Etendue

Etendue or étendue is a property of an optical system, which characterizes how "spread out" the light is in area and angle. The étendue can be defined in several equivalent ways. From the source point of view, it is the area of the source times the solid angle the system's entrance pupil subtends as seen from the source. From the system point of view, the étendue is the area of the entrance pupil times the solid angle the source subtends as seen from the pupil. These definitions are for infinitesimally small "elements" of area and solid angle however, and have to be summed over both the source and the diaphragm as shown below.

Etendue is important because it never decreases in any optical system. A perfect optical system produces an image with the same étendue as the source. The étendue is related to the Lagrange invariant and the optical invariant, which 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 étendue.

The term étendue comes from the French word for extent or spread. The French word for the optical property is étendue géométrique, meaning "geometrical extent". Other names for this property are acceptance, throughput, light-grasp, collecting power, and the AΩ product. Throughput and AΩ product are especially used in radiometry and radiative transfer.

Detailed definition

Variables used in defining étendue.

Consider a light source Σ and a "receiver" S, both of which are extended surfaces rather than mathematical points, and which are separated by a medium which is perfectly transparent. To obtain the étendue, one must consider the contribution of each point on the surface of the source to the illumination of each point on the receiver.

Several definitions are needed, as shown in the figure:

• $d\Sigma$ and $dS$ are two infinitely small surface elements, which can be considered to be tiny flat surfaces tangent to the surfaces of Σ and S, respectively,
• $\Phi$ is the radiant flux emitted by Σ and received by S,
• $d\Phi$ is the radiant flux emitted by $d\Sigma$ and received by S,
• $d^{2}\Phi$ is the radiant flux emitted by $d\Sigma$ and received by $dS$,
• $\mathbf {N} _{\Sigma }$ and $\mathbf {N} _{S}$ are the normal vectors to $d\Sigma$ and $dS$,
• $\alpha _{\Sigma }\,$ and $\alpha _{S}\,$ are the respective angles between the propagation direction and $\mathbf {N} _{\Sigma }$ and $\mathbf {N} _{S}$. The propagation direction is along the line joining $d\Sigma$ and $dS$.
• $d\Omega _{\Sigma }\,$ is the solid angle subtended by $dS$, as viewed from the centre of $d\Sigma$,
• $d\Omega _{S}\,$ is the solid angle subtended by $d\Sigma$, as viewed from the centre of $dS$, and
• $d$ is the distance between the surface elements $d\Sigma$ and $dS$.

Naturally, the surface elements are given by

$d\Omega _{\Sigma }={\frac {dS\cdot \cos {\alpha _{S}}}{d^{2}}}$ and $d\Omega _{S}={\frac {d\Sigma \cdot \cos {\alpha _{\Sigma }}}{d^{2}}}$.

By definition, the étendue of the pencil of light which "connects" the two surface elements is

$d^{2}G=d\Sigma \cdot \cos {\alpha _{\Sigma }}\cdot d\Omega _{\Sigma }={\frac {d\Sigma \cdot dS\cdot \cos {\alpha _{\Sigma }}\cdot \cos {\alpha _{S}}}{d^{2}}}=dS\cdot \cos {\alpha _{S}}\cdot d\Omega _{S}$.

The étendue of the whole system is then

$G=\int _{\Sigma }\int _{S}d^{2}G\ .$

One can show that the radiance of the pencil of light that goes from $d\Sigma$ to $dS$ is given by

$L={\frac {d^{2}\Phi }{d^{2}G}}$.

Refractive index

The conservation of étendue discussed above applies to the case of light propagation in free space, or more generally, in a medium in which the refractive index is constant. In a system with rotational symmetry in which the refractive index changes, the angular spread of an extended source can be decreased by a factor of the square of the relative refractive index. For example, if the surface world is viewed from the bottom of a swimming pool (where the refractive index of water relative to air is about 1.33), the $2\pi$ solid angle of the surface view is compressed into a visual angle of only $1.1\pi$. For a more generalized definition, the étendue of a system with rotational symmetry can be multiplied by the refractive index squared. This quantity is then conserved even in the case of variable refractive index.

References

• Greivenkamp, John E. (2004). Field Guide to Geometrical Optics. SPIE Field Guides vol. FG01. SPIE. ISBN 0-8194-5294-7.

Further reading

• Xutao Sun et al., 2006, "Etendue analysis and measurement of light source with elliptical reflector", Displays (27), 56–61.