Spherical Aberration

In our small angle formulas for spherical thin lenses, our approximations hold to a good degree... but not perfectly. In playing with Occam's Razor, we may have cut ourselves1.

When we take an accurate look at how rays are bent by spherical lenses, the picture is not quite so nice as when we use small angle approximations.

Spherical aberration.

The reader can see that rays coming in from infinity focus at different points along the optical axis of a spherical lens depending on the distance of the ray from the axis. This was something that we alluded to in our treatment of spherical mirrors, where we noted that the focal point of reflected rays moved toward the mirror's surface as the ray strayed from the optical axis. In lenses, we can explain this aberration using our qualitative understanding of Snell's Law.

Recall that when light travels from high index to low index, the ray is bent away from the normal; whereas the reverse occurs when it travels from low index to high index. In our picture above, let the lens have a higher index than the air which surrounds it. Then, going from left to right, as each ray enters the lens, it is bent toward the normal. As each ray leaves the lens, it is bent away from the normal. The ray coming in close to the optical axis makes a small initial angle with the normal, and bends relatively gently, following our small angle approximations nicely. However, the ray far away from the axis makes a large angle with the first normal. By the time it meets the second surface, it makes quite a large angle with the normal; this is compounded when it is refracted away from the normal in travelling from high to low index. The result is that the higher ray gets focussed much closer to the lens.

In short, spherical aberration gives rise to a blurry image because rays from each object point are focussed in a small region instead of a unique image point2.

We saw that parabolic mirrors had an exact focus; in a similar way, aspherical lenses have a sharper focus. For technical reasons, making aspherical lenses is difficult, and the problem spherical aberration is usually resolved by using lenses in conjunction to cancel the difference in focal lengths.

In the next diagram, the index of the converging lens is greater than that for the diverging one; both indices, of course, are larger than the index for air.

Fixing spherical aberration.

Our intuition about Snell's Law tells us that light rays will bend toward, away, and away from the normal as they hit each of the three boundaries. In travelling from the "denser" converging lens to the diverging lens, we are still travelling from a higher index to a lower one; however, the difference in bend between nearer and further rays is not quite so bad as before. In leaving the diverging lens and erupting into air, for a suitable choice of n1, n2 the rays will converge upon the same point on the optical axis.

The reader can imagine how complicated things can get when we have multiple-lens systems, like telescopes. Fortunately, in the case of telescopes, there is a solution to spherical aberration simpler than doubling up every lens. Since telescopes are mainly concerned with observing light from objects far away, parabolic mirrors are perfect for constructing a reflecting telescope. The parabolic mirror catches the light rays to be observed, and focusses them at a common point. A lens can be placed to magnify details.

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