PHYSICAL REVIEW E 68, 026604 ~2003!

Tracking ultrashort pulses through dispersive media: Experiment and theory

H. Gersen,* J. P. Korterik, N. F. van Hulst, and L. Kuipers†

Applied Optics Group, Department of Science & Technology and MESA1 Research Institute, University of Twente, P.O. Box 217,7500 AE Enschede, The Netherlands

~Received 19 March 2003; published 15 August 2003!

We report on the direct visualization of a femtosecond pulse propagating through a dispersive waveguide ata telecom wavelength. The position of a propagating pulse is pinpointed at a particular point in space and timeusing a scanning probe based measurement. The actual propagation of the pulse is visualized by changing thereference time. Our phase-sensitive and time-resolved measurement provides local information on all proper-ties of the light pulse as it propagates, in particular its phase and group velocity. Here, we show that the groupvelocity dispersion can be retrieved from our measurement by developing an analytical model for the mea-surements performed with a time-resolved photon scanning tunneling microscope. As a result, interesting anduseful effects, such as pulse compression, pulse spreading, and pulse reshaping, become accessible in the localmeasurement.

DOI: 10.1103/PhysRevE.68.026604 PACS number~s!: 42.25.2p, 07.60.Ly, 07.79.Fc, 42.65.Tg

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I. INTRODUCTION

Light as an electromagnetic wave is characterized bcombination of time varying electric and magnetic fielpropagating through space. In the past decades the ustime varying optical fields in the form of ultrashort lighpulses has proven to be a valuable tool in the study oftrafast phenomena@1#. The attractiveness of ultrashort lighpulses not only lies in the possibility to trace processestheir ultrafast dynamics but also in the fact that one simcan do things faster. Of primary importance are data tranand data processing utilizing the large attainable bandwiIn this respect, probably one of the most spectacular goathe creation of an all-optical computer.

Optical delay elements capable of transmitting ultrashpulses will be one of the key ingredients for all-optical daprocessing@2#. Photonic crystals~PhCs! in which photonsexperience multiple reflections are one of the promising cdidates for this task, as the group velocity in PhCs canseveral orders of magnitude lower than in bulk materiwith the same refractive index@3–5#. PhCs are materialswith spatial periodicity in their refractive index and havdispersion properties that can be tailor made by selectheir scale and geometry@6#. In this way it becomes possiblto control the group and phase velocity of ultrashort ligpulses in a unique way. The unique dispersion characterisof PhCs may lead to a variety of interesting optical nonlinphenomena such as gap solitons@7,8#, pulse splitting of gen-erated second-harmonic pulses@9#, highly efficient wave-length conversion, and can also be employed for dispermanagement in optical telecommunications.

To integrate ultrashort pulses with various evolving phtonic technologies, it is necessary to analyze the interacof ultrashort optical pulses with these photonic devic

*Corresponding author. Email address: h.gersen@tnw.utwent†Present address: Nanophotonics Group, FOM–Institute

Atomic and Molecular Physics~AMOLF!, Kruislaan 407, 1098 SJAmsterdam, The Netherlands.

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Many investigations on dynamical effects in~non!linear dis-persive media are based on numerical simulations@10–12#.We intend to gain a better understanding of the interactionlight with ~sub!wavelength sized structures by experimenmeans.

Experimental investigations on the propagation oftrashort pulses in~non!linear dispersive media have so falargely been limited to ‘‘black-box–type’’ characterization othe medium@5,13–17#. The known incoming pulse and thmeasured transmitted or reflected pulse are compared wtheoretical model. By cutting slices from the medium~‘‘cut-back’’ method! it becomes possible to study aspects of tinternal pulse development@18#; yet this method is destructive and has several other clear drawbacks. First, not evphotonic structure can be arbitrarily changed in length wiout affecting its properties. Second, if a disagreementtween experiment and theory is found, it may be hard to fithe underlying cause for the discrepancy. Third, it is imptant to realize that the black-box method integrates allpulse propagation effects accumulated in the entire structIf a structure has spatially varying optical properties, onaveraged information is obtained. To overcome these drbacks and obtain fulllocal information on pulse propagatiothroughout a medium, local time-resolved measurementscrucial @15#. An experimental method enabling the observtion of dynamical effects directly inside a photonic structuhas only recently been demonstrated@19#.

In the last few years a so-called interferometric photscanning tunneling microscope~PSTM! has proven itsunique capacity to measure the amplitude and phase dbution of the optical field locally inside photonic structur@20–22#. Here, we report on the nondestructive visualizatiof a femtosecond pulse propagating through a linear dissive waveguide in space and time using an interferomePSTM at infrared wavelengths. The ability to measure awavelength range relevant for telecommunications opensthe possibility to investigate local dynamic behavior of phtonic crystals and integrated optical circuits in the nearture. An advantage of the short laser pulses applied isdifferent effects such as reflection at the end facets~Fabry-

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GERSENet al. PHYSICAL REVIEW E 68, 026604 ~2003!

