Lambert's cosine law

In optics, Lambert's cosine law says that the observed radiant intensity or luminous intensity from an ideal diffusely reflecting surface or ideal diffuse radiator is directly proportional to the cosine of the angle θ between the observer's line of sight and the surface normal; I = I0 cos θ.

It is named after Johann Heinrich Lambert, from his Photometria, published in 1760.

It has the same radiance because, although the emitted power from a given area element is reduced by the cosine of the emission angle, the solid angle, subtended by surface visible to the viewer, is reduced by the very same amount.

Because the ratio between power and solid angle is constant, radiance (power per unit solid angle per unit projected source area) stays the same.

When an area element is radiating as a result of being illuminated by an external source, the irradiance (energy 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.

For example, if the moon were a Lambertian scatterer, one would expect to see its scattered brightness appreciably diminish towards the terminator due to the increased angle at which sunlight hit the surface.

For example, if the sun were a Lambertian radiator, one would expect to see a constant brightness across the entire solar disc.

The fact that the sun exhibits limb darkening in the visible region illustrates that it is not a Lambertian radiator.

The situation for a Lambertian surface (emitting or scattering) is illustrated in Figures 1 and 2.

The length of each wedge is the product of the diameter of the circle and cos(θ).

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·m2·sr) and the number of photons per second emitted into the vertical wedge is I dΩ dA.

Since the wedge size dΩ was chosen arbitrarily, for convenience we may assume without loss of generality that it coincides with the solid angle subtended by the aperture when "viewed" from the locus of the emitting area element dA.

Thus the normal observer will then be recording the same I dΩ dA photons per second emission derived above and will measure a radiance of The observer at angle θ to the normal will be seeing the scene through the same aperture of area dA0 (still corresponding to a dΩ wedge) and from this oblique vantage the area element dA is foreshortened and will subtend a (solid) angle of dΩ0 cos(θ).

In general, the luminous intensity of a point on a surface varies by direction; for a Lambertian surface, that distribution is defined by the cosine law, with peak luminous intensity in the normal direction.

Thus when the Lambertian assumption holds, we can calculate the total luminous flux,

[citation needed] Radians and steradians are, of course, dimensionless and so "rad" and "sr" are included only for clarity.

If its area is 0.1 m2 (~19" monitor) then the total light emitted, or luminous flux, would thus be 31.4 lm.