Filter passband against tilt and f-number
Drag the angle of incidence and the f-number. A narrow bandpass filter’s passband moves to shorter wavelengths with tilt and smears in a converging beam; the curves and the numbers recompute from the effective-index model on every change.
Beam and filter
Transmission against wavelength
Share of pupil area per 0.25 nm of ray centertallest bar 6.1 %
Key numbers
- Center at 10°, collimated
- shift
- Mean ray center
- shift
- Ray-center spread
- steepest ray half angle
- Smeared FWHM
- nominal 3.00 nm
- Smeared peak
- collimated 0.90
- Half-max midpoint
- shift
- T at 532.0 nm, cone
- collimated
- Eq. (2) estimate of the mean shift
- cone term tilt factor
| Center at 10°, collimated | 529.991 nm; shift −2.009 nm |
|---|---|
| Mean ray center | 528.067 nm; shift −3.933 nm |
| Ray-center spread | 11.15 nm; steepest ray 520.850 nm; half angle 14.04° |
| Smeared FWHM | 7.54 nm; nominal 3.00 nm |
| Smeared peak | 0.39; collimated 0.90 |
| Half-max midpoint | 528.697 nm; shift −3.303 nm |
| T at 532.0 nm, cone | 0.25; collimated 0.13 |
| Eq. (2) estimate of the mean shift | −3.914 nm; cone term −1.995 nm; tilt factor 0.955 |
Settings
- Center wavelength: 532.0 nm
- FWHM: 3.00 nm
- Peak transmission: 0.90
- Effective index: 2.00 (incident medium air, n₀ = 1.00)
- Angle of incidence: 10°
- f-number: f/2
- Line to monitor: 532.0 nm
jezdangomez.com/tools/filter-passband/
Presets
Everything runs in your browser; no setting or result is sent anywhere.
Model, limits and references
Model
- Center wavelength from the effective-index relation λ(θ) = λ₀·√(1 − (n₀·sin θ / n_eff)²), one index for unpolarized light. Incident medium air, n₀ = 1.00.
- Converging beam as a uniformly filled circular pupil with half angle α = arctan(1/2N). Each ray is weighted by its share of pupil area; the filter normal is tilted θ₀ from the beam axis.
- Passband as a three-cavity flat-top model, T = T_peak / [1 + ((λ − λ_c)/(FWHM/2))⁶], with the FWHM held constant at every angle.
- Pupil-averaged transmission as the passband convolved with the distribution of ray centers, on a 0.01 nm grid. Smeared FWHM between the half-of-peak crossings, center as their midpoint.
- Hand estimate of the cone term, Δλ_cone ≈ −λ₀·(n₀·α)²/(4·n_eff²) with α in radians, scaled by (cos 2θ₀ + cos²θ₀)/2 for a tilted axis.
Limits
- Above about 25° the s and p passbands separate and their widths change; a single effective index then predicts the center only approximately.
- A Gaussian beam fills the pupil unevenly and smears the passband less than a uniformly filled pupil.
- A real coating has its own passband shape, skirts and blocking. Confirm the final choice against measured curves at the working angle and f-number.
Related pages
- How far does your filter’s passband move at 10 degrees in an f/2 cone?, the NB-532-3-B worked example behind the defaults
- Angle shift calculator, tilt shift, edge rays and the cone term as a form
- Datasheet revision, Rev A to Rev B, a filter datasheet rewritten with the test conditions stated, compared line by line
References
- P. H. Lissberger and W. L. Wilcock, “Properties of all-dielectric interference filters. II. Filters in parallel beams of light incident obliquely and in convergent beams,” J. Opt. Soc. Am. 49, 126 (1959), doi:10.1364/JOSA.49.000126 (opens in a new tab).
- H. A. Macleod, Thin-Film Optical Filters, 5th ed., CRC Press, 2017.
- C. R. Pidgeon and S. D. Smith, “Resolving power of multilayer filters in nonparallel light,” J. Opt. Soc. Am. 54, 1459 (1964), doi:10.1364/JOSA.54.001459 (opens in a new tab).
NB-532-3-B is a fictional filter.
Start with a free review of your technical content
Get a one-page comparison of your technical content with two competitors, three recommended documents or tools, and a price for the first project.