Lagrange invariant

In optics the Lagrange invariant is a measure of the light propagating through an optical system. It is defined by

H=n{\overline {u}}y-nu{\overline {y}}

,

where $y$ and $u$ are the marginal ray height and angle respectively, and $ȳ$ and $ū$ are the chief ray height and angle. $n$ is the ambient refractive index. In order to reduce confusion with other quantities, the symbol $Ж$ may be used in place of $H$ .^[1] $Ж 2$ is proportional to the throughput of the optical system (related to étendue).^[1] For a given optical system, the Lagrange invariant is a constant throughout all space, that is, it is invariant upon refraction and transfer.

The optical invariant is a generalization of the Lagrange invariant which is formed using the ray heights and angles of any two rays. For these rays, the optical invariant is a constant throughout all space.^[2]

References

^ ^a ^b Greivenkamp, John E. (2004). Field Guide to Geometrical Optics. SPIE Field Guides vol. FG01. SPIE. p. 28. ISBN 0-8194-5294-7.
^ Optics Fundamentals, Newport Corporation, retrieved 9/8/2011