Group Velocity and Group Index

Clock a short pulse through 100 m of BK7 glass and compare with the obvious estimate $t=nL/c$ : the pulse arrives 8.2 ns late. The obvious estimate used the wrong index. We take from aq-em-phase-group-velocity the general machinery of wave packets — the two-frequency beat whose envelope moves at $\Delta\omega/\Delta k$ , the continuum statement $v_g=\mathrm{d}\omega/\mathrm{d}k$ (the tangent to the dispersion curve, where the phase velocity of aq-optics-phase-velocity is the chord), the group index, and the superluminal caveat near absorption bands. What this article owns is the material instance that optics actually uses: substituting $k=n(\omega)\,\omega/c$ turns the tangent slope into $v_g=c/n_g$ with $n_g=n+\omega\,\mathrm{d}n/\mathrm{d}\omega=n-\lambda\,\mathrm{d}n/\mathrm{d}\lambda$ — a group index computable from the same Sellmeier data that tabulates $n$ . At the BK7 d-line: $n_d=1.51680$ but $\mathrm{d}n/\mathrm{d}\lambda=-0.04180\ \mathrm{\mu m^{-1}}$ , so $n_g=1.5414$ — the envelope of a pulse crawls at $v_g=c/n_g=1.945\times10^8$ m/s while its crests do $1.9765\times10^8$ m/s. In every transparency window $\mathrm{d}n/\mathrm{d}\lambda<0$ (normal dispersion), so $n_g>n$ universally there: pulses always run late relative to the crest-speed estimate, by 81.9 ps per metre in BK7. That gap is the timing budget of every fibre link, the calibration constant of optical coherence tomography, and the carrier-envelope slip of every femtosecond laser.

Key takeaways

A signal is never monochromatic: it is a wave packet, and its envelope — which carries the energy and the information — moves at the group velocity $v_g=\mathrm{d}\omega/\mathrm{d}k$ , the tangent slope of the dispersion curve at the carrier (recalled from aq-em-phase-group-velocity).
In a medium, $k=n\omega/c$ gives $v_g=c/n_g$ with the group index $n_g=n+\omega\,\mathrm{d}n/\mathrm{d}\omega=n-\lambda\,\mathrm{d}n/\mathrm{d}\lambda$ ( $\lambda$ = vacuum wavelength).
Normal dispersion means $\mathrm{d}n/\mathrm{d}\lambda<0$ , so $n_g>n$ : the envelope always lags the crests in a transparency window.
BK7 at 587.6 nm: $n_d=1.51680$ , $\mathrm{d}n/\mathrm{d}\lambda=-0.04180\ \mathrm{\mu m^{-1}}$ ⇒ $n_g=1.5414$ , $v_g=1.945\times10^8$ m/s — an 81.9 ps/m delay beyond the naive $c/n$ estimate.
Pulse arrival times, fibre latency, and interferometric coherence positions are set by $n_g$ ; refraction angles and phase matching are set by $n$ — two different datasheet columns.
Inside anomalous-dispersion bands $v_g$ can exceed $c$ or turn negative; there it stops describing transport, and the signal front still moves at exactly $c$ (owner's caveat, recalled by name).
Key idea: the carrier moves at the chord $\omega/k=c/n$ , the envelope at the tangent $\mathrm{d}\omega/\mathrm{d}k=c/n_g$ with $n_g=n-\lambda\,\mathrm{d}n/\mathrm{d}\lambda$ — and since $\mathrm{d}n/\mathrm{d}\lambda<0$ in every transparency window, pulses are always slower than crests: time flight with $n_g$ , never with $n$ .

The core insight

A pulse's carrier crests move at the phase velocity $c/n$ , but the envelope — the thing that arrives, carries energy, and can be timed — moves at the group velocity $v_g=\mathrm{d}\omega/\mathrm{d}k=c/n_g$ , where $n_g=n+\omega\,\mathrm{d}n/\mathrm{d}\omega=n-\lambda\,\mathrm{d}n/\mathrm{d}\lambda$ . Normal dispersion makes $n_g>n$ , so in glass the envelope always lags: BK7's $n_d=1.5168$ hides a group index of $1.5414$ .

