The Lorentz Oscillator Model

We take the parallel from the electromagnetic track and then go further. aq-em-absorption-dispersion drives the same bound-electron oscillator through the same resonance on the EM rails — Cauchy fit, Drude limit, ionosphere; this article is the optics-track counterpart, the derivation owner that aq-optics-refractive-index forward-referenced when it promised a microscopic origin for $n(\omega)$ . Here the model is solved from Newton's law in full, and the numbers it mints feed every remaining article of this dispersion chapter. The question is the one the chapter has been circling: why is the refractive index of glass 1.5 rather than 1.05 or 15, and why does it depend on colour? The Lorentz answer is almost embarrassingly simple. Treat each bound electron as a mass on a spring — restoring force $-m\omega_0^2x$ , drag $-m\gamma\dot{x}$ , driven by the field $-eE_0e^{-i\omega t}$ . One differential equation, solved in steady state and summed over $N$ electrons per unit volume, delivers the complex dielectric function $\varepsilon(\omega)=1+\omega_p^2/(\omega_0^2-\omega^2-i\gamma\omega)$ with one collective strength scale, the plasma frequency $\omega_p^2=Ne^2/(m\varepsilon_0)$ . The real part is dispersion; the imaginary part is absorption; they are one function evaluated once. And the model is quantitative. Give it one ultraviolet resonance at $\lambda_0=100$ nm and ask for the static index of glass, $n=1.5$ : it demands $N=1.394\times10^{29}$ electrons per cubic metre — a real solid's valence density — and then returns $n=1.512$ at the sodium D-line, a crown glass to three figures. One spring, two decimal places.

Key takeaways

The Lorentz model treats each bound electron as a damped, driven oscillator: $m(\ddot{x}+\gamma\dot{x}+\omega_0^2x)=-eE$ , with resonance $\omega_0$ (UV for transparent solids) and damping $\gamma$ (the linewidth).
Steady state gives $x(\omega)$ , the polarization $P=-Nex$ , and the dielectric function $\varepsilon(\omega)=1+\omega_p^2/(\omega_0^2-\omega^2-i\gamma\omega)$ , $\chi=\varepsilon-1$ , with plasma frequency $\omega_p^2=Ne^2/(m\varepsilon_0)$ .
The Lorentzian split: $\chi'(\omega)$ (dispersion) and $\chi''(\omega)$ (absorption) share one denominator; at resonance $\chi'(\omega_0)=0$ while $\chi''$ peaks at $\omega_p^2/(\gamma\omega_0)$ .
Static limit $\chi(0)=\omega_p^2/\omega_0^2$ : the model glass with $\chi(0)=1.25$ has $n=1.5$ ; below resonance $\chi'(589\ \mathrm{nm})=1.287$ gives $n=1.512$ .
Anomalous dispersion ( $\chi'$ falling with $\omega$ ) occupies exactly the absorption band $\omega_0\pm\gamma/2$ — dispersion and absorption are one causal response.
Key idea: one driven, damped oscillator per bound electron generates the material's complete linear optical response, the dielectric function $\varepsilon(\omega)=1+\omega_p^2/(\omega_0^2-\omega^2-i\gamma\omega)$ — its real part is refraction and dispersion, its imaginary part is absorption, and $\omega_p^2=Ne^2/(m\varepsilon_0)$ sets the strength.

The core insight

A bound electron responding to light is a driven, damped harmonic oscillator. Its steady-state response, summed over the density $N$ , is the dielectric function $\varepsilon(\omega)=1+\omega_p^2/(\omega_0^2-\omega^2-i\gamma\omega)$ : the real part disperses the wave, the imaginary part absorbs it, and everything is fixed by $\omega_0$ , $\gamma$ , and $\omega_p^2=Ne^2/(m\varepsilon_0)$ .

