Fourier Transform Spectrometry in the Ultraviolet
INSTRUMENTATION Anne P. Thorne Blackett Laboratory Imperial College London SW7 2BZ, U.K.
Sixteen years ago Horlick and Yuen (1) contributed an article to these pages on "Atomic Spectrochemical M e a s u r e ments with a Fourier Transform Spec trometer." Some years later, again in these pages, Faires (2) wrote on "Fouri er Transforms for Analytical Atomic S p e c t r o s c o p y . " Progress in F o u r i e r transform spectrometry (FTS) has ac celerated over the last five years, and the time seems right for a new assess ment of its role in atomic spectrochem istry. FT-IR has, of course, been a wellestablished technique for much longer than 16 years, b u t it is F T - U V t h a t is relevant to analytical atomic spectros copy—both in the emission and the ab sorption modes—and especially in the wavelength region between 250 a n d 190 n m where t h e sensitive lines of most elements are found. Two ques tions therefore arise. First, what are the difficulties and constraints of extend ing F T S into the UV, and to what ex tent have they been overcome? Second, if the technical problems can be or have been solved, does F T S offer significant a d v a n t a g e s over dispersive g r a t i n g 0003-2700/91/0363-057A/$02.50/0 © 1991 American Chemical Society
s p e c t r o m e t r y for analytical a t o m i c 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 t o discuss my view of the proper role of F T S in a t o m i c s p e c t r o m e t r y a t present. Finally, I offer the customary hostage t o fortune by speculating on the future. FTS: A quick guide
This section should be regarded as a sketch m a p t o help those unfamiliar with F T S to reach a vantage point from which they can survey both the useful ness of the technique and the difficul ties of the road ahead to shorter wave
lengths. A fuller t r e a t m e n t is given in standard texts (3-5), and a more spe cific background for the type of highresolution atomic spectroscopy dis cussed here can be found in References 6 and 7. We start with the Michelson inter ferometer, in which a collimated light beam is divided at a beam splitter into two coherent beams of equal amplitude t h a t are incident normally on two plane mirrors. T h e reflected beams recombine coherently at the beam splitter to give circular interference fringes a t in finity, focused by a lens at the plane of the detector. For monochromatic light of wavelength λ() and intensity β ( λ 0 ) , the intensity at the center of the fringe
Figure 1. (a) Symmetric interferogram and (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 · 57 A
INSTRUMENTATION pattern as a function of the optical path difference χ between the two beams is given by the familiar twobeam interference relation 70 = β(λ 0 )[1 + cos (2πχ/\)]
=
B(a 0 )(l + cos27TCT0x)
(1)
where the wavenumber σ is defined by σ = 1/λ = vie, normally measured in cm - 1 (y is the frequency of light in s _ 1 and c is the speed of light in cm s - 1 ). If χ is changed by scanning one of the mir rors, the recorded intensity (the inter ferogram) is a cosine of spatial frequen cy σο. Its temporal frequency is given by /ο = νσο where υ is the rate of change of optical path, or twice the mirror speed. A scan speed of 1 mm/s puts / into the audiofrequency range, and the interferometer can be regarded as a de vice for converting optical frequencies to audiofrequencies: f = υσ = (v/c)v. If the source contains more than one frequency, the detector sees a super position of such cosines. IQ(x) = \ Β(σ)(1 + cos27Tffx)dff Jo
(2)
Subtracting the constant intensity ί^Β(σ)άσ corresponding to the mean value of the interferogram (I(x)) leaves Equation 3.
/(x) = /„(*)-(/(*)> = [ Β(σ) cos {2πσχ)άσ
(3)
Figure 2. (a) Top-hat truncation function of the interferogram extending from 0 to ±L and (b) its transform, the instrument function sine 2aL = sin 2πσυ(2πσΙ.). interferogram is recorded to a finite path difference L rather than to infin ity. Third, the interferogram is actually recorded by sampling it at discrete in tervals Ax. Equation 4 thus becomes
δσ = l^Licm" 1 )
Ν
Β(σ) = Υ
/(ρΔχ)β- 2 , Γ , ν ί , Δ ι
where Ν Αχ = L and p is the index num ber of the sample. The consequence of the first modifi cation 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 dif ference, which effectively multiplies the infinite interferogram by a top-hat (boxcar) function of width 2L, is to con volute the spectrum with the Fourier transform of the top-hat function.
I(x) cos (2πσχ)άχ (4) Jo Actually the Fourier transform not only reproduces Β(σ) but also necessar ily adds a mirror image B(—σ) at nega tive frequencies (Figure 1). This can easily be verified by direct integration for the simple case of the monochro matic source Β(σΰ), and it is under standable because the relation cos 2πσχ = cos 2π(—σ)χ ensures that Β(σ) and Β(—σ) produce identical interferograms. Negative frequencies are un real, but the mirror image enters into the consideration of aliasing below. There are three important modifica tions to these simple relations. First, the interferogram is never totally sym metric about χ = 0, and to recover the full spectral information it is necessary to take the complex rather than the cosine Fourier transform. Second, the
F T
i _T~L \ = 2L(sin 2™L/2™L) Η 2L sine 2aL
(7)
(5)
Jo
The right-hand side of Equation 3 con tains all the spectral information in the source and is the cosine Fourier trans form of the source distribution Β(σ). The latter can therefore be recovered by the inverse Fourier transform
shape recorded from an ideal mono chromatic input is sine 2(σ — OQ)L (Fig ure 2). The first zero of this function is at (σ — σ0) = ± 1/2L; this value defines the resolution of the instrument:
(6)
This sine function is the instrument function of the spectrometer: The line-
The third modification, the discrete sampling, has the effect of replicating the computed spectrum at wavenumber intervals of l/Δχ. To see why, consider Equation 5. If σ is replaced by σ ± l/Δχ, the argument of the exponen tial is increased by 27ri'p, an integral number of 2π, which leaves it un changed. The replication also applies to the negative image Β(—σ), and it can be seen from Figure 3a that the nega tive part of the first replication will overlap the positive part of the original unless 1/Δχ > 2
