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In radiometry, radiance is the radiant flux emitted, reflected, transmitted or received by a given surface, per unit solid angle per unit projected area. 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. These are directional quantities. The SI unit of radiance is the watt per steradian per square metre (W·sr−1·m−2), while that of spectral radiance in frequency is the watt per steradian per square metre per hertz (W·sr−1·m−2·Hz−1) and that of spectral radiance in wavelength is 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). The microflick is also used to measure spectral radiance in some fields.[1][2] Radiance is used to characterize diffuse emission and reflection of electromagnetic radiation, or to quantify emission of neutrinos and other particles. 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.

Contents

1 Description 2 Mathematical definitions

2.1 Radiance 2.2 Spectral radiance

3 Conservation of basic radiance 4 SI radiometry units 5 See also 6 References 7 External links

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 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
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[edit] Radiance[edit] 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[3]

L

e

, Ω

=

2

Φ

e

∂ Ω

∂ A cos ⁡ θ

,

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

where

∂ is the partial derivative symbol; Φe is the radiant flux emitted, reflected, transmitted or received; Ω is the solid angle; A cos θ is the projected area.

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/(∂Ω ∂A) 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[edit] Spectral radiance in frequency of a surface, denoted Le,Ω,ν, is defined as[3]

L

e

, Ω , ν

=

L

e

, Ω

∂ ν

,

displaystyle L_ mathrm e , Omega
Omega
,nu = frac partial L_ mathrm e , Omega
Omega
partial nu ,

where ν is the frequency. Spectral radiance in wavelength of a surface, denoted Le,Ω,λ, is defined as[3]

L

e

, Ω , λ

=

L

e

, Ω

∂ λ

,

displaystyle L_ mathrm e , Omega
Omega
,lambda = frac partial L_ mathrm e , Omega
Omega
partial lambda ,

where λ is the wavelength. Conservation of basic radiance[edit] Radiance of a surface is related to étendue by

L

e

, Ω

=

n

2

Φ

e

∂ G

,

displaystyle L_ mathrm e , Omega
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[4]

L

e

, Ω

=

L

e

, Ω

n

2

displaystyle L_ mathrm e , Omega
Omega
^ * = frac L_ mathrm e , Omega
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. SI radiometry units[edit]

SI radiometry units

v t e

Quantity Unit Dimension Notes

Name Symbol[nb 1] Name Symbol Symbol

Radiant energy Qe[nb 2] joule J M⋅L2⋅T−2 Energy of electromagnetic radiation.

Radiant energy
Radiant energy
density we joule per cubic metre J/m3 M⋅L−1⋅T−2 Radiant energy
Radiant energy
per unit volume.

Radiant flux Φe[nb 2] watt W = J/s M⋅L2⋅T−3 Radiant energy
Radiant energy
emitted, reflected, transmitted or received, per unit time. This is sometimes also called "radiant power".

Spectral flux Φe,ν[nb 3]  or Φe,λ[nb 4] watt per hertz  or watt per metre W/Hz  or W/m M⋅L2⋅T−2  or M⋅L⋅T−3 Radiant flux per unit frequency or wavelength. The latter is commonly measured in W⋅nm−1.

Radiant intensity Ie,Ω[nb 5] watt per steradian W/sr M⋅L2⋅T−3 Radiant flux emitted, reflected, transmitted or received, per unit solid angle. This 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 M⋅L2⋅T−2  or M⋅L⋅T−3 Radiant intensity per unit frequency or wavelength. The latter is commonly measured in W⋅sr−1⋅nm−1. This is a directional quantity.

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 a surface, per unit solid angle per unit projected area. This is a directional quantity. This 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 M⋅T−2  or M⋅L−1⋅T−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".

Irradiance Flux density Ee[nb 2] watt per square metre W/m2 M⋅T−3 Radiant flux received by a surface per unit area. This is sometimes also confusingly called "intensity".

Spectral irradiance Spectral flux density 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 M⋅T−2  or M⋅L−1⋅T−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 (1 Jy = 10−26 W⋅m−2⋅Hz−1) and solar flux unit (1 sfu = 10−22 W⋅m−2⋅Hz−1 = 104 Jy).

Radiosity Je[nb 2] watt per square metre W/m2 M⋅T−3 Radiant flux leaving (emitted, reflected and transmitted by) a surface per unit area. This 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 M⋅T−2  or M⋅L−1⋅T−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 exitance Me[nb 2] watt per square metre W/m2 M⋅T−3 Radiant 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 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 M⋅T−2  or M⋅L−1⋅T−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".

