Astigmatism (optical)

Pekka Buttler, June-October 2021

Introduction

‘Astigmatism’ is not only a flaw with optical lenses, but also affects many people’s eyesight (it does mine). Even so, it is quite often the most misunderstood form of optical flaw or aberration.

Astigmatism is – at times – visually similar to comatic aberration (‘coma’), which may contribute to that these two are often mixed up. This confusion comes in the form that many otherwise knowledgeable bloggers, vloggers, reviewers and other pundits talk about ‘coma’ in images that clearly show astigmatism. In effect, astigmatism is often mistaken for coma. This article aims to try to clear up some of this confusion.

Optical astigmatism also has a not easy-to-understand relationship with field curvature. This also might contribute to some of the apparent misconceptions. Bear with me.

This article is part of a JAPB series of articles on the optical flaws of lenses and you can find the index of the series here.

What is Astigmatism and what causes it?

Note please: Astigmatism is inordinately complex to understand, and even the best schematic illustrations are only limitedly helpful without a thorough understanding of optics and the way optical schematics are intended to be interpreted. Instead of trying to produce better illustrations than the rest of the photo-blogosphere, I’ll try to explain astigmatism verbally. If you want to, you’re welcome to tell me how I managed…

What is ‘astigmatism’?

Astigmatism is – assuming you know the classical languages – actually a very descriptive term. In ancient Greek, stigma (στίγμα) means ‘mark’ or ‘spot’ and a- (α-) means ‘without’ or ‘lack of’. A-stigma-tism therefore indicates an inability to produce spots, because all spots or points are rendered either as (fuzzy) lines or airy discs.

In simple terms, an optical system (a single lens element or a collection of lens elements) that is unable to focus monochromatic light rays emanating from an off-centre point onto a point on the focal plane but instead focuses that light onto a line or symmetrical blob, suffers from astigmatism. Unlike other monochromatic aberrations (e.g. spherical aberration or comatic aberration), astigmatism is typically characterized by that this inability to refocus a point’s rays takes place in one or two dimensions: A lens element may produce sagittal or tangential astigmatism or a mix of both. See illustration below.

Sagittal … ? Tangential …?
While photographers (and – thanks to maths education – most other people) conceive of a 2-dimensional system to be characterised by two axes (namely: x (horizontal) and y (vertical)), optical lenses produce their image according to another coordinate system, most simply described as distance (from optical axis) and angle.

Sagittal astigmatism shows up along a ring (centred on the image centre) and occurs when the lens refocuses a point’s light at the correct distance from the image centre, but is less precise regarding angle, whereas
tangential astigmatism shows up along rays traveling from the image centre outward, and occurs when the angle of refocusing is correct, but distance is less precise.

Note please: One commonplace illustration of how tangential and sagittal astigmatism affect imagery typically involve a cart wheel (rim and spokes) and go on to make the point that sagittal astigmatism leads to sharply defined spokes but a blurry rim, while tangential astigmatism would produce a sharp rim, but that the spokes would be blurry. Not only is this illustration somewhat misleading (for the rim to be sharp all around the circumference, the axis of the cartwheel would have to be in the image centre, at which stage the spokes would only be blurry far away from the image centre), but it is also very simplistic in that it a) disregards that tangential and sagittal astigmatism typically coincide to some degree; and b) overlooks the complex ‘relationship’ ¹ astigmatism has with field curvature. Read on, I hope to be able to clarify.

What causes astigmatism

There are basically two types of astigmatism: astigmatism in eyes (ophthalmic astigmatism) and astigmatism in optical lenses (optical astigmatism). The reason why I’m bringing up ophthalmic astigmatism is that understanding ophthalmic astigmatism actually helps understanding optical astigmatism.

Ophthalmic astigmatism

Ophthalmic astigmatism is typically caused by a lack of symmetry in the cornea (the ‘lens’ of the eye). Instead of the cornea being perfectly circular, those who suffer ophthalmic astigmatism have a cornea that is slightly compressed (instead of circular, it is compressed into an oval. This is referred to as having a ‘cylindrical’ lens).

This lack of symmetry leads to that rays originating from a point cannot be refocused on a point on the eye’s retina. Instead, persons suffering from astigmatism are liable to see points as blurry lines, squares as blurry rectangles or diamonds and will consequently have big trouble making out text (without astigmatism-correcting glasses).

Ophthalmic astigmatism is not dependent on whether the rays originate along or off the optical axis ². When a cornea is asymmetrical this can be corrected by glasses that introduce a corrective asymmetry.

Theoretically optical lenses also may be affected by the same kinds of issues, but for an optical lens to display significant ophthalmic-type astigmatism, either manufacturing standards would have to be exceedingly shoddy or the lens would have to be subject to significant de-centering. In effect, only very old, hand-made optical lenses are liable to systematically suffer from such maladies.

