Instrument Controls
Choose a source and a sample, then move the mirror: drag it in the drawing, use the slider, or press Scan.
The Interferometer
Top view. The beamsplitter sends half of the light to the fixed mirror and half to the movable mirror; the halves come back, recombine and travel through the sample to the detector.
Detector signal
- Recorded signal
- The mirror now
Not to scale.
Key to the interferometer drawing
- Light going out from the source to the mirrors
- Reflected from the fixed mirror
- Reflected from the movable mirror
- Signal bar: full height is the empty holder at zero path difference
Below the beamsplitter, the wave that shifts as the mirror moves came from the movable mirror.
Interferogram
The detector signal against the path difference δ, recorded as the mirror moves. The filled dot is the mirror now.
- Recorded signal
- The mirror now
- Wavenumber 1 alone
- Wavenumber 2 alone
- Empty holder over the same stretch
Show the interferogram as a table
| Path difference, δ (µm) | Detector signal (%) |
|---|
From Interferogram to Spectrum
The Fourier transform turns the interferogram you recorded into a spectrum, one wavenumber at a time.
For each wavenumber, the transform asks how much of the interferogram is a cosine wave of that wavenumber. Multiply the interferogram by cos(2πν̃δ) and add up the product over the whole scan. Where the two keep in step, the product stays positive and the sum is large; where they drift in and out of step, the positive and negative parts cancel. Each sum, the relative product area, is one point of the spectrum: the green bar at the probe in the bottom panel is as tall as that sum. The spectrum starts whole, as the fast Fourier transform (FFT) computes every wavenumber at once; Sweep builds it again from 400 cm⁻¹, one probe wavenumber at a time, and Clear Points empties it so you can build it by moving the probe.
The interferogram here is the recording with its average level subtracted, so it is centred on zero; only the part that goes up and down carries information about wavenumbers. The mirror records only positive path differences, from zero to the end of its travel, and the product's area is taken over that range. An interferogram is symmetric about zero path difference, so the negative side would only add the same area again, and dividing by the background cancels that doubling.
The recording stops abruptly at the ends of the scan. In the spectrum, that sudden edge puts ripples on both sides of every sharp line. Multiplying the interferogram by a window that falls smoothly to zero at the ends (apodization, “removing the feet”) removes most of the ripples, at the cost of somewhat wider lines. The window is drawn dotted around the interferogram; compare the peaks in the spectrum with each window.
400 to 4000 cm⁻¹; every wavenumber the probe passes adds its point to the spectrum
- Relative product area
- Spectrum points
Show the interferogram, probe cosine and product as a table
| Path difference, δ (µm) | Interferogram (% points) | Probe cosine | Product |
|---|
Show the spectrum so far and the background as a table
| Wavenumber (cm⁻¹) | Wavelength (µm) | Relative product area (spectrum so far) | Background (relative) |
|---|
Spectrum
The finished spectrum. Each absorption is marked with its wavenumber; the Absorptions table names the vibration responsible.
Simulated: idealized bands placed at the positions found in reference IR spectra, not a measured spectrum.
Show the spectrum as a table
| Wavenumber (cm⁻¹) | Wavelength (µm) | Relative intensity |
|---|
Absorptions
Simulated: these absorptions are found in the simulated spectrum. Its bands are idealized versions of those in reference IR spectra; a measured spectrum has more bands, with different shapes and heights.
Identify the unknown
The unknown is one of the five liquids. Use the absorptions below, and the vibrations responsible, to decide which.
| Wavenumber (cm⁻¹) | Wavelength (µm) |
|---|
Bands are found where the transmittance dips at least 4 percentage points below the ground on both sides. With detector noise on, a dip must also stand well clear of the noise, which is largest where the source is dim.
About the Model
The interferometer
A Michelson interferometer splits the beam in two at the beamsplitter. One half goes to a fixed mirror and the other to a mirror that moves; both come back and recombine. If the movable mirror is a distance x further away than the fixed one, its half travels an extra 2x, the path difference δ = 2x. Light of wavenumber ν̃ (wavelength λ = 1/ν̃) arrives in step when δ is a whole number of wavelengths and opposite when it is a whole number plus a half, so its brightness at the detector is ½B(ν̃)[1 + cos 2πν̃δ]. The detector adds up every wavenumber in the beam. That sum, plotted against δ, is the interferogram.
Changing the light source, a wavenumber, the sample, the noise or the resolution starts a new recording, so the signal is drawn again as the mirror moves. At zero path difference every wavenumber is in step, so the signal is largest there: the centerburst. Away from it, the different wavenumbers fall in and out of step at different rates and mostly cancel, and the signal settles near half its largest value. A sample in the beam removes light at the wavenumbers it absorbs, which changes the interferogram slightly everywhere.
Transform and transmittance
The spectrum is recovered with the cosine Fourier transform, B(ν̃) ∝ Σ I(δ) cos(2πν̃δ), summed over every recorded δ after the average is subtracted. The From Interferogram to Spectrum panel does this one wavenumber at a time; the fast Fourier transform (FFT) does all of them at once. The interferogram is padded with zeros to 32,768 points before the FFT, which spaces the spectrum's points about 0.48 cm⁻¹ apart without adding any detail.