FIG. 1. ~Color! Schematic representation of a pulse trackiexperiment. The evanescent field of the pulse traveling insidewaveguide is picked up by a fiber probe with subwavelengthmensions. The photon tunneling signal picked up by the probinterferometrically mixed with part of the same pulse that hpropagated through the reference branch. The length of the rence branch, and thus the time taken by the reference pulse to tthrough this branch, is controlled by an optical delay line. Easubsequent measurement shown in this paper is obtained by rscanning the optical probe across the photonic structure whileheight above the structure (,10 nm) is kept constant by a feedbacmechanism. The simultaneous measurement of optics and topophy makes it possible to directly relate optical information to tstructural properties.

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Perot effect! and scattering out of the structure can be dcriminated in the time domain.

The microscope used is based on the heterodyne intermetric PSTM, recently developed by our group@23# and pro-vides the full amplitude and phase information of a putraveling through a waveguide structure. From these timdependent and phase-sensitive measurements, both the

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group and the phase velocity can be determined direcComparison with an analytical model, developed in this pper, shows excellent agreement. The model proves that ethe local group velocity dispersion~GVD! can be measuredinside a photonic structure. As a result effects such as pbroadening, pulse compression, and pulse reshaping efbecome accessible in the local measurement.

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FIG. 2. ~Color! A time-resolved heterodyne interference PSTM measurement on a Si3N4 channel waveguide for a fixed position of thoptical delay~image size of 290.539.38mm2). Linearly polarized light has been coupled in the waveguide to excite the TE00 mode.~a! Thetopography of the waveguide. The measured height and width of the waveguide are 3963 nm and 1.1560.04mm. The slab thickness isdetermined to be 17065 nm. ~b! The raw data from the lock-in amplifier, corresponding to the optical amplitudeA times the cosine of theoptical phase for a single tracked pulse as a function of the lateral position in the plane of the sample.~c! The optical field amplitude of thepulse while propagating through the waveguide as derived from~b!. It is apparent that the amplitude is confined to the waveguide.~d! Thecorresponding cosF of the optical field for the complete measurement.~e! A zoom-in on the boxed area given in~d! to clearly show theindividual wavefronts~image size: 72.639.38mm2). It is clear that the wave fronts in the image are straight, indicating plane wpropagation corresponding to the TE00 mode.

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TRACKING ULTRASHORT PULSES THROUGH . . . PHYSICAL REVIEW E68, 026604 ~2003!

II. EXPERIMENTAL ASPECTS

A schematic overview of the pulse tracking experimendepicted in Fig. 1. Linearly polarized light is coupled intointegrated waveguide structure and propagates throughphotonic structure. In the PSTM, the local evanescent fiabove the structure, with a decay length of typically 10–1nm, is picked up by a fiber probe with subwavelength dimsions. The evanescent field is locally converted into a progating wave, which is coupled into the fiber probe, guidthrough the fiber, and subsequently detected. For each ption on the photonic structure, the phase and time informtion of the evanescent field is the same as that of the progating wave. By picking up the evanescent field, dirinformation is therefore obtained on the propagating ligfield. By raster scanning the probe, using a height feedbmechanism, the optical field distribution is probed in the nfield of the waveguide surface, while simultaneously the sface topography is acquired.

As an electromagnetic field is characterized by an amtude, a phase, and a polarization state, it is essential tocoherent detection methods. Therefore the sample andPSTM have been integrated in one branch of a MaZehnder–type interferometric setup as shown in Fig. 1.each position on the sample surface, the photon tunnesignal picked up by the near-field probe is interferometricamixed with light split from the same laser source that hpropagated along the reference branch of the setup. Thterference between light in the signal and reference brancmeasured with a photodetector. The optical frequency ofreference beam is shifted with the difference frequency ofkHz between two acousto-optic~AO! modulators to allowheterodyne detection of the photon tunneling signal. Thesulting signal, measured with a dual-output lock-in amplifi~LIA !, contains both the optical amplitude and relative phof the local optical field within the sample@23#.

In order to visualize the propagation of a femtosecopulse, we incorporated an optical delay line in the referebranch of the interferometric setup. Optical interference wonly occur when there is temporal overlap between the pin the signal and the pulse in the reference branch atpoint where the branches join again. In the current expment the photon-tunneling signal and the reference sigrecombine in a 50/50 fiber coupler after propagating throudifferent lengths of a single-mode fiber and other bulk opcal glass components. The pulse in the signal branch tra69.760.5 cm through dispersive fibers and other glass coponents~including the sample!, and 23.560.2 cm throughair. For the reference branch these distances are60.5 cm and 75.660.5 cm, respectively. Note that in threference branch the path length in air is changed as a ftion of the position of the optical delay line. The positionthe optical delay determines the optical path length ofreference branch, and with this defines a reference timethe measurement.

