= Collisional phase relaxation and Lorentzian linewidth
{title2=$g^{(1)}(\tau)=e^{-|\tau|/\tau_c}e^{-i\omega_0\tau}$}
If phase-randomizing collisions form a Poisson process of mean interval $\tau_c$, field correlation survives only when no such collision occurs during the delay. Thus its magnitude decays as $e^{-|\tau|/\tau_c}$. The <Wiener-Khinchin theorem> gives the normalized <Lorentzian line shape>
$$
L(\omega)=\frac1\pi\frac{1/\tau_c}{(\omega-\omega_0)^2+(1/\tau_c)^2}.
$$
Its <full width at half maximum> is $2/\tau_c$ in <angular frequency> or $1/(\pi\tau_c)$ in ordinary <frequency>. A population lifetime is a different time constant: lifetime-limited amplitude decay at half the population rate introduces the familiar factor of two. The definition of the supplied lifetime must therefore accompany any numerical linewidth.
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