Détendue

Détendue, or étendue, is a concept in optics that describes how spread out light is in both physical space and angle. It combines the cross-sectional area that a beam occupies with the range of directions the light rays can take, and it is conserved in lossless optical systems. This makes it a fundamental measure of how much light can pass through an optical setup.

Mathematically, the differential etendue is dG = cos θ dA dΩ, where dA is a small area on a surface, θ is the angle between the ray and the surface normal, and dΩ is a differential solid angle. The total étendue of a beam is G = ∫∫ cos θ dA dΩ, with units of m^2 sr. In many practical cases, especially for small angles, G is approximated by G ≈ A Ω, where A is the aperture area and Ω is the associated solid angle.

As a conservation law, étendue cannot be reduced by passive optical elements such as lenses and mirrors. In an ideal imaging system, G is preserved from source to image. This property constrains the throughput of optical instruments: a beam with a given étendue cannot illuminate a system whose exit étendue is smaller, and it informs decisions about source brightness, aperture size, and detector area. Étendue also relates to numerical aperture in fiber optics, where the maximum acceptance angle determines the portion of solid angle that can be guided, thereby bounding the system’s overall étendue.

Détendue is central to radiometry and geometric optics, used to assess and optimize the balance between light sources, optical components, and detectors, and to analyze the efficiency limits of imaging and illumination systems.