spectral resolution and sampling issues in fourier-transform spectral interferometry

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Spectral resolution and sampling issues in Fourier-transform spectral interferometry Christophe 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 Recherche 7639, F-91761 Palaiseau Cedex, France Received January 18, 2000; revised manuscript received May 12, 2000 We 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’s finite 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. © 2000 Optical Society of America [S0740-3224(00)00109-0] OCIS codes: 320.7120, 320.7100, 120.3180, 120.5050 1. INTRODUCTION Spectral interferometry, a technique relying on the use of frequency-domain interferences between two beams of different optical paths, 1,2 has been shown in recent years to be of great use in femtosecond spectroscopy. 311 In- 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 electric field 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 electric field of the unknown pulse, this technique is much more sen- sitive than its nonlinear counterparts 1215 and 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 electric field thus allowed the transposition of two-dimensional nuclear magnetic resonance to the optical domain, 16,17 as well as the time-resolved measurement of photon-echo emissions. 1821 Spectral interferometry has also been used for measuring the linear dispersion of materials, 22 for characterizing the complex dielectric function of semi- conductor nanostructures, 23 and for discriminating be- tween coherent and incoherent radiation in secondary emission from semiconductor quantum wells. 24,25 Fi- nally, spectral interferometry is a key ingredient in a re- cent nonlinear pulse-measurement technique, known as spectral phase interferometry for direct electric field re- construction. This very efficient and noniterative tech- nique makes use of spectral interferences between two frequency-sheared replicas of the unknown pulse. 15,2629 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. 11 Indeed, 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 Section 2 we 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 Section 3 we 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 Section 4 we discuss limitations arising from the finite spectral resolution of the spectrometer, as well as from the use of a detector made of a finite number of pixels. We will show that such effects can be carefully characterized and in most cases corrected for. Finally, we propose in Section 5 an improved FTSI procedure, which relies on the same ex- perimental scheme but involves more careful data pro- cessing. 2. FOURIER-TRANSFORM SPECTRAL INTERFEROMETRY In 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 E 0 ( t ) and E( t ) the time dependence of the two electric fields, E 0 ( v ) and E( v ) their Fourier transforms, and D w ( v ) 5 arg@E(v)# 2 arg@E 0 (v)# the difference in spectral phase that we intend to measure. In a typical spectral interferometry experiment, a relative time delay t is in- Dorrer et al. Vol. 17, No. 10 / October 2000 / J. Opt. Soc. Am. B 1795 0740-3224/2000/101795-08$15.00 © 2000 Optical Society of America

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Dorrer et al. Vol. 17, No. 10 /October 2000 /J. Opt. Soc. Am. B 1795

Spectral resolution and sampling issues inFourier-transform spectral interferometry

Christophe Dorrer, Nadia Belabas, Jean-Pierre Likforman, and Manuel Joffre

Laboratoire d’Optique Appliquee, Ecole Nationale Superieure des Techniques Avancees, Ecole Polytechnique,Centre National de la Recherche Scientifique, Unite Mixte de Recherche 7639, F-91761 Palaiseau Cedex, France

Received January 18, 2000; revised manuscript received May 12, 2000

We investigate experimental limitations in the accuracy of Fourier-transform spectral interferometry, a widelyused technique for determining the spectral phase difference between two light beams consisting of, for ex-ample, femtosecond light pulses. We demonstrate that the spectrometer’s finite spectral resolution, pixelaliasing, 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. © 2000 Optical Society of America[S0740-3224(00)00109-0]

OCIS codes: 320.7120, 320.7100, 120.3180, 120.5050

1. INTRODUCTIONSpectral interferometry, a technique relying on the use offrequency-domain interferences between two beams ofdifferent optical paths,1,2 has been shown in recent yearsto be of great use in femtosecond spectroscopy.3–11 In-deed, spectral interferometry allows the retrieval, in asimple way, of the difference in spectral phase betweentwo time-delayed light pulses. This makes possible themeasurement of the complex transfer function of any lin-ear optical element by use of a broadband light sourcesuch as a femtosecond laser or an incoherent white lamp.It also allows the full characterization of the electric fieldof an unknown pulse, assuming that a well-characterizedreference pulse of appropriate spectrum is available. Be-cause the measured quantity is linear in the electric fieldof the unknown pulse, this technique is much more sen-sitive than its nonlinear counterparts12–15 and can beused for extremely weak pulses, which is one of the mainreasons for its widespread use in femtosecond spectros-copy. A complete measurement of the electric field thusallowed the transposition of two-dimensional nuclearmagnetic resonance to the optical domain,16,17 as well asthe time-resolved measurement of photon-echoemissions.18–21 Spectral interferometry has also beenused for measuring the linear dispersion of materials,22

