Telecentric lens field of view calculation from magnification

A telecentric lens datasheet hands you a magnification, a sensor format and a working distance — and no field of view. That is not an omission. The field of view is set by the camera you mount behind the lens, because the sensor's active area divided by the fixed magnification is the window the optic actually sees, and the same 0.3x lens gives a different window on every sensor format. Below is the one-line formula, a table of the common sensor sizes, worked arithmetic in both directions — sensor to field, part size to magnification — and the image-circle, pixel-size and depth-of-field checks that turn a number into an orderable lens.

Why does a telecentric lens list magnification instead of focal length?

Because an object-side telecentric lens images at one fixed magnification by construction, its datasheet leads with that number — 0.3x, 0.5x, 1x — and the field of view follows from the sensor you choose, not from a focal length you dial in.

An object-side telecentric lens accepts chief rays that run parallel to the optical axis. A part 5 mm nearer or farther therefore images at the same scale, inside the stated depth of field, so the magnification is a design constant printed on the label. On an entocentric (standard) lens the magnification is an accident of focal length and working distance, which is why the usual focal-length sums do not apply here. That scale stability is the whole reason gauging systems pay for telecentric optics; entocentric vs telecentric lenses: when to use each walks the decision, and when you need a telecentric lens for gauging covers whether you need one at all.

What is the field of view formula for a telecentric lens?

Field of view equals sensor active area divided by magnification — FOV = S / m — so a 6.6 mm sensor side at 0.3x gives 6.6 mm / 0.3 = 22.0 mm of object space.

Run the division once per side, because sensors are rectangular and the part may rotate. The same equation, applied to pixel pitch instead of sensor width, gives the object-space pixel size your measurements live on; see calculate object space pixel size from magnification. Nothing in the sum mentions working distance or focal length — that is the telecentric contract.

How large is a 2/3 inch sensor, really?

A 2/3 in sensor has a nominal active area of 8.8 × 6.6 mm with an 11.0 mm diagonal — the fraction is a vacuum-tube-era convention, so never multiply or divide the 2/3 itself.

The inch names come from camera tubes, not silicon, which is why a 1 in sensor measures 12.8 × 9.6 mm; the image sensor format reference tabulates the exact areas. The nominal values you will use most:

Two habits keep this table honest. First, prefer the exact active-area figure from the camera datasheet: sensors sold under one format name differ by a few percent in width and height, and that difference lands directly in the field of view. Second, remember which side you divided — the smaller number limits round parts. Lens sensor format: 1/1.8 inch, 2/3 inch and 1 inch lenses explained goes deeper on matching formats.

How do you calculate the field of view of a 0.3x telecentric lens?

On a nominal 2/3 in camera a 0.3x telecentric lens sees 6.6 / 0.3 = 22.0 mm across the short side and 8.8 / 0.3 = 29.3 mm across the long side — a 22 × 29 mm window.

Work it in four lines:

1. Write down the camera's active area per side — 8.8 × 6.6 mm nominal here, or the exact figures from the camera datasheet.
2. Divide each side by the magnification on the lens label: 6.6 / 0.3 = 22.0 mm and 8.8 / 0.3 = 29.3 mm.
3. Turn the camera so the part's long axis meets the 29.3 mm side; a round part is limited by the 22.0 mm side.
4. Repeat the division with pixel pitch to get measurement granularity: 3.45 µm / 0.3 = 11.5 µm per pixel in object space.

A sensor of that format with 3.45 µm pixels puts about 1913 pixels across that 22.0 mm side (6.6 mm / 3.45 µm), which is why a 0.3x gauge on this format reads edges to a few micrometres rather than to a pixel. At 1x the arithmetic collapses: the field of view is the sensor itself. The LS-TCL-1972 is a 1x object-side telecentric lens with a 440 mm working distance for the matching sensor format:

On that camera the window is then simply the 8.8 × 6.6 mm active area, one sensor-width of object space, and every millimetre on the sensor is a millimetre on the part.

How do you pick the magnification when the part size is known?

Invert the formula, m = S / FOV: for a 20 mm part, using the smaller 6.6 mm sensor side and aiming at a 26.4 mm field gives 6.6 / 26.4 = 0.25, so you order a 0.25x lens.

Four steps from part to part number:

1. Size the field: largest part dimension plus 10-20% for placement scatter and edge lighting. A 20 mm gear with a 20% allowance targets 20 × 1.2 = 24 mm.
2. Divide using the smaller sensor side, 6.6 mm, so coverage holds whichever way the part lies: 6.6 / 24 = 0.275.
3. Step down to the nearest available magnification: 0.25x gives 6.6 / 0.25 = 26.4 mm by 8.8 / 0.25 = 35.2 mm, and the 20 mm part fills 20 / 26.4 = 76% of the narrow field.
4. Test the next step up before committing: at 0.3x the narrow field is 22.0 mm, the part fills 91%, and only 1 mm of margin remains per edge. With fixtures indexing ±1 mm, order 0.25x.

