Abbe sine condition

In optics, the Abbe sine condition is a condition that must be fulfilled by a lens or other optical system in order for it to produce sharp images of off-axis as well as on-axis objects. It was formulated by Ernst Abbe in the context of microscopes.^[1]

The Abbe sine condition says that

the sine of the object-space angle ${\textstyle \alpha _{\mathrm {o} }}$ should be proportional to the sine of the image space angle ${\textstyle \alpha _{\mathrm {i} }}$

Furthermore, the ratio equals the magnification of the system. In mathematical terms this is: $${\displaystyle {\frac {\sin \alpha _{\mathrm {o} }}{\sin \alpha _{\mathrm {i} }}}={\frac {\sin \beta _{\mathrm {o} }}{\sin \beta _{\mathrm {i} }}}=|M|}$$

where the variables ${\textstyle (\alpha _{\mathrm {o} },\beta _{\mathrm {o} })}$ are the angles (relative to the optic axis) of any two rays as they leave the object, and ${\textstyle (\alpha _{\mathrm {i} },\beta _{\mathrm {i} })}$ are the angles of the same rays where they reach the image plane (say, the film plane of a camera). For example, (${\textstyle \alpha _{\mathrm {o} },\alpha _{\mathrm {i} })}$ might represent a paraxial ray (i.e., a ray nearly parallel with the optic axis), and ${\textstyle (\beta _{\mathrm {o} },\beta _{\mathrm {i} })}$ might represent a marginal ray (i.e., a ray with the largest angle admitted by the system aperture). An optical imaging system for which this is true in for all rays is said to obey the Abbe sine condition.

The Abbe sine condition can be derived by Fermat's principle.^[2]

A thin lens satisfies $${\displaystyle {\frac {\tan \alpha _{\mathrm {o} }}{\tan \alpha _{\mathrm {i} }}}={\frac {\tan \beta _{\mathrm {o} }}{\tan \beta _{\mathrm {i} }}}=|M|}$$instead, which means that it does not satisfy Abbe sine condition at large angles. The difference is on the order of ${\displaystyle \alpha _{o}^{3}}$, which corresponds to the coma aberration.