Now that you have reviewed some of the basic terms in optics, it’s time to see what can go wrong in an optical system. Even when an optical system is perfectly made, the image is never quite perfect. The difference between perfection and what is theoretically possible is called aberrations. These can be divided into two categories: chromatic (color) aberrations and monochromatic aberrations. There are only two chromatic aberrations, so let’s start with them.

## Chromatic Aberrations

The two chromatic aberrations are axial chromatic and lateral chromatic. The distinction is that axial chromatic aberration is present on the optical axis, as well as everywhere else. Lateral chromatic aberration occurs only off-axis.

**Axial chromatic** aberration occurs everywhere in the field of view, and is pretty much the same everywhere. It is caused by the fact that the refractive index of glass is different for different wavelengths (colors) of light. The result is that a simple lens (e.g. a magnifying glass) has a different focal length for each wavelength (color) of light. Early telescope users, like Galileo, were troubled by this problem because there was always a colored blur around everything they looked at. Human eyes are most sensitive to green light, so that’s where they focused their telescopes, leaving red and blue out of focus and making a magenta (red + blue) blur. It wasn’t until 1757 that John Dolland discovered that two different types of glass could be combined to greatly reduce this problem. This combination of two lenses of different types of glass is called an achromat (from the Greek ‘*a*‘ (without) and ‘*chromos*‘ color).

**Lateral chromatic** aberration increases linearly with the distance from the optical axis. This means that it is zero on axis because the distance from the axis is zero. Another way to think about lateral color is that it is a chromatic difference in magnification – red objects appear bigger than blue or vice versa. The result is that off-axis images of stars in a telescope appear to be little line segments that are blue at one end and red at the other. The line segments are in a saggital orientation, which means that they lie on lines that pass through the optical axis.

## Monochromatic Aberrations

There are five monochromatic aberrations: spherical aberration, coma, astigmatism, Petzval curvature and distortion. All of them can be present even if the optical system is being used with monochromatic light, e.g. a laser. They differ in appearance and their dependence on F/# and field height (distance from the optical axis). The monochromatic aberrations were discovered over several decades in the late nineteenth century, but they were codified by L. Seidel, and are thus known as the Seidel aberrations.

**Spherical aberration** is the only monochromatic aberration that is present on the optical axis. It is similar to axial chromatic in this regard as well as the fact that it is the same everywhere in the field. On the other hand, the blur due to spherical aberration varies as the cube of the F/#. This means that the blur is 8X as large for a lens at F/2 as for the same lens stopped down to F/4. The characteristic shape of spherical aberration is a circular blur.

**Coma** is not present on axis, but it is the first monochromatic aberration to show up as you move off-axis. This is because it varies linearly with field. For this reason, coma is the second most important aberration in microscopes and telescopes. Ernst Abbe, a man who is responsible for many advances in microscopy came up with a formula called the Abbe sine condition. If a lens corrects spherical aberration and coma, it meets the sine condition and is known as an **aplanat**. The blur size from coma varies as the square of the F/#. The characteristic shape of coma is an ice cream cone.

**Astigmatism** is rarely a concern in telescopes and microscopes, but was the limiting factor in early photographic lenses. The reason for this is that astigmatism increases as the square of the field angle. For small angles, astigmatism is usually much less of a problem than spherical aberration and coma, but it grows very rapidly as you get far from the axis. Astigmatism’s blur size increases linearly with the F/# of the lens. The characteristic shape of astigmatism is more complicated than spherical aberration or coma. At one focal position it appears as a line segment oriented sagittally (on a line passing through the optical axis), while a small distance away it appears as a line tangent to a circle centered on the optical axis. The two surfaces on which the sagittal and tangential line segments appear are known as the sagittal and tangential foci. There is a surface between the sagittal and tangential focal surfaces, called the medial focus, where the image is a more circular blur that is smaller than either of the line segments. A lens that is corrected for spherical aberration, coma and astigmatism is called **anastigmatic** from the Greek ‘*ana*‘ for up from and ‘*stigma*‘ meaning point.

**Petzval curvature** or **field curvature** differs from the previous aberrations; it does not blur the image at all. Rather, it causes the image to lie on a surface that is not a plane. As a first approximation, the surface is spherical. Lenses are typically used to inspect things that lie on a plane, and most detectors, whether CCD, CMOS or film, are also planar. This means that even though Petzval curvature doesn’t blur the image on the Petzval image surface, it does result in a blur on a plane image surface, and the blur increases with the square of the image height. A common ploy is to attempt to introduce Petzval curvature into a lens to flatten out the astigmatic focal surfaces, giving a smaller blur on the image plane. Petzval curvature is completely insensitive to F/#.

The last of the five Seidel aberrations is **distortion**. Like Petzval curvature, it does not blur the image, but unlike Petzval curvature it does not curve the image. Instead, it either compresses the edges of an image, resulting in barrel distortion, or expands them, resulting in pincushion distortion. Distortion, like Petzval curvature, is insensitive to F/#, but it varies as the cube of the distance from axis. For more on this topic, such as the difference between f*tan(Θ) distortion and f*Θ distortion, see our distortion page.

These descriptions really need pictures. Unfortunately, I’ll have to add them later.

## Caveats

Although the aberrations have been listed singly, they normally occur in combinations. Most lenses have all of the above aberrations, as well as chromatic variation of the monochromatic aberration (e.g. the spherical aberration in red is different from the spherical aberration in blue). Things also become more complex as the F/# is reduced and the field angle increased. The Seidel aberrations are also known as third order aberrations, and there are fifth and higher order aberrations in addition. If you want to learn more about this topic, I’d suggest reading Welford’s “Aberrations of Optical Systems”.