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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 yO be the deviation of a point PO in the object from the optic axis. The deviation of the corresponding point PI in the image is denoted as yI. Let C be the center of curvature of the spherical lens. The angle between the line connecting PO and C and the optic axis is denoted as αO. The angle between the line connecting C with PI and the optic axis is denoted as αI. If αO is positive then αI is negative so it is common practice to work with the absolute values of the angles. Let nO and nI be the indices of refraction on the object-side and image-side of the spherical interface.

The sine condition is then

yOnOsin(|αO|) = yInIsin(|αI|)

If there were no distortion in the image the value of yI would be proportional to yO; i.e.,

yI = kyO

where k is a constant independent of yO. Instead

yI = (nO/nI)sin(|αO|)/sin(|αI|)yO

where αO depends upon yO.

If the signs of the angles are taken into account Abbe's sine condition takes the form

yOnOsin(αO) + yInIsin(αI) = 0

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