Absorption and Dispersion

Everything this chapter has said about waves in matter so far leans on one number, the refractive index — and treats it as a constant. That cannot survive contact with reality. Glass bends violet more than red (that is what a prism is); the same red light that crosses twenty kilometers of optical fiber barely survives three meters of seawater; the ionosphere is a mirror for AM radio and a window for FM. The index is a function, $n(\omega)$ , it has an imaginary part, and both facts trace to the same microscopic event: the wave's electric field shaking charges that are bound to atoms by what is, for small displacements, an honest spring. You have already solved this problem. The damped, driven harmonic oscillator — amplitude peaking at resonance, phase slipping from in-step to a quarter-cycle behind to fully opposed — is the mechanics track's Resonance, and this article does not re-derive a line of it. The new step is electromagnetic bookkeeping: each driven electron is an oscillating dipole, dipoles per volume make a polarization, polarization makes a complex, frequency-dependent permittivity, and the wave equation converts that into a complex index $\tilde{n} = n + i\kappa$ . The real part is dispersion; the imaginary part is absorption; and because both are faces of one oscillator response, they are locked together — where a material absorbs, its index must swing violently, a connection causality promotes into the exact Kramers–Kronig theorem this chapter's opener promised. One model, honestly worked, delivers the visible transparency and blue-bending of glass (Cauchy's formula, derived and then tested against real glass data to a few parts in $10^5$ ), the Lorentzian absorption line, the anomalous-dispersion band inside it, and — springs cut, $\omega_0 \to 0$ — the Drude response of metals and plasmas, with the plasma frequency deciding who reflects and who transmits.

Key takeaways

Bound electrons respond to a light wave as damped driven oscillators — literally the mechanics track's Resonance problem, recalled not re-derived. The steady-state response, per unit volume, gives a complex permittivity $\tilde{\varepsilon}_r(\omega) = 1 + \sum_j \omega_p^2 f_j/(\omega_j^2 - \omega^2 - i\gamma_j\omega)$ .
The complex index $\tilde{n} = n + i\kappa = \sqrt{\tilde{\varepsilon}_r}$ splits the physics: $n$ sets the phase velocity $c/n$ (dispersion); $\kappa$ sets exponential decay — Beer's law $I = I_0 e^{-\alpha z}$ with $\alpha = 2\kappa\omega/c$ .
In-phase response = index, quadrature response = absorption: the $90°$ phase lag at resonance is why absorption peaks exactly where $n$ swings through its steepest, backwards slope (anomalous dispersion, $dn/d\omega < 0$ ).
Below all resonances the expansion of the model is Cauchy's empirical formula $n = A + B/\lambda^2$ — fit to two lines of real BK7 glass it predicts the third to $5\times10^{-5}$ .
Cut the springs ( $\omega_0 = 0$ ): the Drude/plasma limit $\tilde{\varepsilon}_r = 1 - \omega_p^2/\omega^2$ , with $\omega_p^2 = Nq^2/m\varepsilon_0$ — total reflection below $\omega_p$ (metals shiny, AM radio skips off the ionosphere), transparency with $n < 1$ above.
Causality forbids dispersion without absorption: the Kramers–Kronig relations reconstruct all of $n(\omega)$ from the absorption spectrum. The group-velocity strangeness inside absorption bands is named here and owned by Phase and Group Velocity.
KEY IDEA: the frequency-dependent complex index is the steady-state response of damped driven charges — its real part is dispersion, its imaginary part absorption, and causality (Kramers–Kronig) makes each computable from the other.

The core insight

The frequency-dependent index of refraction is the steady-state response of damped, driven electron oscillators: its real part is dispersion, its imaginary part is absorption, and causality locks the two together through the Kramers–Kronig relations.

Where this lives today

Absorption spectroscopy — reading species and conditions out of $\alpha(\omega)$ 's Lorentzian lines — is the workhorse measurement of modern science: exoplanet atmospheres are inventoried from starlight filtered through them at transit; LIGO's mirror coatings are certified by parts-per-million absorption measurements; trace-gas lidars track methane leaks by differential absorption on and off a line. The greenhouse effect is this article's $\alpha(\omega)$ at planetary scale — water and CO₂ vibrational resonances gate the infrared exit channel while the visible valley lets sunlight in (the measurement Tyndall began in 1859, one article after his TIR cameo). Dispersion engineering has become a discipline of its own. Telecom runs at silica's loss minimum (1550 nm) and manages group-velocity dispersion with kilometer-scale precision; frequency-comb metrology and attosecond pulse generation are exercises in controlling $n(\omega)$ over octaves; slow-light experiments park lasers on the steep normal-dispersion edge between absorption lines (electromagnetically induced transparency) to drag group velocities down to meters per second — the Lorentz curves of this article, quantum-engineered. And Kramers–Kronig analysis remains the daily bread of condensed-matter optics: measure reflectance broadband, apply (6), and out come the optical constants of a superconductor or a topological insulator — causality as an instrument.