for characterizing the complex dielectric function of semi-conductor nanostructures,23 and for discriminating be-tween coherent and incoherent radiation in secondaryemission from semiconductor quantum wells.24,25 Fi-nally, spectral interferometry is a key ingredient in a re-cent nonlinear pulse-measurement technique, known asspectral phase interferometry for direct electric field re-construction. This very efficient and noniterative tech-nique makes use of spectral interferences between twofrequency-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-

0740-3224/2000/101795-08$15.00 ©

length calibration of the spectrometer.11 Indeed, it wasshown that a calibration accuracy better than one-tenthof the spacing between two pixels is often required toachieve 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 frequencysampling. Both can result in phase measurement distor-tion when not properly taken into account.

In Section 2 we review one of the most common imple-mentations of spectral interferometry, also known asFourier-transform spectral interferometry (FTSI), whichallows the retrieval of the spectral phase from a few Fou-rier transforms of the experimental interference spec-trum. In Section 3 we address the problem of spectraldata 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 pointsare evenly spaced in the frequency domain. In Section 4we discuss limitations arising from the finite spectralresolution of the spectrometer, as well as from the use of adetector made of a finite number of pixels. We will showthat such effects can be carefully characterized and inmost cases corrected for. Finally, we propose in Section 5an improved FTSI procedure, which relies on the same ex-perimental scheme but involves more careful data pro-cessing.

2. FOURIER-TRANSFORM SPECTRALINTERFEROMETRYIn this section we describe how FTSI permits the retrievalof the difference in spectral phase between two lightpulses from their interference spectrum. Let us callE0(t) and E(t) the time dependence of the two electricfields, E0(v) and E(v) their Fourier transforms, andDw(v) 5 arg@E(v)# 2 arg@E0(v)# the difference in spectralphase that we intend to measure. In a typical spectralinterferometry experiment, a relative time delay t is in-

2000 Optical Society of America

1796 J. Opt. Soc. Am. B/Vol. 17, No. 10 /October 2000 Dorrer et al.

troduced between the two beams, which are then recom-bined collinearly with a beam splitter. The total electricfield, E0(t) 1 E(t 2 t), is then spectrally resolved with aspectrometer and a CCD detector. The total frequencyspectrum thus reads

I~v! 5 uE0~v! 1 E~v!exp~ivt!u2

5 uE0~v!u2 1 uE~v!u2 1 E0* ~v!E~v!

3 exp~ivt! 1 c.c., (1)

where c.c. holds for the complex conjugate of its precedingterm. The last two terms result in spectral interferencesthrough a term in cos@Dw (v) 1 vt#, causing a rapidly os-cillating frequency dependence.

The interference pattern therefore strongly depends onthe spectral phase difference, although an experimentalmeasurement of the power spectrum yields only the phasecosine. However, there are a number of ways for retriev-ing the phase from its cosine, e.g., by use of polarizationmultiplexing.6 We are here interested in the techniquethat uses Fourier transforms,6,7 or FTSI, which we brieflyreview below.

Let us call f(t) 5 E0* (2t) ^E(t) the correlation productbetween the two fields. The power spectrum then reads

I~v! 5 uE0~v!u2 1 uE~v!u2 1 f~v!exp~ivt! 1 c.c. (2)

Note that f(v) 5 F.T.f(t) 5 E0* (v)E(v) 5 uE0* (v)E(v)u3 exp@iDw(v)# carries all the information on the spectralphase difference Dw(v) 5 arg@ f (v)#. Therefore extract-ing f(v) from the other terms in Eq. (2) will fulfill ourpurpose. This can be achieved by Fourier transformingEq. (2):

F.T.21I~v! 5 E0* ~2t ! ^ E0~t ! 1 E* ~2t ! ^ E~t !