As a model system for our measurements, we have usSi3N4 planar channel waveguide fabricated in a Si3N4 /SiO2layer system on a Si substrate. The experimentally demined slab thickness, width, and height of the structure

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17065 nm, 1.1560.04mm, and 3963 nm, respectively.Linearly polarized light with a center wavelength of 13062 nm has been coupled into the channel waveguide wipolarization parallel to the sample plane, such that onlyfundamental TE00 mode is excited. This is, as determinedan effective index method, the only mode supported bywaveguide for the measured waveguide parameters. All msurements have been performed by raster scanning ancoated fiber probe with the fast axis along the waveguchannel (line frequency50.098 Hz), for a fixed position ofthe optical delay line in the reference branch.

The pulses launched into the photonic structure are gerated by a Ti:sapphire–pumped optical parametric oscilla~Spectra-Physics Opal!. The laser system has a repetition raof 80 MHz. The arrival time~12.5 ns! between subsequenlaser pulses is such that the pulse picked up by the probeonly interfere with part of the same laser pulse that hpropagated along the reference branch. The pulse duratiothe input field Ẽ(t) is measured by a conventional bacground free intensity autocorrelation technique and yieldFWHM of the pulse amplitude of 12363 fs @1#.

III. PULSE TRACKING IN A WAVEGUIDE

A local time-resolved heterodyne interference measument of a short optical pulse inside our model systempresented in Fig. 2 for a fixed position of the delay linFigure 2~b! shows the raw data obtained with the LIA, whiFig. 2~a! shows the simultaneously acquired topography. Tsignal from the LIA corresponds to the measured optical aplitude A times the cosine of the phase of the optical fie(cosF) of a single tracked pulse, as will be discussed laNote that a scanning probe based technique is inhereslow so that averaging over many individual laser pulstakes place. From this measurement the optical amplitand the cosine of the phase of the optical field of the pucan be separated, as given in Fig. 2~c! and 2~d!, respectively.The pattern in Fig. 2~c! shows a roughly Gaussian shapalong the propagation direction with a full width at hamaximum ~FWHM! of 66.8mm. The profile of the ampli-tude in the direction perpendicular to the propagation dirtion corresponds to the mode profile of an excited TE00mode. The cosF term of the optical field as depicted in Fig2~d! shows a clear fringe pattern as a function of the lateposition in the plane of the sample. To show the individuwavefronts, a zoom in on the boxed area in Fig. 2~d! isdepicted in Fig. 2~e!. It is clear that the wavefronts in thimage are straight, also indicating single-mode propagatFor multiple modes with different effective indices, the smultaneous detection would lead to a beating pattern inA cosF image@20,23#.

The line profile of Fig. 3~a! shows the measuredA cosFalong the dashed line in Fig. 2~b!. A zoom in on a part of theline profile is depicted in Fig. 3~b!, which shows that indi-vidual fringes are clearly resolved. A simple Fourier tranform @Fig. 3~c!# of the data along the waveguide sufficesdetermine which wavelengths are contained in the fringe ptern. We find a central wavelength of 87162 nm inside thestructure associated with the central optical frequencyv0 of

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GERSENet al. PHYSICAL REVIEW E 68, 026604 ~2003!

the spectrum of the femtosecond laser pulse. This walength corresponds to a phase velocityvf of 2.0160.023108 m/s (neff51.4960.01).

We directly measured the velocity at which the carrifrequency cycles move forward locally inside the waveguiHowever, the envelope of the pulse moves with a differvelocity. In Figs. 4~b!–~h! the measured optical amplitudefor seven subsequent measurements with increasing rence times are shown. For each reference time a Gausenvelope is found, which reveals the position of the puthereby pinpointing the pulse position for each referentime. As time passes, the pulse is found further alongwaveguide. The speed at which the pulse envelope progates can be directly determined from the positional chaof the center of the pulse during the known time intervBetween each measurement, the reference time is sh20062 fs by lengthening the reference branch by 6060.6 mm. The linear dependence of the position of the puin the waveguide as a function of the reference time~Fig. 5!shows that the pulse propagates locally with a constant grvelocity. From the slope of the fitted straight line, we findgroup velocityvg of 1.6760.03310

8 m/s.The measured group and phase velocity can be comp

to the values calculated by an effective index method. Thcalculations are based on measured refractive indices foSi3N4 and SiO2 layers grown in our institute in the wavelength range 600–1600 nm. The calculations use the locmeasured width and height of the ridge of 1.1560.04mm

FIG. 3. ~a! The optical amplitude times the cosine of the opticphase for a pulse as measured through the heart of the wavefor the entire scan range@Fig. 2~a!#. ~b! A zoom-in on a part of theline profile of ~a! from which a period of 87162 nm can be deter-mined by a simple Fourier transform as seen in~c!. The periodicityof the wave fronts yields the wavelength inside the structure aciated with the optical carrier-frequency cycles. This wavelencorresponds to a phase velocityvf of 2.0160.02310

8 m/s.

