Copyright Ed Wesly Reproduced with permission

## Spherical Aberration

Explanation: Due to the geometry of spherical surfaces, rays from a particular point on an object passing through the edge of a lens focus nearer to the lens than do the paraxial rays.

In the top half of Figure 1, the ideal ray trace shows that all rays from the same infinite object point arrive at the same focal point. Below that is shown the typical sperically aberrated case, where the marginal rays from the edge of the lens come to a focus nearer to the lens than the centrally located paraxial rays.

This is simply a consequence of applying Snell's Law to all the points of a spherically curved surface; the rays hitting the peripheries of the lens are incident at larger and larger angles, and the beams will be bent much too strongly to match those coming in through the center.

The extent of SA can be measured linearly in two dimensions: along the optical axis, the Longitudinal Spherical Aberration (LSA in Figure 2), which is the distance from the paraxial focus to the focal point of the marginal rays; or at right angles to the optical axis, the Transverse Spherical Aberration (TSA in the figure), which is measured in the focal plane containing the paraxial focus to where the marginal ray intercepts the focal plane. The "miss" angle between the actual path of the marginal ray and its intended path to the paraxial focus is another possible measure of the degree of SA.

An outline of the rays leaving the lens and headed to their respective foci is labelled the caustic curve in Figure 2. Where the marginal rays cross the caustic is the smallest circle of confusion, as after that the marginal rays are expanding the diameter of the circle of confusion. The image at the paraxial focus will look like a bright nucleus with a hazy halo thanks to the marginal rays; the circle of least confusion will have maximum contrast and will be a small patch of light.

Since the marginal rays define the position of the smallest circle of confusion, the position of the best focus will change as a spherically aberrated lens is stopped down. The new marginal rays defined by the aperture meet the caustic closer to the paraxial focus, so it seems that the object has moved closer to the lens since the new focal plane is further from the lens.

Figure 4 compares the SA of positive and negative lenses. Notice that in both cases the marginal rays bend quicker than paraxial rays so that the marginal focal point is shifted closer to the lens. This makes the marginal rays more convergent or more divergent. When the marginal rays are inside the paraxial ones, this condition is termed undercorrected spherical aberration; the "outside" divergent rays of the negative lens are said to be overcorrected.

It can be predicted from the figure that when observing SA in the virtual image of a negative lens, the image spot will move toward the lens as the point of observation moves further off-axis. The reverse happens in the real image of a positive lens, where the marginal rays move the image closer to the lens, away from the viewer.