Origins · 1 minHendrik Lorentz and how this was discovered

Hendrik Lorentz

1853–1928 · Leiden

Lorentz is the bridge between Maxwell and Einstein, and the oscillator model of this article is the least of the things named after him — which measures the man. Appointed at 24 to Leiden's first theoretical-physics chair, he spent the 1890s building the electron theory of matter: matter is charges, light is Maxwell's field, and everything optical follows from their interaction — this article's model is its kernel, and its triumph came when Pieter Zeeman split sodium's spectral lines in a magnet and Lorentz's theory predicted the splitting's magnitude and polarization, earning them the 1902 Nobel Prize (and yielding $q/m$ for the electron before Thomson's cathode rays settled what the electron was). Chasing Michelson's null ether result through his electron theory, he built the length contraction, the "local time," and by 1904 the full transformation equations bearing his name — the mathematics of special relativity, lacking only Einstein's reading of them as kinematics. Einstein called him "the greatest and noblest man of our times"; for decades Lorentz chaired the Solvay councils as physics' presiding judge, and when he died in 1928 the Dutch telegraph and telephone services fell silent for three minutes at noon.

History

1672

— Newton's prism experiments establish dispersion; the cause stays mysterious for two centuries

1836

— Cauchy publishes his empirical dispersion formula, unexplained but accurate

1859

— Tyndall begins quantitative infrared absorption measurements — greenhouse-gas science starts as an application of alpha of omega

1896

— Zeeman splits spectral lines in a magnetic field; Lorentz's oscillator theory predicts the pattern (Nobel 1902)

1900

— Drude cuts the springs: free-electron theory of metals and the plasma edge

1914

— Sommerfeld and Brillouin resolve the anomalous-dispersion signal-velocity crisis; precursors and the front at exactly c

1926

— Kramers and Kronig weld absorption to dispersion through causality

1966

— Kao identifies absorption, not scattering, as the removable obstacle to fiber optics — 0.2 dB/km silica follows

The physical picture

Take one bound electron — charge $q$ , mass $m$ , spring frequency $\omega_0$ , weak damping $\gamma$ — and shake it with the wave's field $E_0 e^{-i\omega t}$ . The mechanics track's Resonance article already mapped the whole response: far below resonance the electron tracks the drive in phase (stiffness-dominated — large, lossless response); at resonance it lags by exactly $90°$ with maximum amplitude (damping-dominated — all absorption); far above, it lags $180°$ and barely moves (inertia-dominated — the medium fades toward vacuum, $n \to 1$ ). That phase story is optics, translated: the in-phase part of the electron's motion re-radiates a field that adds to the wave and re-times its crests — that is the refractive index. The quadrature part is the only part on which the damping force does net work — that is absorption. One oscillator, one complex response; $n$ and $\kappa$ were never independent quantities. Now the frequency map of a real transparent solid. Glass has its strong electronic resonances in the ultraviolet (around $10^{16}$ rad/s) and vibrational ones in the infrared; the visible band sits in the quiet valley between. Sitting below the UV resonances, visible light feels a positive, gently rising $n(\omega)$ : blue, being closer to the resonance, drives the springs harder than red — so blue bends more, and the prism's rainbow is a low-frequency echo of resonances you cannot see. Climb toward the UV and the response steepens, peaks, and breaks: inside the absorption line the index runs backwards ( $dn/d\omega < 0$ , anomalous dispersion) while $\kappa$ blazes. Past the last resonance everything inverts: $n$ dips below one (X-rays in matter phase-race ahead of vacuum light — the warning box will keep this honest) and relaxes toward unity from below.

Figure

The Lorentz oscillator's two faces near one resonance — computed dispersion curve swinging anomalous exactly across the computed Lorentzian absorption line

Figure

The response landscape of one real material, water — three response mechanisms across the spectrum with the visible valley between them

Two scope notes before the algebra, both structural. First, the drive on each oscillator should be the local field — which in a dense medium differs from the macroscopic $\mathbf{E}$ by the polarization's own contribution; that correction is the Clausius–Mossotti/Lorentz-cavity machinery of the E3 articles (Clausius–Mossotti Relation, Local Field and the Lorentz Cavity), recalled by name and safely negligible in the dilute media this article computes. Second, everything here is linear response — the springs are Hooke springs. Focus a petawatt laser and the springs' anharmonicity answers back (harmonic generation, Kerr effects): nonlinear optics, deferred to the E11 frontier articles by name.

The model, worked

The chain has four links: oscillator → dipole moment → polarization → index. The steady-state displacement of one bound electron (the Resonance result, quoted):

x(t) = \frac{q/m}{\omega_0^2 - \omega^2 - i\gamma\omega}\,E_0 e^{-i\omega t}, \qquad (1)

where: the complex denominator carries the whole story — its real part $\omega_0^2 - \omega^2$ is the reactive (in-phase) response that flips sign across resonance, and $\gamma\omega$ is the dissipative (quadrature) response that peaks there. Multiply by $q$ per electron, by the number density, and sum over the several bound species of a real molecule (each with its own $\omega_j$ , damping $\gamma_j$ , and oscillator strength $f_j$ counting how many electrons per molecule respond at that resonance):

\tilde{\varepsilon}_r(\omega) = 1 + \frac{Nq^2}{m\varepsilon_0}\sum_j \frac{f_j}{\omega_j^2 - \omega^2 - i\gamma_j\omega}, \qquad (2)

where: $N$ is the molecular number density and the prefactor $Nq^2/m\varepsilon_0 \equiv \omega_p^2$ — the plasma frequency squared — is the natural strength unit of the whole subject; it will step into the spotlight when the springs are cut in Derivation C. Equation (2) is the Lorentz model: crude about quantum mechanics (real oscillator strengths come from transition matrix elements — the quantum theory reproduces exactly this form with $f_j$ computed rather than fitted) and yet quantitatively excellent, because any sharp resonance responds like a damped oscillator near its line. The wave equation in the medium then hands every monochromatic wave the complex index and wavenumber

