Fourier transform spectroscopy: Definition from Answers.com

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Fourier transform spectroscopy is a measurement technique whereby spectra are collected based on measurements of the coherence of a radiative source, using time-domain or space-domain measurements of the electromagnetic radiation or other type of radiation. It can be applied to a variety of types of spectroscopy including optical spectroscopy, infrared spectroscopy (FT IR, FT-NIRS), Fourier transform (FT) nuclear magnetic resonance^[1], mass spectrometry and electron spin resonance spectroscopy. There are several methods for measuring the temporal coherence of the light, including the continuous wave Michelson or Fourier transform spectrometer and the pulsed Fourier transform spectrograph (which is more sensitive and has a much shorter sampling time than conventional spectroscopic techniques, but is only applicable in a laboratory environment).

1 Conceptual introduction (for FTIR and other absorption spectroscopy)
2 Continuous wave Michelson or Fourier transform spectrograph
- 2.1 Extracting the spectrum
3 Pulsed Fourier transform spectrometer
- 3.1 Examples of Pulsed Fourier transform spectrometry
- 3.2 The Free Induction Decay
4 Fellgett Advantage
5 Converting spectra from time domain to frequency domain
6 See also
7 References and notes
8 Further reading
9 External links

Conceptual introduction (for FTIR and other absorption spectroscopy)

The goal of any absorption spectroscopy (FTIR, Ultraviolet-visible ("UV-Vis") spectroscopy, etc.) is to measure how well a sample absorbs or transmits light at each different wavelength. The most straightforward way to do this is to shine a monochromatic light beam through a sample, measure how much of the light is absorbed, and repeat for each different wavelength. (This is how UV-Vis spectrometers work, for example.)

Fourier transform spectroscopy is a less intuitive way to get the same information. Rather than passing a monochromatic beam of light through the sample, this technique passes a beam containing many different frequencies of light at once, and measures how much of that beam is absorbed by the sample. Next, the beam is modified to contain a different combination of frequencies, giving a second data point. This process is repeated many times. Afterwards, a computer takes all this data and works backwards to infer what the absorption is at each wavelength.

The beam described above is generated by starting with a broadband light source—one containing the full spectrum of wavelengths to be measured. The light shines into a certain configuration of mirrors that allows some wavelengths to pass through but blocks others (due to wave interference). The beam is modified for each new data point by moving one of the mirrors; this changes the set of wavelengths that pass through.

As mentioned, computer processing is required to turn the raw data (light absorption for each mirror position) into the desired result (light absorption for each wavelength). The processing required turns out to be a common algorithm called the Fourier transform (hence the name, "Fourier transform spectroscopy"). The raw data is sometimes called an "interferogram".

Continuous wave Michelson or Fourier transform spectrograph

The Fourier transform spectrometer is just a Michelson interferometer but one of the two fully-reflecting mirrors is movable, allowing a variable delay (in the travel-time of the light) to be included in one of the beams.

The Michelson spectrograph is similar to the instrument used in the Michelson-Morley experiment. Light from the source is split into two beams by a half-silvered mirror, one is reflected off a fixed mirror and one off a moving mirror which introduces a time delay -- the Fourier transform spectrometer is just a Michelson interferometer with a movable mirror. The beams interfere, allowing the temporal coherence of the light to be measured at each different time delay setting, effectively converting the time domain into a spatial coordinate. By making measurements of the signal at many discrete positions of the moving mirror, the spectrum can be reconstructed using a Fourier transform of the temporal coherence of the light. Michelson spectrographs are capable of very high spectral resolution observations of very bright sources. The Michelson or Fourier transform spectrograph was popular for infra-red applications at a time when infra-red astronomy only had single pixel detectors. Imaging Michelson spectrometers are a possibility, but in general have been supplanted by imaging Fabry-Perot instruments which are easier to construct.

Extracting the spectrum

The intensity as a function of the path length difference in the interferometer $p$ and wavenumber $\tilde{\nu} = 1/\lambda$ is ^[2]

$I(p,\tilde{\nu}) = I(\tilde{\nu})[1 + \cos(2\pi\tilde{\nu}p)]$ ,

where $I(\tilde{\nu})$ is the spectrum to be determined. Note that it is not necessary for $I(\tilde{\nu})$ to be modulated by the sample before the interferometer. In fact, most FTIR spectrometers place the sample after the interferometer in the optical path. The total intensity at the detector is

$I(p) = \int_0^\infty I(p,\tilde{\nu}) d\tilde{\nu} = \int_0^\infty I(\tilde{\nu})[1 + \cos (2\pi\tilde{\nu}p)] d\tilde{\nu}.$

This is just a Fourier cosine transform. The inverse gives us our desired result in terms of the measured quantity $I (p)$ :

$I(\tilde{\nu}) = 4 \int_0^\infty [I(p) - \tfrac{1}{2}I(p=0)] \cos (2\pi\tilde{\nu}p) dp.$

Pulsed Fourier transform spectrometer

A pulsed Fourier transform spectrometer does not employ transmittance techniques. In the most general description of pulsed FT spectrometry, a sample is exposed to an energizing event which causes a periodic response. The frequency of the periodic response, as governed by the field conditions in the spectrometer, is indicative of the measured properties of the analyte.

Examples of Pulsed Fourier transform spectrometry

In magnetic spectroscopy (EPR, NMR), an RF pulse in a strong ambient magnetic field is used as the energizing event. This turns the magnetic particles at an angle to the ambient field, resulting in gyration. The gyrating spins then induce a periodic current in a detector coil. Each spin exhibits a characteristic frequency of gyration (relative to the field strength) which reveals information about the analyte.

In FT-mass spectrometry, the energizing event is the injection of the charged sample into the strong electromagnetic field of a cyclotron. These particles travel in circles, inducing a current in a fixed coil on one point in their circle. Each traveling particle exhibits a characteristic cyclotron frequency-field ratio revealing the masses in the sample.

The Free Induction Decay

Pulsed FT spectrometry gives the advantage of requiring a single, time-dependent measurement which can easily deconvolute a set of similar but distinct signals. The resulting composite signal, is called a free induction decay, because typically the signal will decay due to inhomogeneities in sample frequency, or simply unrecoverable loss of signal due to entropic loss of the property being measured.

Fellgett Advantage

Main article: Fellgett's advantage

One of the most important advantages of Fourier transform spectroscopy was shown by P.B. Fellgett, an early advocate of the method. The Fellgett advantage, also known as the multiplex principle, states that a multiplex spectrometer such as the Fourier transform spectroscopy will produce a gain of the order of the square root of m in the signal-to-noise ratio of the resulting spectrum, when compared with an equivalent scanning monochromator, where m is the number of elements comprising the resulting spectrum when the measurement noise is dominated by detector noise.

Converting spectra from time domain to frequency domain

Main article: Fourier transform

$S(t) = \int_{-\infty}^\infty I(\nu) e^{- i\nu 2\pi t}\,d\nu$

The sum is performed over all contributing frequencies to give a signal S(t) in the time domain.

$I(\nu) = 2Re \int_{-\infty}^\infty S(t) e^{2i\pi \nu t}dt$

gives non-zero value when S(t) contains a component that matches the oscillating function.
Remember that

$e^{ix} = \cos x + i\sin x \!$

References and notes

^ Antoine Abragam. 1968. Principles of Nuclear Magnetic Resonance., 895 pp., Cambridge University Press: Cambridge, UK.
^ Peter Atkins, Julio De Paula. 2006. Physical Chemistry., 8th ed. Oxford University Press: Oxford, UK.

External links

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Fourier transform spectroscopy

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