Lorentz oscillator model

Remarks

The steady state is $x = \mathrm{Re}(x_0 e^{-i\omega t})$ with $m(\omega_0^2 - \omega^2 - i\gamma\omega) x_0 = -e E_0$, and $P = -N e x$ gives $\chi$. For $\gamma > 0$ the poles of $\chi$ are $\omega = -i\gamma/2 \pm \sqrt{\omega_0^2 - \gamma^2/4}$, in the lower half-plane, so $\chi$ is analytic in the upper half-plane and satisfies the Kramers–Kronig relations. Several kinds of oscillators give a sum of such terms.

Example: with $\gamma = 0$ and $0 \le \omega < \omega_0$, $\chi(\omega) = \omega_p^2 / (\omega_0^2 - \omega^2)$ is positive and increasing in $\omega$, so the refractive index $\sqrt{\varepsilon_r}$ grows with frequency: this is normal dispersion, as in glass for visible light.

Non-example: a constant susceptibility $\chi \neq 0$ is not of Lorentz form, since every Lorentz susceptibility tends to $0$ as $\omega \to \infty$.

The case $\omega_0 = 0$, free electrons, gives the optical response of the Drude model.