Fourier-transform spectroscopy

An example of a spectrum: The spectrum of light emitted by the blue flame of a butane torch. The horizontal axis is the wavelength of light, and the vertical axis represents how much light is emitted by the torch at that wavelength.

One of the most basic tasks in spectroscopy is to characterize the spectrum of a light source: how much light is emitted at each different wavelength. The most straightforward way to measure a spectrum is to pass the light through a monochromator, an instrument that blocks all of the light except the light at a certain wavelength (the un-blocked wavelength is set by a knob on the monochromator). Then the intensity of this remaining (single-wavelength) light is measured. The measured intensity directly indicates how much light is emitted at that wavelength. By varying the monochromator's wavelength setting, the full spectrum can be measured. This simple scheme in fact describes how some spectrometers work.

Fourier-transform spectroscopy is a less intuitive way to get the same information. Rather than allowing only one wavelength at a time to pass through to the detector, this technique lets through a beam containing many different wavelengths of light at once, and measures the total beam intensity. Next, the beam is modified to contain a different combination of wavelengths, giving a second data point. This process is repeated many times. Afterwards, a computer takes all this data and works backwards to infer how much light there is at each wavelength.

To be more specific, between the light source and the detector, there is 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 can pass through.

As mentioned, computer processing is required to turn the raw data (light intensity for each mirror position) into the desired result (light intensity 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". Because of the existing computer equipment requirements, and the ability of light to analyze very small amounts of substance, it is often beneficial to automate many aspects of the sample preparation. The sample can be better preserved and the results are much easier to replicate. Both of these benefits are important, for instance, in testing situations that may later involve legal action, such as those involving drug specimens.[2]

An "interferogram" from a Fourier-transform spectrometer. This is the "raw data" which can be Fourier-transformed into an actual spectrum. The peak at the center is the ZPD position ("zero path difference"): Here, all the light passes through the interferometer because its two arms have equal length.

The method of Fourier-transform spectroscopy can also be used for absorption spectroscopy. The primary example is "FTIR Spectroscopy", a common technique in chemistry.

In general, the goal of absorption spectroscopy is to measure how well a sample absorbs or transmits light at each different wavelength. Although absorption spectroscopy and emission spectroscopy are different in principle, they are closely related in practice; any technique for emission spectroscopy can also be used for absorption spectroscopy. First, the emission spectrum of a broadband lamp is measured (this is called the "background spectrum"). Second, the emission spectrum of the same lamp shining through the sample is measured (this is called the "sample spectrum"). The sample will absorb some of the light, causing the spectra to be different. The ratio of the "sample spectrum" to the "background spectrum" is directly related to the sample's absorption spectrum.

Accordingly, the technique of "Fourier-transform spectroscopy" can be used both for measuring emission spectra (for example, the emission spectrum of a star), and absorption spectra (for example, the absorption spectrum of a liquid).

The intensity as a function of the path length difference (also denoted as retardation) in the interferometer ${\displaystyle p}$ and wavenumber ${\displaystyle {\tilde {\nu }}=1/\lambda }$ is [3]

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

where ${\displaystyle I({\tilde {\nu }})}$ is the spectrum to be determined. Note that it is not necessary for ${\displaystyle I({\tilde {\nu }})}$ 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

${\displaystyle {\begin{aligned}I(p)&=\int _{0}^{\infty }I(p,{\tilde {\nu }})d{\tilde {\nu }}\\&=\int _{0}^{\infty }I({\tilde {\nu }})[1+\cos(2\pi {\tilde {\nu }}p)]\,d{\tilde {\nu }}.\end{aligned}}}$

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

${\displaystyle I({\tilde {\nu }})=4\int _{0}^{\infty }\left[I(p)-{\frac {1}{2}}I(p=0)\right]\cos(2\pi {\tilde {\nu }}p)\,dp.}$

In magnetic spectroscopy (EPR, NMR), a microwave pulse (EPR) or a radio frequency pulse (NMR) 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 Fourier-transform 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.

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.

Pulsed sources allow for the utilization of Fourier-transform spectroscopy principles in scanning near-field optical microscopy techniques. Particularly in nano-FTIR, where the scattering from a sharp probe-tip is used to perform spectroscopy of samples with nanoscale spatial resolution, a high-power illumination from pulsed infrared lasers makes up for a relatively small scattering efficiency (often < 1%) of the probe.[4]