Spectral resolution and sampling issues in Fourier-transform spectral interferometry

Spectral resolution and sampling issues inFourier-transform spectral interferometryChristophe Dorrer,Nadia Belabas,Jean-Pierre Likforman,and Manuel Joffre Laboratoire d’Optique Applique´e,E´cole Nationale Supe´rieure des Techniques Avance´es,Ecole Polytechnique, Centre National de la Recherche Scientifique,Unite´Mixte de Recherche7639,F-91761Palaiseau Cedex,FranceReceived January18,2000;revised manuscript received May12,2000We investigate experimental limitations in the accuracy of Fourier-transform spectral interferometry,a widely used technique for determining the spectral phase difference between two light beams consisting of,for ex-ample,femtosecond light pulses.We demonstrate that the spectrometer’sfinite spectral resolution,pixel aliasing,and frequency-interpolation error can play an important role,and we provide a new and more accu-rate recipe for recovering the spectral phase from the experimental data.©2000Optical Society of America [S0740-3224(00)00109-0]OCIS codes:320.7120,320.7100,120.3180,120.50501.INTRODUCTIONSpectral interferometry,a technique relying on the use of frequency-domain interferences between two beams of different optical paths,1,2has been shown in recent years to be of great use in femtosecond spectroscopy.3–11In-deed,spectral interferometry allows the retrieval,in a simple way,of the difference in spectral phase between two time-delayed light pulses.This makes possible the measurement of the complex transfer function of any lin-ear optical element by use of a broadband light source such as a femtosecond laser or an incoherent white lamp. It also allows the full characterization of the electricfield of an unknown pulse,assuming that a well-characterized reference pulse of appropriate spectrum is available.Be-cause the measured quantity is linear in the electricfield of the unknown pulse,this technique is much more sen-sitive than its nonlinear counterparts12–15and can be used for extremely weak pulses,which is one of the main reasons for its widespread use in femtosecond spectros-copy.A complete measurement of the electricfield thus allowed the transposition of two-dimensional nuclear magnetic resonance to the optical domain,16,17as well as the time-resolved measurement of photon-echo emissions.18–21Spectral interferometry has also been used for measuring the linear dispersion of materials,22 for characterizing the complex dielectric function of semi-conductor nanostructures,23and for discriminating be-tween coherent and incoherent radiation in secondary emission from semiconductor quantum wells.24,25Fi-nally,spectral interferometry is a key ingredient in a re-cent nonlinear pulse-measurement technique,known as spectral phase interferometry for direct electricfield re-construction.This very efficient and noniterative tech-nique makes use of spectral interferences between two frequency-sheared replicas of the unknown pulse.15,26–29 Despite the widespread use of spectral interferometry, there have been few detailed studies up to now on its re-liability,with the exception of a recent work demonstrat-ing the large sensitivity of the retrieved data on the wave-length calibration of the spectrometer.11Indeed,it was shown that a calibration accuracy better than one-tenth of the spacing between two pixels is often required to achieve the best possible accuracy in the measured spec-tral phase.In this paper we address two other experi-mental limitations affecting the reliability of spectral in-terferometry:spectral resolution and frequency sampling.Both can result in phase measurement distor-tion when not properly taken into account.In Section2we review one of the most common imple-mentations of spectral interferometry,also known as Fourier-transform spectral interferometry(FTSI),which allows the retrieval of the spectral phase from a few Fou-rier transforms of the experimental interference spec-trum.In Section3we address the problem of spectral data sampling:In spectrometers,data are usually avail-able as an array of points evenly spaced in the wave-length domain,while available fast Fourier transform (FFT)algorithms are most efficient when the data points are evenly spaced in the frequency domain.In Section4 we discuss limitations arising from thefinite spectral resolution of the spectrometer,as well as from the use of a detector made of afinite number of pixels.We will show that such effects can be carefully characterized and in most cases corrected for.Finally,we propose in Section5 an improved FTSI procedure,which relies on the same ex-perimental scheme but involves more careful data pro-cessing.2.FOURIER-TRANSFORM SPECTRAL INTERFEROMETRYIn this section we describe how FTSI permits the retrieval of the difference in spectral phase between two light pulses from their interference spectrum.Let us call E0(t)and E(t)the time dependence of the two electric fields,E0(␻)and E(␻)their Fourier transforms,and ⌬␸(␻)ϭarg͓E(␻)͔Ϫarg͓E0(␻)͔the difference in spectral phase that we intend to measure.In a typical spectral interferometry experiment,a relative time delay␶is in-0740-3224/2000/101795-08$15.00©2000Optical Society of Americatroduced between the two beams,which are then recom-bined collinearly with a beam splitter.The total electric field,E0(t)ϩE(tϪ␶),is then spectrally resolved with a spectrometer and a CCD detector.The total frequency spectrum thus readsI͑␻͒ϭ͉E0͑␻͒ϩE͑␻͒exp͑i␻␶͉͒2ϭ͉E0͑␻͉͒2ϩ͉E͑␻͉͒2ϩE0*͑␻͒E͑␻͒ϫexp͑i␻␶͒ϩc.c.,(1) where c.c.holds for the complex conjugate of its preceding term.The last two terms result in spectral interferences through a term in cos͓⌬␸(␻)ϩ␻␶͔,causing a rapidly os-cillating frequency dependence.The interference pattern therefore strongly depends on the spectral phase difference,although an experimental measurement of the power spectrum yields only the phase cosine.However,there are a number of ways for retriev-ing the phase from its cosine,e.g.,by use of polarization multiplexing.6We are here interested in the technique that uses Fourier transforms,6,7or FTSI,which we briefly review below.Let us call f(t)ϭE0*(Ϫt) E(t)the correlation product between the twofields.The power spectrum then reads I͑␻͒ϭ͉E0͑␻͉͒2ϩ͉E͑␻͉͒2ϩf͑␻͒exp͑i␻␶͒ϩc.c.(2) Note that f(␻)ϭF.T.f(t)ϭE0*(␻)E(␻)ϭ͉E0*(␻)E(␻)͉ϫexp͓i⌬␸(␻)͔carries all the information on the spectralphase difference⌬␸(␻)ϭarg͓f(␻)͔.Therefore extract-ing f(␻)from the other terms in Eq.(2)will fulfill our purpose.This can be achieved by Fourier transforming Eq.(2):F.T.Ϫ1I͑␻͒ϭE0*͑Ϫt͒ E0͑t͒ϩE*͑Ϫt͒ E͑t͒ϩf͑tϪ␶͒ϩf͑ϪtϪ␶͒*.(3) f(tϪ␶)is centered on tϭ␶,while the last term is cen-tered on tϭϪ␶.Thefirst two terms,autocorrelation functions of the individualfields,are centered at tϭ0. Therefore,for reasonably well-behaved pulses and for large enough values of␶,f(t)does not overlap with the other terms in Eq.(3)and can be easily extracted from the interference spectrum.30Note that thefirst two terms can also be directly subtracted off in the frequency do-main if two additional measurements are made while one of the two beams is blocked;thus the noninterfering parts are subtracted.This allows the use of smaller values of the time delay␶,which will be shown in the next sections to be a desirable feature.To summarize,FTSI relies on a few simple steps:An inverse Fourier transform of the interference spectrum, followed by a selection of afinite time window so as to keep only the correlation product between the twofields. The time delay must be adjusted so that this truncation is made possible.A Fourier transform back into the fre-quency domain then allows the retrieval of f(␻)ϭE0*(␻)E(␻)and the spectral phase difference⌬␸(␻)ϭarg͓f(␻)͔.In cases in which E0(␻)has been indepen-dently measured with nonlinear phase measurement techniques,this allows the determination of E(␻)and hence E(t)after an inverse Fourier transform.Note that the same information could have been obtained through a direct measurement of the correlation function f(t)with time-domain interferometry,also known as dispersive Fourier-transform spectroscopy.31However,the advan-tage of FTSI lies in the multichannel detection of the whole data by use of CCD detectors,which makes the in-terferometric requirements less difficult to fulfill and the technique more practical than a scanning measurement of the correlation function.However,actual detectors never provide directly the power spectrum I(␻).Obviously,the signal is always spoiled with some amount of electronic and photon noise, an effect of minor importance that is discussed in Appen-dix A.More important,the measured signal is not I(␻) but an array of data points related to I(␻)through the ap-paratus function.Taking this apparatus function into account turns out to be of particular importance in the case of spectral interferometry,as will be shown in Sec-tion4.To demonstrate experimentally the incidence of the ap-paratus function,we used a homemade Ti:sapphire oscil-lator that delivers pulses of duration ranging between20 and50fs,depending on the operating conditions.A se-quence of two nearly identical pulses is obtained with a balanced Michelson interferometer;the time delay be-tween the two pulses is controlled with a step motor. Therefore,in the following,E(t)ϭE0(t)and the mea-sured spectral phase⌬␸(␻)should reflect only the spec-tral dispersion of the interferometer.The interference spectra are recorded with a Jobin–Yvon HR-460spec-trometer followed by an EG&G1024ϫ256CCD detector. Note that in this particular set of experiments,which is intended only to demonstrate the limitations of spectral interferometry rather than actually to measure the elec-tricfield,it was not required to characterize the spectral phase of E0(␻),since it cancels out in the measured spec-tral phase difference⌬␸(␻).For the same reason,iden-tical experimental results would have been obtained if an incoherent white lamp had been used instead of a femto-second laser.3.FREQUENCY SAMPLINGIn this section we address the issue of frequency sam-pling.We will neglect here thefinite spectral resolution of the spectrometer,as such issues will be discussed in Section4.Let us call x the spatial coordinate in the de-tector plane along which the spectrum dispersion occurs. Although x is nearly proportional to wavelength in most spectrometers,this is never exactly the case,so we prefer to use a general calibration function␻(x)that relates the frequency␻to the spatial coordinate x.We develop this calibration law with respect to frequency around the laser center frequency,␻0:␻͑x͒ϭ␻0ϩ␣1xϩ12␣2x2ϩ16␣3x3ϩ¯.(4) As a result of this nonlinear dependence of␻versus x, the frequency values for which the signal is sampled,␻i, are not evenly spaced,since the detector pixels are evenly spaced in x.Because x is roughly proportional to1/␻,the nonlinear terms in Eq.(4)are usually not small in femto-second experiments in which the spectral extent is quitelarge.This causes changes in the frequency step,␻iϩ1Ϫ␻i,by as much asϮ20%over the spectral range of our spectrometer.The noneven frequency sampling of the data might be thought to preclude the use of the ex-tremely efficient Cooley–Tukey FFT algorithm,thus mak-ing the FTSI spectral phase retrieval much more time consuming.In the following,however,we will show that the FFT can still be used.A.Plain Fast Fourier Transform of the DataLet usfirst consider what happens when we ignore the nonlinear calibration law and simply proceed in comput-ing the FFT of the experimental data array͕I(␻i)͖.We obtain an array͕I(k i)͖that actually corresponds to the Fourier transform of͕I(x i)͖,where k is the spatial fre-quency,I͑k͒ϭF.T.Ϫ1I͑x͒ϭ͵I͑x͒exp͑Ϫikx͒d xϭN.I.T.ϩ͵f͓␻͑x͔͒exp͓i␻͑x͒␶͔ϫexp͑Ϫikx͒d xϩc.c.,(5) where N.I.T.stands for noninterferometric terms,which do not depend on␶.The result is plotted in Fig.1as a function of␰ϭk/␣1for three different values of the time delay between the two pulses.Note that if we were to neglect the nonlinear terms in Eq.(4),␰would be the ex-act Fourier conjugate of␻,i.e.,the time t.Indeed,we ob-serve that the data shown in Fig.1peak at␰ϭ␶and␰ϭϪ␶.However,the correlation peak is not simply translated in time as would be expected for f(tϪ␶)but also broadens when␶increases.This can be easily ex-plained by taking into account the calibration law,I͑k͒ϭN.I.T.ϩ͵f͓␻͑x͔͒exp͓i␻͑x͒␶͔ϫexp͑Ϫikx͒d xϩc.c.ϭN.I.T.ϩexp͑i␻0␶͒(F.T.Ϫ1͕f͓␻͑x͔͒ϫexp͓i⌽␶͑x͔͖͒)͑␰Ϫ␶͒ϩ͓c.c.͔͑Ϫ␰Ϫ␶͒ϭN.I.T.ϩexp͑i␻0␶͒f␶͑␰Ϫ␶͒ϩexp͑Ϫi␻0␶͒f␶͑Ϫ␰Ϫ␶͒*,(6) where⌽␶(x)ϭ12␣2␶x2ϩ16␣3␶x3ϩ...is a phase factor resulting from the nonlinear terms in the calibration law. f␶(␰)is the inverse Fourier transform of f͓␻(x)͔exp͓i⌽␶(x)͔.f͓␻(x)͔with respect to x has a shape similar to f(␻)with respect to␻and does not significantly change the Fourier transform time width.In contrast, the phase factor⌽␶(x)gives the main contribution to the broadening in f␶(␰)observed in Fig.1.Keeping only the quadratic term in⌽␶(x),proportional to␣2␶,wefind that the broadening can be essentially interpreted as an arti-ficial linear chirp in the pulse.This yields the asymmet-ric shape observed in Fig.1,in which the spectral shape of our laser pulses can be recognized for large values of the time delay.32Not surprisingly,we conclude that the result obtained in the time domain when there is a failure to take into ac-count the nonlinearity in the calibration law is different from the actual temporal shape of f(t).This is especially true for shorter pulses,for which the frequency-step variation from one end of the spectrum to the other is greater.B.Discrete Fourier TransformOne approach to account for the discrepancy reported in Subsection3.A is to use a Fourier-transform algorithm that can handle nonevenly spaced data points,such as the discrete Fourier transform.This will obviously yield the correct answer;however,none of these algorithms will be as efficient as the Cooley–Tukey FFT in terms of comput-ing time.We will therefore attempt to use other tech-niques in the following,in order to obtain the correct an-swer more efficiently.C.Data InterpolationThe most straightforward approach for using the Cooley–Tukey FFT algorithm,despite an uneven spacing of the data points,isfirst to interpolate the experimental data so as to numerically generate an array of pointsregularly Fig.1.Magnitude of the fast Fourier transform of the experi-mental interference spectrum,͉I(k)͉,plotted as a function of␰ϭk/␣1,for three different values of the timedelay.Fig.2.Fourier transform of the same experimental data as those used in Fig.1,except that a linear interpolation of the fre-quency axis hasfirst been performed to provide the FFT proce-dure with an array of evenly spaced data points in frequency do-main.spaced in the frequency domain.Figure2shows the re-sult obtained with a linear interpolation of the same data as those used in Fig.1.Although a sharp peak is then observed,in contrast with Fig.1,a superimposed back-ground now appears whose magnitude dramatically in-creases with increasing values of the time delay(dashed area).Although such a feature remains small,it does significantly affect the quality of the spectral phase thus retrieved.As demonstrated in more detail in Appendix B,the observed background is a direct consequence of the error resulting from linearly interpolating the experimen-tal data.This error is most important for large values of the time delay,owing to the rapid frequency oscillations of the spectral interferogram.It might be claimed that a more elaborate interpolation scheme would improve the result.However,aiming at pushing the technique to its limits,we would like to be able to use time delays as great as the Nyquist limit,as will be discussed in Section4.This means that the oscil-lation period can be as small as two pixels.In such a case,any local interpolation scheme such as cubic spline is bound to fail and would not provide satisfactory results. There is a global interpolation scheme that does work, however,known as zerofilling.This technique consists infirst performing a FFT of the data to␰space,then in-creasing the␰-window size,e.g.,to4N or8N,where N is the number of detector pixels,filling the new data points with zeroes.A FFT back into x space yields an array with afiner sampling,now making possible a proper in-terpolation of the data.Although this scheme works and uses only FFT’s,it requires larger arrays to handle.We will show in Subsection3.D that similar results can be ob-tained with only arrays of the same size as the number of pixels on the detector.D.Retrieving the Spectral Phase in a First StepThe approach we propose here consists of retrieving the spectral phase with the␰domain instead of the time do-main.We will show below that such a method is possible and that once the spectral phase is retrieved,data inter-polation will be made easier,allowing the retrieval,as a last step,of the electricfield as a function of time.Let usfirst note the similarity between Eq.(2)and Eq.(6).In both cases,we have a sum of a few terms centered on0andϮ␶,either in t space or in␰space.As is evident in Fig.1,although there is a broadening,the relevant term can still be extracted in␰space.Indeed,the broad-ening mentioned in Subsection3.A can be explained by the fact that a given value of␶does not yield a unique␰for all frequency components,as d␻/d x is equal to␣1only at the center of the spectrum,␻0.Therefore this broad-ening cannot exceed afixed fraction of␶,namely,the rela-tive variation of the frequency spacing over the spectrum. As a consequence,such a broadening cannot cause the overlap between components separated by␶.Thus, choosing a value of␶so that the relevant term can be ex-tracted,we obtain,after a FFT back to x space:exp͑i␻0␶͒f͓␻͑x͔͒exp͓i⌽͑x͔͒exp͑i␣1x␶͒ϭf͓␻͑x͔͒exp͓i␻͑x͒␶͔.(7)It is then straightforward to retrieve f(␻)after we sub-tract the phase␻(x)␶.Note,however,that since this lat-ter term does not vary linearly with x,it is important to take into account the exact calibration law␻(x).This approach does allow us to get rid of the background shown in Fig.2that resulted from the interpolation scheme.Figure3shows the spectral phase retrieved by use of the various techniques discussed above for a time delay between the two pulses set to5ps.Curve(a),obtained by ignoring the nonlinear dependence of the calibration law,exhibits a large parabolic spectral phase,directly re-flecting thefirst nonlinear term in the calibration law. This large quadratic phase is consistent with the broad-ening observed in Fig.1.Curve(b)shows the result ob-tained by performing,prior to the FFT,a linear interpo-lation in the frequency axis,as discussed in Subsection 3.C.Although the retrieved phase is more accurate,it exhibits strong oscillations that are due to the interpola-tion error.Such oscillations around the exact value of the phase are due to the fact that the error in the linear interpolation of the cosine function between two points is dependent on their position.Indeed,let us consider the interpolation on evenly spaced points in the frequency do-main of the function cos͓␻␶ϩ⌬␸(␻)͔recorded on points roughly evenly spaced in the wavelength domain.A negative,zero,or positive error is obtained,thus giving a periodic-like structure.The local period is varying be-cause the wavelength interval associated to afixed spec-tral interval depends on the wavelength.In contrast, this oscillating noise is totally absent in curve(c),which has been obtained with the approach discussed in this subsection.This result is exactly identical to that of the zero-filling method(d),despite the smaller number of points used in the calculation.Note that the residual spectral phase observed here results from the dispersion of the interferometer used in these experiments.Finally,to retrieve the electricfield in the time domain, we need to perform a Fourier transform toward the true time domain t,instead of␰.Fortunately,in most cases the amplitude and phase of the unknown electricfield vary slowly with frequency,unlike the spectralinterfero-Fig.3.Spectral phase obtained from the interference spectrum between two pulses separated by5ps.The phase-retrieval tech-niques used are(a)plain FFT,(b)linear interpolation,(c)the technique described in Subsection3.D,and(d)zero-filling inter-polation.The curves have been vertically shifted for clarity.grams we started from.This is true in many cases such as for a short pulse,a highly chirped pulse for which the phase variation is dominated by lower-order terms,four-wave mixing emission,etc.In such cases,it is straight-forward to interpolate linearly ͉E (␻)͉and ␸(␻)so that a FFT can be performed on the interpolated points,which are now evenly sampled.The result is plotted in Fig.4and compared with the other techniques.It appears that this approach performs much better than the linear-interpolation technique,as the background is reduced by one order of magnitude.Furthermore,on these experi-mental data,the result of this technique cannot be distin-guished from that of the a priori more exact zero-filling method.We must mention that there are some pulses for which the technique described in this section would fail to yield the same accuracy as the zero-filling method (for example,when the unknown pulse is a sequence of two pulses sepa-rated by several picoseconds).Then the spectral ampli-tude itself oscillates rapidly with frequency,so that the interpolation required at the latest stage results in sig-nificant errors.In such a case,one would have to resort to the zero-filling technique,as described at the end of Subsection 3.C.4.SPECTRAL RESOLUTION AND ALIASINGIn spectral interferometry,larger values of the time delay result in a smaller fringe spacing,hence in a reduced fringe contrast that is due to the finite spectral resolution of the spectrometer.This is illustrated,for example,in Fig.5(a)in which the spectral fringes almost vanish for a time delay of 8ps.One must therefore compromise when choosing ␶,which must be small enough so that this effect is not too important but large enough so as to make pos-sible the extraction of the correlation product from the Fourier transform of the interferogram.In this section we discuss the incidence of the spectrometer’s finite spec-tral resolution and of the detector’s finite number of pix-els,which will be obviously most evident for large valuesof the time delay ␶.Furthermore,we will show that such effects can be accounted for after the spectrometer’s appa-ratus function has been carefully measured,a task for which interference spectra have been shown to be particu-larly useful.33,34We assume that the spectrometer response can be ap-proximated to a convolution with a response function R (x ),so that the spatial dependence of the intensity in the detector plane is R (x ) I ͓␻(x )͔.R (x )depends on the spectrometer characteristics,such as focal length,dif-fraction grating,and entrance-slit width.Furthermore,when the laser beam is not highly diffracted by the en-trance slit,such as when we deal with low-energy pulses for which no loss can be afforded,R (x )may also depend on the laser spatial profile within the slit area.This light intensity in the detector plane is then integrated over the pixel area,yielding the following expression for the sig-nal,S i ,collected on a given pixel at position x i ,S i ϭ͵x i Ϫa /2x i ϩa /2R ͑x ͒I ͓␻͑x ͔͒d xϭ͵ϪϱϩϱP ͑x Ϫx i ͕͒R ͑x ͒I ͓␻͑x ͔͖͒d xϭ͕P ͑x ͒R ͑x ͒I ͓␻͑x ͔͖͒͑x i ͒,(8)where P (x )is a rectangle function taking the value 1for ͉x ͉Ͻa /2,a being the pixel width.If we now take into account the fact that the signal is sampled for discrete values of the spatial coordinate,x ϭx i ,we find that the actual function we can experimentally access isS ͑x ͒ϭ⌸͑x ͕͒P ͑x ͒R ͑x ͒I ͓␻͑x ͔͖͒,(9)where ⌸(x )is a Dirac comb of period ␦x ,the pixel spac-ing.Note that ␦x уa .The Fourier transform in ␰space readsS ͑␰͒ϭ⌸͑␰͒(P ͑␰͒R ͑␰͒T.F.Ϫ1͕I ͓␻͑x ͔͖͒),(10)Fig.4.Time-domain determination of the correlation function,f (t Ϫ␶),by use of (a)a plain FFT of the data,(b)a linear inter-polation before the FFT,(c)the technique described in Subsec-tion 3.D,and (d)the zero-filling method.(c)and (d)cannot be dis-tinguished because the difference between the two curves is within the linethickness.Fig.5.(a)Blow-up of a particular spectral region of the inter-ference spectra obtained for ␶ϭ3ps (lower curve)and ␶ϭ9ps (upper curve).(b)Amplitude of the FFT of the above data,plot-ted as a function of ␰.The curve corresponding to ␶ϭ9ps has been multiplied by a factor of 10.where⌸(␰)is a Dirac comb of period Tϭ2␲/(␣1␦x),or16 ps for our setup.The convolution with this Dirac comb results in a folding within the Nyquist window,a phe-nomenon also known as aliasing.This is illustrated in Fig.5(a),which shows the spectral interferograms for two values of the time delay,␶ϭ3ps and␶ϭ9ps.In the former case,in which the delay is significantly smaller than T/2,the fringes are properly sampled.In contrast, the latter case corresponds to a time delay of the order of T/2,which yields a spectrum highly undersampled.Fig-ure5(b)shows the FFT of these spectra in␰space.For ␶ϷT/2,half of the broadened pulse is actually folded in the Nyquist window,i.e.,shifted by T.The part of f␶(Ϫ␰Ϫ␶)*,where tϽϪT therefore interferes with the part of f␶(␰Ϫ␶)for which tϽT,resulting in