Phase Correction Methods
Mertz and Forman phase correction run on a simulated single-sided FTIR interferogram with ZPD offset, phase offset, dispersion and noise, compared with the magnitude spectrum.
Spectrum Error, Mertz
0.449 %
RMS vs truth where B ≥ 5 % of peak
Error Without Correction
2.22 %
real part of the ramp-weighted transform
Phase Estimate Error
0.0273 rad
RMS; phase resolution 61.7 cm⁻¹
Resolution 1/L
3.86 cm⁻¹
ZPD found -1 samples from 0
| Method | RMS error | Data used |
|---|---|---|
| No correction | 2.22 % | −M … N |
| Mertz | 0.449 % | −M … N |
| Forman | 0.41 % | −M … N (resolution of N − M) |
| Magnitude | 0.128 % | double-sided −N … N |
Spectrum
- True B(σ)
- No correction
- Mertz
Phase in the frame of the found ZPD
- True phase
- Estimate (±M samples)
Model
I(x) = 2∫ B(σ) cos(2πσx + φ(σ)) dσ, φ(σ) = 2πσεΔx + φ₀ + φ₂((σ − 2000)/2000)²
Mertz: B = 2Δx·Re[C(σ)e^(−iφ̂)], Forman: B = cos-FT[I ⊛ IFT(e^(−iφ̂))], Magnitude: B = Δx|C(σ)|
A broad source with four absorption bands is turned into an interferogram sampled every HeNe wavelength (Nyquist 7901 cm⁻¹), with the ZPD between samples, a constant phase offset and beamsplitter-like dispersion, plus white noise (fixed seed). The record is single-sided: M samples before the largest sample and N after it. Both corrections take the phase φ̂ from the short double-sided part (triangular apodization), so φ̂ has the low resolution 1/(2MΔx) and errs near sharp bands. Mertz weights the record with a ramp over ±M so each pair of points counts once, transforms it and rotates each bin by −φ̂. Forman convolves the record with the transform of e^(−iφ̂), which makes it symmetric, and cosine-transforms the symmetric half. The magnitude spectrum needs no phase but a double-sided scan, and it turns noise into a positive offset where the signal is weak.
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
The algorithms run on a synthetic interferogram of a known spectrum, so the error against the truth is computed, not modelled; the instrument is ideal apart from the phase error and white noise.
- Griffiths P. R., de Haseth J. A. (2007). Fourier Transform Infrared Spectrometry, 2nd ed., chs. 2 and 4. Wiley.
- Mertz L. (1967). Auxiliary computation for Fourier spectrometry. Infrared Phys. 7, 17–23. doi:10.1016/0020-0891(67)90026-7
- Forman M. L., Steel W. H., Vanasse G. A. (1966). Correction of asymmetric interferograms obtained in Fourier spectroscopy. J. Opt. Soc. Am. 56, 59–63. doi:10.1364/JOSA.56.000059