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Anne P. Thorne Blackett Laboratory Imperial College London SW7 282, U.K.
Sixteen years ago Horlick and Yuen (I) contributed an article to these pages on “Atomic Spectrochemical Measurements with a Fourier Transform Spectrometer.” Some years later, again in these pages, Faires (2)wrote on “Fourier Transforms for Analytical Atomic Spectroscopy.” Progress in Fourier transform spectrometry (FTS) has accelerated over the last five years, and the time seems right for a new assessment of its role in atomic spectrochemistry. FT-IR has, of course, been a wellestablished technique for much longer than 16 years, but it is FT-UV that is relevant to analytical atomic spectroscopy-both in the emission and the absorption modes-and especially in the wavelength region between 250 and 190 nm where the sensitive lines of most elements are found. Two questions therefore arise. First, what are the difficulties and constraints of extending FTS into the UV, and to what extent have they been overcome? Second, if the technical problems can be or have been solved, does FTS offer significant advantages over dispersive grating 0003-2700/9 1/0363-057A/$02.50/0 @ 1991 American Chemical Society
spectrometry for analytical atomic spectroscopy? Following a summary of the essential features of the technique, I will try to answer these two questions (in reverse order) before going on to discuss my view of the proper role of FTS in atomic spectrometry a t present. Finally, I offer the customary hostage to fortune by speculating on the future. FTS: A quick guide This section should be regarded as a sketch map to help those unfamiliar with FTS to reach a vantage point from which they can survey both the usefulness of the technique and the difficulties of the road ahead to shorter wave-
lengths. A fuller treatment is given in standard texts (3-5), and a more specific background for the type of highresolution atomic spectroscopy discussed here can be found in References 6 and 7. We start with the Michelson interferometer, in which a collimated light beam is divided a t a beam splitter into two coherent beams of equal amplitude that are incident normally on two plane mirrors. The reflected beams recombine coherently at the beam splitter to give circular interference fringes a t infinity, focused by a lens at the plane of the detector. For monochromatic light of wavelength A0 and intensity B(Ao), the intensity at the center of the fringe
Figure 1. (a) Symmetric interferogramand (b) its cosine transform, consisting of the true spectrum (shaded black in this figure and in Figures 3 and 4) and its mirror image. ANALYTICAL CHEMISTRY, VOL. 63, NO. 2, JANUARY 15, 1991
57A
INS'TRUMEN7A7lON pattern as a function of the optical path difference x between the two beams is given by the familiar twobeam interference relation
I , = B ( h ) [ l + cos (27rx/X,)] = B(ao)(l
+ cos 27raox)
(1)
where the wavenumber a is defined by a = 1/X = v/c, normally measured in cm-l (Y is the frequency of light in s-l and c is the speed of light in cm s-l). If x is changed by scanning one of the mirrors, the recorded intensity (the interferogram) is a cosine of spatial frequency 60. Its temporal frequency is given by fo = uao where u is the rate of change of optical path, or twice the mirror speed. A scan speed of 1 mm/s puts f into the audiofrequency range, and the interferometer can be regarded as a device for converting optical frequencies to audiofrequencies:f = U Q = ( U / C ) Y . If the source contains more than one frequency, the detector sees a superposition of such cosines.
Io(x)=
[B(a)(l + cos 27rax)do
(2)
Subtracting the constant intensity JtB(a)da corresponding to the mean value of the interferogram ( I ( x ) ) leaves Equation 3.
lom
B ( a ) cos (27rax)da (3)
The right-hand side of Equation 3 contains all the spectral information in the source and is the cosine Fourier transform of the source distribution B ( a ) . The latter can therefore be recovered by the inverse Fourier transform
B(a)=
[I ( x ) cos (27rax)dx
interferogram is recorded to a finite path difference L rather than to infinity. Third, the interferogram is actually recorded by sampling it a t discrete intervals Ax. Equation 4 thus becomes N
B(a)=
shape recorded from an ideal monochromatic input is sinc 2(a - a& (Figure 2). The first zero of this function is at (a - a,) = f 1/2L; this value defines the resolution of the instrument: (7)
1
I(pAx)e-2"i"pA" (5)
p=-N
where NAx = L and p is the index number of the sample. The consequence of the first modification is that the recovered spectrum is complex, and a phase correction has to be applied to rotate it back into the real plane. The effect of the finite path difference, which effectively multiplies the infinite interferogram by a top-hat (boxcar) function of width 2L, is to convolute the spectrum with the Fourier transform of the top-hat function.
