Smith–Helmholtz invariant

In optics the Smith–Helmholtz invariant is an invariant quantity for paraxial beams propagating through an optical system. Given an object at height ${\displaystyle {\bar {y}}}$ and an axial ray passing through the same axial position as the object with angle ${\displaystyle u}$, the invariant is defined by[1][2][3]

${\displaystyle H=n{\bar {y}}u}$,

where ${\displaystyle n}$ is the refractive index. For a given optical system and specific choice of object height and axial ray, this quantity is invariant under refraction. Therefore, at the ${\displaystyle i}$th conjugate image point with height ${\displaystyle {\bar {y}}_{i}}$ and refracted axial ray with angle ${\displaystyle u_{i}}$ in medium with index of refraction ${\displaystyle n_{i}}$ we have ${\displaystyle H=n_{i}{\bar {y}}_{i}u_{i}}$. Typically the two points of most interest are the object point and the final image point.

The Smith–Helmholtz invariant has a close connection with the Abbe sine condition. The paraxial version of the sine condition is satisfied if the ratio ${\displaystyle nu/n'u'}$ is constant, where ${\displaystyle u}$ and ${\displaystyle n}$ are the axial ray angle and refractive index in object space and ${\displaystyle u'}$ and ${\displaystyle n'}$ are the corresponding quantities 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.[4]

The Smith–Helmholtz invariant also relates the lateral and angular magnification of the optical system, which are ${\displaystyle y'/y}$ and ${\displaystyle u'/u}$ respectively. Applying the invariant to the object and image points implies the product of these magnifications is given by[5]

${\displaystyle {\frac {y'}{y}}{\frac {u'}{u}}={\frac {n}{n'}}}$

The Smith–Helmholtz invariant is closely related to the Lagrange invariant and the optical invariant. The Smith–Helmholtz is the optical invariant restricted to conjugate image planes.