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

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

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 Lλ and the radiance per unit frequency is written Lν.

## Intensity

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.

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