Image Circle Passed, Corners Still Dark: The Two Numbers That Decide

The short answer: the arithmetic that says an image circle covers a sensor is necessary, and it is not sufficient. When the corners go dark after a move to a larger sensor on the same mount, two numbers decide whether that lens was ever meant for the new format — the relative illumination at the corner image height, read at the aperture you actually run, and the MTF at that same image height against the tolerance your measurement needs. The diameter comparison only tells you whether coverage is geometrically possible. A lens can pass it and still leave the corners unusable, because the coverage claim was written at a different aperture, a different wavelength or a different image height than the one you are now asking it to hold. Before the next order, get both numbers with their conditions attached.

One Diameter, Two Numbers, and Only One of Them Decides

The usual first move is correct as far as it goes: take the sensor’s effective width and height, compute the diagonal, and compare it against the image circle named in the datasheet. That filters out lenses that were never in the running. What it does not do is tell you whether the corners of your image are usable at the aperture, wavelength and working distance your station actually runs. Those are separate claims, and they are usually the ones missing from the quotation.

What you check	What it settles	What it leaves open	The number to ask for
Sensor diagonal from effective area	Whether coverage is geometrically possible at all	Everything about the corner: brightness, colour, sharpness	Effective width and height in millimetres, and the diagonal you compute from them
Nominal image circle	The boundary the lens was designed for	How the lens behaves at that boundary under your conditions	The diameter, plus the conditions it was defined at
Relative illumination at the corner image height	How much light reaches the corner relative to the centre	Absolute sharpness, colour accuracy, mechanical cut-off	A percentage at a named image height, at your working f-number and wavelength
MTF at the corner image height	Whether the corner can carry your measurement tolerance	Whether the corner is evenly lit or was cut by hardware	Modulation at a stated spatial frequency, at that image height, same conjugates
Rear clear aperture and chief ray angle	Whether the barrel or adapter cuts the cone before the sensor sees it	The illumination level itself	Rear clear aperture in millimetres; chief ray angle against image height

The diameter test settles possibility, not usability

Format names are historical shorthand, not dimensions. A lens sold as “one inch” is shorthand for a design intended to cover a sensor of roughly that diagonal; the name is not a measurement, and two lenses carrying the same name can differ in what they actually cover. Coverage belongs to the lens; angle of view belongs to the sensor. Compute from the effective area of your sensor instead. A 1/1.8-inch part with a 7.2 by 5.4 millimetre effective area has a diagonal of 9.0 millimetres, and that is the number the lens has to beat. For a line sensor the same arithmetic runs on the long dimension, which is why line scan coverage is worked out on scan width rather than on a frame diagonal.

There is a second trap. A lens throws a little light past its nominal circle, and that extra ring is exactly the light you are looking at when the corners are “almost” covered. Beyond the nominal circle nothing is specified: the falloff continues, aberrations grow, and no datasheet figure covers that zone. Counting that ring is how a station ends up with corners that are lit but unusable, and with a lens that gets blamed for a decision nobody actually made in writing.

Corner relative illumination at the aperture you run

Relative illumination is the corner’s brightness normalised to the brightest point in the image. It is plotted against distance from the centre, with the diagonal of each sensor format marked on the axis and one curve per aperture. Reading it is a lookup, not an interpretation: find your diagonal, find your working f-number, read the percentage.

Two things about the curve matter more than the number itself. First, it improves as you stop down, because optical vignetting falls away — but stopping down does not enlarge the circle the lens was designed to cover, and it does not open a cone that the barrel, an adapter or a protective window is mechanically cutting. Those are different mechanisms and only one of them responds to the iris. Second, the shape of the curve says something about design quality: a corner figure that rises again before it falls puts a bright ring in the image, which is a harder defect to work around than a smooth decline.

MTF at the corner image height against your tolerance

Centre MTF is the least informative number in a lens quotation. The corner is where a format mismatch shows up as contrast you cannot recover, so the claim has to be stated at the image height you care about: modulation at a stated spatial frequency, at the corner, at the same conjugates and wavelength as the centre figure. The spatial frequency should be tied to your pixel pitch, not copied from the datasheet headline — reading an MTF chart only helps if the axis is in line pairs per millimetre and the field position is labelled.

Then convert it into your own terms. If the corner MTF cannot hold the edge contrast your sub-pixel routine needs, shading correction will not save it: flattening illumination fixes brightness, not lost modulation. A lens that clears the centre and fails the corner is not a lens to calibrate around. It is a lens that was never specified for that area.

