Lambert's Cosine Law - Details of Equal Brightness Effect

Details of Equal Brightness Effect

The situation for a Lambertian surface (emitting or scattering) is illustrated in Figures 1 and 2. For conceptual clarity we will think in terms of photons rather than energy or luminous energy. The wedges in the circle each represent an equal angle dΩ, and for a Lambertian surface, the number of photons per second emitted into each wedge is proportional to the area of the wedge.

It can be seen that the length of each wedge is the product of the diameter of the circle and 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°. In mathematical terms, the radiance along the normal is I photons/(s·cm2·sr) and the number of photons per second emitted into the vertical wedge is I dΩ dA. The number of photons per second emitted into the wedge at angle θ is I cos(θ) dΩ 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 dΩ₀. We can assume without loss of generality that the aperture happens to subtend solid angle dΩ when "viewed" from the emitting area element. This normal observer will then be recording I dΩ dA photons per second and so will be measuring a radiance of

$I_0=\frac{I\, d\Omega\, dA}{d\Omega_0\, dA_0}$ photons/(s·cm2·sr).

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 dΩ₀ cos(θ). This observer will be recording I cos(θ) dΩ dA photons per second, and so will be measuring a radiance of

$I_0=\frac{I \cos(\theta)\, d\Omega\, dA}{d\Omega_0\, \cos(\theta)\, dA_0} =\frac{I\, d\Omega\, dA}{d\Omega_0\, dA_0}$ photons/(s·cm2·sr),

which is the same as the normal observer.

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