Radiance

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For other uses, see Radiance (disambiguation).

In radiometry, radiance and spectral radiance of a surface in a given direction are the radiant flux emitted, reflected, transmitted or received by that surface, per unit solid angle around that direction per unit projected area of that surface along that direction. These are directional quantities. They are used to characterize diffuse emission and reflection of electromagnetic radiation. In astrophysics, radiance is also used to quantify emission of neutrinos and other particles. The SI unit of radiance is the watt per steradian per square metre (W·sr−1·m−2), while that of spectral radiance is the watt per steradian per square metre per hertz (W·sr−1·m−2·Hz−1) or the watt per steradian per square metre, per metre (W·sr−1·m−3)—commonly the watt per steradian per square metre per nanometre (W·sr−1·m−2·nm−1)—, depending on whether the spectrum is taken as a function of frequency or of wavelength.

Description[edit]

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 some 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 (Hz) with SI units W·sr−1·m−2·Hz−1 or wavelength (nm) with SI units W·sr−1·m−2·nm−1 (more common than W·sr−1·m−3). In some fields spectral radiance is also measured in microflicks.[1][2] Radiance is the integral of the spectral radiance over all wavelengths or frequencies.

For radiation emitted by an ideal black body at temperature T, spectral radiance is governed by Planck's law, while the integral of radiance over the hemisphere into which it radiates, in W/m2, is governed by the Stefan-Boltzmann law. There is no need for a separate law for radiance normal to the surface of a black body, in W·sr−1·m−2, since this is simply the Stefan–Boltzmann law 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. More generally the radiance at an angle θ to the normal (the zenith angle) is given by the Stefan–Boltzmann law times (cos θ)/π.

Definitions[edit]

Radiance[edit]

Radiance of a surface in a given direction, denoted Le,Ω ("e" for "energetic", to avoid confusion with photometric quantities, and "Ω" to indicate this is a directional quantity) and measured in W·sr−1·m−2, is given by:

L_{\mathrm{e},\Omega} = \frac{\partial ^2 \Phi_\mathrm{e}}{\partial \Omega\, \partial A \cos \theta}

where

  • ∂ is the partial derivative symbol;
  • Φe is the radiant flux of that surface, measured in W;
  • Ω is the solid angle around that direction, measured in sr;
  • A is the area of the surface, measured in m2;
  • θ is the angle between the surface normal and that direction, measured in rad;
  • A cos θ is the projected area of that surface along that direction.

In general Le,Ω is a function of viewing angle, depending on θ through cos θ, and in general on both 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 angle).

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[edit]

Radiance of a surface in a given direction per unit frequency, denoted Le,Ω,ν and measured in W·sr−1·m−2·Hz−1, is given by:

L_{\mathrm{e},\Omega,\nu} = \frac{\partial L_{\mathrm{e},\Omega}}{\partial \nu}

where ν is the frequency, measured in Hz.

Radiance of a surface in a given direction per unit wavelength, denoted Le,Ω,λ and measured in W·sr−1·m−3 (commonly in W·sr−1·m−2·nm−1), is given by:

L_{\mathrm{e},\Omega,\lambda} = \frac{\partial L_{\mathrm{e},\Omega}}{\partial \lambda}

where λ is the wavelength, measured in m (commonly in nm).

Nomenclature[edit]

Historically, radiance is called intensity and spectral radiance is 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 power per unit area.

See also[edit]

References[edit]

  1. ^ Palmer, James M. "The SI system and SI units for Radiometry and photometry". Archived from the original on August 2, 2012. 
  2. ^ Rowlett, Russ. "How Many? A Dictionary of Units of Measurement". Retrieved 10 August 2012. 

