Coma and astigmatism in vision lenses: off-axis blur explained

The center of a machine vision image is usually sharp; the corners are where lenses run out of correction. Two aberrations cause most of that edge blur: coma, which turns an off-axis point into a comet with a radial tail, and astigmatism, which splits the same point into two focal lines. Both live in every lens built from spherical surfaces, both grow as the feature moves off-axis, and they answer differently to the iris: stopping down suppresses coma quickly and astigmatism barely at all. The scaling laws below come with a worked example and a decision for each case: where on the sensor a measurement belongs, what f-number to set, and when to stop tuning a standard lens and change the design.

What coma and astigmatism do to an off-axis point

Both errors are zero at the exact center of the field and grow toward the edge, which is why the middle of the image looks clean while the last millimeters of the sensor go soft.

Coma hits a point that sits off-axis. Its light arrives at the glass at an angle, and rays through the outer zones of the aperture focus at a different height than rays near the center. The point spreads into a comet-shaped patch with the tail pointing radially outward, away from the field center. For measurement the danger is asymmetry: an edge tool that finds the 50 percent crossing on a comet tail reads a shifted position, biased along the radius.

Astigmatism attacks the same off-axis point by a different route. The lens presents a different effective curvature in the meridional plane (the plane containing the chief ray and the optical axis) than in the sagittal plane at right angles to it, so the two planes focus at different distances. The point images as a line at one focus, a small round circle of least confusion between them, and a line rotated 90 degrees at the other focus. Standard optics texts call these the tangential and sagittal focal lines; the reference treatment of astigmatism in optical systems covers the geometry.

Both errors are built into spherical surfaces. Correction comes from the designer: more elements, glass types chosen so their errors cancel, and aspherical surfaces. That correction effort is what separates an inexpensive board lens from a multi-element FA lens of the same focal length.

The scaling laws that drive the decision

Third-order aberration theory — a good estimate at moderate field angles — gives the proportions that decide every fix on the list:

Read them as two levers:

• The iris. Halve the aperture diameter — two stops, f/2.8 to f/5.6 — and coma falls to a quarter while astigmatism only falls to half. The iris is a strong coma tool and a weak astigmatism tool.
• Field position. Double the image height of the feature and coma doubles, but astigmatism quadruples. At two-thirds of the half-diagonal the astigmatic patch is (2/3)² ≈ 44 percent of the corner value; at half field it is 25 percent. That quadratic growth is why wide fields get expensive fast.

These two sit in a family: spherical aberration falls with the cube of the aperture, and distortion ignores the iris entirely. The whole family and what stopping down does to each member is covered in optical aberrations in machine vision lenses and how stopping down helps.

Why the iris runs out of power: diffraction

Every stop you close removes aberration blur and adds diffraction blur. The floor is the Airy disk, 2.44 λ N; at 550 nm, typical for white or green lighting, that is about 1.34 µm per unit of f-number:

The decision the table forces: on 3.45 µm pixels, work between f/4 and f/5.6. By f/8 the Airy disk spans three pixels, and each further stop deepens the diffraction floor faster than the remaining coma improves. Sensors with 5 µm pixels buy one more stop; f/8 still sits at about two pixels there.

Worked example: a corner edge on a 1.1" sensor with a 16 mm lens

The case: a bracket sits in a fixed nest, the edge to gauge lands near the corner of the sensor, and the station must hold a repeatable edge position.

The lens is the LS-FFL-0567: a C-mount machine vision lens, 16 mm focal length, 1.1" format, 20 MP, F2.8-22, distortion under 0.3%.

Inputs the station fixes: a 1.1" sensor with 3.45 µm pixels filling a 16 mm image circle, so the corner sits 8 mm off-axis; white LED lighting at a 550 nm design wavelength; a nest that cannot move.

Arithmetic, in the order the decisions chain:

1. Semi-field angle (half the field of view) at the corner: tan θ = 8 mm ÷ 16 mm = 0.5, so θ = 26.6 degrees. At two-thirds of the half-diagonal, 5.33 mm, it is arctan(5.33 ÷ 16) = 18.4 degrees.
2. Astigmatism if the feature moves inward: (5.33 ÷ 8)² = 0.44, so the corner-to-two-thirds move cuts the astigmatic patch to 44 percent; coma falls linearly to 67 percent.
3. Iris: two stops from the F2.8 limit is f/5.6. Coma drops to a quarter, astigmatism to half, and the diffraction floor is 2.44 × 0.55 µm × 5.6 = 7.5 µm, or 2.2 pixels.

