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

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

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 ${\overline{y}}_i$ and refracted axial ray with angle $u_i$ in medium with index of refraction $n_i$ we have $H=n_i{\overline{y}}_i{u}_i$. 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 $nu/n^{\prime }{u}^{\prime }$ is constant, where $u$ and $n$ are the axial ray angle and refractive index in object space and $u^{\prime }$ and $n^{\prime }$ are the corresponding quantities in image space. The Smith–Helmholtz invariant implies that the lateral magnification, $y/y^{\prime }$ 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 $y^{\prime }/y$ and $u^{\prime }/u$ respectively. Applying the invariant to the object and image points implies the product of these magnifications is given by

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

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.

See also

• Etendue
• Lagrange invariant
• Abbe sine condition