The interferogram is a function of time and the values outputted by this function of time are said to make up the time domain. The time domain is Fourier transformed to get a frequency domain, which is deconvolved to product a spectrum. Figure 4 shows the Fast Fourier transform from an interferogram of polychromatic light to its spectrum. Figure 4. The first one who found that a spectrum and its interferogram are related via a Fourier transform was Lord Rayleigh.

He made the discover in But the first one who successfully converted an interferogram to its spectrum was Fellgett who made the accomplishment after more than half a century. It has been applied widely to analytical methods such as infrared spectrometry, nuclear magnetic resonance and mass spectrometry due to several prominent advantages which are listed in Table 1.

Table 1. Fourier transform, named after the French mathematician and physicist Jean Baptiste Joseph Fourier, is a mathematical method to transform a function into a new function. The following equation is a common form of the Fourier transform with unitary normalization constants:. The following equation is another form of the Fourier transform cosine transform which applies to real, even functions:.

## Fourier Analysis on Groups

The math description of the Fourier transform can be tedious and confusing. An alternative explanation of the Fourier transform in FTIR spectrometers is provided here before we jump into the math description to give you a rough impression which may help you understand the math description.

The interferogram obtained is a plot of the intensity of signal versus OPD. A Fourier transform can be viewed as the inversion of the independent variable of a function. Thus, Fourier transform of the interferogram can be viewed as the inversion of OPD. Inverse centimeters are also known as wavenumbers. After the Fourier transform, a plot of intensity of signal versus wavenumber is produced. Such a plot is an IR spectrum.

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Although this explanation is easy to understand, it is not perfectly rigorous. The wave functions of the reflected and transmitted beams may be represented by the general form of:. The resultant wave function of their superposition at the detector is represented as:.

The ratio of radiant power transmitted by the sample I relative to the radiant power of incident light on the sample I 0 results in quantity of Transmittance, T. Absorbance A is the logarithm to the base 10 of the reciprocal of the transmittance T :. Step 1 : The first step is sample preparation. About 2 mg of sample and mg KBr are dried and ground. The particle size should be unified and less than two micrometers. Then, the mixture is squeezed to form transparent pellets which can be measured directly. For liquids with high boiling point or viscous solution, it can be added in between two NaCl pellets.

Then the sample is fixed in the cell by skews and measured. Then the solution is injected into a liquid cell for measurement. Gas sample needs to be measured in a gas cell with two KBr windows on each side. The gas cell should first be vacuumed. Then the sample can be introduced to the gas cell for measurement.

Step 2 : The second step is getting a background spectrum by collecting an interferogram and its subsequent conversion to frequency data by inverse Fourier transform. We obtain the background spectrum because the solvent in which we place our sample will have traces of dissolved gases as well as solvent molecules that contribute information that are not our sample. The background spectrum will contain information about the species of gases and solvent molecules, which may then be subtracted away from our sample spectrum in order to gain information about just the sample.

Figure 6 shows an example of an FTIR background spectrum. The background spectrum also takes into account several other factors related to the instrument performance, which includes information about the source, interferometer, detector, and the contribution of ambient water note the two irregular groups of lines at about cm —1 and about cm —1 in Figure 6 and carbon dioxide note the doublet at cm —1 and sharp spike at cm —1 in Figure 6 present in the optical bench. Step 3 : Next, we collect a single-beam spectrum of the sample, which will contain absorption bands from the sample as well as the background gaseous or solvent.

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Step 4 : The ratio between the single-beam sample spectrum and the single beam background spectrum gives the spectrum of the sample Figure 7. Step 5 : Data analysis is done by assigning the observed absorption frequency bands in the sample spectrum to appropriate normal modes of vibrations in the molecules. Despite of the powerfulness of traditional FTIR spectrometers, they are not suitable for real-time monitoring or field use.

So various portable FTIR spectrometers have been developed. Below are two examples. Ahonen et al developed a portable, real-time FTIR spectrometer as a gas analyzer for industrial hygiene use. The instrument consists of an operational keyboard, a control panel, signal and control processing electronics, an interferometer, a heatable sample cell and a detector.

All the components were packed into a cart. To minimize the size of the instrument, the resolution of FTIR spectrometer was sacraficed. But it is good enough for the use of industrial hygiene. Korb et al developed a portable FTIR spectrometer which only weighs about Moreover, the energy source of the instrument is battery so that the mobility is significantly enhanced. Additionally, this instrument resists vibration. It works well in an operating helicopter. Consequently, this instrument is excellent for the analysis of radiation from the surface and atmosphere of the Earth.

The instrument is also very stable. After a three-year operation, it did not lose optical alignment. The reduction of size was implemented by a creative design of optical system and accessory components. Two KBr prisms were used to constitute the interferometer cavity. Optical coatings replaced the mirrors and beam splitter in the interferometer. The optical path is shortened with a much more compact packaging of components. A small, low energy consuming interferometer drive was designed.

It is also mass balanced to resist vibration. The common He-Ne tube was replaced by a smaller laser diode. The first generation IR spectrometer was invented in late s. It utilizes prism optical splitting system. The prisms are made of NaCl. Further more, the scan range is narrow. Additionally, the repeatability is fairly poor. As a result, the first generation IR spectrometer is no longer in use. The second generation IR spectrometer was introduced to the world in s.

It utilizes gratings as the monochrometer.

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The performance of the second generation IR spectrometer is much better compared with IR spectrometers with prism monochrometer, But there are still several prominent weaknesses such as low sensitivity, low scan speed and poor wavelength accuracy which rendered it out of date after the invention of the third generation IR spectrometer. The invention of the third generation IR spectrometer, Fourier transform infrared spectrometer, marked the abdication of monochrometer and the prosperity of interferometer.

With this replacement, IR spectrometers became exceptionally powerful. Consequently, various applications of IR spectrometer have been realized. Figure 2. Michelson Interferometer The Michelson interferometer, which is the core of FTIR spectrometers, is used to split one beam of light into two so that the paths of the two beams are different.

## OSA | Spectral resolution and sampling issues in Fourier-transform spectral interferometry

Schematic of the Michelson interferometer A typical Michelson interferometer consists of two perpendicular mirrors and a beamsplitter. As a result, a maximum intensity signal is observed by the detector. Fourier Transform of Interferogram to Spectrum The interferogram is a function of time and the values outputted by this function of time are said to make up the time domain.

Advantages of Fourier Transform over Continuous-Wave Spectrometry Fourier transform, named after the French mathematician and physicist Jean Baptiste Joseph Fourier, is a mathematical method to transform a function into a new function. The physically measured information recorded at the detector produces an interferogram, which provides information about a response change over time within the mirror scan distance.

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Therefore, the interferogram obtained at the detector is a time domain spectrum. This procedure involves sampling each position, which can take a long time if the signal is small and the number of frequencies being sampled is large. Figure 6. Background IR spectrum The background spectrum also takes into account several other factors related to the instrument performance, which includes information about the source, interferometer, detector, and the contribution of ambient water note the two irregular groups of lines at about cm —1 and about cm —1 in Figure 6 and carbon dioxide note the doublet at cm —1 and sharp spike at cm —1 in Figure 6 present in the optical bench.

V, Bull. France 87 , 65— Eymard, P. France 92 , — Folland, G. Ghandehari, M. Haagerup, U. IMRN , no. Hewitt, E. Cambridge Philos. Kadison, R.