and $T{\mathcal{F}}$ is the fraction of light that can be resonantly coupled into the cavity. $T$ is the transmission of the mirror and ${\mathcal{F}}$ is the fraction of the laser bandwidth that is resonant with the mode structure of the cavity.

In addition to the sample loss, a fraction of light leaks out of the cavity upon reflection at the mirror. The light intensity at the center of the cavity after $i$ single passes is $$I_i{^\text{LIF}}= I_0{^\text{LIF}}\left[ \left(1-{\mathcal{L}}\right) R\right]^i,$$ where $R$ is the reflectivity of the cavity mirrors. Using the summation rule for geometric series, the summed light intensity, $I_n{^\text{LIF}}$, that has crossed the LIF probe volume after $n$ single passes and the integrated intensity in the limit $n\to\infty$, $I{^\text{LIF}}$, are: $$\begin{aligned}
I_n{^\text{LIF}}&= I_0{^\text{LIF}}\sum_{i=0}^n \left[ \left(1-{\mathcal{L}}\right) R\right]^i \nonumber\\
    &= I_0{^\text{LIF}}\frac{{1 - \left[(1 -{\mathcal{L}}) R\right]^{n+1}}}{1 - (1 -{\mathcal{L}}) R}\\
\label{eq:ILIFint}
I{^\text{LIF}}&= \lim_{n\to\infty} I_n{^\text{LIF}}= \frac{I_0{^\text{LIF}}}{1 - (1 -{\mathcal{L}}) R}\end{aligned}$$ Using eq. \[eq:ILIF0\] and the approximations $\mathcal{L}\ll 1$ and $T\approx 1 - R$ shows that $I{^\text{LIF}}\approx I_\text{L}\mathcal{F}$. This means that, through the repeated use of the decaying light pulse inside the cavity, the amount of light that interacts with the sample is the amount incident on the entrance mirror that is resonant with 