Dispersion: Normal and Anomalous

We take two results as given. From the electromagnetic track, aq-em-absorption-dispersion owns the driven-oscillator story of $n(\omega)$ , and within optics aq-optics-lorentz-oscillator-model has just derived $\varepsilon(\omega)=1+\omega_p^2/(\omega_0^2-\omega^2-i\gamma\omega)$ in full; we recall both by name and quote their curves rather than re-derive them. From the geometric-optics track, aq-optics-dispersion-prisms owns the ray-level half of dispersion — prism deviation, the minimum-deviation refractometer, the angular spread $\mathrm{d}\delta/\mathrm{d}\lambda$ . That article converts a slope of $n(\lambda)$ into a fan of angles; this one owns the slope itself: its sign, where that sign lives, and why. The classification is a single derivative. Across every transparency window the index rises toward the blue: $\mathrm{d}n/\mathrm{d}\lambda<0$ , equivalently $\mathrm{d}n/\mathrm{d}\omega>0$ — normal dispersion, the reason a prism deviates violet more than red; for BK7 the Sellmeier curve gives $\mathrm{d}n/\mathrm{d}\lambda=-0.0418\ \mathrm{\mu m^{-1}}$ at the yellow d-line. But cross into an absorption band and the slope flips: over a window about one linewidth wide, straddling the resonance, $n$ falls as frequency rises — anomalous dispersion. The Lorentz curve shows it is no anomaly: the backwards slope sits exactly on top of the absorption peak $\chi''(\omega)$ , because both are faces of one complex response. The name is a fossil. Le Roux saw iodine vapour deviate red more than violet in 1862; Kundt made the effect systematic in 1871–72; in 1904 R. W. Wood photographed sodium vapour fanning light backwards around its D-lines. Nothing anomalous ever happened; only the assumption that $n$ must always rise with frequency was wrong.

Key takeaways

One derivative classifies dispersion: normal means $\mathrm{d}n/\mathrm{d}\lambda<0$ (index rising toward the blue, $\mathrm{d}n/\mathrm{d}\omega>0$ ); anomalous the opposite.
Every transparency window is normal: BK7 gives $\mathrm{d}n/\mathrm{d}\lambda=-0.0418\ \mathrm{\mu m^{-1}}$ at 589 nm, with $n_F=1.52238>n_d=1.51680>n_C=1.51432$ — blue slowed more than red.
Anomalous dispersion lives inside absorption bands: the Lorentz $\chi'(\omega)$ swings backwards between its extrema at detuning $\pm\gamma/2$ — exactly the full width of the $\chi''$ peak.
The coincidence is structural: $\chi'$ and $\chi''$ share one denominator (one causal response), so the index must run backwards where the medium absorbs.
The slope steepens toward a resonance ( $-0.0723\ \mathrm{\mu m^{-1}}$ at BK7's F-line vs $-0.0312$ at C): normal dispersion is the accumulated tail of resonances you cannot see.
Normal dispersion makes $n_g>n$ ( $1.5414>1.5168$ for BK7); inside the anomalous band the group velocity loses meaning as a signal speed — the front velocity, always $c$ , is the limit.
Key idea: the sign of dn/dω maps the spectrum — positive across every transparent window, negative only across absorption lines — because dispersion and absorption are the real and imaginary parts of one causal response, locked by the Lorentz lineshape and, ultimately, by Kramers–Kronig.

The core insight

Normal dispersion ( $\mathrm{d}n/\mathrm{d}\omega>0$ , index rising toward the blue) is the universal behaviour of a transparent window below its resonances; anomalous dispersion ( $\mathrm{d}n/\mathrm{d}\omega<0$ ) occurs only across an absorption line, in a band of full width $\gamma$ , because the dispersive swing of $\chi'$ and the absorption peak of $\chi''$ are two parts of one complex Lorentzian response.