Optical astigmatism

Optical astigmatism on the other hand affects only off-centre rays. The fundamental cause for optical astigmatism is that even a perfectly symmetrical lens element is not symmetrical when viewed from off-axis. The farther off-axis, the more pronounced is the astigmatism (but also the shape of the lens surface plays a role). If you stop to think about it, optical astigmatism means that (unless corrected) one could not design nor manufacture anything but a narrow-angle lens to be sharp corner-to-corner.

But astigmatism has another, especially disagreeable characteristic, namely that it interacts ³ with field curvature. At this stage, if you have not already read the JAPB article on field curvature (or otherwise feel sufficiently confident), I suggest you have a peek.

The thing with astigmatism is that it is not an absolute. It depends. And it depends complexly.
Imagine a point light source (a star in the night sky), that shows up as a weird, distended blob/streak from the image centre towards the outer limits (tangential astigmatism). That streak you see in your image editing software is not an unavoidable trait of your lens, it is simply how your lens renders an object that is at that distance with the focus setting you’ve currently set your lens on.

Should you start tweaking your lens’ focus ring, you might possibly be able to reach a zone where your lens would transition from tangential astigmatism to sagittal astigmatism, and – just for a moment – your lens would be able to render the star free of astigmatism (if you cannot reach such a point it is likely caused by your lens’ infinity hard stop). Problematically, while there might be such a minimum-astigmatism setting, it would not do you any good, because your star would likely not be in focus.

For the lack of a better term, each lens has a minimal astigmatism-plane – a plane defined by (for each angle and distance) the lack of both sagittal and tangential astigmatism. On one side of that plane (e.g. farther away) you will have one form of astigmatism and on the other side of that plane you will have the other form of astigmatism, but on that plane, astigmatism will be negligible to non-existent. That plane is actually (depth-wise) relatively thick in the image centre, but gets narrower towards image borders and corners. And just as with the plane of focus (remember: field curvature), that plane might be straight and level, somewhat curved, strongly curved, or even wavy.

And just to add insult to injury, the plane of minimal astigmatism might be geometrically quite different from the lens’ ‘field curvature’. And then there’s also the plane defined by your camera’s film/sensor to consider.

As a result of this complex ‘interaction’, you end up with situations where your objects (the airplane in the top left corner) are at one distance, your lens’ field curvature dictates that your field of sharpness is somewhere else entirely (e.g. For the top left corner to be sharp, the airplane should be close enough for its wing to decapitate you) and your plane of minimal astigmatism is somewhere else again. One could say, that all would be well if these three planes would ‘interact’, negotiate and reach a compromise and not ignore each other so successfully. But of course they do not.

And because they do not, the lens’ plane of minimal astigmatism and the lens field curvature will not coincide (except, typically, in the image centre). As a result of this, you might be in the situation where – except for the image’s absolute centre – the best you can hope for is to be allowed to choose between
a) being on the field (of sharpness) curvature but suffering one or the other astigmatism,
b) being on ‘minimal astigmatism plane’ but having to deal with being out of focus because you’re far from the field curvature,
c) or trying to balance between the two, hoping they’re not both so far away from either optimum that you end up with a lot of astigmatism that won’t be noticeable because you’re so far from the field of sharpness.

Controlling astigmatism

Astigmatism was the bête noire, the bugbear of early photographic lens designers – especially as astigmatism (unlike many other aberrations) could not be satisfactorily addressed simply by stopping down (stopping down would typically only mask its worst effects). While even early optical theoreticians (Petzval, Seidel and ilk) hypothesised that one could conceivably control astigmatism, the selection of 18^th and 19^th century optical glasses and lens manufcaturing and computation technologies did not allow these theories to be tested.

The solution originated – alike many solutions to optical problems – in the mid-sized town of Jena, Germany in the closing decade of the 19th century (Jena then had a population of roughly 14 000). Back then Jena not only housed the headquarters of Carl Zeiss, but was also the founding home of Schott, the world’s probably first enterprise focusing on the development and manufacture of specialist optical glass. The combination of Schott’s new glass types with the optical competence at Carl Zeiss (especially that of Dr. Paul Rudolph) allowed for the development of the first functioning astigmatism-corrected lens. The Carl Zeiss Protar (1890) ushered in a new age and made astigmatism-correction (lenses branded as ‘anastigmat’) the main selling point of optical lenses for 60 years to come (until the development of lens coatings ushered in the age of selling lenses by the coatings they sported…)

Today virtually all lenses are designed to control astigmatism, but…
Considering that perfect astigmatism correction would…
… Necessitate that you align a lens’ ‘minimal astigmatism plane’ perfectly with the lens’ field curvature (which, in itself is no mean feat), while …
… forcing both to be planar and (nearly) perpendicular to the optical axis, while …
… at the same time not allowing other optical aberrations run rampant…
… well, maybe it’s one of those many ‘easier said than done’ things.