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and 3963 nm, respectively. The calculated effective refrative indicesne f f for the TE00 mode at the wavelength rang600–1600 nm has been fitted by a quadratic function. Tderivative of this function at 1300 nm then gives a reasable approximation to the group velocity vg51.563108 m/s, while the refractive index itself yieldsphase velocityvf51.95310

8 m/s, both in reasonable agreement with the measured values.

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FIG. 4. ~Color! Pulse tracking experiment for different referentimes. Linearly polarized light has been coupled in the waveguto excite the TE00 mode.~a! The simultaneously measured topogrphy ~image size: 262.238.45mm2). ~b!–~h! The optical field am-plitude as measured by the instrument for different positions ofoptical delay line. From~b! to ~h! the optical path length of thereference branch is increased in steps of 6060.6 mm. This resultsin steps of the reference time of 20062 fs. The measuremenshows that the position of the pulse at a reference time canpinpointed in space. The pulses can be seen to propagate thrthe structure as a function of time giving a direct local measuremof the group velocity.

FIG. 5. Measured position of the pulse as a function ofposition of the optical delay line~crosses!. The letters correspond tothe subsequent measurements as given in Fig. 4. The solid strline represents a least-squares fit to the measured points. Theof the line shows a local value of the group velocityvg for the pulseof 1.6760.033108 m/s.

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TRACKING ULTRASHORT PULSES THROUGH . . . PHYSICAL REVIEW E68, 026604 ~2003!

IV. THE EFFECT OF DISPERSION

In the preceding section we have demonstrated howcan locally measure the group and phase velocity of a progating ultrashort pulse. However, we must take into accothat the temporal shape of a femtosecond optical pulsaltered while it propagates through a dispersive sample.‘‘dispersive’’ in this context we mean any linear systemwhich the propagation constantb(v) as a function of fre-quency has any form other than a straight line throughorigin, i.e., b5v/c. Hereb(v) represents the propagatiocharacteristics of the guided mode in the channel wavegand can be written as a Taylor expansion about its valuv0 with the derivativesb8[db/dv and b9[d

2b/dv2

evaluated atv5v0.This b(v), which defines the dispersion characteristics

exactly the property we would like to measure locally. Qunaturally, we can write vg(v0)[1/b8(v0) and vf[v0 /b(v0) for the locally measured group and phase vlocity, respectively. In the following, we will show that alsthe GVD b9 can be retrieved from our measurement by dveloping an analytical model for the measurements pformed with a time-resolved PSTM. As a result, interestand useful effects such as pulse compression, pulse sping, and pulse reshaping become accessible in the local msurement.

Dispersive broadening of optical pulses is a well-studsubject found in text books@24#. Here, we will discuss theeffects of dispersion to have a starting point for modellithe measurements performed with a time-resolved PSThe interferometric PSTM is modeled~see Fig. 6! as a het-erodyne Mach-Zehnder interferometer consisting of tsingle-mode fibers of unequal lengths and an air path inarm to compensate the unequal lengths, while the photstructure under study is present in the other path. In this,

FIG. 6. Mach-Zehnder model for the time-resolved photon scning tunneling microscope. The fiber length differenceDzf iber be-tween the two arms is compensated by the air path length differeDzair . Heterodyne detection is applied, using acousto-optic molation of the light in the reference branch. The resulting interferesignal is measured with a LIA and allows one to extract the amtude and phase of the local optical field: AO stands for acouoptic modulator; and LIA for lock-in amplifier.