Where this lives today

The group index is a line item on real datasheets. Fibre-optic links budget latency as $n_g L/c$ — 4.88 µs per kilometre of silica at 1550 nm — and high-frequency trading and data-centre time synchronization live or die by that number; the next-order coefficient, GVD, is the pulse-broadening budget owned by aq-optics-group-velocity-dispersion-pulse-broadening. Optical coherence tomography and white-light interferometry position their fringe packets at the group-delay match — retinal thickness maps are group-index measurements — and time-of-flight refractometers likewise read $n_g$ . And in every femtosecond oscillator $v_p\neq v_g$ slips the carrier under the envelope each round trip — the carrier-envelope offset that frequency combs lock. At the exotic end, dispersion engineering makes $v_g$ a free parameter. Electromagnetically induced transparency manufactures a hyper-steep normal-dispersion slope inside a transparency window, reaching $n_g\sim10^7$ — 17 m/s in Hau's 1999 sodium experiment — and "stopped light" quantum memories park the pulse in atomic coherence. Gain-assisted anomalous dispersion delivers the opposite regime ( $v_g=-c/310$ in the Wang experiment) as a precision test bed for the information-velocity theorems of aq-em-phase-group-velocity.

The physical picture

Add two sinusoids of slightly different frequencies and watch the sum: a fast carrier oscillation inside a slow beat envelope. The carrier crests move at $\omega/k$ — the phase velocity. The beat maxima sit where the two components agree in phase, and keeping them in agreement as time runs requires matching phase changes, not phases: the envelope therefore moves at the ratio of differences $\Delta\omega/\Delta k$ , which in the limit is the derivative $\mathrm{d}\omega/\mathrm{d}k$ . That two-line argument — owned in general by aq-em-phase-group-velocity and re-run in Derivation A — contains the entire concept: chord versus tangent on the dispersion diagram. A real pulse is a continuum of such pairs, and to first order every feature of its envelope translates rigidly at the common tangent slope $v_g$ .

Figure

The punchline — a two-wave beat in BK7 glass at two instants: the envelope (dashed) advances at v_g = c/n_g = 1.945×10⁸ m/s while the carrier crests (solid; one crest marked blue) advance faster at v_p = c/n = 1.9765×10⁸ m/s, because n_g = 1.5414 exceeds n_d = 1.5168

Where does the material enter? A pulse's spectrum straddles a band of frequencies, and in a dispersive medium each component travels at its own crest speed $c/n(\omega)$ . Blue components lag red ones (normal dispersion), so the point of constructive interference — the envelope peak — drifts backward relative to the crests: envelope slower than carrier, $n_g>n$ . The group index simply packages the two ways a component can be slow: the $n$ term (crests slow) plus the $\omega\,\mathrm{d}n/\mathrm{d}\omega$ term (neighbouring components de-phase). The second figure shows both indices across the visible for BK7: $n_g$ rides above $n$ everywhere in the window, the gap widening toward the blue where dispersion steepens.

Figure

BK7 across the visible, computed from the Schott Sellmeier equation: the group index n_g = n − λ·dn/dλ (upper curve) rides above the phase index n (lower curve); at the d-line, n = 1.5168 and n_g = 1.5414 — a gap of 0.0246 that makes every pulse 81.9 ps/m late relative to the naive c/n estimate

Think of $n_g$ as "the index a stopwatch sees" and $n$ as "the index a protractor sees". Refraction angles, fringe positions and phase matching read $n$ ; arrival times, latency budgets and coherence positions read $n_g$ . They never coincide in a real material — a frequency-independent index would violate the dispersion relations that close this chapter (aq-optics-kramers-kronig-relations).

