Tracking spectral shapes and temporal dynamics along a femtosecond filament

Tracking spectral shapes and temporaldynamicsalong a femtosecond filament

Emilia Schulz,1,2,⋆ Daniel S. Steingrube,1,2 Thomas Binhammer,3

Mette B. Gaarde,4,5 Arnaud Couairon,6 Uwe Morgner,1,2,7

and Milutin Kova cev1,2

1Leibniz Universitat Hannover, Institut fur Quantenoptik, Welfengarten 1, D-30167 Hannover,Germany

2QUEST, Centre for Quantum Engineering and Space-Time Research, Hannover, Germany3VENTEON Laser Technologies GmbH, D-30827 Garbsen, Germany

4Department of Physics and Astronomy, Louisiana State University, Baton Rouge, Louisiana70803-4001, USA

5PULSE Institute, SLAC National Accelerator Laboratory, Menlo Park, California 94025,USA

6Centre de Physique Theorique,Ecole Polytechnique, CNRS, F-91128, Palaiseau, France7Laser Zentrum Hannover e.V., Germany

⋆[email protected]

Abstract: Thespectral evolution of a high-intensity light channel formedby filamentation is investigated in a detailed experimental study. We alsotrack the spatio-temporal dynamics by high-order harmonic generationalong the filament. Both the spectral and temporal diagnostics are performedas a function of propagation distance, by extracting the light pulses directlyfrom the hot filament core into vacuum via pinholes that terminate thenonlinear propagation. We compare the measured spectral shapes to sim-ulations and analyze numerically the temporal dynamics inside the filament.

© 2011 Optical Society of America

OCIS codes: (020.4180) Multiphoton processes; (190.0190) Nonlinear optics; (190.4160)Multiharmonic generation; (320.0320) Ultrafast optics; (320.5520) Pulse compression;(320.6629) Supercontinuum generation.

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#150577 - $15.00 USD Received 5 Jul 2011; revised 23 Aug 2011; accepted 29 Aug 2011; published 22 Sep 2011(C) 2011 OSA 26 September 2011 / Vol. 19, No. 20 / OPTICS EXPRESS 19495

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1. Introduction

Filamentationof ultrashort laser pulses has become a versatile tool for pulse compression inthe few-cycle regime [1,2], for strong-field physics [3,4], atmospheric applications [5–8], highaspect ratio micromachining [9–11], investigation of Hawking radiation [12, 13], and genera-tion of THz radiation [14–16]. It denotes a nonlinear pulse propagation regime with very longinteraction between a medium and an intense laser pulse forminga dynamic structure with anintense narrow core sustained without the help of any external guiding mechanism[17–19]. Aninterplay between nonlinear effects including Kerr self-focusing, nonlinear absorption of light,and de-focusing due to free electrons, preserves this regime over distances much longer thanthe typical diffraction length [20]. Filamentation is notably featured by a dynamical reshapingof the laser pulse in space and time, which leads to the generation of new frequencies by self-phase modulation (SPM) ranging from the ultraviolet into the infrared (IR) spectral region. Thisspectral-broadening effect is used for the temporal compression of multi-cycle driver pulses tothe few-cycle regime [1,18,21].

Experimental investigations of the filamentation process are in most cases based on the char-acterization of the output pulses in the far field after the nonlinear interaction with the medium.Various experimental studies have been performed which investigate for example the influence


of the pressure of the nonlinear medium and the chirp of the fundamental field on the fila-mentationprocess with regard to the spectral broadening and the pulse duration of the outputpulses [3,22–24]. Inside the filament, however, the spatio-temporal dynamics of the pulses arecomplicated and are known so far only from complex numerical calculations [19,25]. To verifysuch theoretical predictions, experimental studies have to be performed which are able to trackthe evolution of the filamentation process in space and time. New diagnostic techniques havebeen developed to monitor the amplitude and phase of complex pulses [26]. Because of the highintensity in the hot narrow core of the filament, it is impossible to put the diagnostics directly inthe filamentation region. A few approaches exist to extract the pulses and spectral shapes fromcertain positions within a filament. In Refs. [27–29], a time resolved shadowgraphic techniqueled to the retrieval of the pulse dynamics in a filament from space and time frames for the re-fractive index and absorption. In Ref. [30], the filamentation process is stopped by a movableaperture in the filamentation tube guiding the beam further in argon. In Ref. [31], a filament inair is tracked by using impulsive vibrational Raman scattering and by measurement of powerspectra along the propagation axis. These methods are either indirect methods, or do not guar-antee undisturbed pulses due to further propagation in the gas medium after the extraction. Tothe best of our knowledge, an experimental investigation of spectral and temporal dynamicsduring filamentation versus propagation has never been demonstrated.

