Lorentz and Drude Models

Bound and free charges in a light field

A material responds to light because its charges move. This page models a group of charges as a mass on a spring, pushed by the electric field and slowed by friction. That picture gives a complex permittivity $\varepsilon(\omega)$ and, from it, the refractive index $n$ and extinction coefficient $k$ at every frequency. The editor below computes it for one oscillator. The thin-film and layer-stack experiments can take the result as a material.

A charge on a spring

Take an electron with charge $-e$ and mass $m$, held near its atom by a restoring force with natural frequency $\omega_0$ and damped at rate $\gamma$. A field $E_0e^{-i\omega t}$ drives it:

After the start-up transient has damped away, the electron oscillates at the drive frequency. With $N$ such electrons per unit volume, the polarization $P=-Nex$ is proportional to the field, $P=\varepsilon_0\chi E$, with

The relative permittivity is $\varepsilon=1+\chi$; when faster responses are lumped into a constant background it becomes $\varepsilon_\infty+\chi$. The numerator is the strength $S$ in the model section, which is why it has units of $\mathrm{s^{-2}}$. Far below resonance the charge follows the field, and $\chi$ is real and positive. Far above it the mass cannot keep up; the displacement ends up opposite to the force, the polarization points against the field and $\chi$ turns negative. At $\omega=\omega_0$ only damping limits the motion, and the response lags the drive by a quarter cycle. Then the velocity is in phase with the force, so the field does work on the charge every cycle. That work, lost to damping, is the absorption.

Reading the curves

The real part $\varepsilon_1$ is the part of the response in step with the field. The imaginary part $\varepsilon_2$ is the part a quarter cycle behind, which takes energy from it; with the $\exp(-i\omega t)$ convention it is positive in a passive material, one that can absorb energy but not supply it. The wave itself is described by $n$ and $k$, and each depends on both parts. For one oscillator the absorption peak has height about $S/(\gamma\omega_0)$ and, when $\gamma\ll\omega_0$, a full width at half maximum close to $\gamma$. Less damping gives a taller, narrower peak, but the weighted area $\int_0^\infty\omega\,\varepsilon_2\,d\omega=\pi S/2$ (not quite the area under the plotted curve) does not change.

The index follows from $n+ik=\sqrt{\varepsilon}$, so $\varepsilon_1=n^2-k^2$ and $\varepsilon_2=2nk$. Wherever $\varepsilon_1$ is negative, $k$ exceeds $n$: the field loses more than a factor $e$ for each radian of phase it gains, so it fades within a fraction of a wavelength.

Removing the spring

Conduction electrons in a metal have no restoring force. Setting $\omega_0=0$ turns the same formula into the Drude permittivity

The strength is now the square of the plasma frequency $\omega_p$, and $1/\gamma$ is the mean time between collisions. In general $\varepsilon_1$ crosses zero at $\omega^2=\omega_p^2/\varepsilon_\infty-\gamma^2$. Below that the material behaves like a metal. Above it the electrons' response is too weak to outweigh the background, $\varepsilon_1$ turns positive, and light propagates with a loss set by $\gamma$.

Three things to try

The bound oscillator. The default example has $\varepsilon_\infty=1$, $\omega_0=3\times10^{15}$ rad/s (477 THz, or 628 nm), $S=9\times10^{30}\,\mathrm{s^{-2}}$ and $\gamma=3\times10^{14}$ rad/s. At low frequency $\varepsilon=1+S/\omega_0^2=2$. At resonance $\varepsilon=1+10i$, and the driven-response panel reports amplitude 10 with a 90° lag. That panel always evaluates $\chi$, without the background, at $3\times10^{15}$ rad/s, so it shows the resonant case only while $\omega_0$ stays at that value. $\varepsilon_1$ rises to 6.26 at 453 THz, drops to −3.76 at 501 THz, and stays negative up to 672 THz. At the top of the chart, 955 THz, $n=0.82$. That is a phase velocity above $c$; no signal travels faster than light.

Damping. Double the damping from 0.3 to 0.6 × 10¹⁵ rad/s. The $\varepsilon_2$ peak halves from 10 to 5 and its width doubles. At the minimum damping, 0.01, the peak reaches 300, but it is narrower than the spacing of the chart's 401 samples, so the plotted curve tops out near 129.

Free carriers. The Drude example keeps the same strength and damping but sets $\omega_0=0$, so $\omega_p=3\times10^{15}$ rad/s and the collision time is about 3.3 fs. $\varepsilon_1$ crosses zero at $2.985\times10^{15}$ rad/s (475 THz). At 159 THz, $\varepsilon=-7.26+2.48i$, with $n=0.45$ and $k=2.73$. The driven response at $3\times10^{15}$ rad/s has amplitude 0.995 and a 174.29° lag: the polarization points almost exactly against the field.