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Derivation

Dispersion and Absorption from the Lorentz Oscillator

D-204 Home PU-206 Threads light · matter · fields Depends on driven-damped-harmonic-oscillator, maxwell-equations-in-media
Statement

For a dilute-to-moderate dielectric of \(N\) identical bound electrons per unit volume, each modelled as a driven, damped harmonic oscillator with resonance \(\omega_0\) and damping rate \(\gamma\), the relative permittivity and complex refractive index for a monochromatic field \(\mathbf{E}\propto e^{-i\omega t}\) are \(\varepsilon_r(\omega)=1+\dfrac{\omega_p^2}{\omega_0^2-\omega^2-i\gamma\omega}\) and \(n(\omega)=\sqrt{\varepsilon_r(\omega)}\equiv n'(\omega)+i\kappa(\omega)\), with \(\omega_p^2=Ne^2/(\varepsilon_0 m)\). The real part \(n'\) governs phase velocity and dispersion; the imaginary part \(\kappa\) governs attenuation through the intensity absorption coefficient \(\alpha=2\kappa\omega/c\).

Why it matters

Almost every optical property of transparent matter — the fact that glass has \(n\approx1.5\), that a prism spreads white light with violet bent most, that a spectral line both absorbs and shifts the phase of light near it — falls out of one mechanical picture: an electron on a spring, pushed by the wave and losing energy to damping. The Lorentz oscillator is the classical bridge between Maxwell's equations in media and the microscopic response of matter.

It also delivers the qualitative shape of every resonance in physics: a real (dispersive) part that swings up then down through zero, and an imaginary (absorptive) part that peaks as a Lorentzian. Recognising this pair — and that they are not independent but linked by causality (Kramers–Kronig) — is a recurring tool from optics to circuit theory to particle scattering.

