grazing-incidence mirrors that exploit total external reflection ( $n$ just below 1); over-the-horizon radio still bounces off the ionospheric plasma, whose $v_p>c$ crests and $v_g<c$ signals obey the $v_p v_g=c^2$ reciprocity of Derivation B; and waveguide phase velocities above $c$ (same hyperbolic dispersion family) are routine bookkeeping in microwave engineering. None of it has ever sent a bit faster than light.

The physical picture

Freeze a single infinite sinusoid in a snapshot, then let time run. To keep staring at the same crest you must walk alongside it so that the phase you see never changes; the walking speed that achieves this is the phase velocity. Nothing material makes the trip — the crest is a moving condition ( $\phi=\text{const}$ ), the way "noon" sweeps westward around the Earth without anything travelling. That is the first figure: two snapshots of the same wave, indistinguishable except for a rigid shift.

Figure

A single monochromatic wave at t = 0 and a moment Δt later — the marked crest has slid forward by v_p·Δt, but the wave itself is indistinguishable from before: a crest is a phase label with no feature to time a signal by

In a medium the picture gains its physics. The driven electrons re-radiate wavelets slightly out of step with the incident field; summed onto the original wave they retard the crests, and the accumulated retardation is exactly the factor $n$ : $v_p=c/n$ with $n>1$ throughout the transparent windows of glass. Drive the electrons above one of their resonances, however, and the re-radiated field flips phase — it now advances the crests, $n$ dips below one, and $v_p$ exceeds $c$ . That is not an exotic corner case: it happens above every resonance a material has, and since X-ray frequencies sit above essentially all of them, $n<1$ is the normal state of matter in the X-ray band (the basis of grazing-incidence X-ray mirrors). The cleanest laboratory for the effect is a free-electron plasma, where the second figure lives.

Figure

The punchline — above a plasma frequency the index n = √(1 − ωₚ²/ω²) lies below one everywhere: at ω = 2ωₚ, n = 0.866 and the crests do v_p = 1.1547c, while v_p·v_g = c² pins the signal to v_g = 0.866c, below c

The crest speed is nevertheless the working currency of wave optics. Because $v_p$ fixes the accumulated phase $kL=n\omega L/c$ , it decides where interference fringes sit, whether two beams in a nonlinear crystal stay phase-locked, and where a Fabry–Pérot cavity resonates. Wavelength itself is phase-velocity bookkeeping: the source fixes the frequency, the medium slows the crests, so the crests pack closer — $\lambda=v_p/\nu=\lambda_0/n$ . Frequency is conserved across an interface; wavelength is not.

The core equations

A wavefront of the monochromatic plane wave (the object recalled from aq-optics-plane-waves-isotropic-media) is a surface of constant total phase. Demanding that the phase not change as the surface moves defines the phase velocity — the definition is kinematic and needs no material input yet.

v_p=\frac{\omega}{k} \qquad (1)

where: $v_p$ — phase velocity (m/s); $\omega$ — angular frequency (rad/s), fixed by the source and continuous across interfaces; $k$ — magnitude of the wave vector (rad/m). On the dispersion diagram $\omega(k)$ this is the slope of the chord from the origin to the operating point. The material enters through the dispersion relation, recalled by name from aq-optics-plane-waves-isotropic-media: in a linear isotropic medium $k=n\omega/c$ . Substituting turns the kinematic definition into the working optical formula, and the wavelength contraction follows at once.

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

where: $c=299\,792\,458$ m/s — vacuum light speed; $n(\omega)$ — refractive index, $n=\sqrt{\varepsilon_r}$ optically; $\lambda_0$ — vacuum wavelength. Because $n$ disperses, each frequency has its own crest speed — one number per colour, which is all a prism needs. In an absorbing medium the index is complex, $\tilde n=n+i\kappa$ (developed in aq-optics-complex-refractive-index-extinction). Only the real part advances phase; the imaginary part decays amplitude. The phase velocity therefore reads the real part alone.