1 f~t 2 t! 1 f~2t 2 t!* . (3)

f(t2t) is centered on t 5 t, while the last term is cen-tered on t 5 2t. The first two terms, autocorrelationfunctions of the individual fields, are centered at t 5 0.Therefore, for reasonably well-behaved pulses and forlarge enough values of t, f(t) does not overlap with theother terms in Eq. (3) and can be easily extracted from theinterference spectrum.30 Note that the first two termscan also be directly subtracted off in the frequency do-main if two additional measurements are made while oneof the two beams is blocked; thus the noninterfering partsare subtracted. This allows the use of smaller values ofthe time delay t, which will be shown in the next sectionsto be a desirable feature.

To summarize, FTSI relies on a few simple steps: Aninverse Fourier transform of the interference spectrum,followed by a selection of a finite time window so as tokeep only the correlation product between the two fields.The time delay must be adjusted so that this truncation ismade possible. A Fourier transform back into the fre-quency domain then allows the retrieval of f(v)5 E0* (v)E(v) and the spectral phase difference Dw(v)5 arg@ f (v)#. In cases in which E0(v) has been indepen-dently measured with nonlinear phase measurementtechniques, this allows the determination of E(v) andhence E(t) after an inverse Fourier transform. Note thatthe same information could have been obtained through a

direct measurement of the correlation function f(t) withtime-domain interferometry, also known as dispersiveFourier-transform spectroscopy.31 However, the advan-tage of FTSI lies in the multichannel detection of thewhole data by use of CCD detectors, which makes the in-terferometric requirements less difficult to fulfill and thetechnique more practical than a scanning measurementof the correlation function.

However, actual detectors never provide directly thepower spectrum I(v). Obviously, the signal is alwaysspoiled 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(v)but an array of data points related to I(v) through the ap-paratus function. Taking this apparatus function intoaccount turns out to be of particular importance in thecase of spectral interferometry, as will be shown in Sec-tion 4.

To demonstrate experimentally the incidence of the ap-paratus function, we used a homemade Ti:sapphire oscil-lator that delivers pulses of duration ranging between 20and 50 fs, depending on the operating conditions. A se-quence of two nearly identical pulses is obtained with abalanced Michelson interferometer; the time delay be-tween the two pulses is controlled with a step motor.Therefore, in the following, E(t) 5 E0(t) and the mea-sured spectral phase Dw(v) should reflect only the spec-tral dispersion of the interferometer. The interferencespectra are recorded with a Jobin–Yvon HR-460 spec-trometer followed by an EG&G 1024 3 256 CCD detector.Note that in this particular set of experiments, which isintended only to demonstrate the limitations of spectralinterferometry rather than actually to measure the elec-tric field, it was not required to characterize the spectralphase of E0(v), since it cancels out in the measured spec-tral phase difference Dw(v). For the same reason, iden-tical experimental results would have been obtained if anincoherent 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 the finite spectral resolutionof the spectrometer, as such issues will be discussed inSection 4. 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 mostspectrometers, this is never exactly the case, so we preferto use a general calibration function v(x) that relates thefrequency v to the spatial coordinate x. We develop thiscalibration law with respect to frequency around the lasercenter frequency, v0 :

v~x ! 5 v0 1 a1x 112 a2x2 1

16 a3x3 1 ¯ . (4)

As a result of this nonlinear dependence of v versus x,the frequency values for which the signal is sampled, v i ,are not evenly spaced, since the detector pixels are evenlyspaced in x. Because x is roughly proportional to 1/v, thenonlinear terms in Eq. (4) are usually not small in femto-second experiments in which the spectral extent is quite

Dorrer et al. Vol. 17, No. 10 /October 2000 /J. Opt. Soc. Am. B 1797

large. This causes changes in the frequency step, v i112 v i , by as much as 620% over the spectral range of ourspectrometer. The noneven frequency sampling of thedata 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 timeconsuming. In the following, however, we will show thatthe FFT can still be used.

