Same lens on a different sensor: how HFOV changes

You spec an M12 board camera around one lens, then the project lands on a different sensor — the 1/4" module becomes 1/3" — and three questions arrive: what happens to the field of view, to the corner brightness and to the distortion? All three have short, computable answers. Below you get the HFOV formula, one 4 mm lens worked through the 1/4", 1/3.2" and 1/3" formats, the image-circle check that decides coverage, and the scaling rule that tells you when to keep the lens and when to change the focal length. It is written for engineers building compact machines around board-level cameras.

What happens to the HFOV when the same lens moves to a different sensor?

The horizontal field of view widens or narrows with sensor width, because the lens projects one fixed image and each sensor is a different-size window on it: the 4 mm example lens shows about 44° on a 1/4" sensor (3.2 mm wide) and 62° on a 1/3" (4.8 mm wide).

The lens does not know what sits behind it. It projects one cone of light with one image scale, fixed by its focal length, and every sensor crops a rectangle out of that projection. Widen the rectangle and the view widens; shrink it and the view narrows. Nothing in the glass changes — focal length, F-number, image circle and distortion curve all stay as printed. That is why one M12 board lens series can serve formats from 1/4" to 1/2": you pick the lens for the projection it makes, and the sensor decides the framing. The three effects bundled into this question separate cleanly: field of view in the next two sections, coverage and roll-off after that, distortion after that.

How do you calculate HFOV from sensor width and focal length?

Compute the horizontal field of view from two numbers you already have — the sensor's active width and the lens focal length: HFOV = 2 · arctan(w / (2f)); the vertical field uses the same formula on the sensor height.

Work from the active area, not from the inch name: the inch labels are nominal and their real diagonals run well below the name. The sizes below are the standard 4:3 frames tabulated for image sensor formats on Wikipedia; when a camera data sheet prints active pixels and pixel pitch, multiply them and use that width instead.

A few boards output a cropped 16:9 frame from a 4:3 die; if yours does, the height shrinks while the width usually stays, so compute the two fields separately.

What HFOV does this focal length give on 1/4", 1/3.2" and 1/3" sensors?

With f = 4 mm, the lens gives 43.6° horizontal on a 1/4" sensor, 59.1° on 1/3.2" and 61.9° on 1/3", because each width enters the same arctan formula while the focal length never moves.

The arithmetic, step by step — the divisor is 2f = 8.0 mm throughout:

• 1/4": 2 · arctan(3.2 / 8.0) = 2 · arctan(0.400) = 2 × 21.8° = 43.6°
• 1/3.2": 2 · arctan(4.54 / 8.0) = 2 · arctan(0.5675) = 2 × 29.6° = 59.1°
• 1/3": 2 · arctan(4.8 / 8.0) = 2 · arctan(0.600) = 2 × 31.0° = 61.9°

The same arithmetic for four common M12 focal lengths:

Vertical fields follow on the heights, and the working distance for a target object width comes from the same relations used in the working distance and field of view formula guide. Run your own lens–sensor pair here:

Will one lens cover every sensor size?

Coverage is a yes-or-no check: the lens's specified image circle must be at least the sensor diagonal — 4.0 mm for 1/4", 5.7 mm for 1/3.2" and 6.0 mm for 1/3" — so any M12 lens rated to an 8 mm circle covers all three with margin.

An image circle is the diameter of usable projection. Inside it, brightness tapers toward the rim; outside it, the corners are not formed at all. Two facts follow. Moving a lens to a bigger sensor is safe only when the circle clears the new diagonal — glass made for a 1/4" die around a 1/3" sensor can leave black corners. The reverse move always works optically: a bigger-circle lens on a smaller die uses the centre of the projection, where roll-off and distortion are lowest.

The M12 range splits into two circle classes. An image circle up to 8 mm covers every format up to 1/2" (8.0 mm diagonal); the 8-11 mm class covers 1/1.8" (8.93 mm diagonal). LS-MBL-0113 is a 12 mm, F2.8, 5 MP lens for 1/1.8" sensors with an 8-11 mm image circle. On a 1/3" board the same optics clear the 6.0 mm diagonal with margin and read 22.6° instead of the 33.3° a 1/1.8" die sees. The class-by-class map, from these M12 circles up to machine vision lenses for 2/3" and larger sensors, is the subject of the lens sensor format guide; the M12 lens selection guide adds the IR-cut check that colour boards need.

A 12 mm pick from the 8-11 mm circle class:

How much does corner brightness drop on the bigger sensor?

