A distance-based formula Rayleigh-scattered light....

[fpsulli3]

Greetings,

I am trying to learn more about Rayleigh scattering. I'd like to be able to calculate the amount of light scattered away from a beam as it travels through a volume of gas. So, I checked out the Wikipedia article:

http://en.wikipedia.org/wiki/Raleigh_scattering

I am not sure I fully understand the concept of a scattering cross-section, but it seems to be something along the lines of "the cross-sectional area of a solid sphere with the same probability of scattering as the particles being considered." The following excerpt gives me a clue on how to use this cross-section quantity:

The fraction of light scattered by a group of scattering particles is the number of particles per unit volume N times the cross-section. For example, the major constituent of the atmosphere, nitrogen, has a Rayleigh cross section of 5.1×10⁻³¹ m² at a wavelength of 532 nm (green light). This means that at atmospheric pressure, where there are about 2×10²⁵ molecules per cubic meter, about a fraction 10⁻⁵ of the light will be scattered for every meter of travel.​

The way this is worded makes it seem like the attenuation is linear. However, intuitively it seems like the attenuation ought to be exponential, for if a particle scatters away 10% of a beam, the beam must travel on with 90% of its original intensity, and then the next particle would scatter away 10% of that, or 9% of the original intensity (I know these numbers are absurdly big, but they're easier to work with).

I suppose the above paragraph could be interpreted as "10⁻⁵ of the light will be scattered by the first meter, and then 10⁻⁵ of the remaining light by the next meter, and so on" but it really doesn't seem like they're saying that. It really just looks like they multiplied the scattering cross-section times the number of molecules in the volume that the light is passing through without computing any sort of integral.

Hyperphysics seems to say the same thing (in the Rayleigh Scattering section):

http://hyperphysics.phy-astr.gsu.edu/hbase/atmos/blusky.html

Their equations for the intensity of the scattered light is linearly proportional to the number of scatterers.

Perhaps I just have the wrong picture in my head. However, if I am a Nitrogen molecule, and I'm hit by a laser beam that has already been attenuated down to 5% by traveling through a large volume of air, it seems like I'm going to scatter much less light than the first Nitrogen molecules that the beam hit when it was at 100% intensity.

 

[Chestermiller]

What it means is that x % of the remaining beam is scattered per meter. So, it's really differential. Here is a reference that may help you model radiation scattering:

Miller, C., Meakin, P., Franks, R.G.E., and Jesson, J.P., The Fluorocarbon-Ozone Theory – V. One Dimensional Modeling of the Atmosphere: The Base Case, Atmospheric Environment, 12, 2481-2500 (1978)

See the Appendix for how to do the calculation.

 

[fpsulli3]

Thank you!