In radiometry, **radiance** is the radiant flux emitted, reflected, transmitted or received by a *surface*, per unit solid angle per unit *projected* area, and **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.^{[3]}^{[4]} 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.

## 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

**Radiance** of a *surface*, denoted *L*_{e,Ω} ("e" for "energetic", to avoid confusion with photometric quantities, and "Ω" to indicate this is a directional quantity), is defined as^{[5]}

where

- ∂ is the partial derivative symbol;
- Φ
_{e}is the radiant flux emitted, reflected, transmitted or received; *Ω*is the solid angle;*A*cos*θ*is the*projected*area.

In general *L*_{e,Ω} is a function of viewing direction, depending on *θ* through cos *θ* and azimuth angle through ∂Φ_{e}/∂*Ω*. For the special case of a Lambertian surface, ∂^{2}Φ_{e}/(∂*Ω* ∂*A*) is proportional to cos *θ*, and *L*_{e,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

**Spectral radiance in frequency** of a *surface*, denoted *L*_{e,Ω,ν}, is defined as^{[5]}

where *ν* is the frequency.

**Spectral radiance in wavelength** of a *surface*, denoted *L*_{e,Ω,λ}, is defined as^{[5]}

where *λ* is the wavelength.

## Conservation of basic radiance

Radiance of a surface is related to étendue by

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

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**

Quantity | Unit | Dimension | Notes | |||||
---|---|---|---|---|---|---|---|---|

Name | Symbol | Name | Symbol | Symbol | ||||

Radiant energy | Q_{e} | joule | J | M⋅L^{2}⋅T^{−2} | Energy of electromagnetic radiation. | |||

Radiant energy density | w_{e} | joule per cubic metre | J/m^{3} | M⋅L^{−1}⋅T^{−2} | Radiant energy per unit volume. | |||

Radiant flux | Φ_{e} | watt | W or J/s | M⋅L^{2}⋅T^{−3} | Radiant energy emitted, reflected, transmitted or received, per unit time. This is sometimes also called "radiant power". | |||

Spectral flux | Φ_{e,ν}or Φ _{e,λ} | watt per hertzorwatt per metre | W/HzorW/m | M⋅L^{2}⋅T^{−2}orM⋅L⋅T^{−3} | Radiant flux per unit frequency or wavelength. The latter is commonly measured in W⋅sr^{−1}⋅m^{−2}⋅nm^{−1}. | |||

Radiant intensity | I_{e,Ω} | watt per steradian | W/sr | M⋅L^{2}⋅T^{−3} | Radiant flux emitted, reflected, transmitted or received, per unit solid angle. This is a directional quantity. | |||

Spectral intensity | I_{e,Ω,ν}or I_{e,Ω,λ} | watt per steradian per hertzorwatt per steradian per metre | W⋅sr^{−1}⋅Hz^{−1}orW⋅sr ^{−1}⋅m^{−1} | M⋅L^{2}⋅T^{−2}orM⋅L⋅T^{−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. | |||

Radiance | L_{e,Ω} | 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 | L_{e,Ω,ν}or L_{e,Ω,λ} | watt per steradian per square metre per hertzorwatt per steradian per square metre, per metre | W⋅sr^{−1}⋅m^{−2}⋅Hz^{−1}orW⋅sr ^{−1}⋅m^{−3} | M⋅T^{−2}orM⋅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 | E_{e} | watt per square metre | W/m^{2} | M⋅T^{−3} | Radiant flux received by a surface per unit area. This is sometimes also confusingly called "intensity". | |||

Spectral irradiance | E_{e,ν}or E_{e,λ} | watt per square metre per hertzorwatt per square metre, per metre | W⋅m^{−2}⋅Hz^{−1}orW/m ^{3} | M⋅T^{−2}orM⋅L^{−1}⋅T^{−3} | Irradiance of a surface per unit frequency or wavelength. The terms spectral flux density or more confusingly "spectral intensity" are also used. Non-SI units of spectral irradiance include Jansky = 10^{−26} W⋅m^{−2}⋅Hz^{−1} and solar flux unit (1SFU = 10^{−22} W⋅m^{−2}⋅Hz^{−1}). | |||

Radiosity | J_{e} | watt per square metre | W/m^{2} | 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 | J_{e,ν}or J_{e,λ} | watt per square metre per hertzorwatt per square metre, per metre | W⋅m^{−2}⋅Hz^{−1}orW/m ^{3} | M⋅T^{−2}orM⋅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 | M_{e} | watt per square metre | W/m^{2} | 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 | M_{e,ν}or M_{e,λ} | watt per square metre per hertzorwatt per square metre, per metre | W⋅m^{−2}⋅Hz^{−1}orW/m ^{3} | M⋅T^{−2}orM⋅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 | H_{e} | joule per square metre | J/m^{2} | M⋅T^{−2} | 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 | H_{e,ν}or H_{e,λ} | joule per square metre per hertzorjoule per square metre, per metre | J⋅m^{−2}⋅Hz^{−1}orJ/m ^{3} | M⋅T^{−1}orM⋅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 | ε_{ν}or ε_{λ} | 1 | Spectral exitance of a surface, divided by that of a black body at the same temperature as that surface. | |||||

Directional emissivity | ε_{Ω} | 1 | Radiance emitted by a surface, divided by that emitted by a black body at the same temperature as that surface. | |||||

Spectral directional emissivity | ε_{Ω,ν}or ε_{Ω,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 | A_{ν}or A_{λ} | 1 | Spectral flux absorbed by a surface, divided by that received by that surface. This should not be confused with "spectral absorbance". | |||||

Directional absorptance | A_{Ω} | 1 | Radiance absorbed by a surface, divided by the radiance incident onto that surface. This should not be confused with "absorbance". | |||||

Spectral directional absorptance | A_{Ω,ν}or A_{Ω,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 | R_{ν}or R_{λ} | 1 | Spectral flux reflected by a surface, divided by that received by that surface. | |||||

Directional reflectance | R_{Ω} | 1 | Radiance reflected by a surface, divided by that received by that surface. | |||||

Spectral directional reflectance | R_{Ω,ν}or R_{Ω,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 | T_{ν}or T_{λ} | 1 | Spectral flux transmitted by a surface, divided by that received by that surface. | |||||

Directional transmittance | T_{Ω} | 1 | Radiance transmitted by a surface, divided by that received by that surface. | |||||

Spectral directional transmittance | T_{Ω,ν}or T_{Ω,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 | μ_{ν}or μ_{λ} | 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 | μ_{Ω} | 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 | μ_{Ω,ν}or μ_{Ω,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.
- Alternative symbols sometimes seen:
*W*or*E*for radiant energy,*P*or*F*for radiant flux,*I*for irradiance,*W*for radiant exitance. - 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.
- Spectral quantities given per unit wavelength are denoted with suffix "λ" (Greek).
- Directional quantities are denoted with suffix "Ω" (Greek).