Where this lives today

The Lorentz oscillator is the standard parameterization of optical response across materials science. Ellipsometry software fits measured spectra with sums of Lorentz (and Drude) oscillators to extract $n$ and $\kappa$ of semiconductors, coatings, and metals — the fit parameters reported in papers are $\omega_j$ , $\gamma_j$ , and oscillator strengths. Time-domain electromagnetic solvers represent every dispersive material — gold in a plasmonic antenna, silicon in a metasurface — as a fitted Lorentz–Drude pole expansion, executing this model billions of times per simulation. In laser physics the two-level atom's weak-field response is exactly a Lorentz oscillator with a quantum-computed strength, which is why gain curves and electromagnetically-induced-transparency windows are discussed in the language of $\chi'$ and $\chi''$ . The fingerprints also mark the largest scales: the infrared resonances of CO₂ and H₂O — nuclear Lorentz oscillators, heavier masses — gate the outgoing thermal radiation of planets: greenhouse physics is $\chi''(\omega)$ applied to an atmosphere. And the multi-resonance, $\gamma\to0$ form of $\varepsilon(\omega)$ is the Sellmeier equation of every glass catalogue — the subject of aq-optics-sellmeier-cauchy-equations.

The physical picture

Push a swing periodically and three regimes appear. Push slowly and the swing follows the hand in phase — a large, lossless, spring-dominated response. Push at the natural rate and the amplitude peaks while the motion lags by a quarter cycle — all the work now goes into the damping. Push much faster and the swing barely moves, half a cycle out of step — inertia wins. A bound electron in a light wave is this swing: the field is the push, $\omega_0$ the natural rate, and the phase of the electron's motion relative to the drive is the entire story of optics in matter. The translation runs through re-radiation. A driven electron is an oscillating dipole, and a dipole radiates. The part of its motion in phase with the field re-radiates a wavelet that adds to the incident wave and re-times its crests — accumulated over billions of atoms, that is the refractive index. The part in quadrature (90° behind) is the only part on which the field does net work per cycle — that is absorption, drained through $\gamma$ into heat and scattered light. Refraction and absorption are the two phase components of one driven motion.

Figure

The Lorentz atom: a bound electron on a spring of natural frequency ω₀, damped at rate γ, driven by E₀e^(−iωt); the driven dipole re-radiates a phase-lagged wavelet — in phase below resonance (refraction), 90° lagged at ω₀ (absorption), 180° above

Now place a real transparent solid on this map. Glass carries its strong electronic resonances in the ultraviolet — around $10^{16}$ rad/s — so the entire visible band sits below resonance, spring-dominated: $\chi'$ positive, essentially lossless, gently rising toward the resonance. That rise is why blue (closer to $\omega_0$ ) is slowed more than red — dispersion is the low-frequency tail of a resonance you cannot see. Climb into the ultraviolet and the response steepens, peaks, and breaks: across the absorption line $\chi'$ swings violently downward (anomalous dispersion) while $\chi''$ blazes; above the line $\chi'$ goes negative and the index dips below one. The second figure is the punchline — the whole map, computed from the locked model numbers.

Figure

The punchline: χ′(ω) and χ″(ω) through the UV resonance — ω₀ = 1.884×10¹⁶ rad/s (λ₀ = 100 nm), γ = 0.02 ω₀; χ″(ω₀) = 62.5 peaks exactly where χ′ crosses zero, the shaded anomalous band spans ω₀ ± γ/2, and far below resonance χ′ → χ(0) = 1.25, i.e. n = 1.5

The core equations

Everything begins with Newton's second law for one bound electron of charge $-e$ and mass $m$ , held by an effective spring and damped at a phenomenological rate. The drive is the optical field at the atom, in the chapter's fixed $e^{-i\omega t}$ convention.

m\left(\ddot{x}+\gamma\dot{x}+\omega_0^2\,x\right)=-e\,E_0\,e^{-i\omega t} \qquad (1)

where: $x$ — displacement of the electron from equilibrium; $\omega_0$ — natural (resonance) frequency of the binding, ultraviolet for transparent solids; $\gamma$ — damping rate (s⁻¹), the inverse lifetime of a free oscillation, bundling radiative decay and collisions; $E_0$ — field amplitude at the atom. In steady state the electron oscillates at the drive frequency with a complex amplitude — magnitude is response strength, phase is the lag. Substituting $x=x_0e^{-i\omega t}$ turns the differential equation into algebra.