The telecentric lenses category filters by magnification band (0.1-0.3x, 0.3-0.6x, 0.6-1x, 1-2x), sensor format and working distance, so the combination you computed is a two-click shortlist, and the lens finder adds mount and image circle. A spread of magnifications across sensor formats makes the pattern concrete:

All 3 share: Mount C-mount. Values as printed in each product's data sheet; open a part for its drawing and full specifications.

Read each magnification-and-format pair off the table, then divide. The 0.5x entry on the 1 in format gives 12.8 / 0.5 = 25.6 mm by 9.6 / 0.5 = 19.2 mm — a mid-size window. The LS-TCL-2194 is that 0.5x lens, with a coaxial light port and a 110 mm working distance for flat, reflective faces. The 0.166x entry gives 12.8 / 0.166 = 77.1 mm by 9.6 / 0.166 = 57.8 mm on the same 1 in format: one third the magnification, three times the window.

When does a standard lens give the same field of view?

A 60 mm focal length at 260 mm working distance reaches the same 0.3x magnification — f = WD × m / (1 + m) = 260 × 0.3 / 1.3 = 60 mm — and covers the same 22 × 29 mm window, but only at one exact distance.

The standard-lens route computes the field from geometry instead of a fixed ratio:

Check it: 60 / (260 − 60) = 0.3, so a 20 mm part fills the same 6.0 mm of sensor that the telecentric fills. Now push the part 5 mm nearer the camera: 60 / (255 − 60) = 0.3077, and the same 20 mm part images at 20 × 0.3077 / 0.3 = 20.5 mm. Half a millimetre of error from 5 mm of part height — pure perspective, no defect. The telecentric lens gives up nothing in size for that 5 mm; it spends depth of field instead. The decision is clean: flat parts at fixed height for presence and count → standard lens, chosen with how to choose a machine vision lens focal length; anything gauged, or parts with height → telecentric. Run the same sum before you buy:

What should you check before ordering?

Four checks turn the arithmetic into an order: the lens image circle must cover the sensor diagonal (11.0 mm for the format above), the working distance must fit the station, object-space pixel size must resolve the tolerance, and the depth of field must swallow the part's height.

• Image circle. The rated image circle band — up to 8, 8-11, 11-16, 16-22 mm in the catalog filters — must equal or exceed the sensor diagonal: 11.0 mm for that format, 16.0 mm for 1 in. A smaller-format lens on a larger sensor vignettes the corners; the reverse combination works and simply leaves part of the circle unused.
• Working distance. A telecentric working distance is a fixed design value, not a focus range. Check fixture height, barrel length and camera clearance against it before the purchase order, not after.
• Object-space pixel size. Pixel pitch divided by magnification: 3.45 µm at 0.25x is 13.8 µm per pixel. Sub-pixel edge finding to about 0.1 pixel puts edge noise near 1.4 µm, comfortably inside a ±50 µm tolerance on a 26.4 mm field.
• Depth of field. Part height and placement scatter consume it; telecentric depth of field: validate measurement accuracy shows the gauge-block test that proves the lens holds its scale across your whole part.
• Calibration. Calibrate pixels to millimetres with a certified target after mounting, and re-check when anything in the optical path changes; micron-level image measurement accuracy: why it drifts and the fix lists the drift sources.

Why is the measured field of view smaller than the calculation?

Almost always because the true active area is smaller than the format table, a region of interest crops the sensor, or the magnification sits at the bottom of its tolerance band — each of these shrinks the window, never grows it.

Work the sensitivity on a 29.3 mm field. A sensor 1% narrower than nominal cuts 0.29 mm. A centered 1600 × 1200 region of interest on a 2448 × 2048 sensor keeps 1600 / 2448 = 65% of the width, leaving 29.3 × 0.6536 = 19.2 mm. And each 1% of magnification error moves the window 0.29 mm, which is why you calibrate with a target after mounting rather than trusting the label; measure object size from a camera image covers that calibration.

One more shrinkage path: refraction. If the optical path passes through a flat window into water, the object-side field narrows by roughly the refractive index of water, about 1.33 — a 22.0 mm field becomes 22.0 / 1.33 = 16.5 mm. The underwater field of view and lens choice guide works that case in full.

Once magnification, format and working distance agree, the configurator walks camera, lens and light as one system: