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Brunel et al. Vol. 24, No. 6 /June 2007/J. Opt. Soc. Am. A 1641

Fractional-order Fourier analysis for ultrashortpulse characterization

Marc Brunel and Sébastien Coetmellec

CORIA UMR 6614, Université de Rouen, Avenue de l’Université Boîte Postale 12,76801 Saint Etienne du Rouvray Cedex, France

Mickael Lelek and Frédéric Louradour

XLIM, UMR 6172, Université de Limoges, France

Received October 20, 2006; accepted December 8, 2006;posted January 4, 2007 (Doc. ID 76303); published May 9, 2007

We report what we believe to be the first experimental demonstration of ultrashort pulse characterization us-ing fractional-order Fourier analysis. The analysis is applied to the interpretation of spectral interferometryresolved in time (SPIRIT) traces [which are spectral phase interferometry for direct electric field reconstruc-tion (SPIDER)-like interferograms]. First, the fractional-order Fourier transformation is shown to naturallyallow the determination of the cubic spectral phase coefficient of pulses to be analyzed. A simultaneous deter-mination of both cubic and quadratic spectral phase coefficients of the pulses using the fractional-order Fourierseries expansion is further demonstrated. This latter technique consists of localizing relative maxima in a 2Dcartography representing decomposition coefficients. It is further used to reconstruct or filter SPIRIT traces.© 2007 Optical Society of America

OCIS codes: 070.2590, 320.7100, 100.5070.

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. INTRODUCTIONractional-order Fourier transformation1–6 (FRFT) hasttracted much interest in the past years. A main domainf applications is the interpretation of diffraction pat-erns. The Huygens–Fresnel integral can indeed be ex-ressed as a fractional-order Fourier transform.4 Conse-uently, the FRFT is a powerful tool for analyzingiffraction patterns.7,8 It could be used successfully inigital holography to restore fields of particles. However,he number of applications is much wider than just inter-retation of diffraction figures or spatial Fourier optics.s the basis functions of the FRFT are linearly chirped

unctions, FRFT is appropriate in each domain wherehirped signals are analyzed, and particularly linearlyhirped functions (i.e., complex exponentials with linearrequency modulation). Each linearly chirped function isndeed a basis function of the fractional-order Fourierransformation. But despite this potentiality, the numberf real experimental applications remains limited. Of par-icular interest is the domain of ultrashort pulseharacterizations.9–15 Such characterizations are essen-ial for controlling ultrashort pulse formation, pulse com-ression, and pulse interaction with matter. The possiblepplications of the FRFT in pulse characterization haveeen mentioned in previous work concerning FRFT.5,16 Noxperimental demonstration has been done. Differentechniques exist to determine the temporal structure ofltrashort pulses. Frequency-resolved optical gating9,10

FROG) involves spectrally resolving the signal beam of

1084-7529/07/061641-6/$15.00 © 2

n autocorrelation measurement. Another family of tech-iques consists of analyzing spectral interferograms.ost common is spectral phase interferometry for direct

lectric field reconstruction12 (SPIDER), but we can fur-her cite spectral interferometry resolved in time14,15

SPIRIT) frequency domain phase measurement. SPIDERnd SPIRIT are based on spectral shearing interferom-try. This measures the interference between a pair ofpectrally sheared replicas of the input pulse. The spec-ral interferogram obtained is then analyzed using a stan-ard Fourier development, and phase retrieval ischieved.17 The aim of our paper is to experimentallyemonstrate the potentiality of fractional-order Fourierransforms in the analysis of chirped ultrashort pulses. Itill be applied to the analysis of SPIRIT interferograms.irst, the fact that a simple FRFT of spectral interfero-rams leads directly to the determination of cubic spec-ral phase coefficients will be demonstrated. A more pre-ise analysis involving fractional-order Fourier seriesxpansion (FRFSE) will then be proposed.18,19 This tech-ique will be shown to allow a precise simultaneous de-ermination of both cubic and quadratic spectral-phaseoefficients. The technique is then used to filter noisyPIRIT trace. Section 2 shows the determination of third-rder spectral-phase coefficients using FRFT. Sensibilityo noise is further discussed. Section 3 presents the prin-iple of FRFSE and its application in the analysis ofPIRIT interferograms (in particular, the simultaneousetermination of quadratic and cubic spectral-phase coef-

007 Optical Society of America

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1642 J. Opt. Soc. Am. A/Vol. 24, No. 6 /June 2007 Brunel et al.

cients of the pulse to be tested). This section demon-trates finally that the technique can be used to analyzend filter noisy experimental interferograms.

. DETERMINATION OF THE CUBICPECTRAL PHASE THROUGHRACTIONAL-ORDER FOURIERRANSFORMATIONhe following definition of the continuous fractional-orderourier transformation will be considered in thisaper1,16,19:

Fa�U�T���Ta� = C�a�exp�i� cot�a�/2�

�2 Ta2�

��−�

+�

U�T�exp�i� cot�a�/2�

�2 T2��exp�− i2�

TaT

�2 sin�a�/2��dT, �1�

here

C�a� =exp�− i��/4 sgn�sin�a�/2��� − a�/4�

��2 sin�a�/2��1/2 . �2�

fractional-order Fourier transformation is applied hereo the function U�T�. C�a� ensures that the energy conser-ation law is satisfied in the fractional domain, a is theractional Fourier order with 0�a�2, Ta is the coordi-ate variable in the fractional domain, and � is an arbi-rary fixed parameter. It has the dimension of variable T.xperimental traces analyzed in this paper are obtained

hrough a SPIRIT apparatus.14 The SPIRIT experiment istechnique based on spectral shearing interferometry. It

s rather similar to SPIDER12 and produces the samepectral interferogram. All of the results presented herean thus be directly applied to some SPIDER trace analy-is. SPIRIT consists of creating two delayed replicas ofhe pulse to be tested. These replicas are then sent with alight angle difference onto a spectroscope that displayswo identical spectra slightly shifted in space. SPIRITetup produces an unstationary interferogram giveny14,15

S��,t� = �E����2 + �E�� + ���2

+ 2�E���E�� + ���cos��� + �t + ��� − �� + ���,

�3�

here E��� is the frequency representation of the pulselectric field, ��� is its spectral phase, � is the amount ofpectral shear, � is the temporal delay between the twoeplicas, and t is time. Note that after some simplifica-ion, the spectral phase difference is expressed as

��� − �� + �� = − �����. �4�

The beating time period is approximately in the rangef a few picoseconds, so the interferogram cannot be mea-ured with a slow integrating detector. We thus record aime sample of the unstationary interferogram using a

onlinear time-gating technique. Numerically, the re-orded spectral interferogram is Fourier transformed, andfter some linear operations, the phase is retrieved fromhe data.

Let us consider pulses that have passed through dis-ersive delay lines or dispersive materials and acquiredecond- and third-order dispersion such that ���12�2���−�0�2+ 1

3�3���−�0�3. In this case, the spectralPIRIT interferogram contains terms of the form cos����t−��2���−�0�−��3���−�0�2�, i.e., fringes with a lin-ar chirp related to the third-order dispersion term. Theres one specific fractional order that transforms the linearhirp into a peak after the FRFT operation. Correspon-ence between the fractional order aopt that needs to bepplied and the cubic term of the spectral phase is

�

�2 tan�aopt�/2�= ��3�, �5�

hich leads directly to the determination of the third-rder dispersion parameter:

�3� =�

��2 tan�aopt�/2�. �6�

We have performed an experimental demonstration ofhe technique. Figure 1 shows the SPIRIT experimentalnterferogram that has been treated. The pulses analyzedre emitted by a Ti:Sa laser emitting 170 fs nJ pulses.he central wavelength of the pulses is 810 nm. The delayetween the pulses is �=2.97 ps, while the spectral shears �=2.87 THz �1 nm�. The pulses have passed through apectral phase mask to acquire a quadratic and a cubicpectral phase that were separately estimated to be�3�=−0.009 ps3, and �2�=−0.055 ps2. We have then cal-

ulated the fractional-order Fourier transformation of thenterferogram. For a specific order, the linear chirp of theignal is transformed into a thin peak. Figure 2 shows theransform obtained for two different orders, a=1.5 and a1.8. The optimum order is a=1.8. A peak is clearly evi-enced. A more precise analysis shows that the peak isell evidenced when 1.8�a�1.86. From relation (6)e can directly deduce that the cubic spectral phase of

he pulses �3� lies within the range of −0.012 ps3

Fig. 1. Experimental SPIRIT interferogram.

