FTIR Resolution Calculator
Spectral resolution from maximum optical path difference (OPD) and apodization function.
Nominal Resolution 1/L
1 cm⁻¹
boxcar ILS FWHM 0.6034 cm⁻¹
ILS FWHM (apodized)
0.6034 cm⁻¹
1 × boxcar
Points, Single-Sided Scan
8,000
step Δx = 1/(2σ_max) = 1.25 µm
Scan Time L/(2v)
0.5 s
modulation at σ_max: 8 kHz
Apodization Functions
| Window | ILS FWHM (× 1/L) | × boxcar | FWHM at this L (cm⁻¹) | Largest sidelobe |
|---|---|---|---|---|
| Boxcar (none) | 0.6034 | 1.000 | 0.6034 | -22 % (-13 dB) |
| Triangular | 0.8859 | 1.468 | 0.8859 | 4.7 % (-27 dB) |
| Happ–Genzel | 0.9076 | 1.504 | 0.9076 | 0.73 % (-43 dB) |
| Norton–Beer weak | 0.7240 | 1.200 | 0.724 | -5.8 % (-25 dB) |
| Norton–Beer medium | 0.8447 | 1.400 | 0.8447 | -1.4 % (-37 dB) |
| Norton–Beer strong | 0.9654 | 1.600 | 0.9654 | 0.37 % (-49 dB) |
| Blackman–Harris 3-term | 1.1370 | 1.884 | 1.137 | -0.029 % (-71 dB) |
| Blackman–Harris 4-term | 1.3332 | 2.210 | 1.333 | 0.0025 % (-92 dB) |
Apodization A(x/L)
- Boxcar
Instrument line shape
- Boxcar (none)
Model
B(σ) = ∫ I(x)·A(x/L)·cos(2πσx) dx over −L … L, ILS(s) ∝ ∫₀¹ A(u) cos(2πsu) du, s = (σ − σ₀)·L
δσ = 1/L, N = 2Lσ_max, t = L/(2v), f = 2vσ
Cutting the interferogram off at the maximum optical path difference L convolves every line with the instrument line shape. Without apodization that is a sinc of FWHM 0.603/L with −22 % sidelobes; an apodization function tapers the interferogram to suppress the sidelobes at the cost of a wider line. The widths and sidelobes in the table are computed from each window, not quoted. Norton–Beer weak, medium and strong were designed for 1.2, 1.4 and 1.6 times the boxcar width.
Sampling at Δx = 1/(2σ_max) puts the Nyquist wavenumber at σ_max; lines above it alias. The optical path changes at twice the mirror speed, so a single-sided scan to L takes L/(2v) and modulates wavenumber σ at 2vσ. Beam divergence, which adds its own line broadening and limits the usable aperture, is not included.
Model and references
Exact. The equations are exact within the stated assumptions, or are a standard's tabulated values. The results are as good as the inputs. About the tiers
Ideal interferometer: the instrument line shape is the cosine transform of the apodization over ±L, computed numerically; beam divergence and phase errors are not included.
- Griffiths P. R., de Haseth J. A. (2007). Fourier Transform Infrared Spectrometry, 2nd ed., chs. 2 and 4. Wiley.
- Norton R. H., Beer R. (1976). New apodizing functions for Fourier spectrometry. J. Opt. Soc. Am. 66, 259–264; errata 67, 419 (1977). doi:10.1364/JOSA.66.000259
- Naylor D. A., Tahic M. K. (2007). Apodizing functions for Fourier transform spectroscopy. J. Opt. Soc. Am. A 24, 3644–3648. doi:10.1364/JOSAA.24.003644
- Harris F. J. (1978). On the use of windows for harmonic analysis with the discrete Fourier transform. Proc. IEEE 66, 51–83. doi:10.1109/PROC.1978.10837