Assumptions
Each electron feels a linear restoring force \(-m\omega_0^2 x\).Drop it and there is no resonance; a free electron (\(\omega_0=0\)) gives the Drude metal, not a dielectric line.
Damping is linear and viscous, \(-m\gamma\dot{x}\), with \(\gamma\ll\omega_0\) for a sharp line.Drop the linearity and the response saturates or generates harmonics — the nonlinear-optics regime — and superposition of Fourier components fails.
The driving field is the applied macroscopic field, i.e. the local field equals \(\mathbf{E}\).Drop it in a dense medium and neighbouring dipoles add a local correction \(\mathbf{P}/3\varepsilon_0\), replacing \(\varepsilon_r\) by the Clausius–Mossotti / Lorentz–Lorenz form and shifting the resonance.
The medium is non-magnetic, \(\mu_r=1\), and spatially non-dispersive (response is local in space).Drop it and \(n=\sqrt{\varepsilon_r\mu_r}\) needs \(\mu_r(\omega)\), and \(k\)-dependence (spatial dispersion) mixes in.
The electron amplitude is small compared with atomic dimensions and speeds are non-relativistic.Drop it and higher multipoles, magnetic-force terms \(\mathbf{v}\times\mathbf{B}\), and anharmonicity enter.
All \(N\) oscillators are identical and independent, with a single resonance \(\omega_0\).Drop it and one needs a sum over transitions with quantum oscillator strengths \(f_j\), \(\sum_j f_j=Z\); the classical single line is the \(f=1\) special case.
Derivation
1
\[ m\ddot{x}+m\gamma\dot{x}+m\omega_0^2 x=-eE_0\,e^{-i\omega t} \]
Newton's second law for one bound electron (charge \(-e\), \(e>0\)) in the wave field, with a linear restoring force and viscous damping. This is the driven-damped oscillator, taken as a prior result. A
2
\[ x(t)=x_0\,e^{-i\omega t},\qquad \big(-\omega^2-i\gamma\omega+\omega_0^2\big)\,x_0=-\frac{eE_0}{m} \]
Steady-state ansatz: the linear equation forces oscillation at the drive frequency. Each \(\partial_t\to-i\omega\). Transients \(\propto e^{-\gamma t/2}\) have decayed. A
3
\[ x_0=-\frac{eE_0}{m}\,\frac{1}{\omega_0^2-\omega^2-i\gamma\omega} \]
Solve the algebraic relation for the complex amplitude. The complex denominator encodes the phase lag of the electron behind the drive. A
4
\[ p_0=-e\,x_0=\frac{e^2E_0}{m}\,\frac{1}{\omega_0^2-\omega^2-i\gamma\omega} \]
The induced electric dipole moment of one electron is \(p=-ex\); the two sign factors of \(-e\) combine to a positive prefactor. A
5
\[ \mathbf{P}=N\mathbf{p}=\frac{Ne^2}{m}\,\frac{1}{\omega_0^2-\omega^2-i\gamma\omega}\,\mathbf{E} \]
Macroscopic polarization is dipole moment per volume, \(N\) dipoles all driven in phase by the same local field (= \(\mathbf{E}\) by assumption). B
6
\[ \chi(\omega)=\frac{P}{\varepsilon_0 E}=\frac{Ne^2}{\varepsilon_0 m}\,\frac{1}{\omega_0^2-\omega^2-i\gamma\omega}\equiv\frac{\omega_p^2}{\omega_0^2-\omega^2-i\gamma\omega} \]
Definition of electric susceptibility \(\mathbf{P}=\varepsilon_0\chi\mathbf{E}\), and definition of the plasma frequency \(\omega_p^2\equiv Ne^2/(\varepsilon_0 m)\). B
7
\[ \varepsilon_r(\omega)=1+\chi(\omega)=1+\frac{\omega_p^2}{\omega_0^2-\omega^2-i\gamma\omega} \]
From Maxwell in media, \(\mathbf{D}=\varepsilon_0\mathbf{E}+\mathbf{P}=\varepsilon_0(1+\chi)\mathbf{E}\), so \(\varepsilon_r=1+\chi\) (prior result). B
8
\[ \nabla^2\mathbf{E}-\frac{\varepsilon_r}{c^2}\frac{\partial^2\mathbf{E}}{\partial t^2}=0\ \Rightarrow\ k^2=\varepsilon_r\,\frac{\omega^2}{c^2},\qquad n\equiv\frac{ck}{\omega}=\sqrt{\varepsilon_r} \]
Substituting the constitutive relation into Maxwell's equations gives a wave equation in the medium; a plane wave \(e^{i(kz-\omega t)}\) fixes the dispersion relation. With \(\mu_r=1\), the refractive index is \(n=\sqrt{\varepsilon_r}\). B
9
\[ n=n'+i\kappa,\qquad k=\frac{n\omega}{c}=\frac{n'\omega}{c}+i\frac{\kappa\omega}{c} \]
\(\varepsilon_r\) is complex, so \(n\) is complex. Split into real and imaginary parts by definition; this separates propagation from attenuation. A
10
\[ E\propto e^{i(kz-\omega t)}=\underbrace{e^{i(n'\omega z/c-\omega t)}}_{\text{phase, speed }c/n'}\ \underbrace{e^{-\kappa\omega z/c}}_{\text{decay}} \]
Insert the complex \(k\). The real part sets phase velocity \(v=c/n'\); the imaginary part attenuates the amplitude exponentially in \(z\). B
11
\[ I\propto|E|^2\propto e^{-2\kappa\omega z/c}\equiv e^{-\alpha z},\qquad \boxed{\alpha=\frac{2\kappa\omega}{c}} \]
Intensity is amplitude squared; the factor of two defines the Beer–Lambert intensity absorption coefficient. This is the physical meaning of the imaginary index. B
12
\[ \varepsilon_r=1+\frac{\omega_p^2\,(\omega_0^2-\omega^2)}{(\omega_0^2-\omega^2)^2+\gamma^2\omega^2}\;+\;i\,\frac{\omega_p^2\,\gamma\omega}{(\omega_0^2-\omega^2)^2+\gamma^2\omega^2} \]
Rationalise: multiply numerator and denominator of \(\chi\) by the complex conjugate \((\omega_0^2-\omega^2+i\gamma\omega)\). The real part is dispersive (odd about \(\omega_0\)); the imaginary part is a positive Lorentzian (absorptive). B
13
\[ \text{dilute: }|\chi|\ll1\Rightarrow\ n'-1\simeq\tfrac12\,\mathrm{Re}\,\chi,\qquad \kappa\simeq\tfrac12\,\mathrm{Im}\,\chi \]
When \(N\) is small (a gas), \(\sqrt{1+\chi}\approx1+\chi/2\) to first order. This isolates the classic single-resonance line shapes for \(n'\) and \(\kappa\). Not valid in dense media or at line centre of a strong line. C
Result
\[ \varepsilon_r(\omega)=1+\frac{\omega_p^2}{\omega_0^2-\omega^2-i\gamma\omega},\qquad n(\omega)=\sqrt{\varepsilon_r}=n'+i\kappa,\qquad \omega_p^2=\frac{Ne^2}{\varepsilon_0 m} \]