\tilde k=\frac{\tilde n\,\omega}{c}\;\Rightarrow\; E\propto e^{-\kappa\omega z/c}\,e^{i(n\omega z/c-\omega t)},\qquad v_p=\frac{\omega}{\mathrm{Re}\,\tilde k}=\frac{c}{n} \qquad (3)

where: $\kappa$ — extinction coefficient, feeding the Beer–Lambert attenuation $\alpha=2\kappa\omega/c$ ; $\mathrm{Re}\,\tilde k=n\omega/c$ — the oscillatory part of the propagation factor. Absorption changes how far the wave gets, not how fast its crests move. The cleanest $n<1$ medium is a collisionless free-electron plasma — the $\omega_0\to0$ , $\gamma\to0$ limit of the bound-electron response owned by aq-optics-lorentz-oscillator-model. Its index is below one for every propagating frequency.

n(\omega)=\sqrt{1-\frac{\omega_p^2}{\omega^2}}\;,\qquad \omega^2=\omega_p^2+c^2k^2 \qquad (4)

where: $\omega_p$ — plasma frequency, $\omega_p^2=N e^2/(\varepsilon_0 m_e)$ with $N$ the electron density. At $\omega=2\omega_p$ : $n=\sqrt{3}/2=0.866$ , so $v_p=1.1547c$ . Below $\omega_p$ , $k$ is imaginary and the wave is evanescent — the ionosphere mirrors such radio waves back down. Finally the exact bridge to the signal speed. Differentiating $k=n\omega/c$ gives the group velocity in terms of the phase velocity — quoted here, derived and spent in aq-optics-group-velocity-group-index. Note the form: the exact relation is a product with $(1+\ldots)$ in the denominator of $v_g$ , not a resummed reciprocal.

\frac{1}{v_g}=\frac{\mathrm{d}k}{\mathrm{d}\omega}=\frac{n_g}{c}=\frac{1}{v_p}\left(1+\frac{\omega}{n}\frac{\mathrm{d}n}{\mathrm{d}\omega}\right),\qquad n_g=n+\omega\frac{\mathrm{d}n}{\mathrm{d}\omega} \qquad (5)

where: $v_g$ — group (envelope/signal) velocity; $n_g$ — group index. For the plasma of (4) the two velocities obey $v_p\,v_g=c^2$ exactly — chord above $c$ , tangent below. In non-dispersive media $\mathrm{d}n/\mathrm{d}\omega=0$ and $v_g=v_p$ . (Some references compress (5) into $1/v_g=(1/v_p)\left(1-\tfrac{\omega}{n}\tfrac{\mathrm{d}n}{\mathrm{d}\omega}\right)^{-1}$ — that is only the first-order Taylor of the exact relation and fails in strong dispersion; the previous edition of this article carried it.)

Derivation APhase velocity from the constant-phase condition

The goal is the full chain from wave to number: define the phase velocity by following a surface of constant phase, then substitute the in-medium dispersion relation $k=n\omega/c$ — recalled by name from aq-optics-plane-waves-isotropic-media, where it is derived from the optical wave equation — to obtain $v_p=c/n$ and the wavelength contraction $\lambda=\lambda_0/n$ , and finally check that in an absorbing medium only the real part of the complex index enters. The derivation is three lines of calculus plus one substitution, which is exactly the point: the phase velocity is kinematics, and all the physics hides in $n(\omega)$ .

Step 1. Write the wave as $E(x,t)=E_0\cos(kx-\omega t+\varphi_0)$ and identify a wavefront as a locus of constant total phase, $\phi=kx-\omega t+\varphi_0=\text{const}$ . Why: "the crest" is by definition a point of fixed phase; any feature of a pure sinusoid — crest, zero, trough — is such a locus, so tracking phase is the only meaningful way to assign the wave a speed.