Where this lives today

The sign of $\mathrm{d}n/\mathrm{d}\omega$ is a design parameter across photonics. Fibre systems live and die by it: silica's normal material dispersion flattens and reverses in the near-infrared, and the zero-dispersion wavelength near 1.31 µm decides where telecom parks its channels and how fast pulses smear (the machinery belongs to the group-velocity-dispersion article). Ultrafast labs manipulate the same sign deliberately, using prism pairs and chirped mirrors to impose anomalous group-delay dispersion that recompresses stretched femtosecond pulses; in the fibre soliton regime, anomalous dispersion balancing self-phase modulation lets a pulse propagate without spreading. The resonant version is now a precision tool. Steep normal dispersion on the edge of a narrow resonance — engineered by electromagnetically induced transparency — has slowed light to metres per second, while the anomalous core has produced "fast light" pulses whose peaks exit early, never carrying information faster than $c$ . Spectroscopists still run the "hook" method, reading a line's strength from its anomalous S-curve; and every Kramers–Kronig inversion of a measured absorption spectrum computes exactly the normal-plus-anomalous index structure this article classifies.

The physical picture

Think of the bound electrons of a glass as springs driven by the light's field — the picture derived in aq-optics-lorentz-oscillator-model. Far below their ultraviolet resonance they follow the field nearly in phase, with an amplitude that grows as the drive nears resonance. Blue light, sitting closer to the UV lines than red, polarizes the medium more strongly and is slowed more: $n$ rises with frequency. That is all normal dispersion is — the low-frequency tail of resonances parked in the ultraviolet, and the reason the slope steepens toward the blue (BK7: $-0.0312\ \mathrm{\mu m^{-1}}$ at the red C-line, $-0.0723$ at the blue F-line, closer to the UV poles). Now push the drive through the resonance. Just below $\omega_0$ the oscillators respond in phase and enormously — $n$ peaks. At exact resonance the response is purely in quadrature: the in-phase part of $\chi$ vanishes and absorption peaks. Just above, the electrons swing toward antiphase and the in-phase response goes negative — the index collapses below its vacuum value. Between peak and trough the index runs downhill as frequency rises: anomalous. The first figure shows the S-shaped swing with the absorption Lorentzian beneath it, sharing one band.

Figure

One Lorentz resonance: n(ω) peaks at detuning −γ/2, runs backwards (anomalous) across the core, troughs at +γ/2 — while the absorption line κ(ω), full width γ, peaks exactly inside that band; n = 1 at ω₀

The second figure is the punchline: both regimes in real materials. Left, BK7 across the visible — a smooth falling $n(\lambda)$ , slope $-0.0418\ \mathrm{\mu m^{-1}}$ at the d-line. Right, what Wood photographed in 1904: sodium vapour near its D-lines, the index tearing through S-swings that rise with wavelength inside each band — the anomalous sign made visible.

Figure

The punchline, two real materials: (left) BK7 across the visible — normal, with the d-line tangent marking dn/dλ = −0.0418 µm⁻¹; (right) sodium vapour near the D-lines (Wood 1904) — inside each line n rises with λ (anomalous), dipping below 1 on the blue side

The core equations

The classification is a sign convention on one derivative, quoted in either variable. Because $\omega=2\pi c/\lambda$ makes $\lambda$ and $\omega$ run oppositely, the two forms carry opposite signs — keeping them straight is half the battle.

\text{normal:}\ \frac{\mathrm{d}n}{\mathrm{d}\lambda}<0\ \Leftrightarrow\ \frac{\mathrm{d}n}{\mathrm{d}\omega}>0;\qquad \text{anomalous:}\ \frac{\mathrm{d}n}{\mathrm{d}\lambda}>0\ \Leftrightarrow\ \frac{\mathrm{d}n}{\mathrm{d}\omega}<0 \qquad (1)

where: $n$ — real refractive index; $\lambda$ — vacuum wavelength; $\omega$ — angular frequency. The bridge is the chain rule with $\mathrm{d}\lambda/\mathrm{d}\omega=-2\pi c/\omega^2<0$ : "index rises toward the blue" is one statement in two notations. The engine behind both regimes is the Lorentz dielectric function, derived in aq-optics-lorentz-oscillator-model. Its dilute split: real part dispersion, imaginary part absorption.

