Lagrange invariant

{{short description|Measure of the light propagating through an optical system}}

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

:$H\; =\; n\backslash overline\{u\}y\; -\; nu\backslash overline\{y\}$,

where {{mvar|y}} and {{mvar|u}} are the marginal ray height and angle respectively, and {{mvar|ȳ}} and {{mvar|ū}} are the chief ray height and angle. {{mvar|n}} is the ambient refractive index. In order to reduce confusion with other quantities, the symbol {{mvar|Ж}} may be used in place of {{mvar|H}}.{{cite book |first=John E. |last=Greivenkamp |year=2004 |title=Field Guide to Geometrical Optics |publisher=SPIE |others=SPIE Field Guides vol. FG01 |isbn=0-8194-5294-7 |page=28}} {{mvar|Ж²}} is proportional to the throughput of the optical system (related to étendue). 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.[http://www.newport.com/Optics-Fundamentals/604533/1033/content.aspx Optics Fundamentals] {{Webarchive|url=https://web.archive.org/web/20160111224020/http://www.newport.com/Optics-Fundamentals/604533/1033/content.aspx |date=2016-01-11 }}, Newport Corporation, retrieved 9/8/2011