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

where: the complex denominator $\omega_0^2-\omega^2-i\gamma\omega$ carries the whole physics — its real part is the reactive (in-phase) response that flips sign across resonance, its imaginary part $\gamma\omega$ the dissipative (quadrature) response that peaks there. Everything downstream inherits this denominator. Each displaced electron is a dipole $p=-ex$ , and $N$ per unit volume make the macroscopic polarization. The definition $P=\varepsilon_0\chi E$ (recalled from aq-optics-dielectric-susceptibility-permittivity) then turns the mechanical solution into the susceptibility.

\chi(\omega)=\frac{Ne^2/m\varepsilon_0}{\omega_0^2-\omega^2-i\gamma\omega}=\frac{\omega_p^2}{\omega_0^2-\omega^2-i\gamma\omega},\qquad \omega_p^2\equiv\frac{Ne^2}{m\varepsilon_0} \qquad (3)

where: $N$ — number density of oscillating electrons; $\omega_p$ — the plasma frequency, the collective strength scale (density and fundamental constants only, no binding). Adding the vacuum's own response gives the central object of this article. The complex, frequency-dependent dielectric function is the material's complete linear optical fingerprint.

\varepsilon(\omega)=1+\chi(\omega)=1+\frac{\omega_p^2}{\omega_0^2-\omega^2-i\gamma\omega} \qquad (4)

where: $\varepsilon(\omega)$ — relative permittivity (dielectric function) at frequency $\omega$ ; the next article's complex index is $\tilde n=\sqrt{\varepsilon}$ . Index, absorption, reflectance, dispersion — every optical property is a functional of this one curve. Rationalizing the denominator splits $\chi$ into its dispersive and absorptive parts. This is the Lorentzian pair the punchline figure plots.

\chi'(\omega)=\frac{\omega_p^2\,(\omega_0^2-\omega^2)}{(\omega_0^2-\omega^2)^2+\gamma^2\omega^2},\qquad \chi''(\omega)=\frac{\omega_p^2\,\gamma\omega}{(\omega_0^2-\omega^2)^2+\gamma^2\omega^2} \qquad (5)

where: $\chi=\chi'+i\chi''$ ; $\chi'$ — in-phase (dispersive) part, positive below resonance, negative above; $\chi''$ — quadrature (absorptive) part, positive at every frequency for a passive medium. The shared denominator is the fingerprint of their common origin in (2). Three landmark evaluations of (5) anchor every later article. They are the locked numbers of the worked examples below.

\chi(0)=\frac{\omega_p^2}{\omega_0^2},\qquad \chi'(\omega_0)=0,\qquad \chi''(\omega_0)=\frac{\omega_p^2}{\gamma\omega_0} \qquad (6)

where: $\chi(0)$ — the static susceptibility, setting the transparent-window index $n=\sqrt{1+\chi(0)}$ ; at exact resonance the response is purely quadrature — the dispersive part vanishes, the absorptive part peaks $\propto1/\gamma$ . Model glass: $\chi(0)=1.25$ , $\chi''(\omega_0)=62.5$ .

Derivation ASolving the driven electron and building the dielectric function

The goal is the chain that owns this chapter: from Newton's law for one bound electron to the macroscopic dielectric function $\varepsilon(\omega)=1+\omega_p^2/(\omega_0^2-\omega^2-i\gamma\omega)$ , making explicit where each assumption enters — steady state, linearity, the dilute-medium identification of the drive with the macroscopic field — so later articles can quote $\varepsilon(\omega)$ knowing what it stands on.

Step 1. Write Newton's second law for the electron: spring force $-m\omega_0^2x$ , drag $-m\gamma\dot{x}$ , driving force $(-e)E_0e^{-i\omega t}$ — equation (1). Why: for small displacements any smooth binding potential is quadratic to leading order — the "spring" is the harmonic expansion of the true potential.

