Bandpass filters, chromatic aberration and image resolution

A lens bends blue light more strongly than red, so a scene lit with broadband light comes to focus at a different depth for every wavelength. On a monochrome sensor with fine pixels, that spread becomes blur you paid for: edges soften and small features merge. The fix is one of the least expensive parts in the station — a bandpass filter that passes a single narrow band, around 470 nm for instance, so the lens focuses one wavelength instead of many. This explainer gives you the physics, the formula that turns chromatic focal shift into pixels of blur, a worked 35 mm example with the arithmetic shown, and the cases where the filter is the wrong move. You will leave able to say whether chromatic aberration is eating your resolution, and to pick the band that fixes it.

The decision, up front

Fit a bandpass filter when chromatic blur at your working aperture is wider than one pixel; leave it off below about half a pixel. The whole test is one line: px = Δf / (2 × f-number × pixel pitch), where Δf is the lens's focal shift across your light's spectrum. Run it for a 35 mm achromatic C-mount lens at f/2.8 on 3.45 µm pixels: broadband white light leaves about 1.2 px of chromatic blur, and a 50 nm band at the example's blue centre takes that down to 0.3 px — that station gets the filter.

The same lens at f/8 carries 0.4 px of chromatic blur while diffraction alone is 10.7 µm, so colour is no longer the limit and the filter buys nothing visible. The rule holds for monochrome cameras; a color camera behind a bandpass loses two of its three channels before it gains anything.

What chromatic aberration does to an image

The refractive index of optical glass falls as wavelength rises, so blue focuses closer to the lens than red. Focus mid-spectrum and the extreme colours sit either side of the sensor plane: that is longitudinal chromatic aberration, and on a monochrome sensor it reads as a soft halo around every high-contrast edge.

Off axis there is a second term. Each wavelength images at a slightly different height, so magnification varies with colour — lateral chromatic aberration. On a color sensor it shows as fringes that grow toward the field edges; on a mono sensor those fringes are plain smear. One wavelength means one focal point and one magnification, which is why a bandpass removes both terms at once.

Designers correct what they can. An achromatic doublet brings two wavelengths — classically the Fraunhofer F and C lines at 486 nm and 656 nm — to a common focus; an apochromat brings three. What remains between the corrected wavelengths is the secondary spectrum. A design also balances spherical aberration at one chosen wavelength, usually green for visible work, so every lens has a colour it was built around.

Silicon sensors answer from roughly 380 nm to beyond 900 nm, while a visible-band lens is corrected only in the visible. When the lighting or the room adds ultraviolet or near-infrared, those wavelengths land as an out-of-focus overlay on top of a sharp image. That is why the gain from a bandpass is largest in broad-spectrum set-ups, where UV and IR ride along with the visible light.

The numbers that drive it

Three rules cover the physics: a thin single element shifts focus by its focal length divided by the Abbe number of its glass (about 64 for a common crown); an achromatic doublet leaves a secondary spectrum near f/2000; and a defocus δ becomes a blur circle of δ/f-number.

Scaled to a 35 mm focal length: a single crown element shifts focus 35/64 ≈ 0.55 mm from 486 nm to 656 nm; an achromat holds the same span to 35/2000 ≈ 17.5 µm. At close working distances, multiply a focal shift by (1 + m)² to get image-plane movement — 1.28 at the magnification used in the worked example below.

Two more terms set the floor. Stopping down halves the chromatic blur with each full stop, but diffraction grows as you stop: the Airy diameter is 2.44 × λ × f-number, about 3.2 µm at f/2.8 and 6.4 µm at f/5.6 for 470 nm light. Your blur budget is a three-way race between chromatic shift, diffraction and pixel pitch — optical aberrations in machine vision lenses: how stopping down helps walks that trade step by step.

The d = δ/f-number relation is also what depth of field is made of. Run your own lens numbers through it:

Worked example: 35 mm lens at 300 mm, white LED versus narrow band

Inputs, each with its source:

• Lens: the LS-FFL-0679, a 35 mm, 2/3", C-mount machine vision lens rated 6 MP. If the focal length itself is still open, start from how to choose a machine vision lens focal length.
• Working distance: 300 mm, a fixed station standoff.
• Sensor: 2/3" monochrome with 3.45 µm pixels — about 2550 × 1910 pixels on the format's nominal 8.8 × 6.6 mm area, the 5 MP class.
• Aperture: f/2.8.
• Light A: white LED spanning roughly 450-650 nm. Light B: bandpass centred at 470 nm, 50 nm wide.

