Lambert's cosine law
From Wikipedia, the free encyclopedia
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 θ
between the direction of the incident light and the surface normal.[1][2] The law is also known as the cosine
emission law[3] or Lambert's emission law. It is named after Johann Heinrich Lambert, from his
Photometria, published in 1760.[4]
A surface which obeys Lambert's law is said to be Lambertian, and exhibits Lambertian reflectance. Such a
surface has the same radiance when viewed from any angle. This means, for example, that to the human eye
it has the same apparent brightness (or luminance). 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 apparent size (solid
angle) of the observed area, as seen by a viewer, is decreased by a corresponding amount. Therefore, its
radiance (power per unit solid angle per unit projected source area) is the same.
Contents
1 Lambertian scatterers and radiators
2 Details of equal brightness effect
3 Relating peak luminous intensity and luminous flux
4 Uses
5 See also
6 References
Lambertian scatterers and radiators
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 would a Lambertian scatterer.
The emission of a Lambertian radiator does not depend upon 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
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
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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.
Figure 2: Observed intensity (photons/(s·cm2·sr))
for a normal and off-normal observer; dA0 is the
area of the observing aperture and dΩ is the solid
angle subtended by the aperture from the viewpoint
of the emitting area element.
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
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
photons/(s·cm2·sr),
which is the same as the normal observer.
Relating peak luminous intensity and luminous flux
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:
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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. 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 314 lm/m2. If its area is 0.1 m2 (~19" monitor) then the total
light emitted, or luminous flux, would thus be 31.4 lm.
Uses
Lambert's cosine law in its reversed form (Lambertian reflection) implies that the apparent brightness of a
Lambertian surface is proportional to the cosine of the angle between the surface normal and the direction of
the incident light.
This phenomenon is, among others, used when creating mouldings, with the effect of creating light- and
dark-shaded stripes on a structure or object without having to change the material or apply pigment. The
contrast of dark and light areas gives definition to the object. Mouldings are strips of material with various
cross-sections used to cover transitions between surfaces or for decoration.
See also
Transmittance
Reflectivity
Passive solar building design
Sun path
References
RCA Electro-Optics Handbook, p.18 ff1.
Modern Optical Engineering, Warren J. Smith, McGraw-Hill, p.228, 2562.
Pedrotti & Pedrotti (1993). Introduction to Optics. Prentice Hall. ISBN 0135015456.3.
Lambert, J H (1760). Photometria, sive de Mensura et gradibus luminis, colorum et umbrae. (https://archive.org
/details/lambertsphotome00lambgoog)
4.
Incropera and DeWitt, Fundamentals of Heat and Mass Transfer, 5th ed., p.710.5.
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