Airy disk size and the diffraction limit: match pixel to feature

A 20 MP camera with 2 µm pixels bolted behind a low-magnification telecentric is the expensive way to learn that the aperture, not the sensor, sets the resolution floor. Diffraction spreads the image of every point into an Airy disk, and nothing downstream — no smaller pixel, no sharpening filter — recovers detail the disk has blurred away. Two numbers decide what an optic can see: the Airy disk radius on the sensor, and the smallest feature size in object space once magnification is applied. You will compute both from your own working f-number, wavelength and magnification, follow a worked 0.052× telecentric example end to end, and finish with the pixel pitches worth paying for and a five-point ordering check.

What diffraction does to a point of light

Light travels as a wave, and a wave passing any aperture spreads at its edges. Even a mathematically perfect lens therefore does not image a point as a point: it forms a bright central disk ringed by faint halos, the Airy disk. The radius from the pattern peak to the first dark ring is the number that matters:

λ is the wavelength of the light you image with, and WF# is the working f-number: the f-number of the cone reaching the sensor at finite magnification, equal to the infinity-conjugate F-number multiplied by (1 + m). Telecentric data sheets quote it directly, and it is the stop to use in every calculation below.

Two points are just separable when their Airy disks overlap by no more than one radius — the Rayleigh criterion. This floor has nothing to do with lens price or polish: a flawless lens obeys it exactly, and a real lens only lands further inside it. The diffraction limit and aperture effects article covers the aperture side in detail.

The radius and the cut-off frequency

Work the formula at WF# 8 in green light (λ = 0.587 µm) and the radius is 1.22 × 0.587 µm × 8 = 5.7 µm. Two resolution statements hang on that one number.

The first is the radius itself: two points closer together than one Airy radius on the sensor cannot be separated, whatever pixel pitch sits behind them.

The second is the cut-off frequency, the finest stripe pattern the lens passes at all:

At WF# 8 in green light that is 1 / (8 × 0.000587 mm) = 213 lp/mm. Contrast falls to exactly zero at the cut-off, and machine-vision algorithms need contrast to work with, so the usable frequency sits below it — read it from the lens transfer curve at the 20 % contrast line.

Wavelength moves both numbers in step. The same WF# 8 stop gives a 6.3 µm radius and a 192 lp/mm cut-off at 650 nm red, and 4.4 µm and 278 lp/mm at 450 nm blue. Shorter wavelength means a smaller disk and a higher ceiling, which is why the wavelength choice is worth making before the lens choice — the worked example below fixes it at 587 nm.

Magnification carries the blur to the object

The Airy radius lives on the sensor; the feature you care about — a pin diameter, a slot width, a scratch — lives in object space, and magnification m converts between them. A feature with two defining edges needs both edges carried separately through the blur, which puts the practical floor at twice the radius:

At WF# 8 in green light, 2rA = 11.4 µm, and the floor by magnification runs:

A different working f-number scales the whole column linearly: halve the stop to WF# 4 and every floor halves. Magnification itself is a fixed data-sheet value on a telecentric and holds through depth of field, unlike an entocentric lens; the companion object-space pixel size calculation gives the matching pixel-to-millimetre conversion.

Telecentric lenses come in magnification steps, and the step you buy is the step that sets this floor. The LS-TCL-1390 is a 0.179× object-side telecentric on a 2/3" sensor, the LS-TCL-1422 a 0.166× bi-telecentric with 258 mm working distance on 1", the LS-TCL-0651 gives 0.66× at 110 mm WD, and the LS-TCL-2118 is a 1× lens at 108 mm WD.

At WF# 8 in green light the two 0.17× class lenses sit at a 64–69 µm object floor, the 0.66× at 17 µm and the 1× at 11 µm — all before a single pixel has been chosen.

Low magnification is not a compromise but the consequence of a large field or a long standoff. Long standoff at full 1× is its own product class: the LS-TCL-1972 holds 1× magnification at 440 mm working distance on a 2/3" sensor.

Worked example: a 0.052× telecentric at WF# 8 in green light

Take the top row of the table as a complete case — large-field gauging, the kind of task where a 20 MP sensor is tempting.

The inputs:

• Working f-number WF# = 8, read from the lens data sheet; 8 is a typical rated stop for a small-magnification telecentric and the assumption behind every number below.
• Wavelength λ = 0.587 µm, green monochrome light, the gauging default.
• Magnification m = 0.052×, typical of a long-working-distance telecentric: the 11 mm diagonal of a 2/3" sensor divided by 0.052 covers a field of about 211 mm.

The rules in words: the Airy radius equals 1.22 times the wavelength times the working f-number; the resolvable feature is twice that radius, divided by magnification to reach object space; the cut-off frequency is one over the f-number times the wavelength.