Step 2. Seek the steady state: after transients decay (in $\sim1/\gamma$ ) the electron oscillates at the drive frequency, $x=x_0e^{-i\omega t}$ . Then $\dot{x}\to-i\omega x$ , $\ddot{x}\to-\omega^2x$ , and (1) collapses to $(\omega_0^2-\omega^2-i\gamma\omega)\,x_0=-eE_0/m$ — equation (2). Why: a linear system driven sinusoidally responds at the same frequency; the complex $x_0$ packages magnitude and phase lag in one number.

Step 3. Convert motion to dipole moment: $p=(-e)x=\dfrac{e^2/m}{\omega_0^2-\omega^2-i\gamma\omega}\,E$ . Why: the two factors of $(-e)$ multiply to $e^2$ , so only $e^2/m$ enters — electrons (smallest $m$ ) dominate the optical response; heavy ion cores contribute mainly in the infrared.

Step 4. Sum to the polarization: $P=Np=\varepsilon_0\chi E$ defines $\chi(\omega)$ of equation (3), with prefactor $\omega_p^2=Ne^2/m\varepsilon_0$ . Why: taking the drive to be the macroscopic $E$ is the dilute-medium approximation — dense matter needs the local-field (Clausius–Mossotti) correction, flagged in the warning below.

Step 5. Add the vacuum term: $\varepsilon(\omega)=1+\chi(\omega)$ — equation (4), complex because the response lags the drive. Check the limits: $\omega\to0$ gives $\varepsilon\to1+\omega_p^2/\omega_0^2$ , real and static; $\omega\to\infty$ gives $\varepsilon\to1-\omega_p^2/\omega^2\to1$ from below — the electrons cannot follow, and the medium fades toward vacuum. Why: the static limit must look like a dielectric constant, and infinite frequency like nothing at all. *Verified: steady-state substitution, sign bookkeeping, both limits, and the symbolic identity of (3) with the driven-oscillator solution (sympy) in verify_aq-optics-lorentz-oscillator-model.py.*

Go deeper · Derivation B · 1 minThe Lorentzian split, the resonance landmarks, and the static limit

Derivation BThe Lorentzian split, the resonance landmarks, and the static limit

The goal is to separate $\chi(\omega)$ into the measurable pair $\chi'$ (dispersion) and $\chi''$ (absorption), extract the landmarks of equation (6) — $\chi(0)=\omega_p^2/\omega_0^2$ , $\chi'(\omega_0)=0$ , $\chi''(\omega_0)=\omega_p^2/\gamma\omega_0$ — and show that near a sharp resonance $\chi''$ is the canonical Lorentzian of full width $\gamma$ , evaluated at the model glass's locked numbers.

Step 1. Rationalize: multiply numerator and denominator of (3) by the conjugate $(\omega_0^2-\omega^2)+i\gamma\omega$ . The denominator becomes the real $D(\omega)=(\omega_0^2-\omega^2)^2+\gamma^2\omega^2$ , and reading off the numerators gives (5). Why: $D>0$ everywhere, so the sign of $\chi'$ is the sign of $\omega_0^2-\omega^2$ , while $\chi''\propto\gamma\omega$ stays positive — a passive medium always absorbs.

Step 2. Static limit: at $\omega=0$ , $\chi'(0)=\omega_p^2/\omega_0^2$ and $\chi''(0)=0$ . For the model glass $\chi(0)=1.25$ , so $n=\sqrt{2.25}=1.5$ . Why: the static susceptibility measures the spring's compliance — stiffer binding means a smaller index, which is why wide-gap materials have low indices and narrow-gap semiconductors huge ones.

Step 3. Exact resonance: at $\omega=\omega_0$ the reactive numerator vanishes, so $\chi'(\omega_0)=0$ , while $\chi''(\omega_0)=\omega_p^2/\gamma\omega_0$ . With the locked $\gamma=0.02\,\omega_0$ : $\chi''(\omega_0)=\chi(0)\,(\omega_0/\gamma)=1.25\times50=62.5$ . Why: on resonance the motion is purely in quadrature — all dissipation, no refraction — and the peak scales as $1/\gamma$ : long-lived oscillators absorb ferociously on their line.