External links[edit]

SI radiometry units
Quantity Unit Dimension Notes
Name Symbol[nb 1] Name Symbol Symbol
Radiant energy Qe[nb 2] joule J ML2T−2 Energy received, emitted, reflected, or transmitted by a system in form of electromagnetic radiation.
Radiant energy density we joule per cubic metre J/m3 ML−1T−2 Radiant energy of a system per unit volume at a given location.
Radiant flux / Radiant power Φe[nb 2] watt W or J/s ML2T−3 Radiant energy of a system per unit time at a given time.
Spectral flux / Spectral power Φe,ν[nb 3]
or
Φe,λ[nb 4]
watt per hertz
or
watt per metre
W/Hz
or
W/m
ML2T−2
or
MLT−3
Radiant power of a system per unit frequency or wavelength. The latter is commonly measured in W⋅sr−1⋅m−2⋅nm−1.
Radiant intensity Ie,Ω[nb 5] watt per steradian W/sr ML2T−3 Radiant power of a system per unit solid angle around a given direction. It is a directional quantity.
Spectral intensity Ie,Ω,ν[nb 3]
or
Ie,Ω,λ[nb 4]
watt per steradian per hertz
or
watt per steradian per metre
W⋅sr−1⋅Hz−1
or
W⋅sr−1⋅m−1
ML2T−2
or
MLT−3
Radiant intensity of a system per unit frequency or wavelength. The latter is commonly measured in W⋅sr−1⋅m−2⋅nm−1. It is a directional quantity.
Radiance Le,Ω[nb 5] watt per steradian per square metre W⋅sr−1⋅m−2 MT−3 Radiant power of a surface per unit solid angle around a given direction per unit projected area of that surface along that direction. It is a directional quantity. It is sometimes also confusingly called "intensity".
Spectral radiance Le,Ω,ν[nb 3]
or
Le,Ω,λ[nb 4]
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. It is a directional quantity. It is sometimes also confusingly called "spectral intensity".
Irradiance Ee[nb 2] watt per square metre W/m2 MT−3 Radiant power received by a surface per unit area. It is sometimes also confusingly called "intensity".
Spectral irradiance Ee,ν[nb 3]
or
Ee,λ[nb 4]
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. The former is commonly measured in 10−22 W⋅m−2⋅Hz−1, known as solar flux unit, and the latter in W⋅m−2⋅nm−1.[nb 6] It is sometimes also confusingly called "spectral intensity".
Radiosity Je[nb 2] watt per square metre W/m2 MT−3 Radiant power leaving (emitted, reflected and transmitted by) a surface per unit area. It is sometimes also confusingly called "intensity".
Spectral radiosity Je,ν[nb 3]
or
Je,λ[nb 4]
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⋅sr−1⋅m−2⋅nm−1. It is sometimes also confusingly called "spectral intensity".
Radiant exitance Me[nb 2] watt per square metre W/m2 MT−3 Radiant power emitted by a surface per unit area. This is the emitted component of radiosity. "Radiant emittance" is an old term for this quantity. It is sometimes also confusingly called "intensity".
Spectral exitance Me,ν[nb 3]
or
Me,λ[nb 4]
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⋅sr−1⋅m−2⋅nm−1. "Spectral emittance" is an old term for this quantity. It is sometimes also confusingly called "spectral intensity".
Radiant exposure He joule per square metre J/m2 MT−2 Irradiance of a surface times exposure time. It is sometimes also called fluence.
See also: SI · Radiometry · Photometry
  1. ^ Standards organizations recommend that radiometric quantities should be denoted with a suffix "e" (for "energetic") to avoid confusion with photometric or photon quantities.
  2. ^ a b c d e Alternative symbols sometimes seen: W or E for radiant energy, P or F for radiant flux, I for irradiance, W for radiant exitance.
  3. ^ a b c d e f Spectral quantities given per unit frequency are denoted with suffix "ν" (Greek)—not to be confused with the suffix "v" (for "visual") indicating a photometric quantity.
  4. ^ a b c d e f Spectral quantities given per unit wavelength are denoted with suffix "λ" (Greek) to indicate a spectral concentration. Spectral functions of wavelength are indicated by "(λ)" in parentheses instead, for example in spectral transmittance, spectral reflectance and spectral responsivity.
  5. ^ a b The two directional quantities, radiant intensity and radiance, are denoted with suffix "Ω" (Greek) to indicate a directional concentration.
  6. ^ NOAA / Space Weather Prediction Center includes a definition of the solar flux unit (SFU).