# Optical Aberrations

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 differences between perfection and what is actually possible are 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 only occurs off-axis.

## Axial Chromatic Aberration

Axial chromatic aberration (ACA) occurs everywhere in the field of view, and is pretty much the same everywhere. It is caused by the fact that the refractive index (nλ) 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 of light as shown in Figure 1.1a below.
ACA, along with other aberrations, can be measured as a distance along the optical axis (longitudinal) or as a distance perpendicular from the optical axis (transverse) also shown in Figure 1.1b. When measuring the transverse and longitudinal ray aberrations it is helpful to designate abbreviations for each aberration. For transverse axial chromatic aberration we will use the abbreviation “T-ACA” and for the longitudinal axial chromatic aberration we will use “L-ACA”. If you would like to learn more about how to measure aberrations, follow this link.

It should be noted that Figure 1.1, as with many sketches on this page, is drawn to aid in the understanding of aberrations and is not to scale. The actual distances between foci (focal lengths) for different wavelengths would be much smaller.

Fig 1.1 Axial Chromatic Aberration

Early telescope users, like Galileo, were troubled by ACA 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 as shown in Figure 1.2. It is easy to see how this image was created by placing an image plane at the green focus in Figure 1.1b and considering the T-ACA from the blue and red light rays combine to form a blurred magenta ring around the central white dot.

Figure 1.2: Magenta Blur Resulting from ACA

## Correcting for Axial Color

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). Generally, the two types of glasses are combined to bring the blue and red foci to the same location leaving only the green light at a different focal length as shown in Figure 1.3.

Fig 1.3 Achromatic Doublet

Using an achromatic doublet, as in Figure 1.3, significantly improves image quality by reducing the blur. To further reduce this blur all three wavelengths can be brought to a common focus. This type of lens is called an apochromat and is generally made of 3 elements.

Another way to reduce ACA is “stopping down” the lens. This just means allowing a smaller diameter of light through the systems. The term “stopping down” comes from the size of the stop in the lens, which is just the limiting aperture in the optical system. For the achromat in Figure 1.3, the stop size is just the lens diameter and is also equal to the entrance pupil diameter used in calculating the F/# (as described in the Basic Optics Terms section.)

It is important to note that stopping down the lens does not change L-ACA but can greatly decrease T-ACA. This is because stopping down the lens does not change the focal length difference between colors, it only decreases the angle between the light rays and the optical axis; and therefore decreases the distance from the optical axis that the out of focus light rays cross the image plane (T-ACA).

## Lateral Chromatic Aberration

Lateral chromatic aberration (LCA) 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. As seen Figure 1.4, the red image height is larger than the green and blue image heights. The difference between the extreme image heights is equivalent to the amount of LCA is in a system.

Figure 1.4: Lateral Chromatic Aberration

LCA is the result of dispersion of the chief ray, thus when a lens exhibits LCA, light is affected very similarly to light traveling through a dispersive prism (Figure 1.5). If we imagine the prism in Figure 1.5 as the tip of a lens we can see how the phenomenon is created in a lens like Figure 1.4. White light is incident on the first surface of the prism (or lens) it is bent according to Snell’s law and since the angle of refraction is dependent on the wavelength of light, the optical paths for different wavelengths diverge and we get dispersion – (separation of white light into all wavelengths across the visible spectrum). Since the index for blue light (nBLUE) is higher, it is bent more strongly than green and red light.

Fig 1.5 Dispersive Prism

The result of LCA in telescopes is that off-axis images of stars appear to be little line segments that are blue at one end and red at the other as seen in Figure 1.6.  The line segments are in a sagittal orientation, which means that they lie on lines that pass through the optical axis.

Figure 1.6: Separation of Colors due to LCA

## Correcting for Lateral Color:

If ACA is present in a system, LCA is linear with respect to stop position. It is therefore helpful to choose a stop position where the LCA is zero. Once LCA is corrected, ACA can be corrected by achromatizing (changing elements to achromats) elements that are not close to the stop. Once ACA is corrected, the stop can be moved to its original position since LCA depends on the presence of ACA in a lens.

Symmetry can also be utilized to correct for LCA. If a lens system is designed in such a way that it is close to symmetric about the stop, LCA will be decreased.

## Monochromatic Aberrations

There are five monochromatic aberrations: spherical aberration, coma, astigmatism, Petzval field 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

Spherical aberration (SA) 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. Spherical aberration occurs when light rays at or near the edge (or margin) of the lens focus at a different location than those that enter the lens at or near the center as seen in Figure 1.7. Just as ACA was measured, SA can also be measured as a longitudinal or transverse aberration. Longitudinal spherical aberration is designated as LSA, and transverse spherical is designated as TSA as shown below.

