Radiance

For other uses, see Radiance (disambiguation).

Radiance and spectral radiance are radiometric measures that describe the amount of radiation such as light or radiant heat that passes through or is emitted from a particular area, and falls within a given solid angle in a specified direction. They are used to characterize both emission from diffuse sources and reflection from diffuse surfaces. The SI unit of radiance is watts per steradian per square metre (W·sr⁻¹·m⁻²), while that of spectral radiance is W·sr⁻¹·m⁻²·Hz⁻¹.

Description

Radiance characterizes total emission or reflection. Radiance is useful because it indicates how much of the power emitted by an emitting or reflecting surface will be received by an optical system looking at the 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 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 units of W·sr⁻¹·m⁻²·nm⁻¹ (more common than W·sr⁻¹·m^-3). Radiance is the integral of the spectral radiance over all wavelengths.

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/m²/sr, 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.

Definition

Radiance is defined by

$L = \frac{\mathrm{d}^2 \Phi}{\mathrm{d}A\,\mathrm{d}{\Omega} \cos \theta} \approx \frac{\Phi}{\Omega A \cos \theta}$

where

L is the observed or measured radiance (W·m⁻²·sr⁻¹), in the direction θ,

Φ is the total radiant flux or power (W) emitted

θ is the angle between the surface normal and the specified direction,

A is the area of the surface (m²), and

Ω is the solid angle (sr) subtended by the observation or measurement.

The approximation only holds for small A and Ω where cos θ is approximately constant.

In general, L is a function of viewing angle through the cos θ term in the denominator as well as the θ, and potentially azimuth angle, dependence of dΦ / dΩ. For the special case of a Lambertian source, L is constant such that dΦ / dΩ is proportional to cos θ.

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 at 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.

The spectral radiance (radiance per unit wavelength) is written L_λ and the radiance per unit frequency is written L_ν.

Intensity

See also: Intensity (disambiguation) and Intensity (heat transfer)

Radiance is often, confusingly, called intensity in other areas of study, especially heat transfer, astrophysics and astronomy. Intensity has many other meanings in physics, with the most common being power per unit area. The distinction lies in the area rather than the subtended angle of the observer, and relative area of the source.

See also

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

External links

• International Lighting in Controlled Environments Workshop

SI radiometry unitsv · d · e

Quantity	Symbol[nb 1]	SI unit	Symbol	Dimension	Notes
Radiant energy	Qe[nb 2]	joule	J	M⋅L2⋅T−2	energy
Radiant flux	Φe[nb 2]	watt	W	M⋅L2⋅T−3	radiant energy per unit time, also called radiant power.
Spectral power	Φeλ[nb 2][nb 3]	watt per metre	W⋅m−1	M⋅L⋅T−3	radiant power per wavelength.
Radiant intensity	Ie	watt per steradian	W⋅sr−1	M⋅L2⋅T−3	power per unit solid angle.
Spectral intensity	Ieλ[nb 3]	watt per steradian per metre	W⋅sr−1⋅m−1	M⋅L⋅T−3	radiant intensity per wavelength.
Radiance	Le	watt per steradian per square metre	W⋅sr−1⋅m−2	M⋅T−3	power per unit solid angle per unit projected source area. confusingly called "intensity" in some other fields of study.
Spectral radiance	Leλ[nb 3] or Leν[nb 4]	watt per steradian per metre3 or watt per steradian per square metre per hertz	W⋅sr−1⋅m−3 or W⋅sr−1⋅m−2⋅Hz−1	M⋅L−1⋅T−3 or M⋅T−2	commonly measured in W⋅sr−1⋅m−2⋅nm−1 with surface area and either wavelength or frequency.
Irradiance	Ee[nb 2]	watt per square metre	W⋅m−2	M⋅T−3	power incident on a surface, also called radiant flux density. sometimes confusingly called "intensity" as well.
Spectral irradiance	Eeλ[nb 3] or Eeν[nb 4]	watt per metre3 or watt per square metre per hertz	W⋅m−3 or W⋅m−2⋅Hz−1	M⋅L−1⋅T−3 or M⋅T−2	commonly measured in W⋅m−2⋅nm−1 or 10−22W⋅m−2⋅Hz−1, known as solar flux unit.[nb 5]
Radiant exitance / Radiant emittance	Me[nb 2]	watt per square metre	W⋅m−2	M⋅T−3	power emitted from a surface.
Spectral radiant exitance / Spectral radiant emittance	Meλ[nb 3] or Meν[nb 4]	watt per metre3 or watt per square metre per hertz	W⋅m−3 or W⋅m−2⋅Hz−1	M⋅L−1⋅T−3 or M⋅T−2	power emitted from a surface per wavelength or frequency.
Radiosity	Je or Jeλ[nb 3]	watt per square metre	W⋅m−2	M⋅T−3	emitted plus reflected power leaving a surface.
Radiant exposure	He	joule per square metre	J⋅m−2	M⋅T−2
Radiant energy density	ωe	joule per metre3	J⋅m−3	M⋅L−1⋅T−2
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 emittance.
3. ^ ^{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, reflectance and responsivity.
4. ^ ^{a b c} 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.
5. ^ NOAA / Space Weather Prediction Center includes a definition of the solar flux unit (SFU).