A. Plain Fast Fourier Transform of the DataLet us first consider what happens when we ignore thenonlinear calibration law and simply proceed in comput-ing the FFT of the experimental data array $I(v i)%. Weobtain an array $I(ki)% that actually corresponds to theFourier transform of $I(xi)%, where k is the spatial fre-quency,

I~k ! 5 F.T.21I~x ! 5 E I~x !exp~2ikx !dx

5 N.I.T. 1 E f @v~x !#exp@iv~x !t#

3 exp~2ikx !dx 1 c.c., (5)

where N.I.T. stands for noninterferometric terms, whichdo not depend on t. The result is plotted in Fig. 1 as afunction of j 5 k/a1 for three different values of the timedelay between the two pulses. Note that if we were toneglect the nonlinear terms in Eq. (4), j would be the ex-act Fourier conjugate of v, i.e., the time t. Indeed, we ob-serve that the data shown in Fig. 1 peak at j 5 t and j5 2t. However, the correlation peak is not simplytranslated in time as would be expected for f(t 2 t) butalso broadens when t increases. This can be easily ex-plained by taking into account the calibration law,

I~k ! 5 N.I.T. 1 E f @v~x !#exp@iv~x !t#

3 exp~2ikx !dx 1 c.c.

5 N.I.T. 1 exp~iv0t!(F.T.21$f @v~x !#

3 exp@iFt~x !#%)~j 2 t! 1 @c.c.#~2j 2 t!

5 N.I.T. 1 exp~iv0t!ft~j 2 t!

1 exp~2iv0t!ft~2j 2 t!* , (6)

where Ft(x) 512 a2t x2 1

16 a3t x3 1 ... is a phase factor

resulting from the nonlinear terms in the calibration law.ft(j) is the inverse Fourier transform off @v(x)#exp@iFt(x)#. f @v(x)# with respect to x has a shapesimilar to f(v) with respect to v and does not significantlychange the Fourier transform time width. In contrast,the phase factor Ft(x) gives the main contribution to thebroadening in ft(j) observed in Fig. 1. Keeping only thequadratic term in Ft(x), proportional to a2t, we find thatthe 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 shapeof our laser pulses can be recognized for large values ofthe time delay.32

Not surprisingly, we conclude that the result obtainedin the time domain when there is a failure to take into ac-

count the nonlinearity in the calibration law is differentfrom the actual temporal shape of f(t). This is especiallytrue for shorter pulses, for which the frequency-stepvariation from one end of the spectrum to the other isgreater.

B. Discrete Fourier TransformOne approach to account for the discrepancy reported inSubsection 3.A is to use a Fourier-transform algorithmthat can handle nonevenly spaced data points, such as thediscrete Fourier transform. This will obviously yield thecorrect answer; however, none of these algorithms will beas 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 thedata points, is first to interpolate the experimental dataso as to numerically generate an array of points regularly

Fig. 1. Magnitude of the fast Fourier transform of the experi-mental interference spectrum, uI(k)u, plotted as a function of j5 k/a1 , for three different values of the time delay.

Fig. 2. Fourier transform of the same experimental data asthose used in Fig. 1, except that a linear interpolation of the fre-quency axis has first been performed to provide the FFT proce-dure with an array of evenly spaced data points in frequency do-main.

1798 J. Opt. Soc. Am. B/Vol. 17, No. 10 /October 2000 Dorrer et al.

spaced in the frequency domain. Figure 2 shows the re-sult obtained with a linear interpolation of the same dataas those used in Fig. 1. Although a sharp peak is thenobserved, in contrast with Fig. 1, a superimposed back-ground now appears whose magnitude dramatically in-creases with increasing values of the time delay (dashedarea). Although such a feature remains small, it doessignificantly affect the quality of the spectral phase thusretrieved. As demonstrated in more detail in AppendixB, the observed background is a direct consequence of theerror resulting from linearly interpolating the experimen-tal data. This error is most important for large values ofthe time delay, owing to the rapid frequency oscillationsof the spectral interferogram.

It might be claimed that a more elaborate interpolationscheme would improve the result. However, aiming atpushing the technique to its limits, we would like to beable to use time delays as great as the Nyquist limit, aswill be discussed in Section 4. This means that the oscil-lation period can be as small as two pixels. In such acase, any local interpolation scheme such as cubic splineis bound to fail and would not provide satisfactory results.There is a global interpolation scheme that does work,however, known as zero filling. This technique consistsin first performing a FFT of the data to j space, then in-creasing the j-window size, e.g., to 4N or 8N, where N isthe number of detector pixels, filling the new data pointswith zeroes. A FFT back into x space yields an arraywith a finer sampling, now making possible a proper in-terpolation of the data. Although this scheme works anduses only FFT’s, it requires larger arrays to handle. Wewill show in Subsection 3.D that similar results can be ob-tained with only arrays of the same size as the number ofpixels on the detector.

