Radiance

Radiance describes light leaving or crossing a surface in a particular direction: the power $d^2\Phi$ emitted from a small area $dA$ into a small solid angle $d\Omega$ at angle $\theta$ to the surface normal,

in W/(m²·sr). It is the quantity a camera pixel or the eye responds to when viewing an extended source, and it is what "brightness" means in radiometry. Unlike irradiance, which falls with distance from a source, the radiance of an extended source is the same at any distance: a wall looks equally bright from near and far, because the smaller solid angle it fills is exactly offset by the larger area each detector element sees.

Conservation

In a lossless optical system, $L/n^2$ is conserved along a ray, where $n$ is the local refractive index. This is the radiance theorem, equivalent to the conservation of étendue: lenses and mirrors can concentrate power onto a smaller area only by spreading it over a wider solid angle, so they can never make an image brighter than its source. It sets the limit on solar concentration, on how much LED light can be coupled into a fiber, and on the irradiance any lens can produce from a given lamp.

Typical values

Sunlight above the atmosphere delivers about 1361 W/m², and the Sun's disk subtends about 6.8 × 10⁻⁵ sr, so its average radiance is about 2.0 × 10⁷ W/(m²·sr). A 1 mW helium-neon laser with a 0.4 mm waist radius and a 0.50 mrad divergence emits from an area of about 5 × 10⁻⁷ m² into a solid angle of about 8 × 10⁻⁷ sr, a radiance of about 2.5 × 10⁹ W/(m²·sr), roughly 125 times the Sun's. Laser light is distinguished by this high radiance, which lets a milliwatt beam be focused to an irradiance that ordinary lamps cannot reach.

A Lambertian surface, a perfect diffuser, has the same radiance in all directions, and its exitance (total power emitted per unit area) is $\pi L$. White paper and ground glass are approximately Lambertian; the factor $\pi$ rather than $2\pi$ comes from the $\cos\theta$ weighting.

Photometric counterpart and measurement

Weighted by the eye's response, radiance becomes luminance, in candela per square metre, covered under brightness. Radiance is measured with a radiometer that has a defined field of view and aperture, so that both the area and the solid angle it accepts are known, or with a calibrated imaging camera; the instrument is calibrated against a standard source such as an integrating sphere of known radiance.

References: R. W. Boyd, Radiometry and the Detection of Optical Radiation (Wiley, 1983); ISO 80000-7:2019, Quantities and units: Light and radiation; G. Kopp, J. L. Lean, Geophys. Res. Lett. 38, L01706 (2011).