n(\omega)-1\approx\frac{\omega_p^2}{2}\,\frac{\omega_0^2-\omega^2}{(\omega_0^2-\omega^2)^2+\gamma^2\omega^2},\qquad \kappa(\omega)\approx\frac{\omega_p^2}{2}\,\frac{\gamma\omega}{(\omega_0^2-\omega^2)^2+\gamma^2\omega^2} \qquad (2)

where: $\omega_0$ — resonance frequency; $\gamma$ — damping rate (the line's full width); $\omega_p^2=Ne^2/(m\varepsilon_0)$ — plasma frequency squared; valid for $|\chi|\ll1$ so $\tilde n\approx1+\chi/2$ . The two formulas share one denominator — the fingerprint that dispersion and absorption are one response — and $\omega_0^2-\omega^2$ flips sign at resonance: index raised below, depressed above. Near the line the response takes its canonical detuning form, from which the anomalous window can be read off exactly.

n(\omega)-1\approx\frac{\omega_p^2}{4\omega_0}\,\frac{\Delta}{\Delta^2+(\gamma/2)^2},\qquad \Delta\equiv\omega_0-\omega \qquad (3)

where: $\Delta$ — detuning from line centre ( $\gamma\ll\omega_0$ assumed). This odd function has extrema at $\Delta=\pm\gamma/2$ ; between them — a band of full width $\gamma$ , exactly the absorption line's width — $\mathrm{d}n/\mathrm{d}\omega$ is negative. Anomalous dispersion is confined to the absorption band by algebra alone. Away from resonances the window behaviour is packaged by the Sellmeier fit (owned by aq-optics-sellmeier-cauchy-equations), which is how the real-glass slope is computed.

n^2(\lambda)=1+\sum_j\frac{B_j\lambda^2}{\lambda^2-C_j}\ \Rightarrow\ \frac{\mathrm{d}n}{\mathrm{d}\lambda}=-\frac{1}{n}\sum_j\frac{B_jC_j\,\lambda}{(\lambda^2-C_j)^2} \qquad (4)

where: $B_j,C_j=\lambda_j^2$ — fitted strengths and squared resonance wavelengths. With the working band below every dominant resonance, each term of the derivative sum is positive, so $\mathrm{d}n/\mathrm{d}\lambda<0$ identically: normal dispersion in a transparency window is a theorem, not a habit. For BK7 at the d-line the value is $-0.04180\ \mathrm{\mu m^{-1}}$ . Finally, the derivative classified here is the one the next articles consume. The group index inherits $\mathrm{d}n/\mathrm{d}\omega$ directly, so its sign decides how pulses travel.

n_g=n+\omega\frac{\mathrm{d}n}{\mathrm{d}\omega}=n-\lambda\frac{\mathrm{d}n}{\mathrm{d}\lambda},\qquad v_g=\frac{c}{n_g} \qquad (5)

where: $n_g$ — group index; $v_g$ — group velocity, both owned by aq-optics-group-velocity-group-index. Normal dispersion makes $n_g>n$ (BK7: $1.5414>1.5168$ ), slowing pulses below the crest speed; a strongly negative $\mathrm{d}n/\mathrm{d}\omega$ inside an anomalous band can drive $n_g$ below 1 or negative — where $v_g$ stops describing signal transport.

Derivation AThe sign of the slope from the Lorentz response

The goal is to extract from the recalled Lorentz $\chi'(\omega)$ the complete sign map of $\mathrm{d}n/\mathrm{d}\omega$ — positive in the wings on both sides of a resonance, negative only across the core — and to locate the boundaries of the backwards band exactly, at detuning $\pm\gamma/2$ . Strategy: reduce the dilute real index (2) to its near-resonance S-curve (3), differentiate once, and read off the sign changes; the model itself is quoted from aq-optics-lorentz-oscillator-model, not re-derived.