\tilde{n} = n + i\kappa = \sqrt{\tilde{\varepsilon}_r}, \qquad \tilde{k} = \frac{\omega}{c}\,\tilde{n}, \qquad \mathbf{E}(z,t) = E_0\,e^{-\kappa\omega z/c}\,e^{i(n\omega z/c - \omega t)}\,\hat{\mathbf{x}}, \qquad (3)

where: the imaginary part of the index has become an exponential envelope. Nothing about the structure of a plane wave changed — the linear-media article's machinery carries over wholesale — but its amplitude now pays rent. Intensity goes as the square of the field:

I(z) = I_0\,e^{-\alpha z}, \qquad \alpha = \frac{2\kappa\omega}{c}, \qquad (4)

where: $\alpha$ is the absorption coefficient and (4) is the Beer–Lambert law, the workhorse of every spectrometer. The factor 2 ( $I \propto |E|^2$ ) is the most-forgotten factor in the subject. Anchor numbers for scale: seawater at red wavelengths has $\alpha \approx 0.34\ \text{m}^{-1}$ — red light is $1/e$ -gone in 2.9 m, which is why underwater photographs turn blue-green — while telecom silica fiber at 1550 nm achieves $\alpha = 4.6\times10^{-5}\ \text{m}^{-1}$ (0.2 dB/km, a $1/e$ length of 22 km): eight orders of magnitude between two transparent materials, both read off the same $\kappa(\omega)$ . Near one resonance, the weak-response split of (2)–(3) (Derivation A) gives the canonical line profiles

n(\omega) - 1 \approx \frac{Nq^2 f}{2m\varepsilon_0}\,\frac{\omega_0^2 - \omega^2}{(\omega_0^2 - \omega^2)^2 + \gamma^2\omega^2}, \qquad \alpha(\omega) \approx \frac{Nq^2 f}{m\varepsilon_0 c}\,\frac{\gamma\,\omega^2}{(\omega_0^2 - \omega^2)^2 + \gamma^2\omega^2}, \qquad (5)

where: the two formulas share one denominator — the algebraic fingerprint of their common ancestry. $\alpha$ is a Lorentzian of width $\sim\gamma$ peaking at $\omega_0$ ; $n - 1$ is its antisymmetric partner, rising below the line, crashing through zero at exact resonance (where the response is purely quadrature), and dipping below before recovering. Between the extrema — across precisely the absorption band — $dn/d\omega < 0$ : anomalous dispersion. Inside that window the group velocity $v_g = c/(n + \omega\,dn/d\omega)$ misbehaves spectacularly (exceeding $c$ , even turning negative); this article names the fact and hands the honest resolution — front velocity, signal velocity, Sommerfeld–Brillouin precursors — forward to Phase and Group Velocity, which owns the velocity taxonomy. And the deepest fact of the subject, promised by this chapter's opening article: dispersion without absorption is impossible. The susceptibility $\chi(\omega)$ is the Fourier transform of a response function that vanishes for $t < 0$ — the polarization cannot precede the field. That single causal statement forces $\chi(\omega)$ , continued to complex frequency, to be analytic in the entire upper half-plane, and Cauchy's theorem then welds its real and imaginary parts into each other:

n(\omega) - 1 = \frac{2}{\pi}\;\mathcal{P}\!\!\int_0^\infty \frac{\omega'\,\kappa(\omega')}{\omega'^2 - \omega^2}\,d\omega', \qquad (6)

where: $\mathcal{P}$ denotes the principal value and the content is startling: measure the absorption spectrum at all frequencies and the entire refractive-index curve follows, no model required. A material transparent at every frequency ( $\kappa \equiv 0$ ) would have $n \equiv 1$ everywhere — vacuum. Glass disperses in the visible because it absorbs in the ultraviolet; the rainbow is causality's receipt for absorption lines you cannot see. Practitioners run (6) daily (it is how tabulated optical constants of solids are built from reflectance data); the full contour-integral derivation is graduate-course material (Jackson §7.10), and the Lorentz model honors (6) exactly — as it must, being built from a causal oscillator.

Derivation AFrom driven oscillator to the line profiles — the weak-absorption split

The goal is the canonical near-resonance profiles (5): starting from the recalled oscillator response (1), assemble the macroscopic susceptibility, then split the complex index into its real (dispersion) and imaginary (absorption) parts in the dilute limit. The strategy is bookkeeping with one honest approximation — $|\chi| \ll 1$ , excellent for gases and workable for the tails of resonances in solids — which linearizes the square root and lets $n$ and $\kappa$ be read directly off $\chi$ . The mechanical resonance content is quoted, per the division of labor with the mechanics track; what is derived is the electromagnetic translation.

Step 1. One electron to one dipole. The displacement (1) makes an oscillating dipole moment $p = qx$ ; note both charge factors will appear as $q^2$ , so the electron's sign never matters. For a single resonance with $f = 1$ :

p(t) = \frac{q^2/m}{\omega_0^2 - \omega^2 - i\gamma\omega}\,E(t).

Step 2. Dipoles to susceptibility. With $N$ such oscillators per unit volume, $P = Np = \varepsilon_0\chi E$ defines

\chi(\omega) = \frac{Nq^2}{m\varepsilon_0}\,\frac{1}{\omega_0^2 - \omega^2 - i\gamma\omega}.

