# Spherical aberration and coma

#### Doug Kerr

##### Well-known member
Two of the classical aberrations of a lens are called spherical aberration and coma. They are closely related.

Spherical aberration

Spherical aberration results from the fact that in a lens with a spherical surface (or two spherical surfaces), the rays from a point on the object that pass through the lens at different distances from its center (we often speak of them as passing through separate "zones" of the lens, where a "zone" is a ring-shaped region of the lens, usually considered infinitesimal in radial thickness) do not converge at a single point (as we would wish).

We see that here:

The upper panel shows what we would like (this lens presumably does not have a spherical surface). The lower panel shows the result of spherical aberration (this lens presumably does have a spherical surface).

If we place film, ground glass or a sensor at some point in image space (not shown on the figure - use your imagination), the "hollow cones" of rays passing through the different zones of the lens lay down individual rings of light (of infinitesimal radial thickness) on the film, except perhaps for one, which lays down a point. (Which one that is depends on exactly where we place the film.) These rings are all concentric.

Formally, spherical aberration is defined as an aberration affecting the imaging of an on-axis point. We will see why shortly.

Coma, spherical aberration's evil twin

Now again consider a practical lens with spherical surface(s), but consider an off-axis object point.

Here is an example:

From Wikimedia Commons​

[There is a flaw in this figure which I will discuss in a moment; I'll look for a better one.]​

Again, if we place film at a certain location (suggested by this figure) each of the "hollow cones" will deposit on the film a ring of light of infinitesimal radial thickness. The overall result will be a "comet-like" blur figure (thus the name of the aberration, coma).

The main portion of the figure suggests that all the hollow cones come to a focus on the film plane, as if spherical aberration itself has been corrected, perhaps by the use of a non-spherical lens. That is perfectly possible, but not likely. But that is in any case not consistent with the illustration of the resulting blur figure to the right. (That blur figure would be a line.)​

In reality, we typically have two different aspects of the aberration:

• The rays through the different zones of the lens come to a focus at different distances from the lens (as we saw before). [Not as the figure shows!]

• The rays through the different zones of the lens have different transverse magnification, so the axes of their hollow cones make different angles with the axis, and the circles of light they deposit are at different distances from the axis.

This latter is actually true for a point on the axis, but we do not see it, since the axes of the hollow cones are all at zero distance from the axis; it doesn't matter if one of them is 1.1 times are far from the axis as another - they both are at zero distance.​
We can see the distinction between the two situations here:

From Smith, F. Graham et al, Optics and Photonics: An Introduction, figure 2.27​

On the left ("spherical aberration"), we see six illustrative circles created on the focal plane by the hollow cones of rays from six zones (of infinitesimal radial thickness) on the lens.

On the right ("coma"), we see eight illustrative circles created on the focal plane by the hollow cones of rays from eight zones (of infinitesimal radial thickness) on the lens.

The semantics

So, isn't coma just the manifestation of the general case of spherical aberration for off-axis object points? Yes. But it is not the custom to say that. Rather, it is the custom to call it "coma".

Or perhaps spherical aberration is just a special case of coma, for an on-axis object point - where the discrepancy in lateral magnification for rays through different zones of the lens probably occurs but has no effect on the result.

Sure. But it is not the custom to say that. Rather, it is the custom to call it "spherical aberration".

Spherical aberration worse off-axis

We sometime hear that the impact of spherical aberration is worse for off-axis object points than fir on-axis points.

Almost certainly this usually refers to the phenomenon that is formally called "coma". And we would certainly expect the blur figure resulting from coma to have a larger envelope that that of the blur figure for (on-axis) spherical aberration.

But in fact some authors point out (as I did above) that this phenomenon comprises two aspects:

• The rays through the different zones of the lens come to a focus at different distances from the lens. They often refer to this as the "spherical aberration" aspect of coma.

• The rays through the different zones of the lens have different transverse magnification. They often refer to this as the "inconsistent lateral magnification" (or some such) aspect of coma.

Now, is the former of those two components, by itself, typically "greater" for off-axis points than for on-axis points? I don't know.

This figure shows coma without correction of spherical aberration, which does result in the "teardrop" shaped blur figure, but it's not easy to follow. I will try to adapt it in a later version of this note.

from Wikemedia Commons​

Best regards,

Doug

#### Doug Kerr

##### Well-known member
To synopsize this complicated discussion:

A simple lens with spherical surface(s) has these two shortcomings (among others):

A. With a point object, the "hollow cones of rays" passing through different "zones" of the lens (concentric rings) do not come to a focus at the same distance from the lens. Thus they cannot all cooperate to produce focus at a single point

B. The hollow cones passing through different zones of the lens do not experience the same lateral magnification. Thus for an object point off axis by some angle, the different hollow cones will not leave the lens with their axes at the same angle.

Now, for an object point on-axis, phenomenon B has no effect (since the angle of the axes of the incoming cones are all zero, and regardless of how that angle is "magnified" at the output, it will still be zero.