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photonic structure is a dispersive element which can hdispersive properties different from the single mode fibeBy incorporating the single-mode fibers into the model,take into account that dispersive components other thansample are present in both the signal and reference branc

A. Heterodyne detection

To begin, let us consider a Mach-Zehnder interferomewith heterodyne detection. The input fieldẼ(t) is split by thebeam splitter into the fields that travel through the two arof the interferometer. The field through the sample is desnated Ẽs(z,t), and the delayed field through the referenbranch is calledẼr(t2t). In this thez dependence is in-cluded to represent the fact that the near-field probe is ming along the sample, and thus includes additional dispermedium as a function of position, whilet represents theoptical delay in the reference branch in the usual fashion.use a complex representation of the fields and separatetwo interfering pulses into their respective amplitudesA andphasesB,

Ẽs~zs ,t !5As~zs ,t !exp@ iBs~zs ,t !#,

Ẽr~ t,t!5Ar~ t,t!exp~ iDvt !exp@ iBr~ t,t!#, ~1!

where the term exp(iDvt) is introduced to account for thefrequency shift in the reference branch due to the AO molators. At the output of the interferometer the interferencethe two fields is averaged over the response timeTresp of thedetector,

I ~zs ,t!}1

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2~zs ,t !12Ar~ t2t!

3As~zs ,t !cos$Br~ t2t!2Bs~zs ,t !1Dvt%#dt.

~2!We see that the local intensity on the detector will conta

in addition to the average intensitiesuAr u2 and uAsu2 associ-ated with the two beams separately, an interference termportional to the product of the amplitudes in the respectbranches. To measure this interference term, the detectednal I (zs ,t) is passed through a dual output lock-in amplifiewith a phase delay of 90° between both outputs, usingAO frequency shift ofDv as a reference. The phase varition due to this frequency shift is very small compared toresponse time of the detector, in other wordsTrespDv!2p, so thatDv can be considered constant when evaluing the integral. This, combined with the fact that for shopulsesTresp@tp holds, yields

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GERSENet al. PHYSICAL REVIEW E 68, 026604 ~2003!

for the two LIA output signals. These two signals are usedextract both the amplitude and the phase of the local optfield independently, as will be discussed in Sec. IV C. Nthat although not explicitly stated above, a train of pulsarrives on the detector during the integration time oflock-in amplifier, which as a whole generates the signal afrequency ofDv. As a result, averaging takes place ovmany individual laser pulses that are assumed to be ident

B. Pulse propagation through a dispersive medium

We will analyze some of the effects that can arise in pupropagation through linear systems in terms of Gausspulses. Such pulses are simple, mathematically tractable,clearly exhibit all the essential physical features@24#. Westart with an optical pulse with a carrier frequencyv0 and acomplex Gaussian envelope written in the form

Ẽ0~ t !5e2G̃0t

21 iv0t⇔Ẽ0~v!5ApG̃0

e2(v2v0)2/4G̃0, ~4!

where G̃05a2 ib is related to the initial pulse widthtp5@2ln(2)/a#1/2 through the parametera, while the parameterb is a measure for the frequency chirp on the pulse.

The effect of dispersion is best described in the frequedomain to ensure that the total spectral content of the premains the same. The pulse intensities used in the exment are low enough to prevent nonlinear processes sucself-phase modulation. Effects of frequency-dependent gor loss in the medium will be neglected in this analysis. Aresult, the spectral content of the pulses does not changeither of the branches of the interferometer. The output puspectrumẼ(v) after propagating a distancez through a dis-persive medium will be the input spectrumẼ0(v) multipliedby the frequency-dependent propagation constantb(v)through the system

Ẽ~z,v!5Ẽ0~v!exp@2 ib~v!z#. ~5!

Note that in our case different dispersive media~sample andfiber! are present in the sample arm of the interferomeThis can simply be included in the model by multiplying thoriginal spectrum with the different frequency-dependpropagation constants for the respective media. The resusimply a summation over the different propagation constawhich is not explicitly written down in the following analysis.

In order to allow for an analytical calculation of thpropagation effects, the propagation constantb(v) is writtenas a Taylor expansion about its value atv0. With this expan-sion the pulse spectrum can be written as

Ẽ~z,v!5ApG̃0

expF2 ib~v0!z2 ib8z~v2v0!2H 1

4G̃01

ib9z

2 J ~v2v0!2G . ~6!

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The inverse Fourier transform of Eq.~6! describes the timeevolution of the electric field of the pulse as a function of tlength of the dispersive medium@24#. We carry out thistransformation explicitly and obtain the output pulse aftraveling an arbitrary distancez through a dispersive medium

Ẽ~z,t !5AG̃~z!G̃0

expF iv0S t2 zvfD2G̃~z!S t2 zvgD2G ,

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where 1/G̃(z)51/G̃012ib9z is an altered pulse propagatioparameter. The output pulse is still a Gaussian pulse, but

an altered pulse propagation parameterG̃(z) at the output ofthe system. From Eq.~7! it is clear that the carrier-frequenccycles within the pulse propagate at the phase velocityvf ,while the pulse envelope itself propagates at the grouplocity vg evaluated at the center of the pulse spectrum. Tpulse envelope changes in shape with distance because oGVD b9, which is basically a variation of group velocitwith frequency.

