Étendue

Étendue measures how spread out light is, in position and in direction together. For light crossing an area $A$ within a cone of half angle $\theta$ in a medium of index $n$, filled uniformly as from a Lambertian source,

in units of m²·sr or mm²·sr. In general it is the integral of $n^2\cos\theta\,dA\,d\Omega$ over the beam. The quantity $n\sin\theta$ is the numerical aperture, so for a fiber or a lens the étendue it accepts is $\pi A\,\mathrm{NA}^2$.

Conservation

In a lossless optical system the étendue of a beam cannot decrease. A lens can concentrate light onto a smaller area only by spreading it over a larger solid angle, and the reverse. The same principle, stated for a single dimension of a laser beam, is the conservation of the beam parameter product, and it is equivalent to the statement that passive optics cannot make an image brighter, in radiance, than the source. Étendue can grow, through scattering or aberrations, and it can be cut by an aperture, at the cost of power.

What it limits

Coupling a source into a fiber or onto a detector is efficient only when the source's étendue fits within what the receiver accepts. A multimode fiber with a 50 µm core and an NA of 0.22 accepts $3.0 \times 10^{-4}$ mm²·sr. A 1 mm² Lambertian LED emitting into a full hemisphere has an étendue of $\pi$ mm²·sr, about 10,500 times larger, so at most about one ten-thousandth of its light can enter the fiber however it is imaged, and less into a single-mode fiber. A single transverse mode, by contrast, occupies an étendue of order $\lambda^2$, which is why a laser beam can be focused into a single-mode fiber with high efficiency; the étendue of a multimode fiber divided by $\lambda^2$ gives, to within a factor of order one, the number of modes it guides.

Illumination optics, solar concentrators and projectors are all designed around étendue. The maximum concentration of sunlight, for instance, follows from the sun's angular size, and a projector lamp's étendue sets how much of its light reaches the screen through a small imaging chip.

Measurement

Étendue is computed rather than measured directly: from the area of the source or aperture and the angular spread of its emission. The angular distribution comes from a goniometer or a far-field camera, and the area from a near-field image of the emitting surface.

References: R. Winston, J. C. Miñano, P. Benítez, Nonimaging Optics (Elsevier, 2005), Ch. 2; W. T. Welford, R. Winston, High Collection Nonimaging Optics (Academic Press, 1989).