# How is "Band Intensity" related to absorption coefficient

I am interested in the linear absorption of $762\,\rm nm$ light near a transition of molecular oxygen. I need to find some experimental numbers that will tell us how far the $762\,\rm nm$ light will propagate before getting absorbed. Specifically, I want to know the e-folding length, $\gamma^{-1}$ (the length over which the intensity will drop by $e^{-1}$). I believe this is also called the optical depth when using Beer-Lambert law.

My main problem is that I do not know the definitions of experimentally measured quantities and how they relate to the e-folding length. I was reading "Atmospheric Propagation of Radiation" by Frederick Smith and page 61 says that for the inverse wavelength $\lambda^{-1}=13\,120.909\,\rm cm^{-1}$ the Band Intensity is $1.95\times10^{-22}\,\rm cm$. In "Laser Remote Chemical Analysis" they call it the integrated band intensity for this line but with units of cm-molecule (basically the same thing).

Does anyone know how the band intensity relates to the e-folding or absorption length?

Our best guess based on physical and dimensional arguments is that the e-folding length will go like $\gamma^{-1} \propto 1 / (B N \Delta\lambda)$ where $B$ is the band intensity with units of $\rm cm$, $N$ is the number density with units of $\rm cm^{-3}$, and $\Delta\lambda$ is the line width of the transition with units of $\rm cm$.

What the question refers to as "band intensity" is also referred to a "line strength" $S$. To calculate an absorption coefficient $k$ from $S$, a line shape function $f(\nu - \nu_0)$, where $\nu_0$ is the center of the line.
$$k = Sf(\nu - \nu_0)$$
Then "optical depth" = $ku$, where $u$ is called "path length" but is really a measure of the absorbing substance in the path.
• Well I thought that an absorption coefficient should have units like $\text{cm}^{-1}$. If $S$ has units of $\text{cm}$ as it does in the tables I have been looking at then $f(\nu-\nu_0)$ must be dimensionless. But I believe it is typical that line shape functions are defined such that $\int f(\nu-\nu_0) d\nu = 1$. This would suggest that $f(\nu-\nu_0)$ has units of $\text{cm}^{-1}$ or $\text{sec}$ (depending if you integrate over frequency or wavenumber). Any thoughts? Commented Apr 23, 2014 at 14:19
• @LasersMatter From table 5.2 of the Georgia Tech reference, you can see that absorption coefficient can have various units. In your question, the units for $S$ are $cm$ (like line 3 of the table), so the units for absorption coefficient would be $cm^2$ and the units of $f$ would need to be $cm$. Commented Apr 23, 2014 at 15:05
• So the "path length" $u=N L$ where $N$ is the number density of the absorbing molecules and $L$ is the path length of interest. This is assuming that the density is constant. Commented Apr 23, 2014 at 20:38