## 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 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 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 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 Le,Ω ("e" for "energetic", to avoid confusion with photometric quantities, and "Ω" to indicate this is a directional quantity), is defined as[5]

${displaystyle L_{mathrm {e} ,Omega }={frac {partial ^{2}Phi _{mathrm {e} }}{partial Omega ,partial Acos theta }},}$

where

In general Le,Ω is a function of viewing direction, depending on θ through cos θ and azimuth angle through ∂Φe/∂Ω. For the special case of a Lambertian surface, 2Φe/(∂ΩA) is proportional to cos θ, and Le,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 Le,Ω,ν, is defined as[5]

${displaystyle L_{mathrm {e} ,Omega ,nu }={frac {partial L_{mathrm {e} ,Omega }}{partial nu }},}$

where ν is the frequency.

Spectral radiance in wavelength of a surface, denoted Le,Ω,λ, is defined as[5]

${displaystyle L_{mathrm {e} ,Omega ,lambda }={frac {partial L_{mathrm {e} ,Omega }}{partial lambda }},}$

where λ is the wavelength.

Radiance of a surface is related to étendue by

${displaystyle L_{mathrm {e} ,Omega }=n^{2}{frac {partial Phi _{mathrm {e} }}{partial G}},}$

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

${displaystyle L_{mathrm {e} ,Omega }^{*}={frac {L_{mathrm {e} ,Omega }}{n^{2}}}}$

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.

QuantityUnitDimensionNotes
NameSymbolNameSymbolSymbol
Spectral fluxΦe,ν
or
Φe,λ
watt per hertz
or
watt per metre
W/Hz
or
W/m
ML2T−2
or
MLT−3
Radiant flux per unit frequency or wavelength. The latter is commonly measured in W⋅sr−1⋅m−2⋅nm−1.
Spectral intensityIe,Ω,ν
or
Ie,Ω,λ
or
W⋅sr−1⋅Hz−1
or
W⋅sr−1⋅m−1
ML2T−2
or
MLT−3
Radiant intensity per unit frequency or wavelength. The latter is commonly measured in W⋅sr−1⋅m−2⋅nm−1. This is a directional quantity.
RadianceLe,Ωwatt per steradian per square metreW⋅sr−1⋅m−2MT−3Radiant flux emitted, reflected, transmitted or received by a surface, per unit solid angle per unit projected area. This is a directional quantity. This is sometimes also confusingly called "intensity".
or
Le,Ω,λ
watt per steradian per square metre per hertz
or
watt per steradian per square metre, per metre
W⋅sr−1⋅m−2⋅Hz−1
or
W⋅sr−1⋅m−3
MT−2
or
ML−1T−3
Radiance of a surface per unit frequency or wavelength. The latter is commonly measured in W⋅sr−1⋅m−2⋅nm−1. This is a directional quantity. This is sometimes also confusingly called "spectral intensity".
Flux density
Eewatt per square metreW/m2MT−3Radiant flux received by a surface per unit area. This is sometimes also confusingly called "intensity".
Spectral flux density
Ee,ν
or
Ee,λ
watt per square metre per hertz
or
watt per square metre, per metre
W⋅m−2⋅Hz−1
or
W/m3
MT−2
or
ML−1T−3
Irradiance of a surface per unit frequency or wavelength. This is sometimes also confusingly called "spectral intensity". Non-SI units of spectral flux density include Jansky = 10−26 W⋅m−2⋅Hz−1 and solar flux unit (1SFU = 10−22 W⋅m−2⋅Hz−1=104Jy).
RadiosityJewatt per square metreW/m2MT−3Radiant flux leaving (emitted, reflected and transmitted by) a surface per unit area. This is sometimes also confusingly called "intensity".
or
Je,λ
watt per square metre per hertz
or
watt per square metre, per metre
W⋅m−2⋅Hz−1
or
W/m3
MT−2
or
ML−1T−3
Radiosity of a surface per unit frequency or wavelength. The latter is commonly measured in W⋅m−2⋅nm−1. This is sometimes also confusingly called "spectral intensity".
Radiant exitanceMewatt per square metreW/m2MT−3Radiant flux emitted by a surface 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 exitanceMe,ν
or
Me,λ
watt per square metre per hertz
or
watt per square metre, per metre
W⋅m−2⋅Hz−1
or
W/m3
MT−2
or
ML−1T−3
Radiant exitance of a surface 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".
Spectral exposureHe,ν
or
He,λ
joule per square metre per hertz
or
joule per square metre, per metre
J⋅m−2⋅Hz−1
or
J/m3
MT−1
or
ML−1T−2
Radiant exposure of a surface per unit frequency or wavelength. The latter is commonly measured in J⋅m−2⋅nm−1. This is sometimes also called "spectral fluence".
Hemispherical emissivityε1Radiant exitance of a surface, divided by that of a black body at the same temperature as that surface.
Spectral hemispherical emissivityεν
or
ελ
1Spectral exitance of a surface, divided by that of a black body at the same temperature as that surface.
Directional emissivityεΩ1Radiance emitted by a surface, divided by that emitted by a black body at the same temperature as that surface.
Spectral directional emissivityεΩ,ν
or
εΩ,λ
1Spectral radiance emitted by a surface, divided by that of a black body at the same temperature as that surface.
Hemispherical absorptanceA1Radiant flux absorbed by a surface, divided by that received by that surface. This should not be confused with "absorbance".
Spectral hemispherical absorptanceAν
or
Aλ
1Spectral flux absorbed by a surface, divided by that received by that surface. This should not be confused with "spectral absorbance".
Directional absorptanceAΩ1Radiance absorbed by a surface, divided by the radiance incident onto that surface. This should not be confused with "absorbance".
Spectral directional absorptanceAΩ,ν
or
AΩ,λ
1Spectral radiance absorbed by a surface, divided by the spectral radiance incident onto that surface. This should not be confused with "spectral absorbance".
Hemispherical reflectanceR1Radiant flux reflected by a surface, divided by that received by that surface.
Spectral hemispherical reflectanceRν
or
Rλ
1Spectral flux reflected by a surface, divided by that received by that surface.
Directional reflectanceRΩ1Radiance reflected by a surface, divided by that received by that surface.
Spectral directional reflectanceRΩ,ν
or
RΩ,λ
1Spectral radiance reflected by a surface, divided by that received by that surface.
Hemispherical transmittanceT1Radiant flux transmitted by a surface, divided by that received by that surface.
Spectral hemispherical transmittanceTν
or
Tλ
1Spectral flux transmitted by a surface, divided by that received by that surface.
Directional transmittanceTΩ1Radiance transmitted by a surface, divided by that received by that surface.
Spectral directional transmittanceTΩ,ν
or
TΩ,λ
1Spectral radiance transmitted by a surface, divided by that received by that surface.
Hemispherical attenuation coefficientμreciprocal metrem−1L−1Radiant flux absorbed and scattered by a volume per unit length, divided by that received by that volume.
Spectral hemispherical attenuation coefficientμν
or
μλ
reciprocal metrem−1L−1Spectral radiant flux absorbed and scattered by a volume per unit length, divided by that received by that volume.
Directional attenuation coefficientμΩreciprocal metrem−1L−1Radiance absorbed and scattered by a volume per unit length, divided by that received by that volume.
Spectral directional attenuation coefficientμΩ,ν
or
μΩ,λ
reciprocal metrem−1L−1Spectral radiance absorbed and scattered by a volume per unit length, divided by that received by that volume.