Dispersion from the Lorentz Oscillator Model

Derivation

Dispersion from the Lorentz Oscillator Model

D-174 Home PU-204 Threads waves · matter · light Depends on Waves in Conductors and Skin Depth, Electric Dipole Radiation

Statement

Modelling each bound electron in a dielectric as a damped harmonic oscillator of natural frequency \(\omega_0\) and damping rate \(\gamma\), driven by the electric field of a monochromatic wave \(\vec{E}=\vec{E}_0e^{-i\omega t}\), we derive the complex electric susceptibility \(\chi(\omega)=\omega_p^2/(\omega_0^2-\omega^2-i\gamma\omega)\) with plasma frequency \(\omega_p^2=Ne^2/\varepsilon_0 m\), hence the frequency-dependent permittivity \(\varepsilon_r(\omega)=1+\chi(\omega)\) and complex refractive index \(\tilde{n}=n+i\kappa=\sqrt{\varepsilon_r}\). We show that \(n(\omega)\) rises with frequency (normal dispersion) everywhere except inside the absorption line of full width \(\gamma\) centred on \(\omega_0\), where \(dn/d\omega<0\) (anomalous dispersion), and that absorption \(\kappa(\omega)\) is a Lorentzian peaking exactly where the dispersion is anomalous.

Why it matters

This single classical calculation explains why glass bends blue light more than red, why every transparent material has ultraviolet and infrared absorption bands flanking its window of transparency, why metals reflect below their plasma frequency yet transmit above it, and why the refractive index of matter for X-rays is slightly less than one — enabling grazing-incidence X-ray mirrors and telescopes. It is the microscopic origin of every Sellmeier and Cauchy formula used in lens design.

It is also the great survivor of the quantum revolution: quantum mechanics replaces the classical oscillators by transitions weighted with oscillator strengths \(f_j\), but the functional form of \(\varepsilon_r(\omega)\) is untouched. Linear response of any atom is exactly a sum of Lorentzians, which is why a model built from springs in 1878 still fits modern ellipsometry data. The pairing of absorption with anomalous dispersion it predicts is a theorem of causality (Kramers–Kronig), not an accident of the model.

Assumptions

Each electron is bound by a linear restoring force \(-m\omega_0^2\vec{r}\) (small displacements).In strong fields (lasers) the potential's anharmonicity matters: the response acquires terms in \(E^2,E^3\), producing harmonic generation and intensity-dependent refractive index — nonlinear optics. The linear \(\chi(\omega)\) below is only the first term of that expansion.

Damping is linear in velocity, \(-m\gamma\dot{\vec{r}}\), with a single rate \(\gamma\).\(\gamma\) lumps together radiation reaction (computable from electric-dipole radiation) and collisions. If collisions dominate and are strongly frequency-dependent, or Doppler broadening dominates in a gas, the line is no longer a simple Lorentzian (Voigt profile) and \(\varepsilon''\) changes shape.

The field driving each electron is the macroscopic field \(\vec{E}\) (dilute medium).In dense media each dipole also feels the field of its neighbours; the local field exceeds \(\vec{E}\) by \(\vec{P}/3\varepsilon_0\), and \(\chi\) must be corrected to the Clausius–Mossotti / Lorentz–Lorenz form \((\varepsilon_r-1)/(\varepsilon_r+2)=N\alpha/3\varepsilon_0\). Our \(\varepsilon_r=1+\chi\) then overestimates resonance shifts.

Dipole approximation: the field is uniform over the electron's orbit, and the magnetic force is dropped.Both are controlled by small parameters — atomic size over wavelength \(a/\lambda\sim10^{-3}\) and electron speed over light speed \(v/c\). Kept, they produce spatial dispersion (\(\varepsilon\) depending on \(\vec{k}\) as well as \(\omega\)), quadrupole absorption, and magneto-optical effects.

The medium is isotropic, unmagnetised, and the response classical and stationary.A static magnetic field splits the response into circular polarisations (Faraday rotation, Zeeman splitting of \(\omega_0\)); anisotropic binding gives a tensor \(\chi_{ij}\) and birefringence; intense resonant driving saturates real atoms (Rabi physics), which no linear oscillator can capture.

