In
opticsOptics is the branch of physics which involves the behavior and properties of light, including its interactions with matter and the construction of instruments that use or detect it. Optics usually describes the behavior of visible, ultraviolet, and infrared light...
,
Lambert's cosine law says that the
radiant intensityIn radiometry, radiant intensity is a measure of the intensity of electromagnetic radiation. It is defined as power per unit solid angle. The SI unit of radiant intensity is watts per steradian . Radiant intensity is distinct from irradiance and radiant exitance, which are often called intensity...
observed from a
Lambertian surfaceIf a surface exhibits Lambertian reflectance, light falling on it is scattered such that the apparent brightness of the surface to an observer is the same regardless of the observer's angle of view. More technically, the surface luminance is isotropic...
or a Lambertian radiator is directly proportional to the cosine of the angle θ between the observer's line of sight and the
surface normalA surface normal, or simply normal, to a flat surface is a vector that is perpendicular to that surface. A normal to a nonflat surface at a point P on the surface is a vector perpendicular to the tangent plane to that surface at P. The word "normal" is also used as an adjective: a line normal to a...
. A Lambertian surface is also known as an ideal
diffusely reflectingDiffuse reflection is the reflection of light from a surface such that an incident ray is reflected at many angles rather than at just one angle as in the case of specular reflection...
surface. The law is also known as the
cosine emission law or
Lambert's emission law. It is named after
Johann Heinrich LambertJohann Heinrich Lambert was a Swiss mathematician, physicist, philosopher and astronomer.Asteroid 187 Lamberta was named in his honour.Biography:...
, from his
Photometria, published in 1760.
An important consequence of Lambert's cosine law is that when a Lambertian surface is viewed from any angle, it has the same
radianceRadiance and spectral radiance are radiometric measures that describe the amount of radiation such as light or radiant heat that passes through or is emitted from a particular area, and falls within a given solid angle in a specified direction. They are used to characterize both emission from...
. This means, for example, that to the human eye it has the same apparent brightness (or
luminanceLuminance is a photometric measure of the luminous intensity per unit area of light travelling in a given direction. It describes the amount of light that passes through or is emitted from a particular area, and falls within a given solid angle. The SI unit for luminance is candela per square...
). 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
irradianceIrradiance is the power of electromagnetic radiation per unit area incident on a surface. Radiant emittance or radiant exitance is the power per unit area radiated by a surface. The SI units for all of these quantities are watts per square meter , while the cgs units are ergs per square centimeter...
(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
moonThe Moon is Earth's only known natural satellite,There are a number of nearEarth asteroids including 3753 Cruithne that are coorbital with Earth: their orbits bring them close to Earth for periods of time but then alter in the long term . These are quasisatellites and not true moons. For more...
were a Lambertian scatterer, one would expect to see its scattered brightness appreciably diminish towards the
terminatorA terminator, twilight zone or "grey line" is a moving line that separates the illuminated day side and the dark night side of a planetary body...
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 angleOblique angle can refer to:*An angle which is not a multiple of 90°*Another word for "Dutch angle" in cinematography...
s 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
sunThe Sun is the star at the center of the Solar System. It is almost perfectly spherical and consists of hot plasma interwoven with magnetic fields...
were a Lambertian radiator, one would expect to see a constant brightness across the entire solar disc. The fact that the sun exhibits
limb darkeningLimb darkening refers to the diminishing of intensity in the image of a star as one moves from the center of the image to the edge or "limb" of the image...
in the visible region illustrates that it is not a Lambertian radiator. A
black bodyA black body is an idealized physical body that absorbs all incident electromagnetic radiation. Because of this perfect absorptivity at all wavelengths, a black body is also the best possible emitter of thermal radiation, which it radiates incandescently in a characteristic, continuous spectrum...
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
photonIn physics, a photon is an elementary particle, the quantum of the electromagnetic interaction and the basic unit of light and all other forms of electromagnetic radiation. It is also the force carrier for the electromagnetic force...
s rather than
energyIn physics, energy is an indirectly observed quantity. It is often understood as the ability a physical system has to do work on other physical systems...
or
luminous energyIn photometry, luminous energy is the perceived energy of light. This is sometimes also called the quantity of light.Luminous energy is not the same as the radiant energy, the corresponding objective physical quantity. This is because the human eye can only see light in the visible spectrum and has...
. The wedges in the
circleA circle is a simple shape of Euclidean geometry consisting of those points in a plane that are a given distance from a given point, the centre. The distance between any of the points and the centre is called the radius....