Step 4. Near-resonance shape: for $\gamma\ll\omega_0$ , write $\omega_0^2-\omega^2\approx-2\omega_0\Delta$ with detuning $\Delta=\omega-\omega_0$ . Then $\chi''\approx\dfrac{\omega_p^2}{4\omega_0}\,\dfrac{\gamma}{\Delta^2+(\gamma/2)^2}$ — a Lorentzian of FWHM exactly $\gamma$ — and $\chi'\approx-\dfrac{\omega_p^2}{2\omega_0}\,\dfrac{\Delta}{\Delta^2+(\gamma/2)^2}$ , its antisymmetric partner. Why: linewidth and lifetime are Fourier conjugates — an amplitude ring-down $e^{-\gamma t/2}$ is a Lorentzian of width $\gamma$ in frequency.

Step 5. Locate the anomalous band: the extrema of $\chi'$ sit at $\Delta=\mp\gamma/2$ with values $\pm\omega_p^2/(2\gamma\omega_0)=\pm\chi''(\omega_0)/2$ ( $\approx\pm31.2$ here). Between them — precisely across the absorption line — $\chi'$ falls with increasing $\omega$ : anomalous dispersion, developed in aq-optics-normal-anomalous-dispersion. Why: the swing of $\chi'$ through the line is the reactive shadow of the absorption peak. *Verified: the split (5) symbolically (sympy re/im), $\chi(0)=1.25$ , $\chi'(\omega_0)=0$ , $\chi''(\omega_0)=62.5$ , the Lorentzian FWHM $=\gamma$ , and the $\pm\chi''(\omega_0)/2$ extrema in verify_aq-optics-lorentz-oscillator-model.py.*

Limiting cases

Case	Result	Meaning
Static / far below resonance ( $\omega\ll\omega_0$ )	$\chi\to\omega_p^2/\omega_0^2$ , real	transparent dielectric, $n=\sqrt{1+\chi(0)}$ ; the visible window of glass
Below resonance, approaching	$\chi'$ rises with $\omega$	normal dispersion — blue slowed more than red; the prism regime
On resonance ( $\omega=\omega_0$ )	$\chi'=0$ , $\chi''=\omega_p^2/\gamma\omega_0$ peak	pure quadrature: maximum absorption, zero dispersive part
Inside the line ( $\vert\omega-\omega_0\vert\lesssim\gamma/2$ )	$\chi'$ falls with $\omega$	anomalous dispersion, coincident with the absorption band (owned by aq-optics-normal-anomalous-dispersion)
Far above resonance ( $\omega\gg\omega_0,\gamma$ )	$\varepsilon\approx1-\omega_p^2/\omega^2$	binding irrelevant — free-electron/X-ray behaviour, $n$ just below 1
Springs cut ( $\omega_0\to0$ )	$\varepsilon=1-\dfrac{\omega_p^2}{\omega^2+i\gamma\omega}$	Drude limit: metals and plasmas — mirror below $\omega_p$ , window above
Many resonances, $\gamma_j\to0$	$n^2=1+\sum_j\dfrac{B_j\lambda^2}{\lambda^2-C_j}$	the Sellmeier equation (owned by aq-optics-sellmeier-cauchy-equations)

Common trap"The plasma frequency is the frequency at which the medium resonates — just another name for ω₀."

