Interference filter wavelength shift: angle and temperature

A narrow band-pass interference filter passes its rated wavelength only for light that arrives close to straight-on, at the temperature it was coated for. Tilt the rays and the passband slides toward shorter wavelengths; warm the glass and it drifts toward longer ones. Behind a wide-angle lens the slide at the image corner can exceed the filter's entire half width, and the corner goes dark exactly where you needed signal. This explainer gives you both shift rules, the arithmetic that turns a field angle into nanometers of shift, and a committed decision for each half width. Treat filter width and focal length as one decision, and neither corner shading nor a dead laser line will surprise you at commissioning.

How the coating fixes a wavelength

An interference filter is a stack of thin dielectric layers on clear glass, each layer an optical quarter-wave at the design wavelength. At every boundary between materials of different refractive index, part of the beam reflects; the partial beams recombine, canceling outside the passband and reinforcing inside it. Where the passband sits is decided by the optical thickness the layers present to the ray. Two things change that thickness after the filter leaves the coating chamber: the angle at which light crosses the layers, and the temperature of the stack. If you are still choosing coating technology, settle colored glass versus interference filters first; absorption glass shows only a faint version of these shifts, interference coatings at full strength.

Incidence shifts the band short, on a cosine

Tilt the ray and the layers present a smaller effective spacing along it, so every wavelength the stack passes scales with the cosine of the angle inside the coating. Refraction bends an external incidence angle θ into a smaller internal angle set by the stack's effective refractive index: n_eff = 1.5 is the planning value for hard-coated band-pass designs, and stacks near 2.0 shift about half as much.

The small-angle form says doubling the angle roughly quadruples the shift: at 10° the loss is 5.7 nm, and doubling again to 20° costs 22.4 nm, a factor of 3.9 for twice the angle. Computed with the full formula for an 850 nm filter at n_eff = 1.5:

One limit applies regardless of width: past roughly 25° incidence the flat passband distorts, the s and p polarization components split apart, and peak transmission falls. Keep narrow filters under about 25° even when their width would allow more.

Temperature shifts the band long, linearly

Heat expands the layers and raises their refractive indices, so optical thickness grows and the central wavelength drifts longer in proportion to the temperature change. Hard-coated interference filters run 0.002-0.005 nm/°C, and some coating designs reach about 0.014 nm/°C in the visible. Plan at 0.015 nm/°C and every published value is covered with margin.

Put numbers on it: an enclosure climbing from 20°C at commissioning to 50°C in production moves the band by 30°C × 0.014 nm/°C = 0.42 nm, under 2% of the worked example's half width below. Angle delivers tens of nanometers, temperature tenths. It earns a budget line only at laser-line widths: a ±2 nm filter next to a heat source swinging 100°C drifts 1.4 nm, most of its passband. Wide filters ignore temperature; very narrow ones stay out of hot airflow.

Worked example: an 850 nm band-pass behind an 8 mm lens

The set-up is a near-IR LED line, a 2/3" camera with an 8.8 mm image width, an 8 mm lens, and a ±25 nm band-pass. For the line itself, our lens accessories range includes a band-pass filter for the near-IR line with an M30.5 x 0.5 thread:

Half width is a choice you make within the band-pass series; this example runs ±25 nm, a common LED-line width. The lens is from the catalog too. The LS-FFL-1582 is an 8 mm, 2/3" C-mount FA lens with a 57.6° field of view.

Corner angle. The printed 57.6° is the horizontal field on the rated sensor, so the corner chief ray travels at half of it: 28.8°. From first principles, tan u = 4.4 mm / 8 mm = 0.55, u = 28.8°. A filter in front of the lens sees corner rays at about this angle; exit-pupil position moves it a few degrees, and this first-order budget is what you need at ordering time.

Shift. sin 28.8° = 0.482, inside the stack 0.482 / 1.5 = 0.321, and the cosine factor is √(1 - 0.321²) = 0.947. The corner wavelength is 850 nm × 0.947 = 805 nm, a shift of -45 nm.

Verdict. The band spans 825-875 nm, and the corner now peaks at 805 nm, 20 nm beyond the edge: bright center, dark corners, classic band-pass shading. A high-index stack does not save it either, because at n_eff = 2.0 the corner sits at 825 nm, exactly on the band edge with zero margin.

Two committed fixes:

1. Widen to ±50 nm. The bare limit is 30.5°, so the 28.8° corner passes with 1.7° to spare and nothing behind it; exit-pupil geometry and stack index both eat into that. If the scene allows a 12 mm lens, ±50 nm holds a full reserve at 20.1°. With an LED peak inside the band, more skirt light is the normal trade, the same logic as blocking ambient light with a band-pass on a red LED.
2. Keep the narrow width and fit a 16 mm lens. The corner angle drops to 15.4°, the shift to -13 nm, leaving 12 nm of half width unused. The field narrows to 30.8°, so double the standoff to cover the same scene.