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
diameterIn geometry, a diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints are on the circle. The diameters are the longest chords of the circle...
of the circle and cos(
θ). It can also be seen that the maximum rate of photon emission per unit
solid angleThe solid angle, Ω, is the twodimensional angle in threedimensional space that an object subtends at a point. It is a measure of how large that object appears to an observer looking from that point...
is along the normal and diminishes to zero for
θ = 90°. In mathematical terms, the
radianceRadiance and spectral radiance are radiometric measures that describe the amount of radiation such as light or radiant heat that passes through or is emitted from a particular area, and falls within a given solid angle in a specified direction. They are used to characterize both emission from...
along the normal is
I photons/(s·cm
^{2}·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
dA_{0} and the area element
dA will subtend a (solid) angle of
dΩ_{0}. 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·cm
^{2}·sr).
The observer at angle
θ to the normal will be seeing the scene through the same aperture of area
dA_{0} and the area element
dA will subtend a (solid) angle of
dΩ_{0} 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·cm
^{2}·sr),
which is the same as the normal observer.
Relating peak luminous intensity and luminous flux
In general, the
luminous intensityIn photometry, luminous intensity is a measure of the wavelengthweighted power emitted by a light source in a particular direction per unit solid angle, based on the luminosity function, a standardized model of the sensitivity of the human eye...
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 fluxIn photometry, luminous flux or luminous power is the measure of the perceived power of light. It differs from radiant flux, the measure of the total power of light emitted, in that luminous flux is adjusted to reflect the varying sensitivity of the human eye to different wavelengths of...
,
, from the peak
luminous intensityIn photometry, luminous intensity is a measure of the wavelengthweighted power emitted by a light source in a particular direction per unit solid angle, based on the luminosity function, a standardized model of the sensitivity of the human eye...
,
, by integrating the cosine law:


and so
where
is the determinant of the Jacobian matrix for the
unit sphereIn mathematics, a unit sphere is the set of points of distance 1 from a fixed central point, where a generalized concept of distance may be used; a closed unit ball is the set of points of distance less than or equal to 1 from a fixed central point...
, and realizing that
is luminous flux per
steradianThe steradian is the SI unit of solid angle. It is used to describe twodimensional angular spans in threedimensional space, analogous to the way in which the radian describes angles in a plane...
. Similarly, the peak intensity will be
of the total radiated luminous flux. For Lambertian surfaces, the same factor of
relates
luminanceLuminance is a photometric measure of the luminous intensity per unit area of light travelling in a given direction. It describes the amount of light that passes through or is emitted from a particular area, and falls within a given solid angle. The SI unit for luminance is candela per square...
to luminous emittance,
radiant intensityIn radiometry, radiant intensity is a measure of the intensity of electromagnetic radiation. It is defined as power per unit solid angle. The SI unit of radiant intensity is watts per steradian . Radiant intensity is distinct from irradiance and radiant exitance, which are often called intensity...
to
radiant fluxIn radiometry, radiant flux or radiant power is the measure of the total power of electromagnetic radiation...
, and
radianceRadiance and spectral radiance are radiometric measures that describe the amount of radiation such as light or radiant heat that passes through or is emitted from a particular area, and falls within a given solid angle in a specified direction. They are used to characterize both emission from...
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/m
^{2} (= 100 nits, typical PCscreen) seen from the front, will (if it is a perfect Lambert emitter) emit a total luminous flux of 314 lm/m
^{2}. If it's a 19" screen (area ≈ 0.1 m
^{2}), the total light emitted 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 cosine of the angle between the surface normal and the direction of the incident light.
This phenomenon is among others used when creating
moldingsMolding or moulding is a strip of material with various profiles used to cover transitions between surfaces or for decoration. It is traditionally made from solid milled wood or plaster but may be made from plastic or reformed wood...
, which are a means of applying light and dark shaded stripes to a structure or object without having to change the material or apply
pigmentA pigment is a material that changes the color of reflected or transmitted light as the result of wavelengthselective absorption. This physical process differs from fluorescence, phosphorescence, and other forms of luminescence, in which a material emits light.Many materials selectively absorb...
. 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.
See also
 Transmittance
In optics and spectroscopy, transmittance is the fraction of incident light at a specified wavelength that passes through a sample. A related term is absorptance, or absorption factor, which is the fraction of radiation absorbed by a sample at a specified wavelength...
 Reflectivity
In optics and photometry, reflectivity is the fraction of incident radiation reflected by a surface. In general it must be treated as a directional property that is a function of the reflected direction, the incident direction, and the incident wavelength...
 Passive solar building design
In passive solar building design, windows, walls, and floors are made to collect, store, and distribute solar energy in the form of heat in the winter and reject solar heat in the summer...
 Sun path
Sun path refers to the apparent significant seasonalandhourly positional changes of the sun as the Earth rotates, and orbits around the sun. The relative position of the sun is a major factor in the heat gain of buildings and in the performance of solar energy systems...