It is a different kind of quantity altogether. The resonance frequency $\omega_0$ belongs to one electron and its binding — it says where the medium responds. The plasma frequency $\omega_p=\sqrt{Ne^2/m\varepsilon_0}$ belongs to the collection — density only — and says how strongly ( $\chi(0)=\omega_p^2/\omega_0^2$ is the ratio of the two scales, squared). In the model glass they are numerically close ( $1.884$ vs $2.106\times10^{16}$ rad/s) — a pure coincidence of $\chi(0)$ being of order one. Only in the Drude limit $\omega_0\to0$ does $\omega_p$ acquire a spectral meaning of its own, as the edge between metallic reflection and transparency. Companion trap: equation (3) drove each atom with the macroscopic field $E$ — a dilute-medium approximation. In dense matter each dipole also feels its neighbours, $E_{\text{loc}}=E+P/3\varepsilon_0$ , and repairing the bookkeeping gives the Clausius–Mossotti / Lorentz–Lorenz relation: $\chi(0)$ shifts, though the Lorentzian shape survives.

Worked examples

1 · The plasma frequency and electron density of a model glass (confidence builder)

A transparent solid is modelled by one electronic resonance at $\lambda_0=100$ nm (deep UV). Its visible-window index is $n=1.5$ , so $\chi(0)=n^2-1=1.25$ . Find $\omega_0$ , $\omega_p$ , and the electron density $N$ the model requires — and check that $N$ is sensible.

Step 1. Resonance frequency: $\omega_0=2\pi c/\lambda_0=2\pi(2.998\times10^8)/(100\times10^{-9})=\mathbf{1.884\times10^{16}\ rad/s}$ (a 12.4 eV transition — genuinely deep-UV, the right scale for a wide-gap insulator).

Step 2. Plasma frequency from the static limit (equation 6): $\chi(0)=\omega_p^2/\omega_0^2$ , so $\omega_p=\omega_0\sqrt{\chi(0)}=1.884\times10^{16}\times\sqrt{1.25}=\mathbf{2.106\times10^{16}\ rad/s}$ .

Step 3. Electron density from $\omega_p^2=Ne^2/m\varepsilon_0$ : $N=\omega_p^2\,\varepsilon_0 m/e^2=(2.106\times10^{16})^2(8.854\times10^{-12})(9.109\times10^{-31})/(1.602\times10^{-19})^2=\mathbf{1.394\times10^{29}\ m^{-3}}$ .

Step 4. Sanity check: a solid has of order $10^{28}$ – $10^{29}$ atoms per m³ with a few valence electrons each, so $1.4\times10^{29}$ oscillating electrons per m³ is exactly right. Fed two measured numbers, the model returns a density that agrees with the material's actual electron count. ω₀ = 1.884×10¹⁶ rad/s, ωₚ = 2.106×10¹⁶ rad/s, N = 1.394×10²⁹ m⁻³.

2 · The model glass at the sodium D-line

With the same parameters ( $\omega_0$ , $\omega_p$ , $\gamma=0.02\,\omega_0$ ), evaluate $\chi'$ at the sodium D-line, $\lambda=589.3$ nm, and the resulting index. How large is the absorptive part there?

Step 1. Drive frequency: $\omega=2\pi c/\lambda=3.197\times10^{15}$ rad/s — a factor 5.9 below resonance, deep in the normal-dispersion window.

Step 2. Dispersive part from (5): the damping term $\gamma^2\omega^2$ is negligible this far out, so $\chi'\approx\dfrac{\omega_p^2}{\omega_0^2-\omega^2}=\dfrac{4.435\times10^{32}}{3.446\times10^{32}}=\mathbf{1.287}$ .

Step 3. Index: $n=\sqrt{1+\chi'}=\sqrt{2.287}=\mathbf{1.512}$ . The static fit gave $n=1.500$ ; moving up to the D-line raised the index by $0.012$ — the model predicts normal dispersion of the correct size for a crown glass (measured BK7: $n_d=1.5168$ ), from nothing but the resonance position.

Step 4. Absorptive part: $\chi''(589.3\ \mathrm{nm})=4.5\times10^{-3}$ — only $0.35\%$ of $\chi'$ , and even this overstates a real glass, whose visible-band damping is far below the plot-friendly $\gamma=0.02\,\omega_0$ . Far from resonance the response is essentially lossless. χ′(589.3 nm) = 1.287, n = 1.512, χ″/χ′ ≈ 0.35%.

