
Lagrange invariant
The Lagrange invariant H is defined by the expression

where n is the refractive index and the ray coordinates are u (the angle of the ray) and h (the height of the ray). Ray coordinates without a bar are marginal ray quantities and ray coordinates with a bar are chief ray quantities. (See the ABCD matrix page for definitions of these quantities). Using ray coordinate q=h and ray momentum p=nu, we have

The Lagrange invariant is proportional to the area of the phase plane occupied by rays. This quantity is an invariant under propagation and refraction, because these operations correspond to shearing of the phase space. Away from the paraxial regime the shearing is non-uniform, but the conservation of the occupied phase space area remains true. In the figure below, the marginal ray (M) and chief ray (C) define an occupied region of phase space (brown). After propagation and refraction operations, we have rays M' and C' and the occupied region of phase space has the same volume (blue).

The conservation of the Lagrange invariant is a very important concept, closely related to the conservation (for lossless optical systems) of brightness. The consequence of the invariance of the Largrange invariant is that an optical system that makes a bundle of rays narrower must make it more divergent, and an optical system that improves the collimation of a bundle of rays must at the same time make the bundle wider.