The core equations

The definition first, recalled by name from aq-em-phase-group-velocity, which owns the general wavepacket construction. Expanding the dispersion relation $\omega(k)$ about the carrier $k_0$ , the envelope of any narrow-band packet translates at the tangent slope — evaluated at the carrier.

v_g=\left.\frac{\mathrm{d}\omega}{\mathrm{d}k}\right|_{k_0} \qquad (1)

where: $v_g$ — group velocity, the translation speed of the envelope (and, in transparent media, of energy and information); $k_0$ , $\omega_0$ — carrier wavenumber and frequency. The material form follows by differentiating the in-medium dispersion relation $k=n(\omega)\,\omega/c$ (from aq-optics-plane-waves-isotropic-media). The derivative of the product defines the group index, the single most-used number in pulse optics.

n_g\equiv\frac{c}{v_g}=c\,\frac{\mathrm{d}k}{\mathrm{d}\omega}=n+\omega\frac{\mathrm{d}n}{\mathrm{d}\omega} \qquad (2)

where: $n_g$ — group index; $n(\omega)$ — phase index; $\omega\,\mathrm{d}n/\mathrm{d}\omega$ — the dispersive correction: positive in normal dispersion ( $\mathrm{d}n/\mathrm{d}\omega>0$ ), so $n_g>n$ there. Where dispersion vanishes, $n_g=n$ and envelope rides with crests. Tabulated glass data come as $n(\lambda)$ with $\lambda$ the vacuum wavelength, so the practical form trades the frequency derivative for a wavelength derivative via $\omega=2\pi c/\lambda$ (Derivation B).

n_g=n-\lambda\frac{\mathrm{d}n}{\mathrm{d}\lambda} \qquad (3)

where: $\lambda$ — vacuum wavelength; $\mathrm{d}n/\mathrm{d}\lambda$ — the Sellmeier slope, negative throughout every transparency window, which is why the correction $-\lambda\,\mathrm{d}n/\mathrm{d}\lambda$ is positive. BK7 at $\lambda=587.6$ nm: $n=1.51680$ , $\mathrm{d}n/\mathrm{d}\lambda=-0.04180\ \mathrm{\mu m^{-1}}$ ⇒ $n_g=1.51680+0.58756\times0.04180=1.5414$ . What the group index buys is a clock. The envelope of a pulse crossing a length $L$ of medium arrives after the group delay — the formula behind every fibre latency budget and time-of-flight refractometer.

\tau=\frac{L}{v_g}=\frac{n_g\,L}{c} \qquad (4)

where: $\tau$ — envelope transit time; $L$ — geometric path. Per metre of BK7: $n_g/c=5.141$ ns/m versus the naive $n/c=5.060$ ns/m — the pulse runs 81.9 ps/m late. Using $n$ where $n_g$ belongs is the classic timing bug. One order deeper sits the variation of the group velocity across the pulse's own spectrum: the group-velocity dispersion $\beta_2=\mathrm{d}^2k/\mathrm{d}\omega^2=\mathrm{d}(1/v_g)/\mathrm{d}\omega$ , which broadens and chirps pulses rather than merely delaying them. It is named here for the map and owned by aq-optics-group-velocity-dispersion-pulse-broadening.

k(\omega)\approx k_0+\frac{\omega-\omega_0}{v_g}+\tfrac12\,\beta_2\,(\omega-\omega_0)^2 \qquad (5)

where: the first-order term translates the envelope at $v_g$ (this article); the $\beta_2$ term spreads it (the GVD article). Truncating at first order is legitimate when the pulse is narrow-band and the medium transparent.

Derivation AThe envelope of a two-frequency beat moves at dω/dk

The goal is the group velocity from first principles in its cleanest setting: superpose two monochromatic waves straddling a carrier, factor the sum into carrier × envelope, read off the two speeds, and pass to the limit. The construction is recalled from aq-em-phase-group-velocity, which owns the general wavepacket and stationary-phase machinery; the optical extension — each wavenumber supplied by a refractive index, $k=n(\omega)\,\omega/c$ — is what turns the tangent slope into $c/n_g$ and makes the result computable from glass catalogues.

Step 1. Superpose equal amplitudes at $(\omega\pm\Delta\omega,\ k\pm\Delta k)$ , the wavenumbers fixed by the medium through $k(\omega)=n(\omega)\,\omega/c$ :

E=E_0\cos\big[(k+\Delta k)z-(\omega+\Delta\omega)t\big]+E_0\cos\big[(k-\Delta k)z-(\omega-\Delta\omega)t\big].