2L s i n c 2 d
(6)
This sinc function is the instrument function of the spectrometer: The line-
The third modification, the discrete sampling, has the effect of replicating the computed spectrum a t wavenumber intervals of l/Ax. To see why, consider Equation 5. If a is replaced by a f l/Ax, the argument of the exponential is increased by 27rip, an integral number of 27r, which leaves it unchanged. The replication also applies to the negative imageB(-a), and it can be seen from Figure 3a that the negative part of the first replication will overlap the positive part of the original unless l/Ax > 2um, where a, is the highest frequency in the source. The condition for avoiding overlap is Ax I 1/20,, or Ax IXJ2, which is the Nyquist sampling theorem. The replication is known as aliasing. Figure 3a can be interpreted as the unfolding of a stack of Z-fold paper (Figure 3b), each sheet of which has width Aa where
(4)
Actually the Fourier transform not only reproduces B ( a )but also necessarily adds a mirror image B(-a) at negative frequencies (Figure 1). This can easily be verified by direct integration for the simple case of the monochromatic source B(ao), and it is understandable because the relation cos 27rux = cos 27r(-a)x ensures that B ( a ) and B ( - a ) produce identical interferograms. Negative frequencies are unreal, but the mirror image enters into the consideration of aliasing below. There are three important modifications to these simple relations. First, the interferogram is never totally symmetric about x = 0, and to recover the full spectral information it is necessary to take the complex rather than the cosine Fourier transform. Second, the 58 A
Flgure 2. (a) Top-hat truncation function of the interferogram extending from 0 to f~ and (b) its transform, the instrument function sinc 2aL sin PaaLl(2aaL).
Figure 3. (a) Spectral band of maximum wavenumber a,,,,together with its negative image, replicated at wavenumber intervals I/Ax and (b) representation of this replication by folding the spectrum at intervals of the free spectral range Aa = 112A.x.
ANALYTICAL CHEMISTRY, VOL. 63, NO. 2, JANUARY 15, 1991
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INSTRUMENTATION AU = 1/2Ax (cm-l) In Equation 8, Au is known as the alias width and is equivalent to the free spectral range. Any real information on, say, page 3 (third alias) of the stack will be indistinguishable from the replica on that page of the spectrum on page 1 (first alias). Suppose, however, that there is actually no real signal at low frequencies, perhaps because the detector is insensitive to long wavelengths. As illustrated in Figure 4, it is then permissible to undersample the interferogram, knowing that the information on page 3 really belongs there and cannot have come from page 1.One must, of course, be sure that pages 4, 5 . . . are also blank, and it is also necessary to use a bandpass filter to avoid folding in the noise from all these other pages. The quantity N introduced in Equation 5 is a number of some significance. As defined there ( N = L/Ax), it is evidently the number of independent points in the interferogram (because in principle 0 to -L contains the same information as 0 to +L). Using Equations 7 and 8 gives N = Au/6u, so N is also the number of resolution elements (independent spectral points) in the free spectral range. If the maximum wavenumber um comes at the end of the nth alias, n N is numerically equal to the resolving power R
N = L/Ax = u/6u
(9)
and nN = nAu/6u = u,/6u = R
(10)
Grating versus FT spectrometry
Equation 10 brings out a similarity to grating spectrometry: n N is the theoretical resolving power of a grating of N rulings used in the nth order. This re-
sult is not too surprising, for N is simply the number of samples taken by the grating of the incident wavefront. Why build an interferometer to go through the motions of ruling a grating every time one wants a spectrum, instead of buying a grating ruled by someone else in the first place? Of course it is convenient to be able to make a free choice of N, but there must be better reasons. The reasons normally applicable in the IR are the multiplex (Fellgett’s) and throughput (Jacquinot’s) advantages, both of which significantly improve the signal-to-noise ratio (S/N). The multiplex advantage is a consequence of IR detector noise limitations and does not apply in the UV or visible regions (nor, usually, in the near-IR) where noise is not detector limited. I t can sometimes become a disadvantage, as we shall see. The throughput advantage is the consequence of division of amplitude (by the beam splitter) rather than division of wavefront (by the grating). The interferometer can use a circular entrance aperture with an area some 2 orders of magnitude greater than the slit area of a grating spectrometer of the same resolution. When photon shot noise is the dominant source of noise, so that noise is equal to the square root of the signal, this larger signal can lead to an improvement in S/N of -1 order of magnitude. Less well known is the wavenumber (Connes’) advantage. The wavenumbers in the transform are derived directly from the sampling intervals, which in turn are determined from the interference fringes of a He-Ne laser following the same path as the signal beam. This results in an accurately linear wavenumber scale with a reproducibility that depends only on the stability of the laser. The alignment of the laser can introduce a scaling factor of the order of 1/R,so that reference stan-
Figure 4. A band-limited spectrum at high frequency. (a) The interferogram is undersampled so that the band falls in the third alias (Ax = 3/20,,,).(b) The folded spectrum shows that there is no ambiguity provided that no real spectral information exists in the regions u < 2Aa and a 3Aa.
