Working f-number and magnification: light, DOF, resolution

You set the iris to f/2.8, focus on the part, and the frame comes back darker and softer than the exposure math promised. Nothing is broken: the ring value is defined at infinity, where magnification is zero, and at a close working distance the lens runs slower than the dial says. The number that governs exposure, depth of field and diffraction is the working f-number, (f/#)w ≈ (1 + m) × f/#, where m is magnification. This guide works the factor through a 0.5x example with the arithmetic shown, then turns it into three budgets: light lost, focus gained, and where diffraction starts to blur small pixels. It is for anyone sizing a close-range station — inspection under 200 mm, OCR at 0.3x, measurement at 0.5x.

What the f-number on the ring sets

The marking on the iris ring is the ratio of focal length to effective aperture diameter: f/# = focal length / aperture diameter. A 25 mm lens at f/2.8 opens a 25 / 2.8 = 8.9 mm aperture with an area of π × (4.45 mm)² ≈ 62 mm². One stop down to f/4 halves the diameter to 6.3 mm, quarters the area to 31 mm² and halves the light. That is why ring markings step by √2 and each full stop is a factor-of-two exposure step (the F-number convention).

The definition assumes the object sits at infinity. Your part sits much closer, and that assumption is what fails — nearly every ordinary fixed-focal machine vision lens quotes its f/# at that infinite-conjugate condition.

The working f-number: what the sensor actually sees

At finite magnification the image-side cone slows by the factor (1 + m):

Here m is image height over object height, which equals sensor width over field of view once the framing is set (how to read it from a calibration). The formula treats the lens as a single unit; real designs follow it within a few percent across the range machine vision uses.

Below 0.1x the correction is under a third of a stop, so ignore it. From 0.3x up it is most of a stop or more, and every exposure, focus and diffraction figure should be budgeted on (f/#)w from there.

Worked example: a 0.5x set-up on an f/2.8 dial

Take a 25 mm lens on a 2/3" camera (8.8 mm wide sensor) framing a 17.6 mm field of view, so m = 8.8 / 17.6 = 0.5, dial at f/2.8, white light at 550 nm.

• Working f-number: (f/#)w = 1.5 × 2.8 = f/4.2.
• Exposure: throughput falls with the square of the working f-number, (2.8 / 4.2)² = 0.44. The frame is 2.25× darker than the dial promised — 1.2 stops. A 1 ms budget becomes 2.25 ms, or you supply 2.25× the light.
• Diffraction: the spot on the sensor is 2.44 × λ × (f/#)w = 2.44 × 0.55 μm × 4.2 = 5.6 μm (6.8 μm under 660 nm red light). Referred to the object through m = 0.5, that is 11.3 μm of smallest resolvable detail, against 7.6 μm if the dial value had been true.
• Depth of field: with a 10 μm circle of confusion (about 3 pixels on a 3.45 μm camera), 2 × 4.2 × 0.01 / 0.25 = 0.34 mm. Closing the dial one stop to f/4, which works f/6.3, gives 2 × 6.3 × 0.01 / 0.25 = 0.50 mm.

Fixed-magnification lenses print m on the data sheet, so this whole chain becomes a lookup rather than a bench measurement: the LS-FFL-1023 is specified 0.5x at a 150 mm working distance, F13.4.

The three-way trade: light, focus, diffraction

Every working stop buys depth of field and costs light and resolution at once, which is why no single best aperture exists. Normalized to (f/#)w = 2.8 at 550 nm:

The light column is what corners fast-belt and line-scan work into the fastest usable stop. The focus column is what saves applications with part-height variation. The Airy column is the quiet cost: past about two pixels of spot, edges lose contrast faster than focus range grows (contrast and resolution).

The spot rule worth keeping: hold 2.44 × λ × (f/#)w at or below two pixels. On a 3.45 μm pixel that is a ceiling of 6.9 / 1.342 ≈ f/5.1 — work at f/5 or faster for clean edges, and stop down further only when focus range outranks the smallest defect. The diffraction limit in numbers, small pixel camera selection and when sharp previews mislead about focus cover each column of that trade in depth.

Numerical aperture: the same budget from the cone angle

f/# and numerical aperture describe one cone of light from two directions: NA = 1 / (2 × f/#). The worked example's working stop f/4.2 is NA = 1 / (2 × 4.2) = 0.12; the dial's f/2.8 corresponds to NA 0.18. A fast lens is a wide cone — short exposures, thin focus. Microscopy quotes NA in object space; machine vision inherited f-numbers from photography. When a discussion moves between the two conventions, convert through the working value, not the dial, or the comparison is off by the whole (1 + m) factor.

Choosing the working f-number: three cases

Exposure-limited (motion to freeze). Choose the fastest working stop your lighting supports and read the dial backwards: at m = 0.5 you need a dial f/2.8 to work f/4.2, and a lens whose dial bottoms out at f/4 only works f/6. If exposure then runs long, the fix is more light or a lower-f/# lens, not gain.

Depth-of-field-limited (part height varies). Size the stop from the height range: (f/#)w ≈ DOF × m² / (2c). For 0.50 mm of focus range at m = 0.5 with a 10 μm circle of confusion: (f/#)w = 0.5 × 0.25 / 0.02 = 6.3, which is a dial f/4.2 at this magnification.

Resolution-limited (small defects, small pixels). Hold (f/#)w ≤ 2p / (2.44 × λ). For 3.45 μm pixels the ceiling is f/5.1; for 2.5 μm pixels it drops to f/3.7, which at 0.5x means the dial cannot close past f/2.5 without softening the image.

Aperture range is then a selection criterion in its own right. The LS-FFL-0567 covers F2.8-22 with <0.3% distortion on a 1.1" sensor.

Across that dial span at 0.5x the working stops run from f/4.2 to f/33 — enough range to trade light for focus inside one lens, once you read it through (1 + m).

When the dial already is the working value

Two situations suspend the correction. Fixed-magnification lenses, telecentric or not, are specified and measured at their printed magnification, so the printed f/# is the working value: a lens printed 0.4x, F11 runs at about f/11 at its rated distance, not f/15. The LS-FFL-0914 is 0.3x at 168 mm WD, and the LS-TCL-0574 is 0.4x at 227 mm WD.

The second situation is any set-up where m moves. Extension tubes push an ordinary lens into the high-magnification region where (f/#)w climbs steeply (extension tubes for close-up machine vision), and a floating working distance changes m from shot to shot, which is where autofocus optics or a refocus stage move the working stop for you (liquid lens autofocus for changing working distances). Recompute (f/#)w whenever the framing changes.

Checks before you order

1. Write down m from field of view and sensor width; check it against a measured target where the part allows.
2. Multiply the dial value by (1 + m) and carry that working f-number through every exposure, focus and diffraction calculation.
3. Confirm 2.44 × λ × (f/#)w ≤ 2 × pixel size, or accept the focus-for-contrast trade deliberately.
4. At 0.3x and above, shortlist a fixed-magnification lens beside the ordinary fixed-focal one: its printed f/# removes the guesswork.
5. Re-run the numbers whenever the working distance moves, because m moves with it.

Focal length sets what you see (how to choose a machine vision lens focal length); the working f-number decides whether the image is usable. Size the chain in the lens finder or start at the lens step of the configurator; the lenses topic page gathers the rest of the optics guides. If light, focus and spot size will not close at once, send the field of view, sensor and part-height budget through the contact page and we will size the stop with you.