Reading. A gas of tiny charged springs responds to light by oscillating with a frequency-dependent amplitude and a phase lag. Below resonance the electrons follow the field nearly in phase and stiffen the medium, giving \(n'>1\) that rises with \(\omega\) — normal dispersion. Right at \(\omega_0\) the real part passes through its background value while absorption \(\kappa\) peaks; just around it \(n'\) falls with increasing \(\omega\) — anomalous dispersion inside the absorption band. Far above resonance the electrons cannot keep up, lag by \(\pi\), and \(n'<1\). The single complex function \(\varepsilon_r(\omega)\) carries both effects at once: its real part is dispersion, its imaginary part is absorption.

Units check. \(\omega_p^2=Ne^2/(\varepsilon_0 m)\) has units \(\mathrm{m^{-3}\cdot C^2}/(\mathrm{C^2 J^{-1}m^{-1}\cdot kg})=\mathrm{J\,kg^{-1}m^{-2}}=(\mathrm{m^2 s^{-2}})\,\mathrm{m^{-2}}=\mathrm{s^{-2}}\), matching \(\omega_0^2\) and \(\omega^2\) in the denominator, so \(\chi\) and \(\varepsilon_r\) are dimensionless and \(n\) is dimensionless. The absorption coefficient \(\alpha=2\kappa\omega/c\) has units \(\mathrm{s^{-1}/(m\,s^{-1})=m^{-1}}\), correct for a decay length.

Limiting cases
  • Far below resonance (\(\omega\ll\omega_0\), \(\gamma\) negligible): \(\varepsilon_r\to1+\omega_p^2/\omega_0^2\), a real static-like constant; \(n\) is real, \(>1\), and increases with \(\omega\) — the transparent, normal-dispersion window of glass.
  • Free electron (\(\omega_0\to0\)): \(\varepsilon_r\to1-\dfrac{\omega_p^2}{\omega^2+i\gamma\omega}\), the Drude model of a metal/plasma; \(n\) becomes imaginary below \(\omega_p\) (reflection), real above.
  • At line centre (\(\omega=\omega_0\)): \(\mathrm{Re}\,\chi=0\) so \(n'=1\) (dilute), while \(\mathrm{Im}\,\chi=\omega_p^2/(\gamma\omega_0)\) is maximal — pure absorption, no phase advance.
  • Far above resonance (\(\omega\gg\omega_0\)): \(\varepsilon_r\to1-\omega_p^2/\omega^2<1\); electrons lag by \(\pi\), \(n'<1\) (phase velocity exceeds \(c\), signal velocity does not). This is the X-ray regime.
  • Zero damping (\(\gamma\to0\)): \(\mathrm{Im}\,\varepsilon_r\to\pi\)-like spike (a delta line), \(\mathrm{Re}\,\varepsilon_r=1+\omega_p^2/(\omega_0^2-\omega^2)\) diverges at \(\omega_0\) — the idealised lossless dispersion, unphysical exactly on resonance.
Breaks when
  • Strong fields / line centre of a strong line. Large amplitudes push the electron off the linear part of the potential: the restoring force becomes anharmonic, the response saturates, harmonics appear, and \(\chi\) is no longer field-independent. The whole linear-response construction collapses (nonlinear optics).
  • Dense media. The field acting on an atom is not the macroscopic \(\mathbf{E}\) but includes the field of its neighbours' dipoles. One must use the local field \(\mathbf{E}+\mathbf{P}/3\varepsilon_0\), giving the Lorentz–Lorenz relation and a red-shifted, renormalised resonance; the naive \(\varepsilon_r=1+\chi\) with \(\chi\) above is quantitatively wrong.
  • Metals and conductors at low frequency. With \(\omega_0=0\) and mobile carriers, static conduction (\(\sigma_0\)) dominates; the bound-oscillator picture must be replaced by Drude transport, and \(\varepsilon_r\) diverges as \(\omega\to0\).
  • Near overlapping or quantum-degenerate resonances. A single classical \(\omega_0\) cannot capture real spectra; one needs a sum \(\sum_j f_j\omega_p^2/(\omega_j^2-\omega^2-i\gamma_j\omega)\) with quantum oscillator strengths \(f_j\), and near-degeneracy makes the classical amplitudes unreliable.
Failure modes
  • Damping sign flip. Writing \(+i\gamma\omega\) instead of \(-i\gamma\omega\) in the denominator (or using \(e^{+i\omega t}\) without flipping every \(i\)) makes \(\mathrm{Im}\,\varepsilon_r<0\), i.e. spurious gain instead of absorption. The sign of \(\kappa\) must give decay, not growth.
  • Confusing \(n'\) and \(\kappa\). Students report the resonance peak of \(n'\) — but \(n'\) does not peak at \(\omega_0\); it has an S-shaped wiggle there. It is \(\kappa\) (absorption) that peaks at \(\omega_0\).
  • Dropping the factor of 2 in \(\alpha\). \(\alpha=2\kappa\omega/c\) is for intensity; the amplitude decays as \(\kappa\omega/c\). Forgetting the 2 halves the predicted attenuation.
  • Using the dilute \(n'-1\approx\chi/2\) form in a solid. For \(n\approx1.5\), \(\chi\approx1.25\) is not small; the square root must be taken exactly, and local-field corrections applied.
  • Treating \(\varepsilon_r\) as real near \(\omega_0\). Setting \(\gamma=0\) "to simplify" removes all absorption and makes \(n'\) diverge — a mathematically singular, physically meaningless answer on resonance.
  • Miscounting \(N\). Using number of atoms rather than number of bound electrons per unit volume (or ignoring oscillator strength) misestimates \(\omega_p^2\), often by \(Z\).
Discussion