C. Analytical model of a time-resolved PSTM

In this section we derive an analytical model for the uof coherent detection methods to locally measure the tevolution of a short light pulse. In our experiment unchirp

Fourier limited pulses are coming from the laser, so thatG̃0PR. To write the signal measured by the lock-in amplifi@Eq. ~3!# in terms of short laser pulses, we start the analyby separating Eq.~7! for a pulse propagating through a dipersive medium in its respective amplitude and phase,

Ẽ~z,t !51

A4 11~2b9zG0!2expF2 G0

11~2b9zG0!2

3S t2 zvgD2GexpF i2arctan~22b9zG0!

1 iv0S t2 zvfD1 2ib9zG02

11~2b9zG0!2 S t2 zvgD

2G .~8!

From this equation it can clearly be seen that as a resuthe GVD, the pulse is broadened upon propagation throthe dispersive medium. In fact, the factor@11(2b9zGo)

2#1/2 corresponds to the broadening of thFWHM of pulses upon propagation through a dispersive mdium. The phase, i.e., the imaginary part in Eq.~8!, containsa quadratic time dependence which corresponds to a linfrequency chirp in the pulse due to the GVD. Note that

contrast to Eq.~7! the above equation is only valid forG̃0PR as this simplifies the mathematics considerably;though Eq.~8! could as well be derived for a complexG0,this is not needed for our unchirped input pulses.

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TRACKING ULTRASHORT PULSES THROUGH . . . PHYSICAL REVIEW E68, 026604 ~2003!

By substituting the expression for a pulse propagatthrough a dispersive medium in Eq.~3! for the respectivebranches of the interferometer, we obtain

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3cos$Y1k rG r~ t2j r !21ksGs~ t2js!

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Y52arctan~k r !1arctan~ks!

22v0S t1 zrvfr 2 zsvfsD .

Here, A corresponds to an amplitude withAr and As con-stants that correct for the different optical power densitythe two branches,G r ,Gs to the real part of a modified pulspropagation parameter,j r ,js to an optical path length,k r ,ksto a GVD dependent parameter, andY is simply a time-independent variable in the optical phase. To interpretmathematical result, we carry out the integration expliciusing the following standard integral@25#:

E2`

`

exp@2~ax212bx1c!#cos~px212qx1r !dx

5Ap

A4 a21p2expFa~b22ac!2~aq222bpq1cp2!

a21p2G

3cosH arctan~p/a!22

p~q22pr !2~b2p22abq1a2r !

a21p2J @a.0#. ~10!

For the other channel of the LIA, the cosine in Eqs.~9a! and~10! can simply be replaced by the sine term@25#. For ourpurposes it is convenient to write this equation directlyterms of its respective magnitudeA and phaseF in thefollowing manner:

02660

g

is

A5AV1~zs ,t!21V2~zs ,t!2

5Ap

A4 a21p2expFa~b22ac!2~aq222bpq1cp2!

a21p2G ,

~11!

and

F5arccosFV1~zs ,t!A G5H arctan~p/a!2 2p~q22pr !2~b2p22abq1a2r !a21p2 J ,

~12!

where theV1(zs ,t),V2(zs ,t) are the two LIA output chan-nels and show the relation with the measured signals. Sstituting

a5G r1Gs , b52G rj r2Gsjs ,

c5G rj r21Gsjs

2 , p5G rk r1Gsks , ~13!

q52G rj rk r2Gsjsks , r 5G rj r2k r1Gsjs

2ks1Y

in Eqs. ~11! and ~12! and some minor rewriting yields thexpressions for the amplitude and phase of the pulse as msured by the time-resolved PSTM,

A~zs ,t!5ArAsAp/G0

A4 41~k r1ks!2expF22G0~j r2js!2

41~k r1ks!2 G ~14!

and

F~zs ,t!5Y11

2arctanH G rk r1GsksG r1Gs J

2H G0~j r2js!2@k r1ks#41~k r1ks!

2 J , ~15!where the variables with the subscripts are the ones thachange as a function of the position of the probe. Note tthe optical delay line is kept at a fixed position for eaindividual measurement, so thatt is, in fact, constant.