Derivation

\[ m\ddot{\vec{r}}=-m\omega_0^2\vec{r}-m\gamma\dot{\vec{r}}+(-e)\left(\vec{E}+\dot{\vec{r}}\times\vec{B}\right) \]

Newton's second law for one bound electron (charge \(-e\), \(e>0\)): harmonic restoring force, phenomenological viscous damping, and the Lorentz force of the wave. A

\[ \frac{|\dot{\vec{r}}\times\vec{B}|}{|\vec{E}|}\sim\frac{v}{c}\ll1,\qquad \vec{E}(\vec{r},t)\approx\vec{E}_0\,e^{-i\omega t} \]

For a plane wave \(|\vec{B}|=|\vec{E}|/c\), so the magnetic force is smaller by \(v/c\) and is dropped; and since the electron excursion (\(\sim10^{-10}\,\mathrm{m}\)) is tiny compared with optical wavelengths, the spatial phase \(e^{i\vec{k}\cdot\vec{r}}\) is evaluated at the atom's position — the dipole approximation. B

\[ \ddot{\vec{r}}+\gamma\dot{\vec{r}}+\omega_0^2\vec{r}=-\frac{e}{m}\vec{E}_0\,e^{-i\omega t} \]

Divide by \(m\). This is exactly the driven damped oscillator equation, with the drive per unit mass \(-e\vec{E}/m\). A

\[ \vec{r}(t)=\vec{r}_0\,e^{-i\omega t}\ \Rightarrow\ \left(-\omega^2-i\gamma\omega+\omega_0^2\right)\vec{r}_0=-\frac{e}{m}\vec{E}_0 \]

Steady-state ansatz at the driving frequency; with the \(e^{-i\omega t}\) convention each \(\tfrac{d}{dt}\to-i\omega\). Legal because the equation is linear with constant coefficients and transients decay as \(e^{-\gamma t/2}\). B

\[ \vec{r}_0=\frac{-e/m}{\omega_0^2-\omega^2-i\gamma\omega}\,\vec{E}_0 \]

Solve the algebraic equation. The complex denominator carries both the resonance and the phase lag of the electron behind the field. A

\[ \vec{p}=(-e)\,\vec{r}=\frac{e^2/m}{\omega_0^2-\omega^2-i\gamma\omega}\,\vec{E} \]

The induced dipole moment of one atom. The two factors of \(-e\) (force and moment) cancel: the dipole is parallel to \(\vec{E}\) below resonance regardless of the sign of the carrier's charge. A

\[ \vec{P}=N\vec{p}\equiv\varepsilon_0\chi(\omega)\vec{E}\ \Rightarrow\ \chi(\omega)=\frac{\omega_p^2}{\omega_0^2-\omega^2-i\gamma\omega},\qquad \omega_p^2\equiv\frac{Ne^2}{\varepsilon_0 m} \]

With \(N\) oscillators per unit volume the polarisation density defines the susceptibility; the combination \(Ne^2/\varepsilon_0 m\) is named the plasma frequency squared — the only place the density enters. A

\[ \varepsilon_r(\omega)=1+\chi(\omega)=1+\frac{\omega_p^2}{\omega_0^2-\omega^2-i\gamma\omega} \]

From \(\vec{D}=\varepsilon_0\vec{E}+\vec{P}=\varepsilon_0(1+\chi)\vec{E}\). The medium is nonmagnetic, \(\mu_r=1\). A

\[ \varepsilon_r=\varepsilon'+i\varepsilon'',\qquad \varepsilon'=1+\frac{\omega_p^2(\omega_0^2-\omega^2)}{D},\qquad \varepsilon''=\frac{\omega_p^2\,\gamma\omega}{D},\qquad D\equiv(\omega_0^2-\omega^2)^2+\gamma^2\omega^2 \]

Multiply numerator and denominator by the complex conjugate \((\omega_0^2-\omega^2)+i\gamma\omega\). Note \(\varepsilon''>0\) for all \(\omega>0\): the medium always absorbs, never spontaneously amplifies. B

\[ k=\tilde{n}\,\frac{\omega}{c},\qquad \tilde{n}=n+i\kappa=\sqrt{\varepsilon_r}\ \Rightarrow\ E\propto e^{-\kappa\omega z/c}\,e^{i\omega(nz/c-t)},\qquad \alpha=\frac{2\kappa\omega}{c} \]

