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The Abbe Sine Condition in Optics

In geometric optics relationships are derived assuming that the light rays are paraxial; i.e. they lie close to the
optical axis of the system and the angle between them and the axis is small. Strictly speaking the relationships derived in
geometric optics are true in the limit as the deviations from the optical axis go to zero. Since this is clearly a severe
restriction on the validity of optical theory it was a great accomplishment to find a relationship that did not presume paraxial
conditions. Ernst Karl Abbe, a very able German mathematical physicist, found such a condition. He had been hired by
Carl Zeiss to provide analyses useful for improving optical instruments. Abbe was working out the analysis of microscopes
when he discovered the sine conditions.

Consider an image of an object formed by a spherical lens interface. Let y_{O} be the deviation of a point P_{O}
in the object from the optic axis. The deviation of the corresponding point P_{I} in the image is
denoted as y_{I}. Let C be the center of curvature of the spherical lens. The angle between the line connecting
P_{O} and C and the optic axis is denoted as α_{O}. The angle between the line connecting C with
P_{I} and the optic axis is denoted as α_{I}. If α_{O} is positive then α_{I} is
negative so it is common practice to work with the absolute values of the angles.
Let n_{O} and n_{I} be the indices of refraction on the object-side and image-side of the
spherical interface.

The sine condition is then

y_{O}n_{O}sin(|α_{O}|) = y_{I}n_{I}sin(|α_{I}|)

If there were no distortion in the image the value of y_{I} would be proportional to y_{O}; i.e.,

y_{I} = ky_{O}

where k is a constant independent of y_{O}.
Instead