But phenomenon A, for any given location of the focal plane, means that each hollow cone deposits a circle of different diameter, the collection of which makes a finite-diameter circular* spot on the focal plane (not the point image we would desire).
*We assume a circular entrance pupil.​
Because this result is one of several that results from the use of a lens with spherical surface(s), this result is called spherical aberration.

Now, for an object point off axis, as before, phenomenon A, for any given location of the focal plane, means that each hollow cone deposits a circle of different diameter, the collection of which makes a finite-diameter spot on the focal plane.

But here, phenomenon B has an effect, which is that those circles are deposited at different distances from the optical axis. Their combination thus produces a "comet-like" blur figure (with its tail toward the optical axis). Because of that, this result is called coma. (Yes, it is a result of the use of a lens with spherical surface(s) ).

(Hey, I didn't get to name these things.)

In more complicated lenses, these two phenomena still occur, but differently in detail from the simple lens case. Various design steps may be taken minimize one or the other.

One result is often that for an on-axis object point, the spot formed on the focal plane may be very small.

But perhaps for an object point off-axis, a finite-sized spot may be formed, perhaps not even "comet-shaped" (with the tail toward the optical axis) but maybe even "bat-shaped" (with the wings extended to either side of a line through the optical axis).

This same odd result as to the shape of the spot may often be seen when, for an off-axis spot not in focus, a substantially-larger spot is formed.

Best regards,

Doug

#### Doug Kerr

##### Well-known member
To synopsize even further:

Strictly speaking:

A. spherical aberration refers to the fact that the hollow ray cones passing through different zones of the lens do not come to a focus at the same distance from the lens.

B. Coma refers to the fact that the magnification is different for rays passing through different zones of the lens.

If both A and B are present (as is the case for a classical spherical lens, and for most other lenses):

1. For an on-axis point, B has no impact, and A results in a blur circle of finite diameter. We speak of this effect as spherical aberration (rightly so).

2. For an off-axis point, both A and B have impacts, resulting in a "teardrop-shaped" blur figure. We speak of this effect as coma. But it is in fact the result of the combination of spherical aberration and coma.

Conceptually we could have B but not A (coma but not spherical aberration). The result would be a blur figure that was a radial line.

Best regards,

Doug

#### Doug Kerr

##### Well-known member
This figure (I saw a clearer version of it yesterday but can't find it now!) shows the formation of the blur spot for an off-axis point for a thin lens with spherical surfaces, which exhibits "classical" spherical aberration and coma:

from handprint.com​

The ray passing through the "center zone" of the lens (P) lands on the focal plane at point I (I'm not sure why that is not labeled P; it is on the other figure).

The rays passing through a "small-diameter" zone of the aperture (a-d) land around a circle on the focal plane, because of spherical aberration; that circle is displaced from I, because of coma.
The fact that the two "a" dots (pink and cyan) are not shown as coincident is just so they can be seen as distinct; they would actually coincide.​
The rays passing through a "larger-diameter" zone of the aperture (A-D) land around a (larger) circle on the focal plane, because of spherical aberration; that circle is displaced from I (by a greater distance), because of coma.

The overall result from all the rays through the entire aperture is the familiar "teardrop" shaped figure (we might describe its boundary as approximately I-b-B-A-D-d-I).

The term "coma" comes from similarity of that figure to the shape of a comet. Yet, formally, that figure requires both spherical aberration and coma to create! With only the phenomenon of coma (as it is formally defined), the blur figure would be a line running upward from I to the center of the larger circle.

************
I note with horror that this is my 6000th affront to the sensibilities of the members of OPF.

Best regards,

Doug

#### Doug Kerr

##### Well-known member
The matter of the terms sagittal and tangential (and various synonyms) continues to confuse me. In fact, to understand how it works with respect to astigmatism, I have to refer to my own article (which explains it quite well, I think).

In any case, I recently suggested that the terms sagittal coma and tangential coma referred to the "fatness" of the off-axis blur figure in the radial and circumferential directions (not necessarily in that order).

But that's wholly wrong.

Refer again to this figure:

Sagittal coma refers to the "vertical" displacement, in the image, of the points c and C from I (their "ideal" landing point). The reason is that those points are said to lie in the sagittal plane as it crosses the aperture.

Tangential coma refers to the "vertical" displacement, in the image, of the points a and A from I (their "ideal" landing point). The reason is that those points are said to lie in the tangential plane as it crosses the aperture.

It still makes my head hurt.

Best regards,

Doug

#### Doug Kerr

##### Well-known member
After further study of the "textbooks", I find that I was in error regarding the relationship of spherical aberration and coma.

