3. Fourier Transform Spectroscopy

Before attempting to measure the line shape of a Fourier spectrometer or investigate the effect that variations in it may have on vertical profile retrieval, it is necessary to have some basic understanding of the principles and practicalities of Fourier transform infrared spectroscopy. Various aspects of this shall now be discussed (following Chamberlain [1979]) with reference to the Bruker IFS 120M spectrometer where appropriate.

3.1 Basic Principles of Fourier Transform Spectroscopy

According to Chamberlain, Fourier transform spectroscopy is

"...the technique whereby a spectrum is determined by the explicit application of a Fourier transformation to the output of an optical apparatus - generally a two-beam interferometer."

A frequently used optical arrangement is that of the Michelson interferometer.

Figure 3-1. Schematic representation of a Michelson interferometer (section 3.1).

The technique is to split the collimated, incident beam (having power B₀) into two equal parts, introduce into one of the parts a known, variable phase delay relative to the other; recombine the two beams and then measure the power of the recombined beam as a function of the phase delay introduced. In the case of the Michelson interferometer, this is accomplished by moving one of the mirrors to introduce a difference in the optical path taken by each part of the split beam.

As the mirror moves, the detector at the output of the optics will record a series of bright and dark fringes, the temporal spacing of which is related to the difference in optical path length and hence, to the spatial position of the moving mirror. This spatial fringe pattern, which is the measured quantity in Fourier spectroscopy, is known as the interferogram. Since it is a record of power as a function of frequency or wavelength that is the desired quantity, rather than power as a function of phase delay, the interferogram must be subjected to a Fourier transform as shall be shown below.

Consider a simple spectrometer, such as is illustrated in figure 3-1, being used to measure a perfectly monochromatic source. We assume, for current purposes, that it is perfect in every respect including that the two partial beams are identical, each having a power of ¹/₂B₀, and that the moving mirror has infinite travel; some measure of realism shall be introduced later. We shall consider the case of a 'single sided' measurement, that is, one in which the interferogram is recorded for path differences on one side of the zero path difference point only. While it would be possible, given our assumption of perfection, to record a fully infinite interferogram, this is not necessary since the interference function is essentially even about the point of zero path difference.

As illustrated, one of the mirrors may be moved so as to change the length of the optical path in one arm, and hence introduce a phase delay into one of the partial beams relative to the other. If the path difference incurred between the two partial beams is x, this phase delay will be given by 2ps₀x, where s₀ is the frequency¹ of the radiation. The two partial beams are recombined at the beam splitter and the interference pattern that is recorded at the detector is governed by the aforementioned phase delay and may be represented thus (we shall consider only that radiation which reaches the detector):

I(x) = 1/2(B0 + B0.cos2ps0x).

(3.1)

I(x) is referred to as the interference function and has the familiar form of cosine-squared fringes:

I(x) = B0.cos2ps0x.

(3.2)

We see that the interference function has a maximum value, I_max, of B₀ and a minimum, I_min, of 0 for all values of x, as a result of our insistence that the incoming radiation be perfectly monochromatic, and that the fringes have a period of 1/s.

Suppose now that the source has a finite, albeit small, bandwidth ds. The fringes will retain their basic cosine nature but as the optical path difference increases the different frequencies (giving rise to fringes with differing periods) will interfere in an increasingly destructive manner such that the interference fluctuations first die away and I(x) tends to ¹/₂B₀ at approximately x = 1/ds; this is known as the extinction path difference.

For most purposes, a broad band source is required. At x=0 (zero path difference) all components add constructively, giving rise to a 'white light' fringe (so called after its equivalent in visible spectroscopy), but for greater values of x the maximum intensity of the fringes falls off very rapidly, almost to zero. The interference function for a broad band source may be found by replacing B₀ in equation (3.1) with B(s) and summing from zero to infinity with respect to frequency:

(3.3)

It can be seen that the interference function has two distinct parts; the variable term of the interference function is known as the interferogram, and shall henceforth be considered in isolation.

