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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 optical instruments, instruments that use or Photodetector, detect it. Optics usually describes t ...
, Lambert's cosine law says that the observed
radiant intensity In radiometry, radiant intensity is the radiant flux emitted, reflected, transmitted or received, per unit solid angle, and spectral intensity is the radiant intensity per unit frequency or wavelength, depending on whether the spectrum is taken ...
or
luminous intensity In photometry, luminous intensity is a measure of the wavelength-weighted 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 huma ...
from an ideal diffusely reflecting surface or ideal diffuse radiator is directly proportional to the
cosine In mathematics, sine and cosine are trigonometric functions of an angle. The sine and cosine of an acute angle are defined in the context of a right triangle: for the specified angle, its sine is the ratio of the length of the side opposite that ...
of the angle ''θ'' between the observer's line of sight and the
surface normal In geometry, a normal is an object (e.g. a line, ray, or vector) that is perpendicular to a given object. For example, the normal line to a plane curve at a given point is the infinite straight line perpendicular to the tangent line to the ...
; .RCA Electro-Optics Handbook, p.18 ffModern 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 Johann Heinrich Lambert (; ; 26 or 28 August 1728 – 25 September 1777) was a polymath from the Republic of Mulhouse, at that time allied to the Switzerland, Swiss Confederacy, who made important contributions to the subjects of mathematics, phys ...
, from his ''
Photometria ''Photometria'' is a book on the measurement of light by Johann Heinrich Lambert published in 1760.Lambert, Johann Heinrich, Photometria, sive de mensura et gradibus luminis, colorum et umbrae', Augsburg: Eberhard Klett, 1760. It established a com ...
'', published in 1760. A surface which obeys Lambert's law is said to be ''Lambertian'', and exhibits
Lambertian reflectance Lambertian reflectance is the property that defines an ideal "matte" or diffusely reflecting surface. The apparent brightness of a Lambertian surface to an observer is the same regardless of the observer's angle of view. More precisely, the ref ...
. Such a surface has a constant
radiance In radiometry, radiance is the radiant flux emitted, reflected, transmitted or received by a given surface, per unit solid angle per unit projected area. Radiance is used to characterize diffuse emission and reflection of electromagnetic radiati ...
/
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 wit ...
, regardless of the angle from which it is observed; a single human eye perceives such a surface as having a constant brightness, regardless of the angle from which the eye observes the surface. 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 In radiometry, irradiance is the radiant flux ''received'' by a ''surface'' per unit area. The SI unit of irradiance is the watt per square metre (symbol W⋅m−2 or W/m2). The CGS unit erg per square centimetre per second (erg⋅cm−2⋅s−1) ...
(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 Orbit of the Moon, orbits around Earth at Lunar distance, an average distance of (; about 30 times Earth diameter, Earth's diameter). The Moon rotation, rotates, with a rotation period (lunar ...
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 can refer to a number of concepts relating to the intersection of two straight lines at a point. Formally, an angle is a figure lying in a plane formed by two rays, called the '' sides'' of the angle, sharing ...
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 The Sun is the star at the centre of the Solar System. It is a massive, nearly perfect sphere of hot plasma, heated to incandescence by nuclear fusion reactions in its core, radiating the energy from its surface mainly as visible light a ...
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 Limb darkening is an optical effect seen in stars (including the Sun) and planets, where the central part of the disk appears brighter than the edge, or '' limb''. Its understanding offered early solar astronomers an opportunity to construct mode ...
in the visible region illustrates that it is not a Lambertian radiator. A
black body A black body or blackbody is an idealized physical body that absorbs all incident electromagnetic radiation, regardless of frequency or angle of incidence. The radiation emitted by a black body in thermal equilibrium with its environment is ...
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 particles that can ...
s rather than
energy Energy () is the physical quantity, quantitative physical property, property that is transferred to a physical body, body or to a physical system, recognizable in the performance of Work (thermodynamics), work and in the form of heat and l ...
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 point (geometry), points in a plane (mathematics), plane that are at a given distance from a given point, the Centre (geometry), centre. The distance between any point of the circle and the centre is cal ...
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 centre of the circle and whose endpoints lie on the circle. It can also be defined as the longest Chord (geometry), chord of the circle. Both definitions a ...
of the circle and cos(''θ''). The maximum rate of photon emission per unit
solid angle In geometry, a solid angle (symbol: ) is a measure of the amount of the field of view from some particular point that a given object covers. That is, it is a measure of how large the object appears to an observer looking from that point. The poin ...
is along the normal, and diminishes to zero for ''θ'' = 90°. In mathematical terms, the
radiance In radiometry, radiance is the radiant flux emitted, reflected, transmitted or received by a given surface, per unit solid angle per unit projected area. Radiance is used to characterize diffuse emission and reflection of electromagnetic radiati ...
along the normal is ''I'' photons/(s·m2·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''Ω0, 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 : I_0=\frac photons/(s·m2·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''Ω0 cos(''θ''). This observer will be recording photons per second, and so will be measuring a radiance of : I_0=\frac =\frac photons/(s·m2·sr), which is the same as the normal observer.


