Auxiliary computation for Fourier spectrometry

InfraredPhysica Vol. 7. pp. 17-23.PmgamonPrcv Ltd. 1967.Rintod in Omt Britain

AUXILIARY

COMPUTATION

FOR FOURIER SPECTROMETRY L. MJZRTZ

Block Associates, Inc., Blackstone Street, Cambridge, Mass. (Received 8 October 1%6) --A variety of computational procedures are presented to correct for some common instrumental imperfections of Fourier spectrometry. Included are three general techniques for phase correction, which corxcts for failure to register a sample at zero path Merence, and also for intrinsic non-symmetry of the interferogram. Two general techniques for correct@ non-uniform sampling are also presented. Some of the techniques remain to be demonstrated in practice.

in Fourier spectrometry 8 perfectly symmetrical interferogram is acquired, with a sample registered on the center and sampled perfectly uniformly. A simple cosine Fourier transform reveals the desired spectrum. In practice all the conditions are difhcult to achieve; symmetry, an on-center sample, or uniform sampling. Auxiliary computation serves to alleviate these failures by shifting the hardware problems to software. Symmetry and sample alignment are repaired by phase correction. Remedies for nonuniform sampling are treated in a subsequent section. Phase correction is ultimately based on the redundancy of the right and left sides of an interferogram. For perfectly symmetric (or antisymmetric) interferograms such redundancy is obvious. It turns out that neither perfect condition is essential to the redundancy; the very existence of a central fringe is IDEALLY

sufilcient .

A central fringe is of large amplitude because all the component fringes of various frequencies add constructively in phase. This location may be called a place of stationary phase. The right-left redundancy prevails whether or not that phase be zero, which happens only to be a further condition for symmetry (42 for antisymmetry). A more severe form of non-symmetry develops when the stationary phase location becomes a function of wavelength. This situation is referred to as a chirp interferogramo) and if the function is sufficiently gradual, the right-left redundancy is not impaired. The essence of phase correction is to exclude any components which are in quadrature with the proper phase. Such quadrature components may be random due to noise or may be systematic. Operations which oppose the right-left redundancy of the interferogram upset the stationary phase condition and introduce systematic quadrature components. In particular, if one side of the interferogram is omitted (tantamount to assuming zeros there) then even an intrinsically symmetric interferogram must develop sine components. In quantitative terms non-symmetric apodization (truncation) of the interferogram introduces quadrature components according to the antisymmetric part of the apodizing function. The proper phase components arise from the symmetric part of the apodizing function.* *Roof of these relations may be found in L. Mmrrz ~ansformur&v~ in Optics, p. 26, Wiley (1965). A perGnent article by SAKN and VANKSE J. opt. Sot. Am. 56, 131 (1%6) proves the special case for intrinsically symmetric interferograms with off-center boxcar truncation. Their statement of the resulting relations is given in equation (17). 17 ll

18

L.

hfBRT2

At this stage the situation may be graphically clarilied by example. Figure. 1 shows the amplitude and phase spectra from an approximately on-center interferogram. It happens to be the infrared spectrum of the star x Cygni between 4000 and 10,000 cm-l (abscissa range O-14,200 cm-i) The first thing to note is that where there is appreciable signal-tonoise in the amplitude spectrum, the phase spectrum is well defined and approximately horizontal. The phase spectrum pertains to the location of the largest sample in the interferogram, i.e. approximately the central fringe. Although initial calculation of phases

Fro. 1. Amplitude and phase spectrafrom on-cater interferogram.Source, x Cygni. Abscissa range, 0 -142akm-1 at half sampling frequel?cy.

ordinarily pertains to either the middle location or the extreme left location, the simple addition of a gradient of one cycle per sample frequency, per sample point, to the initial phase spectrum shifts the pertinent reference to any desii location. Where the phase spectrum is perfectly horizontal the phase is accurately referred to the location of the central fringe. In the example of Fig. 1 the phase spectrum is slightly curved which means that the interferogram is slightly chirped. The accurate location of the central fringe shifts slightly with wavelength. Figure 2 shows an off-center interferogram of the same source, This was multiplied by the truncating function shown by the dashed line prior to calculation of the amplitude and phase spectra shown in Fig. 3. The lirst thing to notice is that Fig. 3 exhibits higher resolution than Fig. 1 in spiteof the fact that the same number of input samples was used for calculation. This is a result of including information farther from the central fringe in the interferogram, even though only on one side. The motive for the slope of the dashed line in Fig. 2 is that if it were not for the sloped region, the neighborhood of the central fringe would have been counted twice with respect to the more distant fringes. As a consequence the contrast of narrow absorption lines would have been reduced by almost a factor of two. The symmetric part (around the central fringe) of the dashed function is a simple boxcar. Although a simple step could equally replace the sloped region, a small error in the central fringe location would lead to an error as shown by the gray region of Fig. 4(a) for the symmetric part, whereas using the slope leads to error as shown by the gray region of Fig. 4(b). I feel the latter type of error