The arithmetic:

• rA = 1.22 × 0.587 µm × 8 = 5.7 µm on the sensor
• 2rA = 11.4 µm
• object floor = 11.4 µm / 0.052 = 219 µm, call it 220 µm
• cut-off = 1 / (8 × 0.000587 mm) = 213 lp/mm, a 4.7 µm pattern period

Result: the smallest feature this optic resolves is about 220 µm on the object, and nothing finer than a 4.7 µm-period pattern survives on the sensor with any contrast at all.

Decision: a 2 µm pixel is wasted here — the 11.4 µm Airy diameter spans it 5.7 times and the radius 2.8 times, so every extra pixel re-samples blur the lens has already merged. Anything under 220 µm is unresolvable at any pixel count. To reach a 100 µm feature, halve the stop to WF# 4, floor 110 µm, or raise magnification to 0.21×, floor 54 µm. For feature sizes in the tens of microns, step up the magnification class: the LS-TCL-2306 is a 1.5× telecentric at 150 mm WD on a 1/3" sensor.

At 1.5× the same WF# 8 stop puts the floor at 8 µm on the object, and a 3.45 µm pixel reads 2.3 µm per pixel there.

How small a pixel is worth paying for

Set the pixel pitch p between half and one Airy radius:

At WF# 8 in green light the window runs from rA/2 to rA, that is from 2.8 µm up to one Airy radius. The logic: the smallest separable image-space feature is 2rA, so two pixels across it allow a pitch up to rA; below rA/2 the blur, not the pitch, sets the smallest detail, and extra pixels add only data volume, read-out time and shot noise per pixel.

Set common pitches against that window. A 3.45 µm 5 MP sensor lands mid-window at WF# 8 — matched. A 2 µm pitch oversamples close to three to one and pays in sensitivity, because a pixel half as wide gathers a quarter of the photons. A 6.45 µm legacy 1.4 MP pitch leaves the sensor as the bottleneck. The whole-system floor is set by the worse of the two, roughly 2 × max(rA, p), divided by magnification exactly as before — and the picture flips with the iris: at WF# 2.8 the radius drops to 2.0 µm and even a 3.45 µm pixel becomes the coarse element. Match the pair deliberately, or the Nyquist limit between sensor and lens decides for you.

Size the sensor against your field and feature with the pixel calculator:

When the formula is not the whole answer

Aberrations. The formula is a ceiling, not a promise. Real glass adds spherical aberration, coma, astigmatism and field curvature, so at wide stops a lens lands below its diffraction limit, and the honest number is the 20 % contrast frequency on that lens's own transfer curve. Closing the stop suppresses aberrations and walks the system up toward the diffraction ceiling — one reason moderately stopped-down telecentrics behave so predictably.

Spectrum. The formula holds for the wavelength you image with. White light makes the lens form three slightly different Airy patterns at once, and chromatic aberration widens the effective blur. Monochrome green light collapses the spectrum to a single λ — the same reason green is the recommended gauging wavelength for visible-range telecentrics, whose correction is centered there — and a band-pass filter buys most of the same benefit when the lighting is fixed.

Contrast and defocus. Resolution and contrast are different currencies: a 300 µm defect at 5 % contrast can fail on an optic whose floor is 220 µm, because the transfer function at that feature's frequency has eaten the modulation. Closing the stop for depth of field doubles rA per stop and quarters the light, so the DOF-versus-resolution trade has hard numbers on both sides. Defocus blur in a telecentric stays symmetric through depth, which keeps measurements repeatable while contrast falls — the mechanism behind telecentric symmetric blur. For gauging the floor is the whole point: see when a telecentric lens earns its place for gauging, and the micron-level image measurement accuracy guide for what drifts once the optics are right.

Checks before you order

1. Compute 2rA/m for the candidate magnification and stop, and require the floor at a third of the smallest feature or below — 2rA/m is the zero-margin limit, and contrast is already fading before you reach it.
2. Fix the wavelength before the lens: green monochrome (0.587 µm) for gauging; blue only when the last 23 % of radius matters and the optic is corrected for it.
3. Place the pixel pitch between rA/2 and rA at the rated WF# — half to one Airy radius at WF# 8 in green — and stop paying below that window.
4. Read the lens transfer curve at the feature's frequency, 1/(2t) for image-side width t: the 11.4 µm floor of the worked example lives at 44 lp/mm, far under the 213 lp/mm cut-off.
5. Budget the exposure: from WF# 8 to WF# 16 the light drops fourfold, so check the light head has the headroom before committing to a closed stop.

Where the numbers land badly — feature under the floor, pixel window empty, exposure too long — move magnification or the stop, not the sensor: those two levers shift the floor linearly and the pixel never does. The telecentric lenses and machine vision lenses pages filter by magnification and working distance, the lens finder and lens configurator carry the same filters, and the contact page is where a feature size plus field of view gets this computation checked before you order.