Etendue

Etendue or étendue (/ˌeɪtɒnˈduː/; French pronunciation: ​) 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 optics. Etendue or étendue (/ˌeɪtɒnˈduː/; French pronunciation: ​) 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 optics. From the source point of view, it 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 is important because it 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 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. The term étendue comes from the French étendue géométrique, meaning 'geometrical extent'. Other names for this property are acceptance, throughput, light grasp, light-gathering or -collecting power, optical extent, geometric 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. 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 area of dS projected in the direction of the light propagation is dS cos θ. The etendue of this light crossing dS is defined as Because angles, solid angles, and refractive indices are dimensionless quantities, etendue has units of area (given by dS). 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 increasing entropy, which only can be reverted if a priori knowledge is used to reconstruct a phase-matched wave-front such as with phase conjugated mirrors. 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. 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). 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.