Fourier transform spectroscopy Michelson interferometer

It is the self-coherence function that is measured in Fourier transform spectroscopy. Writing the measured on-axis intensity at the output of the Michelson interferometer as... 

Thus, we need only take a one-sided interferogram. From Eq. 13 we can also see why spectroscopy associated with the Michelson interferometer is called Fourier transform spectroscopy. Movement of the mirror which corresponds to a change in retardation provides a signal which is a function of distance. This signal is then decoded by a Fourier transform to give a spectrum which is a function in the reciprocal space. This is why wavenumbers (cm-1) are such a convenient unit to use with this type of spectroscopy. 

The problem is how to convert the interferogram I (s) obtained with a Michelson interferometer into the spectrum I (i>). Problems of this kind are met with in many areas of physics and technology for example, the problem of determining the spectrum of harmonics for a musical instrument (flute or violin). At audiofrequencies the problem is easily solved with an appropriate set of electronic circuits that performs a so-called Fourier analysis. In Fourier transform spectroscopy the solution is obtained by mathematical treatment of the interferogram 7(s). In order to illustrate the principle of this treatment in a simple way, let us go back to the case of a single narrow laser line, i.e. monochromatic radiation. 

If I (5) is other than a few discrete narrow lines, the tool to evaluate / (5) from I[s) is the Fourier transform, where I ) is the interferogram measured with a two-beam (Michelson) interferometer. This is the fundamental idea of Fourier transform spectroscopy. We have left aside the question of whether the Fourier integral Eq. (2.12) exists and whether it is meaningful or not. For the mathematical requirements on I (s), the reader is referred to the literature 34). it is sufficient to say here that, for all physically and experimentally reasonable interferograms I s), these requirements are usually met. 

Turning back to the Michelson interferometer and Fourier transform spectroscopy, let us first consider the interferogram of a continuous spectrum. Each spectral element of infinitesimal width rf and intensity I (v) gives rise to the same interferogram pattern as a narrow line [see Eq. (2.6)], and the actual interferogram is the superposition of all these... 

Summarizing the results of our discussion of the practice of Fourier transform spectroscopy, we start with the presumption that the equipment for most routine spectroscopic investigations consists of a Fourier spectrometer with a Michelson interferometer and a digital computer. In other words, the advantages of the lamellar grating used as a two-beam interferometer, and of phase modulation, for example, have been utilized only for certain special applications in the extreme far-infrared. All commercial Fourier spectrometers are available with a computer attached, which in most cases not only performs the Fourier transform but is also programmed to control the instrument. Commercial instruments have a remote switch for the selection of the different spectral ranges, and the filters and beams... 

The other principal advantage which applies to Fourier transform spectroscopy is the multiplex or "Fellgett" advantage 21,64) n yas P. Fellgett who first pointed out that there is an advantage when the data in all elements of a spectrum are obtained simultaneously instead of being measured for each element successively. In Fourier transform spectroscopy, the radiation in the Michelson interferometer is not separated into spectral elements. The interferogram contains... 

This technique is quite similar to Fourier-transform spectroscopy, where the phase difference between the two beams in a Michelson interferometer are generated by changing the path difference as a function of time. In the dual comb spectroscopy the phase difference it generated by the different repetition frequencies of the two... 

There are many other types of two-beam interferometers besides the one originally described by Michelson (see Chapter 5). Many of these interferometers do not vary the path difference between two beams by a single mirror moving at constant velocity. Except for stationary interferometers used for Fourier transform spectroscopy (Section 5.6), an optical element or combination of optical elements is moved so that the optical path difference is changed at a certain rate, known as the optical velocity or OPD velocity, V. For the Michelson interferometer, V = 2V. In general, the Fourier frequency for radiation of wavenumber v is given by... 

In the mid-IR, routine infrared spectroscopy nowadays almost exclusively uses Fourier-transform (FT) spectrometers. This principle is a standard method in modem analytical chemistry45. Although some efforts have been made to design ultra-compact FT-IR spectrometers for use under real-world conditions, standard systems are still too bulky for many applications. A new approach is the use of micro-fabrication techniques. As an example for this technology, a miniature single-pass Fourier transform spectrometer integrated on a 10 x 5 cm optical bench has been demonstrated to be feasible. Based upon a classical Michelson interferometer design, all... 

A Fourier transform infrared spectroscopy spectrometer consists of an infrared source, an interference modulator (usually a scanning Michelson interferometer), a sample chamber and an infrared detector. Interference signals measured at the detector are usually amplified and then digitized. A digital computer initially records and then processes the interferogram and also allows the spectral data that results to be manipulated. Permanent records of spectral data are created using a plotter or other peripheral device. 

This Fourier transform process was well known to Michelson and his peers but the computational difficulty of making the transformation prevented the application of this powerful interferometric technique to spectroscopy. An important advance was made with the discovery of the fast Fourier transform algorithm by Cooley and Tukey 29) which revived the field of spectroscopy using interferometers by allowing the calculation of the Fourier transform to be carried out rapidly. The fast Fourier transform (FFT) has been discussed in several places 30,31). The essence of the technique is the reduction in the number of computer multiplications and additions. The normal computer evaluation requires n(n — 1) additions and multiplications whereas the FFT method only requires (n logj n) additions and multiplications. If we have a 4096-point array to Fourier transform, it would require (4096) (4095) or 16.7 million multiplications. The FFT allows us to reduce this to... 

Fourier transform (FT) IR spectroscopy is one of several nondispersive optical spectroscopies based on interferometry. A two-beam interferometer first proposed by Michelson is the basis of most modern FT-IR spectrometers, as exemplified by the schematic of the Bruker Equinox 55 spectrometer (Bruker Optik, Ettlingen, Germany) in Fig. 2. Simply described, the interferometer comprises a beam splitter and two mirrors. A collimated beam of IR energy is split at the beam splitter into equal halves. Half of the energy travels through the beam splitter to one of the mirrors, which is positioned at a fixed distance away from the beam splitter. The reflected beam travels perpendicular to the incident beam to a moving mirror. IR radiation reflects off the fixed and moving mirrors and recombines at the beam splitter. The recombined IR beam projects from the interferometer towards the detector on an optical path perpendicular to the source beam. 

Fourier transform infrared (FTIR) spectroscopy has been extensively developed over the past decade and provides a number of advantages. The main part of FTIR spectrophotometer is the Michelson interferometer. Radiation containing all IR wavelengths (e.g., 4000-400 cm 1) is emitted by source of infrared radiation (Globar) and is split into two beams. One beam is of fixed length, and the other is of variable length (movable mirror). 

In FTIR spectroscopy, an interference wave interacts with the sample in contrast to a dispersive instrument where the interacting energy assumes a well-defined wavelength range. The interference wave is produced in an interferometer (Fig. 4.1.1), the most common of which is the Michelson interferometer. A computer is used to control the interferometer, to collect and store data, and to perform the Fourier transformation. In addition, the computer performs post-spectroscopic operations such as spectral presentation, resolution enhancement, calibration, and calculation of correlation equations. 

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