With Eq. ~14!, we have an analytical expression for thmeasured pulse amplitude. It can be seen that the measlength of the pulse~see Fig. 2! in the waveguide is influ-enced by both the original pulse length as well as byGVD in the two branches of the interferometer. To getmore qualitative feeling for the influence of the GVD on thmeasured FWHM, we depicted the optical amplitudes callated by Eq.~14! for different values of the GVD rangingfrom 0 to 150 ps2/km in Fig. 7. In this calculation the measured fiber lengths in both branches, as given earlier, wused as an input. In Fig. 7 we see Gaussian pulse envelofor which the FWHM increases as function of the amountGVD in the system. From the resulting Gaussian shapecan conclude that the pulse shape is almost completely

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GERSENet al. PHYSICAL REVIEW E 68, 026604 ~2003!

termined by the exponent in Eq.~14!. Therefore, we canwrite the following for the FWHM of the Gaussian pulse:

tpulse~zs!5vgsAln 2@41~k r1ks!2#G0 , ~16!measured along the propagation direction. From this expsion it is clear that the measured FWHM is given by toriginal pulse length corrected for the GVD in the twbranches of the interferometer. Note that thek r and ks asgiven by Eq.~9b! are opposite in sign. So if the optical fibein the two branches are equal in length and have the sGVD, their effect on the measured FWHM of the amplituand phase cancels, as is obvious from Eqs.~14! and~15!. Theterm in front of the exponential in Eq.~14! is found to inducea slight asymmetry in the Gaussian shape, only visible wvery large values for the GVD are used as input forcalculations. In the experimental setup, an additional incsion of 300mm equivalent to the total scanrange in tsample branch yields a negligible reduction of the prefacof only 0.0395% for a GVD of 75 ps2/km.

However, as can be seen in Fig. 7, the maximum visiamplitude of the pulses reduces as a function of the Gdue to this prefactor. To get a feeling for this dependenFig. 8 shows the calculated visibility of the fringe pattern fthe fiber length difference between the two branches usdifferent values of the GVD ranging from 0 to 300 ps2/km.The amplitude at the center of the pulse when no dispermedium is present in either of the two branches is define100% visibility. Plotted is the amplitude at the center of tpulse for different amounts of dispersive medium. It is clethat the stretching of the pulses in the two branches hastrong influence on the visibility of the fringes. Note thatequal amount of dispersive medium in both branches ha

FIG. 7. ~Color! Calculated optical amplitude@Eq. ~14!# for ourexperimental setup for different values of the GVD in our systeThe GVD parameter ranges from 0 to 150 ps2/km in steps of25 ps2/km. Other experimental parameters used in this calculaare given in the text. A Gaussian pulse shape can be observedwhich the FWHM strongly depends on the group velocity dispsion in the two branches of the interferometer.

02660

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influence on the detected signal. For optimal detection eciency, the dispersive media in the two branches shoclearly be balanced.

As we are interested in the local optical properties ofphotonic structure, we started this discussion with the fthat the photonic structure is present in the sample arm ofinterferometer. The different material contributions duethe b(v) can simply be added as can be seen from Eq.~5!.So if the two fibers and other dispersive elements~excludingthe sample itself! are equal in the two branches, then the on

.

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FIG. 8. ~Color! The imbalance between the amount of dispersmedia in the two branches of the interferometer has a strong inence on the visibility of the fringe pattern as can be seen incalculation. Depicted is the calculated visibility of the fringe patteas a function ofDzf iber ~see Fig. 6!, using values of the GVDranging from 0 to 300 ps2/km. For optimal detection efficiency, thdispersive media in the two branches should clearly be balanc

FIG. 9. ~Color! Direct comparison between the measured~solidblack line! and calculated~red line! optical amplitude. An excellentagreement is obtained for a GVD parameter of 72.4 ps2/km as canbe seen here. The calculated phase@Eq. ~15!# neglecting the rapidoscillating term aroundv0 for this GVD parameter shows thatsmall chirp in the fringes is to be expected in the measurement.right axis represents the amount of periods changed due to theear chirp. This expected chirp is small compared to the currnonlinearity in the probe movement.

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TRACKING ULTRASHORT PULSES THROUGH . . . PHYSICAL REVIEW E68, 026604 ~2003!

remaining contribution due to the GVD is from the sampitself. Therefore, the GVD of the structure under study cdirectly and locally be determined from the locally measuamplitude and optical phase of the pulse. Even if brancare not compensated~excluding the sample!, a referencemeasurement without the sample would yield the disperscharacteristics of the setup such that a direct fit using~14! would yield the local group velocity dispersion of thsample.

V. DISCUSSION

In the experiment the 12363 fs pulse travels through dispersive fibers and other glass components in the two armthe interferometer. In Eq.~16! we have seen that the FWHMof the pulse along the waveguide is determined by theferences between the two branches and the original plength. In the experiment we have measured a FWMH althe propagation direction of 66.8mm @see Fig. 2~c!#. Underthe assumption of a homogeneous GVD in our system, githe difference in fiber length of'30 cm, we can reproducthe measured results with a GVD value of 72.4 ps2/km. Adirect comparison for the measured and calculated pulseplitude is given in Fig. 9. The black line in this graph showthe measured optical amplitude through the heart ofwaveguide given in Fig. 2~c!, while the red line representthe calculated amplitude. It is clear that an excellent agrment is obtained between experiment and theory. The msured GVD value is in the range expected for single-mofibers for infrared wavelengths, although the value appearbe high, as typical values are smaller then 20 ps2/km @26#.This high value could be explained by the fact that othoptical components with unknown dispersion characterisare also presents in both the branches.

As a result of the GVD, the pulses in the reference asample branches have been stretched considerably. Theculated value for the GVD shows that the nearly Fourlimited 12363 fs laser pulses have lengthened to a 1144signal pulse and a 640 fs reference pulse by the time tinterfere with each other in the fiber coupler of the interfemetric PSTM. The stretching of the pulses has a large inence on the visibility of the signal as demonstrated by FigIn the experiment the pulses propagate through 660.5 cm and 38.560.5 cm of dispersive fibers and otheglass components in the two branches, respectively. Basethe theoretical model, the visibility in the experiment wtherefore reduced to about 65% at the time of measuremeBalancing the dispersive media in the two branches wclearly lead to an increase of the visibility.

Until now, we have mostly discussed the measured amtude of the pulses. However, our detection scheme is finterferometric, so information about both amplitude and otical phase is obtained. As demonstrated, the distancetween the fringes is a direct measure for the local wavelenand therefore yields the phase velocity. The expression@Eq.~15!# for the optical phase of the pulse, however, containquadratic term corresponding to a linear frequency chirpthe pulse. In Fig. 9 we also plotted the calculated phase@Eq.~15!# divided by 2p while neglecting the rapid oscillating

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term aroundv0 from Y @see Eq.~9b!#. The total phasechange within the pulse duration for the experimental cfiguration is roughly 0.1 periods over the measured FWHof 66.8mm which corresponds to 77 periods of the opticfield. In other words, the periodicity in the center of the puwith respect to the wings would differ by only 2–4 nm. Tdirectly measure the linear chirp on the optical pulse, a smdifference of 0.1–0.3 % in the periodicity must be resolveIn principle, such small differences can be directly measurhowever, a highly linear scan system is required. At tstage the linearity of the probe movement in our setup, whhas a position feedback system to compensate for thelinearity of the piezoelectric scanner, is of the same ordemagnitude~less than 0.2%).

It is tempting to consider the information obtained by ttime-resolved measurements performed in this paper aconventional cross correlation of the optical fields in the rerence branch and the photonic structure, similar to eawork using phase-sensitive time-resolved interferome@1,15#. However, this notion is too simplistic. The pulse evelope in the waveguide propagates at a speed different fthe phase information, and therefore the position on the ptonic structure cannot be translated to a single time delatfor the cross-correlation function. This is in contrast to pupropagation in air, for which both the group and phaselocities are the same. The difference between movingprobe or the optical delay can also be seen in the theoremodel in Eqs.~14! and~15!. A movement of the probe movement leads to an increase ofks . However, by moving onlythe optical delay line,k r will remain constant as no additional dispersion is included as a function of the movemeThis clearly demonstrates the effect of including additiondispersive medium on the measured optical signal as a fution of the probe movement with respect to the measuments performed by moving the optical delay line in air, i.changingt as in conventional phase-sensitive time-resolvinterferometry.

VI. CONCLUSION

In conclusion, the propagation in time of a nearly Fourlimited laser pulse propagating through a channel waveguhas been visualized by an interferometric PSTM. The loamplitude and phase of the pulse have been retrieved soboth the phase and group velocity could be measured locThe observed length of the measured pulse envelope isplained by comparison with an analytical model derivedthis paper. The observed FWHM of the measured pushape can be attributed to the group velocity dispersionthe fibers that are unequal in length for the two branchesthe interferometer.

The model shows that by balancing the amount of dispsive medium in the two branches of the interferometerbecomes possible to locally measure the GVD of the strture under study. Even if branches are not compensatereference measurement makes it possible to measure g

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GERSENet al. PHYSICAL REVIEW E 68, 026604 ~2003!

velocity dispersion locally in a sample. As a result, intereing and useful effects such as pulse compression, pspreading, and pulse reshaping become accessible in thcal measurement.

It is expected that the time-resolved interferometPSTM will, in the near future, be used for the local expemental investigation of predicted physical phenomena ins~non!linear dispersive media, such as integrated wavegustructures and photonic crystals.

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ACKNOWLEDGMENTS

This research was part of the strategic Research Orietion on Advanced Photonic Structures of the MESA1 Re-search Institute. Furthermore, this work was part of thesearch program of the Stichting voor FundamentOnderzoek der Materie@FOM, financially supported by theNederlandse Organisatie voor Wetenschappelijk Onderz~NWO!#.

v.

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