Step 1. Start from the dilute real part, $n-1=\tfrac{\omega_p^2}{2}\,\dfrac{\omega_0^2-\omega^2}{(\omega_0^2-\omega^2)^2+\gamma^2\omega^2}$ , and linearise near the line: $\omega_0^2-\omega^2\approx2\omega_0\Delta$ with $\Delta=\omega_0-\omega$ , and $\gamma\omega\approx\gamma\omega_0$ . Why: a narrow line ( $\gamma\ll\omega_0$ ) only samples the response within a few linewidths of centre, so the profile becomes a universal function of $\Delta/\gamma$ .

Step 2. The index becomes the odd S-curve $n-1\approx\dfrac{\omega_p^2}{4\omega_0}\,\dfrac{\Delta}{\Delta^2+(\gamma/2)^2}$ — equation (3). Why: odd in $\Delta$ means the index is raised below resonance and depressed above by equal amounts, crossing $n=1$ exactly at line centre, where the response is purely in quadrature.

Step 3. Differentiate with respect to $\Delta$ : $\dfrac{\mathrm{d}(n-1)}{\mathrm{d}\Delta}\propto\dfrac{(\gamma/2)^2-\Delta^2}{[\Delta^2+(\gamma/2)^2]^2}$ . The numerator vanishes at $\Delta=\pm\gamma/2$ : the extrema of $n(\omega)$ — peak on the low-frequency side, trough on the high. Why: locating the extrema is locating the boundary of the anomalous band; everything between them runs backwards.

Step 4. Translate the sign. Since $\Delta=\omega_0-\omega$ , $\mathrm{d}n/\mathrm{d}\omega=-\mathrm{d}n/\mathrm{d}\Delta$ : for $|\Delta|<\gamma/2$ , $\mathrm{d}n/\mathrm{d}\Delta>0$ hence $\mathrm{d}n/\mathrm{d}\omega<0$ — anomalous over a band of full width $\gamma$ ; outside, the sign reverts — normal in both wings. Why: this is the classification of equation (1) derived rather than declared — the anomalous band is a window exactly as wide as the absorption line.

Step 5. Check the far wings. Far below, $n-1\to\omega_p^2/2\omega_0^2>0$ with a gently rising slope — the window regime Sellmeier packages. Far above, $n\to1-\omega_p^2/2\omega^2<1$ , rising toward 1 from below — normal again. Why: the model hands back both everyday limits — glass in the visible, and the X-ray regime where every material has $n$ a whisker under one. *Verified: S-curve extrema at $\Delta=\pm\gamma/2$ (sympy, exact); core slope sign; wing sign; far-above $n<1$ ; numerically $\mathrm{d}n/\mathrm{d}\omega<0$ across the locked solid's core and $>0$ at 550 nm in verify_aq-optics-normal-anomalous-dispersion.py.*

Go deeper · Derivation B · 2 minWhy the anomalous band must sit on the absorption peak

Derivation BWhy the anomalous band must sit on the absorption peak

The goal is to show that the coincidence of the backwards index and the absorption maximum is structural — a property of any causal resonant response, not a quirk of the Lorentz model — by comparing the widths and centres of the two curves and naming the general theorem. Strategy: put $\chi'$ and $\chi''$ over their common denominator, observe that one parameter $\gamma$ controls both shapes, and confirm the geometry.

Step 1. Write the recalled susceptibility over one denominator: $\chi(\omega)=\dfrac{\omega_p^2\,[(\omega_0^2-\omega^2)+i\gamma\omega]}{(\omega_0^2-\omega^2)^2+\gamma^2\omega^2}$ . Why: real and imaginary parts are two numerators over one shared denominator — they cannot be shaped independently, being one complex function on the real axis.

Step 2. Read the shapes near resonance. The imaginary part is the even Lorentzian $\chi''\propto\dfrac{\gamma/2}{\Delta^2+(\gamma/2)^2}$ , peaked at $\Delta=0$ with FWHM $\gamma$ ; the real part is the odd S-curve of Derivation A, extrema at $\Delta=\pm\gamma/2$ . Why: the same $\gamma$ that sets the line's width sets how far apart the index's peak and trough sit — one parameter, two curves.