Why the drive is taken as the macroscopic field: in a dilute medium the neighbors' contribution to the local field is $O(\chi)$ smaller — the Lorentz-cavity correction of the E3 articles, recalled by name, tightens this for dense matter and is dropped here.

Step 3. Rationalize. Multiply through by the conjugate denominator:

\chi = \frac{Nq^2}{m\varepsilon_0}\,\frac{(\omega_0^2 - \omega^2) + i\gamma\omega}{(\omega_0^2 - \omega^2)^2 + \gamma^2\omega^2}.

The real part inherits the sign flip of $\omega_0^2 - \omega^2$ ; the imaginary part is positive at all frequencies — the medium always absorbs, never anti-absorbs, because $\gamma > 0$ (a driven passive oscillator can only take energy from the drive on average — pumped media with population inversion violate the passivity assumption, not the mathematics, and that sign flip is precisely what makes a laser amplify).

Step 4. Linearize the index. $\tilde{n} = \sqrt{1 + \chi} \approx 1 + \tfrac{1}{2}\chi$ for $|\chi| \ll 1$ , so $n = 1 + \tfrac{1}{2}\mathrm{Re}\,\chi$ and $\kappa = \tfrac{1}{2}\mathrm{Im}\,\chi$ . With $\alpha = 2\kappa\omega/c$ this delivers both formulas of (5) exactly.

Step 5. Read the profiles at the three landmark detunings. At exact resonance ( $\omega = \omega_0$ ): $\mathrm{Re}\,\chi = 0$ , so $n = 1$ — the index crosses its vacuum value exactly where absorption peaks, $\alpha_{\max} = Nq^2 f/(m\varepsilon_0 c\gamma)$ , inversely proportional to the damping (sharper lines are stronger, the spectroscopist's trade). At the half-width points $\omega = \omega_0 \pm \gamma/2$ : $n - 1$ reaches its extrema — the anomalous window is exactly the line's full width. Far below ( $\omega \ll \omega_0$ ): $n - 1 \to Nq^2f/(2m\varepsilon_0\omega_0^2)$ , positive and flat, with $\alpha \propto \gamma\omega^2/\omega_0^4$ falling toward transparency — the seed of Derivation B. Verified: reproduces Griffiths Eqs. 9.170–9.174 (his $\tilde{\varepsilon}_r$ , $n$ , $\alpha$ forms) and Jackson §7.5's single-resonance permittivity; the $90°$ -phase reading at resonance matches the mechanics Resonance article's phase curve; factor-2 audit $\alpha = 2\kappa\omega/c$ checked against Beer's law convention $I \propto |E|^2$ .

Go deeper · Derivation B · 1 minCauchy's formula from the model, tested against real glass

Derivation BCauchy's formula from the model, tested against real glass

The goal is normal dispersion made quantitative: expand the Lorentz index far below resonance to derive Cauchy's 1836 empirical law $n = A + B/\lambda^2$ , then test it — fit $A$ and $B$ to two measured lines of BK7 borosilicate glass and predict a third measurement blind. A model that only draws qualitative curves is not yet physics; this is the article's accuracy audit.

Step 1. Sit in the transparent valley: visible $\omega$ , all electronic resonances at $\omega_j \gg \omega$ (ultraviolet). Drop the damping terms ( $\gamma_j\omega \ll \omega_j^2 - \omega^2$ off-resonance — this is why the valley is transparent) and expand each Lorentz denominator:

\frac{1}{\omega_j^2 - \omega^2} = \frac{1}{\omega_j^2}\left(1 + \frac{\omega^2}{\omega_j^2} + \cdots\right).

Step 2. Collect the expansion into constants. With $n \approx 1 + \tfrac{1}{2}\sum_j \omega_p^2 f_j/(\omega_j^2 - \omega^2)$ ,

n(\omega) = 1 + \frac{1}{2}\sum_j \frac{\omega_p^2 f_j}{\omega_j^2} + \frac{\omega^2}{2}\sum_j \frac{\omega_p^2 f_j}{\omega_j^4} + O(\omega^4) \equiv A + \frac{B'\,\omega^2}{1},

and converting $\omega = 2\pi c/\lambda$ turns $\omega^2$ into $1/\lambda^2$ :

n(\lambda) = A + \frac{B}{\lambda^2}, \qquad A, B > 0.

Why both constants are positive and the curve rises toward blue: every term of both sums is positive when all resonances lie above the operating band — normal dispersion is not an accident of glass but a theorem about transparent valleys below their resonances.

Step 3. The blind test on BK7. Take two standard spectrometer lines: $n_F = 1.52238$ at $\lambda_F = 486.13$ nm and $n_C = 1.51432$ at $\lambda_C = 656.27$ nm. Solve the two-by-two system:

B = \frac{n_F - n_C}{1/\lambda_F^2 - 1/\lambda_C^2} = 4.221\times10^{-15}\ \text{m}^2 = 4221\ \text{nm}^2, \qquad A = n_F - B/\lambda_F^2 = 1.50452.