Step 2. Take the total differential of the constraint: $k\,\mathrm{d}x-\omega\,\mathrm{d}t=0$ , hence $v_p=\mathrm{d}x/\mathrm{d}t=\omega/k$ . Why: this is the speed an observer must move at for the phase at their location to stay frozen — the rigid-slide speed of the whole crest pattern, the chord slope $\omega/k$ on the dispersion diagram.

Step 3. Insert the dispersion relation of the medium, $k=n\omega/c$ : $v_p=\omega/(n\omega/c)=c/n$ , and with $\nu$ fixed by the source, $\lambda=v_p/\nu=\lambda_0/n$ . Why: the medium's entire influence on a monochromatic wave is the single number $n(\omega)$ ; the crests slow by $n$ and therefore pack together by $n$ , while the frequency — set by the boundary condition at the source — cannot change.

Step 4. For a lossy medium put $\tilde n=n+i\kappa$ into $\tilde k=\tilde n\omega/c$ : the field factors as $E\propto e^{-\kappa\omega z/c}\,e^{i(n\omega z/c-\omega t)}$ , an oscillation at $\mathrm{Re}\,\tilde k=n\omega/c$ inside a decaying envelope. The constant-phase condition applies to the oscillatory factor, so $v_p=\omega/\mathrm{Re}\,\tilde k=c/n$ still. Why: absorption removes amplitude but does not reschedule the zero-crossings; speed and decay separate cleanly into real and imaginary parts of one complex index. *Verified: $v_p=\omega/k$ from the differentiated phase, $v_p=c/n$ with $k=n\omega/c$ , $\lambda=\lambda_0/n$ (500 nm → 333.3 nm at $n=1.5$ ), and the real/imaginary split of $\tilde k$ — symbolically and numerically in verify_aq-optics-phase-velocity.py.*

Go deeper · Derivation B · 2 minAn index below one is causal — crests carry no information

Derivation BAn index below one is causal — crests carry no information

The goal is the $n<1$ story done honestly: show that a real physical response — the free-electron plasma, the $\omega_0\to0,\ \gamma\to0$ limit of the Lorentz oscillator recalled by name from aq-optics-lorentz-oscillator-model — drives the index below one over an entire frequency range, compute the superluminal crest speed there, and then demonstrate with the same dispersion relation that the quantity able to carry information moves below $c$ . The general envelope machinery and the front-velocity theorem belong to aq-em-phase-group-velocity; here we recall them by name and run the material numbers.

Step 1. Free electrons respond to $E\,e^{-i\omega t}$ with displacement $x=eE/(m_e\omega^2)$ opposite in phase to a bound electron below resonance; the resulting susceptibility is $\chi=-\omega_p^2/\omega^2$ , so $\varepsilon_r(\omega)=1-\omega_p^2/\omega^2$ and $n=\sqrt{1-\omega_p^2/\omega^2}<1$ for all $\omega>\omega_p$ . Why: with no restoring force the electrons oscillate 180° out of phase with the drive, so the displacement lies opposite the driving force and the re-radiation advances rather than retards the transmitted crests — the microscopic reason the index dips below one above any resonance.

Step 2. Evaluate at $\omega=2\omega_p$ : $n=\sqrt{1-\tfrac14}=\sqrt{3}/2=0.8660$ , hence $v_p=c/0.866=1.1547c$ . Why: a definite number makes the paradox sharp — these crests genuinely sweep past at 15% over light speed, and any theory that forbade it would be wrong, since radio crests above the ionosphere and X-ray crests in glass do exactly this.

Step 3. Ask what could carry a message at that speed — and find nothing. A monochromatic wave occupies all of time: its value at any future instant is already determined, so observing crests conveys nothing new. To signal you must modulate — switch the wave on, imprint a pulse — and modulation means a band of frequencies whose interference envelope moves at $v_g=\mathrm{d}\omega/\mathrm{d}k$ , the beat/envelope construction owned by aq-em-phase-group-velocity and instantiated for materials in aq-optics-group-velocity-group-index. Why: information is by definition a departure from the predictable; a pure sinusoid has none, so its crest speed is causally empty.