Step 3. Overlay them. The index runs backwards precisely for $|\Delta|<\gamma/2$ ; the absorption exceeds half its peak on the same interval. The anomalous band is the absorption line, with no free parameter left to separate them. Why: this is the observable — Wood's sodium vapour swings exactly around the D-lines, and the spectroscopic "hook" method uses the S-curve to locate and weigh a line.

Step 4. Name the general reason. Causality — the polarization cannot precede the field — makes $\chi(\omega)$ analytic in the upper half-plane, and the Kramers–Kronig relations then reconstruct $\chi'$ everywhere from $\chi''$ ; a dispersive swing without an absorption peak to generate it is impossible. The framing is owned by aq-sm-kramers-kronig-relations, the optical dispersion relations by aq-optics-kramers-kronig-relations — both recalled by name, not re-derived. Why: the Lorentz model exhibits the lock because it is causal; KK says every causal medium, whatever its microphysics, is locked the same way — anomalous dispersion is absorption's shadow. *Verified: shared denominator $(\omega_0^2-\omega^2)^2+\gamma^2\omega^2$ ; extrema at $\pm\gamma/2$ against FWHM $\gamma$ (sympy); the locked solid's $\chi''(\omega_0)=62.5$ peak coincides with its negative-slope core in verify_aq-optics-normal-anomalous-dispersion.py.*

Limiting cases

Case	Result	Meaning
Far below resonance ( $\omega\ll\omega_0$ )	$n\approx1+\omega_p^2/2\omega_0^2$ , $\mathrm{d}n/\mathrm{d}\omega>0$	transparency window: normal dispersion, Sellmeier/Cauchy territory
Approaching the line from below	$\|{\mathrm{d}n/\mathrm{d}\lambda}\|$ : 0.031 → 0.042 → 0.072 µm⁻¹ (BK7 C → d → F)	slope steepens as the drive nears resonance — blue sits closer to the UV poles
On resonance ( $\omega=\omega_0$ )	$n=1$ (dilute), $\kappa$ maximal, slope most negative	pure quadrature response: the index crosses its vacuum value at peak absorption
Anomalous window ( $\|\omega-\omega_0\|<\gamma/2$ )	$\mathrm{d}n/\mathrm{d}\omega<0$ over full width $\gamma$	the backwards band coincides with the absorption line; group velocity loses meaning
Far above resonance ( $\omega\gg\omega_0$ )	$n\approx1-\omega_p^2/2\omega^2<1$ , $\mathrm{d}n/\mathrm{d}\omega>0$	plasma-like tail: index below one but rising — normal again, the X-ray regime

Common trap“Anomalous dispersion is abnormal, and the superluminal group velocities inside it break relativity.”

Both halves are wrong. "Anomalous" is a fossil from the 1860s–70s, coined when $n$ rising with wavelength contradicted every known glass; the Lorentz model shows it is the inevitable behaviour of any causal resonant response inside its absorption line. And the velocities: inside the band the steep negative $\mathrm{d}n/\mathrm{d}\omega$ can push the group index below 1 or negative, giving $v_g>c$ or backwards-peaking pulses. Nothing signals faster than light there, because in a strongly absorbing, reshaping band the group velocity ceases to describe energy or information transport — the pulse is rebuilt from its leading edge, and the front velocity, pinned at exactly $c$ , is the only true limit (Sommerfeld–Brillouin, owned by the group-velocity article). The safe rule: in transparent regions the group velocity is meaningful and subluminal; inside an absorption band, quote no group velocity — and never call the physics abnormal.

Worked examples

1 · The slope of real glass (the everyday sign)

Using the Schott Sellmeier coefficients for BK7 ( $B=[1.03961212,\ 0.231792344,\ 1.01046945]$ , $C=[0.00600069867,\ 0.0200179144,\ 103.560653]\ \mathrm{\mu m^2}$ ), find $\mathrm{d}n/\mathrm{d}\lambda$ at the yellow d-line (587.56 nm) and classify the dispersion. How does the slope change across the visible?

