# Linear Absorption & Scattering

So long as the laser beam is not too intense, the amount of light that is scattered or absorbed from the beam as it travels through air or any other transparent medium is proportional to the power of the beam. When this proportionality holds, the absorption or scattering is said to be linear. When the loss of light from the beam is linear, the following relation gives the power P as a function of the range R
P(R) = P0 EXP(-R/R0)
where P0 is the power of the beam as it leaves the laser and R0 is the attenuation length of light in the medium. The attenuation length depends on the scattering length Rs and the absorption length Ra
1/R0 = 1/Rs + 1/Ra.
In air, there is almost no absorption of visible light, and all losses are by scattering, whether from suspended aerosols (dust, pollen, lint, fog, smog, smoke, or clouds) or from the air itself. In clear dry air at sea level, the scattering length in meters from the air molecules is given by
Rs = 0.96×1030 m-3 L4
where L is the wavelength in meters. When aerosols are present, the scattering length is roughly independant of wavelength, and can be estimated by the distance it takes for something to appear noticably hazy. For non-visible wavelengths, the absorption by air can be important; in many cases so important that it makes that frequency useless as a weapon. The graph below shows the approximate attenuation of light in air. The data for the infrared, visible, and near UV is strongly dependant on atmospheric condition (humidity, aerosols, trace gasses); the graph gives only rough values for air at sea level (cobbled together from several different data sets, which occasionally contradict each other due to being taken under different atmospheric conditions).
The attenuation length in air is inversely proportional to the density of the air, so attenuation is less of an issue as the altitude increases or on alien planets with low pressure atmospheres. The attenuation in the infrared will depend strongly on the composition of the atmosphere, so exotic mixes such as ammonia, methane, and carbon dioxide will give quite different absorption lengths than what is shown on the graph above in the 1.0e-6 m to 1.5e-5 m range.

In water, all wavelengths except for visible wavelength are absorbed almost immediately, only blue or green light can travel farther than a few meters. The graph below gives the attenuation and absorption lengths of light from the far infrared to the gamma ray part of the spectrum. In the visible light region, the absorption is so small (or, equivalently, the absorption length so large) that the attenuation is dominated by impurities and suspended particulates and thus the attenuation of pure water in the visible region is somewhat uncertain. Two different data sets are shown to indicate the uncertainty, in almost any real world situation, the attenuation length of visible light will be smaller than what is shown. In the vacuum ultraviolet, there was a lack of reliable data, so the existing data was extrapolated with a discontinuity at the oxygen K-edge (absorption spectra often show such sharp features where certain electronic transitions become possible). Clearly, however, if you want your death ray to go through more than ten meters or so of water, you should have it lase at wavelengths between 0.4 microns to 0.5 microns (visible blue to visible green).

Whenever there is absorption, the medium the laser travels through will be heated. This can cause problems if it heats up too much. Water that boils will cause bubbles in the beam, scattering all the light away. Air that heats up will become less dense, so that it acts as a lens to defocus the beam (this is called thermal blooming). Small amounts of thermal blooming can be corrected by focusing the beam in front of the target, so the defocusing effect brings the focal point to the target. Very short pulses can also counteract these effects by passing through before the medium has time to respond - water cannot boil nor air expand fast enough to affect a nanosecond or shorter pulse. If you know the absorption length, you can find the power deposited in the medium per unit length Va

Va = P(R)/Ra.
For example, if a P = 1 megawatt beam going through water has Ra = 5 meters, the beam is depositing 1/5 megawatt/meter into the water.

When you have scattering, the beam will glow in the color of the light used by the laser - a blue beam will glow blue, a red beam will glow red, and an infrared beam will glow infrared (so you could not see it with the naked eye, but could detect it with special sensors). The power scattered per unit length Vs is

Vs = P(R)/Rs.
A 1000 W beam of 0.5 micron (5×10-7 m) green light traveling through clean, dry air at sea level with a scattering length of 60,000 meters will scatter 1/60th of a watt per meter. Compare this to fluorescent tubes, which emit roughly 10 watts per meter. The beam from this laser would be obvious at night, noticable indoors, but faint and easily overlooked in broad daylight. Under more realistic atmsopheric conditions, green light may scatter some 20 times as much, giving a scattered power of 1/3 of a watt per meter.

### References

[1] H. Kaplan, "Practical Applications of Infrared Thermal Sensing and Imaging Equipment," SPIE Optical Engineering Press, Bellingham.
[2] L. F. Fortson, J. W. Fowler, R. A. Ong, C. L. Pryke, http://nacho.princeton.edu/fowler/Papers/icrc99_corsika.ps.gz