Wednesday, February 22, 2012

Lambert Radiators

We derived some of the radiometric laws by assuming a uniformly radiating point source and considering a flux tube linking it to a illuminated surface. To handle an extended radiating surface we can divide the surface into differential elements which will approximate a point source. A complication is that we have to use Lambert's cosine law in order to determine the projected areas at both ends of the flux tube since the surface areas of the source and the illuminated object are not necessarily normal to the connecting "ray." To find the total flux emitted by the surface element we have to sum or integrate over the hemisphere exterior to the surface. We can divide the hemisphere that receives the flux into elements of solid angle

dΩ = sin(θ)dθdφ.

The sides of the element of solid angle are dθ and sin(θ)dφ. For the flux in the direction of angle θ relative to the surface normal we have,

dΦ = L cos(θ) dAdΩ

where L is the Luminance. The total flux for the hemisphere is,

The assumptions we have made are those of geometrical optics which just considers the rays involved in the illumination process. The more general case which treats light as waves and takes interference into account is physical optics. It is used by astronomical optical interferometry to study the emission from stellar sources.

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