What is Lambert's Cosine Law?

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- GoPhotonics

Nov 21, 2022

Lambert’s cosine law states that the radiant intensity from the ideal diffusely reflecting surface and cosine of the angle θ between the direction of incident light and surface normal are directly proportional. Mathematically,

E (illumination from a surface) ∝ Cos Ɵ

When a surface is radiating as a result of being illuminated by an external source, the irradiance (energy or photons radiated per unit time per unit area) landing on that surface will be proportional to the cosine of the angle between the illuminating source and the normal. This phenomenon has been proposed by Lambert and hence named Lambert’s cosine law. Diffuse reflection can be defined as the type of reflection of light or an incident ray where scattering happens at many angles and not just at one angle as shown in the figure below.

This law is named after Johann Heinrich Lambert and was formulated in 1760. It is also known as Lambert’s emission law or cosine emission law. A surface that obeys Lambert's law is said to be Lambertian and exhibits Lambertian reflectance. Lambertian reflectance is defined as that property of substances due to which they appear equally bright when from any angle. E.g., the human eye has the same apparent 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 the surface visible to the viewer, is also reduced by the very same amount. Because the ratio between power and solid angle is constant, radiance stays the same. This law is typically used in radiometry and computer graphics.

Derivation of Lambert's cosine law

Case I – Consider a surface that is normal to the luminous flux as shown in the figure above. Then, as per Lambert's cosine law, we get, Illumination on the surface A is given by,

Case II – Consider a surface inclined to the flux in such a way that the normal to the surface makes an angle (θ) to the flux axis, as shown below.

Then, by Lambert's cosine law, we get,

Case III – Consider a point 'P' on a plane surface and the distance between the source of light (S) and the point 'P' is 'r' meters. The source (S) is located at a height of 'h' meters from the surface and its luminous intensity is 'I' candle power as shown in figure below.

Then, by the cosine law of illumination, we get, the illumination at point 'P' as,

Therefore, the illumination at point 'P' is obtained by replacing the value of r as

Where,is the illumination at any point located directly below the source of light.