Smearing of interferograms in Fourier transformspectroscopy

Thomas G. Kyle and Ronald Blatherwick

The effects of changing spectral radiance during the scan of an interferometer on the spectra obtained fromthe interferogram are considered. For well-behaved variations the integrated radiance and base line are thesame as at the zero path difference of the interferometer. The changes in the appearance of spectral fea-tures are hardly changed beyond a few resolution elements from the changing features.

1. Introduction

The spectra of gases involved in a dynamic processmay be difficult to interpret when obtained by use of aconventional spectrometer because of the changestaking place, but it is known that the measured signalat each wavelength corresponds to the incoming radi-ance at a particular time. The situation is more con-fused in the case of a Fourier transform instrumentwhere the spectrum is related to the Fourier transformof the changing signal.1 The situation involves morethan there being no time resolution shorter than thescan time. The dynamic changes in the radiance causeconcern that spectral features might be greatly distortedby the transformation process. Chemical reactions canbe a source of the spectral variations, 2 but the temporalvariations are a particular problem for atmosphericspectra taken from satellites3 or balloons.4

The central difficulty involved in using a Fouriertransform spectrometer to obtain spectra of time-varying events is that the instrument does not trulymeasure the Fourier transform of the spectrum unlessthe spectrum is independent of time during the scan.

In the present work it is assumed that the spectra areobtained from a single-sided spatial scan of length L,which is then treated as being symmetric about the zeropath difference point at the beginning of the scan. Thismeans the transform involves the cosine rather than thecomplex exponential function. The time and the in-

When this work was done both authors were with University ofDenver, Physics Department, Denver, Colorado 80203; T. G. Kyle isnow with Oklahoma State University, Computer & Information Sci-ences Department, Stillwater, Oklahoma 74078.

Received 26 August 1983.0003-6935/84/020261-03$02.00/0.C 1984 Optical Society of America.

terferometer path difference are taken to be equivalentvariables represented by x. The time-varying radiancecan thus be expressed as B(w,x), where w is the angularfrequency of the radiance. The interferometer is as-sumed to be ideal with no dispersion or phase error inits response. If the interferogram is given by I(x), onehas

c LI(x) = c |, B(co,x) cos(x)dw, (1)

where c = (2wr)-1 .As mentioned above, Eq. (1) is not a proper Fourier

transform because B is a function of x and . Thepresent work investigates the characteristics of thespectrum obtained by transforming I(x) and the effectof the time-varying radiance on the integrated radianceobtained from the spectrum.

11. Representing the Time Variation

The most easily treated case of B (w,x) is the separablecase of

B(w,x) = b(w)g(x). (2)

This corresponds to only the source brightness changingwith no variation in the relative spectral intensity.Such variations would normally be associated withmechanical events,' such as the partial closing of anaperture, or electrical effects, such as a variation in theelectronic gain. It can be seen that b(x) can be takenoutside the integral in Eq. (1) giving a result which isequivalent to there being an apodizing function whichcan enhance or distort particular spectral features5 inthe same way as convolving the spectrum with thetransform of g(x). This being the case, when g(O) isunity, it is clear that the integral of the spectrum ob-tained by transforming the interferogram correspondingto B(ct),x) is equal to the measured radiance at the zeropath difference of the interferogram if the dimensionalunits are properly treated.

15 January 1984 / Vol. 23, No. 2 / APPLIED OPTICS 261

This case is included to emphasize that only in theseparable case can the spectral variation be treated asan apodizing function. Noncritical reading of Park3

could leave the impression that general spectral varia-tions can be thought of as apodizing effects.

More useful conclusions can be reached if the re-stricted case of B(w,x) being properly represented bya Taylor series expansion of N terms for 0 < x < L isconsidered. Although this case is not completely gen-eral, it includes all cases where the radiance at eachfrequency can be represented by its own power series.Let the series be

B(x) = B(w,O) + E Zn d nBn(,O))n= ni dXn

Then the interferogram becomes

I(x) = C f. B(co,O) cos(wx)dw

+ C x" d"1(w,0) cos(wx)dw. (4)n=1 fO n! dXn

The result of cosine transforming I(x) is B'(w), where

B'(o) = c f B(o',O)Lfo[(w - co')Ldw'

N n+ 1 dnB(c,',o)

n=1 n! dXn

X fn [(o - w')L]do', (5)

with

fn [(o - c')L] = X+ xn cos[(w - o')xjdx. (6)

Ill. Conclusions

The first term in Eq. (5) indicates that the spectrumwhich was present at the zero path difference is presentin the resulting spectrum B'(w) with degraded resolu-tion just as would have been the case if no variation hadtaken place in the spectrum during the interferometerscan.

Table I lists the expressions for fn (Z) with Z = (w -w')L for values of n up to 4. It can be seen that fo is thesinc function, sin(Z)/Z, as would be expected on thebasis of Eq. (5) being the regularly expected spectrumwhen each of the different derivatives of B is zero.Evaluating the expression for f[n (0) indicates that

fA() = /(n + 1).

