IN18 · Many colors, one detector

Fourier Transform Infrared Spectrometer

How can a detector that sees total brightness recover an entire spectrum? Moving the mirror changes how far the two light paths differ. Each infrared wavelength then contributes its own bright-dark rhythm to the total detector signal. A Fourier transform separates these rhythms into a spectrum. Comparing sample and reference spectra reveals absorption where the reference supplies enough light.

Move a Michelson mirror and recover infrared absorption from saved reference and sample detector counts.

Brightness has many rhythms

Preview half OPD $0.205\,\mathrm{cm}$

Window first-zero spacing $4.88\,\mathrm{cm}^{-1}$

Recovered band maximum $1549\,\mathrm{cm}^{-1}$

Recovered peak absorbance $0.682$

IlluminationScan pointsOptical path sampling step

h/\mathrm{\mu m}

Ideal sample peak absorbance

0.7

Common-output sample

Drag the mirror to inspect a detector sample, or move the sample holder out of the recombined beam.

Window, calibration and model

Transform windowDisplay zero paddingFirst source line

\sigma_1/\mathrm{cm}^{-1}

1250

Source-line separation

\Delta\sigma/\mathrm{cm}^{-1}

12

Total source power

P_0/\mathrm{mW}

1.5

Encoder scale error

\epsilon/\mathrm{ppm}

150

Unknown zero-path offset

\delta_0/\mathrm{\mu m}

0.3

Detector gain factor

g_d

1

Bounded detector noise

n_C

1

Acquisition seed

s_0

191

A separate known-power exposure calibrates detector gain. A chosen line calibrates encoder scale and zero path within a coarse zero bound; a second line checks the result. No laser encoder is drawn or assumed perfect.

The sample follows recombination. Infrared colors and propagation markers are schematic; mirror travel is enlarged thirty-five times. The source continuum is numerically integrated in one-inverse-centimetre bins, plus two narrow lines. Finite-window spectra are broadened; zero padding does not improve resolution.

The optical band ends at2300 inverse centimetres. Saturation, insufficient sampling or weak reference masks absorption. The model omits dispersion, angular spread, multiple reflections, polarization and source drift.

Model assumptions

An ideal balanced, normal-incidence air Michelson feeds one detector through a sample in the common recombined output. Different optical frequencies add incoherent powers. A chosen continuum plus two lines probes one synthetic absorption band; this is not a molecular database. Separate dark/span and known-line scans calibrate detector gain, encoder scale and zero path; a second line is withheld. The real phase-corrected Fourier density comes from raw counts and saved encoder positions. A finite window broadens features; zero padding only changes displayed sampling. Absorption is withheld for saturation, alias risk, weak background or incompatible calibration. Infrared colors, propagation speed and mirror travel magnification are schematic. Dispersion, angular spread, window reflections, atmospheric lines, polarization and time-dependent source drift are omitted.

Saved scans retain their positions and calibration

The two raw interferograms

$x_{\mathrm{encoder}}/\mathrm{mm}\quad;\quad C/10^3$

Turquoise is the reference; violet is the sample. The moving marker replays a saved point only when the apparatus matches this record.

Spectra recovered from these counts

$\sigma/\mathrm{cm}^{-1}\quad;\quad S_\sigma/[10^{-6}\,\mathrm W/(\mathrm{cm}^{-1})]$

Absorbance after background division

$\sigma/\mathrm{cm}^{-1}\quad;\quad A=-\log_{10}T$

Violet is recovered from the raw pair. Dashed grey is the separate ideal sample; finite resolution can change the measured band depth. Unresolved bins are omitted.

Calibration, checks and retained pairs

Saved reference/sample pairs
Pair	Points	Calibration	Window	Valid

Exports include both raw count arrays, every encoder position, independent dark/span and line observations, the withheld check, processing, model version and seed. Ideal source and sample data are explicitly separate.

Start with the essentials

Focus question: How can a detector that sees total brightness recover an entire spectrum?
One-sentence intuition: Different wavelengths acquire phase at different rates as the mirror moves.

Core mathematical model

One bright output after a common-path sample

P_+(\delta)=\frac12\int S_\sigma(\sigma)T(\sigma)\left[1+\cos(2\pi\sigma\delta)\right]\,\mathrm d\sigma,\qquad\delta=2x

Wavenumber is in inverse centimetres and optical path difference in centimetres. The mirror travels half the optical path difference. The sample follows recombination, so its intensity transmission multiplies the whole output at each frequency. Different optical frequencies add powers.

Divide two recovered spectra where background is resolved

\widehat T(\sigma)=\frac{\widehat S_{\mathrm{sample}}(\sigma)}{\widehat S_{\mathrm{reference}}(\sigma)},\qquad\widehat A=-\log_{10}\widehat T

Both spectra are phase-corrected real Fourier transforms of saved counts, with the same finite window. Their ratio is an instrument-broadened estimate; it need not equal the ideal local sample transmission exactly. Weak-reference, alias-risk and saturated data cannot support absorbance.

Sample spacing and maximum path difference do different jobs

\sigma_{\mathrm{Nyq}}=\frac{1}{2h},\qquad\Delta\sigma_{\mathrm{box}}=\frac{1}{2D}

The calibrated optical path step sets Nyquist. The nominal half range is $D$ ; the box-window first zero is the stated resolution criterion. A triangular window doubles that first-zero width. Zero padding adds displayed bins without narrowing it.

Common difficulties

A broad absorption band needs broad illumination

Typical misconceptionTwo isolated source lines reveal the whole sample band.

Better mental modelOnly illuminated wavenumbers carry sample information. The default adds a continuum to the two narrow source lines.

An interferogram is already an intensity measurement

Typical misconceptionSquare its Fourier transform to obtain the measured light spectrum.

Better mental modelThe positive-frequency transform is proportional to spectral intensity. Squaring it again changes peak ratios and units.

Run the experiment

01
Inspect an actual mirror position
Drag the scanning mirror and watch the detector count and current interferogram point.
What to observe: The physical mirror and double-pass path share the selected encoder position. Motion replays the preview scan; it does not rewrite a saved scan.
02
Insert the common-path sample
Drag the sample holder out of the recombined output, then back in.
What to observe: Both arm fields have already recombined before they enter the sample. Its band reduces only the corresponding spectral contributions.
03
Compare the two scan lengths
Compare Two colors beat together and A short scan blurs the pair.
What to observe: Shortening maximum optical path difference broadens the instrument line shape. More zero padding leaves that line shape unchanged.
04
Retain before changing processing
Keep this pair, change the window in the collapsed group and export the saved records.
What to observe: Old reference/sample counts, encoder coordinates and calibration remain fixed. The preview uses a new processing choice until explicitly kept.

Fourier Transform Infrared Spectrometer

Saved scans retain their positions and calibration

The two raw interferograms

Spectra recovered from these counts

Absorbance after background division

The detector sees a sum of fringe rhythms

Core mathematical model

One bright output after a common-path sample

Divide two recovered spectra where background is resolved

Sample spacing and maximum path difference do different jobs

Common difficulties

A broad absorption band needs broad illumination

An interferogram is already an intensity measurement

Run the experiment

Inspect an actual mirror position

Insert the common-path sample

Compare the two scan lengths

Retain before changing processing

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