Herschel's condition

Remarks

The proof is the axial analogue of the proof of the Abbe sine condition. Moving the object point along the axis by $\delta z$ changes the optical path length along a ray by $n \, \delta z \cos\theta$ to first order, and similarly in the image space. Requiring the change of path difference to be the same for all rays, and comparing with the axial ray, gives $n \, \delta z (1 - \cos\theta) = n' \, \delta z' (1 - \cos\theta')$, which is the condition since $1 - \cos\theta = 2\sin^2(\theta/2)$.

The Abbe sine condition governs sharp imaging of a small area perpendicular to the axis. Except in special cases, a system cannot satisfy both conditions for all aperture angles, so it cannot sharply image a small volume around $O$.

Example: for small angles, $\sin^2(\theta/2) \approx \theta^2/4$, and paraxial optics gives $\theta' = n\theta / (n' m)$, where $m$ is the lateral magnification. Then the condition reads $m_z = n' m^2 / n$, which is the standard paraxial relation between longitudinal and lateral magnification. So paraxial rays always satisfy it, and it restricts only the rays at finite aperture.

Going deeper

For a student

Herschel's condition and the sine condition together show that a stigmatic image of a small three-dimensional region is impossible for most magnifications. In practice, systems corrected for the sine condition, such as microscope objectives, have a very small depth of sharp focus at large aperture.