The Lorentz model's power is that one denominator, \(\omega_0^2-\omega^2-i\gamma\omega\), organises all of linear optics. Its real part changes sign across the resonance and generates dispersion; its imaginary part is a positive-definite Lorentzian that always describes absorption. The two are two faces of one complex response function, and they are the analogue of the reactance and resistance of a driven RLC circuit — the electron on a spring is an RLC resonator for light.

Normal versus anomalous dispersion is not a special feature of exotic materials: every medium is normally dispersive in its transparency windows (between resonances) and anomalously dispersive inside each absorption band. The rainbow of a prism is normal dispersion in the visible, far from glass's UV resonances; the strong bending and colour of a dye near its absorption line is anomalous dispersion. The apparent \(n'<1\) above resonance (superluminal phase velocity) is not a relativity violation — energy and information travel at the group and front velocities, which stay below \(c\).

Deeper still, the real and imaginary parts are not independent. Because the polarization cannot respond before the field arrives (causality: the response function vanishes for \(t<0\)), \(\chi(\omega)\) is analytic in the upper half \(\omega\)-plane, and \(\mathrm{Re}\,\chi\) and \(\mathrm{Im}\,\chi\) are Hilbert transforms of each other — the Kramers–Kronig relations. Dispersion and absorption are therefore mathematically inseparable: a medium that absorbs anywhere must disperse everywhere. The Lorentz oscillator is the simplest function that manifestly satisfies these relations, which is why it is the universal fitting form for optical spectra. Quantum mechanically, the classical \(\omega_p^2\) is replaced by \(\sum_j f_j\,\omega_{p}^2\) with transition oscillator strengths obeying the Thomas–Reiche–Kuhn sum rule \(\sum_j f_j=Z\), so the classical single-oscillator result is exact in its total integrated strength.

Common misconceptions. (i) "Absorption and refraction are separate phenomena" — no, they are the imaginary and real parts of one \(n(\omega)\), tied by causality. (ii) "\(n<1\) means faster-than-light signalling" — only phase velocity exceeds \(c\); no information does. (iii) "The refractive index peaks at resonance" — absorption peaks there; the index makes an anomalous downturn.

