The concept of etendue starts with the idea that it is really hard to stuff a lot of light into a small hole. Etendue is the number that tells us how big the hole is.
With that basic idea, let’s start diving into the details. The amount of light is quantified as energy and is normally measured in Watts or lumens. It is represented by the symbol Φ (pronounced fie, rhymes with pie). The equation relating the amount of light to etendue is
Φ = L A Ω
where L is the luminance or radiance of the source, A is the area of the source and Ω (omega) is the solid angle of the beam of light. Luminance or radiance is a physical property of a light source; it is something you have to look up. Solid angle is the 3D equivalent of the angles we learned in geometry. It is measured in steradians (sr). If the beam of light is a cone, its solid angle is given by
Ω = 2π sin²θ
where θ (theta) is the half-angle of the cone (angle between the center and the edge). A hemisphere has a solid angle of 2π sr, and an f/1 cone (30° half angle) has a solid angle of .π/2 sr.
An important principle to remember is conservation of energy. If we transmit light from a source to an image, unless light is absorbed, reflected or scattered out of the beam, it will get to the image. Similarly, radiance/luminance is conserved. And if both energy and radiance/luminance are conserved, so must etendue be. From conservation of etendue, we conclude that if we collect all of the light from a source with area A1 that is emitting light into a hemisphere and transmit it to an image, that is illuminated from a hemisphere, the area of the image must be also be A1. On the other hand, if the image is illuminated from less than a hemisphere (Ω at image < Ω at source), then the area of the image must be proportionally greater. Specifically, if the image is illuminated by an f/1 cone, the area of the image must be 4*A1.
Do you need expert advice on illumination system design? Eckhardt Optics can help. Contact us or call us at (651)315-8249 for a consultation with our engineers.