Why: two components are the minimal spectrum with any structure — the simplest object that can carry a timeable feature, since one component alone is featureless (the lesson of the phase-velocity article).

Step 2. Apply the sum-to-product identity:

E=2E_0\,\cos(\Delta k\,z-\Delta\omega\,t)\,\cos(kz-\omega t).

Why: the sum factors exactly — a fast carrier $\cos(kz-\omega t)$ multiplied by a slow envelope $\cos(\Delta k\,z-\Delta\omega\,t)$ ; no approximation has been made yet.

Step 3. Read off the two speeds by the constant-phase argument applied to each factor: carrier crests at $v_p=\omega/k$ , envelope maxima where $\Delta k\,z-\Delta\omega\,t=\text{const}$ , i.e. moving at $v_{\text{env}}=\Delta\omega/\Delta k$ . Why: the beat maximum sits where the two components agree in phase; keeping the agreement as time advances matches phase changes, so the envelope speed is a difference quotient — the finite-difference tangent — while the carrier speed is the chord.

Step 4. Shrink the detuning: $v_g=\lim_{\Delta k\to0}\Delta\omega/\Delta k=\mathrm{d}\omega/\mathrm{d}k$ , evaluated at the carrier. A general narrow-band pulse is a Fourier integral of such pairs, and to first order in the expansion of $k(\omega)$ every pair's envelope moves at this same slope — the packet translates rigidly at $v_g$ (the continuum upgrade is owned by aq-em-phase-group-velocity). With $k=n\omega/c$ , the slope is $\mathrm{d}k/\mathrm{d}\omega=(n+\omega\,\mathrm{d}n/\mathrm{d}\omega)/c$ , so $v_g=c/n_g$ . Why: the derivative is the same for every nearby pair only because the expansion was truncated at first order — the second-order spread of tangent slopes is precisely GVD, deferred by name to the pulse-broadening article. *Verified: the sum-to-product factorization and the envelope speed $\Delta\omega/\Delta k$ symbolically (sympy), plus $v_g=c/n_g$ with BK7 numbers, in verify_aq-optics-group-velocity-group-index.py.*

Go deeper · Derivation B · 1 minThe group index in frequency and wavelength forms

Derivation BThe group index in frequency and wavelength forms

The goal is the working formula pair $n_g=n+\omega\,\mathrm{d}n/\mathrm{d}\omega=n-\lambda\,\mathrm{d}n/\mathrm{d}\lambda$ , derived by differentiating the dispersion relation and then converting to the vacuum-wavelength variable in which every glass catalogue is written — ending with the real BK7 evaluation that anchors this article. The two forms are one identity connected by the chain rule; sign discipline (which variable, which slope) is where all practical mistakes live, so each sign is tracked explicitly.

Step 1. Differentiate $k(\omega)=n(\omega)\,\omega/c$ by the product rule:

\frac{\mathrm{d}k}{\mathrm{d}\omega}=\frac{1}{c}\left(n+\omega\frac{\mathrm{d}n}{\mathrm{d}\omega}\right)\equiv\frac{n_g}{c},\qquad v_g=\left(\frac{\mathrm{d}k}{\mathrm{d}\omega}\right)^{-1}=\frac{c}{n_g}.

Why: the tangent slope of $\omega(k)$ is the reciprocal of $\mathrm{d}k/\mathrm{d}\omega$ ; the bracket is defined as the group index because it plays for envelopes exactly the role $n$ plays for crests.

Step 2. Change variable to vacuum wavelength with $\omega=2\pi c/\lambda$ , so $\mathrm{d}\omega/\mathrm{d}\lambda=-2\pi c/\lambda^2$ and hence $\mathrm{d}n/\mathrm{d}\omega=\big(\mathrm{d}n/\mathrm{d}\lambda\big)\big(\mathrm{d}\lambda/\mathrm{d}\omega\big)=-\frac{\lambda^2}{2\pi c}\frac{\mathrm{d}n}{\mathrm{d}\lambda}$ . Why: catalogues tabulate $n(\lambda)$ , not $n(\omega)$ ; the minus sign appears because frequency and wavelength run in opposite directions, and it is the sign that makes normal dispersion come out with $n_g>n$ in both variables.

