I am sure that many chemists who, having decided that they find spectroscopy an interesting field in which to work, have shared my experience when attending a lecture to learn about the Fourier transform infrared spectrophotometer. The speaker starts with a description of the Michelson interfer- ometer, which seems fairly understandable when restricted to a monochromatic beam, but much less so when the beam is made polychromatic. Then the complication of sample absorption is added, followed by the concept of the Fourier transform. The majority of the audience is thoroughly lost by then, hut is tremendously relieved to learn that a computer will take care of all these difficult calculations and decides that, therefore, it is really not imperative to understand them. The mathematical part of most papers on the theory of F T spectrophotometers is generally equally difficult to follow.

Of course, this is not a very satisfactory state of affairs. Undoubtedly, the chemist who has some understanding of what the computer will he doing would he able to make much better use of the instrument if he or she should have the op- portunity. The following account of the explanation I suc- ceeded in developing for myself will, I hope, he helpful to some who have had the same difficulties. I t is based in part on a paper by Frank and Gorke? Those readers interested in such other aspects of the subject as the relative advantages and disadvantages of F T spectrophotometers over dispersive in- struments and descrintions of commercial FT instruments are

Frederick C. Strong, Il l Faculdade de Engenharia de Alimentos e

Agrlcola Universidade Estadual de Campinas

Campinas, S.P., Brasil

rcft:rrtvl to the ~ L S V U S S ~ U ~ S hy I.ow in (::~len Kwing's culumn on 'Tl~pics in ('hemicnl lnsrr~~mentatit~n" in t h i s Journul

How the Fourier Transform Infrared Spectrophotometer Works

Beer's Law Familiarity with the notation for the absorption process and

Beer's law is an essential starting poink3 Let monochromatic radiation (e.g., from a laser) pass through a blank and reach a detector with radiant power PO,. When passed through a sample in a cell that is identical to that for the blank and containing an equal amount of solvent in the beam, let its radiant power, reduced hy absorption, be P,. By definition, transmittance ( T ) and absorbance (A) have the relation:

T = PJPk = lo-" (1)

If, instead of a laser, a polychromatic beam, a very narrow entrance slit, a monochromator, and a very narrow exit slit are employed, the beam will he approximately monochromatic and approximately infinitesimal in radiant powec4

T = dPldPO I lo-" (2)

Since we will he considering the infrared region, we will use wavenumher (a), rather than wavelength (A). We recognize the dispersion of the radiation source with respect to wave- number as its emission spectrum. Dispersion is a deriva- tive,"

1 Frank, W., and GBrke, K., Optik, 45,381 (1976). "ow, M. J. D., d. CHEM. EDUC., 47 (3), A163; (4),A255; (5),A349

119701. , ~ . .~ :'Strung, F. C., 111, Anal. Chem., 24,338,2013 (1952).

Hememher, though, that an FT spectrophotometer does not contain a monochromator.

F g.m 1 . Oplm (I agams of me lhrm modes of operation of a Farrim wanshxm inlured rpeclropnolorneter (Wil ey Model 3185)

so the infinitesimal spectral bandwidth, do, of the exit slit will pass

d P = P ? d c (4)

through the blank and

dP=Pbda (5)

through the sample. In stating Beer's law, we can emphasize that absorptivity

is a function of o by writing a(o):

Volume 56, Number 10, October 1979 1 681

Figure 2. lnterfwograrn of a l a m beam in an FT infrared spectrophotmter Figure 3 Typical aooperoion cuwe far the reference beam of an FT lnharea spactropholometer v lh scannmg morror a1 posntlon x

Periodiciiy of the Equation P,,, = %%(I + cos2rxlA

%PA1 + x xIX 2rx IA eos2rxIA 1 + cas2rx/X cos2rxIX)

0 0 0 1 2 p", A12 112 li -1 o 0

X 1 2 a 1 2 p i 3Xl2 312 3 r -1 0 0

A = o(o)bc (6)

and

T = l o - d c l b c (7)

FT Instrument Construction: interferometer and Laser There are a number of commercial F T infrared spectro-

photometers on the markeL2 Figure 1 shows a set of optical diagrams for the Model 318s Spectrophotameter of the WiUey ~ o i ~ o r a t i o n . They are especially convenient for our discus- sion. The diagrams show: (top) the He-Ne laser measuring beam (middle) and sample heam, and (bottom) the reference beam. The most important component of the instrument is the Michelson interferometer, repeated in each diagram in the upper left square. It is best understood by considering the path of the He-Ne laser beam (top diagram).

The beam leaves the laser on the left, is reflected from three mirrors, and passes through a hole in the diagonal mirror. I t then strikes the beam splitter (a partially transmitting- partially reflecting material). The part transmitted is reflected from the upper fixed mirror and from the upper side of the beam splitter to the detector. The other part of the initial beam is reflected from the beam splitter to the scanning mirror on the left, then reflected hack, passes through the beam splitter, and reaches the detector through a hole in the colli- mator.

The two reunited beams will interfere constructively or destructively, depending on the relation between their path difference (x) and the wavelength, as in the case of a diffrac- tion grating. For a monochromatic heam, the radiant power (P,) a t the detector can he represented by

P, = llzPO(l+ cos2rxIX) (8)

This is a periodic function that varies between Pi and 0 (see the table). I t is plotted in Figure 2. Since A for the laser is known, the interferogram (Fig. 2) can be used to measure x, the distance of travel (or position of the scanning mirror). From here on, it will he convenient to replace 1/A with u.