D. Retrieving the Spectral Phase in a First StepThe approach we propose here consists of retrieving thespectral phase with the j domain instead of the time do-main. We will show below that such a method is possibleand that once the spectral phase is retrieved, data inter-polation will be made easier, allowing the retrieval, as alast step, of the electric field as a function of time.

Let us first note the similarity between Eq. (2) and Eq.(6). In both cases, we have a sum of a few terms centeredon 0 and 6t, either in t space or in j space. As is evidentin Fig. 1, although there is a broadening, the relevantterm can still be extracted in j space. Indeed, the broad-ening mentioned in Subsection 3.A can be explained bythe fact that a given value of t does not yield a unique jfor all frequency components, as dv/dx is equal to a1 onlyat the center of the spectrum, v0 . Therefore this broad-ening cannot exceed a fixed fraction of t, namely, the rela-tive variation of the frequency spacing over the spectrum.As a consequence, such a broadening cannot cause theoverlap between components separated by t. Thus,choosing a value of t so that the relevant term can be ex-tracted, we obtain, after a FFT back to x space:

exp~iv0t!f@v~x !#exp@iF~x !#exp~ia1xt!

5 f@v~x !#exp@iv~x !t#. (7)

It is then straightforward to retrieve f(v) after we sub-tract the phase v(x)t. Note, however, that since this lat-ter term does not vary linearly with x, it is important totake into account the exact calibration law v(x). Thisapproach does allow us to get rid of the backgroundshown in Fig. 2 that resulted from the interpolationscheme.

Figure 3 shows the spectral phase retrieved by use ofthe various techniques discussed above for a time delaybetween the two pulses set to 5 ps. Curve (a), obtainedby ignoring the nonlinear dependence of the calibrationlaw, exhibits a large parabolic spectral phase, directly re-flecting the first 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 Subsection3.C. Although the retrieved phase is more accurate, itexhibits strong oscillations that are due to the interpola-tion error. Such oscillations around the exact value ofthe phase are due to the fact that the error in the linearinterpolation of the cosine function between two points isdependent on their position. Indeed, let us consider theinterpolation on evenly spaced points in the frequency do-main of the function cos@vt 1 Dw (v)# recorded on pointsroughly evenly spaced in the wavelength domain. Anegative, zero, or positive error is obtained, thus giving aperiodic-like structure. The local period is varying be-cause the wavelength interval associated to a fixed spec-tral interval depends on the wavelength. In contrast,this oscillating noise is totally absent in curve (c), whichhas been obtained with the approach discussed in thissubsection. This result is exactly identical to that of thezero-filling method (d), despite the smaller number ofpoints used in the calculation. Note that the residualspectral phase observed here results from the dispersionof the interferometer used in these experiments.

Finally, to retrieve the electric field in the time domain,we need to perform a Fourier transform toward the truetime domain t, instead of j. Fortunately, in most casesthe amplitude and phase of the unknown electric fieldvary slowly with frequency, unlike the spectral interfero-

Fig. 3. Spectral phase obtained from the interference spectrumbetween two pulses separated by 5 ps. The phase-retrieval tech-niques used are (a) plain FFT, (b) linear interpolation, (c) thetechnique described in Subsection 3.D, and (d) zero-filling inter-polation. The curves have been vertically shifted for clarity.

Dorrer et al. Vol. 17, No. 10 /October 2000 /J. Opt. Soc. Am. B 1799

grams we started from. This is true in many cases suchas for a short pulse, a highly chirped pulse for which thephase variation is dominated by lower-order terms, four-wave mixing emission, etc. In such cases, it is straight-forward to interpolate linearly uE(v)u and w(v) so that aFFT can be performed on the interpolated points, whichare now evenly sampled. The result is plotted in Fig. 4and compared with the other techniques. It appears thatthis approach performs much better than the linear-interpolation technique, as the background is reduced byone 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-fillingmethod.