Step 3. Substitute into the bracket: $\omega\,\dfrac{\mathrm{d}n}{\mathrm{d}\omega}=\dfrac{2\pi c}{\lambda}\cdot\left(-\dfrac{\lambda^2}{2\pi c}\dfrac{\mathrm{d}n}{\mathrm{d}\lambda}\right)=-\lambda\dfrac{\mathrm{d}n}{\mathrm{d}\lambda}$ , giving $n_g=n-\lambda\,\mathrm{d}n/\mathrm{d}\lambda$ . Why: the $2\pi c$ cancels identically — the wavelength form is exact, not approximate, and both forms are the same number for the same wave.

Step 4. Evaluate for BK7 at the d-line from the Schott Sellmeier fit: $n(0.58756\ \mathrm{\mu m})=1.51680$ and $\mathrm{d}n/\mathrm{d}\lambda=-0.04180\ \mathrm{\mu m^{-1}}$ , so

n_g=1.51680-0.58756\times(-0.04180)=1.51680+0.02456=1.5414,\qquad v_g=\frac{c}{1.5414}=1.945\times10^8\ \mathrm{m/s}.

Why: the correction term $+0.0246$ is three times the F-to-C index spread of the same glass (0.00806) — a group-index effect larger than the dispersion optical designers already track to five decimals. *Verified: the chain-rule identity $n+\omega\,\mathrm{d}n/\mathrm{d}\omega=n-\lambda\,\mathrm{d}n/\mathrm{d}\lambda$ symbolically (sympy), the analytic Sellmeier derivative against a finite difference, and $n_g=1.5414$ , $v_g=1.945\times10^8$ m/s in verify_aq-optics-group-velocity-group-index.py.*

Limiting cases

Case	Result	Meaning
Non-dispersive medium	$\mathrm{d}n/\mathrm{d}\omega=0$ ⇒ $n_g=n$ , $v_g=v_p$	vacuum (exactly); envelope rides with crests
Normal dispersion (glass window)	$\mathrm{d}n/\mathrm{d}\lambda<0$ ⇒ $n_g>n$ , $v_g<v_p$	BK7: 1.5414 vs 1.5168 — pulses run 81.9 ps/m late
Plasma / hyperbolic family	$v_p v_g=c^2$ ⇒ $n_g=1/n>1$	crests above $c$ , envelope below — pinned product
Anomalous band (on a resonance)	$n_g<1$ , $\le0$ possible ⇒ $v_g>c$ or negative	envelope reshapes; not a transport speed there
Steep normal dispersion (EIT)	$n_g\sim10^7$ ⇒ $v_g\sim30$ m/s	slow light: bicycle-speed pulses in cold vapours
High frequency $\omega\to\infty$	$n\to1$ , $n_g\to1$	every medium turns vacuum-like; fronts at exactly $c$

Common trapA pulse crosses a glass slab in time L·n/c

It does not — that is the phase index doing the group index's job, and it is wrong in every real medium, because a frequency-independent $n$ does not exist. The envelope of the pulse — the thing your detector triggers on — arrives after $\tau=n_g L/c$ with $n_g=n-\lambda\,\mathrm{d}n/\mathrm{d}\lambda>n$ throughout any transparency window. In BK7 the error is 81.9 ps per metre (1.62%); over a 100 m path it is 8.2 ns — enormous against photodiode jitter, and fatal to a fibre latency budget (silica at 1550 nm: $n=1.4440$ but $n_g=1.4626$ ). The complementary trap is inverted: never hand $n_g$ to Snell's law or a Fresnel coefficient — those are phase phenomena and read $n$ . And keep the jurisdiction in mind: inside an anomalous-dispersion band $n_g$ can drop below 1 or negative, $v_g$ then exceeds $c$ or reverses, and the group delay formula stops describing transport altogether — the packet reshapes, and the causal front still arrives at exactly $c$ (the owner's superluminal caveat, aq-em-phase-group-velocity).