In fact, the phenomenon of coma by itself (absent any spherical aberration) produces the well-known "teardrop" (or "comet") blur figure, as seen here:

From Wyant and Creath, Basic Wavefront Aberration Theory for Optical Metrology, excerpted from fig. 26.
(Used here under the doctrine of fair use.)​

Here, W131 is the Seidel coefficient of coma, which quantifies "how much coma" there is. R is the nominal radius of the wavefront as it emerges from the exit pupil (the nominal distance to where the rays converge). h is the radius of the exit pupil. x0 is the position of the off-axis object point (assumed to be vertically displaced). Rho (looks like "p") is the radial distance from the center of the exit pupil to the "zone" whose rays make the particular circle component whose radius and center position are given by the equations.

The location of the center is reckoned from the point marked "Gaussian image", which refers to the point where rays passing through the center of the exit pupil would converge.

I think that the vertical axis of the figure should be labeled "x" rather than "x0".

The label "image plane" does not refer to the horizontal line at the bottom, which is the y axis (not labeled here). It is the label for this "panel", in which we are looking at the image plane.​
If there were also spherical aberration, that would "pile on" in influencing the overall shape of the blur figure.

Note that:

• If W131 were zero (there was "no" coma), then all the circles would just degenerate to dots at the Gaussian focus; the "blur figure" would be the ideal point image.

• For the rays passing through the center of the exit pupil (rho = 0), we have a zero radius, zero displacement "circle" (the "point" of the blur figure at the Gaussian image location).

My apologies for having "barked up a wrong tree" in this matter. You suffer from watching me learn.

Best regards,

Doug

#### Maris Rusis

##### Member
Thanks Doug for that illuminating dissertation.

Spherical aberration is a regular interest of mine in its contribution to the characteristics of soft focus lenses. I wonder if there is an aspherical surface that minimises both spherical aberration and coma and what would its blur figures look like.

The Beach Multi-focal and Struss Pictorial soft focus lenses of the 1920s and 1930s were reputed to include hand-polished aspheres to give an even soft focus effect over the entire plate rather than sharper in the middle and fuzzier in the corners. Or is that just legend?

#### Doug Kerr

##### Well-known member
Hi, Maris,

Thanks Doug for that illuminating dissertation.

Spherical aberration is a regular interest of mine in its contribution to the characteristics of soft focus lenses. I wonder if there is an aspherical surface that minimises both spherical aberration and coma and what would its blur figures look like.

As I understand it, there is no simple lens whose spherical surface minimizes both spherical aberration and coma. Doing that requires a more complex lens design (in the course of which other aberrations may be mitigated as well).

The Beach Multi-focal and Struss Pictorial soft focus lenses of the 1920s and 1930s were reputed to include hand-polished aspheres to give an even soft focus effect over the entire plate rather than sharper in the middle and fuzzier in the corners. Or is that just legend?

As I understand it, the Struss Pictorial lens used a single meniscus lens (which gives a lot of spherical aberration and coma) with a stop placed in front of the element which served to reduce the coma (even though coma is not ordinarily affected by aperture). I have no good details at hand. I assume that the lens has spherical surfaces, but who knows.

This lens is discussed some in this thesis of W. R. Young III:

http://research-repository.st-andrews.ac.uk/bitstream/10023/505/6/W Russell Young PhD thesis.pdf

I'm not sure about the Beach Multi-focal (it is mentioned in that thesis but there is no detailed discussion). It might be a lens in which the aperture stop has multiple small holes across its span.

I hear about hand-polishing of the Struss lens, but I don't think that means to distress the surface to provide soft-focus. But I really have no idea.

One wrinkle in this area is that these soft-focus lenses were designed for use with orthochromatic film, with which their chromatic aberrations contributed to the soft focus effect. With the advent of panchromatic film, this nice recipe got screwed up, and so narrow-band filters often had to be used with these lenses to maintain their intended performance.

There is a lot of neat stuff!

Best regards,

Doug

#### Jerome Marot

##### Well-known member

This is quite an interesting document. Thank you.

#### Doug Kerr

##### Well-known member
Hi, Jerome,

This is quite an interesting document. Thank you.
Glad you like it. I find it quite valuable.

Best regards,

Doug

#### Jerome Marot

##### Well-known member
As I understand it, the Struss Pictorial lens used a single meniscus lens (which gives a lot of spherical aberration and coma) with a stop placed in front of the element which served to reduce the coma (even though coma is not ordinarily affected by aperture). I have no good details at hand. I assume that the lens has spherical surfaces, but who knows.

That lens may correspond to patent US 1 347 673

I hear about hand-polishing of the Struss lens, but I don't think that means to distress the surface to provide soft-focus. But I really have no idea.

You may be interested in patent US 2 233 591. Although it is from a later era, it shows a method on how to grind a meniscus lens with variable curvature for soft-focus effect. Earlier manufacturers may have used similar methods and kept them a trade secret.

One wrinkle in this area is that these soft-focus lenses were designed for use with orthochromatic film, with which their chromatic aberrations contributed to the soft focus effect.

Indeed.

#### Doug Kerr

##### Well-known member
Hi, Jerome,

Thanks.

Both very interesting.

Best regards,

Doug