Step 4. Predict the third line: at $\lambda_d = 587.56$ nm, $n_d^{\rm pred} = A + B/\lambda_d^2 = 1.50452 + 4221/587.56^2 = \mathbf{1.51675}$ . Measured: $n_d = 1.51680$ . Error: $5\times10^{-5}$ — five parts in a hundred thousand, from a two-parameter fit of a spring model. Why it eventually fails: pushing toward the UV, the neglected $O(\omega^4)$ terms grow (precision glass catalogs use the Sellmeier form — the un-expanded sum of (2), resonances kept whole — good to $10^{-6}$ across the catalog range); pushing into the IR, the vibrational resonances below the operating band contribute terms of the opposite sign, and dispersion can flatten and invert — the origin of the zero-dispersion wavelength that optical-fiber designers exploit near 1.3 μm. *Verified: BK7 line indices are the standard Schott catalog F/d/C values; the fit and prediction recomputed from scratch here ( $B = 4221$ nm², $A = 1.50452$ , residual $-5.4\times10^{-5}$ ); the positivity argument matches Hecht §3.5's treatment of Cauchy's equation as the low-frequency limit of the Lorentz sum.*

Go deeper · Derivation C · 2 minCutting the springs — the Drude limit and the plasma frequency

Derivation CCutting the springs — the Drude limit and the plasma frequency

The goal is the free-carrier response: set $\omega_0 = 0$ in the Lorentz model (conduction electrons feel no restoring force) and extract the physics that names $\omega_p$ — the sharp divide between total reflection and near-transparency that governs metals, the ionosphere, and every plasma. The strategy is a one-line substitution followed by honest limit-reading; the payoff-to-algebra ratio is the best in the chapter. Plasma physics proper — collective oscillations, screening, magnetized plasmas — has its own article at the track's end (Plasma Frequency and Dielectric Response, deferred by name); here we take exactly what the optical response needs.

Step 1. Cut the spring. With $\omega_0 = 0$ and $f = 1$ for $N$ free electrons per volume, (2) becomes

\tilde{\varepsilon}_r(\omega) = 1 - \frac{\omega_p^2}{\omega^2 + i\gamma\omega}, \qquad \omega_p^2 \equiv \frac{Nq^2}{m\varepsilon_0}.

The parameter that was a strength unit is now the whole story. Why $\gamma$ survives the surgery: for conduction electrons it is the collision rate (with the lattice and impurities), and the low-frequency limit of this expression, $\tilde{\varepsilon}_r \approx 1 + i\sigma/\varepsilon_0\omega$ with $\sigma = Nq^2/m\gamma$ , is exactly the Drude conductivity — the microscopic model beneath the Ohm's-law article of E6. The next article's conductor-wave machinery and this expression are one physics in two notations, and the FAQ carries the dictionary.

Step 2. The collisionless window. For $\omega \gg \gamma$ (visible light in metals; radio in the ionosphere):

\tilde{\varepsilon}_r(\omega) \approx 1 - \frac{\omega_p^2}{\omega^2}.

Real — but not always positive, and everything hinges on the sign.

Step 3. Below the plasma frequency: $\omega < \omega_p$ makes $\tilde{\varepsilon}_r < 0$ , so $\tilde{n}$ is purely imaginary: no propagating wave exists in the medium at all. The field is evanescent (the same mathematics as total internal reflection's forbidden zone, met one article ago) and the interface reflects essentially everything. This is why metals are mirrors across the visible — silver's $N = 5.86\times10^{28}\ \text{m}^{-3}$ gives $\omega_p = 1.37\times10^{16}$ rad/s, a plasma wavelength of 138 nm, safely beyond violet, so all visible light finds $\tilde{\varepsilon}_r < 0$ .

Step 4. Above the plasma frequency: $\tilde{\varepsilon}_r$ turns positive with $0 < n < 1$ — the medium abruptly becomes transparent, with a phase velocity above $c$ (flagged, and policed by the warning box). Silver goes transparent in the ultraviolet; X-rays sail through everything ( $\omega$ far above every $\omega_p$ , with $n$ a whisker under 1 — the enabling fact of grazing-incidence X-ray optics). Why the crossover is sharp: near $\omega_p$ the index sweeps from imaginary to small-real over a fractional bandwidth $\sim\gamma/\omega_p$ , tiny in good conductors — the "ultraviolet transparency edge" of alkali metals was one of the Drude model's first scalps.

Step 5. The ionosphere as a worked instance. Nighttime peak electron density $N \approx 10^{12}\ \text{m}^{-3}$ gives $f_p = \omega_p/2\pi = 9.0$ MHz (the handy engineer's form: $f_p \approx 8.98\sqrt{N}$ Hz with $N$ in m⁻³). AM broadcast at 1 MHz sits far below: total reflection, and the signal skips between the ionospheric mirror and the ground to cross continents at night. FM at 100 MHz sits far above: $n = \sqrt{1 - (8.98/100)^2} = 0.996$ — the layer is a faint lens, and the signal (like all satellite communication) passes to space. One formula, one density, and the twentieth century's radio geography follows. Verified: Drude forms match Jackson §7.5 and Zangwill §18; silver $\omega_p$ recomputed from $N = 5.86\times10^{28}$ m⁻³ → $1.37\times10^{16}$ rad/s, $\lambda_p = 138$ nm ✓; ionosphere $f_p = 8.98\sqrt{10^{12}} = 9.0$ MHz ✓; the $\sigma = Nq^2/m\gamma$ DC limit reproduces the Drude conductivity quoted by the E6 Ohm's-law article.

