In optics the SmithâÂÂHelmholtz invariant is an invariant quantity for paraxial beams propagating through an optical system. Given an object at height and an axial ray passing through the same axial position as the object with angle , the invariant is defined by
where 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 th conjugate image point with height and refracted axial ray with angle in medium with index of refraction we have . Typically the two points of most interest are the object point and the final image point.
The SmithâÂÂHelmholtz invariant has a close connection with the Abbe sine condition. The paraxial version of the sine condition is satisfied if the ratio is constant, where and are the axial ray angle and refractive index in object space and and are the corresponding quantities in image space. The SmithâÂÂHelmholtz invariant implies that the lateral magnification, is constant if and only if the sine condition is satisfied.
The SmithâÂÂHelmholtz invariant also relates the lateral and angular magnification of the optical system, which are and respectively. Applying the invariant to the object and image points implies the product of these magnifications is given by
The SmithâÂÂHelmholtz invariant is closely related to the Lagrange invariant and the optical invariant. The SmithâÂÂHelmholtz is the optical invariant restricted to conjugate image planes.