Jezdan Gomez
Jezdan Gomez

FWHM is a collimated number

The bandwidth on a filter datasheet is measured in a collimated beam. In a converging beam each ray meets the coating at its own angle and sees its own center wavelength, so the passband behind the filter is wider, lower and moved to the blue.

The number

A filter datasheet gives the full width at half maximum (FWHM) of the passband, for example 3.0 nm at 532 nm. It is measured in a collimated beam a few millimeters wide, usually at normal incidence. The figure describes the passband one ray sees at one angle of incidence.

Lens-coupled instruments rarely give the filter that beam. An f/2 cone has a half angle of 14.04°, and a filter tilted 10° to keep its reflection out of the source receives rays from 0° to 24.04°.

What it means

The center wavelength of an interference filter moves to shorter wavelengths with the angle of incidence θ\theta. For light from air, Lissberger and Wilcock (1959) give the law with one effective index neffn_\mathrm{eff} for the coating:

λ(θ)=λ01−(sin⁡θneff)2\displaystyle \lambda(\theta) = \lambda_0 \sqrt{1 - \left(\frac{\sin\theta}{n_\mathrm{eff}}\right)^2}

A beam of working f-number NN has the cone half angle α=arctan⁡(1/2N)\alpha = \arctan(1/2N). With the axis tilted by θ\theta, the rays arrive between max⁡(θ−α,0)\max(\theta - \alpha, 0) and θ+α\theta + \alpha, and their centers spread over

Δλspread=λ(max⁡(θ−α, 0))−λ(θ+α)\displaystyle \Delta\lambda_\mathrm{spread} = \lambda\big(\max(\theta - \alpha,\, 0)\big) - \lambda(\theta + \alpha)

For a uniformly filled circular pupil the small-angle estimate of the mean center adds a cone term to the tilt shift, scaled by (cos⁡2θ+cos⁡2θ)/2(\cos 2\theta + \cos^2\theta)/2 for a tilted axis:

λˉ−λ0≈λ(θ)−λ0−λ0 α24 neff2⋅cos⁡2θ+cos⁡2θ2\displaystyle \bar{\lambda} - \lambda_0 \approx \lambda(\theta) - \lambda_0 - \frac{\lambda_0\,\alpha^2}{4\,n_\mathrm{eff}^2} \cdot \frac{\cos 2\theta + \cos^2\theta}{2}

For 532 nm, effective index 2.0, f/2 and 10° of tilt the ray centers run from 520.850 to 532.000 nm: a spread of 11.15 nm, 3.7 times the 3.0 nm FWHM. The estimated mean shift is −3.91 nm, within 0.02 nm of the exact pupil average.

Compute it

FWHM in a cone calculator

Range 300 to 2000 nm.

Range 0.3 to 100 nm.

Range 1.3 to 2.5.

Range 0.5 to 64.

Range 0 to 45 °.

Spread of ray centers
11.15nm
Spread against FWHM
3.72×
Mean shift, estimate
−3.91nm

The spread is wider than the FWHM: specify the filter in collimated space or widen the band past the spread.

The collimated passband is drawn at the axis tilt. The band below it spans the centers seen by the steepest and the shallowest ray; the exact smeared passband is the collimated one convolved with the pupil’s distribution of centers.

α = arctan(1 / 2N) = 14.036°

λ(θ + α) = 520.850 nm

λ(max(θ − α, 0)) = 532.000 nm

Δλspread = 532.000 − 520.850 = 11.150 nm

Δλmean ≈ (−2.009) + (−1.905) = −3.914 nm

Where it misleads

The spread is not the width of the passband behind the filter. Each ray carries its own FWHM-wide passband, and the receiver adds them, so the result is the collimated passband convolved with the distribution of ray centers. The angle law is flat near normal incidence, so most of the pupil sits near the red end of the spread and the smeared curve is lopsided.

The exact smeared width and peak need the pupil integral. For the f/2, 10° example it gives a half-maximum width of 7.54 nm and a peak transmission of 0.39 against 0.90 collimated. The filter passband tool computes that integral, and the article on a tilted f/2 cone works through the example in full.

One effective index for both polarizations predicts the centers to a few hundredths of a nanometer below about 25° of incidence. The s and p widths separate earlier, and the mean center alone does not say where the half-maximum midpoint of a skewed curve sits.

What to specify

  • The working f-number and the tilt of the beam axis at the filter, and the pupil fill when it is not uniform.
  • Center wavelength and FWHM in the working geometry, or collimated values with the effective index and the polarization it was fitted for.
  • Measured transmission curves at normal incidence and at the working angle.
  • A bandwidth wider than the ray-center spread when the filter must sit in the cone, or a collimated section of the beam when it need not.

References

  • Lissberger, P. H. and Wilcock, W. L. (1959). Properties of all-dielectric interference filters. II. Filters in parallel beams of light incident obliquely and in convergent beams. J. Opt. Soc. Am. 49, 126–130. doi:10.1364/JOSA.49.000126 (opens in a new tab)
  • Pidgeon, C. R. and Smith, S. D. (1964). Resolving power of multilayer filters in nonparallel light. J. Opt. Soc. Am. 54, 1459–1466. doi:10.1364/JOSA.54.001459 (opens in a new tab)
  • Macleod, H. A. (2017). Thin-Film Optical Filters, 5th ed. CRC Press, Boca Raton.

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