Step 1. Differentiate the Sellmeier form (equation 4): $\dfrac{\mathrm{d}n}{\mathrm{d}\lambda}=-\dfrac{1}{n}\sum_j\dfrac{B_jC_j\lambda}{(\lambda^2-C_j)^2}$ , and evaluate at $\lambda=0.58756\ \mathrm{\mu m}$ with $n_d=1.51680$ . The two UV terms dominate and contribute negatively; the IR term adds a small positive amount. Total: $\mathrm{d}n/\mathrm{d}\lambda=\mathbf{-0.04180\ \mu m^{-1}}$ — a finite-difference check agrees to five decimals.

Step 2. Classify: negative slope, so normal dispersion — $n$ rises toward the blue ( $n_F=1.52238>n_d=1.51680>n_C=1.51432$ ), and by the chain rule $\mathrm{d}n/\mathrm{d}\omega>0$ . Blue sits closer to BK7's ultraviolet resonances and drives the bound electrons harder.

Step 3. Track the steepening: the same derivative gives $-0.0312\ \mathrm{\mu m^{-1}}$ at the red C-line but $-0.0723\ \mathrm{\mu m^{-1}}$ at the blue F-line — more than twice as steep. dn/dλ(589 nm) = −0.0418 µm⁻¹, normal; the slope steepens from −0.031 to −0.072 µm⁻¹ across the visible. That gradient is $\mathrm{d}^2n/\mathrm{d}\lambda^2$ — the curvature that becomes group-velocity dispersion in the fibre articles, and the reason a prism's blue end is stretched.

2 · The backwards slope on a resonance (exam level)

The chapter's locked Lorentz solid has a UV resonance at $\lambda_0=100$ nm ( $\omega_0=1.884\times10^{16}\ \mathrm{rad/s}$ ), $\chi(0)=1.25$ , and $\gamma=0.02\,\omega_0=3.767\times10^{14}\ \mathrm{rad/s}$ . Evaluate $n=\mathrm{Re}\sqrt{1+\chi}$ at the edges and centre of the anomalous band and confirm the sign of $\mathrm{d}n/\mathrm{d}\omega$ there.

Step 1. This solid is dense ( $\chi$ of order 60 on resonance), so use the full complex square root. At the band edges and centre: $n(\omega_0-\gamma/2)=\mathbf{6.24}$ , $n(\omega_0)=\mathbf{5.64}$ , $n(\omega_0+\gamma/2)=\mathbf{2.57}$ . (At centre $\chi=62.5\,i$ is purely imaginary, giving $n\approx\kappa\approx5.6$ — peak absorption, $\kappa(\omega_0)=5.55$ .)

Step 2. The slope across the core: $\dfrac{\Delta n}{\Delta\omega}=\dfrac{2.57-6.24}{3.767\times10^{14}}=\mathbf{-9.7\times10^{-15}\ s}$ — the index falls by 3.67 while the frequency climbs one linewidth: anomalous, coinciding with the $\chi''$ peak as Derivation B requires.

Step 3. Contrast with the same solid in its transparency window: at 550 nm the slope is $\mathrm{d}n/\mathrm{d}\omega=+8.5\times10^{-18}\ \mathrm{s}>0$ — normal, and a thousand times gentler. dn/dω = −9.7×10⁻¹⁵ s (anomalous) across the band versus +8.5×10⁻¹⁸ s (normal) in the visible. One resonance supplies both regimes; where you sit decides which you see.

Physics of the portrait

The portrait is the effect that forced optics to take its response functions seriously: the index that runs the wrong way.