The Claim That Makes the Number Usable

A coverage number without conditions is a marketing sentence. When you ask for the corner figures, ask for the frame they sit in, and refuse the number if the frame is missing. At minimum: the aperture or aperture range, the wavelength band, the working distance and conjugates, the image height in millimetres, the relative illumination percentage, the MTF value with its spatial frequency, the rear clear aperture, the chief ray angle against image height, and the pass criterion the maker applies to its own measurement.

Ask for the same numbers in the batch documentation rather than in a one-off email. Production batches ship with test reports covering the agreed family parameters — typically MTF, distortion, back focal length and depth of field — together with the batch number, the test conditions, the inspection coverage and an explicit pass or fail call. A corner row at your image height belongs in that report, because that is the document you will still have when a second order arrives. If a maker can state corner performance in correspondence but not in the batch report, the claim has not yet become part of the product.

Ask also for the family ladder around your part. A maker that builds formats from 2/3 inch up to 1.1 inch can tell you what the next size up delivers at the same working distance and field of view, which turns a reorder into a comparison instead of a gamble. Our own FA C-mount range is laid out that way, so the step from a smaller format to a larger one can be read as a family decision rather than a fresh design project.

Working It Out on Your Station

1. Compute the true diagonal. Use the effective area from your sensor documentation, not the format name and not the marketing “type”. Write the number down; it is the reference for everything below.
2. Find the coverage claim and its conditions. If the datasheet gives a diameter and nothing else, treat the figure as unqualified until the maker supplies the frame described above.
3. Measure a flat field. Illuminate a uniform white target with the station’s own lighting, focus at the working distance, then read pixel values at the centre and at several radii out to the corners. Repeat one stop down. The difference between the two runs separates optical vignetting from everything else.
4. Isolate mechanical cut-off. Remove the protective window, the filter, the spacer ring and any adapter, then repeat. Check the rear clear aperture against the cone your sensor needs: a mount that fits is not a mount that passes light. If the corners improve with hardware off, you have found a mechanical cause and the lens may be innocent.
5. Repeat at working conditions. Working aperture, working distance, and your illumination band. A near-infrared band can move both focus and throughput, so a visible-light corner test does not transfer to it.
6. Compare against tolerance, not against the centre. Put the corner MTF beside the contrast your measurement needs, and the corner illumination beside the exposure headroom your lighting leaves. Draw both lines before you look at the data.

Questions That Come Back After the First Measurement

Does stopping down bring the corners back?

Partly, and usually not enough. Closing the iris reduces optical vignetting, which lifts corner illumination until the vignetting is gone, and it can sharpen the corner slightly by trimming aberrations. It does not enlarge the designed circle, and it does not clear a mechanical obstruction. It also costs light and depth of field, and past a point diffraction takes back what the aberrations gave up. If the corners only become acceptable at an aperture your station cannot afford in exposure time or in depth, the honest reading is that the lens does not cover the format.

The datasheet says one inch — why are the corners still dark?

Three usual reasons. The sensor may not be the format the name suggests: a 1/1.2-inch sensor sits between 2/3 inch and 1 inch on diagonal, which is exactly the gap where lenses get chosen by name instead of by number. The claim may have been made at a smaller aperture or a shorter image height than yours. Or something in the optical path — a protective window, a filter stack, a spacer, an adapter with a narrow bore — is cutting the cone before the sensor sees it. Test in that order: the first two are specification problems, the third a mounting problem you can fix this week.

Crop to the usable circle, or change the lens?

Cropping is legitimate while the remaining area still meets your tolerance. Recompute millimetres per pixel on the reduced area, recheck the measurement uncertainty and the field of view the station was specified for, then write the crop into the as-built record so the next engineer does not rediscover it. Cropping stops being legitimate when the reduced field no longer covers the part, or when the corner you keep is still below your contrast line. At that point the honest options are a larger-format lens from the same family, a different sensor, or a change in station geometry — and the format-by-format lens data you gathered is what makes that conversation short. The distinction is worth keeping separate from colour fringing at the field edge: uneven coverage shows up as brightness and contrast falling away, while colour shading is a different problem with a different fix.

Conclusion

The diameter arithmetic earns its place as a filter, and it should never be the last word. What decides a reorder is whether the corner of your image, at your aperture, your wavelength and your image height, still carries the illumination and the contrast your measurement needs. Ask for those two numbers with their conditions attached, ask for them inside the batch report, and treat an unqualified diameter as an unanswered question. A maker who can put corner relative illumination and corner MTF beside your sensor diagonal has answered the question you were actually asking. A datasheet that offers only the circle has answered a different one.