Maxwell's equations in a linear medium give the dispersion relation \(k^2=\varepsilon_r\,\omega^2/c^2\) — the same route that yields the skin depth in conductors. The real part of \(\tilde{n}\) sets the phase velocity \(c/n\); the imaginary part attenuates the field, and the intensity \(\propto|E|^2\) decays with absorption coefficient \(\alpha\). B

\[ |\chi|\ll1:\quad \tilde{n}=\sqrt{1+\chi}\approx1+\tfrac{1}{2}\chi\ \Rightarrow\ n-1=\frac{\omega_p^2}{2}\,\frac{\omega_0^2-\omega^2}{D},\qquad \kappa=\frac{\omega_p^2}{2}\,\frac{\gamma\omega}{D} \]

Binomial expansion, valid for gases and any dilute medium where the susceptibility is small (checked a posteriori in the examples). Refraction is governed by the in-phase (real) part of the electron's response, absorption by the quadrature (imaginary) part. B

\[ \Delta\equiv\omega-\omega_0,\ \ |\Delta|,\gamma\ll\omega_0:\qquad n-1\approx-\frac{\omega_p^2}{4\omega_0}\,\frac{\Delta}{\Delta^2+\gamma^2/4},\qquad \kappa\approx\frac{\omega_p^2\,\gamma}{8\omega_0}\,\frac{1}{\Delta^2+\gamma^2/4} \]

Near resonance write \(\omega_0^2-\omega^2=(\omega_0+\omega)(\omega_0-\omega)\approx-2\omega_0\Delta\) and \(D\approx4\omega_0^2(\Delta^2+\gamma^2/4)\). The absorption \(\kappa\) is a Lorentzian of full width \(\gamma\) at half maximum; \(n-1\) is its dispersive (odd) partner. C

\[ \frac{d(n-1)}{d\Delta}=-\frac{\omega_p^2}{4\omega_0}\,\frac{\gamma^2/4-\Delta^2}{(\Delta^2+\gamma^2/4)^2}<0\ \ \text{for}\ |\Delta|<\frac{\gamma}{2} \]

Differentiate the dispersive lineshape. Between the extrema at \(\Delta=\pm\gamma/2\) — exactly the half-power points of the absorption line — \(n\) falls with increasing frequency: anomalous dispersion. Outside this band the derivative is positive: normal dispersion. The extreme values are \((n-1)_{\pm}=\pm\,\omega_p^2/(4\gamma\omega_0)=\pm\tfrac12\kappa_{\max}\). C

\[ \varepsilon_r(\omega)=1+\omega_p^2\sum_j\frac{f_j}{\omega_j^2-\omega^2-i\gamma_j\omega},\qquad \sum_j f_j=Z \]

Real atoms have many resonances. Quantum mechanically each transition \(0\to j\) contributes a Lorentzian weighted by its oscillator strength \(f_j\), and the Thomas–Reiche–Kuhn sum rule fixes the total weight to the electron number \(Z\) per atom. The classical form is exact term by term; only the weights are quantum. C

Result

\[ \varepsilon_r(\omega)=1+\frac{\omega_p^2}{\omega_0^2-\omega^2-i\gamma\omega},\qquad \omega_p^2=\frac{Ne^2}{\varepsilon_0 m},\qquad \tilde{n}=n+i\kappa=\sqrt{\varepsilon_r},\qquad n-1\approx\frac{\omega_p^2}{2}\frac{\omega_0^2-\omega^2}{(\omega_0^2-\omega^2)^2+\gamma^2\omega^2},\quad \kappa\approx\frac{\omega_p^2}{2}\frac{\gamma\omega}{(\omega_0^2-\omega^2)^2+\gamma^2\omega^2} \]

Reading. Below resonance the electrons oscillate in phase with the field, their re-radiated waves retard the transmitted phase, and \(n>1\), growing as \(\omega\) approaches \(\omega_0\) — this is the normal dispersion that makes prisms work. Crossing the resonance the electron's phase flips by \(\pi\); within one linewidth of \(\omega_0\) the index plunges with frequency (anomalous dispersion) while absorption peaks, and above resonance \(n<1\): the electrons now respond as if free, opposing the applied field. Absorption is confined to a Lorentzian of width \(\gamma\); everywhere else the medium is transparent and normally dispersive.

Units check. \(\omega_p^2=Ne^2/\varepsilon_0 m\): \(\mathrm{m^{-3}\,C^2/[(C^2\,N^{-1}\,m^{-2})\,kg]}=\mathrm{N\,m^{-1}\,kg^{-1}}=\mathrm{s^{-2}}\) — a frequency squared, as required. In \(\chi\), numerator and denominator are both \(\mathrm{s^{-2}}\), so \(\chi\), \(\varepsilon_r\), \(n\) and \(\kappa\) are dimensionless. The absorption coefficient \(\alpha=2\kappa\omega/c\) has units \(\mathrm{s^{-1}/(m\,s^{-1})=m^{-1}}\), an inverse length.

