Smith-Helmholtz invariant

In optics the Smith-Helmholtz invariant is an invariant quantity for light propagating through an optical system. It is defined by[1][2]

${\displaystyle H=nyu}$,

where ${\displaystyle u}$ is the marginal ray angle, ${\displaystyle y}$ is the chief ray height, and ${\displaystyle n}$ is the refractive index. For a given optical system, this quantity is invariant under refraction, and implies a variety of other interesting relationships.

It is closely connected to the Abbe sine condition. The paraxial version of the sine condition is satisfied if the ratio ${\displaystyle u/u'}$ is constant, where ${\displaystyle u'}$ is the marginal ray angle in image space. The Smith-Helmholtz invariant implies that the lateral magnification, ${\displaystyle y/y'}$ is constant if and only if the sine condition is satisfied.[3]