Lambert's cosine law

From Academic Kids

Lambert's cosine law is the statement that the total power observed from a "Lambertian" surface is directly proportional to the cosine of the angle θ made by the observer's line of sight and the line normal to the surface. The law is also known as the cosine emission law or Lambert's emission law. It is named after Johann Heinrich Lambert, from his Photometria, published in 1760.

An important consequence of Lambert's cosine law is that when an area element on the surface is viewed from any angle, it has the same apparent brightness. This is because although the emitted intensity from an area element is reduced by the cosine of the emission angle, the observed size (solid angle) of the area element is also reduced by that same amount, so that while the area element appears smaller, its brightness is the same. For example, in the visible spectrum, the Sun is almost a Lambertian radiator, and as a result the brightness of the Sun is almost the same everywhere on an image of the solar disk. Also, a perfect black body is a perfect Lambertian radiator.

Lambertian reflectors

When an area element is radiating as a result of being illuminated by an external source, the flux (energy/time/area or photons/time/area) landing on that area element will be proportional to the cosine of the angle between the illuminating source and the normal. A Lambertian reflector will then reflect this light according to the same cosine law as a Lambertian emitter. This will mean that, although the apparent brightness of the surface will depend on the angle from the normal to the illuminating source, it will not depend on the angle from the normal to the observer. For example, if the moon were a Lambertian reflector, one would expect to see its reflected brightness appreciably diminish towards the outer edge, or limb. The fact that it does not illustrates that the moon is not a Lambertian reflector, and in fact tends to reflect more light into the oblique angles than a Lambertian reflector.

Explanation of equal brightness effect

Missing image
LambertCosineLaw1.png
Figure 1: Emission rate (photons/s) in a normal and off-normal direction. The number of photons/sec directed into any wedge is proportional to the area of the wedge.
Missing image
LambertCosineLaw2.png
Figure 2: Observed intensity (photons/(scm2sr)) for a normal and off-normal observer; dA0 is the area of the observing aperture and is the solid angle subtended by the aperture from the viewpoint of the emitting area element.

This situation is illustrated in Figures 1 and 2, which illustrate the problem for two dimensions. For conceptual clarity we will think in terms of photons rather than energy. The wedges in the circle each represent an equal angle and the number of photons per second emitted into each wedge is proportional to the area of the wedge.

It can be seen that the height of each wedge is the diameter of the circle times cos(θ). It can also be seen that the maximum rate of photon emission per unit solid angle is along the normal and diminishes to zero for θ=90 degrees. In mathematical terms, the intensity along the normal is I photons/(scm2sr) and the number of photons per second emitted into the vertical wedge is I dA. The number of photons per second emitted into the wedge at angle θ is I cos(θ) dA.

Figure 2 represents what an observer sees. The observer directly above the area element will be seeing the scene through an aperture of area dA0 and the area element dA will subtend a (solid) angle of 0. We can assume without loss of generality that the aperture happens to subtend solid angle when "viewed" from the emitting area element. This normal observer will then be recording I  dA photons per second and so will be measuring an intensity of

<math>

I_0=\frac{I d\Omega dA}{d\Omega_0 dA_0} <math> photons/(scm2sr). The observer at angle θ to the normal will be seeing the scene through the same aperture of area dA0 and the area element dA will subtend a (solid) angle of 0 cos(θ). This observer will be recording I cos(θ)  dA photons per second, and so will be measuring an intensity of

<math>

I_0=\frac{I \cos(\theta) d\Omega dA}{d\Omega_0 \cos(\theta) dA_0} =\frac{I d\Omega dA}{d\Omega_0 dA_0} <math> photons/(scm2sr), which is the same as the normal observer.

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