The Michelson Interferometer

If a collimated beam of monochromatic radiation of wavelength X is passed into an ideal beamsplitter, 50% of the incident radiation will be reflected to one of the mirrors and 50% will be transmitted to the other mirror. The two beams are reflected from these mirrors, then returning to the beamsplitter where they recombine and interfere. Fifty per cent of the beam reflected from the fixed mirror is 

The moving mirror produces an optical path difference between fre t o ms of the interferometer For path d08erences of (n4-l/2)X,lhe i ijeiaras inH] fe de in the case of the transmitted beam  

The optical path difference (OPD) between the beams that travel to the fixed and movable mirror and back to the beamsplitter is called retardation, 8. When the path length on both arms of the interferometer are equal, the position of the moving mirrors is referred to as the position of zero retardation or zero path difference (ZPD). The two beams are perfectly in phase on recombination at the beamsplitter, where the beams interfere constructively and the intensity of the beam passing to the detector is the sum of the intensities of the beams passing to the fixed and movable mirrors. Therefore, all the light from the source reaches the detector at this point and none returns to the source. To understand why no radiation returns to the source at ZPD one has to consider the phases on the beam splitter. 

A beam that is reflected by a mirror at normal incidence undergoes a phase change of r. However, the phase difference between the reflected and the transmitted beams from a beam splitter is tt/2 the beam that is reflected undergoes a phase change of IX jl while the phase of the transmitted beam remains unchanged (Lawson 2000). In this section this statement is proved. 

First consider an ideal, thin and symmetric beam splitter. The relative amphtude and phase shift of the reflected wavefront are r and 5, and the relative amplitude and phase shift of the transmitted wavefront are t and 8, respectively. 

The source emits a beam with amplitude which is split into a reflected complex 

The corresponding intensities of the incident beam (/q) and the emerging beam (Id and /s) are 

The heart of the optical hardware in a FT spectrometer is the interferometer. Nowadays, the most common set-up used is the classic two-beam Michelson interferometer shown schematically in Fig. 5.1. It consists of two mutually perpendicular plane mirrors, a fixed mirror Ml and a movable one M2. A semi-reflecting mirror, the beam splitter, bisects the planes of these two mirrors. A beam emitted by a source S is split in two by the beam splitter. The reflected part of the beam travels to the fixed mirror Ml through the distance L, is reflected there and hits the beam splitter again after the total path length of 2L. The same happens to the transmitted radiation. However, as the mirror M2 is not fixed at the same position L but can be moved very precisely back and forth around L by a distance x, the total path length of the transmitted part is accord- 

IR and Raman Spectroscopy Fundamental Processing. Siegfried Wartewig Copyright 2003 WILEY-VCH Verlag GmbH Co. KGaA, Weinheim ISBN 3-527-30245-X 

In the case of a monochromatic ray, the two bundles of radiation interfere constructively if their optical retardation is a multiple of the wavelength X, i.e. 

The beam, modulated by the movement of the mirror M2, leaves the interferometer and is finally focussed on the detector. The signal actually registered by the detector is thus the radiation intensity of the combined beams as a function of the retardation 5. For a source of monochromatic radiation with frequency v (or wavenumber v = l/X = vie), the intensity at the detector is given by the equation 

As the retardation is a function of time, the interferogram is a function defined in the time domain. If the movable mirror is scanned at a constant velocity v, the retardation is simply 

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The superposition principle is illustrated further with the Michelson interferometer. Light is divided between two arms at a beamsphtter, recombined and the resulting intensity is observed. For a monochromatic source, the on-axis intensity is a superposition of the two recombined beams, and varies cosinu-soidally with the difference in path lengths Az... 

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... 

This interferometric dilatometer consists of a rather simple and small Michelson interferometer, in which the two arms are parallel, and of a 4He cryostat, in which the sample to be measured is hold. The sample is cooled to 4 K, and data are taken during the warm up of the cryostat. The optical path difference between the two arms depends on the sample length hence a variation of the sample length determines an interference signal. The Michelson interferometer consists of a He-Ne stabilized laser (A = 0.6328 xm), two cube corner prisms, a beam splitter, three mirrors and a silicon photodiode detector placed in the focal plane of a 25 mm focal length biconvex lens (see Fig. 13.1). 

Based on the way the interferometer is configured, CCMI sensors can be categorized into two groups, namely the Mach-Zehnder interferometer (MZI) type and the Michelson interferometer (MI) type. The MZI sensor works in transmission mode, i.e., the transmitted interference signal is detected. The MI sensor works in reflection mode, where the light passes the interferometer twice and the reflected interference signal is detected. 

One of these devices that is typically used in infrared spectroscopy is the Michelson interferometer (Fig. 6.22). This device works by splitting the beam into two components perpendicular to each other. Then each beam gets reflected by minors in such a way that the reflected beams recombine again at the beam splitter. In one of these beams a path difference is introduced by moving the mirror on which it reflects. This... 

FT-IR utilizes the Michelson interferometer rather than the grating or prism of the dispersive system. The Michelson interferometer has two mutually perpendicular arms. One arm of the interferometer contains a stationary, plane mirror the other arm contains a moveable mirror. Bisecting the two arms is a beamsplitter which splits the source beam into two equal beams. These two light beams travel their respective paths in the arms of the interferometer and are reflected back to the beam splitter and on to the detector. The two reunited beams will interfere constructively or destructively, depending on their path differences and the wavelengths of the light. When the path lengths in the two arms are the same, all of the frequencies... 

Fig. 7.—The Michelson Interferometer (BS = beam splitter, FM = fixed mirror, and MM = movable mirror).

In Fourier transform spectrometry, the wavelength components of light are not physically separated. Instead, the light is analyzed in the time frame of reference (the time domain) by passing it through a Michelson interferometer. The Michelson interferometer is so constructed that light is separated into two beams by a beamsplitter. One beam strikes a stationary mirror and is reflected back to the beamsplitter. 

Optical System. As shown in Figure 11, the optical section of the instrument consists of a beamsplitter, two optical wedge mirrors and a detector section with associated detector collection mirror. This is basically the Michelson interferometer technique except that the end mirrors have been replaced by optical wedges mirrored on the back side. The two windows are necessary only to maintain an ambient pressure in the Interferometer section, and a vacuum in the detector section. The window on the detector can be replaced with an optical filter if only a selected spectral region is to be investigated. 

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). 

A schematic diagram of a Michelson interferometer, the heart of the FTIR system, is shown in Figure 1 (2). The Michelson interferometer modulates each wavelength in the infrared region at a different frequency in the audio range. 

A Michelson interferometer is generally the device used in an FTIR spectrometer. The Michelson interferometer is composed of two mirrors and a beam splitter positioned at an angle of 45° to the fixed and moving mirrors. 

The Michelson interferometer is used in most FTIR spectrometers. However, some FTIRs use a cube-corner interferometer. 

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. 

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. 

A collimated light beam from the 1R source is directed to the Michelson interferometer where it is divided by the beam splitter. One half of the beam is reflected from a fixed mirror and the other half from a moving mirror. The two light beams recombine after returning from the mirrors and give rise to a reconstructed beam which is optically an interference wave. The interference light beam passes through the sample and is modified by its interaction with the... 

Although standard IR spectrometers are used for studying the amide bands, FTIR spectrometers are more accurate and reliable. FT-IR spectrophotometers are based upon the Michelson interferometer. A typical instrument (Fig. 7.1) comprises an optical bench housing the interferometer, sample, infrared source and detector, coupled to a computer, which controls the spectral scanning, analysis and data processing (for review see Griffiths, 1980). 

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