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Brunel et al. Vol. 24, No. 6 /June 2007/J. Opt. Soc. Am. A 1643

��3��−0.008 ps3. This domain is in good accordanceith the expected value. To evaluate the sensibility of theethod, we have analyzed different traces, all related to

he same pulse. This is easily carried out from a 2DPIRIT that can record several interferograms related tohe same pulse for different delays of the gate pulse. Weave thus carried the same analysis for two other traces.ubic spectral phase �3� is obtained within the range of

−0.011,−0.007 ps3� and �−0.010,−0.006 ps3�. We can seehat these results confirm the robustness of the analysis.n summary, a simple fractional-order Fourier transformf the SPIRIT interferogram allows for a direct evaluationf cubic spectral phase. We will now show that with aore complex tool (FRFSE), the analysis can becomeore precise and further lead to the evaluation of a qua-

ratic spectral phase.

. ANALYSIS OF THE SPIRIT TRACE WITHHE FRACTIONAL-ORDER FOURIERERIES EXPANSION. Decomposition of Functions on the Basis of ainearly Chirped Function, Mathematical Principle ofecomposition

n analogy to the Fourier series expansion (FSE), it is pos-ible to develop a fractional-order FSE technique. TheRFSE is a powerful tool to analyze or filter chirped sig-als. It is based on the FRFT, and we will preserve theefinition of the FRFT already introduced in the previousection.1,16,19 The FRFSE technique requires first the de-ermination of basis functions. They are noted as n,a�T�,nd can be obtained from an inverse fractional-order Fou-ier transformation of the Dirac function18:

n,a�T� = F−a���Ta − n�0���T�, �7�

here �0 is the central frequency and n= ±1, ±2, . . . de-nes the expansion orders (we will see later some caseshere the orders n are not integers). a is the fractional

rder. Using the definition of the kernel of the con-inuous fractional-order Fourier transformation,e obtain an orthonormal basis of functions

Fig. 2. Two different FRFTs of the SPIRIT interferogram.

. . . ,−1,aN0 �T� ,0,a

N0 �T� ,1,aN0 �T� , . . . �, where each function

n,aN0 �T� is given by

n,aN0 �T� =

C�− a�

�C�− a��

� T

Texp�− i�

cot�a�/2�

�2

��T2 + �n�2 sin�a�/2�

T �2� + i2�nT

T � . �8�

0,aN0 �T� is a linearly chirped function, while other func-

ions are expansions of this ‘‘mother’’ function. N0 giveshe number of expansions, such that �n � �N0. Consideringn aperiodic signal U�T� defined on the interval− T /2 , T /2�, it has been demonstrated in Ref. 18 that itan be decomposed on this basis: The FRFSE of the signal�T� can be written as

U�T� = n=−�

�

CU�n,a�n,aNO�T�,

CU�n,a� =�− T/2

T/2

U�T�n,aNO�T�dT. �9�

¯0,aNO�T� is the complex conjugate of 0,a

NO�T�.n practical cases, reconstruction is made using anite number of expansion functions. We call U�T�he reconstructed function, which is defined by˜ �T�= n=−N

N CU�n ,a�n,aNO�T�. The error of reconstruction

an be computed through the ratio �� T �U�T�U�T��2dT� / �� T �U�T��2dT�. This decomposition can bearried out for each fractional order a �0�a�2�. Decom-osition coefficients CU�n ,a� can indeed be calculated, re-ardless of the value of a. However, when the analyzedignal is composed of linear chirps, only some specificractional orders aopt are related to each of these chirps.his process of decomposition is of particular interest in

heir analysis. Typically, in the case of a linearly chirpedignal, FRFSE is first applied to determine the optimizedractional order that describes this linear chirp. Then, theechnique can be applied to filter noisy signals or inter-erograms.

ig. 3. (Color online) Cartography of the decomposition coeffi-ients �C �n ,a��.

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1644 J. Opt. Soc. Am. A/Vol. 24, No. 6 /June 2007 Brunel et al.

. Analysis of the Experimental SPIRIT Interferogramith Fractional-Order Fourier Series Expansionet us consider again the SPIRIT interferogram of Fig. 1.he FRFSE decomposition technique has been applied tohis trace. As in the previous subsection, the technique isarticularly appropriate to investigate the linear chirp co-fficient, i.e., K=−��3�, and thus the third-order disper-ion term. From a technical point of view, the origin of the

axis is first centered at �0=2327 THz. The first steponsists of determining the decomposition coefficientsS�n ,a� of S��� from Eqs. (8) and (9). They are computednd presented graphically in Fig. 3 for different values ofhe fractional order a between 0.1 and 2. For each value of, the number of decomposition orders is 2N0+1, with0=50 for this plot. The �CS�n ,a��2 coefficients are pre-

ented in color (online) on a 3D plot. The x and y axes rep-esent parameters n and a. The maximum is located atnopt=0, aopt=1). This maximum is due to the envelope ofhe SPIRIT trace. It does not contain any informa-ion about the spectral cubic phase. Two relativeaxima of energy are located at (nopt=−30, aopt=0.2) and

nopt= +30, aopt=1.8). These maxima are linked to the lin-ar chirp function [i.e., the term cos�K�2

¯ �], and thus to

Fig. 4. Determination of the optimum fractional order.

ig. 5. Decomposition coefficients in the optimized basis aaopt.

he cubic spectral phase coefficient �3�. The presence ofwo maxima is not surprising, as the trace contains a co-ine term, and not only a complex exponential. Each rela-ive maximum can be associated with each of the twoomplex conjugate exponentials. Actually, a rapid simula-ion of a SPIRIT trace using relation (3) shows that thealue 1.8 is the one that leads to the right chirp on thePIRIT trace (i.e., higher frequency oscillations for higherulsations), while the other value 0.2 would lead to a posi-ive chirp on the SPIRIT interferogram and is not appro-riate. Figure 4 shows �CS�30,a��2 versus a. We can seehat an optimal fractional order emerges when a=aopt1.8. According to relations (8) and (9), this value of aopt

s obtained when the linear chirp of CS�n ,a� vanishes.rom the expression of n,aopt

No �T�, it leads tocot�aopt� /2� /�2−��3�=0, which is equivaient to rela-

ion (6). We then present in Fig. 5 the decomposition co-fficients for this fractional order aopt=1.8 versus n. Wean observe the emergence of the local peak located atopt=30. As the dimension parameter � is �=20 THz, wealculate with the previous relation that the third-orderispersion parameter is −0.009 ps3, as was estimated inection 2. It must be noted that this determination doesot depend on the second-order dispersion coefficient �2�

or on the time delay between the two replicas of the in-ident pulse. The sole influential parameters are the cubicpectral phase coefficient �3� and the spectral shear �.

The determination of the quadratic dispersion coeffi-ient �2� is linked to the value of coefficient nopt. If thenterferometric component of the SPIRIT trace containednly the linear chirp, i.e., cos�−��3���−�0�2� [without thergument ��−��2���−�0�], the local peak that describeshe linear chirp in the chart would be observed for nopt0, with aopt=1.8 unchanged. The carrier frequency ����2�� of the trace induces a shift. Using the relation (8),

he shift of nopt is directly linked to the carrier frequencyhrough

� − ��2� −2�nopt

�= 0, �10�

here � is the support of the initial trace. In the case ofur experimental interferogram, the local peak is ob-

ig. 6. Decomposition coefficients in the optimized basis aa with higher precision.

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Brunel et al. Vol. 24, No. 6 /June 2007/J. Opt. Soc. Am. A 1645

erved at nopt=30. Support of the trace is �=60.04 THz,pectral shear is �=2.87 THz, and the delay between thewo replicas is �=2.97 ps. We obtain that the second-orderispersion parameter is �2�= �1/����−2�nopt / ��−0.06±0.03 ps2. As the determination of �2� suffers an

mportant imprecision here (it is due to the fact that therder of decomposition n takes only discrete values), theecomposition technique has been carried out using non-nteger orders of decomposition n=k n, where k is an in-eger and n is the n increment that is not necessarily 1it can be 0.1, for example). The use of a noninteger n in-rement just affects the calculus of the coefficientsS�n ,a� to satisfy the Parseval equality. Figure 6 shows

he detail of the decomposition coefficients for the frac-ional order aopt=1.8 versus n. The n increment is nown=0.1, and N0=50 is unchanged. (The number of decom-osition orders is 2N0 / n+1=1001.) A zoom of the peakocated near nopt=30 (see Fig. 7) shows that the maxi-

um of this peak is 29.9. With better precision, we obtainow: �2�=−0.055±0.003 ps2. Note finally that the tech-ique is robust. It gives similar results when applied tother traces related to the same pulse (as in Section 2ith a simple FRFT).