19

Auxiliary computation for Fourier spwtromtry

r------------_----

-----

--

----7 f

I

I 1. I

PATH

DIFFERENCE

FM. 2. Off-center interferogram of x Cygni. with two (dashed and dotted) truncation functions.

Ro. 3. Amplitude and phase spectra from Fig. 2 with dashed mtion.

to be more acceptable. Furthermore the sloped function is readily developed by a versatile program which calculates an arbitrary trapezoidal truncation, based on four address locations: the vertex locations of the trapezoid. With this program the right hand cutoff of the truncation function can also be sloped in order to reduce sidelobes in the final instrument profile. The same program is also used to form the truncation function shown by the dotted line in Fig. 2, and this truncation is used to obtain a reference phase spectrum. The dotted apodization is triangular, including 10 samples each side of the central fringe. Triangular truncation is preferable to boxcar because the former’s instrument proBe contains no negative (MOO0 phase shift) sidelobes, as the latter’s does.

20

L.

0

A

MWTZ

0

o

FIG. 4. Uncertainties in the even part of A stepped truncation, B sloped truncation.

Using the dotted apodization the low resolution, high signal-to-noise spectra are obtained and shown in Fig. 5. Phase correction can now proceed to find that portion of the amplitudes in Fig. 3 which is in phase with the phase given by Fig. 5. The phase spectrum of Fig. 3 is subtracted from that of Fig. 5 and the cosine of each resulting angle multiplied by the corresponding amplitude of Fig. 3. The result is the phase corrected spectrum shown in Fig. 6. Notice that in regions where no signal exists the spectrum fluctuates around zero, rather than above zero. The systematically positive contribution of r.m.s. noise (especially at low signal) has been eliminated. Second, notice that most of the correction pertains to the wings of deep absorption bands. It is in these regions that systematic quadrature components were generated by nonsymmetric truncation. Third, notice that spurious frequencies (in this case pickup at 180 c/s and a deliberately superimposed signal at 75 c/s) can be violently altered during phase correction. Only by chance would such spurious frequencies have the proper phase.

FREQUENCY

Fro. 5. Amplitude and phase spectra from Fig. 2 with dotted truncation.

The interferogram shown in Fig. 2 is not severely chirped, and so the apodization function (dashed) is suitable for the entire spectral range. In the case of deliberate severe chirping, where the central fringe location is dispersed over an appreciable fraction of the interferogram, then each wavelength will require its own apodizing function. Unfortunately this two dimensional apodization precludes the use of the efficient Cooley-Tukey Fourier transform program.(ss 8) The less efficient recursion techniques must then be adopted. The phase correction procedure just described is only one of several possible methods. For example, another method due to Forman,@) is to construct a correcting function

21

Auxiliary computation for Fourier spectrometry

having the phase spectrum of Fig. 5, and an amplitude spectrum corresponding to a desired filter function. The filter function will generally have unit amplitude over the desired spectral region and zero amplitude outside.

r

. FREQUENCY

FIO. 6. Phase corrected amplitude spectrum from Fig. 2.

The correcting function is then convolved with the interferogram, thereby symmetrizing the interferogram. With a symmetrical interferogram the simple use of cosine Fourier transformation directly omits the quadrature sine components. It is also possible to avoid constructing a correction function by using the short section of interferogram already available, which led to the spectra of Fig. 5. This short section is convolved with the full interferogram and so the result is a modified autocorrelation, predominantly symmetrical. Subsequent cosine Fourier transformation results in a phase corrected spectrum whose amplitudes have been weighted by the amplitude spectrum of Fig. 5. Equalization may then be performed by dividing by that amplitude spectrum, although the result may exaggerate the noise in low signal regions. In the convolution phase correction procedure just described it is unnecessary to use the apodixation which was necessary for the previously described multiplication procedures. With a symmetrical interferogram an accurate center is available to omit the left hand side so that the central fringe region is not unduly counted twice. A third class of phase correction based solely on amplitude spectra is also possible in principle. It has never been tried in practice although it may have modest application for analogue computation and emission line spectra. As stated before, in the complex plane:

