étendue

Étendue is a concept in radiometry and geometric optics that quantifies how much light can propagate through an optical system. It is defined as the product of the cross-sectional area of a beam and the solid angle through which the beam occupies space, effectively measuring the beam’s “throughput.” For a differential element, the differential étendue is dG = dA cos θ dΩ, where dA is a surface area element, θ is the angle between the ray direction and the surface normal, and dΩ is the differential solid angle. The total étendue of a beam is the integral G = ∫∫ dA cos θ dΩ over the light distribution.

In lossless, passive optical systems, étendue is conserved: the output étendue cannot exceed the input étendue. This conservation is a geometric-optics analogue of Liouville’s theorem and is closely tied to the invariance of radiance (brightness) along rays in such systems. Radiance L, defined as flux per unit area per unit solid angle, remains constant along an optical path, and the relation dG = dA cos θ dΩ underpins throughput limitations in imaging and beam shaping.

Applications of étendue include assessing and bounding the maximum flux transmitted through apertures and angular acceptances, comparing the brightness efficiency of instruments, and informing design trade-offs in telescopes, cameras, fiber optics, and photolithography. For simple geometries, the étendue of a circular aperture of area A viewed through a cone of half-angle α is G = A Ω, with Ω = 2π(1 − cos α).

Limitations arise from real-world losses such as vignetting, scattering, and absorption, which reduce étendue; nevertheless, passive optical systems cannot increase étendue beyond what the input beam provides.