Step 4. Differentiate the plasma dispersion relation $\omega^2=\omega_p^2+c^2k^2$ : $2\omega\,\mathrm{d}\omega=2c^2k\,\mathrm{d}k$ , so $v_p v_g=(\omega/k)(\mathrm{d}\omega/\mathrm{d}k)=c^2$ exactly. At $\omega=2\omega_p$ : $v_g=c^2/1.1547c=0.866c<c$ . The strict bound is sharper still: the front of any switched-on signal probes $\omega\to\infty$ , where $n\to1$ , and moves at exactly $c$ (Sommerfeld–Brillouin, recalled by name from aq-em-phase-group-velocity). Why: the same dispersion curve that lifts the chord above $c$ lowers the tangent below it — superluminal $v_p$ and subluminal $v_g$ are one geometric fact, and causality is policed at the front, not at the crests. *Verified: $\chi=-\omega_p^2/\omega^2$ from the free-electron equation of motion, $n(2\omega_p)=0.8660$ , $v_p=1.1547c$ , $v_p v_g=c^2$ hence $v_g=0.8660c$ , and $n\to1^-$ as $\omega\to\infty$ (sympy series) in verify_aq-optics-phase-velocity.py.*

Limiting cases

Case	Result	Meaning
Vacuum / non-dispersive	$n=1$ , $v_p=c$ , $v_g=v_p$	chord and tangent coincide; pulses keep shape
Transparent glass (normal dispersion)	$n>1$ , $\mathrm{d}n/\mathrm{d}\omega>0$ ⇒ $v_p<c$ , $v_g<v_p$	visible light in glass: blue crests slower than red
Above a resonance / X-ray band	$n\approx1-\omega_p^2/2\omega^2<1$ ⇒ $v_p>c$	routine superluminal crests; grazing-incidence X-ray mirrors
Plasma, $\omega>\omega_p$	$n=\sqrt{1-\omega_p^2/\omega^2}$ , $v_p v_g=c^2$	ionospheric radio: $v_p>c$ , $v_g<c$ , product pinned
At cutoff $\omega\to\omega_p$	$k\to0$ : $v_p\to\infty$ , $v_g\to0$	crest speed diverges while energy stalls; below cutoff, evanescent
Absorbing medium	$\tilde n=n+i\kappa$ : $v_p=c/n$ only	$\kappa$ sets decay length, never the crest speed

Common trapIf crests move faster than light, a fast shutter should let me signal faster than light

It should not, and seeing why is the whole point of the phase/group distinction. The moment you operate a shutter you no longer have a monochromatic wave: chopping the wave imprints sidebands, and what emerges is a wave packet whose envelope travels at the group velocity $v_g=\mathrm{d}\omega/\mathrm{d}k$ — which in the very media where $v_p>c$ (plasma, X-ray band) sits below $c$ , pinned by $v_p v_g=c^2$ for the plasma family. The un-chopped wave, meanwhile, carries nothing: it has been running since $t=-\infty$ and its future is its past, so watching its crests race by at $1.1547c$ tells you nothing you did not already know. And the sharpest feature of all — the switch-on front itself — is built from the $\omega\to\infty$ components, for which every medium has $n\to1$ : the front propagates at exactly $c$ , always (Sommerfeld–Brillouin, recalled from aq-em-phase-group-velocity). Superluminal crests, subluminal envelopes, luminal fronts — three speeds, one dispersion curve, zero paradoxes.

Worked examples

1 · Crest speeds in glass and above the ionosphere (confidence builder)

Find the phase velocity (a) of 500 nm light in crown glass, $n=1.5$ ; (b) of an 18.0 MHz shortwave broadcast crossing an ionospheric layer with electron density $N=1.0\times10^{12}\ \mathrm{m^{-3}}$ .