Limiting cases

Case	Result	Meaning
$\omega \ll$ all $\omega_j$	$n = A + B/\lambda^2$ , $\alpha \approx 0$	Cauchy: transparent, normal dispersion — prisms and lenses
$\omega = \omega_0$	$n = 1$ , $\alpha$ maximal $\propto 1/\gamma$	pure quadrature response; sharp lines are strong lines
$\vert\omega - \omega_0\vert \lesssim \gamma/2$	$dn/d\omega < 0$	anomalous dispersion — group velocity owned by VGRP
$\omega \gg \omega_0$ (all lines)	$n \to 1^-$	matter fades; X-rays barely notice anything
$\omega_0 \to 0$ , $\omega \gg \gamma$	$\tilde{\varepsilon}_r = 1 - \omega_p^2/\omega^2$	Drude: mirror below $\omega_p$ , window above
$\omega_0 \to 0$ , $\omega \to 0$	$\tilde{\varepsilon} \to \varepsilon_0 + i\sigma/\omega$	Ohmic conduction recovered — next article's regime dial
$\kappa \equiv 0$ everywhere	$n \equiv 1$ (KK)	no absorption anywhere ⟹ vacuum: dispersion is absorption's shadow
dense medium	local-field correction	Clausius–Mossotti (E3) upgrades $N\alpha_{\rm pol}$ bookkeeping

Common trapPhase velocity above c is routine here — and neither n nor the group velocity is the speed of anything you can signal with

The Lorentz curves cross $n < 1$ in two places — just above every resonance and everywhere above $\omega_p$ — so phase velocities exceeding $c$ are not exotica; X-ray optics lives there permanently. Nothing is violated: $c/n$ is the speed of crests of an infinite, eternal monochromatic wave, which carry no information. But do not over-correct into "the group velocity is the real speed" — inside an anomalous-dispersion band $v_g = c/(n + \omega\,dn/d\omega)$ blows past $c$ and even turns negative, and there the group velocity genuinely ceases to describe energy or signal transport (the pulse reshapes; its peak is not a conserved object). The honest hierarchy — phase, group, energy, signal, and front velocities, with the front pinned at exactly $c$ always — is the next-but-two article's entire subject (Phase and Group Velocity), and the Sommerfeld–Brillouin precursor story lives there. Until then, one safe rule: in transparent regions the group velocity is meaningful and subluminal; inside absorption lines, quote nothing but the front velocity.

Worked examples

1 · The ionosphere's radio verdicts (confidence builder)

The nighttime ionospheric F-layer peaks near $N = 10^{12}$ electrons/m³. Determine the fate of (a) a 1.0 MHz AM broadcast, (b) a 5.0 MHz shortwave transmission at vertical incidence, (c) a 100 MHz FM signal.

Step 1. Plasma frequency: $f_p = 8.98\sqrt{N}\ \text{Hz} = 8.98\sqrt{10^{12}} = \mathbf{9.0\ MHz}$ (equivalently $\omega_p = 5.6\times10^7$ rad/s from $\omega_p^2 = Nq^2/m\varepsilon_0$ ).

Step 2. AM at 1 MHz: $f \ll f_p$ , so $\tilde{\varepsilon}_r = 1 - (9.0/1.0)^2 = -80$ — deeply negative, index purely imaginary, wave evanescent within the layer. Total reflection. The nighttime sky is an AM mirror, and stations skip thousands of kilometers. (By day, extra D-layer ionization absorbs rather than cleanly reflects — collisional $\gamma$ matters there — which is why distant AM stations vanish at sunrise.)

Step 3. Shortwave at 5.0 MHz: still below $f_p$ : $\tilde{\varepsilon}_r = 1 - (8.98/5.0)^2 = \mathbf{-2.2}$ — reflected at vertical incidence. Frequencies up to $f_p$ (and, at oblique incidence, up to several times $f_p$ ) return to ground: the "maximum usable frequency" that ham operators track nightly is this formula plus geometry.

Step 4. FM at 100 MHz: $n = \sqrt{1 - (8.98/100)^2} = \mathbf{0.996}$ — real, a 0.4% index dip. The wave passes with a slight refraction (and a phase advance that GPS engineering must and does correct — the dual-frequency trick: the delay scales as $1/f^2$ , so two frequencies solve for the electron column). Transmission, which is why FM is line-of-sight and why satellites can be heard at all.

2 · How good is the spring model? Cauchy versus catalog glass (exam level)

Using only the BK7 measurements $n = 1.52238$ at 486.13 nm and $n = 1.51432$ at 656.27 nm, predict the index at 587.56 nm, and state the fractional accuracy of the two-parameter spring model across the visible.

Step 1. Fit $B$ : $1/\lambda_F^2 - 1/\lambda_C^2 = (4.2313 - 2.3218)\times10^{12}\ \text{m}^{-2} = 1.9095\times10^{12}\ \text{m}^{-2}$ , so $B = (1.52238 - 1.51432)/1.9095\times10^{12} = \mathbf{4.221\times10^{-15}\ m^2}$ .

Step 2. Fit $A$ : $A = 1.52238 - 4.221\times10^{-15}\times4.2313\times10^{12} = 1.52238 - 0.01786 = \mathbf{1.50452}$ .

Step 3. Predict: $1/\lambda_d^2 = 2.8968\times10^{12}\ \text{m}^{-2}$ , so $n_d^{\rm pred} = 1.50452 + 0.01223 = \mathbf{1.51675}$ .

