létendue

L'étendue, often written étendue in optics literature, is a property of light beams that describes the trade-off between the beam’s cross-sectional area and the range of directions it occupies. In geometric optics and radiometry, it is defined as the product of the beam’s cross-sectional area A and the solid angle Ω it subtends, E = AΩ. For a small, well-collimated ray bundle this reduces to E ≈ AΩ, with units of square meters steradians (m^2 sr). The concept captures how much light can pass through an optical system given both its size and angular spread.

Etendue is invariant in lossless optical systems: for the portion of light that is transmitted without clipping, the etendue entering a system equals the etendue exiting it. This stems from Liouville’s theorem in phase space and underpins fundamental limits on imaging brightness and optical concentration. In practice, optical elements that clip rays, such as stops or vignetting, can reduce etendue by reducing either A or Ω.

Applications and implications of étendue include setting limits on the achievable combination of spatial resolution and angular acceptance in imaging, and defining how much a light source can be concentrated or how efficiently light can be coupled into an instrument or fiber. It is a useful design metric in illumination, projection, and optical coupling, where a source with high etendue cannot be efficiently mapped into a system with smaller etendue.

Relation to radiance: etendue is connected to radiance, as E = ∫∫ L dA dΩ, illustrating that brightness is conserved in passive optical paths.