Lagrange invariant

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 u ¯ y − n u y ¯ {\displaystyle 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] Ж2 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]

See also

  • Etendue
  • Smith-Helmholtz invariant
  • Abbe sine condition

References

  1. ^ a b Greivenkamp, John E. (2004). Field Guide to Geometrical Optics. SPIE Field Guides vol. FG01. SPIE. p. 28. ISBN 0-8194-5294-7.
  2. ^ Optics Fundamentals Archived 2016-01-11 at the Wayback Machine, Newport Corporation, retrieved 9/8/2011
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