Temperature in this example adds +0.42 nm over a 30°C rise at the worst coefficient, noise next to -45 nm.

How much incidence your lens produces

Two angles feed the budget. The first is the corner field angle, u = arctan(half image width / focal length); any printed field of view halves into it. On a 2/3" sensor, 8 mm gives 28.8°, and lengthening the lens works down the scale: 16 mm gives 15.4°, 35 mm gives 7.2°. Bigger sensors steepen everything: a 1" sensor, 12.8 mm wide, reaches arctan(6.4 mm / 8 mm) = 38.7° on the same 8 mm lens. The lens focal length calculator converts sensor and field of view into focal length on this geometry, the same one behind the working distance and field of view formula.

The second is the aperture cone around every ray, set by the f-number: half-cone ≈ arctan(1 / (2N)). At f/2.8 that is 10.1°, at f/8 it is 3.6°. A filter close to the sensor sees this spread on top of the field angle, and an f/2.8 cone alone spans ±5.7 nm of shift at the line. Stop down to tighten the band, and buy the light back from the LED rather than from camera gain.

Matching half width to focal length

Set the allowed incidence where the shift reaches 80% of the half width, which keeps a reserve for exit-pupil geometry and stack-index spread:

Exact to better than half a degree for half widths up to ±50 nm at this wavelength; shifts scale with wavelength, so the same width at 525 nm tolerates about 27° instead of 21°. Decisions for a 2/3" sensor at 850 nm, using focal lengths the lens finder filters by:

• A wide scene on an 8 mm-class lens: ±50 nm is the minimum width, and even it passes at 28.8° without reserve. Prefer 12 mm with ±50 nm. The narrower LED-line width on an 8 mm lens shades the corners; that is arithmetic, not bad luck.
• 16-25 mm lenses: the LED-line width holds its reserve across 15.4°-10.0°.
• ±10 nm and narrower, laser-line and fluorescence work: a 25-50 mm lens from machine vision lenses, or an object-side telecentric lens whose chief rays arrive near 0°. Narrower bands also transmit less peak light, so plan exposure around it, and on a color sensor mind what a narrow band does to RGB bit depth.
• A ±2 nm filter near anything hot: budget the thermal drift above and move it out of the airflow.

Two C-mount lenses for a 2/3" sensor bracket the trade, an 8 mm wide and a 35 mm narrow:

The long one earns its place: the LS-FFL-0679 is a 35 mm, 2/3", 6 MP C-mount machine vision lens. At 7.2° corner incidence it clears every width in the table down to ±5 nm.

Checks before you order

1. Write down the central wavelength, half width and thread of the candidate filter; adapter rings bridge any thread gap to the lens.
2. Compute the corner incidence: half the printed field of view, or arctan(half sensor width / focal length).
3. Read the shift at that angle from the table or the formula, and require it to stay inside 80% of the half width.
4. If the filter sits near the sensor, add the aperture cone, arctan(1 / (2N)), to the field angle.
5. For ±5 nm and narrower, fix the telecentric or long lens now, and keep the filter out of hot airflow.
6. Center the passband on the LED's hot-state peak, not on a round number, and leave room for the LED's own drift with current and temperature.
7. Confirm focus at the working wavelength; a lens focused in white light sits off its best focus at 850 nm.

When the usual rules mislead

• Plain cos θ over-predicts, because it skips refraction into the stack: at 28.8° it claims -105 nm where the stack delivers -45 nm. Budgeting with cos θ buys more width than you need.
• High-index stacks soften the rule: n_eff = 2.0 halves the 28.8° shift to -25 nm. When a data sheet prints angle-of-incidence curves, those beat any estimate.
• Cut-off and daylight filters do not need this budget. A 400 nm edge moves about 2.6% at 20°, near 10 nm, against a passband hundreds of nanometers wide; only a cut-off sitting hard against your LED band deserves the arithmetic.
• Blocking travels with the stack: a tilted or warmed filter loses signal at its edges long before it loses deep UV or far IR rejection.
• At 850 nm, focus shifts across wavelengths; use an IR-corrected design or refocus at the line. The narrow band pays you back by cutting chromatic aberration and resolution loss, but only while it stays on the line.

Run the system configurator with the focal length the table gives you, and pick the filter at the accessories step. If the line, the width or the lens is already fixed and the budget will not close, send us the numbers and we will size the pairing.