Abbe sine condition

Remarks

The condition says that the ratio $\sin\theta / \sin\theta'$ is the same for all rays, not only for paraxial ones. One proof compares the optical path lengths from a point near $O$ in the object plane to its ideal image, along different rays. To first order in the object height $h$, the path length changes by $n h \sin\theta - n' h' \sin\theta'$, and sharp imaging requires this change to be the same for all rays, hence zero.

A system with no spherical aberration that satisfies the sine condition is called aplanatic; it is free of coma near the axis.

Example: a microscope objective of numerical aperture $n \sin\theta = 0.9$ and magnification $m = -40$ in air on the image side needs $\abs{\sin\theta'} = 0.9 / 40 = 0.0225$ at the edge of the aperture.

Without the sine condition the image of an off-axis point is blurred into a comet-shaped spot, the aberration called coma.

Going deeper

For a student

The sine condition is a design criterion for aplanatic objectives and for wide-aperture systems. It is the large-angle replacement for the paraxial relation $n \theta = m n' \theta'$, which follows from the Lagrange invariant. For the axial direction, the analogous condition for sharp imaging along the axis is Herschel's condition, and in general the two cannot both hold.