In optics the Smith–Helmholtz invariant is an invariant quantity for paraxial beams propagating through an optical system.
and an axial ray passing through the same axial position as the object with angle
For a given optical system and specific choice of object height and axial ray, this quantity is invariant under refraction.
th conjugate image point with height
and refracted axial ray with angle
in medium with index of refraction
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
are the axial ray angle and refractive index in object space and
are the corresponding quantities in image space.
The Smith–Helmholtz invariant implies that the lateral magnification,
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
The Smith–Helmholtz is the optical invariant restricted to conjugate image planes.
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