Limiting cases

Static limit \(\omega\to0\): \(\varepsilon_r(0)=1+\omega_p^2/\omega_0^2\), a real constant \(>1\) — the static dielectric constant of the electronic response. Stiffer binding (larger \(\omega_0\)) means smaller polarisability.
Transparent regions, \(\gamma\to0\) off resonance: \(n^2=1+\omega_p^2/(\omega_0^2-\omega^2)\) — the one-term Sellmeier equation; expanding for \(\omega\ll\omega_0\) gives Cauchy's formula \(n\approx A+B/\lambda^2\). Lossless normal dispersion.
Exactly on resonance \(\omega=\omega_0\): \(\varepsilon_r=1+i\,\omega_p^2/\gamma\omega_0\): the response is pure quadrature, \(n\approx1\) (dilute) and \(\kappa=\kappa_{\max}=\omega_p^2/2\gamma\omega_0\). Maximum power transfer to the medium, as for the driven oscillator at \(\omega_0\).
High frequency \(\omega\gg\omega_0\): \(\varepsilon_r\to1-\omega_p^2/\omega^2\): binding and damping become irrelevant and every material responds as a free-electron plasma. Hence \(n<1\) for X-rays (total external reflection at grazing incidence) and \(\varepsilon_r\to1\) as \(\omega\to\infty\) — no medium responds instantaneously.
Free electrons \(\omega_0\to0\): \(\varepsilon_r=1-\omega_p^2/(\omega^2+i\gamma\omega)\) — the Drude model of a conductor, with AC conductivity \(\sigma(\omega)=\varepsilon_0\omega_p^2/(\gamma-i\omega)\); at low frequency this reproduces the skin-depth result for waves in conductors, and above \(\omega_p\) the metal becomes transparent (alkali metals in the UV).

Breaks when

The field is strong enough to probe the anharmonicity of the binding potential. At laser intensities the expansion \(F=-m\omega_0^2 r\) fails; the polarisation acquires \(\chi^{(2)}E^2+\chi^{(3)}E^3+\cdots\), producing second-harmonic generation, Kerr lensing and self-phase modulation. Linear \(\varepsilon_r(\omega)\) no longer describes propagation.
Resonant driving of real (quantum) atoms saturates the transition. A two-level atom can absorb at most one photon per lifetime; strong resonant fields cause Rabi oscillation and power broadening. The classical oscillator, which absorbs linearly without limit, gets both the saturation intensity and the lineshape wrong in this regime.
The medium is dense, \(N\alpha/\varepsilon_0\not\ll1\). The local field differs from the macroscopic field; \(\varepsilon_r=1+\chi\) must be replaced by Clausius–Mossotti, resonances shift (Lorentz redshift), and near sharp dense resonances light mixes with the polarisation wave into polaritons — the index is no longer a single-valued function of \(\omega\) computed atom by atom.
Above an ionisation or absorption edge. Once \(\hbar\omega\) exceeds a binding energy the electron leaves for the continuum; discrete Lorentzians must be replaced by a continuous oscillator-strength density, and \(\varepsilon''\) shows edges (X-ray absorption spectra) rather than isolated lines.
In gases at optical frequencies where Doppler broadening exceeds \(\gamma\). The observed line is the Lorentzian convolved with the Maxwellian velocity distribution (Voigt profile); the measured width is set by temperature, not by the oscillator's \(\gamma\).