Radiant exposure He joule per square metre J/m2 M⋅T−2 Radiant energy
Radiant energy
received by a surface per unit area, or equivalently irradiance of a surface integrated over time of irradiation. This is sometimes also called "radiant fluence".

Spectral exposure He,ν[nb 3]  or He,λ[nb 4] joule per square metre per hertz  or joule per square metre, per metre J⋅m−2⋅Hz−1  or J/m3 M⋅T−1  or M⋅L−1⋅T−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 emissivity ε

1 Radiant exitance of a surface, divided by that of a black body at the same temperature as that surface.

Spectral hemispherical emissivity εν  or ελ

1 Spectral exitance of a surface, divided by that of a black body at the same temperature as that surface.

Directional emissivity εΩ

1 Radiance emitted by a surface, divided by that emitted by a black body at the same temperature as that surface.

Spectral directional emissivity εΩ,ν  or εΩ,λ

1 Spectral radiance emitted by a surface, divided by that of a black body at the same temperature as that surface.

Hemispherical absorptance A

1 Radiant flux absorbed by a surface, divided by that received by that surface. This should not be confused with "absorbance".

Spectral hemispherical absorptance Aν  or Aλ

1 Spectral flux absorbed by a surface, divided by that received by that surface. This should not be confused with "spectral absorbance".

Directional absorptance AΩ

1 Radiance absorbed by a surface, divided by the radiance incident onto that surface. This should not be confused with "absorbance".

Spectral directional absorptance AΩ,ν  or AΩ,λ

1 Spectral radiance absorbed by a surface, divided by the spectral radiance incident onto that surface. This should not be confused with "spectral absorbance".

Hemispherical reflectance R

1 Radiant flux reflected by a surface, divided by that received by that surface.

Spectral hemispherical reflectance Rν  or Rλ

1 Spectral flux reflected by a surface, divided by that received by that surface.

Directional reflectance RΩ

1 Radiance reflected by a surface, divided by that received by that surface.

Spectral directional reflectance RΩ,ν  or RΩ,λ

1 Spectral radiance reflected by a surface, divided by that received by that surface.

Hemispherical transmittance T

1 Radiant flux transmitted by a surface, divided by that received by that surface.

Spectral hemispherical transmittance Tν  or Tλ

1 Spectral flux transmitted by a surface, divided by that received by that surface.

Directional transmittance TΩ

1 Radiance transmitted by a surface, divided by that received by that surface.

Spectral directional transmittance TΩ,ν  or TΩ,λ

1 Spectral radiance transmitted by a surface, divided by that received by that surface.

Hemispherical attenuation coefficient μ reciprocal metre m−1 L−1 Radiant flux absorbed and scattered by a volume per unit length, divided by that received by that volume.

Spectral hemispherical attenuation coefficient μν  or μλ reciprocal metre m−1 L−1 Spectral radiant flux absorbed and scattered by a volume per unit length, divided by that received by that volume.

Directional attenuation coefficient μΩ reciprocal metre m−1 L−1 Radiance absorbed and scattered by a volume per unit length, divided by that received by that volume.

Spectral directional attenuation coefficient μΩ,ν  or μΩ,λ reciprocal metre m−1 L−1 Spectral radiance absorbed and scattered by a volume per unit length, divided by that received by that volume.

See also: SI · Radiometry · Photometry

^ Standards organizations recommend that radiometric quantities should be denoted with suffix "e" (for "energetic") to avoid confusion with photometric or photon quantities. ^ 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. ^ a b c d e f g Spectral quantities given per unit frequency are denoted with suffix "ν" (Greek)—not to be confused with suffix "v" (for "visual") indicating a photometric quantity. ^ a b c d e f g Spectral quantities given per unit wavelength are denoted with suffix "λ" (Greek). ^ a b Directional quantities are denoted with suffix "Ω" (Greek).

See also[edit]

Étendue Light field Sakuma–Hattori equation Wien displacement law

References[edit]

^ Palmer, James M. "The SI system and SI units for Radiometry and photometry" (PDF). Archived from the original (PDF) on August 2, 2012.  ^ Rowlett, Russ. "How Many? A Dictionary of Units of Measurement". Retrieved 10 August 2012.  ^ a b c "Thermal insulation — Heat transfer
Heat transfer
by radiation — Physical quantities and definitions". ISO 9288:1989. ISO catalogue. 1989. Retrieved 2015-03-15.  ^ William Ross McCluney, Introduction to Radiometry and Photometry, Artech House, Boston, MA, 1994 ISBN 978-0890066782

External links[edit]

International Lighting in Controlled Environme

.