Fundamentals of Fourier transform spectroscopy

Fourier transform methods have revolutionized many fields in physics and chemistry, and applications of the technique are to be found in such diverse areas as radio astronomy [52], nuclear magnetic resonance spectroscopy [53], mass spectroscopy [54], and optical absorption/emission spectroscopy from the far-infrared to the ultraviolet [55-57]. These applications are reviewed in several excellent sources [1, 54,58], and this section simply aims to describe the fundamental principles of FTIR spectroscopy. A more theoretical development of Fourier transform techniques is given in several texts [59-61], and the interested reader is referred to these for details. 

The signal seen at the detector for a given value of the optical path difference (OPD), given by the symbol 6, is dependent upon the wavelengths, amplitudes, and phases of the components of the radiation. Constructive interference for all components occurs only at S = 0, where the maximum signal is observed (often referred to as the centerburst or central maximum). The signal that is seen at the detector as a function of S, 1(5) for an ideal interferometer, is given by 

The cosine Fourier transform given by Eq. (2) is only applicable if the interferogram is perfectly symmetrical about 5 = 0. In practice additional wavenumber-dependent phase shifts are present, owing to beamsplitter characteristics or refraction effects, and cause the interferogram to appear partially asymmetric. The modulated part of Eq. (1) then becomes 

The optical path difference over which the interferogram can be digitized is limited by the dimensions of the interferometer. 

Replacement of Fourier integrals by summations has two important ramifications of practical importance  

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Smith, B.C. (1995) Fundamentals of Fourier Transform Spectroscopy. Boca Raton, FL CRC Press. 

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