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In
radiometry Radiometry is a set of techniques for measuring electromagnetic radiation, including visible light. Radiometric techniques in optics characterize the distribution of the radiation's power in space, as opposed to photometric techniques, which cha ...
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 spec ...
emitted, reflected, transmitted or received by a given surface, 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 po ...
per unit projected area. Radiance is used to characterize diffuse emission and reflection of
electromagnetic radiation In physics, electromagnetic radiation (EMR) consists of waves of the electromagnetic (EM) field, which propagate through space and carry momentum and electromagnetic radiant energy. It includes radio waves, microwaves, infrared, (visible) lig ...
, and to quantify emission of
neutrino A neutrino ( ; denoted by the Greek letter ) is a fermion (an elementary particle with spin of ) that interacts only via the weak interaction and gravity. The neutrino is so named because it is electrically neutral and because its rest mass ...
s and other particles. The
SI unit The International System of Units, known by the international abbreviation SI in all languages and sometimes pleonastically as the SI system, is the modern form of the metric system and the world's most widely used system of measurement. ...
watt The watt (symbol: W) is the unit of power or radiant flux in the International System of Units (SI), equal to 1 joule per second or 1 kg⋅m2⋅s−3. It is used to quantify the rate of energy transfer. The watt is named after James Wat ...
per steradian per square metre (). It is a ''directional'' quantity: the radiance of a surface depends on the direction from which it is being observed. The related quantity spectral radiance is the radiance of a surface per unit frequency or wavelength, depending on whether the spectrum is taken as a function of frequency or of wavelength. Historically, radiance was called "intensity" and spectral radiance was called "specific intensity". Many fields still use this nomenclature. It is especially dominant in
heat transfer Heat transfer is a discipline of thermal engineering that concerns the generation, use, conversion, and exchange of thermal energy (heat) between physical systems. Heat transfer is classified into various mechanisms, such as thermal conduction, ...
,
astrophysics Astrophysics is a science that employs the methods and principles of physics and chemistry in the study of astronomical objects and phenomena. As one of the founders of the discipline said, Astrophysics "seeks to ascertain the nature of the hea ...
and
astronomy Astronomy () is a natural science that studies celestial objects and phenomena. It uses mathematics, physics, and chemistry in order to explain their origin and evolution. Objects of interest include planets, moons, stars, nebulae, galaxies ...
. "Intensity" has many other meanings in
physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which relat ...
, with the most common being power per unit area.

Description

Radiance is useful because it indicates how much of the power emitted, reflected, transmitted or received by a surface will be received by an optical system looking at that surface from a specified angle of view. In this case, the solid angle of interest is the solid angle subtended by the optical system's entrance pupil. Since the
eye Eyes are organs of the visual system. They provide living organisms with vision, the ability to receive and process visual detail, as well as enabling several photo response functions that are independent of vision. Eyes detect light and conve ...
is an optical system, radiance and its cousin
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 ...
are good indicators of how bright an object will appear. For this reason, radiance and luminance are both sometimes called "brightness". This usage is now discouraged (see the article
Brightness Brightness is an attribute of visual perception in which a source appears to be radiating or reflecting light. In other words, brightness is the perception elicited by the luminance of a visual target. The perception is not linear to luminance, ...
for a discussion). The nonstandard usage of "brightness" for "radiance" persists in some fields, notably
laser physics Laser science or laser physics is a branch of optics that describes the theory and practice of lasers. Laser science is principally concerned with quantum electronics, laser construction, optical cavity design, the physics of producing a populat ...
. The radiance divided by the index of refraction squared is invariant in
geometric optics Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ca ...
. This means that for an ideal optical system in air, the radiance at the output is the same as the input radiance. This is sometimes called ''conservation of radiance''. For real, passive, optical systems, the output radiance is ''at most'' equal to the input, unless the index of refraction changes. As an example, if you form a demagnified image with a lens, the optical power is concentrated into a smaller area, so 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 (W⋅m−2). The CGS unit erg per square centimetre per second (erg⋅cm−2⋅s−1) is often used ...
is higher at the image. The light at the image plane, however, fills a larger solid angle so the radiance comes out to be the same assuming there is no loss at the lens. Spectral radiance expresses radiance as a function of frequency or wavelength. Radiance is the integral of the spectral radiance over all frequencies or wavelengths. For radiation emitted by the surface of an ideal
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 ...
at a given temperature, spectral radiance is governed by
Planck's law In physics, Planck's law describes the spectral density of electromagnetic radiation emitted by a black body in thermal equilibrium at a given temperature , when there is no net flow of matter or energy between the body and its environment. A ...
, while the integral of its radiance, over the hemisphere into which its surface radiates, is given by the Stefan–Boltzmann law. Its surface is Lambertian, so that its radiance is uniform with respect to angle of view, and is simply the Stefan–Boltzmann integral divided by π. This factor is obtained from the solid angle 2π steradians of a hemisphere decreased by integration over the cosine of the zenith angle.

Mathematical definitions

Radiance of a ''surface'', denoted ''L''e,Ω ("e" for "energetic", to avoid confusion with photometric quantities, and "Ω" to indicate this is a directional quantity), is defined as :$L_ = \frac,$ where *∂ is the
partial derivative In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). Par ...
symbol; *Φe is the
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 spec ...
emitted, reflected, transmitted or received; *Ω is the
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 po ...
; *''A'' cos ''θ'' is the ''projected'' area. In general ''L''e,Ω is a function of viewing direction, depending on ''θ'' through cos ''θ'' and azimuth angle through . For the special case of a Lambertian surface, is proportional to cos ''θ'', and ''L''e,Ω is isotropic (independent of viewing direction). When calculating the radiance emitted by a source, ''A'' refers to an area on the surface of the source, and Ω to the solid angle into which the light is emitted. When calculating radiance received by a detector, ''A'' refers to an area on the surface of the detector and Ω to the solid angle subtended by the source as viewed from that detector. When radiance is conserved, as discussed above, the radiance emitted by a source is the same as that received by a detector observing it.

Spectral radiance in frequency of a ''surface'', denoted ''L''e,Ω,ν, is defined as :$L_ = \frac,$ where ''ν'' is the frequency. Spectral radiance in wavelength of a ''surface'', denoted ''L''e,Ω,λ, is defined as :$L_ = \frac,$ where ''λ'' is the wavelength.

Radiance of a surface is related to étendue by :$L_ = n^2 \frac,$ where *''n'' is the
refractive index In optics, the refractive index (or refraction index) of an optical medium is a dimensionless number that gives the indication of the light bending ability of that medium. The refractive index determines how much the path of light is bent, o ...
in which that surface is immersed; *''G'' is the étendue of the light beam. As the light travels through an ideal optical system, both the étendue and the radiant flux are conserved. Therefore, ''basic radiance'' defined byWilliam Ross McCluney, ''Introduction to Radiometry and Photometry'', Artech House, Boston, MA, 1994 :$L_^* = \frac$ is also conserved. In real systems, the étendue may increase (for example due to scattering) or the radiant flux may decrease (for example due to absorption) and, therefore, basic radiance may decrease. However, étendue may not decrease and radiant flux may not increase and, therefore, basic radiance may not increase.

* Étendue *
Light field The light field is a vector function that describes the amount of light flowing in every direction through every point in space. The space of all possible '' light rays'' is given by the five-dimensional plenoptic function, and the magnitude of e ...
* Sakuma–Hattori equation * Wien displacement law

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