Exponential Attenuation

Consider attenuation by absorption of a monodirectional beam of monochromatic radiation by an optically homogeneous medium. By optically homogeneous is meant homogeneous on the scale of the wavelength; no medium is absolutely homogeneous (continuous). By monodirectional is meant that the radiation is confined to a narrow set of directions, and by monochromatic is meant that it is confined to a narrow set of wavelengths. Again, a truly monodirectional, monochromatic beam does not exist. Even a laser beam has a finite angular divergence and range of wavelengths.

The beam is directed along the x-axis and its irradiance, radiant energy crossing unit area per unit time (Sec. 4.2) at x = 0 is F0. We can imagine the region from 0 to an arbitrary distance x to be subdivided into N identical slices each of thickness Ax = x/N (Fig. 2.1). Upon transmission from 0 to Ax the irradiance of the incident beam is reduced (attenuated) from F0 to F because of absorption within Ax. It is reasonable to assume that if Ax is sufficiently small attenuation is proportional to Ax and to F0, where the proportionality constant (absorption coefficient) is denoted by k:

Fundamentals of Atmospheric Radiation: An Introduction with 400 Problems. Craig F. Bohren and Eugene E. Clothiaux Copyright © 2006 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 3-527-40503-8

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