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TitleLambert's Cosine Law
TagsOptics Radiation Physical Sciences Quantity Light
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Total Pages4
Table of Contents
                            Lambert's cosine law
	Lambertian scatterers and radiators
	Details of equal brightness effect
	Relating peak luminous intensity and luminous flux
Document Text Contents
Page 1

Lambert's cosine law 1

Lambert's cosine law
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 observer's
line of sight and the surface normal.[1][2] The law is also known as the cosine emission law or Lambert's emission
law. It is named after Johann Heinrich Lambert, from his Photometria, published in 1760.
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.

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

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.

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.

Page 2

Lambert's cosine law 2

Figure 2: Observed intensity (photons/(s·cm2·sr))
for a normal and off-normal observer; dA

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.

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


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


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:

and so

where is the determinant of the Jacobian matrix for the unit sphere, and realizing that is luminous flux
per steradian.[3] 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.

Page 3

Lambert's cosine law 3

Lambert's cosine law in its reversed form (Lambertian reflection) implies that the apparent brightness of a
Lambertian surface is proportional to cosine of the angle between the surface normal and the direction of the incident
This phenomenon is among others used when creating moldings, which are a means of applying light and dark
shaded stripes to 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. Moldings are strips of material with various cross sections used to cover
transitions between surfaces or for decoration.

[1][1] RCA Electro-Optics Handbook, p.18 ff
[2][2] Modern Optical Engineering, Warren J. Smith, McGraw-Hill, p.228, 256
[3] Incropera and DeWitt, Fundamentals of Heat and Mass Transfer, 5th ed., p.710.

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