Failure modes

Mixing time conventions. With \(e^{-i\omega t}\) the denominator is \(\omega_0^2-\omega^2-i\gamma\omega\) and absorption means \(\varepsilon''>0\), \(\kappa>0\); with the engineers' \(e^{+j\omega t}\) all imaginary parts flip sign. Mixing conventions silently turns an absorber into a gain medium.
Taking \(n=\sqrt{\varepsilon'}\). The refractive index is the real part of \(\sqrt{\varepsilon'+i\varepsilon''}\), not the square root of the real part. Near resonance, where \(\varepsilon'\) can pass through zero or go negative, the two differ grossly.
Using the dilute formula \(n-1=\chi/2\) for dense media. For glass or water \(\chi\sim1\); the binomial expansion and the neglect of local-field corrections both fail. Density-proportionality of \(n-1\) holds only for gases.
Concluding that \(v_p=c/n>c\) (when \(n<1\)) or \(v_g>c\) (anomalous region) violates relativity. Phase and group velocities are not signal velocities; the front velocity is exactly \(c\) (Sommerfeld–Brillouin). No information travels faster than \(c\).
Confusing the widths. \(\gamma\) is the FWHM of the \(\kappa\) (and \(\varepsilon''\)) Lorentzian in angular frequency near resonance; the separation of the extrema of \(n\) is also \(\gamma\); but the FWHM of the intensity transmission dip depends on optical depth. Quoting linewidths without saying which curve is a standing source of factor-of-2 errors.
Dropping the \(\omega\) in \(\gamma\omega\). Writing the denominator as \(\omega_0^2-\omega^2-i\gamma\omega_0\) is a common near-resonance shortcut; used far from resonance it corrupts the low- and high-frequency limits of \(\varepsilon''\).

Discussion

The physics is interference. Each driven dipole re-radiates a secondary wavelet at the driving frequency (the mechanism quantified by electric-dipole radiation), and the transmitted field is the coherent sum of the incident wave and all these wavelets. Below resonance the electrons respond in phase with the field, the forward-scattered wavelets lag the primary wave by the \(90^\circ\) inherent in dipole re-radiation plus the small oscillator lag, and the summed field is phase-retarded — it travels effectively slower, \(n>1\). Above resonance the oscillator's response has flipped phase by \(\pi\), the interference advances the phase, and \(n<1\). Refraction is therefore not light "slowing down in a denser medium" in any mechanical sense: it is the incident wave being progressively replaced by phase-shifted scattered light (the Ewald–Oseen extinction picture), with the vacuum wave extinguished and rebuilt at phase velocity \(c/n\).

The damping constant deserves respect. Even a perfectly isolated atom has \(\gamma\neq0\), because an oscillating charge radiates: equating the dipole radiation power \(P=e^2\omega_0^4 r_0^2/12\pi\varepsilon_0c^3\) to the dissipation \(\tfrac12 m\gamma\omega_0^2r_0^2\) gives the radiative rate \(\gamma_{\text{rad}}=e^2\omega_0^2/6\pi\varepsilon_0mc^3\). For visible light this evaluates to a few times \(10^7\ \mathrm{s^{-1}}\) — strikingly close to measured natural linewidths of strong atomic lines, one of the classical model's quiet triumphs. In dense matter collisions add to \(\gamma\), and in the Drude limit \(\gamma\) becomes the inverse of the electron's momentum-relaxation time, tying this derivation directly to DC conductivity and skin depth.

The deepest content of the result is analytic structure. Viewed as a function of complex \(\omega\), \(\chi(\omega)=\omega_p^2/(\omega_0^2-\omega^2-i\gamma\omega)\) has both poles at \(\omega=\pm\sqrt{\omega_0^2-\gamma^2/4}-i\gamma/2\), in the lower half-plane. That placement encodes causality — the polarisation cannot precede the field — and by contour integration forces the Kramers–Kronig relations: \(n(\omega)-1\) is the Hilbert transform of \(\kappa(\omega)\). Consequently anomalous dispersion inside an absorption line is not a peculiarity of springs; any causal medium that absorbs must show it, and conversely the entire refractive-index curve of a material can be reconstructed from its absorption spectrum alone. The sum rule \(\int_0^\infty\omega\,\varepsilon''(\omega)\,d\omega=\tfrac{\pi}{2}\omega_p^2\) is exact for any \(N\) electrons per unit volume, whatever forces bind them: total absorption strength counts electrons, which is why \(\omega_p\) alone controls the universal high-frequency limit \(\varepsilon_r\to1-\omega_p^2/\omega^2\).

Common misconceptions. \(n<1\) does not mean energy travels faster than light — the phase pattern does, the signal front does not. Anomalous dispersion is not "anomalous" in the sense of rare: every material exhibits it inside every absorption line; it merely cannot be seen where the medium is too opaque to transmit. And the model is not obsolete because atoms are quantum: quantum linear-response theory reproduces the Lorentzian form exactly, with \(f_j\) as the only quantum fingerprint.

Worked examples

Estimate the refractive index of air at \(\lambda=589\ \mathrm{nm}\), treating each molecule as one electron bound with a UV resonance at \(\lambda_0=100\ \mathrm{nm}\). Take \(N=2.5\times10^{25}\ \mathrm{m^{-3}}\) (ideal gas at \(20^\circ\mathrm{C}\), 1 atm) and neglect \(\gamma\) (we are far from resonance).