The integrals of fn [(w - w')L] over all w yield a zeroresult except for fo which yields 7r/2. When this factoris taken into account, it can be seen that the integral ofthe radiance B'(w), over a region wide enough to permitthe effects of the end points of the integrals to be ne-glected, is equal to the integral of B (w,O). That is, theintegrated radiance from the resulting spectrum is equalto the radiance reaching the interferometer at the timeof zero path difference. Furthermore, by consideringregions where the derivatives of B with respect to x arezero and letting some regions have zero or nonzero ra-diance, it can be seen that the values of B(w) in suchregions are equal to the radiance which would have beenobserved if no spectral variations had occurred in anypart of the spectrum. This means the base line or zeroradiance value and the radiance scaling factor for thespectrum obtained by transforming the smeared in-terferogram are unaffected by the smearing.

It can also be seen from Eq. (5) that the spectrumB'(co) is composed of the sum of the spectrum at x = 0degraded by a rectangular apodizing function and thedifferent derivatives of the radiance also evaluated atthe zero path difference degraded by the functions f,&listed in Table I. Those spectral features which are notchanging with the time will be precisely the same as ifno spectral smearing had occurred during the interfer-ogram scan provided the features are well separated infrequency from any spectral feature which is changing.This means it is not possible to represent the spectrumobtained from the smeared interferogram as being thatobtained by a particular apodizing function becausesome spectral features will be altered by the smearingand some will not.

Strictly speaking, by multiplying an interferogramby a particular function g(x), which can be called anapodizing function, it is possible to alter the values of

1.2

1.0 go

0.8 - --.

0.8

0.4

(7)

Table 1. Evaluating Eq. (6) and Replacing (w - W')L by Z Give theListed Functions; these are the Resolution Functions for the nth Derivative

of the Radiance with Respect to X

n fn(Z)

0 [sin(Z)]VZ

1 [cos(Z) - 1 + Z sin(Z)]/Z 2

2 [2Z cos(Z) + (Z 2- 2) sin(Z)]/Z 3

3 X3Z2 - 6) cos(Z) + 6 + (Z3- 6Z) sin(Z)]/Z 4

4 [(4Z3 - 24Z) cos(Z) + (Z4- 12Z2 + 24) sin(Z)]/Z 5

0.2

0.0

-0.2

-0.40.0 2.0 4.0 8.0 8.0 10.0 12.0

Fig. 1. Instrument response functions for the derivative componentsof the smeared spectrum compared to the zeroth-order responsefunction Ao = sin(z)/z. The central peaks of fn have the value 1/(n+ 1), but fn becomes approximately equal to fo for large Z. The in-

tegral of fn(Z) over all Z is zero except for fo.

262 APPLIED OPTICS / Vol. 23, No. 2 / 15 January 1984

,Z

the interferogram in a way that will cause it to transforminto a completely different spectrum. Assume it isdesired to generate a particular interferogram I'(x) fromI"(x). Except for the zeros of I'(x) this can be done bymultiplying I"(x) by

h(x) = I'(x)/I"(x). (8)

Such h (x) do not correspond to what is usually calledan apodizing function and normally lack physicalmeaning and will not be considered here.

The functions f,, (z) which are convolved to degradethe resolution of the derivatives of the spectral radianceare plotted in Fig. 1. It can be seen that the distortionsin the spectrum obtained from the smeared interfero-gram will be localized to the vicinity of the changingspectral features. The different f, (Z) are approxi-mately equal for Z > 27r, the second zero of the sincfunction f(Z). This comes about because each f has'a sin(Z)/Z term and other terms involving a sine or co-sine of the same frequency divided by a higher than firstpower of Z, which become small for larger values of Z.Recall also that Eq. (5) shows the f being divided byn! before being convolved with the derivatives of B, andthis further reduces the distortions in B' due to spectral

changes occurring while the interferogram is being re-corded.

The effects of the type of interferogram smearingtreated here can be important within a few resolutionelements of the changing spectral features, but there areno other longer range effects. In particular, the inte-grated radiance remains the same as would have beenmeasured at the time corresponding to x = 0 whenfeatures within a few resolution elements of the ends ofthe integration regions are unchanging. There are noscaling or zero level shifts in the spectrum obtained fromthe smeared interferogram. The net effects of smearingcan be significant but are not nearly so troublesome asmight have been expected.

References1. L. Mertz, Infrared Phys. 7, 17 (1967).2. G. Guelachvili, "Distortions in Fourier Spectra and Diagnosis,"

Spectroscopic Techniques, Vol. 2, G. A. Vanasse, Ed. (Academic,New York, 1981).

3. J. H. Park, Appl. Opt. 21, 1356 (1982).4. A. Goldman, F. J. Murcray, and E. Niple, Appl. Opt. 22, 3721

(1980).5. E. Codding and G. Harlick, Appl. Spectrosc. 27, 85 (1972).

W. T. Silfvast of AT&T Bell Laboratories, Holmdel, photographedby W. J. Tomlinson, also of AT&T Bell, at IQEC XII, Munich, June

1982.

15 January 1984 / Vol. 23, No. 2 / APPLIED OPTICS 263