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.

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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). Imagining the prism in figure 1.5 is the tip of a lens shows us how the phenomenon occurs in a lens like one from figure 1.4. White light that is incident on the first surface of the prism (or lens) bends according to Snell’s law. 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 bends more strongly than green and red light.

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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. Figure 1.9 shows an example of lens bending and splitting.

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, known as an artificially flattened field, results in acceptable but not exceptional image quality.

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

Axial chromatic aberration (ACA) occurs everywhere in the field of view, and is pretty much the same everywhere. The fact that the refractive index (nλ) of glass is different for different wavelengths causes this. 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.

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We can think of distortion 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.

Another option is to add a negative field-flattener lens to the system placed close to the image surface. This lens’ design allows it 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.

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.

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It is important to note that stopping down the lens does not change L-ACA but instead 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. Therefore, it decreases the distance from the optical axis that the out of focus light rays cross the image plane (T-ACA).

If the lens needs additional correction, making the lens aspherical can further reduce SA. Lens designers can use combinations of these techniques and more to achieve the level of correction needed.

ACA, along with other aberrations, can be measured two ways. The first as a distance along the optical axis (longitudinal) and the second as a distance perpendicular from the optical axis (transverse). These are shown in figure 1.1b. It’s helpful, when measuring the transverse and longitudinal ray aberrations, to designate abbreviations for each aberration. For transverse axial chromatic aberrations we will use the abbreviation “T-ACA” and “L-ACA” for the longitudinal axial chromatic aberration. If you would like to learn more about how to measure aberrations, follow this link.

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Once a lens is corrected for spherical aberration, coma and astigmatism we call it anastigmatic. The name is derived from the Greek ‘ana’, meaning ‘up from’, and ‘stigma’, meaning ‘point’.

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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.

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Using a combination of several thin lenses to adjust the Petzval curvature can also be an option. Generally, there are no limits on reducing the size of the curvature if there are equal amounts of negative and positive powered lenses in the system.

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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, we can reduce coma by using a close to symmetric system.

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Although we list these aberrations individually, 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. You can also visit our How to Measure Aberrations page to get an introduction to some different methods.

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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.

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.

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. If you’re looking for another way to think about lateral color, consider this: lateral color 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.

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.

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, using a thick meniscus can work.

We can see examples of both pin-cushion and barrel distortion 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.

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 where the sagittal and tangential line segments appear are 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.

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 aberrations only occurs off-axis.

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

“Stopping down” the lens is another way to reduce ACA. 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.)

Note that figure 1.1 functions as an 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.

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It wasn’t until 1757 that John Dolland discovered combining two different types of glass could reduce this problem. This technique, called an achromat (from the Greek ‘a‘ (without) and ‘chromos‘ color), brings the blue and red foci to the same location and leaves the green light at a different focal length. See figure 1.3 for an example.

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.

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Early telescope users, like Galileo, were troubled by ACA because there was always a colored blur around everything they observed. Human eyes are most sensitive to green light, so that’s where they focused their telescopes, leaving red and blue out of focus. This resulted in a magenta (red + blue) blur as shown in figure 1.2. It’s easy to see how this image results in a blur. First place an image plane at the green focus in figure 1.1b. Then consider how the T-ACA from the blue and red light rays combine, forming a blurred magenta ring around the central white dot.

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.

Since SA varies with the cube of the entrance pupil diameter, stopping down the lens will greatly reduce SA. Changing the glass type to one with a higher index can help reduce the curvature needed to bend the light and thereby reduce SA.

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.

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If ACA is present in a system, LCA is linear with respect to stop position. Thus it’s 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.

If the object and image are interchanged, the sign of distortion will flip. We can use this principle 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.

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If we image stars through a lens with astigmatism, as shown in figure 1.14, the image shape will depend on which plane of focus we selected as the image plane. If we position the image plane 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 we position the image plane at the medial focus, the off-axis image will simply look out of focus when compared to the un-aberrated image. When positioning the image plane at the sagittal focus, the image of the stars will resemble short line segments that rest on lines that go through the optical axis.

A custom lens designer can use a few techniques to reduce spherical aberration. One technique called “lens bending” requires adjusting 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 faces 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.

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. Like measuring ACA, SA is sometimes measured as a longitudinal or transverse aberration. We refer to longitudinal spherical aberration as LSA and transverse spherical as TSA, as shown below.

Since the blur from Petzval curvature on a planar image surface increases with the square of field, decreasing the field is one option to minimize it. 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.

We can also use symmetry to correct for LCA. Designing a lens system so that it is close to symmetric about the stop will decrease LCA.

There are five monochromatic aberrations: spherical aberration, coma, astigmatism, Petzval field curvature, and distortion. Each 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). Discovered over several decades in the late nineteenth century and codified by L. Seidel, these monochromatic aberrations are thus known as the Seidel aberrations.

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. We call this an aplanat.