Lambert Beer

The main relationship for modeling light extinction across a medium is Lambert-Beer law, also known as Bouguer law. It behaves reasonably well for low-concentration media, like gases and low turbidity water.

Its derivation is quite easy: x axis being along light direction, let us define an infinitesimal layer of medium of length dx and surface S. The intensity (or irradiance, W m~ ) of light I is diminished passing through the layer by the quantity dl proportional to dx dl = - k ■ I ■ dx where k (m_1) is called extinction coefficient and can be empirically determined. It is very important to notice that k is dependent on medium characteristics, position in the medium (if heterogeneous), direction, and (most important) wavelength.

Integrating over a finite given layer L to account for extinction in all the thin layers and assuming homogeneous extinction properties, it is possible to compute the intensity of the light flux leaving the medium:

where Iin and Iout are the intensities of light fluxes entering and quitting the layer respectively (e.g., flux entering atmosphere and reaching ground). The ratio Iout/Iin is called transmittance. The quantity k ? L is called optical depth and is a measure of the ability of the finite layer to block light.

In stratified heterogeneous (with respect to light extinction) systems, in order to obtain precise results, sometimes it can be useful to apply this relationship separately to different layers, using different k values according to layer properties (Figure 1).

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