Relating peak luminous intensity and luminous flux

In general, the
luminous intensity In photometry, luminous intensity is a measure of the wavelength-weighted 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 huma ...
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 that ...
, F_\text, from the peak
luminous intensity In photometry, luminous intensity is a measure of the wavelength-weighted 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 huma ...
, I_, by integrating the cosine law: \begin F_\text &= \int_0^ \int_0^ \cos(\theta) \, I_\, \sin(\theta)\,d\theta\,d\phi\\ &= 2\pi\cdot I_\int_0^\cos(\theta)\sin(\theta)\,d\theta \\ &= 2\pi\cdot I_\int_0^\frac\,d\theta \end and so :F_\text=\pi\,\mathrm\cdot I_ where \sin(\theta) 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. If this matrix is square, that is, if the number of variables equals the number of component ...
for the
unit sphere In mathematics, a unit sphere is a sphere of unit radius: the locus (mathematics), set of points at Euclidean distance 1 from some center (geometry), center point in three-dimensional space. More generally, the ''unit -sphere'' is an n-sphere, -s ...
, and realizing that I_ is luminous flux per
steradian The steradian (symbol: sr) or square radian is the unit of solid angle in the International System of Units (SI). It is used in three-dimensional geometry, and is analogous to the radian, which quantifies planar angles. A solid angle in the fo ...
.Incropera and DeWitt, ''Fundamentals of Heat and Mass Transfer'', 5th ed., p.710. Similarly, the peak intensity will be 1/(\pi\,\mathrm) of the total radiated luminous flux. For Lambertian surfaces, the same factor of \pi\,\mathrm 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 wit ...
to luminous emittance,
radiant intensity In radiometry, radiant intensity is the radiant flux emitted, reflected, transmitted or received, per unit solid angle, and spectral intensity is the radiant intensity per unit frequency or wavelength, depending on whether the spectrum is taken ...
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 ...
, and
radiance In radiometry, radiance is the radiant flux emitted, reflected, transmitted or received by a given surface, per unit solid angle per unit projected area. Radiance is used to characterize diffuse emission and reflection of electromagnetic radiati ...
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 100π 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.


See also

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Transmittance Electromagnetic radiation can be affected in several ways by the medium in which it propagates.  It can be Scattering, scattered, Absorption (electromagnetic radiation), absorbed, and Fresnel equations, reflected and refracted at discontinui ...
*
Reflectivity The reflectance of the surface of a material is its effectiveness in Reflection (physics), reflecting radiant energy. It is the fraction of incident electromagnetic power that is reflected at the boundary. Reflectance is a component of the respon ...
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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 Sun path, sometimes also called day arc, refers to the diurnal motion, daily (sunrise to sunset) and seasonal arc (geometry), arc-like path that the Sun appears to follow across the sky as the Earth Earth's rotation, rotates and Earth's orbi ...


References

{{DEFAULTSORT:Lambert's Cosine Law Eponymous laws of physics Radiometry Photometry 3D computer graphics Scattering