For other uses, see Radiance (disambiguation).

## 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 (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

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.

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

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.