Fourier Transform Spectroscopy

The obtained spectral resolution is limited by the maximum optical path length difference. This is easy to see considering the properties of discrete Fourier transforms, or simply recognizing that the range of path length differences determines the number of oscillation cycles which can be counted. Quantitatively, the resolution in terms of spectroscopic wavenumber is the inverse of the maximum optical path length difference. Simple instruments may work with only a few centimeters of path length difference, achieving spectral resolutions of somewhat better than 1 cm⁻¹, while high precision spectrometers work with much longer path length differences, e.g. several meters.

On the other hand, the maximum wavenumber is half the inverse spatial resolution of the measured path length difference. Therefore, a not particularly high spatial resolution is required for instruments working only with relatively long optical wavelengths, while UV instruments are more demanding in that respect. The spatial accuracy, however, should be much higher – see below.

For calculations, note that the variation of pass length difference in a Michelson interferometer is twice the amount of movement of a retroreflector.

For spectrometers as used in infrared spectroscopy, often uses a very broadband light source for measuring optical properties of samples in a wide wavelength region. The light source should of course have a sufficiently high spectral flux and emit continuously with stable optical properties throughout the interferometer scan. For the near infrared, incandescent lamps are suitable, but there emission is limited to wavelengths below roughly 5 μm by the transmissivity of the bulb glass. For longer wavelength regions, and therefore uses for millimeters not requiring a glass bulb – for example, Nernst glowers based on an electrically heated rod made of zirconium/yttrium ceramics. Silicon carbide rods can even be used up to about 40 μm. Also there are mercury vapor lamps.

For the interferometer to work properly, one requires a light beam with high enough spatial coherence. This is because different spatial components of a beam can produce different contributions to the interferogram, effectively washing out the pattern.

Ideally, one would have a Gaussian beam from a laser source. In practice, however, one often deals with incoherent sources, where the light has to be spatially filtered, accepting some loss of optical power. However, the possible power throughput is still substantially better than for a grating monochromator as used in other forms of spectroscopy, where light needs to be fed through a narrow optical slit. This is called the Jacquinot advantage, named after Pierre Jacquinot who identified it.

The optical components of the interferometer should of course properly work over the full spectral region of interest. The most substantial challenge arises from the beam splitter, which would ideally exhibit a 50:50 splitting ratio for all relevant wavelengths. That is not strictly required, but it should at least not lead to highly asymmetric splitting or introduce high power losses e.g. by absorption in a substrate. In infrared spectroscopy, one often uses beam splitters with calcium fluoride (CaF₂) substrates for wavelengths up to 8 μm. KBr-based beam splitters with a germanium-based coating can be used up to 25 μm wavelength, but that material is hygroscopic and must therefore be carefully protected against moisture. For the far infrared, one often uses polymer films.

Broadband mirrors are more easily obtained; one often uses metal-coated first surface mirrors, because dielectric mirrors could hardly provide the required reflection bandwidth.

The interferometer arm length difference is usually varied with a motorized drive, which can normally not be trusted to provide sufficiently accurate variations of the position. Therefore, one often simultaneously records a second interferogram, using light from a narrow-linewidth laser with sufficiently stable wavelength. One can then computationally correct the data for any deviations of the movement from a perfectly linear movement.

Note that it is not sufficient only to have a positional accuracy which allows one to clearly resolve the oscillations of an interferogram. This is because random position errors also limit the signal-to-noise ratio of the obtained spectra. Therefore, it is essential to realize Fourier transform spectrometers with highly accurate opto-mechanics and an accurate reference interferometer. That also provides a very high wavelength accuracy – better than in dispersive instruments.

A simple Fourier transform applied to the raw data will generally not deliver a calibrated optical spectrum, mostly because the responsivity of the used photodetector and the reflectivity of the beam splitter are wavelength-dependent; further influences can come from other optical elements of the setup. Such influences do not matter in absorption spectroscopy, because one only compares spectra with and without an absorbing sample, and the obtained intensity ratios are not affected; one only requires sufficiently strong signals for all relevant wavelengths.

When measuring optical spectra of sources, however, one needs to apply a calibration. It may be done, for example, by comparing with the recorded spectrum of a light source with known spectral shape. In the infrared, one often uses black body radiation for that calibration.

In some cases, one may even calibrate a spectrometer for obtaining absolute values of the power spectral density. This is often not easy, however, for example because of influences of the required spatial filtering of the input beam (see above).

Discrete Fourier transforms can quite easily and efficiently be computed, using a Fast Fourier Transform (FFT) algorithm. In the simplest form, such an algorithm works with a number of data points which is a power of 2. Even on a relatively simple microprocessor, the FFT computation usually takes much less time than the acquisition of the raw data.

Interference-based methods of spectroscopy have been used already in the early days of optics, for example by Hippolyte Fizeau, who resolved the doublet of the yellow sodium fluorescence line in the 19^th} century (→ Fizeau interferometers). However, computations of optical spectra based on Fast Fourier transform have been implemented only from the middle of the 20^th century on, when computers became available; first commercial devices appeared in the 1960s.

For the kind of interferometer as explained above, the sensitivity to mechanical noise (vibrations and shocks or inaccuracies of an optical delay line) is quite high. That sensitivity can be massively reduced by using a common-path interferometer based on birefringence. This can be realized, for example, with a simple optical beam path where two polarization components finally interfere at a polarizer. The optical delay between the two polarization components can be adjusted by moving a wedged birefringent crystal [7]. Because that does not only greatly reduce the sensitivity to vibrations, but also allows very accurate scanning of the delay range, it is particularly suitable for Fourier transform spectroscopy in relatively short wavelength regions.

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