where E is the spectrum, C and S are the cosine and sine transforms of the truncating function, * denotes convolution and J_ means “is perpendicular to”. Because of the perpendicularity of the above terms their vector addition is according to Pythagoras’ theorem.* *The perpendicularity depends on the alignment of the divisiin location which separates even and odd parts of the truncating function, with the central fringe (stationary phase location). Otherwise, perpendicularity is not assured and so vector addition is not according to Pythagoras.

22

L.MEnIz

The amplitude spectrum A is thus:

A=

NE*c)2+(E*s)Bl*

and we seek to evaluate E * C from knowledge of A, C and S. Initially it should be possible to use A as an approximation of E. Thus E*Cw

[@--(A

+S)a]‘,Az

We may note that S is antisymmetric and A * S resembles a smoothed derivative of A. Further iteration would be A,, = [A2 -

(As-1 *sy14

which is expected to converge, although convergence has not been proven. This is perhaps not farfetched because in the limit if limA,= -CO

E*C

then E*C=

[Aa-(E*C*S)2]4

Under the circumstance that C is a sine function, then C f S = S, so that E*C=

[AZ--(E*S)a]i

whichis the original equation. The practicality of this class of phase correction techniques remains to be demonstrated. Furthermore the influence of noise on this technique (and vice-versa) is not clear. We can now turn our attention to calculations to repair sampling non-uniformities. The class of non-uniformities which will be treated is those which are reproducible, mild, and smooth. Interferometer drive mechanisms based on simple electromagnetic actuation or on mechanical linkages tend to give this class of non-uniform sampling. Measurements are made of a monochromatic source to serve as a basis for correction. The resulting interferogram is a pure sinusoid, though sampled at non-uniform intervals. If any d.c. component is present, that should be eliminated, and also the sinusoid should be normahxed in amplitude prior to further consideration. Also it will prove optimal that the monochromatic fringes have a frequency of roughly l/6 sample frequency. A relatively easy way to accurately ascertain path difference along the sampled sinusoid is to select and count all positive going zero crossings. The count is interpolated by the arcsin of the samples adjacent to those zero crossings. At those samples we have an accurate measure of retardation versus sample number, T (m). Retardation at the in between samples may be found by interpolation. Finally, the samples are weighted by the derivative T’(m) in order to compensate variations of sample density, and a Fourier synthesis carried out by recursion. Unfortunately the efficient Cooley-Tukey program is not suitable for this application involving non-uniformly spaced input samples. Another possible method is based on instrument profiles. Sidebands resulting from f.m. modulation are purely antisymmetric whereas those from a.m. modulation are purely

Auxiliary~mpu~tion for Fourierspectrometry

23

symmetric. Therefore if we obtain an empirical instrument profile as the tentative spectrum of the measured monochromatic source, then any antisymmetric part is ascribable to non-uniformities in sampling. (It is tacitly assumed that previously described quadrature effects are absent or have been phase corrected, and fmher that the profile is not perfore completely positive.) If we now separate the measured instrument profile into even and odd parts, C and S respectively, then the measured spectrum I will be related to the real spectrum E by:

where v is the frequency (v equals 1 at the monochromatic frequency). The factor v in the last term results from f.m. sideband strength being proportional to frequency, although this is only valid for weak modulation. Just as in the third class of phase correction methods we can use I as a first approximation of E and iterate I,=r-v(Ii&-l

*s>

Again, convergence (even speedy convergence) is hoped for, and it may be necessary that C be a sine fur&ion. 1. 2. 3. 4.

REFERENCES Mwrz, L,, &esSpeclresinfrarooiypesdes Awes, p. 120, (Colloquc)W. of Lii (1964). COOLEY, J. W. and J. W. TUKEY,Maths Comput. 19,297 (1965). FORMAN, M. L., .I. opt. Sot. Am. !!6,978 (1966). FORMAN, M. L., W. H. STELand G. A. VANASSE,J. opt. Sot. Am. !%,59 (1966).