>
60A
ANALYTICAL CHEMISTRY, VOL. 63, NO. 2, JANUARY 15, 1991
dards are still required for absolute wavelengths more accurate than, say, l:lO5. In principle, however, only one such reference is required to determine the scaling factor, in contrast to the set of standards spaced out over the wavelength range that is required to calibrate a grating spectrometer. The high resolution attainable has already been mentioned. Simply by extending the scan length, the width of the instrumental function can be reduced to the point where it has a negligible effect on the observed line profile. In most of the sources used in atomic spectroscopy the true linewidths are determined by Doppler broadening, and in practice u/6uo (or X/GXo) is usually in the range lo5 (light element at -6000 K) to lo6 (heavy element at room temperature). Allowing two to three resolution points per Doppler width, this calls for a resolving power R from 200000 to 2 million. The lower end of this range would require a large grating spectrometer or an echelle with a predisperser, whereas the upper end is simply unattainable with any form of grating. It can easily be shown (8) that when the instrumental width is reduced to below half the Doppler width, the resultant convolved profile is almost indistinguishable from a pure Gaussian, and the undesirable ringing shown in Figure 2 is damped down to less than 0,001of the peak intensity. As a matter of fact it is not necessary to live with the sinc function even if the line is not fully resolved: FTS offers the ability to change the instrument profile by manipulating (mathematically) the way in which the interferogram is truncated at the ends of the scan, and it is possible to suppress the ringing at the expense of an increase in linewidth, a process known as apodising. The last important feature of FTS is that it necessarily gives a complete spectral record over the entire spectral bandpass seen by the detector, rather like a vastly superior photographic plate. Obviously this can be a mixed blessing. If one is interested only in half a dozen spectral lines, nearly all the information is redundant; but in a number of applications-and I will give some examples below-this global record is invaluable. Problems and solutions for the UV The loss of the multiplex advantage and the reduced benefit of the throughput advantage were for very many years taken as adequate reasons for not venturing to shorter wavelengths. A stronger deterrent was the tightening of the optical and mechanical toler-
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Figure 5. Optical arrangement of the Imperial College interferometer. A is the entrance aperture; B is the beam splitter; C,and C2 are the catseye retroreflectors: D, and D2are the detectors: MI is the collimating mirror: M2and M3 are the focusing mirrors: M4, M5. and M6 are folding mirrors: L is the controlling laser; and LD is the laser detector.
