Distortion free interferograms in Fourier transformspectroscopy with nonlinear detectors
Guy Guelachvili
A new method for removing the most important distortions from Fourier transform spectra recorded withnonlinear detectors is reported and illustrated with practical examples. The removal of these distortions isobtained with a two-output interferometer configuration. This essentially works because the resultinginterferogram is made up of the difference between the signals of the two detectors and because these signalshave the same amplitude and opposite phases. The proposed method leaves the detectors unchanged andmakes use of their distorted signals. The correction is not restricted to the nonlinear effects of the onlydetectors but also of the entire electronic system amplifying each detector signal. It easily applies toabsorption, emission, low and high resolution spectra, as well as spectra of short and long lifetime phenomena.The benefit of the method is not restricted to Fourier transform spectroscopy. It is even not restricted totechniques making use of detectors. Its advantages are available for any data treatment where the importantinformation is made up of the difference between nonlinear functions (or signals) having similar amplitudesand opposite phases, whatever the number of pairs of such distorted functions (or signals) may be.
1. Introduction
The two most important features of Fourier trans-form spectroscopy, namely, the multiplex and the6tendue advantages, respectively, proposed by Fell-gett' and Jacquinot,2 each imply that a great numberof photons reach the detectors when recording theinterferogram. It is often hopeless to obtain linearmeasurements from such overilluminated detectors,and this affects seriously the quality of the measure-ment of the intensities in the transformed spectra.The only usual solution is to decrease the energy re-ceived from the source by either reducing its intensity,restricting the recorded spectral range, or both. Inother words it amounts to taking only a partial benefitof the main advantages of Fourier transform spectros-copy.
This paper proposes a method for removal of themost important systematic errors in intensity due tothe nonlinearity affecting the detectors. This methodis applicable when the two outputs of the Michelsoninterferometer are used to record the interferogram.Curiously it could require sending additional photonsonto the detectors.
The author is with Ohio State University, Physics Department,Columbus, Ohio 43210.
The principle is given in the next section which firststresses the essential differences between the one- andtwo-output interferometer configurations from the en-ergy point of view. Section III reports experimentalresults illustrating the distortions and efficiency oftheir correction. The remaining part of the paperproposes several solutions to achieve practical use ofthe method.
II. Principle
The demonstration given here is general and appliesto emission and absorption spectra. However, thedistortions discussed in this paper are often more im-portant and always more insidious in absorption spec-tra.
A. One-Output Interferometer
Let us consider a classical Michelson interferometer,the output of which is recorded on one detector. IfI(A) is the amount of energy available at the pathdifference A, the data corresponding to the sample ofthe recorded interferogram at this path difference willbe the result of the activated electrical signal deliveredby the detector, amplified, filtered, and integrated byan electronic device. It is supposed that only thedetector is nonlinear and not the rest of the recordingsystem.
Let G, go, g, and g2 be, respectively, the gain of thewhole amplification and the zero-, first-, and second-order nonlinearity terms of the detector. The energyI(A) falling on the detector is actually made up of aconstant part Ic and a part IM(A) modulated as afunction of A. This latter is usually the only one which
is recorded and transformed to get the calculated spec-trum. Of course, the detector reacts to I(A) and notmerely to IM(A). Due to its nonlinearity, it will insteadof I(A) give a signal proportional to ID(A), which maybe written
ID(A) = I(A) [g0 + g1 I(A) + g22(A)], (1)
where
I(A) = IC + IM(A) - (2)
Substituting I(A) from Eq. (2) into Eq. (1), ID(A) be-comes
ID(A) = gJIC + g1I2C + g241+ {[go + 2g 1Ic + 3g2
2C]IM(A)}
+ [g1 + 3g2jCj1M(A)j + g2IM1(A)) .
After the electronic filtering, the constant part cor-responding to the first term of Eq. (3) will disappear.The remaining part of Eq. (3) multiplied by G andintegrated is Fourier transformed. This gives the ac-tual spectrum which replaces the expected correct one,namely, the Fourier transform of IM(A).
The intensity distortion effects will come out essen-tially from the third and fourth terms of Eq. (3), sincethe second part of it is strictly proportional to thecorrect interferogram IM(A). Because of the impor-tance of (g1 + 3g2IC) relative to g2, the third term of Eq.(3) is the cause of the most troublesome features locat-ed around the wavenumber zero and at twice the wa-venumbers of the actual spectrum. Data samplingaccording to the sampling theorem will make the side
features overlap and then distort the spectrum. Toavoid this overlapping, oversampling may be a solutionbut only when the spectral range of interest is suffi-
ciently narrow. More precisely, with aB and SE defin-ing its limits this would be only for 2
aB 2 aE, which is in
most cases a serious restriction.
B. Two-Output Interferometer
1. How to Get the Interferogram
Now let us suppose that the interferometer has twodifferent outputs as, for example, in the Connes-typeinstrument shown in Fig. 1. Unlike in the precedingexample the whole energy available from the interfer-ometer is now a constant Io. The detectors A and B,
respectively, receive energies IA(A) and IB(A); there-fore,
IO IA(A) + IB(A) - (4)
As in the above one-output case, each right-handside term of Eq. (4) is made up of a constant part and a
part modulated as a function of A. More explicitly,
IA(A) = IC,A + IM,A(A)),(5)
IB(A) =IC,B + IM,B(A) -
The relative values of the constants IC,A and IC,B
depend essentially on the reflection and transmissioncoefficients R and T of the beam splitter and also of its
(3)
Interferometer DetectionFig. 1. With a two-output interferometer, the most important non-
linearities in the detectors A and B are experimentally removed by
appropriately balancing the gains GA and GB-
absorption coefficient which will be without losing anygenerality, supposed equal to 0 in the rest of the discus-sion. The sum of the constant terms is always equal tothe whole energy available Io, which may be written
IO = IC,A + C,B -
This is easily understood in the two extreme situa-tions where the beam splitter is either a mirror ortotally removed. In both cases the whole energy wouldbe reflected back to detector B and IC,A 0 and ICB =Io. If now the beam splitter is supposed ideal (4RT = 1instead of 0), it will give
IC,A =ICB =IO/2-
Actual beam splitters with 0 < 4RT < 1 will then leadto
ICA < ICB,
which states that the mean energy falling on eachdetector will always be smaller on detector A thandetector B located on the same side of the interferome-ter as the source. From Eqs. (4), (5), and (6) thefollowing relationship is readily obtained:
IM,A(A) + IM,B(A) = 0 -
It simply expresses the well-known statement thatthe modulated signals at each output have the sameamplitude and opposite phases. As a consequence,the resulting interferogram must be made up of thedifference between the signals, respectively, given bydetectors A and B. This is schematically representedin Fig. 1.
2. Interferogram with Nonlinear DetectorsLet us see what happens now from the nonlinearity
behavior of the detectors. The detector A which re-ceives the energy IA(A) will instead deliver a signal
proportional to ID,A(A) given by
ID,A(A) = IA(A)[g0A + gAIA(A) + g2A1A(A)] -
The same occurs for detector B:
IDB(A) = IB(A) [cOB + g1BIB(A) + g2BIB(A)]
Equations (9) and (10) are similar to Eq. (1). Theoutputs ID,A(A) and ID,B(A) of detectors A and B are,
3. How to Get a Distortion-Free InterferogramEquation (11) is similar to Eq. (3). Its first term
represents the constant part, but in this two-detectorcase it vanishes almost entirely depending on the rela-tive contributions of detectors A and B. Ideally itshould not be necessary to use electronic filtering toremove this component. The second term of Eq. (11)is the correct interferogram within a constant factor.The two remaining terms are responsible for the dis-torting effects. Unlike in the one-detector situation,the third term which represents the most importantdistortions in Eq. (11) may now vanish if Eq. (12) issatisfied:
GA 9_~1B + 3 2BICBGA = GB X l32BC91A + 32AICA
This condition may be easily obtained using appro-priate values for gains GA and GB. Then the recordedinterferogram is free from the most important intensi-ty distortions.
The removal of the strongest perturbing effects asexpressed above may even be done with two detectorshaving different nonlinearity behaviors. It could bebetter anyway to have them identical. In this case g1A=91B and2A = g2B. If additionallyICA = ICB (i.e., withan ideal beam splitter) to get the correct compensa-
tion, it is only needed to have the two gains GA and GBequal to each other.
Usually the two detectors are rather similar, and thebeam splitter is not perfect. If identical gains are usedagain on both channels, the cancellation of the intensi-ty distortions may be obtained by the following proce-dure. Indeed Eq. (12) may be satisfied only by makingICA = ICB. These values are naturally not equal to eachother according to Eq. (7). To obtain this equality,additional light must consequently be sent on to detec-tor A. This paradoxically means that, like in the for-est, fire is used to fight fire, here intensity distortionsdue to the use of too much light may be corrected byaddition of light. In practice, as said above, it is,however, much more easily done by adjusting gains GAand GB-
Ill. Experimental Evidence of the Distortion Effects andof their Correction
i I I0 cr 2o'Fig. 2. Parasitic first harmonic features increase with the sourceemitted energy. Their signs are different from one output of the
interferometer to the other.
respectively, multiplied by gain factors GA and GB andintegrated. The difference [GAID,A(A) - GBID,B(A)],the constant term of which is removed by appropriatefiltering, is the actual interferogram replacing the ex-pected correct one [GAIA(A) - GBIB(A)].
Taking Eq. (8) into account and using IM(A) in placeOf IM,A(A) and -IM,B(A), from Eqs. (5), (9), and (10) theactual interferogram may be writtenGAIDA(A) - GBIDB(A)
A. One-Output InterferometerFigure 2 gives actual spectra recorded with only one
of the two outputs of the high information interferom-eter of Laboratoire d'Infrarouge at Orsay, equippedwith He-cooled Cu-doped Ge detectors. The source isa globar, the emission intensity of which is convenient-ly controlled through the variation of the voltage of itspower supply. To easily see the distortion effectscommented on above, the recorded spectral range hasbeen taken narrow enough using an appropriate opti-cal filter, and the interferograms have been oversam-pled. As explained in the previous section, the sideparasitic features are then clearly coming out. Theresolution for all the spectra displayed on the samewavenumber scale from 0 to 4000 cm-1 is of the order of
Fig. 3. Intermediate spectrum is calculated from a distortion-compensated interferogram.
0.5 cm-'. All the spectra which were calculated at the
CNRS Computer Center (CIRCE) located at Orsayhave been normalized. Consequently, their absoluteintensities cannot be compared. On the other hand,
the relative importance of the side features is with sucha presentation directly readable.
Six different spectra derived from detector A aregiven for six different globar temperatures obtained by
varying the electrical current intensity i from 3 to 13 A.This temperature variation is clearly seen from the
average slopes of the upper part of the spectra whichreveal when i increases the displacement toward the
visible of the location of the wavenumber of maximumemission of the blackbodylike globar. The 500-cm-'wide spectrum is located around af = 1650 cm-'. Two
sideband features appear next to 0 and around 2c at-3300 cm-'. Since a sine Fourier transform is usedhere to obtain the spectra, the left-hand side featurevanishes at the origin of the wavenumber axis becauseof overlapping with the negative frequencies image.
As expected, the relative importance of these har-monics is increasing from the first trace to the lowerone. It is proportional to [g1A + 3g2AICA]IM2(A)- Itsexact variation as a function of the globar current i isnot determined here. If it were, it should have beennecessary not to forget, first, that g1A has the oppositesign of g2A and, second, that neither i nor i2 are linear
energy scales. The upper trace of Fig. 2 is the equiva-
lent of the lowest one but now with detector B. The
sign of the parasitic features has clearly changed.
B. Two-Output Interferometer
The intermediate spectrum of Fig. 3 is calculatedfrom the distortion-free interferogram obtained underthe general disposition of Fig. 1 with a globar currentequal to 9 A. It is to be compared to the lower onewhich is taken from Fig. 2. Although the energy usedfor this compensated spectrum is at least twice, thespectral distortions due to the nonlinearity in bothdetectors have disappeared. These distortions areeven more apparent in the upper spectrum of the fig-ure obtained again from detector A but with only 3 Athrough the globar.
In this particular case, the practical determinationof the gains GA and GB which satisfy Eq. (12) and thenremove the distortion has been done by preliminarytrials using low resolution test spectra as the interme-diate spectrum in Fig. 3.
IV. Practical Procedures for Applying the Correction
The interferogram may be obtained as shown in Fig.
1 where only the difference between the signals given
by detectors A and B is recorded. It is also possible torecord separately the signal of each detector and tomake their difference afterward.
In the first case, GA and GB must be determinedbefore the actual interferogram is recorded. Low reso-lution tests may be performed. For this purpose, thefinal spectrum should include one or two sufficientlywide ranges without signal. This may be done with the
free recording spectral range slightly larger than theoptical filter. The zero signal regions should for theappropriate relative values of GA and GB remain locat-ed at the zero level. High resolution checks may alsobe performed if some absorption reference lines arepresent in the spectra. The tops of these lines shouldthen be located at the zero energy level (see Fig. 2 inRef. 3) when the right adjustment between GA and GBis found. Low and high resolution tests can be donerapidly. For example, the low resolution spectra asshown in Figs. 2 and 3 can be obtained within 1 min orless. The high resolution tests need more time. How-ever, one fast recording, i.e., poor signal-to-noise con-ditions should be enough to see clearly whether thetops of the reference lines are well located at zero. Tospend even less time for these preliminary tests, when-ever possible a lowering of the required resolution mayoften be obtained by oversaturating the profiles of thereference lines.
When the signal of each detector is recorded sepa-rately, predetermination of the gains GA and GB isuseless. The good balance between these two gainsmay be done after the recording of the data, makinguse of the above low and high resolution tests. GA andGB would no longer be gains of electronic amplifiersbut simply multiplicative constants. Their relative
values may be even automatically determined by ade-quate fitting computer programs. Recording sepa-rately the data from channels A and B requires moreimportant data storage capabilities. On the otherhand, the a posteriori balance procedure may be morepractical. Additionally it would allow full applicationof the proposed method to the spectra of the manysources which are only observable during limited peri-ods of time.
This study was partly done at the Department ofPhysics of Ohio State U. during the author's tenure as aDistinguished Visiting Professor for 1986-87, on leavefrom Universit6 Paris XI, Laboratoire d'Infrarouge.The author acknowledges K. Narahari Rao for his helpon various aspects of this work.
References1. P. B. Fellgett, "Three Concepts Make a Million Points," Infrared
Phys. 24, 95 (1984).2. P. Jacquinot, "How the Search for a Throughput Advantage led
to Fourier Transform Spectroscopy," Infrared Phys. 24, 99(1984).
3. A. Levy, N. Lacome, and G. Guelachvili, "Measurement of N 20Line Strengths from High Resolution Fourier Transform Spec-tra," J. Mol. Spectrosc. 103, 160 (1984).
Books continued from page 4585The main drawback to buying the book is that the material isthirteen years old. Although the translation contains a few refer-
ences to recent work, 90% of the references are to work before theoriginal edition written in 1972. Recent work is not incorporatedinto the text. Much has been learned recently about subjects suchas the Mori equation and the Green's function method. It is too badthey waited so long to translate such a good monograph.
G. D. MAHAN
Coherent Radiation Sources. Edited by A. W. SAENZ and H.UBERALL. Springer-Verlag, New York, 1985. 235 pp. $29.50.
A title such as this one, COHERENTRADIATION SOURCESimmediately brings to mind lasers and masers, but this book hasabsolutely nothing to do with such sources. Instead, it is aboutradiation produced by highly relativistic, charged particles as theypenetrate various types of target material, namely, coherent brems-strahlung, channeled radiation, and transition radiation. So in allcases the radiation is high energy gamma radiation. The purpose ofthe book is to review the present state-of-the-art involving theseradiations. It is quite thorough, discussing the theoretical andexperimental aspects of producing and measuring these radiationsas well as examining their uses. Since the discovery of these phe-nomena occurred about 20 years ago, a review of this sort thatcollects and summarizes what is known about these types of radia-tion is a reasonable and worthwhile endeavor. The editors, authors,and publisher have done an excellent job.
The book has seven chapters plus a brief introduction. The firsttwo chapters deal with coherent bremsstrahlung. Then there arefour chapters on channeling and a concluding chapter on transitionradiation. Although the seven main chapters are written by differ-
ent sets of authors the book flows smoothly. In fact, it is veryreadable. And, at least one of the chapters is a translation fromRussian, but if you were not told that you would never realize it.The chapters on theory involve several simple models of crystals towhich quantum mechanics is applied. The theory is quite simpleand convincing. It was fascinating for a nonexpert in the field to seethe application of quantum mechanics to somewhat esoteric sys-tems. In general, the experimental verification offered for the the-ory fits well. The discussion of applications is the least satisfactoryaspect of the book. Many applications are optimistically suggested,but for most of them there is little evidence that they will eventuallyprove to by as useful as hoped. For example, it is suggested thatchanneled radiation may be used to study crystal defects. However,so far most experiments have been restricted to diamond, silicon,and germanium because of the great perfection and minimal thermaleffects required to perform the experiments. Also, sample prepara-tion seems restrictive. Finally, sources of the high energy, highintensity charged particle beams necessary for these experiments areexpensive to operate and are not widely available, to say the least.However a discussion of applications is important and adds to theoverall value of the book.
Several other features of the book that should not be overlookedare the many and excellent figures and graphs throughout the book,the index for easy reference, and the many references cited in all thechapters. This reviewer only detected three trivial typos that didnot obscure the meaning of the text. In fact, the only detractingfeature of the book is the simplifying practice of letting m = c = h = 1in the third and eighth chapters.
The reviewer found the applications of quantum mechanics in thisbook fascinating. Also the book was readily understandable. ThusCOHERENTRADIATIONSOURCES, although probably of great-est use to the specialist, should be of interest to many outside thefield. It would be a good addition to any technical library.
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