Physics of the portrait

The portrait here is not a person but the object the model constructs: the dielectric function $\varepsilon(\omega)$ — one complex curve that is the material's entire linear optical identity.

Origins · 1 mindielectric function and how this was discovered

dielectric function

ε(ω) = 1 + χ(ω), the material's complete linear optical response · electromagnetic optics of media

The dielectric function is what the dielectric constant became once physics looked closely. Faraday's capacitor experiments defined a single number per material; Maxwell's 1865 theory promised $n=\sqrt{\varepsilon}$ — and promptly embarrassed itself on water, whose static $\varepsilon\approx80$ predicts $n\approx9$ against the measured $1.33$ . The resolution is this article's content: $\varepsilon$ is a function of frequency, and water's polar-molecule response, enormous at DC, cannot follow an optical field. Sellmeier (1871) built the first resonance-based dispersion formula; Lorentz, from 1878, grounded it in charged oscillators inside matter, and his electron theory — vindicated by Zeeman's 1896 line-splitting — made $\varepsilon(\omega)$ the first computed material response in physics. Quantum mechanics kept the functional form, re-deriving the oscillator strengths from transition matrix elements (Ladenburg; Thomas–Reiche–Kuhn). Today $\varepsilon(\omega)$ is the lingua franca of optical materials science: ellipsometers measure it, density-functional codes compute it, photonics solvers ingest it as fitted Lorentz–Drude poles — every "optical constant" in every database is a sample of this one function.

History

1836

— Cauchy publishes his empirical dispersion formula; accurate but unexplained

1871

— Sellmeier models dispersion as mechanical resonances of bound particles, pre-electron

1878

— Lorentz connects the refractive index to the density and response of molecular charges

1892

— Lorentz's electron theory: matter as charges in Maxwell's field, this model its kernel

1896

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

1900

— Drude cuts the restoring force: the free-electron optics of metals and plasmas

1921

— Ladenburg ties the classical oscillator strengths to quantum transition probabilities

1925

— Thomas, Reiche and Kuhn: the f-sum rule Σfⱼ = Z that quantum mechanics imposes on the model

Going further

Texts. Hecht, Optics, 5th ed., §3.5 (Dispersion) · Foot, Atomic Physics, §7.5 · Loudon, The Quantum Theory of Light, 3rd ed., Ch. 1–2 · Saleh & Teich, Fundamentals of Photonics, 3rd ed., §5.5 · Jackson, Classical Electrodynamics, 3rd ed., §7.5 · Demtröder, Laser Spectroscopy, Vol. 1.

Provenance. Equations of motion, susceptibility, and permittivity verified against Hecht (Optics, 5th ed., §3.5), Jackson (Classical Electrodynamics, 3rd ed., §7.5, eqs. 7.51), and Foot (Atomic Physics, §7.5). A verifier should re-derive $\chi(\omega)$ from the driven-oscillator solution, confirm $\omega_p^2=Ne^2/m\varepsilon_0$ , check the Lorentzian FWHM equals $\gamma$ , and verify the Drude limit and TRK sum rule $\sum_jf_j=Z$ . The locked model-glass numbers ( $\omega_0=1.884\times10^{16}$ rad/s; $\omega_p=2.106\times10^{16}$ rad/s; $N=1.394\times10^{29}$ m⁻³; $\chi''(\omega_0)=62.5$ ; $\chi'(589.3\ \mathrm{nm})=1.287\Rightarrow n=1.512$ ) are recomputed from scratch in verify_aq-optics-lorentz-oscillator-model.py. The EM-track parallel is aq-em-absorption-dispersion (recalled by name, both canon); this article is the optics-track derivation owner.

Prerequisites. Electric Polarization and the Displacement Field · Dielectric Susceptibility and Permittivity · Maxwell's Equations in Dielectric Media. Leads to. Complex Refractive Index and the Extinction Coefficient · Dispersion: Normal and Anomalous · Sellmeier and Cauchy Dispersion Equations · Kramers–Kronig Relations.

Why does a classical spring model work for quantum atoms at all?

