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 $$n u / n' u'$$ 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.