Editing Radiance

{{short description|Physical quantity in radiometry}}
{{Other uses}}

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 [[diffuse reflection|reflection]] of [[electromagnetic radiation]], and to quantify emission of [[neutrino]]s and other particles. The [[International System of Units|SI unit]] of radiance is the [[watt]] per [[steradian]] per [[square metre]] ({{nobreak|W·sr<sup>−1</sup>·m<sup>−2</sup>}}). 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 [[Spectral radiometric quantity|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]], [[astrophysics]] and [[astronomy]]. "Intensity" has many other meanings in [[physics]], with the most common being [[intensity (physics)|power per unit area]] (so the radiance is the intensity per solid angle in this case).

==Description==
[[File:photometry_radiometry_units.svg|thumb|upright=1.5|Comparison of photometric and radiometric quantities]]
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 [[human eye|eye]] is an optical system, radiance and its cousin [[luminance]] 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]] for a discussion). The nonstandard usage of "brightness" for "radiance" persists in some fields, notably [[laser physics]].

The radiance divided by the index of refraction squared is [[Invariant (physics)|invariant]] in [[geometric optics]]. 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]] 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]] at a given temperature, spectral radiance is governed by [[Planck's law]], while the integral of its radiance, over the hemisphere into which its surface radiates, is given by the [[Stefan–Boltzmann law]]. Its surface is [[Lambert's cosine law|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 [[Stefan–Boltzmann law#Integration of intensity derivation|integration over the cosine of the zenith angle]].

==Mathematical definitions==
===Radiance===
'''Radiance''' of a ''surface'', denoted ''L''<sub>e,Ω</sub> ("e" for "energetic", to avoid confusion with photometric quantities, and "Ω" to indicate this is a directional quantity), is defined as<ref name="ISO_9288-1989">{{cite web|url=http://www.iso.org/iso/home/store/catalogue_tc/catalogue_detail.htm?csnumber=16943|title=Thermal insulation — Heat transfer by radiation — Physical quantities and definitions|work=ISO 9288:1989 | publisher=[[International Organization for Standardization|ISO]] catalogue|year=1989|access-date=2015-03-15}}</ref>
:<math>L_{\mathrm{e},\Omega} = \frac{\partial^2 \Phi_\mathrm{e}}{\partial \Omega\, \partial(A \cos \theta)},</math>
where
*∂ is the [[partial derivative]] symbol;
*Φ<sub>e</sub> is the [[radiant flux]] emitted, reflected, transmitted or received;
*Ω is the [[solid angle]];
*''A'' cos ''θ'' is the ''projected'' area.

In general ''L''<sub>e,Ω</sub> is a function of viewing direction, depending on ''θ'' through cos ''θ'' and [[azimuth angle]] through {{nobreak|∂Φ<sub>e</sub>/∂Ω}}. For the special case of a [[Lambertian reflectance|Lambertian surface]], {{nobreak|∂<sup>2</sup>Φ<sub>e</sub>/(∂Ω ∂''A'')}} is proportional to cos ''θ'', and ''L''<sub>e,Ω</sub> 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===
{{main|Spectral radiance}}
'''Spectral radiance in frequency''' of a ''surface'', denoted ''L''<sub>e,Ω,ν</sub>, is defined as<ref name="ISO_9288-1989" />
:<math>L_{\mathrm{e},\Omega,\nu} = \frac{\partial L_{\mathrm{e},\Omega}}{\partial \nu},</math>
where ''ν'' is the frequency.

'''Spectral radiance in wavelength''' of a ''surface'', denoted ''L''<sub>e,Ω,λ</sub>, is defined as<ref name="ISO_9288-1989" />
:<math>L_{\mathrm{e},\Omega,\lambda} = \frac{\partial L_{\mathrm{e},\Omega}}{\partial \lambda},</math>
where ''λ'' is the wavelength.

==Conservation of basic radiance==
Radiance of a surface is related to [[étendue]] by
:<math>L_{\mathrm{e},\Omega} = n^2 \frac{\partial \Phi_\mathrm{e}}{\partial G},</math>
where
*''n'' is the [[refractive index]] 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 by<ref>William Ross McCluney, ''Introduction to Radiometry and Photometry'', Artech House, Boston, MA, 1994 {{ISBN|978-0890066782}}</ref>
:<math>L_{\mathrm{e},\Omega}^* = \frac{L_{\mathrm{e},\Omega}}{n^2}</math>
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.

==SI radiometry units==
{{SI radiometry units}}

==See also==
*[[Étendue]]
*[[Light field]]
*[[Sakuma–Hattori equation]]
*[[Wien displacement law]]

==References==
{{reflist}}

==External links==
*[https://web.archive.org/web/20080124230143/http://ncr101.montana.edu/Light1994Conf/4_2_Sliney/Sliney%20Text.htm International Lighting in Controlled Environments Workshop]

[[Category:Physical quantities]]
[[Category:Radiometry]]