Step 1. Glass: $v_p=c/n=299\,792\,458/1.5=\mathbf{1.9986\times10^8\ m/s}$ — two-thirds of $c$ . The wavelength inside is $\lambda=\lambda_0/n=500/1.5=333.3$ nm; the frequency $\nu=c/\lambda_0=6.0\times10^{14}$ Hz is unchanged.

Step 2. Ionosphere: the plasma frequency is $f_p=\tfrac{1}{2\pi}\sqrt{Ne^2/(\varepsilon_0 m_e)}$ . With $N=1.0\times10^{12}\ \mathrm{m^{-3}}$ : $f_p=\mathbf{8.98\ MHz}$ , so $2f_p=17.96$ MHz — the 18.0 MHz broadcast sits within a quarter percent of $\omega=2\omega_p$ .

Step 3. Evaluate at $\omega=2\omega_p$ : $n=\sqrt{1-\tfrac14}=\mathbf{0.866}$ . The crests move at $v_p=c/n=1.1547c=\mathbf{3.462\times10^8\ m/s}$ — faster than light in vacuum, as they genuinely do over every shortwave path that grazes the ionosphere. v_p = 1.9986×10⁸ m/s in glass; v_p = 1.1547c = 3.462×10⁸ m/s at ω = 2ωₚ in the ionospheric plasma.

2 · The signal audit — v_p·v_g = c²

For the 18 MHz wave of example 1 ( $n=0.866$ at $\omega=2\omega_p$ ), find the group velocity and group index, and confirm no signal outruns light.

Step 1. For the plasma dispersion relation $\omega^2=\omega_p^2+c^2k^2$ , differentiating gives $v_p v_g=c^2$ (Derivation B). So $v_g=c^2/v_p=c\cdot n=0.866c=\mathbf{2.596\times10^8\ m/s}$ — an actual radio pulse crosses the layer slower than light, even as its crests outrun light.

Step 2. The group index is $n_g=c/v_g=1/n=\mathbf{1.1547}$ . Check against $n_g=n+\omega\,\mathrm{d}n/\mathrm{d}\omega$ : for the plasma, $\omega\,\mathrm{d}n/\mathrm{d}\omega=\omega_p^2/(\omega^2 n)=0.25/0.866=0.2887$ , and $0.866+0.2887=1.1547$ ✓. Note $n<1$ but $n_g>1$ : the dispersive term always rescues causality.

Step 3. Sanity-check the product: $v_p v_g=(1.1547c)(0.866c)=1.0000\,c^2$ ✓. v_g = 0.866c = 2.60×10⁸ m/s, n_g = 1.1547 — the signal stays below c.

Physics of the portrait

The phase velocity is the oldest wave speed in optics — it is what Huygens' expanding wavefronts propagate at, what Foucault measured in water, and what the refractive index has silently encoded since Snel — yet its modern meaning was fixed only when dispersion theory forced physicists to say precisely which speed a wave has.

Origins · 1 minphase velocity and how this was discovered

phase velocity

the crest speed v_p = ω/k = c/n — the speed the refractive index measures · electromagnetic optics of media

The phase velocity is the speed light was always assumed to have — one wave, one speed — until precision demanded better. Huygens' Traité de la Lumière (1690) already propagated wavefronts at a single speed per medium, and his construction required light to travel slower in denser media, where the rival corpuscular theory required faster. Foucault's rotating-mirror measurement of 1850 — light slower in water than in air, by just the factor $n$ — settled that duel in the wave's favour, and Maxwell's 1865 theory then computed the speed from first principles: $v=1/\sqrt{\varepsilon\mu}=c/n$ . The trouble began when dispersion was taken seriously: if $n$ depends on frequency, a pulse has no single speed, and near resonances the crest speed misbehaves spectacularly — Lorentz's electron theory predicted, and X-ray optics confirmed (Compton's total external reflection, 1923), whole spectral regions with $n<1$ and crests above $c$ . The resolution — that the crest speed is a phase label, while signals ride the envelope and fronts travel at exactly $c$ — was nailed down by Sommerfeld and Brillouin in 1914, splitting "the speed of light in a medium" into a taxonomy of which the phase velocity is the first and simplest member: the one that sets wavelengths, fringes, and phase matching, and the one number every optical datasheet still quotes as $n$ .