Fig 1.7 Spherical Aberration

The blur due to TSA varies as the cube of the F/#. The F/#, as stated on our Basic Optics Terms page, is the focal length of the lens divided by the entrance pupil diameter, or in the case of a single lens, the diameter of the lens (F/# = f/d). So if the focal length is held constant and the entrance pupil diameter is increased the blur from spherical aberration will also increase (as the F/# decreases). Considering this, the cube of the F/# is therefore inversely proportional to TSA. For example, 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 and it creates a haziness across the entire image as seen in Figure 1.8.

Figure 1.8: Spherical Aberration on Star Cluster

## Correcting for Spherical Aberration

When designing a custom lens, a lens designer can use a few techniques to reduce spherical aberration. One of them is called “lens bending” and refers to the adjusting of the lens curvatures while keeping the lens power the same. This effectively adjusts the shape of the lens to minimize SA. For systems used with infinite conjugates (object or image at infinity), it is best to bend the lens so the greatest curvature is toward infinity. Looking at Figure 1.7, this is not the case. The greatest curve is faced away from the incident light. This causes a large SA as discussed above. Referring now to Figure 1.9, the upper lens layout shows the same lens flipped around so the greatest curve of the lens faces the incident light from infinity. This greatly reduces the SA.
Another technique called “lens splitting” is when a single lens is split into multiple lenses in close proximity that have a total power of the original single lens. This is effective because SA is highly dependent on angle of incidence, therefore if the lens can be split into multiple lenses, we can decrease the angles of incidence while keeping the same power. An example of lens bending and splitting is shown in Figure 1.9 below.
Since SA varies with the cube of the entrance pupil diameter, stopping down the lens will greatly reduce SA. The glass type can also be changed to a glass with a higher index to help reduce the curvature needed to bend the light and thereby reduce SA.
If additional correction is needed, the lens can be made aspheric to further reduce SA. Lens designers can use combinations of these techniques and more to achieve the level of correction needed.

Figure 1.9: SA Reduction by Lens Bending and Splitting

## Coma

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, after spherical, is the second most important aberration in microscopes and telescopes. The blur size from coma varies as the square of the F/#, so it is still highly dependent on entrance pupil diameter. The characteristic shape of coma is an ice cream cone, or, a comet for which it is named.
As can be seen in the ray layout in Figure 1.10 below, coma can be described as a variation in magnification with pupil. We see that the chief ray crosses the image plane at one location, but as we move farther out in the pupil, the image height increases. This is called “outer coma” since the comet shape tail points away from the optical axis. It is also possible to get “inner coma” which would result in the tail pointing toward the optical axis.

Figure 1.10 Ray Layout of Lens Suffering from Coma

An image suffering from purely coma will have good image quality in the center, but as you move farther off-axis the image will degrade linearly with field position. Looking at Figure 1.11 we can see the difference between a un-aberrated image (a) and an image from a lens suffering from coma (b). If the object was something other than a random point field such as stars, like an image of a sample in a microscope, we would see a sharp focus in the center and a linearly increasing blur as we get closer to the edge of the image.

Figure 1.11 Effect of Coma on Off-Axis Stars

## Correcting for Coma

Choosing the correct shape of the lens can be very helpful in reducing coma. Using a convex-planar lens with the convex side facing infinity will help to minimize coma. This is close to the bending needed to minimize spherical aberration. Also, since the amount of blur is proportional to the square of the entrance pupil diameter, stopping down the lens will help minimize coma.
Shifting the stop is also helpful in correcting coma if spherical is present in the system. This is similar to how stop shift corrects for LCA if ACA is present. Also like LCA, coma can be reduced by using a close to symmetric system.

Ernst Abbe, a man who is responsible for many advances in microscopy, came up with a formula called the Abbe sine condition. The sine condition has two forms as shown in Figure 1.12. The first is for the case of finite object and image distances. In this case, if the ratio of the sine of the object angle to the sine of the image angle is constant for all rays, the condition is met. In the case of an infinite object, if the ratio of object ray height to the sine of the image angle is constant for all rays, the condition is met. If a lens corrects spherical aberration and coma, it meets the sine condition and is known as an aplanat.

Figure 1.12 Ernst Abbe Sine Condition

## Astigmatism

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 with 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.
As seen in Figure 1.13, there are two focal positions. At one focal position the image of an off-axis point 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 position between the sagittal and tangential focal planes, called the medial focus, where the image is a more circular blur that is smaller in diameter than either of the line segments.

