Smith–Helmholtz invariant

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

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 ${\displaystyle {\overline{y}}_i}$ and refracted axial ray with angle ${\displaystyle u_i}$ in medium with index of refraction ${\displaystyle n_i}$ we have ${\displaystyle H=n_i{\overline{y}}_i{u}_i}$. 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 ${\displaystyle nu/n^{\prime }{u}^{\prime }}$ is constant, where ${\displaystyle u}$ and ${\displaystyle n}$ are the axial ray angle and refractive index in object space and ${\displaystyle u^{\prime }}$ and ${\displaystyle n^{\prime }}$ are the corresponding quantities in image space. The Smith–Helmholtz invariant implies that the lateral magnification, ${\displaystyle y/y^{\prime }}$ is constant if and only if the sine condition is satisfied.

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

${\displaystyle \frac{y^{\prime }}{y}\frac{u^{\prime }}{u}=\frac{n}{n^{\prime }}}$

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.