the mode structure of the cavity. We will discuss the value of ${\mathcal{F}}$ for our setup in section \[sec:CELIF\_characteristics\].

Similarly, we can derive the integrated CRD intensity, $I{^\text{CRD}}$, at the exit mirror taking into account that, following the initial single pass, light is only collected after every round trip: $$\begin{aligned}
I_0{^\text{CRD}}&= I_\text{L} T^2 {\mathcal{F}}\left( 1 - {\mathcal{L}}\right)\label{eq:ICRD0}\\
I_n{^\text{CRD}}&= I_0{^\text{CRD}}\sum_{j=0}^{n/2} \left[ \left(1-{\mathcal{L}}\right) R\right]^{2j} \nonumber \\
    &= I_0{^\text{CRD}}\frac{{1 - \left[(1 -{\mathcal{L}})^2 R^2\right]^{n+1}}}{1 - (1 -{\mathcal{L}})^2 R^2}\\
I{^\text{CRD}}&= \lim_{n\to\infty} I_n{^\text{CRD}}= \frac{I_0{^\text{CRD}}}{1 - (1 -{\mathcal{L}})^2 R^2}\label{eq:ICRD}\end{aligned}$$

In the limit of an empty cavity (${\mathcal{L}}= 0$) and $T=1-R$, the measured, time-integrated light intensity imparting on the detector is $$\label{eq:transmission}
I{^\text{CRD}}= I_\text{L} T{\mathcal{F}}/2$$ where $T{\mathcal{F}}/2$ is the cavity transmission function. The comparison of the integrated LIF and CRD intensities yields: $$\label{eq:ILIF_accurate}
I{^\text{LIF}}= I{^\text{CRD}}\frac{[1+(1-{\mathcal{L}})R](1-{\mathcal{L}}/2)}{T(1-{\mathcal{L}})}$$ In the limit of small loss ${\mathcal{L}}\ll 1$ and additionally $R \approx 1$, $I{^\text{LIF}}$ can be approximated as $$\label{eq:ILIF_approx}
I{^\text{LIF}}\approx I{^\text{CRD}}\frac{1+R}{T} \approx I{^\text{CRD}}\frac{2}{T}$$ As an example, for a relatively poor mirror reflectivity of $R = 0.998$ and a large loss of ${\mathcal{L}}= 0.001$, which would be amenable to a single-pass absorption measurement, the relative error of this approximation is smaller than $10^{-3}$. This error is dominated by the mirror reflectivity and reduces to 