In practical terms, the best optical designers can achieve is to try to modify lens element shapes, materials and distances in such a way that field curvature, the minimal astigmatism-plane and the sensor plane roughly coincide. And because astigmatism is distance-dependent, this optimisation can really be done only for one focusing distance. This is especially obvious when you push a lens far beyond its design parameters (e.g. by mounting a general purpose standard lens onto a stack of extension tubes).

I don’t shoot astro, or night-time vistas and I don’t do photolithography. Do I need to care about astigmatism?

The short answer is: yes. Not necessarily more than other aberrations, but care you should. The reason is simple.

While you need a wide-open shot of a night-time vista or a starlit sky to clearly see the effect of sagittal/tangential astigmatism (thus helping you diagnose that the lens, indeed, suffers from astigmatism), that does not mean that astigmatism might not affect every shot⁴. Just as with all other aberrations that become clearly visible in specific circumstances (like spherical aberration or coma in point-light sources or chromatic aberration in high-contrast transitions), astigmatism is not just a pain in extreme-contrast shots. Instead, if the lens evidences astigmatism in starlit nights, it will have astigmatism also in regluar, low contrast situations. Astigmatism, just as spherical aberration, leads to an overall lessening of contrast and definition, with two important qualifications:

Firstly, given that astigmatism does not occur with rays that come in along the optical axis, astigmatism cannot occur in the centre of the frame (spherical aberration, however, can). At the same time, for rays to be liable to produce astigmatism (in combination with ‘suitable’ lens elements), those rays do not need to be especially oblique – even tele lenses and telescopes can suffer astigmatism.

Secondly, while one needs to photograph a starlit sky (or similar extreme contrast setting with point-light sources) to be able to distinguish between coma, astigmatism, spherical aberration and field curvature with certainty, tangential astigmatism – especially when strong – is liable to not only produce a general lessening of definition and contrast, but may (similarly to coma and field curvature) contribute to an image that looks like the corners are stretched-out or smeared, while significant sagittal astigmatism may produce a similar but radial smeared effect⁴.

Dealing with astigmatism in the field.

With most aberrations, the number one remedy is simple: stop down. This works best with a long list of aberrations (spherical aberration, longitudinal chromatic aberration, coma) and even significantly helps address vignetting as well as helping to mask mild field curvature through deepening the depth of field.

And certainly stopping down can also affect astigmatism (especially with those lenses that are relatively well but not perfectly corrected), but it may not be enough. Considering that the main issue causing astigmatism is that a circular object such as a lens element is not circular from the perspective of an off-axis ray, simply making that circular object smaller through closing down the aperture does not address the problem entirely. Closing down may in fact produce a situation that makes astigmatism more visible (as other aberrations recede quicker). Even so, if your aim is a crisp image free of visible aberrations, closing down a few stops is always the way to proceed.

Otherwise, the only real piece of advice one can give is to – if astigmatism is a real issue with a specific lens – not to use that lens. I remind you that astigmatism correction is typically neither perfect nor universal. A lens that might offer brilliant and astigmatism-free pictures at a close range may likely not be well suited for landscape or astro work (and vice versa). It is therefore important to know your lenses.

Footnotes

¹ Bloggers and pundits (yours truly included) tend to do a weird thing when we try to explain complex issues (like optics and optical aberrations) in laymen’s terms: we tend to phrase everything in human-scope terms. We talk about two forces of nature ‘interacting’ (they’re not, they are merely overlapping and we with our feeble brains are struggling and thinking of it like it was a negotiation, while its not); we speak of things having ‘relationships’ even when the things don’t even know of each others’ existence (and how could they?), and we raunch up lens comparisons (e.g. here) by making be like it was a competition among human (or even: animal) competitors. On one level this anthropomorphising of lenses and optics is pitiful. On another, it may just be an indicator for how languages tend to be most expressive when one’s using terminology intended for describing intra-human affairs.

² To be precise, ophthalmic astigmatism typically gets worse farther from the optical axis, but as a normal person’s eyes’ optical axis is always directed towards what the person is trying to decipher, off-axis astigmatism in human eyes is largely irrelevant.

³ To be honest, astigmatism does not interact with anything, because astigmatism just is a fact of how the universe works and has no agency nor the ability to ‘interact’ or decide not to… But the result (in your photography) is determined by the relative interactions compound effects of field curvature, astigmatism (and all other aberrations).

4 Again, to be precise, one should say “… affect every shot at that focusing distance and object distance”, because, as already indicated, astigmatism reacts to focusing distance and object distance. Therefore a lens that show obvious astigmatism in a starlit sky need not show any astigmatism in a close-up shot.

⁵ This also highlights one of the ‘advantages’ of astigmatism, namely that (in the case of some lenses) the relative geometries of field curvature may contribute to producing interesting bokeh effects, such as the famed swirly bokeh.

Read more:

• Astigmatism (optical systems) on Wikipedia
• Paul van Walree on Astigmatism (archived)
• Some slides from an optics course at the Friedrich Schiller Universität (Pay special attention to slides 21-22
• Interesting visualisations on the causal mechanism of Astigmatism at FSU.