the ob-served time-domain fringes(b)and also in a characteris-tic beating in the frequency domain(a).A situation in which␶ϾT/2should therefore be avoided.More precisely,the condition max(␦␻␶)Ͻ1/2 must be fulfilled to avoid the occurrence of any aliasing, where␦␻is the(nonconstant)frequency separation be-tween two adjacent pixels.However,even when this is the case,the signal will still be distorted through the mul-tiplication by P(␰)R(␰).This term,characterizing the spectral resolution of our setup,must now be measured. For our purpose,one may use either an atomic narrow spectral line or spectral interferometry itself.In thefirst case,I(␻)can be approximated to a Dirac distribution,so that the FFT of the spectrum yields P(␰)R(␰),or rather its aliased version,⌸(␰) ͓P(␰)R(␰)͔.In the second ap-proach,we record a series of interference spectra for dif-ferent values of the time delays.35The spectral resolu-tion can clearly be deduced from the decrease in the fringe contrast.More precisely,we plot on Fig.6(a)the FFT of the data,which shows a decrease of the signal as␶increases.However,as was mentioned in Section3,a large part of this decay results from the uneven sampling of the data.More quantitatively,the signal at the peak for a given value of␶reads P(␶)R(␶)f␶(0),where f␶(0)Ͻf(0)owing to the broadening resulting from⌽␶(x).This can be taken into account by numerically generating interference spectra from the experimental laser spec-trum,i.e.,multiplying the experimental͉E0͓␻(x)͔͉2by cos͓␻(x)␶͔and computing the Fourier transform for theexperimental values of␶.The decay thus observed inFig.6(b)is now entirely due to f␶(0),since the calculation was not limited by the spectrometer resolution.By divid-ing the maxima of Fig.6(a)by those of Fig.6(b),we obtain P(␶)R(␶)for several values of␶,from which we can de-duce the entire response function through interpolation, as this function is slowly varying with␰.To compare these two approaches,we plot in Fig.7the function P(␰)R(␰)obtained with this technique,which we compare with the data derived from a narrow spectral line.Provided that we add the contribution from the in-tervals͓Ϫ3T/2,ϪT/2͔and͓T/2,3T/2͔to the data ob-tained with spectral interferometry,we reach a good agreement between the two techniques,except for small values of␰for which spectral interferometry is not valid. However,only the second approach yields the unaliased P(␰)R(␰),which can then be directly used to correct the experimental interference spectra.5.IMPROVED FOURIER-TRANSFORM SPECTRAL INTERFEROMETRYSCHEME AND CONCLUSIONTo summarize,we have investigated some instrumental limitations in FTSI,which,to our knowledge,have not been reported up to now.Our results lead us to propose an improved FTSI procedure that allows a partial com-pensation for instrumental limitations.First,the spec-tral calibration must be performed with great care,fol-lowing the technique reported previously.11The spectral resolution of the spectrometer should then be character-ized with the technique described in Section4,yielding the product P(␰)R(␰).After data acquisition,the spec-tral interferograms are Fourier transformed into␰space by use of a Cooley–Tukey FFT,where the data can be di-vided by P(␰)R(␰).After truncation,a FFT back into the wavelength domain allows recovery of the spectral phase difference between the two pulses.Finally,the data can be obtained in time domain after proper interpolation so as to generate an array of evenly spaced frequencies,as described in Subsection3.D.Fig.6.(a)FFT computed from a series of experimental interfer-ence spectra obtained for different values of␶.(b)FFT com-puted from a series of numerically computed interference spectra obtained for different values of␶and by use of the experimental laserspectrum.Fig.7.␰-domain apparatus function of the spectrometer ob-tained with a narrow spectral line(thin solid curve)or with spec-tral interferences(dashed curve).The thick solid curve shows the aliased apparatus function deduced after periodization(i.e., after adding the dotted curve),thus simulating the convolution product with P(␰).The apparatus function that should be used for correction of interference spectra is the dashed curve.。