Fig. 7. Zoom of the peak.

Fig. 8. Reconstructed SPIRIT interferogram with N0=50.

In summary, the FRFSE allows a direct determinationf the quadratic and cubic spectral phase coefficients ofltrashort pulses directly from our experimental SPIRIT

nterferogram. This technique consists of localizing rela-ive maxima in a 2D cartography representing fractional-rder Fourier decomposition coefficients. It is thus a veryowerful tool for the analysis of SPIRIT interferograms.

. Filtering of the SPIRIT Interferograme now present a last potentiality of the technique: the

ltering of spectral interferograms. The different coeffi-ients CS�n ,aopt� of the experimental SPIRIT trace haveeen calculated in Subsection 3.B (see Fig. 5). Using rela-ions (8) and (9), it is now possible to reconstruct thePIRIT trace. Figure 8 shows the reconstructed function.he mother basis function is 0,aopt

N0 �T� with N0=50. Thiseans that reconstruction has been made, taking into ac-

ount the coefficients between n=−50 and n=50. The n in-rement is n=1. We can see that some noise present onhe trace has been removed. Oscillations are more regu-ar. In summary, the reconstruction process allows noiseltering. This can, however, be avoided if necessary. Weave carried the reconstruction process using 201 ordersf decomposition �N0=100�. Reconstruction is then excel-ent with an error coefficient as low as 0.78%. In this case,he reconstruction leads to a trace identical to the originalrace (not reported here), with all defaults reproduced.

In conclusion, depending on the number of orders thatre used, it is possible to filter the SPIRIT interferogram.ote that in the present case, the number of orders (101

r 201) does not affect the determination of spectral coef-cients. Let us mention, however, that the techniqueould be of particular interest in the case of highly noisynterferograms. It is well known that the noise and theow-intensity signal present in the wings of SPIRIT orPIDER interferograms can induce some errors in phaseetrieval. Using a relatively low number of decompositionoefficients, the central part of the SPIRIT interferograman be reconstructed faithfully, while noise present in theings is attenuated. Using reconstruction with theRFSE technique, it should be possible to take into ac-ount less of the noise present in the low-intensity signal,hus improving the accuracy of phase retrieval. This wille investigated in our future work. In the previous sec-ions, the FRFSE technique had been demonstrated to al-ow for precise analysis of SPIRIT interferograms. It isurther well adapted to filter and reconstruct noisy ex-erimental interferograms.

. CONCLUSIONn conclusion, we have presented, to the best of ournowledge, the first experimental demonstration ofractional-order Fourier analysis in ultrashort pulse char-cterization. SPIRIT interferograms have been consid-red. The fractional-order Fourier transformation haseen shown first to naturally allow the determination ofubic spectral phase coefficients of the pulses to be ana-yzed. Fractional-order Fourier series expansion has theneen demonstrated to allow a simultaneous determina-ion of both cubic and quadratic spectral phase coeffi-ients of the pulses. This latter technique consists of lo-

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1646 J. Opt. Soc. Am. A/Vol. 24, No. 6 /June 2007 Brunel et al.

alizing relative maxima in a 2D cartographyepresenting fractional-order Fourier decomposition coef-cients. It can be further used to filter noisy experimentalPIRIT traces. This should be of particular interest to im-rove the accuracy of phase retrieval. Fractional-orderourier analysis should become a powerful numerical tool

or the analysis of ultrashort pulses. It could complete ef-ciently existing numerical tools, particularly when sig-ificant noise is present.

CKNOWLEDGMENTShe authors thank T. Mansuryan and L. Mouradian fromhe Ultrafast Optics Laboratory (YSU, Armenia) for thessistance that they brought during SPIRIT experimentalecordings, and D. Lebrun for encouraging this work.

M. Brunel’s e-mail address is marc.brunel@coria.fr.

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