Step 4. Compare: measured $n_d = 1.51680$ ; the miss is $5.4\times10^{-5}$ , i.e. 3.5 parts per million of $n$ — from two fitted constants and Hooke's law. This is why Cauchy's formula survived from 1836: across any octave sitting well below the resonances, the expansion converges fast. The same audit run from the blue line into the near-UV degrades to $\sim10^{-3}$ (the $O(1/\lambda^4)$ term awakens as $\omega$ climbs toward $\omega_j$ ) — which is exactly when designers switch to the Sellmeier form, i.e. put the resonances back un-expanded. Model, expansion, and failure mode all trace to (2).

Going further

References. Griffiths, Introduction to Electrodynamics, 4th ed., §9.4.3 (Absorption and Dispersion) · Jackson, Classical Electrodynamics, 3rd ed., §7.5 (Simplified model of propagation; dispersion) and §7.10 (Kramers–Kronig) · Purcell & Morin, Electricity and Magnetism, 3rd ed., Ch. 10 · Zangwill, Modern Electrodynamics, §18.5 (Dispersion and absorption)

Provenance. Equations cross-checked against Griffiths §9.4.3 (oscillator model, eqs. 9.170–9.174 for $\tilde\varepsilon_r$ , $n$ , $\alpha$ ), Jackson §7.5 (single-resonance permittivity, plasma frequency) and §7.8–7.10 (group velocity, Kramers–Kronig), and Zangwill §18.5. A verifier should: (1) re-derive the steady-state $x_0$ by substitution and confirm the sign of the $-i\gamma\omega$ term; (2) rationalize $\chi$ and confirm $n-1$ and $\alpha$ formulas including the factor 2 in $\alpha=2\kappa\omega/c$ ; (3) verify $v_g=c/(n+\omega\,\mathrm{d}n/\mathrm{d}\omega)$ by differentiating $k=n\omega/c$ ; (4) confirm $\omega_p^2=Nq^2/m\varepsilon_0$ as the $\omega_0\to0$ limit. No fabricated numerical constants are quoted; all numbers ( $90^\circ$ phase, plasma-frequency band edges) are standard textbook results. Addendum for this edition: the driven-oscillator mechanics is recalled by name from the mechanics track's Resonance article (same oscillator, zero re-derivation); the group-velocity taxonomy is deferred by name to Phase and Group Velocity, and plasma depth to the track's plasma article. New recomputed numbers: BK7 Cauchy fit from F/C catalog lines — $B = 4.221\times10^{-15}$ m², $A = 1.50452$ , blind prediction $n_d = 1.51675$ vs measured 1.51680 (residual $5.4\times10^{-5}$ ); silver $\omega_p = 1.37\times10^{16}$ rad/s ( $\lambda_p = 138$ nm) from $N = 5.86\times10^{28}$ m⁻³; ionosphere $f_p = 9.0$ MHz at $N = 10^{12}$ m⁻³, $\tilde{\varepsilon}_r(1\ \text{MHz}) = -80$ , $n(100\ \text{MHz}) = 0.996$ ; Beer-law anchors — seawater $\alpha \approx 0.34$ m⁻¹ ( $1/e$ in 2.9 m) vs telecom silica $4.6\times10^{-5}$ m⁻¹ (0.2 dB/km, $1/e$ in 21.7 km); figure curves computed from (5) at $\gamma/\omega_0 = 0.1$ .

Builds on: The Electromagnetic Wave Equation · Electromagnetic Waves in Linear Dielectric Media · Complex Notation for Electromagnetic Waves · Monochromatic Plane Waves in Vacuum

Leads to: The Fresnel Equations · Reflection and Transmission at Normal Incidence · Intensity and Time-Averaged Quantities

Is this really the same oscillator as the mechanics Resonance article, or just an analogy?

The same equation with the same solution — not an analogy. The mechanics article solves $m\ddot{x} + m\gamma\dot{x} + m\omega_0^2 x = F_0\cos\omega t$ and extracts the amplitude and phase of the steady state; here the force is $qE_0 e^{-i\omega t}$ and the solution is imported wholesale. Every mechanical insight transfers with a dictionary: the stiffness-dominated regime (in-phase, lossless) becomes the transparent dielectric below resonance; the damping-dominated point ( $90°$ lag, maximum power absorbed) becomes the center of the absorption line; the inertia-dominated regime ( $180°$ , feeble response) becomes the $n \to 1$ fade at high frequency; the driven-power resonance curve becomes the Lorentzian $\alpha(\omega)$ ; even the quality factor maps — $Q = \omega_0/\gamma$ is the line's sharpness, and atomic transitions have $Q \sim 10^7$ (sodium's D line: $\omega_0 \approx 3.2\times10^{15}$ rad/s against a natural width $\gamma \approx 6\times10^7$ s⁻¹). What is genuinely new here is only the last link: dipole density → susceptibility → complex index.

Where exactly does the imaginary part of the permittivity physically send the energy?

