Radiance

Radiance is a radiometric quantity that describes how much radiant energy travels through or is emitted by a given area in a particular direction. It is defined by the relation dΦ = L dA cos θ dΩ, where dΦ is the differential radiant flux, dA is the area element, θ is the angle between the area normal and the direction of travel, and dΩ is the differential solid angle. The radiance L has units of watts per square meter per steradian (W m^-2 sr^-1). Spectral radiance Lλ includes wavelength dependence and is measured per unit wavelength.

Radiance is distinct from irradiance, which measures radiant flux per unit area incident on a surface, and from luminance, which is the photometric analogue used in human vision. Radiance can be integrated over wavelength to yield total radiance, or evaluated as spectral radiance Lλ. The irradiance on a surface is E = ∫∫ L cos θ dΩ, illustrating how radiance and geometry determine perceived brightness. In optics, radiance is conserved along a ray in lossless systems, a consequence of etendue preservation; this means radiance remains constant through ideal lenses and free-space propagation that involves no absorption or scattering.

In practical terms, radiance is central to imaging, rendering, and remote sensing. In rendering, the outgoing radiance from a surface in a given direction is governed by the BRDF, which relates incoming radiance to reflected radiance. For a Lambertian surface, the radiance is independent of viewing direction and depends on the surface’s emission or reflected energy. Radiance concepts underpin imaging design, telescope throughput, and energy balance in illumination studies.