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 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. Figure "etendue in refraction" shows an infinitesimal surface dS on the xy plane separating two media of refractive indices nΣ and nS. The normal to dS points in the direction of the z axis. Incoming light is confined to a solid angle dΩΣ and reaches dS at an angle θΣ to its normal. Refracted light is confined to a solid angle dΩS and leaves dS at an angle θS to its normal. The directions of the incoming and refracted light are contained in a plane making an angle φ to the x axis, defining these directions in a spherical coordinate system. With these definitions, Snell's law of refraction can be written as and its derivative relative to θ multiplied by each other result in where both sides of the equation were also multiplied by dφ which does not change on refraction. This expression can now be written as and multiplying both sides by dS we get that is showing that the etendue of the light refracted at dS is conserved. The same result is also valid for the case of a reflection at a surface dS, in which case and θΣ = θS. Brightness theorem A consequence of the conserveration of etendue is the brightness theorem, which states that no optical system can increase the brightness of the light emitted from a source to a higher value than the brightness of the surface of that source (where "brightness" is defined as the optical power emitted per unit solid angle per unit emitting or receiving area). Conservation of basic radiance Radiance of a surface is related to étendue by: where is the radiant flux emitted, reflected, transmitted or received; n is the refractive index in which that surface is immersed; G is the étendue of the light beam. As the light travels through an ideal optical system, both the étendue and the radiant flux are conserved. Therefore,