History

1690

— Huygens' Traité de la Lumière propagates light as wavefronts, slower in denser media

1850

— Foucault measures light slower in water than in air, deciding for the wave picture

1865

— Maxwell's electromagnetic theory fixes the wave speed at v = 1/√(εμ) = c/n

1914

— Sommerfeld and Brillouin separate phase, group, signal and front velocities in dispersive media

1923

— Compton demonstrates total external reflection of X-rays — the n below one regime made experimental

Going further

Texts. Hecht, Optics, 5th ed., §2.4, §3.5 (harmonic waves, phase and group velocity) · Saleh & Teich, Fundamentals of Photonics, 2nd ed., Ch. 5 (electromagnetic optics, phase velocity) · Loudon, The Quantum Theory of Light, 3rd ed., Ch. 1 (dispersion, phase and group velocity) · Foot, Atomic Physics, §7 (light–atom interaction, refractive index) · Demtröder, Laser Spectroscopy, Vol. 1, Ch. 2 (dispersion of the refractive index)

Provenance. Definitions and derivations follow Hecht (Optics, §2.4, §3.5), Saleh & Teich (Ch. 5), and Loudon (Ch. 1). A verifier should reproduce $v_p=\omega/k=c/n$ from the constant-phase condition and from the dielectric wave equation, confirm $v_g=c/(n+\omega\,\mathrm{d}n/\mathrm{d}\omega)$ , and check the worked numbers ( $c=299\,792\,458$ m/s, $v_p=2.0\times10^8$ m/s and $\lambda=333$ nm for $n=1.5$ ). No non-standard numerical values are asserted. Addendum for this edition: the compact reciprocal form $1/v_g=(1/v_p)\,(1-(\omega/n)\,\mathrm{d}n/\mathrm{d}\omega)^{-1}$ carried by the previous edition is only the first-order Taylor of the exact relation and has been replaced by $1/v_g=(n+\omega\,\mathrm{d}n/\mathrm{d}\omega)/c$ (inequality of the two forms checked symbolically in verify_aq-optics-phase-velocity.py); the previous edition's limiting-case table also placed the $n<1$ region "below resonance" — it lies above a resonance. Plasma anchors ( $n=0.866$ , $v_p=1.1547c$ , $v_g=0.866c$ at $\omega=2\omega_p$ ; $f_p=8.98$ MHz for $N=10^{12}\ \mathrm{m^{-3}}$ ) are recomputed in the same script.

Builds on: Plane Waves in Homogeneous Isotropic Media · The Refractive Index · Maxwell's Equations in Dielectric Media · Dielectric Susceptibility and Permittivity Leads to: Group Velocity and Group Index · Dispersion: Normal and Anomalous · Complex Refractive Index and the Extinction Coefficient · Sellmeier and Cauchy Dispersion Equations

Is anything physical actually moving at the phase velocity?

No — and that is the honest answer, not a technicality. The crest is a condition (a point where the phase equals a particular value), and conditions can sweep through space at any speed without transporting matter, energy or information, the way the tip of a searchlight beam can sweep a distant cloud faster than $c$ . In a transparent, weakly dispersive medium the energy happens to travel near $v_p$ (and exactly at $v_p$ if dispersion vanishes), which is why the distinction stays invisible in everyday optics. It becomes visible precisely where $v_p$ misbehaves — near resonances and in $n<1$ bands — and there the energy and signal speeds detach and stay below $c$ .