Worked examples

1 · The group index of BK7 at the d-line (confidence builder)

From the Schott Sellmeier data for BK7 — $n_d=1.51680$ at $\lambda=587.56$ nm, $\mathrm{d}n/\mathrm{d}\lambda=-0.04180\ \mathrm{\mu m^{-1}}$ — find the group index and group velocity, and compare $v_g$ with $v_p$ .

Step 1. Wavelength form of the group index (equation 3), with $\lambda$ in µm to match the slope's units: $n_g=n-\lambda\,\mathrm{d}n/\mathrm{d}\lambda=1.51680-0.58756\times(-0.04180)$ .

Step 2. The correction is $0.58756\times0.04180=0.02456$ , positive because the slope is negative: $n_g=1.51680+0.02456=\mathbf{1.5414}$ .

Step 3. Speeds: $v_g=c/n_g=299\,792\,458/1.5414=\mathbf{1.945\times10^8\ m/s}$ , against $v_p=c/n_d=1.9765\times10^8$ m/s. The envelope moves 1.6% slower than the crests ( $v_p/v_g=n_g/n=1.016$ ): each crest is born at the rear of a pulse, slides forward through it, and dies at the front. n_g = 1.5414, v_g = 1.945×10⁸ m/s.

2 · Timing a pulse through 100 m of glass — the naive index fails

A pulsed rangefinder fires through $L=100$ m of BK7 (round trip through a monolithic delay block). Compute the transit time predicted by the naive estimate $\tau=nL/c$ and by the correct $\tau=n_g L/c$ , and the resulting timing error.

Step 1. Naive (phase-index) estimate: $\tau_p=n_d L/c=1.51680\times100/(2.99792458\times10^8)=\mathbf{505.95\ ns}$ .

Step 2. Correct (group-index) transit: $\tau_g=n_g L/c=1.54136\times100/(2.99792458\times10^8)=\mathbf{514.14\ ns}$ .

Step 3. The pulse is late by $\Delta\tau=\tau_g-\tau_p=(n_g-n)L/c=0.02456\times100/c=\mathbf{8.19\ ns}$ — i.e. 81.9 ps for every metre of glass. Read as distance, a rangefinder calibrated with $n$ would misplace its target by $c\,\Delta\tau/(2n)\approx0.8$ m over this path. The same accounting at telecom wavelengths (silica: $n=1.4440$ , $n_g=1.4626$ at 1550 nm) sets the latency of every fibre span at 4.879 µs/km — always from $n_g$ . Δτ = 8.19 ns over 100 m; group delay 5.141 ns per metre of BK7.

Physics of the portrait

The group velocity is the youngest of the classical wave speeds — it had to wait for physicists to admit that "the" velocity of light in a medium was ill-posed, and it was born not in optics but in the mathematics of water waves.

Origins · 1 mingroup velocity and how this was discovered

group velocity

the envelope speed v_g = dω/dk = c/n_g — the speed a pulse and its information actually travel · electromagnetic optics of media

The group velocity entered physics sideways. William Rowan Hamilton, reworking optics and mechanics into one variational structure, noted in 1839 that in a dispersive medium the velocity of a group of waves differs from the velocity of its individual crests — an observation that sat unused for decades (Stokes revived it as an examination question in 1876). It was Rayleigh's Theory of Sound (1877) that made $v_g=\mathrm{d}\omega/\mathrm{d}k$ standard equipment, tying it to the energy transport of a wave train and supplying the practical forms used ever since; the deep-water case — crests at twice the group speed, visibly overtaking their own packet — made the abstraction undeniable. Optics inherited the concept wholesale and gave it its sternest test: near absorption lines the group velocity exceeds $c$ or turns negative, and Sommerfeld and Brillouin's 1914 dissection of a signal entering a Lorentz medium established what survives there (nothing kinematic — the front arrives at exactly $c$ ). The modern era turned $v_g$ from a diagnosis into a design variable: fibre engineers budget latency with the group index, ultrafast laboratories manage the carrier-envelope slip $v_p\neq v_g$ cycle by cycle, and electromagnetically induced transparency has parked $n_g$ at $10^7$ — light at bicycle speed — en route to stopping it altogether.