Arithmetic, step by step:

1. Magnification. m ≈ f/(WD − f) = 35/(300 − 35) = 0.132. Field of view = 8.8 mm / 0.132 ≈ 67 mm wide and 6.6 mm / 0.132 ≈ 50 mm high. One pixel covers 3.45 µm / 0.132 ≈ 26 µm at the part — the same relation shown in calculate object space pixel size from magnification.
2. Image-plane gain. At m = 0.132, the sensor plane moves (1 + m)² = 1.28 times any focal-length shift.
3. Broadband. Δf ≈ 35/2000 = 17.5 µm; image shift = 17.5 × 1.28 = 22.4 µm; blur d = 22.4/(2 × 2.8) = 4.0 µm; that is 4.0/3.45 = 1.2 pixels.
4. Bandpass. 50 nm is a quarter of the 200 nm white span, so Δf ≈ 4.4 µm; d = 4.4 × 1.28 / 5.6 = 1.0 µm; that is 0.3 pixels. Diffraction at f/2.8 is the 3.2 µm figure from above, now the largest term left.
5. Decision. A 4.0 µm broadband blur exceeds the 3.45 µm pixel pitch, so chromatic shift is the biggest single error in this station. Fit the bandpass and refocus at the band's centre wavelength; after that, sharpness belongs to the lens's monochromatic correction and to diffraction, not to colour.

The lens used for these numbers:

Filter alone, or a matched narrowband LED?

A bandpass over white light fixes the wavelength problem and throws away most of the photons. The production answer pairs the filter with an LED whose line sits inside the passband.

An IR-cut filter alone removes the infrared a visible lens cannot focus, but the visible-span shift stays: 4.0 µm either way in this example. What an IR-cut does and does not fix is covered in color camera near-infrared imaging: IR-cut filter options.

The matched pair beats either part alone. A visible LED line runs 15-30 nm wide, well inside a 50 nm passband, so almost nothing you paid for is blocked — and daylight or hall lighting falls outside the band, so the camera sees your scene and little else. Why a single wavelength wins in general is laid out in monochromatic lighting in machine vision: why one wavelength wins; choosing that wavelength for contrast is its own question, answered in which LED colour gives the best contrast in machine vision.

Choosing the band: centre wavelength and bandwidth

Centre the band on the lens unless contrast says otherwise. A visible-band lens is balanced near one design wavelength, usually green, where both its spherical correction and its achromatic pair sit. If the part only differs from its background at one specific wavelength, take the contrast: a fraction of a pixel of extra sharpness never rescued a ten-grey-level difference.

Match the bandwidth to the source. For an LED-driven station, a passband two to three times the LED's 15-30 nm line holds the line inside the filter even when the centre drifts. Interference filters shift shorter as rays tilt — about 1.5% in wavelength at a 15° ray angle, near 7 nm here — so wide-angle optics and short working distances want the extra width. And every halving of bandwidth halves the chromatic shift along with the signal, so exposure or gain has to double to hold grey level.

Below 400 nm, treat the source as a UV hazard: enclose the beam path and wear UV-rated eye protection. The bandpass passes its own band by design, so the filter is never the eye shield — the enclosure is.

Checks before you order

Monochrome sensor. A bandpass on a color camera starves two of three channels before it sharpens anything; the case for mono is made in monochrome camera with optical filter: why contrast beats color.

Thread and holder. Filters screw into the lens front thread or sit in a holder on the lens accessories shelf — adapters and holders live there too. A near-infrared example in an M30.5 x 0.5 thread:

Light in band. Confirm the source actually emits inside the passband before ordering glass; for methods, see measuring light wavelength without optical filters.

Exposure budget. Recompute exposure and gain for in-band power. If you also stack neutral density to tame a bright LED, neutral density filter optical density: sizing attenuation sizes that loss.

Placement on telecentrics. On a telecentric lens, where the filter sits changes more than transmission — telecentric lens filters: where to place them and what they change gives the rules.

Prove the gain. Shoot a resolution target before and after at the production aperture. A dot grid works well — for example the LS-LXR-0029, 0.5 mm dots on a 76 x 76 mm board — and reading contrast off the shots is covered in lens contrast and resolution: evaluating fine detail.

When the usual answer is wrong

Color imaging. Color work needs the full spectrum; a bandpass hands you a one-channel image. Choose filters for contrast instead, per machine vision optical filters: choosing contrast over color.

Small apertures. At f/8 the example lens carries 0.4 px of chromatic blur against 10.7 µm of diffraction. The filter cannot reveal what diffraction hides; open the aperture first, then filter.

Uncorrected lenses. A single crown element still shifts focus about 0.23 mm across a 50 nm band near this blue centre — about 15 px of blur at f/2.8 even after filtering. A bandpass removes a corrected lens's residual; it does not correct a lens.

Apochromatic glass. A lens that already holds three wavelengths at one focus has a residual well under a quarter pixel across a 50 nm band. Spend the money on light, not glass.

Starved stations. If exposure already sits at the motion limit, losing most of a white LED's output costs more than one pixel of blur. Switch to a matched LED rather than filtering broadband light.

A short checklist

• Monochrome camera behind the filter — color sensors lose two channels.
• Band centred on the design wavelength unless contrast lives elsewhere.
• Passband two to three times the LED line width, wider for tilted rays.
• Filter thread matched, rays near normal incidence.
• Exposure and gain recomputed for in-band power.
• Lens refocused at the working wavelength after fitting.
• Before-and-after target shots at the production aperture.

To pick the part, work the accessories finder or let the configurator at the extras step place the filter in a full station — band-pass, IR-cut and UV-cut filters sit side by side there.

If the numbers above leave a borderline case, bring them to the contact page and we will run the arithmetic with you.