Michelson Interferometer
Interferogram → spectrum via Fourier transform. Core of FTIR spectroscopy.
Resolution δσ = 1/L
100 cm⁻¹
0.01 µm⁻¹; Δλ = λ²/L = 10 nm
Resolving Power L/λ
100
ILS FWHM 60.34 cm⁻¹ (Boxcar (none))
Nyquist Wavenumber
25600 cm⁻¹
step 0.1953 µm; λ_min = 0.3906 µm
Fringe Visibility ½ at
220.6 µm
OPD where a line's fringes fade to half
Interferogram
- I(x)
Recovered spectrum
- Recovered B(σ)
Two-beam fringes at the central wavelength
Optical Path Difference
20 µm
2ΔL (light crosses each arm twice)
Fringe Order OPD/λ
20
phase 0° past a bright fringe
Detector: Visibility V
0.9943
I = 0.8974 of the input; beams 0.225 and 0.225
Coherence Factor |γ|
0.9943
Gaussian line of the FWHM above, at this OPD
Model
I(x) = Σ cos(2πσᵢx)·exp(−(πΓx)²/(4 ln 2))·A(x/L), B(σ) = Δx[I₀ + 2Σ Iₙ cos(2πσnΔx)]
δσ = 1/L, σ_Nyquist = 1/(2Δx), Δx = L/N
Each Gaussian line of FWHM Γ (cm⁻¹) contributes a cosine fringe whose envelope is the Fourier transform of its profile, so broad lines lose fringe contrast within a short OPD. The single-sided interferogram, sampled at N points from zero path difference to L, is cosine-transformed (FFT of the mirrored record, zero-padded for a smooth curve) to recover the spectrum, convolved with the instrument line shape of the chosen apodization. Lines closer than about 1/L merge; lines above the Nyquist wavenumber fold back. The interferometer is ideal: no beam divergence, phase error or detector response. The Fellgett (multiplex) advantage comes from measuring all wavenumbers in every sample.
I = I₁ + I₂ + 2√(I₁I₂)|γ| cos(2π·OPD/λ), V = 2√(I₁I₂)|γ|/(I₁ + I₂)
Two-beam fringes use lossless splitters. In the Michelson both beams meet the splitter once in reflection and once in transmission (I₁ = R T R₁, I₂ = T R R₂), so the splitter ratio sets the throughput but not the visibility. In the Mach-Zehnder output A receives Rₐ T_b and Tₐ R_b and output B receives Rₐ R_b and Tₐ T_b, with complementary fringes that sum to the input. |γ| is the coherence of one Gaussian line of the FWHM set above.
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 two-beam interferometer with Gaussian lines, sampled single-sided from zero path difference; no beam divergence, phase error or detector response. Fringes: lossless splitters and plane waves, visibility from the beam powers and the line's coherence.
- Griffiths P. R., de Haseth J. A. (2007). Fourier Transform Infrared Spectrometry, 2nd ed., chs. 2 and 4. Wiley.
- Born M., Wolf E. (1999). Principles of Optics, 7th ed., ch. 7 (two-beam interference) and ch. 10 (partial coherence). Cambridge University Press. doi:10.1017/CBO9781139644181
- Hariharan P. (2003). Optical Interferometry, 2nd ed., ch. 2. Academic Press.