Set up symbolically: far from resonance \(n-1=\dfrac{\omega_p^2}{2(\omega_0^2-\omega^2)}\). A

\[ \omega_p^2=\frac{Ne^2}{\varepsilon_0 m}=\frac{(2.5\times10^{25})(1.602\times10^{-19})^2}{(8.854\times10^{-12})(9.109\times10^{-31})}=7.96\times10^{28}\ \mathrm{s^{-2}} \]

Only the electron density enters; \(e^2/\varepsilon_0m=3.18\times10^{3}\ \mathrm{m^3\,s^{-2}}\) is a handy constant. A

\[ \omega_0=\frac{2\pi c}{\lambda_0}=\frac{2\pi(2.998\times10^{8})}{1.00\times10^{-7}}=1.88\times10^{16}\ \mathrm{rad\,s^{-1}},\qquad \omega=\frac{2\pi c}{\lambda}=3.20\times10^{15}\ \mathrm{rad\,s^{-1}} \]

Convert both wavelengths to angular frequencies; \(\omega_0^2=3.55\times10^{32}\ \mathrm{s^{-2}}\), \(\omega^2=1.02\times10^{31}\ \mathrm{s^{-2}}\). A

\[ n-1=\frac{\omega_p^2}{2(\omega_0^2-\omega^2)}=\frac{7.96\times10^{28}}{2(3.55\times10^{32}-1.02\times10^{31})}=\frac{7.96\times10^{28}}{6.89\times10^{32}}=1.2\times10^{-4} \]

Substitute. Check the dilute assumption: \(\chi=2(n-1)\approx2.3\times10^{-4}\ll1\). Consistent. B

\[ n-1\approx1.2\times10^{-4}\quad(\text{measured for air: }2.9\times10^{-4}) \]

Reading. One classical electron per molecule lands within a factor of \(\sim2.5\) of the measured value — the shortfall is the oscillator strength of the several electrons per molecule that actually respond (step 14). The model also predicts the right dispersion: since \(\omega^2\) subtracts in the denominator, blue light (\(\omega\) larger) sees a larger \(n\) than red. Units check. \(\mathrm{s^{-2}/s^{-2}}\): dimensionless, as an index must be.

Dilute sodium vapour, \(N=1.0\times10^{17}\ \mathrm{m^{-3}}\), illuminated exactly on the D-line resonance \(\lambda_0=589\ \mathrm{nm}\). (a) Predict \(\gamma\) from radiation damping alone. (b) Find the peak extinction \(\kappa\), the absorption coefficient \(\alpha\), and the \(1/e\) intensity penetration depth.

On resonance \(\kappa_{\max}=\dfrac{\omega_p^2}{2\gamma\omega_0}\) and \(\alpha=2\kappa\omega_0/c\); first we need \(\gamma\). A

\[ \gamma_{\text{rad}}=\frac{e^2\omega_0^2}{6\pi\varepsilon_0mc^3},\qquad \omega_0=\frac{2\pi c}{\lambda_0}=3.20\times10^{15}\ \mathrm{rad\,s^{-1}}\ \Rightarrow\ \gamma_{\text{rad}}=\frac{(1.602\times10^{-19})^2(3.20\times10^{15})^2}{6\pi(8.854\times10^{-12})(9.109\times10^{-31})(2.998\times10^{8})^3}=6.4\times10^{7}\ \mathrm{s^{-1}} \]

Radiative damping rate from the electric-dipole radiation power balance. The measured natural width of the Na D2 line is \(6.15\times10^{7}\ \mathrm{s^{-1}}\) — the classical estimate is good to a few percent here. B

\[ \omega_p^2=\frac{Ne^2}{\varepsilon_0m}=(1.0\times10^{17})(3.18\times10^{3})=3.18\times10^{20}\ \mathrm{s^{-2}},\qquad \kappa_{\max}=\frac{\omega_p^2}{2\gamma\omega_0}=\frac{3.18\times10^{20}}{2(6.4\times10^{7})(3.20\times10^{15})}=7.8\times10^{-4} \]

Peak extinction from step 12 at \(\Delta=0\). Dilute check: \(|\chi|=2\kappa_{\max}\approx1.6\times10^{-3}\ll1\). Consistent. B

\[ \alpha=\frac{2\kappa_{\max}\omega_0}{c}=\frac{2(7.8\times10^{-4})(3.20\times10^{15})}{2.998\times10^{8}}=1.7\times10^{4}\ \mathrm{m^{-1}},\qquad \ell=\frac{1}{\alpha}\approx60\ \mathrm{\mu m} \]

