Re: Laser distance-diffusion equation when related to output power?

When you talk about the divergence of the beam, what you mean is the 
increase of w per length unit from the laser, measured in mrad. In 
reality, this only applies at a longer distance from the laser, but 
generally the divergence is theta = lambda/ (pi * wo), where wo is the w-
radius at the point where the beam leaves the laser.

The example we will use is that of a 10mW (Io) He-Ne laser operating at a 
wavelength (lambda) of 633nm and having an wo of 0.50mm.  We will compute 
the value of the intensity (I) of the beam at 100m.(L)
	w(L) = Theta * L = lambda * L / (pi * wo) = 40mm
	A = pi * w(L)^2 = 0.0051 m^2
	I = Io / A = 1.96 W/m^2
In comparison, the intensity of Sunlight on the surface of the Earth is 
about 1000 W/m^2.

As far as computing the loss of the laser beam's intensity in relation to 
the material (fog, glass, etc.) that is passes through is far more 
complicated.  The intensity of the beam that makes it through the 
materials is given by:
	I = Io * (1-R)^2 * e^(-b*l)
Where, I is the intensity of the beam when it emerges from the material, 
Io is the intensity of the beam before it enters the material, R is the 
reflectivity of the material the beam is passing into, e is the 
exponential function, b is the absorption coefficient of the material the 
beam is passing through, and l is the thickness of the beams path from 
where it enters the material to where it exits the material.

Thus, knowing the reflectivity and absorption coefficient for the material 
in question will allow the beam's lose of intensity to be calculated for a 
given distance of travel.

Finally, in terms of the human eye's ability to see the laser beam after 
it has begun to diverge, in the wavelength region between 450nm and 900nm, 
the human eye is highly transparent (total absorption 5-10%) and tolerant 
to intensities of at least 200 mW/cm^2.

References:
1) Siegman, Lasers, Copyright 1986 by University Science Books.  

2) R. M. Rose, L. A. Shepard, and J. Wulff, The Structure and Properties 
of Materials, Vol. 4, Electronic Properties. Copyright 1966 by John Wiley 
& Sons, New York.