Worked examples

Example 1 — Refractive index and normal dispersion of a transparent dielectric below resonance. A solid dielectric has \(N=1.0\times10^{28}\ \mathrm{m^{-3}}\) bound electrons and a single UV resonance \(\omega_0=1.5\times10^{16}\ \mathrm{rad\,s^{-1}}\); damping is negligible in the visible. Find \(n'\) at green (\(\lambda=500\ \mathrm{nm}\)) and at blue (\(\lambda=419\ \mathrm{nm}\)) and confirm normal dispersion.

1
\[ \omega_p^2=\frac{Ne^2}{\varepsilon_0 m}=\frac{(1.0\times10^{28})(1.602\times10^{-19})^2}{(8.854\times10^{-12})(9.109\times10^{-31})} \]
Plasma frequency from the electron density. Symbols first, then numbers. A
2
\[ \omega_p^2=\frac{2.566\times10^{-10}}{8.066\times10^{-42}}=3.18\times10^{31}\ \mathrm{s^{-2}}\quad(\omega_p=5.64\times10^{15}\ \mathrm{rad\,s^{-1}}) \]
Arithmetic; units \(\mathrm{s^{-2}}\) as checked above. A
3
\[ \omega=\frac{2\pi c}{\lambda}:\quad \omega_{500}=3.77\times10^{15},\qquad \omega_{419}=4.50\times10^{15}\ \mathrm{rad\,s^{-1}} \]
Convert wavelengths to angular frequencies. A
4
\[ \varepsilon_r\simeq1+\frac{\omega_p^2}{\omega_0^2-\omega^2},\qquad \omega_0^2=2.25\times10^{32}\ \mathrm{s^{-2}} \]
Negligible damping in the transparency window, so \(\varepsilon_r\) is real. B
5
\[ \varepsilon_r(500)=1+\frac{3.18\times10^{31}}{2.25\times10^{32}-1.42\times10^{31}}=1+0.151=1.151 \]
Insert \(\omega_{500}^2=1.42\times10^{31}\). A
6
\[ \varepsilon_r(419)=1+\frac{3.18\times10^{31}}{2.25\times10^{32}-2.03\times10^{31}}=1+0.155=1.155 \]
Insert \(\omega_{419}^2=2.03\times10^{31}\). Higher \(\omega\) shrinks the denominator. A
7
\[ n'(500)=\sqrt{1.151}=1.073,\qquad n'(419)=\sqrt{1.155}=1.075 \]
Exact square root (not the dilute approximation, since \(\chi\sim0.15\)). B
\[ n'(500\,\mathrm{nm})=1.073<n'(419\,\mathrm{nm})=1.075 \]

Reading. The index rises as frequency rises (wavelength falls): blue is bent more than green — textbook normal dispersion, driven by the UV resonance the visible light sits below. Units. \(n'\) dimensionless.

Example 2 — Absorption at line centre for a dilute gas. A rarefied atomic vapour has \(N=1.0\times10^{20}\ \mathrm{m^{-3}}\), a visible resonance \(\omega_0=4.0\times10^{15}\ \mathrm{rad\,s^{-1}}\) (\(\lambda_0=471\ \mathrm{nm}\)), and radiative damping \(\gamma=1.0\times10^{9}\ \mathrm{rad\,s^{-1}}\). Find \(\kappa\) and the intensity absorption coefficient \(\alpha\) exactly on resonance.