Decision: set f/5.6 and assign the corners to detection — presence, gross defects, anything with a pixel or two of tolerance. If the gauged edge must stay at 8 mm image height, the arithmetic above closes the iris route: astigmatism halves at best while the Airy floor doubles. Move the nest so the feature lands inside about 5.3 mm of the axis — two-thirds of this half-diagonal — or change the optics. The rule generalizes: corners detect, the central two-thirds measures, and the outer third of the field belongs to corrected designs.

The four fixes compared

Two stops closed is a quarter of the photons. Funding it with exposure time on a moving part is the classic way machine vision motion blur enters an otherwise sharp station; funding it with gain costs noise; funding it with lighting costs the light source.

The longer-lens row: 35 mm on the same 16 mm image circle works at arctan(8 ÷ 35) = 12.9 degrees, half the obliquity of 16 mm. You hold the same field of view by standing off in proportion; the lens focal length calculator turns sensor, field of view and working distance into the focal length.

Smaller format plus longer glass compounds the gain. The LS-FFL-0679 is a 35 mm C-mount lens rated 6 MP for 2/3" sensors.

A 2/3" sensor carries an 11 mm diagonal, 5.5 mm to the corner, so 35 mm works at arctan(5.5 ÷ 35) = 8.9 degrees — a third of the 26.6 degrees above. The field of view shrinks with the sensor unless the station stands off further, so budget the station length. For the focal-length decision itself, see how to choose a machine vision lens focal length.

When a telecentric lens is the better fix

When the feature cannot be repositioned and the measurement must live at the edge of a wide field, stop asking the iris and change the architecture. A telecentric lens images the whole field with chief rays parallel to the axis, so the obliquity that feeds coma and astigmatism is near zero by construction, and the design is corrected over one bounded field at one working distance. Magnification stays constant when part height varies, which removes the height-to-scale error of ordinary lenses at the same time.

The LS-TCL-1422 is a bi-telecentric C-mount lens: 0.166x magnification, 258 mm working distance, listed for 1" sensors.

A 1" sensor carries a 16 mm diagonal, so at 0.166x the field of view is 16 ÷ 0.166 ≈ 96 mm diagonal: fit for bracket-sized parts, not for a whole tray. The boundary between the two architectures is drawn in entocentric vs telecentric lenses: when to use each, and the tolerance case that decides it in when you need a telecentric lens for gauging. Browse telecentric lenses when the station tolerates a fixed working distance.

Checks before you commit

1. Corner focus sweep. Put a dot near the corner, sweep focus through best center focus, and read the signature: comet = coma, line–circle–rotated line = astigmatism, corner sharpening while center softens = field curvature.
2. Image-circle margin. The lens's image circle must exceed the sensor diagonal with margin; a lens that just covers softens and darkens the corners. Calibration target field of view and lens coverage margins shows how much margin to hold.
3. Iris and light budget together. A lens that reaches f/5.6 with room to f/8 is only half the requirement; the four-times light bill is the other half.
4. Format labels. Trust the printed image circle, not the format name; lens sensor format explained untangles the 1/1.8", 2/3" and 1" naming.
5. Wide-angle sanity. Semi-field angles beyond about 30 degrees (fields of view beyond 60 degrees) are wide-angle territory; expect corner compromise and plan measurements inward.

When the usual answer is wrong

"Stop it down" has a fence around it, drawn two sections above: past f/8 on small pixels the cure costs more than the disease, and astigmatism — the dominant corner error — only halves per two stops anyway.

"Soft corners cannot measure" is half wrong. Symmetric blur, including the circle of least confusion of astigmatism, keeps edge positions repeatable because it spreads equally either side of the true edge; telecentric symmetric blur and why defocus keeps measurements repeatable works through the argument. Coma is the exception: its asymmetric tail drags the 50 percent crossing outward along the radius, and the bias survives a scale recalibration.

"A 20 MP lens resolves 20 MP" needs a where. Resolution classes describe on-axis capability; a lens can be honest about its class and still soften the corners your gauge uses. Test on your own sensor at your working distance before the purchase order.

Quick checklist

• Gauging features inside two-thirds of the half-diagonal on standard FA glass; corners for detection.
• f/4 to f/5.6 on 3.45 µm pixels; f/8 acceptable from 5 µm pixels.
• Two stops down means one quarter of the light: fund it with lighting, not gain, on moving parts.
• Comet tail at the corner means coma, and the iris helps; line–circle–rotated line means astigmatism, and position or design helps.
• Edge gauging at the field edge with sub-pixel tolerance means a corrected or telecentric lens at a fixed working distance.

Run the corner sweep before the purchase order: it costs ten minutes and chooses between a fixture move, an iris setting and a corrected lens. Then filter machine vision lenses by sensor format and image circle in the lens finder, or start at the lens step of the vision system configurator.