In
optics
Optics is the branch of physics that studies the behaviour and properties of light, including its interactions with matter and the construction of instruments that use or detect it. Optics usually describes the behaviour of visible, ultrav ...
, 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; I = I0cos(''θ'').
[RCA Electro-Optics Handbook, p.18 ff][Modern Optical Engineering, Warren J. Smith, McGraw-Hill, p. 228, 256] 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
Luminance 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, is emitted from, or is reflected from a particular area, and falls with ...
). 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
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
The Moon is Earth's only natural satellite. It is the fifth largest satellite in the Solar System and the largest and most massive relative to its parent planet, with a diameter about one-quarter that of Earth (comparable to the width of ...
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 angle
In Euclidean geometry, an angle is the figure formed by two rays, called the '' sides'' of the angle, sharing a common endpoint, called the ''vertex'' of the angle.
Angles formed by two rays lie in the plane that contains the rays. Angles ...
s 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
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
photon
A photon () is an elementary particle that is a quantum of the electromagnetic field, including electromagnetic radiation such as light and radio waves, and the force carrier for the electromagnetic force. Photons are massless, so they alwa ...
s rather than
energy
In physics, energy (from Ancient Greek: ἐνέργεια, ''enérgeia'', “activity”) is the quantitative property that is transferred to a body or to a physical system, recognizable in the performance of work and in the form of ...
or
luminous energy
In photometry, luminous energy is the perceived energy of light. This is sometimes called the quantity of light.[circle
A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is cons ...](_blank)
each represent an equal angle ''d''Ω, of an arbitrarily chosen size, and for a Lambertian surface, the number of photons per second emitted into each wedge is proportional to the area of the wedge.
The length of each wedge is the product of the
diameter
In geometry, a diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints lie on the circle. It can also be defined as the longest chord of the circle. Both definitions are also valid f ...
of the circle and cos(''θ''). 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·m
2·sr) and the number of photons per second emitted into the vertical wedge is . The number of photons per second emitted into the wedge at angle ''θ'' is .
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 ''d''Ω
0, which is a portion of the observer's total angular field-of-view of the scene. Since the wedge size ''d''Ω was chosen arbitrarily, for convenience we may assume without loss of generality that it coincides with the solid angle subtended by the aperture when "viewed" from the locus of the emitting area element dA. Thus the normal observer will then be recording the same photons per second emission derived above and will measure a radiance of
:
photons/(s·m
2·sr).
The observer at angle ''θ'' to the normal will be seeing the scene through the same aperture of area ''dA''
0 (still corresponding to a ''d''Ω wedge) and from this oblique vantage the area element ''dA'' is foreshortened and will subtend a (solid) angle of ''d''Ω
0 cos(''θ''). This observer will be recording photons per second, and so will be measuring a radiance of
:
photons/(s·m
2·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
In 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 electromagnetic radiation (including infrared, ultraviolet, and visible light), in tha ...
,
, from the peak
luminous intensity,
, by integrating the cosine law:
and so
:
where
is the determinant of the
Jacobian matrix
In vector calculus, the Jacobian matrix (, ) of a vector-valued function of several variables is the matrix of all its first-order partial derivatives. When this matrix is square, that is, when the function takes the same number of variable ...
for the
unit sphere, and realizing that
is luminous flux per
steradian.
[Incropera and DeWitt, ''Fundamentals of Heat and Mass Transfer'', 5th ed., p.710.] Similarly, the peak intensity will be
of the total radiated luminous flux. For Lambertian surfaces, the same factor of
relates
luminance
Luminance 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, is emitted from, or is reflected from a particular area, and falls with ...
to
luminous emittance
In photometry, illuminance is the total luminous flux incident on a surface, per unit area. It is a measure of how much the incident light illuminates the surface, wavelength-weighted by the luminosity function to correlate with human brightne ...
,
radiant intensity to
radiant flux
In radiometry, radiant flux or radiant power is the radiant energy emitted, reflected, transmitted, or received per unit time, and spectral flux or spectral power is the radiant flux per unit frequency or wavelength, depending on whether the spe ...
, 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/m
2 (= 100 nits, typical PC monitor) will, if it is a perfect Lambert emitter, have a luminous emittance of 100π lm/m
2. If its area is 0.1 m
2 (~19" monitor) then the total light emitted, or luminous flux, would thus be 31.4 lm.
See also
*
Transmittance
*
Reflectivity
The reflectance of the surface of a material is its effectiveness in reflecting radiant energy. It is the fraction of incident electromagnetic power that is reflected at the boundary. Reflectance is a component of the response of the electronic ...
*
Passive solar building design
In passive solar building design, windows, walls, and floors are made to collect, store, reflect, and distribute solar energy, in the form of heat in the winter and reject solar heat in the summer. This is called passive solar design because, unli ...
*
Sun path
References
{{DEFAULTSORT:Lambert's Cosine Law
Radiometry
Photometry
3D computer graphics
Scattering