Figure 1.13 Ray Layout of a Lens Suffering from Astigmatism

If stars are imaged through a lens with astigmatism, as shown in Figure 1.14, the image shape will depend on which plane of focus was selected as the image plane. If the image plane is positioned on the tangential focus, the off-axis image of the stars would resemble lines tangent to an imaginary circle centered on the optical axis. When the image plane is positioned at the medial focus the off-axis image will simply look out of focus when compared to the un-aberrated image. As for the case when the image plane is positioned at the sagittal focus, the image of the stars will resemble short line segments that are positioned on lines that go through the optical axis.

Figure 1.14 Tangential, Sagittal and Medial Foci

## Correcting for Astigmatism

If spherical and/or coma is present in a system suffering from astigmatism, then shifting the stop can be helpful in minimizing the blur from astigmatism. Lens bending can also be utilized to help to minimize astigmatism. Since astigmatism is proportional to the square of the field angle and increases linearly with aperture, a smaller field angle or pupil size can be chosen if astigmatism is too great. In the case that stop shift, bending, field angle and pupil size are insufficient, additional lenses may be added to the system that contribute the opposite sign astigmatism to cancel it out.

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

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 (as shown in Figure 1.15), and the blur increases with the square of the image height. The radius of the Petzval surface is completely insensitive to F/#, but the blur on a flat image plane caused by the Petzval curvature increases linearly with entrance pupil diameter.

Fig 1.15 Petzval Image Surface

If our familiar example object of stars were imaged through a system with Petzval field curvature the image would look in focus in the center of the image, with the focus falling off near the edges as shown in Figure 1.16. This is very similar to the image at the medial focus of a system suffering from astigmatism.

Fig 1.16 Petzval Field Curvature

## Correcting for Field Curvature:

Since the blur from Petzval curvature on a planar image surface increases with the square of field, one option to minimize it is to decrease the field. Although this can help correct the blur, it is not usually acceptable, as most lenses have a specific field angle that is desirable for their application.

Another option is to add a negative field-flattener lens to the system which is placed close to the image surface. This lens is designed to have the opposite field curvature of the system to cancel it out. Due to its position near the focal plane it will not affect other aberrations greatly.

Petzval curvature will be zero for a meniscus lens with equal radii. The power of the lens is proportional to the thickness, so if one desires to change the power without adding to the Petzval curvature, a thick meniscus may be used.

A combination of several thin lenses can also be used to adjust the Petzval curvature. It can generally be made as small as desired if there is approximately equal amounts of negative and positive powered lenses in the system.

Astigmatism is closely related to field curvature and when both are present in a system the result is two image surfaces as shown in Figure 1.17. The tangential and sagittal image surfaces will converge on the Petzval surface if astigmatism is corrected for. This can be done by lens bending, stop shift, moving elements or changing optical glasses in a system.

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. This is known as an artificially flattened field and results in acceptable but not exceptional image quality.

Fig 1.17 Tangential, Sagittal, and Petzval Image Surfaces

## Distortion

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 can be thought of as varying transverse magnification with field as seen in Figure 1.18. Magnification is equal to the ratio of the image height to the object height. In a distortion-less system, this ratio would be the same across the field, but when distortion is present this ratio is variable. In Figure 1.18, the transverse magnification is greater for the larger field position (blue line) and is an example of pincushion distortion. Barrel distortion would result in the larger field position having a smaller magnification.

Distortion is completely insensitive to F/#, and it varies as the cube of the distance from axis.

Fig 1.18 Distortion Ray Layout

Examples of both pin-cushion and barrel distortion can be seen in Figure 1.19 using a grid of lines as the object. Pin-cushion distortion is characterized by increased magnification with increased field (distance from the optical axis). This is shown by the magnification of the red arrow being approximately equal to the un-aberrated image magnification and the blue arrow having a larger magnification than the un-aberrated image. Barrel distortion is just the opposite, characterized by decreased magnification with increased field, as shown.

Fig 1.19 Magnification of Pincushion and Barrel Distortion

## Correcting for Distortion

Since the magnification difference from distortion increases with the cube of field, one option to minimize it is to decrease the field. Although decreasing the field of view can help correct distortion, it is not usually acceptable, as most lenses have a specific field angle that is desirable for their application.
If the object and image are interchanged, the sign of distortion will flip. This principle can be used to create a distortion-less system. If two identical lenses are used and a stop is placed midway between them, distortion will be zero as seen in Figure 1.20. If it is not possible to utilize exact symmetry, another option is to add more elements to the system in an attempt to “balance” the distortion seen such that the sum of all element contributions to distortion = zero.
Shifting the stop will also affect distortion as well as lens bending and glass selection.

Fig 1.20 Symmetric System to Cancel Distortion

For more on this topic, such as the difference between f*tan(?) distortion and f*? distortion, see our distortion page.

## 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”.