In _{0}cos(''θ'').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

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 ^{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
:$I\_0=\backslash frac$ 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
:$I\_0=\backslash frac\; =\backslash frac$ photons/(s·m^{2}·sr),
which is the same as the normal observer.

^{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.

{{DEFAULTSORT:Lambert's Cosine Law
Radiometry
Photometry
3D computer graphics
Scattering

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, ultraviol ...

, Lambert's cosine law says that the radiant intensity 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 human eye ...

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 = IJohann Heinrich Lambert
Johann Heinrich Lambert (, ''Jean-Henri Lambert'' in French; 26 or 28 August 1728 – 25 September 1777) was a polymath from the Republic of Mulhouse, generally referred to as either Swiss or French, who made important contributions to the subject ...

, 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 technically, the su ...

. Such a surface has the same 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 radia ...

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 withi ...

). 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, theirradiance 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 (W⋅m−2). The CGS unit erg per square centimetre per second (erg⋅cm−2⋅s−1) is often us ...

(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 o ...

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 a ...

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 center of the Solar System. It is a nearly perfect ball of hot plasma, heated to incandescence by nuclear fusion reactions in its core. The Sun radiates this energy mainly as light, ultraviolet, and infrared radi ...

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
A black body or blackbody is an idealized physical body that absorbs all incident electromagnetic radiation, regardless of frequency or angle of incidence. The name "black body" is given because it absorbs all colors of light. A black body ...

is an example of a Lambertian radiator.
Details of equal brightness effect

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 always ...

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 heat ...

or luminous energy. The wedges in the 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 con ...

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 for ...

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 poi ...

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 radia ...

along the normal is ''I'' photons/(s·mRelating peak luminous intensity and luminous flux

In general, theluminous 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 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 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 th ...

, $F\_\backslash text$, 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 human eye ...

, $I\_$, by integrating the cosine law:
$$\backslash begin\; F\_\backslash text\; \&=\; \backslash int\_0^\; \backslash int\_0^\; \backslash cos(\backslash theta)\; \backslash ,\; I\_\backslash ,\; \backslash sin(\backslash theta)\backslash ,d\backslash theta\backslash ,d\backslash phi\backslash \backslash \; \&=\; 2\backslash pi\backslash cdot\; I\_\backslash int\_0^\backslash cos(\backslash theta)\backslash sin(\backslash theta)\backslash ,d\backslash theta\; \backslash \backslash \; \&=\; 2\backslash pi\backslash cdot\; I\_\backslash int\_0^\backslash frac\backslash ,d\backslash theta\; \backslash end$$
and so
:$F\_\backslash text=\backslash pi\backslash ,\backslash mathrm\backslash cdot\; I\_$
where $\backslash sin(\backslash theta)$ is the determinant of the Jacobian matrix for the unit sphere
In mathematics, a unit sphere is simply a sphere of radius one around a given center. More generally, it is the set of points of distance 1 from a fixed central point, where different norms can be used as general notions of "distance". A uni ...

, 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. Whereas an angle in radia ...

.Incropera and DeWitt, ''Fundamentals of Heat and Mass Transfer'', 5th ed., p.710. Similarly, the peak intensity will be $1/(\backslash pi\backslash ,\backslash mathrm)$ of the total radiated luminous flux. For Lambertian surfaces, the same factor of $\backslash pi\backslash ,\backslash mathrm$ 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 withi ...

to luminous emittance, 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 sp ...

, 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 radia ...

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/mSee also

* Transmittance * Reflectivity * Passive solar building design * Sun pathReferences