Origins · 1 minanomalous dispersion and how this was discovered

anomalous dispersion

the backwards index slope dn/dω < 0 inside an absorption band — dispersion running the "wrong way" exactly where the medium absorbs · optics of dispersive media

Anomalous dispersion earned its name by breaking a rule that had held since Newton: in every glass anyone had measured, the index rose toward the blue. In 1862 François Le Roux found iodine vapour deviating red more than violet — so contrary to expectation that the result was set aside. Christian Christiansen rediscovered the effect in 1870 in fuchsine solution, and August Kundt, in a systematic series from 1871–72, established the law of the phenomenon: the backwards slope is always attached to a strong absorption band, normal dispersion resuming on either side. That empirical lock was explained within the decade by Sellmeier's resonance formula and the damped-oscillator treatments, and it became the decisive test of the electron theory of dispersion: the model demanded an S-shaped index swing centred on each absorption line, and in 1904 Robert W. Wood's sodium-vapour prism made the demanded curve visible in a single photograph — light fanned backwards around the D-lines. The modern reading is deeper: causality (Kramers–Kronig) makes the coincidence of anomalous dispersion and absorption a theorem binding every material. The name survives as a fossil — the physics it labels is the signature of resonance itself.

History

1862

— Le Roux observes iodine vapour deviating red more than violet, the first (misread) sighting of anomalous dispersion

1870

— Christiansen finds the backwards index slope in fuchsine solution, tying it to a strong absorption

1871

— Kundt begins his systematic study: anomalous dispersion always straddles an absorption band, with Sellmeier's resonance formula appearing the same year

1878

— Lorentz grounds resonance dispersion in the electron theory, predicting the S-curve through every line

1904

— R. W. Wood photographs the dramatic anomalous dispersion of sodium vapour around the D-lines with a vapour prism

Going further

Texts. Hecht, Optics, 5th ed., §3.5 · Foot, Atomic Physics, §7.5–7.6 · Loudon, The Quantum Theory of Light, 3rd ed., Ch. 1 · Saleh & Teich, Fundamentals of Photonics, 3rd ed., §5.5 · Demtröder, Laser Spectroscopy, Vol. 1, §3.1 · Jackson, Classical Electrodynamics, 3rd ed., §7.5.

Provenance. The equations are the standard Lorentz-model and Kramers–Kronig results in Hecht §3.5, Foot §7.5–7.6, Loudon ch. 1, Saleh & Teich §5.5, Jackson §7.5/§7.10. A verifier should reconfirm: (1) the complex index $\tilde n^2=1+(Ne^2/m\varepsilon_0)/(\omega_0^2-\omega^2-i\gamma\omega)$ from the driven oscillator (owned by aq-optics-lorentz-oscillator-model and aq-em-absorption-dispersion, recalled by name); (2) that the extrema of $n(\omega)$ occur at detuning $\pm\gamma/2$ so the anomalous band width equals $\gamma$ ; (3) $n_g=n+\omega\,\mathrm{d}n/\mathrm{d}\omega$ and $\alpha=2\omega\kappa/c$ ; (4) the KK integral pairing (framing owned by aq-sm-kramers-kronig-relations). New recomputed numbers: BK7 slope $-0.04180\ \mathrm{\mu m^{-1}}$ at 587.56 nm (analytic and finite-difference), $-0.0723$ at F, $-0.0312$ at C; locked Lorentz solid $n(\omega_0\mp\gamma/2)=6.24/2.57$ , $n(\omega_0)=5.64$ , core slope $-9.7\times10^{-15}$ s versus $+8.5\times10^{-18}$ s at 550 nm. The prism/angular-spread side is owned by aq-optics-dispersion-prisms.

Prerequisites. The Refractive Index · The Lorentz Oscillator Model · Dielectric Susceptibility and Permittivity · Complex Refractive Index and the Extinction Coefficient. Leads to. Group Velocity and Group Index · Kramers–Kronig Relations · Sellmeier and Cauchy Dispersion Equations · Absorption Coefficient and Beer–Lambert Law.

If it's not anomalous, why do we still call it anomalous dispersion?

