Radiance

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.^[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.

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 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 or wavelength. Radiance is the integral of the spectral radiance over all frequencies or wavelengths. For radiation emitted by an ideal black body at a given temperature, spectral radiance is governed by Planck's law, while the integral of radiance over the hemisphere into which it radiates, is governed by the Stefan–Boltzmann law. There is no need for a separate law for the radiance of a black body, 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. A black body being Lambertian, its radiance does not depend on 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^[3]

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

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, ∂²Φ_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^[3]

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

where ν is the frequency.

Spectral radiance in wavelength of a surface, denoted L_e,Ω,λ, is defined as^[3]

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

where λ is the wavelength.

Conservation of basic radiance

Radiance of a surface is related to étendue by

$L_{\mathrm{e},\Omega} = n^2 \frac{\partial \Phi_\mathrm{e}}{\partial G},$

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^[4]

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

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.

SI radiometry units

SI radiometry units

Quantity		Unit		Dimension	Notes
Name	Symbol[nb 1]	Name	Symbol	Symbol
Radiant energy	Qe[nb 2]	joule	J	M⋅L^2⋅T^−2	Energy of electromagnetic radiation.
Radiant energy density	we	joule per cubic metre	J/m^3	M⋅L^−1⋅T^−2	Radiant energy per unit volume.
Radiant flux	Φe[nb 2]	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	Φe,ν[nb 3] or Φe,λ[nb 4]	watt per hertz or watt per metre	W/Hz or W/m	M⋅L^2⋅T^−2 or M⋅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	Ie,Ω[nb 5]	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	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⋅L^2⋅T^−2 or M⋅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	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	Ee[nb 2]	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	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/m^3	M⋅T^−2 or M⋅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	Je[nb 2]	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	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/m^3	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/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	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/m^3	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/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	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/m^3	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

1. Standards organizations recommend that radiometric quantities should be denoted with suffix "e" (for "energetic") to avoid confusion with photometric or photon quantities.
2. Alternative symbols sometimes seen: W or E for radiant energy, P or F for radiant flux, I for irradiance, W for radiant exitance.
3. 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.
4. Spectral quantities given per unit wavelength are denoted with suffix "λ" (Greek).
5. Directional quantities are denoted with suffix "Ω" (Greek).

See also

• Étendue
• Light field
• Sakuma–Hattori equation
• Wien displacement law

References

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

External links

• International Lighting in Controlled Environments Workshop