Lambert's cosine law
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In optics, Lambert's cosine law says that the radiant intensity or luminous intensity observed from an ideal diffusely reflecting surface or ideal diffuse radiator is directly proportional to the cosine of the angle
A surface which obeys Lambert's law is said to be Lambertian, and exhibits Lambertian reflectance. Such a surface has the same radiance/luminance when viewed from any angle. This means, for example, that to the human eye it has the same apparent brightness. 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.
Lambertian scatterers and radiators
[edit]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. A Lambertian scatterer will then scatter this light according to the same cosine law as a Lambertian emitter. This means that although the radiance of the surface depends 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 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. The fact that it does not diminish illustrates that the moon is not a Lambertian scatterer, and in fact tends to scatter more light into the oblique angles than a Lambertian scatterer.
The emission of a Lambertian radiator does not depend on the amount of incident radiation, but rather from radiation originating in the emitting body itself. 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. A black body is an example of a Lambertian radiator.
Details of equal brightness effect
[edit]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 d
The length of each wedge is the product of the diameter of the circle and cos(
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 d
- photons/(s·m2·sr).
The observer at angle
- photons/(s·m2·sr),
which is the same as the normal observer.
Relating peak luminous intensity and luminous flux
[edit]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, , from the peak luminous intensity, , by integrating the cosine law: and so
where is the determinant of the Jacobian matrix for the unit sphere, and realizing that is luminous flux per steradian.[5] Similarly, the peak intensity will be of the total radiated luminous flux. For Lambertian surfaces, the same factor of relates luminance to luminous emittance, radiant intensity to radiant flux, and radiance to radiant emittance.[citation needed] Radians and steradians are, of course, dimensionless and so "rad" and "sr" are included only for clarity.
Example: A surface with a luminance of say 100 cd/m2 (= 100 nits, typical PC monitor) will, if it is a perfect Lambert emitter, have a luminous emittance of 100
See also
[edit]References
[edit]- ^ RCA Electro-Optics Handbook, p.18 ff
- ^ Modern Optical Engineering, Warren J. Smith, McGraw-Hill, p. 228, 256
- ^ Pedrotti & Pedrotti (1993). Introduction to Optics. Prentice Hall. ISBN 0135015456.
- ^ Lambert, Johann Heinrich (1760). Photometria, sive de mensura et gradibus luminis, colorum et umbrae. Eberhard Klett.
- ^ Incropera and DeWitt, Fundamentals of Heat and Mass Transfer, 5th ed., p.710.