History

1839

— Hamilton distinguishes group from phase propagation in dispersive media

1877

— Rayleigh's Theory of Sound systematizes v_g = dω/dk and links it to energy transport

1914

— Sommerfeld and Brillouin dissect signal arrival in a Lorentz medium: precursors, signal velocity, front at exactly c

1999

— Hau slows light to 17 m/s in a cold sodium cloud via electromagnetically induced transparency

2000

— Wang, Kuzmich and Dogariu measure v_g = −c/310 in gain-assisted caesium; the front still arrives at c

Going further

Texts. Hecht, Optics, 5th ed., §7.2.2 (Group Velocity) · Foot, Atomic Physics, §7.4 · Loudon, The Quantum Theory of Light, 3rd ed., Ch. 1 · Saleh & Teich, Fundamentals of Photonics, 2nd ed., §5.6 · Demtröder, Laser Spectroscopy, Vol. 1, Ch. 3 · Jackson, Classical Electrodynamics, 3rd ed., §7.8 (Superposition and Group Velocity)

Provenance. Definitions $v_p=\omega/k$ , $v_g=\mathrm{d}\omega/\mathrm{d}k$ and the group-index identities $n_g=n+\omega\,\mathrm{d}n/\mathrm{d}\omega=n-\lambda\,\mathrm{d}n/\mathrm{d}\lambda$ follow directly from $k=n\omega/c$ (Hecht §7.2.2; Saleh & Teich §5.6). A verifier should reproduce the two-wave beat derivation and recompute the glass example from standard Sellmeier coefficients; the anomalous-dispersion inequalities and the front-velocity result ( $v_{\text{front}}=c$ ) are standard (Loudon Ch. 1; Jackson §§7.8, 7.11), recalled here from aq-em-phase-group-velocity rather than re-derived. Addendum for this edition: the previous edition's fused-silica anchor ( $n=1.458$ , $\mathrm{d}n/\mathrm{d}\lambda\approx-1.7\times10^{-5}\ \mathrm{nm^{-1}}$ , $n_g\approx1.468$ , 4.89 ns/m) is not reproducible from the Malitson Sellmeier fit, which gives $\mathrm{d}n/\mathrm{d}\lambda(589\ \mathrm{nm})=-3.50\times10^{-5}\ \mathrm{nm^{-1}}$ and $n_g=1.4790$ (delay 4.933 ns/m) — the slope was understated by half. This edition anchors on BK7 instead ( $n_d=1.51680$ , $\mathrm{d}n/\mathrm{d}\lambda=-0.04180\ \mathrm{\mu m^{-1}}$ , $n_g=1.5414$ , $v_g=1.945\times10^8$ m/s), with both glasses recomputed analytically and by finite difference in verify_aq-optics-group-velocity-group-index.py.

Builds on: Phase Velocity · Plane Waves in Homogeneous Isotropic Media · The Refractive Index · Dispersion: Normal and Anomalous Leads to: Sellmeier and Cauchy Dispersion Equations · Kramers–Kronig Relations · The Lorentz Oscillator Model · Absorption Coefficient and Beer–Lambert Law

Why is the envelope speed a derivative rather than the ratio ω/k?

Because an envelope is an agreement between components, and agreements move at the rate their differences dictate. The beat maximum of two superposed waves sits where their phases coincide; a moment later each phase has advanced by its own $\omega\,\mathrm{d}t$ , and the coincidence point must shift by whatever $\mathrm{d}z$ keeps the phase difference zero: $\Delta k\,\mathrm{d}z=\Delta\omega\,\mathrm{d}t$ , i.e. speed $\Delta\omega/\Delta k$ . The ratio $\omega/k$ answers a different question — how fast one component's own phase pattern slides.

