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What is the cause of spherical aberrations in lenses? I know that it's due to the differing width of the lens, but I don't understand/know the physics of why the lens width affects the focal distance of the light ray. I've seen a lot of optics before, but presently about all I know is that we posit that lenses obey $$\frac{1}{f} = \frac{1}{O} + \frac{1}{I}\tag{1}$$ where $f$ is the focal distance, $I$ is the image distance, and $O$ is the object distance.

So how do we see that the lens width should have an effect on the focal distance?

Qmechanic
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Bob
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2 Answers2

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Because the lens varies in width, some rays travel through more glass than others. Now take into account that travelling in any medium (index of refraction > 1) increases the effective distance which the rays travel, and you get that the rays which have traveled through thicker glass, have effectively traveled more distance. It follows that if all rays came from the same point, but have traveled different distances, then the focal point of each ray is slightly different.

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Yuval Weissler
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To add to Youval Weissler's Correct Answer, one can indeed choose the shape of the surface of a thick lens to eliminate spherical aberration for one configuration of input rays. For example, with a collimated input beam parallel to the optical axis at one given wavelength, one can choose the shape of the lens to eliminate all aberration and all the rays in Yuval's diagram would converge to a point. But in general there would be spherical aberration: if the collimated beam were tilted to shift the focus to an off-axis point for the system designed to be aberration free for the on-axis collimated beam, there in general will be a small aberration (including coma). To focus an on-axis collimated beam without aberration calls for a plano-convex lens (as in Youval's answer) whose curved surface is nearest the focus and which is a hyperbola of revolution about the optic axis.

In general, one can design to eliminate all aberration for the image of one point source at one wavelength. With the use of stacks of different dispersion behavior glasses, one can also eliminate aberration at the same point at several wavelengths. Very few systems give aberration free imaging for point sources on a whole object surface; the Maxwell Fisheye Lens is a rare example of such a system, as is an Aplanatic Sphere. Most such examples are lenses whose refractive index is a smoothly varying, complicated function of position within the lens.

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Selene Routley
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