We may represent the interferogram thus:

(3.4)

This equation has the form of a cosine Fourier integral and relates the spectrum of the source to the measured interferogram. It follows that if the interferogram is subjected to a Fourier transformation we find that

(3.5)

which describes the original spectrum in terms of the measured interferogram.

It will be noted that since the foregoing integrals are semi-infinite there is the implicit suggestion that we require a knowledge of the interferogram in infinite detail over a semi-infinite range of optical path differences and, indeed, if the final spectrum is to be a perfect measurement of the source in every respect, this is so. In practice, however, this is not the case; we do not require infinitely high resolution and we can obtain satisfactory results by sampling the interferogram at discrete intervals over a finite length. The interferogram will, in any case, be truncated by physical constraints on the maximum optical path difference of the instrument, and further degraded at greater path differences by the presence of skew rays in the interferometer; these effects shall be considered in section 3.4.3. In addition to these, errors may be introduced by such factors as dispersion in the optics or analogue electronics of the instrument, non-linearities in the detectors and amplifiers, or if the true position of the white light fringe is not properly ascertained. These, with the exception of the last (section 3.5), shall not be explicitly considered as their understanding is not essential to the work described herein, this being concerned with the measurement of the effects of instrumental imperfections rather than their correction.

3.2 The Optical Arrangement of the Bruker IFS 120M

The optical configuration of the Bruker IFS 120M is shown in figure 3-2. The instrument is based on a Michelson interferometer with a maximum optical path difference of 257 cm. The collimated input beam is incident on an off-axis parabolic mirror that focuses the radiation on to the chosen field stop which can be any one of twelve apertures drilled in a disk mounted on stepper motor. The beam then expands onto a collimating paraboloid, identical to the input mirror, and enters the actual interferometer. The beam splitter directs half the radiation into the reference arm of the instrument to a fixed cube corner mirror and half into the scan arm to the moving mirror. This mirror, another cube corner, is mounted on a carriage drawn along the arm by a wire loop. A piezo electric stack is located between the moving mirror and its carriage to facilitate fine and rapid of adjustment of the position of the mirror.

In order to cover the full range of the 120M (approximately 750 cm^-1 to 8000 cm^-1), two beam splitters and two detectors are required; their properties are summarised in table 3-1 (below):

3.3 Relevant Aspects of Data Acquisition in the Bruker IFS 120M

From the discussion in section 3.1 it is obviously of the utmost importance that the optical path difference giving rise to each of the fringes in the interferogram be recorded accurately, and so any usable system must have the facility to measure the position of the moving mirror at each sample point. In the case of the 120M (and other systems) this is accomplished by means of a 633 nm helium:neon 'counting' laser.

The laser beam is introduced into the interferometer parallel to, and at the edge of, the infrared beam². It therefore follows the same path as the infrared radiation, and one part of the split beam will acquire the same optical path difference. After recombination, the red beam is extracted and sent to a separate detector which monitors the interferogram of this monochromatic source. Since this laser itself is stabilised to 3.3x10^-5 cm^-1/min (Jewkes (private communication)), counting the fringes of its interferogram provides an accurate measure of optical path difference and a means of controlling the speed of the moving mirror, and timing samples of the infrared interferogram.

Figure 3-2 Schematic drawing of the optical layout of the Bruker IFS 120M (section 3.2).

Rather than taking the minimum number of samples required by the sampling theorem, the Bruker samples the interferogram at every second amplitude zero-crossing of the interferogram of the counting laser (putting an upper frequency limit of 7900 cm^-1 on measurements). If the maximum required frequency is lower than 7900 cm^-1, the recorded interferogram is digitally filtered and resampled before being transformed to produce the desired spectrum.

Summary
Acknowlegements
Contents
Chapters: 1, 2, 3, 4, 5, 6, 7, 8
Appendices
References