Etendue (source code)
= Etendue
{title2=$G=n^2A\Omega$}
<Etendue> measures how much position-and-direction space a bundle of <light rays> occupies. A small spot accepting a wide range of directions can have the same <etendue> as a large spot accepting a narrow range. This makes <etendue> useful for deciding whether an <optical instrument> can deliver an extended source's entire admitted <radiant flux> to a small receiver. For forward crossings of a surface, the <geometrical optics> definition is
$$
G=\int_{\mathcal B}n^2\cos\theta\,dA\,d\Omega,
\qquad 0\leq\theta<\frac{\pi}{2},
$$
where $\mathcal B$ is the occupied set of surface positions and directions, $n$ is the local <refractive index>, $dA$ is actual surface <area>, $d\Omega$ is <solid angle>, and $\theta$ is the <angle> to the forward <unit normal>. The <cosine> supplies the <projected area>. Opposite crossing directions must be counted separately with positive <projected area>, rather than cancelling them in a signed <integral>. <Étendue (Photon Snacks) by R. John Koshel>[Étendue, p. 1, equation (1.1): refractive-index-weighted projected-area integral]{ref}
The shorthand $G\simeq n^2A\Omega$ assumes constant <refractive index>, a narrow nearly normal bundle, and the same admitted directions across the whole <area> $A$. At larger <angles>, use the <cosine>-weighted <integral>; a position-dependent angular acceptance cannot generally be replaced by the product of two independent extents.
Back to article page
