basic radiance defined as: is also conserved. In real systems, the étendue 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, étendue may not decrease and radiant flux may not increase and, therefore, basic radiance may not increase. Etendue as a volume in phase space 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 n at point r. The optical momentum of the ray at that point is defined by where . The geometry of the optical momentum vector is illustrated in figure "optical momentum". In a spherical coordinate system p may be written as from which 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 which is an infinitesimal volume in phase space x, y, p, q. Conservation of etendue in phase space is the equivalent in optics to Liouville's theorem in classical mechanics. Etendue as volume in phase space is commonly used in nonimaging optics. Maximum concentration Consider an infinitesimal surface dS, immersed in a medium of refractive index n crossed by (or emitting) light inside a cone of angle α. The etendue of this light is given by Noting that n sin α is the numerical aperture NA, of the beam of light, this can also be expressed as Note that dΩ is expressed in a spherical coordinate system. Now, if a large surface S is crossed by (or emits) light also confined to a cone of angle α, the etendue of the light crossing S is The limit on maximum concentration (shown) is an optic with an entrance aperture S, in air () collecting light within a solid angle of angle 2α (its acceptance angle) and sending it to a smaller area receiver Σ immersed in a medium of refractive index n, whose points are illuminated within a solid angle of angle 2β. From the above expression, the etendue of the incoming light is and the etendue of the light reaching the receiver is Conservation of etendue then gives where C is the concentration of the optic. For a given angular aperture α, of the incoming light, this