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.
The mathematical condition is as follows:
where the variables u, U are the angles (relative to the optic axis) of any two rays as they leave the object, and u’, U’ are the angles of the same rays where they reach the image plane (say, the film plane of a camera). For example, (u,u’) might represent a paraxial ray (i.e., a ray nearly parallel with the optic axis), and (U,U’') might represent a marginal ray (i.e., a ray with the largest angle admitted by the system aperture); the condition is general, however, and does not only apply to those rays.
Put in words, the sine of the output angle should be proportional to the sine of the input angle.
Read more about Abbe Sine Condition: Magnification and The Abbe Sine Condition
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