ances required. The normal requirement for two-beam interferometry is a maximum discrepancy of X/4 on the recombining wavefronts. This puts an upper limit of kX/8 on each of the optical surfaces involved, or f 2 5 nm a t 200 nm. The requirement that the scanning mirror stay aligned to this tolerance during the scan translates to an angular tolerance (for a 20-mm beam) of -0.5 arc second. Most instruments built for the visible and the near-UV have tackled the angular problem by replacing the plane mirrors by retroreflectors, in the form of catseyes. The incoming and outgoing beams are then always parallel, but tilting the catseye introduces a shear into the outgoing beam. The allowable shear tolerance of about 10 pm at 200 nm sets a limit of some four arc seconds on the tilt. This same shear tolerance applies to the guidance system, which constrains the motion of the catseye along the optic axis of the interferometer. The one advantage of going to shorter wavelengths is that the scan distance becomes shorter: A resolving power of 2 million a t 200 nm requires a maximum optical path difference of 20 cm, or a physical displacement of A10 cm from zero path difference. Finally in this list come the requirements of the sampling system. To work unaliased to below 200 nm requires a sampling interval of not more than 90 nm, or one-seventh of a He-Ne laser fringe (Equation 8). In practice it is frequently possible and desirable to work in the second or a higher alias (using, e.g., a solar-blind detector insensitive above 300 nm), but it is still necessary to subdivide the laser fringes rather than to use whole fringes or multiples of them as is done in the IR. Random errors in the sampling steps ap62 A
pear as noise in the transformed spectrum, and in practice sampling steps must be accurate to about one part in 1000 (about 0.1 nm) if one wants to keep the sampling noise a t a negligible level. In addition to these formidable optical and mechanical constraints, the handling of the data is not a trivial problem. The number of data points is, as we have seen, numerically equal to the resolving power, and there is little incentive for attempting UV-FTS if one cannot exploit the full useful resolving power of a million or so. (Such high resolving powers are not normally required, or indeed attainable, in the IR.) However, computer technology has advanced so much in recent years that data acquisition, manipulation, and storage can no longer be regarded as a limitation on UV-FTS; the time taken for a million-point FFT (fast Fourier transform), for example, has dropped over the last decade from a few hours on a minicomputer to a few minutes on a PC, and a spectrum that once filled all of a demountable 16-in. hard disk now resides happily in a small fraction of the PC disk. The feasibility of high-resolution FTS in the visible and near-UV was first demonstrated on the large, highperformance instruments at the National Solar Observatory (NSO) in Kitt Peak, AZ, and at Orsay, France; References 9 and 10 describe these instruments, and Reference 10 gives details about the earlier Orsay instruments. At Imperial College we set out to design and build a compact high-resolution FTS specifically for the UV down to 180 nm (11);this has been operating, with more or less continuous improvements, since 1986. Figure 5 shows the optical arrangement: It is an f/25 spectrometer o ~ l y1.5 m long with maxi-
ANALYTICAL CHEMISTRY, VOL. 63, NO. 2, JANUARY 15, 1991
mum resolution 0.025 cm-l (resolving power 2 million a t 200 nm). Over about the same period a very large FT spectrometer, modeled on the NSO instrument but designed for high resolution from 20 pm to 200 nm, was built by Los Alamos National Laboratory (12). This instrument has in fact been little, if a t all, used a t its short wavelength end. In the commercial sphere, Bomem and Bruker have extended the ranges of their IR-vis instruments to shorter wavelengths, and Chelsea Instruments makes a commercial version of the Imperial College FT spectrometer. UV-FTS is therefore feasible and practicable, but not yet readily available. Given a spectrometer that can resolve true lineshapes and measure enormous numbers of wavelengths with unprecedented accuracy over a wide spectral range, to what practical problems should it be applied? UV-FTS for analytical chemistry For the physicist there are several obvious answers to this question: hyperfine structure and isotope shifts in atomic spectra, rotational structure of electronic bands in molecular spectra, generation of accurate secondary wavelength standards, high-quality wavelength measurements for term analysis, relative intensity measurements for branching ratios and hence transition probabilities, lineshape investigations for study of collisional processes, and so on. For the analytical chemist the answers are less obvious. For atomic emission spectroscopy the excellent resolution and wavelength reproducibility are clearly advantageous, particularly for the line-rich spectra of the rareearth and actinide elements, where there are often severe spectral interferences. The wide spectral coverage allows a choice of analytical lines and a choice that can be made after the data have been recorded. There is, however, a well-known unfavorable feature of FTS that has been discussed in a number of papers (1,2,8, 13, 14). It is commonly known as the “multiplex disadvantage,” but i t should really be regarded as a “Fourier disadvantage,” which can be potentially enhanced by the true multiplex disadvantage. The origin of the problem is that when random noise is added to the interferogram on the left-hand side of Equation 3, the transform of the noise is added to the spectrum on the righthand side of the equation. The Fourier transform of random noise is “white noise,” which has the same mean intensity at all frequencies. The noise from every line in the spectral band contrib-
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