Difference between lambertian and isotropic

Posted on May 5, 2007 By : Mizake Laziaf
What is the difference between lambertian and isotropic?

Both terms mean "the same in all directions" and are unfortunately sometimes used interchangeably.

Isotropic implies a spherical source that radiates the same in all directions, i.e., the intensity (W/sr) is the same in all directions. We often hear about an "isotropic point source." There can be no such thing; because the energy density would have to be infinite. But a small, uniform sphere comes very close. The best example is a globular tungsten lamp with a milky white diffuse envelope, as used in dressing room lighting. From our vantage point, a distant star can be considered an isotropic point source.

Lambertian refers to a flat radiating surface. It can be an active surface or a passive, reflective surface. Here the intensity falls off as the cosine of the observation angle with respect to the surface normal (Lambert's law). The radiance (W/m2-sr) is independent of direction. A good example is a surface painted with a good "matte" or "flat" white paint. If it is uniformly illuminated, like from the sun, it appears equally bright from whatever direction you view it. Note that the flat radiating surface can be an elemental area of a curved surface.

The ratio of the radiant exitance (W/m2) to the radiance (W/m2-sr) of a lambertian surface is a factor of p or (pi) and not 2p or 2pi . We integrate radiance over a hemisphere, and find that the presence of the factor of cos(q) or (cos teta) in the definition of radiance gives us this interesting result. It is not intuitive, as we know that there are 2p steradians in a hemisphere.

A lambertian sphere illuminated by a distant point source will display a radiance which is maximum at the surface where the local normal coincides with the incoming beam. The radiance will fall off with a cosine dependence to zero at the terminator. If the intensity (integrated radiance over area) is unity when viewing from the source, then the intensity when viewing from the side is 1/p . Think about this and consider whether or not our Moon is lambertian. I'll have more to say about this at a later date in another place!

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