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 ${\bar {y}}$ and an axial ray passing through the same axial position as the object with angle $u$, the invariant is defined by

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

where $n$ 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 $i$th conjugate image point with height ${\bar {y}}_{i}$ and refracted axial ray with angle $u_{i}$ in medium with index of refraction $n_{i}$ we have $H=n_{i}{\bar {y}}_{i}u_{i}$. 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 $nu/n'u'$ is constant, where $u$ and $n$ are the axial ray angle and refractive index in object space and $u'$ and $n'$ are the corresponding quantities in image space. The Smith–Helmholtz invariant implies that the lateral magnification, $y/y'$ 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 $y'/y$ and $u'/u$ respectively. Applying the invariant to the object and image points implies the product of these magnifications is given by

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

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.

See also

• Etendue
• Lagrange invariant
• Abbe sine condition