Transmittance is the sample's single-beam spectrum divided by the background (the same scan with the holder empty), T = I ÷ I0, and absorbance is A = −log10 T. Wavelength and wavenumber are related by λ (µm) = 10,000 ÷ ν̃ (cm⁻¹).
Sampling and resolution
A real instrument reads the detector each time a helium–neon reference laser's interference fringe passes, every 0.6328 µm of path difference. The page does the same, so the highest wavenumber the transform can report is 1 ÷ (2 × 0.6328 µm), about 7,900 cm⁻¹, comfortably above the 4,000 cm⁻¹ shown. Two lines closer together than about 1 ÷ δmax cannot be told apart, where δmax is the largest path difference recorded. The Resolution setting chooses the mirror's travel to match, and puts the mirror back at zero path difference: 0 to 6.25 µm at 800 cm⁻¹ and 0 to 12.5 µm at 400 cm⁻¹, enough to watch the longest wavelength (20 µm) go through its interference cycles, and short enough for the slider to move smoothly through every fringe. Travels this short give broad spectra: bands closer together than the resolution merge into one. For recording spectra, 4 cm⁻¹ moves the mirror from 0 to 1,250 µm, δmax = 0.25 cm, as a laboratory FTIR does; choosing it sets the mirror speed to 1,000 µm/s so a scan takes a few seconds, and choosing a low resolution again puts the speed back.
Noise and averaging
With Noise on, each reading gets random detector noise of 0.04 % of the largest signal, and averaging n scans divides it by √n, so 4 scans halve the noise and 64 scans cut it eightfold. Because a scan takes so little time, an FTIR can average many scans in the time a dispersive instrument takes for one; this is the multiplex advantage. The background is taken as noise-free, so all the noise in the transmittance comes from the sample's scan. The noise figure under the Spectrum is the difference from the same transform without noise. With noise, the Absorptions table lists a dip only where it stands at least seven times the noise clear of its surroundings and of 100 %: noise alone makes dips four or five times its size many times across a spectrum, but almost never seven. One scan at 4 cm⁻¹ is noisy enough to hide the weaker bands; averaging 16 scans brings nearly all of them out.
Unknowns
Unknown A, B and C are three of the five liquids, chosen at random each time the page loads or New Unknowns is pressed. They are drawn as liquid cells, so the holder gives nothing away.
Simplifications
- The interferogram is ideal unless Noise is on: no stray light, a perfect 50:50 beamsplitter, and a mirror that moves perfectly straight.
- The broadband source is a blackbody at 1,500 K, cut off below about 440 cm⁻¹ and above about 4,300 cm⁻¹ by the beamsplitter and detector.
- Each sample's spectrum is a set of idealized IR bands at the positions found in reference IR spectra of the liquid, film or gas, with rounded widths and heights; Raman-only vibrations are left out. Liquids are drawn as thin films between potassium bromide (KBr) windows, which pass infrared down to 400 cm⁻¹, polystyrene as a film on a card, and carbon dioxide as a gas cell; each carbon dioxide band is drawn as one smooth band, without the rotational structure a real gas shows.
- The fingerprint region is shaded from 600 to 1,200 cm⁻¹.
- Scan works as a real instrument does: each scan is a new recording; the mirror returns quickly to zero path difference without recording, then sweeps at constant speed to the end of its travel, read at every reference-laser fringe. The mirror's travel starts at zero path difference, so the recording is one side of the interferogram (single-sided); an ideal interferogram is symmetric, so the transform takes the other side as its mirror image and the spectrum is the same as from both sides. A real instrument's interferogram is slightly lopsided, from small phase errors in the beamsplitter and electronics, so an instrument recording one side also records a short stretch on the other side of zero and uses it to correct them; this idealized model has no phase errors. Moving the mirror by hand records every fringe it passes. The mirror speed runs from 0.5 µm/s, slow enough to watch each wave fall in and out of step, to 1,000 µm/s; a real mirror sweeps much faster, so a real scan takes about a second and many scans are averaged.
- The drawing of the instrument magnifies the waves (3 drawing units per micrometre of wavelength) and, at 800 and 400 cm⁻¹, the mirror's movement twice as much, so the reflected wave moves with the mirror while keeping its true phase; at 4 cm⁻¹ the travel is drawn smaller and the beam as a line. The waves are frozen in place, and in each arm the wave going out and the wave reflected back overlap. With two wavelengths, wavenumber 2's wave is drawn in proportion to the square root of its brightness, because a wave's brightness goes as its height squared: at 30 % it is about half as tall as wavenumber 1's, which keeps its height. Between the readings, every 0.6328 µm of path difference, the charts draw the smooth curve the readings describe (band-limited, windowed-sinc interpolation), as the readings hold every wavenumber up to about 7,900 cm⁻¹.
Keyboard
Tab moves between controls; arrow keys move the sliders. The wavenumber and Mirror position boxes take an exact value (Enter applies it, Escape abandons the edit); a wavenumber box also takes a wavelength, such as 6 µm. The Detector Signal chart's description gives the present reading, and every value is in the tables. Alt + Shift + P (Option + Shift + P on a Mac) switches presentation mode on and off, and Escape leaves it.