Pure historical inertia. When Le Roux (1862), Christiansen (1870) and Kundt (1871–72) found media in which red deviated more than violet, the behaviour contradicted every glass ever measured, so it was "anomalous" in the plain sense of against-the-rule. Within a decade Sellmeier's and Lorentz's resonance theories showed the backwards slope is the guaranteed behaviour of any absorbing resonance — every material has anomalous bands, usually parked in the UV and IR where nobody was looking. By then the name had stuck. Modern usage keeps it as a technical label for $\mathrm{d}n/\mathrm{d}\omega<0$ (and, in fibre optics, for the anomalous-GVD regime); everyone agrees it is regular physics.

How can the index be less than 1 just above a resonance — doesn't that mean light travels faster than c?

It means the phase velocity $c/n$ exceeds $c$ , which is routine and harmless. The crests of a perfectly monochromatic wave carry no information — such a wave has been everywhere forever — so their speed is not bounded by relativity. Any actual signal has a front, and that front propagates at exactly $c$ in every medium regardless of $n$ . Just above every absorption line, and throughout the X-ray region (where all matter has $n$ slightly below 1, enabling grazing-incidence X-ray mirrors), phase velocities exceed $c$ with no paradox. The velocity taxonomy — phase, group, signal, front — belongs to the phase- and group-velocity articles, recalled by name.

Is the anomalous band exactly as wide as the absorption line, or is that an approximation?

For a single narrow Lorentz line it is exact in the half-width sense: the extrema of $n(\omega)$ sit at detunings $\pm\gamma/2$ , and $\gamma$ is precisely the FWHM of the $\chi''$ Lorentzian, so the backwards window and the line's half-maximum band coincide. In real materials three things blur this: lines overlap, so neighbouring S-curves partially cancel; Doppler and collisional broadening reshape gas lines into Voigt profiles without a single $\gamma$ ; and in dense media the dilute approximation fails, shifting the extrema (the dense worked example swings $n$ from 6.2 to 2.6). The qualitative lock — backwards slope confined to the absorbing core — survives all of these, because it follows from causality, not from the model.

Why does the slope of n(λ) get steeper toward the blue even in a perfectly transparent glass?

Because the transparency window is the tail of resonances living in the ultraviolet, and a tail steepens as you approach its source. In the Sellmeier derivative every UV term contributes $\propto\lambda/(\lambda^2-C_j)^2$ , growing rapidly as $\lambda^2$ approaches $C_j$ ; BK7's $|\mathrm{d}n/\mathrm{d}\lambda|$ more than doubles from the red C-line ( $0.0312\ \mathrm{\mu m^{-1}}$ ) to the blue F-line ( $0.0723$ ). That is why a prism's spectrum is stretched at its violet end, and why the second derivative $\mathrm{d}^2n/\mathrm{d}\lambda^2$ — the input to group-velocity dispersion — is positive across the visible for every ordinary glass.

What did Wood actually see in the 1904 sodium experiment?

Wood filled a long tube with sodium vapour, denser at the bottom than the top, so the vapour formed a prism whose refracting power depended on the gas's index. Away from the D-lines that index is essentially 1 and white light passes undeviated. But near 589.0 and 589.6 nm the index tears through its anomalous S-curves — well above 1 on the long-wavelength side of each line, well below 1 on the short side — so the "prism" deviated neighbouring colours in opposite directions, splaying the spectrum into curved flares around each line. The photograph is the Lorentz model's $n(\omega)$ traced by light itself — the canonical image of anomalous dispersion.

Between normal and anomalous there must be points where dn/dω = 0 — what happens there?

Two different "zeros" deserve separating. At the extrema of the S-curve around a line ( $\Delta=\pm\gamma/2$ ) the first derivative vanishes momentarily as the slope changes sign; nothing dramatic is observable there because absorption is already substantial. Far more useful is the zero of the second derivative in a broad transparency window: silica's $\mathrm{d}^2n/\mathrm{d}\lambda^2$ crosses zero near 1.27–1.31 µm, where the UV-resonance tail and the IR tail curve in opposite senses and cancel. There the group-velocity dispersion vanishes and pulses spread minimally — why early telecom operated at 1.31 µm. The full story ( $D$ , $\beta_2$ , dispersion-shifted fibre) belongs to the group-velocity-dispersion article, recalled by name.