The lnterferometer and the Reference Beam

Consider next the reference beam (bottom of Fig. 1). The radiation leaving the source (upper right) is polycbromatic, so we will choose an infinitesimal bandwidth, do, of the spectrum that is monochromatic:

dP(r)O = P: da (9)

After passage through the interferometer, the radiation will he modified by interference, depending on the value of x:

dP(x)O = 'I2P:(1 + codrxn) da (10)

The meaning of eqn. (10) can be seen hy considering Figure 3. The upper, dashed curve is P:, the dispersion curve or spectrum of the source in wavenumbers. I t is multiplied by the factor %(I + cos2rxo), where x has a specific, fixed value. Let n be an integer. For each value of u such that

and

P(x)b = 'lzP'(1 + cadrn) (12)

= '/&'(1+ 1) = P: (13) giving the upper points on the curve. For

.xu = (n + ll~) (14)

P(x)b = '/zP:[l + codr(n + lI2)] (15) = '/8:(1 - 1) = 0 (16)

giving points along the abscissa axis a t (n + l/z)lx. In between these points, the curve will he approximately as shown. The curve for Pb is the enuelope for the saw-toothed curve. Of course, as x changes, the points where P(x)b equals zero will change.

We can obtain the radiant power of the polychromatic beam of the source that reaches the detector by integrating with respect to u over the emission range, UI to 02, keeping x con- stant:

P(X)O = + xlq~:(l + cos2rxa) do (17)

Integrating term by term,

P(x)O = 1 112~: do + 1:~: COS2rXo da (18) 2 The first integral is one half the total radiant power (PO) of the source. When x is 0, the cosine is 1 and

P(0)" PO12 + P I 2 = P (19) Since i t is only a t x = 0 that there will be constructive inter- ference for all values of u, it is called the "white light position" or the "origin." We can then write

(20)

Figure 4 is an interferogram of the reference beam, i.e., an experimental recording of eqn. (20).

I 0 x

Figure 4. lnterferogram of the reference beam

682 1 Journal of Chemical Education

Figure 5. Typical dispersion curve for the sample beam of an FT inhared spectrophotometer with scanning mirror at position x

Figure 6. lnterferogram of a sample of Daw Saran Wrap

The Interferometer and the Sample Beam

Consider, finally, the sample beam (Fig. 1, middle section). Here, Pbn is decreased by the factor ~O-"("*C, the transmit- tance of the sample, and dP(x) 'J becomes dP(x). Introducing the factor into eqn. (lo),

d P ( x ) = %P: 10-"'"Ibc(l + cos2?rxo) d a (21)

The result is illustrated by the lowest curve in Figure 5. The area beneath it will be the integral

P ( x ) = i S o l p : 10-a(albC(l + cos2s.z) d a 0 ,

(22)

Integrating term by term and evaluating the result a t x = 0, as in eqns. (17) through (20), gives

1 P ( r ) = - P(0 ) + - P: 10-'("bC cos2axo d a (23)

2 ; 1," Figure 6 is an experimental recording of eqn. (23) for a sample of Dow Saran Wrap.

The Fourier Transform

After the detector system measures P(x) and P(x)O as functions of x , the Fourier transform must he carried out to yield the absorption spectrum. The principle of the transform is that the two variables, u and x , and their two functions can he interchanged. Transposing 'Id'(0) in eqns. (20) and (23) and interchanging variables, they become

P: = i 1:' [P(x)n - 1/2P(0)o] C O S ~ T C X d z (24)

and

p: 10-olclbc = - J:' [ P ( z ) - 1/2P(0)] eos2na.x dx (25)

where XI and xz are the limits of travel of the interferometer mirror. Figures 7 and 8 are single-beam spectra transformed by computer and plotted. They represent eqns. (24) and (25), respectively. Note the absorptions of COz and water vapor in each.

Dividing eqn. (25) by eqn. (24) gives the absorption spec- trum of the sample:

Figure 7. Single-beam spectrum obtained from the Fourier transform of the reference beam interferogram in Figure 4

Figure 8. Slngle-beam absorption spectrum of Dow Saran Wrap obtained from the Fourier transform of the interferogram in Figure 6

Figure 9. Doublpbeam absorption spectrum of Dow Saran Wrap obtained from the ratio of the spectra in Figures 7 and 8

Volume 56. Number 10, October 1979 1 683

l: [P(x) - %P(O)] cos2aar dr specializing in optics would undoubtedly feel i t is all ele- 1 0 - 0 1 d b ~ = mentary and unnecessary. However, the author hopes it will

(26' serve to provide some chemists with a basic understanding of [ P ( X ) ~ - %P(O)"] COS~TCX d l the principles of a Fourier transform infrared spectropho-

Experimentally, the spectrum is obtained by calculating the tometer that they did have before.

point-by-point ratio of Figures 7 and 8, yielding Figure 9.

Final Comment Acknowledgment The author recognizes that the above presentation omits The interferograms and spectra shown were very kindly

many sophisticated points that are required to produce high provided by R. R. Willey of the Willey Corporation, Mel- quality spectra. A theoretical mathematician would probably bourne, Florida, and were recorded on the Model 318.5 F T be horrified by the loose use made of calculus, and a physicist Infrared Spectrophotometer.

684 / Journal of Chemical Education