We divide the path of the laser beam into 1mm cubes.
There is a probability, p, that a photon will encounter a colloid particle and be reflected in some other direction.
Thus, the probability that the photon will make it to the next cube in line is 1 - p.
The probability that a photon will make it through a line of n cubes is (1-p)n
We will assume that a photon colliding with a colloid particle can be reflected in any direction, ie. that the collision is an omnidirectional radiator. (That is at odds with how light scattering in colloidal dispersions is conventionally described, especially for small particle sizes. Unless I've made an error, it seems to work as a first approximation. It doesn't work for a detailed calculation of signal strength. For that, you have to use MIE scattering.)
The probability that the photon will be refleted toward the aperture of the camera is small. It's a function of the area of the aperture and the distance between the camera and the reflecting colloid particle. It will be treated as a constant and ignored for the time being.
|Note that each probability has a maximum value. In this case it is around 225.|
|We find a maximum DN of 246 for p = 0.056. We normalize the various probability curves such that their maxima lie on this curve.|