Etendue
Other names for etendue include acceptance, throughput, light grasp, light-gathering power, optical extent,[1] and the AΩ product.
These definitions must be applied for infinitesimally small "elements" of area and solid angle, which must then be summed over both the source and the diaphragm as shown below.
[5] A perfect optical system produces an image with the same etendue as the source.
The radiance of an optical system is equal to the derivative of the radiant flux with respect to the etendue.
An infinitesimal surface element, dS, with normal nS is immersed in a medium of refractive index n. The surface is crossed by (or emits) light confined to a solid angle, dΩ, at an angle θ with the normal nS.
The etendue of an infinitesimal bundle of light crossing dS is defined as
Etendue is the product of geometric extent and the squared refractive index of a medium through which the beam propagates.
However, it can alternatively be expressed in units of area (square meters) multiplied by solid angle (steradians).
[1][6] Consider a light source Σ, and a light detector S, both of which are extended surfaces (rather than differential elements), and which are separated by a medium of refractive index n that is perfectly transparent (shown).
[7][better source needed] According to the definition above, the etendue of the light crossing dΣ towards dS is given by:
Similarly, the etendue of the light crossing dS coming from dΣ is given by:
showing that etendue is conserved as light propagates in free space.
If both surfaces dΣ and dS are immersed in air (or in vacuum), n = 1 and the expression above for the etendue may be written as
Integration on dΣ and dS results in G = πΣ FΣ→S which allows the etendue between two surfaces to be obtained from the view factors between those surfaces, as provided in a list of view factors for specific geometry cases or in several heat transfer textbooks.
For example, a magnifying glass can increase the intensity of sunlight onto a small spot, but does so because, viewed from the spot that the light is concentrated onto, the apparent size of the sun is increased proportional to the concentration.
As shown below, etendue is conserved as light travels through free space and at refractions or reflections.
It is then also conserved as light travels through optical systems where it undergoes perfect reflections or refractions.
Etendue can then remain constant or it can increase as light propagates through an optic, but it cannot decrease.
[2][page needed] From the perspective of thermodynamics, etendue is a form of entropy.
This increases entropy due to heat, allowing a corresponding decrease in etendue.
[8][9] The conservation of etendue in free space is related to the reciprocity theorem for view factors.
The conservation of etendue discussed above applies to the case of light propagation in free space, or more generally, in a medium of any refractive index.
[2][page needed] Figure "etendue in refraction" shows an infinitesimal surface dS on the x-y plane separating two media of refractive indices nΣ and nS.
where As the light travels through an ideal optical system, both the etendue and the radiant flux are conserved.
and therefore, for an infinitesimal area dS = dx dy on the xy-plane immersed in a medium of refractive index n, the etendue is given by
Conservation of etendue in phase space is the equivalent in optics to Liouville's theorem in classical mechanics.
[2][page needed] Etendue as volume in phase space is commonly used in nonimaging optics.
Consider an infinitesimal surface dS, immersed in a medium of refractive index n crossed by (or emitting) light inside a cone of angle α.
Noting that n sin α is the numerical aperture NA, of the beam of light, this can also be expressed as
If the optic were a collimator instead of a concentrator, the light direction is reversed and conservation of etendue gives us the minimum aperture, S, for a given output full angle 2α.