Corner brightness falls roughly with the fourth power of the cosine of the half-field angle, so the same example lens reads about 74% of centre brightness at the 1/4" edge (21.8° half-angle) and about 54% at the 1/3" edge (31.0°).

The geometric estimate, worked out at f = 4 mm:

• 1/4" edge: θ = arctan(1.6 / 4.0) = 21.8°, cos 21.8° = 0.928, 0.928⁴ = 0.74 → about 74% of centre, −0.4 stop
• 1/3" edge: θ = arctan(2.4 / 4.0) = 31.0°, cos 31.0° = 0.857, 0.857⁴ = 0.54 → about 54% of centre, −0.9 stop

These are the cos⁴ numbers for a rectilinear design; mechanical shading near the image-circle rim adds on top of them. That is the whole reason a lens specified for the larger format stays flat where a small-format design slumps: its roll-off was corrected out to the bigger diagonal. The difference matters most where the measurement needs the corners — locating edges on uniform surfaces — and least in code reading, where a 30% corner dip rarely stops a decode. Measure real shading once with a flat white target at fixed exposure. For more total light through the same aperture, the lever is a lower F-number at the same focal length, worked through in the lower f-number guide.

Does barrel or pincushion distortion change when the sensor changes?

Distortion does not move with the sensor: barrel and pincushion are a fixed mapping from field angle to image height, so each radius of the lens's projection always bends the same way, and a smaller sensor simply reads the flatter inner part of that fixed curve.

Illustrative example: a data sheet prints 3% distortion at 4.0 mm image height and 1% at 2.0 mm. A 1/4" die, half-width 1.6 mm, reads under 1% at its edge; a 1/2" die, half-width 3.2 mm, reads somewhere between 1% and 3%. The lens never changed — the frame just reaches further out on the curve. This is why the same barrel that looks mild on a 1/3" board looks aggressive on 2/3", and why shrinking the crop is the cheapest route to tighter geometry.

Field curvature behaves the same way: it is specified per image height, so the smaller crop keeps the flatter centre. Where the residual must be corrected pixel-wise, profile the exact lens–camera pair as in the board-camera distortion profile guide.

What else changes when you swap sensors under the same lens?

Three more things change besides the framing: the lens's resolution class must match the new pixel pitch, the focus must be re-set and locked on the new board, and the IR-cut arrangement on colour sensors must be re-checked.

• Resolution class. A lens that feeds 5 MP cleanly on 1/1.8" is not automatically good for 12 MP on the same class; contrast at the new pixel pitch decides, and the Nyquist matching guide gives the arithmetic.
• Focus. The new board's flange height and the lens's thread travel decide where focus lands, so re-set focus on the camera you will actually ship, judge the corners, then lock — the M12 focusing procedure is the checklist.
• IR cut. On colour boards, confirm the IR-cut filter still sits at the right distance behind the mount; if the filter drops out of the optical path, near-IR leaks in and colours drift even though the geometry looks fine.

How do you keep the same field of view on a different sensor?

To hold the same object width when the sensor changes, scale the focal length by the width ratio: f_new = f_old × w_new / w_old, so the example lens on a 1/4" sensor becomes a 6 mm lens on 1/3" for the same 43.6°.

Worked check: f_new = 4 mm × 4.8 / 3.2 = 6.0 mm, and 2 · arctan(4.8 / 12) = 2 · arctan(0.400) = 43.6° — identical framing, from glass whose circle still clears the 6.0 mm diagonal. Going down in format shrinks the focal length by the same ratio: the same lens on 1/3" calls for roughly 2.7 mm on 1/4".

Decision: reuse the lens when the new framing still covers the part with margin; rescale the focal length by the width ratio when the framing is the spec. Before ordering, run the checks:

• image circle ≥ new diagonal — 6.0 mm for 1/3", 8.93 mm for 1/1.8"
• HFOV recomputed from the real active width, never the inch name
• resolution class at or above the new pixel count
• focus re-set on the shipped camera, corners inspected, lock tightened
• IR-cut filter confirmed in place for colour boards
• corner shading measured if the format grew

From wide to narrow, the usual M12 ladder:

All 3 share: Aperture F2.8. Values as printed in each product's data sheet; open a part for its drawing and full specifications.

Read the ladder against the HFOV table above: the 2.8 mm entry runs past 90° on its 1/2.5" format (2 · arctan(5.76 / 5.6) = 91.6°), the 12 mm entry holds 33.3° on 1/1.8", and the 25 mm entry narrows to 16.3°. To pair any of them with a camera and a working distance, start at the lens finder or the lens configurator; the rest of the lens library sits on the lenses topic page.