1
\[ \omega_p^2=\frac{Ne^2}{\varepsilon_0 m}=\frac{(1.0\times10^{20})(1.602\times10^{-19})^2}{(8.854\times10^{-12})(9.109\times10^{-31})}=3.18\times10^{23}\ \mathrm{s^{-2}} \]
Same formula, lower density (a factor \(10^{8}\) below Example 1). A
2
\[ \mathrm{Im}\,\chi\big|_{\omega=\omega_0}=\frac{\omega_p^2\,\gamma\omega_0}{(0)^2+\gamma^2\omega_0^2}=\frac{\omega_p^2}{\gamma\omega_0} \]
On resonance \(\omega_0^2-\omega^2=0\); the Lorentzian imaginary part reaches its maximum. B
3
\[ \mathrm{Im}\,\chi=\frac{3.18\times10^{23}}{(1.0\times10^{9})(4.0\times10^{15})}=\frac{3.18\times10^{23}}{4.0\times10^{24}}=0.0795 \]
Arithmetic; dimensionless, and \(\ll1\), so the dilute expansion is valid here. A
4
\[ \kappa\simeq\tfrac12\,\mathrm{Im}\,\chi=0.0397,\qquad n'\simeq1+\tfrac12\,\mathrm{Re}\,\chi=1\ (\text{since }\mathrm{Re}\,\chi=0) \]
Dilute first-order relations; real part vanishes at line centre, so pure absorption. B
5
\[ \alpha=\frac{2\kappa\omega_0}{c}=\frac{2(0.0397)(4.0\times10^{15})}{3.0\times10^{8}} \]
Definition of the intensity absorption coefficient. Symbols before numbers. B
6
\[ \alpha=\frac{3.18\times10^{14}}{3.0\times10^{8}}=1.06\times10^{6}\ \mathrm{m^{-1}} \]
Arithmetic; units \(\mathrm{m^{-1}}\). A
\[ \kappa\approx0.040,\qquad \alpha\approx1.1\times10^{6}\ \mathrm{m^{-1}}\quad(\text{decay length }1/\alpha\approx0.9\ \mu\mathrm{m}) \]

Reading. Even a very thin vapour is essentially opaque at exact line centre — intensity falls by \(1/e\) in under a micron — while the phase index is unshifted (\(n'=1\)). This is why atomic absorption lines are black in a spectrum. Units. \(\kappa\) dimensionless, \(\alpha\) in \(\mathrm{m^{-1}}\).