If a phase velocity above c is harmless, why can I not encode a message in the crests?

Because the crests of a monochromatic wave are perfectly predictable. The wave $\cos(kx-\omega t)$ is specified for all time by three numbers ( $E_0$ , $\omega$ , $\varphi_0$ ); once a receiver has locked onto it, every future crest arrival is known in advance, so no crest can surprise — and information is exactly the capacity to surprise. Any attempt to make the crests informative — turning the wave on and off, shifting its phase, modulating its amplitude — necessarily spreads the spectrum over a band $\Delta\omega$ , and the imprinted feature then propagates as an envelope at $v_g=\mathrm{d}\omega/\mathrm{d}k$ , not at $\omega/k$ . The faster-than-light thing is precisely the component of the wave on which nothing can be written.

How is the phase velocity measured if nothing rides on it?

Through phase-sensitive observables rather than time-of-flight. Refraction is the classic route: Snell's law reads the crest-speed ratio directly, $\sin\theta_1/\sin\theta_2=v_{p1}/v_{p2}=n_2/n_1$ , so a prism spectrometer is a phase-velocity meter. Interferometry is the precision route: inserting a sample of thickness $L$ shifts the fringe pattern by the extra optical phase $(n-1)\omega L/c$ , giving $v_p$ to parts in $10^8$ . And the wavelength contraction $\lambda=\lambda_0/n$ — measurable with a standing wave — is phase velocity in disguise. Time-of-flight experiments, by contrast, clock the group velocity; Foucault's water measurement worked because in water at optical frequencies $v_g$ and $v_p$ differ by only about 1.5%, far below his precision.

What exactly happens at the plasma frequency, where v_p diverges?

At $\omega=\omega_p$ the wavenumber $k\to0$ : the whole medium oscillates in phase — an infinite-wavelength collective oscillation, not a travelling wave. The "infinite phase velocity" is then the statement that all crests occur simultaneously everywhere, which transports nothing; consistently, the group velocity $v_g=c^2k/\omega\to0$ , so energy transport shuts down completely. Just below $\omega_p$ , $k$ becomes imaginary and the field is evanescent — it penetrates a skin depth and is reflected, which is why the ionosphere is a mirror for frequencies below its plasma frequency (a few to ~10 MHz) and transparent above. The divergence of $v_p$ at cutoff is thus the least paradoxical superluminal effect of all: it marks the edge where propagation itself ceases.

Does the phase velocity ever fall below the group velocity?

Yes — whenever dispersion is anomalous, $\mathrm{d}n/\mathrm{d}\omega<0$ , the group index $n_g=n+\omega\,\mathrm{d}n/\mathrm{d}\omega$ drops below $n$ and $v_g>v_p$ . This happens inside absorption bands (the regime mapped in aq-optics-normal-anomalous-dispersion), where the group velocity can exceed $c$ or turn negative — at which point it stops describing energy transport altogether, as the group-velocity article details. In the transparent windows of ordinary glasses dispersion is always normal ( $\mathrm{d}n/\mathrm{d}\omega>0$ ), so there the ordering $v_g<v_p<c$ holds and both velocities behave.

Why keep the phase velocity at all if signals use the group velocity?

Because half of optics is not about sending signals but about keeping waves in step. Interference, diffraction, resonance and nonlinear frequency conversion all depend on relative phase, and relative phase accumulates at the phase velocity: a Fabry–Pérot resonates where the round-trip phase $2n\omega L/c$ hits $2\pi m$ ; second-harmonic generation dies unless $n(2\omega)=n(\omega)$ ; an interferometer arm converts $\delta(nL)$ straight into fringes. The refractive index itself — the most-quoted number in optics — is the phase velocity in disguise, $n=c/v_p$ . The group velocity answers "when does the pulse arrive"; the phase velocity answers "how do the waves add when it does" — and the second question is asked at least as often.