Intensity attenuation coefficient and penetration depth. B

\[ \gamma\approx6.4\times10^{7}\ \mathrm{s^{-1}},\qquad \kappa_{\max}\approx7.8\times10^{-4},\qquad \alpha\approx1.7\times10^{4}\ \mathrm{m^{-1}},\qquad \ell\approx60\ \mathrm{\mu m} \]

Reading. A vapour a hundred-billion times more dilute than air stops resonant light within a tenth of a millimetre — resonance concentrates the entire interaction strength into a band of width \(\gamma/\omega_0\sim2\times10^{-8}\) of the optical frequency. Detune by a few linewidths and the vapour is essentially transparent. (The real D2 line has oscillator strength \(f\approx0.64\), scaling \(\kappa\) and \(\alpha\) down by that factor; Doppler broadening in a warm vapour spreads the same integrated absorption over a wider line.) Units check. \(\alpha=\kappa\omega/c\): \(\mathrm{s^{-1}/(m\,s^{-1})=m^{-1}}\); \(\ell\) in metres.

Problems

Aluminium has free-electron density \(N=1.81\times10^{29}\ \mathrm{m^{-3}}\). Compute its plasma frequency \(\omega_p\) and the vacuum wavelength below which the metal becomes transparent (\(\omega_0\to0\) limit, neglect \(\gamma\)).
Solution
\(\omega_p^2=Ne^2/\varepsilon_0m=(1.81\times10^{29})(3.18\times10^{3})=5.76\times10^{32}\ \mathrm{s^{-2}}\), so \(\omega_p=2.40\times10^{16}\ \mathrm{rad\,s^{-1}}\). In the Drude limit \(\varepsilon_r=1-\omega_p^2/\omega^2\): for \(\omega<\omega_p\), \(\varepsilon_r<0\), \(\tilde n\) is purely imaginary and waves are evanescent (total reflection); for \(\omega>\omega_p\), \(n\) is real and the metal transmits. Cutoff wavelength \(\lambda_p=2\pi c/\omega_p=2\pi(2.998\times10^{8})/(2.40\times10^{16})=7.8\times10^{-8}\ \mathrm{m}\approx78\ \mathrm{nm}\) — deep ultraviolet, consistent with aluminium's measured plasma edge near \(83\ \mathrm{nm}\) and with its excellent visible reflectivity.
Model atomic hydrogen gas (\(N=2.5\times10^{25}\ \mathrm{m^{-3}}\)) with a single resonance at the Lyman-\(\alpha\) wavelength \(\lambda_0=121.6\ \mathrm{nm}\). Find the static susceptibility \(\chi(0)\) and the low-frequency refractive index.
Solution
\(\omega_0=2\pi c/\lambda_0=2\pi(2.998\times10^{8})/(1.216\times10^{-7})=1.549\times10^{16}\ \mathrm{rad\,s^{-1}}\), \(\omega_0^2=2.40\times10^{32}\ \mathrm{s^{-2}}\). \(\omega_p^2=(2.5\times10^{25})(3.18\times10^{3})=7.96\times10^{28}\ \mathrm{s^{-2}}\). \(\chi(0)=\omega_p^2/\omega_0^2=7.96\times10^{28}/2.40\times10^{32}=3.3\times10^{-4}\). Then \(n(0)\approx1+\chi(0)/2=1.00017\). (Measured for H\(_2\) gas: \(n-1\approx1.4\times10^{-4}\) per atom-equivalent — right order; the exact quantum answer spreads the oscillator strength over the whole Lyman series and continuum.)
For 10 keV X-rays in silicon (electron density \(N_e=Z\,n_{\text{atom}}=14\times5.0\times10^{28}=7.0\times10^{29}\ \mathrm{m^{-3}}\)), all binding frequencies satisfy \(\omega\gg\omega_j\). Show that \(n=1-\delta\) with \(\delta=\omega_p^2/2\omega^2\), evaluate \(\delta\), and find the critical grazing angle for total external reflection.
Solution
