---
title: "Fourier Transform Infrared Spectrometer Interactive Lab | Sandbox Physics"
description: "How can a detector that sees total brightness recover an entire spectrum? Moving the mirror changes how far the two light paths differ."
canonical_url: "https://sandboxphysics.com/labs/ftir-spectrometer/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/ftir-spectrometer/"
last_modified: "2026-11-03"
---

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.

> Interactive simulation: open this page in a web browser to change the parameters and run the model.

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.

Physics tutorial

## The detector sees a sum of fringe rhythms

**Background** A beam splitter sends infrared light down two paths. A moving mirror changes the return distance in one arm. Each wavelength alternately reinforces and cancels at a different displacement rhythm.

**Why it matters** A broadband detector records total brightness, yet a position-tagged scan can retain enough information to recover a spectrum.

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 misconception** Two isolated source lines reveal the whole sample band.

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

#### An interferogram is already an intensity measurement

**Typical misconception** Square its Fourier transform to obtain the measured light spectrum.

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

### Run the experiment

1. 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.

2. 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.

3. 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.

4. 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.

Physical conclusion

Keep reference and sample detector counts with their actual encoder grid, independent calibration and chosen transform window.

## Related experiments and articles

How physics instruments measure · Reading path

### How physics instruments measure

Articles on this path that use this experiment, followed by the other experiments they suggest.

1. [Article **How total brightness reveals many infrared colors**](https://sandboxphysics.com/stories/instruments/ftir-spectrometer/)

Waves & optics · Starter path

## From one ray to interference measurement

Start with refraction, find where the ray picture stops working, then use interference to measure tiny distances.

[Start here → **Optical Boundary Explorer** Turn the incident angle and watch reflected and transmitted power trade off. Then switch to the critical-angle scene. Observe · 11 min](https://sandboxphysics.com/labs/optical-boundary-explorer/)

1. [1 Optical Boundary Explorer](https://sandboxphysics.com/labs/optical-boundary-explorer/)
2. [2 Rainbow Caustic Observatory](https://sandboxphysics.com/labs/rainbow-laboratory/)
3. [3 Camera Obscura](https://sandboxphysics.com/labs/camera-obscura/)
4. [4 Coherent Wave Composer](https://sandboxphysics.com/labs/coherent-wave-composer/)
5. [5 Young Double-Slit Bench](https://sandboxphysics.com/labs/young-double-slit-bench/)
6. [6 Michelson Precision Lab](https://sandboxphysics.com/labs/michelson-precision-lab/)

[View the full path →](https://sandboxphysics.com/topics/waves-and-optics/#starter-path)