Problems
  1. Show that the frequency at which \(n'\) is maximal (dilute limit) is approximately \(\omega_0-\gamma/2\), and that \(n'\) is minimal near \(\omega_0+\gamma/2\), so the anomalous-dispersion region has width \(\sim\gamma\).
    Solution In the dilute limit \(n'-1\simeq\tfrac12\mathrm{Re}\,\chi=\tfrac12\dfrac{\omega_p^2(\omega_0^2-\omega^2)}{(\omega_0^2-\omega^2)^2+\gamma^2\omega^2}\). Near resonance write \(\omega_0^2-\omega^2\approx2\omega_0(\omega_0-\omega)=2\omega_0\Delta\) with detuning \(\Delta=\omega_0-\omega\), and \(\gamma\omega\approx\gamma\omega_0\). Then \(n'-1\propto\dfrac{2\omega_0\Delta}{4\omega_0^2\Delta^2+\gamma^2\omega_0^2}=\dfrac{2\Delta}{\omega_0(4\Delta^2+\gamma^2)}\). Differentiate w.r.t. \(\Delta\) and set to zero: \(\dfrac{d}{d\Delta}\dfrac{\Delta}{4\Delta^2+\gamma^2}=\dfrac{(4\Delta^2+\gamma^2)-\Delta(8\Delta)}{(4\Delta^2+\gamma^2)^2}=0\Rightarrow\gamma^2-4\Delta^2=0\Rightarrow\Delta=\pm\gamma/2\). The maximum of \(n'\) is at \(\Delta=+\gamma/2\) i.e. \(\omega=\omega_0-\gamma/2\), the minimum at \(\omega=\omega_0+\gamma/2\); between them (\(|\omega-\omega_0|<\gamma/2\)) \(n'\) falls with \(\omega\) — anomalous dispersion of full width \(\gamma\).
  2. For the vapour of Worked Example 2, find the detuning \(\omega-\omega_0\) at which the absorption coefficient falls to half its line-centre value.
    Solution \(\mathrm{Im}\,\chi(\omega)=\dfrac{\omega_p^2\gamma\omega}{(\omega_0^2-\omega^2)^2+\gamma^2\omega^2}\). Near resonance, with \(\omega_0^2-\omega^2\approx2\omega_0(\omega_0-\omega)=-2\omega_0\delta\) (\(\delta=\omega-\omega_0\)) and \(\omega\approx\omega_0\): \(\mathrm{Im}\,\chi\approx\dfrac{\omega_p^2\gamma\omega_0}{4\omega_0^2\delta^2+\gamma^2\omega_0^2}=\dfrac{\omega_p^2\gamma/\omega_0}{4\delta^2+\gamma^2}\). This is a Lorentzian in \(\delta\) with HWHM where \(4\delta^2=\gamma^2\), i.e. \(\delta=\pm\gamma/2\). Numerically \(\gamma/2=5.0\times10^{8}\ \mathrm{rad\,s^{-1}}\), so the absorption is halved at \(\omega=\omega_0\pm5.0\times10^{8}\ \mathrm{rad\,s^{-1}}\) (full linewidth \(\gamma=1.0\times10^{9}\ \mathrm{rad\,s^{-1}}\), i.e. \(\Delta\nu=\gamma/2\pi\approx1.6\times10^{8}\ \mathrm{Hz}\)).
  3. A metal is modelled with \(\omega_0=0\), \(\gamma=0\). Show \(\varepsilon_r=1-\omega_p^2/\omega^2\) and find, for \(\omega_p=1.4\times10^{16}\ \mathrm{rad\,s^{-1}}\) (aluminium), the wavelength below which the metal becomes transparent.
    Solution With \(\omega_0=0,\ \gamma=0\): \(\varepsilon_r=1+\dfrac{\omega_p^2}{-\omega^2}=1-\dfrac{\omega_p^2}{\omega^2}\). For \(\omega<\omega_p\), \(\varepsilon_r<0\Rightarrow n\) purely imaginary \(\Rightarrow\) total reflection (evanescent). For \(\omega>\omega_p\), \(\varepsilon_r>0\Rightarrow n\) real \(\Rightarrow\) transparent. The onset is \(\omega=\omega_p\): \(\lambda_p=\dfrac{2\pi c}{\omega_p}=\dfrac{2\pi(3.0\times10^{8})}{1.4\times10^{16}}=1.35\times10^{-7}\ \mathrm{m}=135\ \mathrm{nm}\). Aluminium reflects visible/UV and becomes transparent in the vacuum-UV below \(\approx135\ \mathrm{nm}\).
  4. Estimate the low-frequency (static) refractive index of the solid in Worked Example 1 and compare with its value at \(500\ \mathrm{nm}\). Comment on the sign of the dispersion.
    Solution As \(\omega\to0\): \(\varepsilon_r\to1+\dfrac{\omega_p^2}{\omega_0^2}=1+\dfrac{3.18\times10^{31}}{2.25\times10^{32}}=1+0.1413=1.141\), so \(n'(0)=\sqrt{1.141}=1.068\). At \(500\ \mathrm{nm}\), \(n'=1.073\) (Example 1). Since \(n'(0)=1.068<n'(500)=1.073\), the index rises monotonically from DC toward the resonance: positive \(dn'/d\omega\) throughout the transparency window — normal dispersion, consistent with the medium sitting below its lowest resonance.
  5. Verify by direct differentiation that at \(\omega=\omega_0\) (dilute limit) the real part \(n'\) has zero slope through unity is false — instead show \(dn'/d\omega<0\) there, quantifying the anomalous slope in terms of \(\omega_p,\omega_0,\gamma\).
    Solution From Problem 1, near resonance \(n'-1\approx\dfrac{\omega_p^2}{2}\dfrac{2\Delta}{\omega_0(4\Delta^2+\gamma^2)}\) with \(\Delta=\omega_0-\omega\), so \(d\Delta/d\omega=-1\). At \(\omega=\omega_0\) (\(\Delta=0\)): \(n'-1\approx\dfrac{\omega_p^2\Delta}{\omega_0\gamma^2}\). Then \(\dfrac{dn'}{d\omega}=\dfrac{\omega_p^2}{\omega_0\gamma^2}\dfrac{d\Delta}{d\omega}=-\dfrac{\omega_p^2}{\omega_0\gamma^2}<0\). The slope is negative (anomalous), not zero, and is steepest for small \(\gamma\). For Worked Example 2: \(\dfrac{dn'}{d\omega}=-\dfrac{3.18\times10^{23}}{(4.0\times10^{15})(1.0\times10^{9})^2}=-\dfrac{3.18\times10^{23}}{4.0\times10^{33}}=-8.0\times10^{-11}\ \mathrm{s}\). Negative, confirming anomalous dispersion at line centre; note the value is large in magnitude per unit \(\omega\) because \(\gamma\) is small.