thumb|Conservation of etendue
Etendue or étendue () is a property of light in an optical system, which characterizes how "spread out" the light is in area and angle. It corresponds to the beam parameter product (BPP) in Gaussian beam optics. Other names for etendue include acceptance, throughput, light grasp, light-gathering power, optical extent, and the AΩ product. Throughput and AΩ product are especially used in radiometry and radiative transfer where it is related to the view factor (or shape factor). It is a central concept in nonimaging optics. The term étendue comes from French, where it means "extent".
From the source point of view, etendue is the product of the area of the source and the solid angle that the system's entrance pupil subtends as seen from the source. Equivalently, from the system point of view, the etendue equals the area of the entrance pupil times the solid angle the source subtends as seen from the pupil. 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. Etendue may be considered to be a volume in phase space.
Etendue never decreases in any optical system where optical power is conserved. A perfect optical system produces an image with the same etendue as the source. The etendue is related to the Lagrange invariant and the optical invariant, which also share the property of being constant in an ideal optical system. The radiance of an optical system is equal to the derivative of the radiant flux with respect to the etendue.
Definition
right|thumb|400px|Etendue for a [[differential element|differential surface element in 2D (left) and 3D (right).]]
An infinitesimal surface element, , with normal is immersed in a medium of refractive index . The surface is crossed by (or emits) light confined to a solid angle, , at an angle with the normal . The area of projected in the direction of the light propagation is . The etendue of an infinitesimal bundle of light crossing is defined as
<math display="block">\mathrm{d}G = n^2\, \mathrm{d}S \cos \theta\, \mathrm{d}\Omega\,.</math>
Etendue is the product of geometric extent and the squared refractive index of a medium through which the beam propagates.
In free space
right|thumb|300px|Etendue in free space.
Consider a light source , and a light detector , both of which are extended surfaces (rather than differential elements), and which are separated by a medium of refractive index that is perfectly transparent (shown). To compute the etendue of the system, one must consider the contribution of each point on the surface of the light source as they cast rays to each point on the receiver.
According to the definition above, the etendue of the light crossing towards is given by:
<math display="block">\mathrm{d}G_\Sigma = n^2\, \mathrm{d}\Sigma \cos \theta_\Sigma\, \mathrm{d}\Omega_\Sigma = n^2\, \mathrm{d}\Sigma \cos \theta_\Sigma \frac{\mathrm{d}S \cos \theta_S}{d^2}\,,</math>
where is the solid angle defined by area at area , and is the distance between the two areas. Similarly, the etendue of the light crossing coming from is given by:
<math display="block">\mathrm{d}G_S = n^2\, \mathrm{d}S \cos \theta_S\, \mathrm{d}\Omega_S = n^2\, \mathrm{d}S \cos \theta_S \frac{\mathrm{d}\Sigma \cos \theta_\Sigma}{d^2}\,,</math>
where is the solid angle defined by area . These expressions result in
<math display="block">\mathrm{d}G_\Sigma = \mathrm{d}G_S\,,</math>
showing that etendue is conserved as light propagates in free space.
The etendue of the whole system is then:
<math display="block">G = \int_\Sigma\!\int_S \mathrm{d}G\,.</math>
If both surfaces and are immersed in air (or in vacuum), and the expression above for the etendue may be written as
<math display="block">\mathrm{d}G = \mathrm{d}\Sigma\, \cos \theta_\Sigma\, \frac{\mathrm{d}S\, \cos \theta_S}{d^2} = \pi\, \mathrm{d}\Sigma\,\left(\frac{\cos \theta_\Sigma \cos \theta_S}{\pi d^2}\, \mathrm{d}S\right) = \pi\, \mathrm{d}\Sigma\, F_{\mathrm{d}\Sigma \rarr \mathrm{d}S}\,,</math>
where is the view factor between differential surfaces and . Integration on and results in which allows the etendue between two surfaces to be obtained from the view factors between those surfaces.
Conservation
The etendue of a given bundle of light is conserved: etendue can be increased, but not decreased in any optical system. This means that any system that concentrates light from some source onto a smaller area must always increase the solid angle of incidence (that is, the area of the sky that the source subtends). 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. However, if light was to hit, say, a diffuser, its solid angle would increase, increasing the etendue. Etendue can then remain constant or it can increase as light propagates through an optic, but it cannot decrease. This is a direct result of the fact that entropy must be constant or increasing.
Conservation of etendue can be derived in different contexts, such as from optical first principles, from Hamiltonian optics or from the second law of thermodynamics.
The conservation of etendue in free space is related to the reciprocity theorem for view factors.
In refractions and reflections
right|thumb|300px|Etendue in refraction.
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. In particular, etendue is conserved in refractions and reflections.
Conservation of basic radiance
Radiance of a surface is related to etendue by:
<math display="block">L_{\mathrm{e},\Omega} = n^2 \frac{\partial \Phi_\mathrm{e{\partial G}\,,</math>
where
- is the radiant flux emitted, reflected, transmitted or received;
- is the refractive index in which that surface is immersed;
- is the étendue of the light beam.
As the light travels through an ideal optical system, both the etendue and the radiant flux are conserved. Therefore, basic radiance defined as:
<math display="block">L_{\mathrm{e},\Omega}^* = \frac{L_{\mathrm{e},\Omega{n^2}</math>
is also conserved. In real systems, the etendue may increase (for example due to scattering) or the radiant flux may decrease (for example due to absorption) and, therefore, basic radiance may decrease. However, etendue may not decrease and radiant flux may not increase and, therefore, basic radiance may not increase.
As a volume in phase space
200px|thumb|right|Optical momentum.
In the context of Hamiltonian optics, at a point in space, a light ray may be completely defined by a point , a unit Euclidean vector indicating its direction and the refractive index at point . The optical momentum of the ray at that point is defined by
<math display="block">\mathbf{p} = n(\cos \alpha_X, \cos \alpha_Y, \cos \alpha_Z) = (p, q, r)\,,</math>
where . The geometry of the optical momentum vector is illustrated in figure "optical momentum".
In a spherical coordinate system may be written as
<math display="block">\mathbf{p} = n\!\left(\sin \theta \cos \varphi, \sin \theta \sin \varphi, \cos \theta \right)\,,</math>
from which
<math display="block">\mathrm{d}p\, \mathrm{d}q = \frac{\partial(p, q)}{\partial(\theta, \varphi)} \mathrm{d}\theta\, \mathrm{d}\varphi = \left(\frac{\partial p}{\partial \theta} \frac{\partial q}{\partial \varphi} - \frac{\partial p}{\partial \varphi} \frac{\partial q}{\partial \theta}\right) \mathrm{d}\theta\, \mathrm{d}\varphi = n^2 \cos \theta \sin \theta\, \mathrm{d}\theta\, \mathrm{d}\varphi = n^2 \cos \theta\, \mathrm{d}\Omega\,,</math>
and therefore, for an infinitesimal area on the -plane immersed in a medium of refractive index , the etendue is given by
<math display="block">\mathrm{d}G = n^2\, \mathrm{d}S \cos \theta\, \mathrm{d}\Omega = \mathrm{d}x\, \mathrm{d}y\, \mathrm{d}p\, \mathrm{d}q\,,</math>
which is an infinitesimal volume in phase space . Conservation of etendue in phase space is the equivalent in optics to Liouville's theorem in classical mechanics.