Inradiometry,radiance is theradiant flux emitted, reflected, transmitted or received by a given surface, per unitsolid angle per unit projected area. Radiance is used to characterize diffuse emission andreflection ofelectromagnetic radiation, and to quantify emission ofneutrinos and other particles. TheSI unit of radiance is thewatt persteradian persquare metre (W·sr−1·m−2). It is adirectional quantity: the radiance of a surface depends on the direction from which it is being observed.
The related quantityspectral radiance is the radiance of a surface per unitfrequency orwavelength, depending on whether thespectrum 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 inheat transfer,astrophysics andastronomy. "Intensity" has many other meanings inphysics, with the most common beingpower per unit area (so the radiance is the intensity per solid angle in this case).
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'sentrance pupil. Since theeye is an optical system, radiance and its cousinluminance 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 articleBrightness for a discussion). The nonstandard usage of "brightness" for "radiance" persists in some fields, notablylaser physics.
The radiance divided by the index of refraction squared isinvariant ingeometric 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 calledconservation of radiance. For real, passive, optical systems, the output radiance isat 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 theirradiance 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 idealblack body at a given temperature, spectral radiance is governed byPlanck's law, while the integral of its radiance, over the hemisphere into which its surface radiates, is given by theStefan–Boltzmann law. Its surface isLambertian, 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 byintegration over the cosine of the zenith angle.
Radiance of asurface, denotedLe,Ω ("e" for "energetic", to avoid confusion with photometric quantities, and "Ω" to indicate this is a directional quantity), is defined as[1]
where
In generalLe,Ω is a function of viewing direction, depending onθ through cosθ andazimuth angle through∂Φe/∂Ω. For the special case of aLambertian surface,∂2Φe/(∂Ω ∂A) is proportional to cosθ, andLe,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 asurface, denotedLe,Ω,ν, is defined as[1]
whereν is the frequency.
Spectral radiance in wavelength of asurface, denotedLe,Ω,λ, is defined as[1]
whereλ is the wavelength.
Radiance of a surface is related toétendue by
where
As the light travels through an ideal optical system, both the étendue and the radiant flux are conserved. Therefore,basic radiance defined by[2]
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.
| Quantity | Unit | Dimension | Notes | ||
|---|---|---|---|---|---|
| Name | Symbol[nb 1] | Name | Symbol | ||
| Radiant energy | Qe[nb 2] | joule | J | M⋅L2⋅T−2 | Energy of electromagnetic radiation. |
| Radiant energy density | we | joule per cubic metre | J/m3 | M⋅L−1⋅T−2 | Radiant energy per unit volume. |
| Radiant flux | Φe[nb 2] | watt | W = J/s | M⋅L2⋅T−3 | Radiant energy emitted, reflected, transmitted or received, per unit time. This is sometimes also called "radiant power", and calledluminosity in astronomy. |
| Spectral flux | Φe,ν[nb 3] | watt perhertz | W/Hz | M⋅L2⋅T −2 | Radiant flux per unit frequency or wavelength. The latter is commonly measured in W⋅nm−1. |
| Φe,λ[nb 4] | watt per metre | W/m | M⋅L⋅T−3 | ||
| Radiant intensity | Ie,Ω[nb 5] | watt persteradian | W/sr | M⋅L2⋅T−3 | Radiant flux emitted, reflected, transmitted or received, per unit solid angle. This is adirectional quantity. |
| Spectral intensity | Ie,Ω,ν[nb 3] | watt per steradian per hertz | W⋅sr−1⋅Hz−1 | M⋅L2⋅T−2 | Radiant intensity per unit frequency or wavelength. The latter is commonly measured in W⋅sr−1⋅nm−1. This is adirectional quantity. |
| Ie,Ω,λ[nb 4] | watt per steradian per metre | W⋅sr−1⋅m−1 | M⋅L⋅T−3 | ||
| Radiance | Le,Ω[nb 5] | watt per steradian per square metre | W⋅sr−1⋅m−2 | M⋅T−3 | Radiant flux emitted, reflected, transmitted or received by asurface, per unit solid angle per unit projected area. This is adirectional quantity. This is sometimes also confusingly called "intensity". |
| Spectral radiance Specific intensity | Le,Ω,ν[nb 3] | watt per steradian per square metre per hertz | W⋅sr−1⋅m−2⋅Hz−1 | M⋅T−2 | Radiance of asurface per unit frequency or wavelength. The latter is commonly measured in W⋅sr−1⋅m−2⋅nm−1. This is adirectional quantity. This is sometimes also confusingly called "spectral intensity". |
| Le,Ω,λ[nb 4] | watt per steradian per square metre, per metre | W⋅sr−1⋅m−3 | M⋅L−1⋅T−3 | ||
| Irradiance Flux density | Ee[nb 2] | watt per square metre | W/m2 | M⋅T−3 | Radiant fluxreceived by asurface per unit area. This is sometimes also confusingly called "intensity". |
| Spectral irradiance Spectral flux density | Ee,ν[nb 3] | watt per square metre per hertz | W⋅m−2⋅Hz−1 | M⋅T−2 | Irradiance of asurface per unit frequency or wavelength. This is sometimes also confusingly called "spectral intensity". Non-SI units of spectral flux density includejansky (1 Jy =10−26 W⋅m−2⋅Hz−1) andsolar flux unit (1 sfu =10−22 W⋅m−2⋅Hz−1 =104 Jy). |
| Ee,λ[nb 4] | watt per square metre, per metre | W/m3 | M⋅L−1⋅T−3 | ||
| Radiosity | Je[nb 2] | watt per square metre | W/m2 | M⋅T−3 | Radiant fluxleaving (emitted, reflected and transmitted by) asurface per unit area. This is sometimes also confusingly called "intensity". |
| Spectral radiosity | Je,ν[nb 3] | watt per square metre per hertz | W⋅m−2⋅Hz−1 | M⋅T−2 | Radiosity of asurface per unit frequency or wavelength. The latter is commonly measured in W⋅m−2⋅nm−1. This is sometimes also confusingly called "spectral intensity". |
| Je,λ[nb 4] | watt per square metre, per metre | W/m3 | M⋅L−1⋅T−3 | ||
| Radiant exitance | Me[nb 2] | watt per square metre | W/m2 | M⋅T−3 | Radiant fluxemitted by asurface per unit area. This is the emitted component of radiosity. "Radiant emittance" is an old term for this quantity. This is sometimes also confusingly called "intensity". |
| Spectral exitance | Me,ν[nb 3] | watt per square metre per hertz | W⋅m−2⋅Hz−1 | M⋅T−2 | Radiant exitance of asurface per unit frequency or wavelength. The latter is commonly measured in W⋅m−2⋅nm−1. "Spectral emittance" is an old term for this quantity. This is sometimes also confusingly called "spectral intensity". |
| Me,λ[nb 4] | watt per square metre, per metre | W/m3 | M⋅L−1⋅T−3 | ||
| Radiant exposure | He | joule per square metre | J/m2 | M⋅T−2 | Radiant energy received by asurface per unit area, or equivalently irradiance of asurface integrated over time of irradiation. This is sometimes also called "radiant fluence". |
| Spectral exposure | He,ν[nb 3] | joule per square metre per hertz | J⋅m−2⋅Hz−1 | M⋅T−1 | Radiant exposure of asurface per unit frequency or wavelength. The latter is commonly measured in J⋅m−2⋅nm−1. This is sometimes also called "spectral fluence". |
| He,λ[nb 4] | joule per square metre, per metre | J/m3 | M⋅L−1⋅T−2 | ||
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