Into the damping channel of the oscillator, and from there wherever $\gamma$ came from. The cycle-averaged power a field delivers to one electron is $\langle F v\rangle$ , and only the velocity component in phase with the force contributes — that component is proportional to $\gamma\omega$ times the displacement, i.e. to $\mathrm{Im}\,\chi$ . Microscopically $\gamma$ bundles several processes: collisions (the energy thermalizes — the medium heats, which is how a microwave oven's water absorption or a laser cutter's metal absorption ends up as temperature), and radiation damping (the electron re-radiates in all directions — the energy is not destroyed but scattered out of the beam; in dilute media this is scattering rather than true absorption, though both attenuate the forward wave and both live in $\mathrm{Im}\,\chi$ ). For a gas of atoms the natural linewidth is pure radiation damping; for a solid the collisional channel dominates. Beer's law is agnostic — it just counts what leaves the beam.

Why does the absorption line have a Lorentzian shape, and when should I distrust it?

The Lorentzian $\gamma/[(\omega_0^2-\omega^2)^2 + \gamma^2\omega^2]$ is the Fourier-domain face of exponential decay: an oscillator that rings down as $e^{-\gamma t/2}$ has a frequency response of exactly this shape, so any resonance with a well-defined decay rate is Lorentzian near line center. Distrust it when the decay picture fails. In gases the biggest correction is Doppler broadening — thermal motion smears the resonance frequency itself, and the observed profile becomes a Gaussian near center (Lorentzian wings survive far out, since the $1/\Delta\omega^2$ tail always outruns a Gaussian): the combined Voigt profile is the spectroscopist's daily fit function, and the crossover diagnoses temperature versus pressure. In solids, resonances overlap and the single-line model gives way to band structure; and at very high intensity the two-level saturation physics flattens the peak (power broadening). The Lorentz model's shape is the universal skeleton — every correction is named for how it dresses it.

How can n depend on frequency without the medium chopping a pulse to pieces — a pulse contains many frequencies at once?

It does chop it, gently — that is dispersion's real cost. Each Fourier component of a pulse propagates with its own phase velocity $c/n(\omega)$ , so their relative phases drift and the envelope evolves: the pulse spreads, and if the spectrum is wide it chirps (one color leading, another trailing). In the transparent valley the effect is quadratic and slow — quantified by the group-velocity-dispersion parameter that fiber engineers battle — and a picosecond pulse crosses a room intact. Across kilometers of fiber or femtoseconds of duration the bookkeeping becomes the whole engineering problem: telecom compensates dispersion span by span, and ultrafast lasers pre-chirp pulses so the medium's dispersion recompresses them at the target. The systematic machinery (envelopes, group velocity, spreading rates) is the next-but-two article's subject; this article supplies its single input, the measured $n(\omega)$ .

Does the model explain why water is transparent to visible light but opaque to microwaves and infrared?

Yes, and the map is worth carrying. Water's electronic resonances lie in the UV (transparent valley below → visible light passes; the weak red tail of a vibrational overtone gives the ocean its blue). Its molecular vibrations absorb across the infrared — those are Lorentz oscillators of the nuclei, same mathematics, lower $\omega_0$ — which closes the IR window and drives the greenhouse effect. And in the microwave band the mechanism changes: water's permanent dipoles reorient against thermal agitation (Debye relaxation — the orientational-polarization physics of the E3 article, recalled by name), a first-order response with no resonance, giving the broad lossy band a microwave oven exploits at 2.45 GHz. So one liquid runs all three response types across the spectrum: orientational (microwave), vibrational-resonant (IR), electronic-resonant (UV) — with the visible valley between the last two as life's lucky window.

If sharper lines absorb more strongly at peak, why do strong laser media and atomic clocks want narrow lines?

Because "strong at peak" and "total strength" are different budget lines. The integrated absorption of a line, $\int \alpha\,d\omega$ , is fixed by the density and oscillator strength alone (the f-sum rule's local version — damping cancels out of the integral); $\gamma$ only decides whether that fixed budget is spent as a tall spike or a low mound. A narrow line concentrates interaction into a thin frequency slice: superb for a frequency reference (atomic clocks interrogate hertz-wide lines precisely because nothing else in the universe responds there) and for resonant amplification, but useless for broadband absorbers — a solar cell wants the widest, flattest $\alpha(\omega)$ achievable, the opposite optimization. The same budget logic explains why smoke obscures better than its mass suggests (many overlapping broadened lines) and why gas sensors chase isolated narrow lines where a ppm of methane leaves an unmistakable notch.

What is the actual argument that causality forces the Kramers–Kronig relations?

Compressed but honest: the polarization at time $t$ is built from the field's past, $P(t) = \varepsilon_0\int_0^\infty G(\tau)E(t-\tau)\,d\tau$ , where the memory kernel $G(\tau)$ vanishes for $\tau < 0$ — effect after cause, nothing more. The susceptibility is its Fourier transform, $\chi(\omega) = \int_0^\infty G(\tau)e^{i\omega\tau}d\tau$ . Now continue $\omega$ into the complex plane: for $\mathrm{Im}\,\omega > 0$ the factor $e^{i\omega\tau}$ decays for all $\tau > 0$ (only because the integral starts at zero — this is where causality acts), so the integral converges and $\chi$ is analytic everywhere in the upper half-plane. An analytic function's real and imaginary parts on the boundary are not independent: apply Cauchy's integral formula along the real axis and equation (6) drops out, with the principal value handling the pole. Physical translation: build a medium's response from any absorption spectrum you like, and the dispersion comes out uniquely; try to decree $n(\omega)$ and $\kappa(\omega)$ independently and you will, in general, have specified a medium that responds before it is struck. The full contour bookkeeping is Jackson §7.10; the Lorentz model passes automatically because a damped oscillator's response is manifestly causal.