achieved. With cavity lengths of the order of 1m the effective pathlengths are of the order of many kilometers. The absorption of light by a sample inside the cavity causes an additional loss and consequently a shorter ring-down time. The reduced ring-down time, $\tau$, is directly linked to the absorption coefficient $$\label{eq:alpha_CRD}
  \alpha = \sigma \rho = \frac{1}{c} \left(\frac{1}{\tau} - \frac{1}{\tau_0}\right),$$ where $\sigma$ is the absorption cross section and $\rho$ is the number density of the sample, leading to a photon loss per pass of $\alpha d$. As the ring-down time is the measurable, the technique is immune to power fluctuations of the incident laser source leading to an absolute and self-calibrated measurement.

Eq. \[eq:alpha\_CRD\] assumes that the entire cavity is filled by the sample which is the case in a typical cell experiment. If the sample volume is localized, *e.g.* inside a molecular beam, then the right-hand side of eq. \[eq:alpha\_CRD\] needs to be multiplied by the ratio of the cavity and sample lengths, $d/s$, to account for the higher density in the smaller sample volume.

In cavity ring-down spectroscopy, the absorbance is measured over the pathlength through the sample (integrated column density) and, in contrast to LIF, is 