We must mention that there are some pulses for whichthe technique described in this section would fail to yieldthe 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 theinterpolation required at the latest stage results in sig-nificant errors. In such a case, one would have to resortto the zero-filling technique, as described at the end ofSubsection 3.C.

4. SPECTRAL RESOLUTION ANDALIASINGIn spectral interferometry, larger values of the time delayresult in a smaller fringe spacing, hence in a reducedfringe contrast that is due to the finite spectral resolutionof the spectrometer. This is illustrated, for example, inFig. 5(a) in which the spectral fringes almost vanish for atime delay of 8 ps. One must therefore compromise whenchoosing t, which must be small enough so that this effectis not too important but large enough so as to make pos-sible the extraction of the correlation product from theFourier transform of the interferogram. In this sectionwe 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 values

Fig. 4. Time-domain determination of the correlation function,f(t 2 t), 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 iswithin the line thickness.

of the time delay t. Furthermore, we will show that sucheffects can be accounted for after the spectrometer’s appa-ratus function has been carefully measured, a task forwhich interference spectra have been shown to be particu-larly useful.33,34

We assume that the spectrometer response can be ap-proximated to a convolution with a response functionR(x), so that the spatial dependence of the intensity inthe detector plane is R(x) ^ I@v(x)#. R(x) depends onthe 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 pulsesfor which no loss can be afforded, R(x) may also dependon the laser spatial profile within the slit area. This lightintensity in the detector plane is then integrated over thepixel area, yielding the following expression for the sig-nal, Si , collected on a given pixel at position xi ,

Si 5 Exi2a/2

xi1a/2

R~x ! ^ I@v~x !#dx

5 E2`

1`

P~x 2 xi!$R~x ! ^ I@v~x !#%dx

5 $P~x ! ^ R~x ! ^ I@v~x !#%~xi!, (8)

where P(x) is a rectangle function taking the value 1 foruxu , a/2, a being the pixel width. If we now take intoaccount the fact that the signal is sampled for discretevalues of the spatial coordinate, x 5 xi , we find that theactual function we can experimentally access is

S~x ! 5 P~x !$P~x ! ^ R~x ! ^ I@v~x !#%, (9)

where P(x) is a Dirac comb of period dx, the pixel spac-ing. Note that dx > a. The Fourier transform in jspace reads

S~j! 5 P~j! ^ (P~j!R~j!T.F.21$I@v~x !#%), (10)

Fig. 5. (a) Blow-up of a particular spectral region of the inter-ference spectra obtained for t 5 3 ps (lower curve) and t 5 9 ps(upper curve). (b) Amplitude of the FFT of the above data, plot-ted as a function of j. The curve corresponding to t 5 9 ps hasbeen multiplied by a factor of 10.

1800 J. Opt. Soc. Am. B/Vol. 17, No. 10 /October 2000 Dorrer et al.

where P(j) is a Dirac comb of period T 5 2p/(a1dx), or 16ps for our setup. The convolution with this Dirac combresults in a folding within the Nyquist window, a phe-nomenon also known as aliasing. This is illustrated inFig. 5(a), which shows the spectral interferograms for twovalues of the time delay, t 5 3 ps and t 5 9 ps. In theformer case, in which the delay is significantly smallerthan T/2, the fringes are properly sampled. In contrast,the latter case corresponds to a time delay of the order ofT/2, which yields a spectrum highly undersampled. Fig-ure 5(b) shows the FFT of these spectra in j space. Fort ' T/2, half of the broadened pulse is actually folded inthe Nyquist window, i.e., shifted by T. The part offt(2j 2 t)* , where t , 2T therefore interferes with thepart of ft(j 2 t) for which t , T, 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 t . T/2 should therefore beavoided. More precisely, the condition max(dvt) , 1/2must be fulfilled to avoid the occurrence of any aliasing,where dv is the (nonconstant) frequency separation be-tween two adjacent pixels. However, even when this isthe case, the signal will still be distorted through the mul-tiplication by P(j)R(j). This term, characterizing thespectral resolution of our setup, must now be measured.For our purpose, one may use either an atomic narrowspectral line or spectral interferometry itself. In the firstcase, I(v) can be approximated to a Dirac distribution, sothat the FFT of the spectrum yields P(j)R(j), or ratherits aliased version, P(j) ^ @P(j)R(j)#. In the second ap-proach, we record a series of interference spectra for dif-ferent values of the time delays.35 The spectral resolu-tion can clearly be deduced from the decrease in thefringe contrast. More precisely, we plot on Fig. 6(a) theFFT of the data, which shows a decrease of the signal as tincreases. However, as was mentioned in Section 3, alarge part of this decay results from the uneven samplingof the data. More quantitatively, the signal at the peakfor a given value of t reads P(t)R(t)ft(0), where ft(0), f(0) owing to the broadening resulting from Ft(x).This can be taken into account by numerically generatinginterference spectra from the experimental laser spec-trum, i.e., multiplying the experimental uE0@v(x)#u2 bycos@v(x)t# and computing the Fourier transform for theexperimental values of t. The decay thus observed inFig. 6(b) is now entirely due to ft(0), since the calculationwas not limited by the spectrometer resolution. By divid-ing the maxima of Fig. 6(a) by those of Fig. 6(b), we obtainP(t)R(t) for several values of t, from which we can de-duce the entire response function through interpolation,as this function is slowly varying with j.

To compare these two approaches, we plot in Fig. 7 thefunction P(j)R(j) obtained with this technique, which wecompare with the data derived from a narrow spectralline. Provided that we add the contribution from the in-tervals @23T/2,2T/2# and @T/2, 3T/2# to the data ob-tained with spectral interferometry, we reach a goodagreement between the two techniques, except for smallvalues of j for which spectral interferometry is not valid.However, only the second approach yields the unaliasedP(j)R(j), which can then be directly used to correct theexperimental interference spectra.

5. IMPROVED FOURIER-TRANSFORMSPECTRAL INTERFEROMETRYSCHEME AND CONCLUSIONTo summarize, we have investigated some instrumentallimitations in FTSI, which, to our knowledge, have notbeen reported up to now. Our results lead us to proposean 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.11 The spectralresolution of the spectrometer should then be character-ized with the technique described in Section 4, yieldingthe product P(j)R(j). After data acquisition, the spec-tral interferograms are Fourier transformed into j spaceby use of a Cooley–Tukey FFT, where the data can be di-vided by P(j)R(j). After truncation, a FFT back into thewavelength domain allows recovery of the spectral phasedifference between the two pulses. Finally, the data canbe obtained in time domain after proper interpolation soas to generate an array of evenly spaced frequencies, asdescribed in Subsection 3.D.

Fig. 6. (a) FFT computed from a series of experimental interfer-ence spectra obtained for different values of t. (b) FFT com-puted from a series of numerically computed interference spectraobtained for different values of t and by use of the experimentallaser spectrum.

Fig. 7. j-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 showsthe aliased apparatus function deduced after periodization (i.e.,after adding the dotted curve), thus simulating the convolutionproduct with P(j). The apparatus function that should be usedfor correction of interference spectra is the dashed curve.

Dorrer et al. Vol. 17, No. 10 /October 2000 /J. Opt. Soc. Am. B 1801

For some implementations of FTSI (for example, inspectral phase interferometry for direct electric field re-construction), the extent of the time-dependent correla-tion product is sufficiently small so that the response ofthe spectrometer does not significantly modify this quan-tity. In such cases, there is no need to correct data in thej domain, which saves one step in the above procedure.

APPENDIX A: INFLUENCE OFEXPERIMENTAL NOISEIn this appendix we discuss the influence of experimentalnoise on the spectral phase retrieved with FTSI. We as-sume that a frequency-dependent noise, N(v), is added tothe spectral intensity, I(v), so that the total detected sig-nal reads

I~v! 1 N~v! 5 uE0~v!u2 1 uE~v!u2 1 f~v!

3 exp~ivt! 1 c.c. 1 N~v!. (A1)

Applying FTSI, as defined in Section 2, we compute theinverse Fourier transform of the above expression andmultiply by a window function, H(t), to extract f(t 2 t).Assuming that H(t) does not overlap with the nonrel-evant terms and that it is equal to 1 when f(t 2 t) is non-zero, we obtain f(t 2 t) 1 H(t)N(t). Finally, a Fouriertransform yields f(v) 1 F.T.@H(t 1 t)N(t 1 t)#. For agiven value of the frequency v, let us write this expres-sion as a exp(iDw) 1 b exp(iu). The error on the extractedspectral phase is then arg@a exp(iDw) 1 b exp(iu)#2 arg@a exp(iDw)#, which simplifies to arg$1 1 (b/a)3 exp@i(u 2 Dw)#% and then to arctan$(b/a)sin(u 2 Dw)/@1 1 (b/a)cos(u 2 Dw)#%. For a large value of the signal-to-noise ratio ua/bu, the phase error can be written as(b/a)sin(u 2 Dw), whose magnitude is always smallerthan ub/au 5 uH(v) ^ N(v)u/uE0(v)E(v)u.

To summarize, we find that the noise in spectral phaseis simply equal to the ratio of the filtered noise to theproduct of the electric field spectral amplitudes.

APPENDIX B: PHASE ERROR RESULTINGFROM LINEAR INTERPOLATIONIn this appendix we compute the error in spectral phaseresulting from linear interpolation in frequency of the ex-perimental data. Let us first recall that for a functiong(v) linearly interpolated on point v between points vaand vb , the interpolated value is

g inter~v! 5 g~va! 1 @g~vb! 2 g~va!#

3 ~v 2 va!/~vb 2 va!. (B1)

The interpolation error then reads

g inter~v! 2 g~v! ' 21

2~v 2 va!~v 2 vb!

]2g

]v2

5 k~v!]2g

]v2 , (B2)

where k(v) 5 2(v 2 va)(v 2 vb)/2.Let us first consider the procedure consisting of the lin-

ear interpolation of the interferogram followed by the ex-

traction of the phase, as discussed in Subsection 3.C. Inthis case, the interferometric part in the measured spec-trum reads

g~v! 5 uE0~v!uuE~v!ucos@vt 1 Dw~v!#. (B3)

We assume here that the variation of g(v) is due onlyto the cosine term in the interval in which interpolation isperformed, i.e., that the spectrum does not change signifi-cantly in this small interval. In this case, the second de-rivative of g(v) reads

]2g

]v2 ' uE0~v!uuE~v!u]2 cos@vt 1 Dw~v!#

]v2

5 2uE0~v!uuE~v!u H S ]w

]v1 t D 2

cos@vt 1 Dw~v!#

1]2w

]v2 sin@vt 1 Dw~v!#J . (B4)

Owing to the large value of the delay t, the first term inthe above sum is dominant. This shows that on a point vat which the interpolated interferogram is obtained be-tween points va and vb , the error is

N~v! 5 2uE0~v!uuE~v!ucos@vt 1 Dw~v!#

3 S ]w

]v1 t D 2

k~v!. (B5)

It is the product of the interferometric part by the func-tion L(v) 5 (]w/]v 1 t)2k(v). As a result, the correla-tion product in the temporal domain is the sum of theerror-free product and the convolution of this product bythe function L(t). This explains why the superimposedbackground that is due to the interpolation of the inter-ferogram is moving with the correlation product when thedelay is varied, as observed in Fig. 2. A direct conse-quence is that this noise cannot be filtered in the tempo-ral domain when the interferometric component is ex-tracted. The level of this noise is roughly proportional tot2. There is thus a dramatic parasitic effect as the delaybetween the two interfering pulses is increased. This er-ror then leads to an error in the retrieved spectral phase,whose magnitude can be deduced by applying Appendix Ato the value of N(v) obtained in Eq. (B5). This noise isthus directly related to the function L(v), and its level isalso roughly proportional to t2.

In contrast, when the interpolation is performed on thespectral phase according to the procedure discussed inSubsection 3.D, the interpolation noise takes muchsmaller values. Indeed, the error resulting from linearlyinterpolating the extracted spectral phase vt 1 w(v) isproportional to ]2w/]v2k(v), according to relation (B2).It is then independent of delay, since the linear interpo-lation of a straight line does not introduce any error.

The corresponding author, M. Joffre, can be reached atthe address on the title page, by fax at 33 1 69 31 99 96, orby e-mail at [email protected].

Note added in proof. It recently came to our attentionthat the use of filtering in the j domain as developed inSubsection 3.D of this paper has been also reported by Jo-nas and coworkers.36

1802 J. Opt. Soc. Am. B/Vol. 17, No. 10 /October 2000 Dorrer et al.

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