Because near any sharp resonance, every linear response looks like a damped driven oscillator — the pole structure, not the microscopic dynamics, dictates the shape. Quantum mechanics changes only the bookkeeping: a real atom has many transition frequencies $\omega_j$ , each contributing a Lorentz term weighted by an oscillator strength $f_j$ computed from the transition dipole matrix element, with the Thomas–Reiche–Kuhn sum rule $\sum_jf_j=Z$ guaranteeing the total counts exactly the $Z$ electrons the classical model assumed. The quantum result is the classical formula with calibrated weights — which is why Sellmeier fits built on this model reproduce measured glass indices to parts in $10^6$ .

What is the damping rate γ physically — friction on an electron?

It is the total decay rate of the oscillation, bundling distinct channels. An isolated atom's dipole re-radiates in all directions — radiation damping — giving natural optical linewidths of order $10^7$ – $10^8$ s⁻¹ ( $Q=\omega_0/\gamma\sim10^7$ : sodium's D-line rings ten million times before decaying). In dense matter, collisions and lattice coupling dominate and $\gamma$ is much larger. In gases, thermal motion adds Doppler broadening — not a decay but a smearing of the resonance frequency itself, giving the Voigt profile. Whatever its origin, $\gamma$ is what the linewidth measures: linewidth and lifetime are Fourier conjugates.

Why is χ″ always positive — could a medium have negative absorption?

For a passive medium the model guarantees it: in (5), $\chi''\propto\gamma\omega$ with $\gamma>0$ , and physically the quadrature response is the component on which the field does positive net work each cycle — energy flows from wave to medium, always. Negative $\chi''$ would mean the medium does net work on the wave: amplification. That is not impossible, but it requires an energy source — a laser gain medium with population inversion is a collection of oscillators pumped until the response sign flips. The sign is also convention-locked: with the chapter's $e^{-i\omega t}$ factor, absorption means $\chi''>0$ and $\kappa>0$ ; the opposite convention flips both.

The model drives the electron with the macroscopic field E — when is that wrong?

Whenever the medium is dense enough that the field at an atom differs from the volume-averaged $E$ . Each atom sits among polarized neighbours whose dipole fields add to the drive: $E_{\text{loc}}=E+P/3\varepsilon_0$ for isotropic surroundings. Repairing equation (3) with this local field gives the Clausius–Mossotti / Lorentz–Lorenz relation, which shifts the effective resonance and rescales $\chi(0)$ . For gases ( $\chi\sim10^{-4}$ ) the correction is negligible; for glass ( $\chi\sim1$ ) it changes the quantitative fit — the $N$ of worked example 1 is an effective value — but the Lorentzian structure and every scaling here survive intact.

What is the relation between this article and aq-em-absorption-dispersion?

Deliberate parallels, not duplicates. aq-em-absorption-dispersion develops the same driven-oscillator physics inside the electromagnetism track — importing the mechanical resonance solution, then spending its depth on the Drude limit, the ionosphere's radio verdicts, and a Cauchy fit to glass data. This article is the optics track's derivation owner: it solves the oscillator from Newton's law in full, mints the dielectric function and the locked model-glass numbers, and feeds them forward to the complex index, normal/anomalous dispersion, Sellmeier/Cauchy, and Kramers–Kronig articles. Both treatments are canon; each recalls the other by name.

Above the resonance the model gives n below 1 — is that a problem?

No — it is a correct prediction. For $\omega\gg\omega_0$ , $\varepsilon\approx1-\omega_p^2/\omega^2$ is slightly less than one: the electrons respond 180° out of phase, opposing the vacuum term, and the phase velocity $c/n$ exceeds $c$ . Nothing physical travels at that speed — a monochromatic crest carries no information, and signals ride the group velocity, which stays below $c$ here. This is the everyday physics of the X-ray band, where $n$ sits a few parts in $10^6$ below unity for all matter (enabling grazing-incidence X-ray mirrors), and of plasmas above $\omega_p$ . The full velocity taxonomy is owned by the phase- and group-velocity articles later in this chapter.