In n_g = n − λ·dn/dλ, do I use the vacuum wavelength or the wavelength inside the glass?

The vacuum wavelength — always. The formula was derived by substituting $\omega=2\pi c/\lambda$ , which is the vacuum relation, and Sellmeier coefficients are fitted with $\lambda$ meaning vacuum (or air) wavelength. Feeding in the medium wavelength $\lambda_0/n$ double-counts the index and corrupts the correction term by a factor of order $n$ . The confusion arises because Rayleigh's other form, $v_g=v_p-\lambda_{\text{med}}\,\mathrm{d}v_p/\mathrm{d}\lambda_{\text{med}}$ , genuinely uses the in-medium wavelength — a different formula in different variables. Safe practice: work from catalogue $n(\lambda_{\text{vac}})$ data and convert nothing.

Can the group index be measured directly, or only computed from Sellmeier fits?

Directly, and routinely. A white-light or low-coherence interferometer locates the position where the envelope of the fringe packet is centred — the group-delay match — so inserting a sample of thickness $L$ shifts the packet by $(n_g-1)L$ , giving $n_g$ without any differentiation of data. Time-of-flight measurement of a picosecond pulse through a known length does the same cruder and faster. Comparing the interferometric $n_g$ with the refractometric $n$ for the same sample is a classic laboratory exercise: for BK7 at the d-line you expect $1.5414$ versus $1.5168$ — an extra 24.6 µm of optical path per millimetre of glass, about 42 fringes, which the apparatus resolves easily.

Why is the group index larger than the phase index in every transparency window, for every material?

Because transparency windows sit between resonances, and there the index always climbs with frequency. The Lorentz-oscillator response (owned by aq-optics-lorentz-oscillator-model) makes $\mathrm{d}n/\mathrm{d}\omega>0$ wherever absorption is negligible — normal dispersion is not a habit of glass but a theorem about lossless response functions, formalized by the Kramers–Kronig relations that close this chapter. Positive $\mathrm{d}n/\mathrm{d}\omega$ feeds straight into $n_g=n+\omega\,\mathrm{d}n/\mathrm{d}\omega>n$ . The inequality fails only where transparency fails: inside an absorption band the slope turns negative and $n_g$ dives below $n$ — even below zero.

If v_g can exceed c near a resonance, what actually limits how fast a message crosses the medium?

The front velocity — and it is always exactly $c$ . A genuine message must contain a surprise — a switch-on, a bit flip — and such a feature has spectrum out to arbitrarily high frequency, where every medium's electrons are too sluggish to respond: $n\to1$ and the leading edge propagates as in vacuum. Sommerfeld and Brillouin made this exact in 1914 (recalled from aq-em-phase-group-velocity): ahead of $z=ct$ the field is identically zero, whatever $n_g$ is doing. Measured superluminal and negative group velocities — Wang's $-c/310$ — are real envelope reshaping of smooth, band-limited pulses whose "information" already resided in their leading tails; put a sharp front on the pulse and it arrives at $c$ , every time.

How do I get n_g from Sellmeier coefficients in practice?

Differentiate analytically, never numerically from tabulated points. With $n^2(\lambda)=1+\sum_j B_j\lambda^2/(\lambda^2-C_j)$ , one differentiation gives $\mathrm{d}n/\mathrm{d}\lambda=-\frac{\lambda}{n}\sum_j\frac{B_j C_j}{(\lambda^2-C_j)^2}$ — manifestly negative between resonances, which re-proves $n_g>n$ there. Then $n_g=n-\lambda\,\mathrm{d}n/\mathrm{d}\lambda$ with $\lambda$ in the same units as $\sqrt{C_j}$ (µm for glass catalogues). For BK7 this yields $-0.04180\ \mathrm{\mu m^{-1}}$ and $n_g=1.5414$ at the d-line; for fused silica, $-0.0350\ \mathrm{\mu m^{-1}}$ and $n_g=1.4790$ at 589 nm — note the widely copied value 1.468 comes from an understated slope and does not survive the differentiation.