For \(\omega\gg\omega_j\) every term of step 14 reduces to \(-f_j\omega_p^2/\omega^2\) and the sum rule gives \(\varepsilon_r=1-\omega_p^2/\omega^2\) with \(\omega_p^2\) built from the total electron density; since \(\omega_p^2/\omega^2\ll1\), \(n\approx1-\omega_p^2/2\omega^2\equiv1-\delta\). Numbers: \(\omega_p^2=(7.0\times10^{29})(3.18\times10^{3})=2.23\times10^{33}\ \mathrm{s^{-2}}\); \(\hbar\omega=10\ \mathrm{keV}\Rightarrow\omega=(10^4\times1.602\times10^{-19})/(1.055\times10^{-34})=1.52\times10^{19}\ \mathrm{rad\,s^{-1}}\), \(\omega^2=2.31\times10^{38}\ \mathrm{s^{-2}}\). \(\delta=2.23\times10^{33}/(2\times2.31\times10^{38})=4.8\times10^{-6}\). Since \(n<1\), X-rays entering from vacuum can totally reflect at grazing incidence: Snell gives \(\cos\theta_c=n\Rightarrow\theta_c\approx\sqrt{2\delta}=\sqrt{9.6\times10^{-6}}=3.1\times10^{-3}\ \mathrm{rad}\approx0.18^\circ\). This is the operating principle of grazing-incidence X-ray telescope mirrors.
Using the near-resonance forms of step 12, show that the extrema of \(n(\omega)\) sit at \(\omega=\omega_0\pm\gamma/2\) with \((n-1)_{\pm}=\pm\,\omega_p^2/4\gamma\omega_0=\pm\tfrac12\kappa_{\max}\), and evaluate the peak index excursion for the sodium vapour of worked example 2.
Solution
\(n-1=-\dfrac{\omega_p^2}{4\omega_0}\dfrac{\Delta}{\Delta^2+\gamma^2/4}\). Setting \(d(n-1)/d\Delta=0\) requires \(\gamma^2/4-\Delta^2=0\), i.e. \(\Delta=\pm\gamma/2\). Substituting: \((n-1)_{\pm}=\mp\dfrac{\omega_p^2}{4\omega_0}\dfrac{\pm\gamma/2}{\gamma^2/2}\Rightarrow|n-1|_{\text{ext}}=\dfrac{\omega_p^2}{4\gamma\omega_0}\), with the maximum on the low-frequency side (\(\Delta=-\gamma/2\)). Since \(\kappa_{\max}=\omega_p^2/2\gamma\omega_0\), the extreme index excursion is exactly \(\kappa_{\max}/2\). Sodium numbers: \(|n-1|_{\text{ext}}=3.18\times10^{20}/[4(6.4\times10^{7})(3.20\times10^{15})]=3.9\times10^{-4}\) — five times the index contrast of air, from a vapour \(10^{8}\) times thinner, but only within \(\pm\gamma/2\approx\pm3\times10^{7}\ \mathrm{rad\,s^{-1}}\) (\(\pm5\ \mathrm{MHz}\)) of line centre.
The ionosphere is a Drude medium (\(\omega_0=0\), \(\gamma\) negligible) with electron density \(N\approx1.0\times10^{12}\ \mathrm{m^{-3}}\). Find the plasma frequency in Hz, then the refractive index, phase velocity and group velocity for a 15 MHz shortwave signal, and verify \(v_pv_g=c^2\).
Solution
\(\omega_p^2=(1.0\times10^{12})(3.18\times10^{3})=3.18\times10^{15}\ \mathrm{s^{-2}}\Rightarrow\omega_p=5.64\times10^{7}\ \mathrm{rad\,s^{-1}}\), \(f_p=\omega_p/2\pi=9.0\ \mathrm{MHz}\). Below 9 MHz waves reflect (the basis of over-the-horizon radio); at \(f=15\ \mathrm{MHz}\): \(n=\sqrt{1-f_p^2/f^2}=\sqrt{1-(9.0/15)^2}=\sqrt{1-0.36}=0.80\). Phase velocity \(v_p=c/n=3.75\times10^{8}\ \mathrm{m\,s^{-1}}>c\). Group velocity: from \(\omega^2=\omega_p^2+c^2k^2\), \(2\omega\,d\omega=2c^2k\,dk\Rightarrow v_g=d\omega/dk=c^2k/\omega=cn=2.40\times10^{8}\ \mathrm{m\,s^{-1}}<c\). Product: \(v_pv_g=(c/n)(cn)=c^2\). Energy and information travel at \(v_g<c\); the superluminal \(v_p\) is a pattern speed carrying no signal.