In this paper, we present a novel concept for analyzing pulses directly from different posi-tions within a filament. We have applied two different types of diagnostics. First, the IR spectrafrom the filament are analyzed versus the position of extraction and serve as a linear diag-nostics. Second, high-order harmonics which are generated by intensity spikes inside the fila-ment [32,33] serve as a highly nonlinear diagnostics. Both techniques rely on the semi-infinitegas cell (SIGC) geometry [34, 35] using a double differential pumping scheme for realizinghigh pressure gradients. Atmospheric pressure in the SIGC enables filamentation, while a steeptransition to vacuum ensures the abrupt ending of all nonlinear effects to the pulse. By scan-ning the vacuum transition through the filament, both IR pulses and high-order harmonics fromdifferent positions in the nonlinear interaction region are extracted and analyzed. Using thelinear diagnostics, we investigate the spectral broadening of the IR pulses versus propagationthat gives information about the spectral reshaping of the time- and space-integrated pulse dur-ing the filamentation process. The nonlinear diagnostics using high-order harmonics serves totrack the local intensity inside the filament and to estimate the pulse duration of the pulseswhich give rise to high-order harmonic generation (HHG). By implementing these methods forthe spatially resolved analysis of the filament along its propagation axis, detailed comparisonsof experiments and simulations of the direct measurements of spectral and temporal dynamicswithin a filament allow us to confirm theoretical predictions.

2. Experimental setup

A chirped-pulse-amplification system (Dragon, KM-Labs Inc.) delivers 35-fs-pulses centeredat 780 nm with energies of 1.3 mJ at 3 kHz repetition rate (see Fig. 1). The beam quality ismeasured to have an M2 of 1.4. An aperture with diameter of 7.5 mm is placed behind theamplifier and transmits about 80% of the power. This corresponds to∼ 27 GW of peak power.The 1.0 mJ pulses are then focused with a concave silver mirror of 2 m focal length througha 1-mm-thick entrance window (CaF2) into a 1-m-long SIGC filled with argon at differentpressures. The entrance window is placed far enough from the focus region to avoid nonlineareffect in the material and in the air before the cell. Because of the slight divergence of thebeam the linear focus position is shifted to about 2.4 m. In order to initiate a filament in theSIGC the critical powerPcr ∝ n−1

2 ∝ p−1 must be exceeded [36], which is reached for ourlaser parameters at a pressurep above 250 mbar, estimated with the nonlinear index coefficient


f = 2 m

translation stage

SIGCentrance window

argon inlet

exit window

vacuum

beam dump

spectrometer

metal plate 2

gas

vacuum

filament

cell entrance

semi-infinite gas cellpump

pre-vacuum

~ 1 cm

IR XUV

metal plate 1

< 10 mbar-1

< 5.0 x 10 mbar-4

CPA - system

780 nm

35 fs

1.3 mJ

3 kHz

aperture

CM

Fig. 1. (Color online) Experimental setup. Pulses from a chirped-pulse amplifier are fo-cusedby a curved mirror (CM) into a semi-infinite gas cell (SIGC) filled with argon. Thefilament formed in the argon gas is truncated by a laser-drilled pinhole with subsequentpropagation in vacuum. The generated white-light is coupled out with an exit window forspectral analysis.

n2 = 1.74×10−19cm2/W in 1 atm argon [37,38]. At the end of the SIGC, an abrupt transitionto vacuum is realized by a laser-drilled pinhole (diameter∼ 700µm) in a metal plate. A secondlaser-drilled pinhole (diameter∼ 500µm) is installed at 1 cm distance from the first one. Thecombination of both pinholes establish a double differential pumping stage with a pressuregradient from atmospheric pressure in the SIGC to< 5×10−4 mbar in the vacuum chamber. Inthe intermediate chamber, the gas is diluted below 10−1 mbar, measured in a distance of about10 cm from the interaction region. Thus, nonlinear effects on the pulses after the extractioncan be neglected. A delay line between the entrance window and the focusing mirror allowsfor displacement of the whole filament over a range of 30 cm. While moving the filament withrespect to the exit pinhole of the SIGC, it is truncated at different lengths and pulses fromdifferent evolution stages of the filament are extracted. The pulses propagate 1 m in vacuumbefore they are transmitted through an exit window (2 mm CaF2) to avoid nonlinear effectsin the material. The scattered light is recorded through a multi-mode fiber by a spectrometer(Avantes, Ava-Spec-2048-SPU, 273-1100 nm). For the analysis of high-order harmonics, theexit window is replaced by further propagation in vacuum to an extreme ultraviolet (XUV)spectrometer (McPherson, with 300 lines/mm and CCD DH420A-F0, ANDOR Technology)passing a 200-nm-thick aluminum foil [33].

3. Numerical methods

We performed simulations based on an unidirectional propagation equation along thez-direction for the envelopeE (r,z,t) of the laser fieldE(r,z,t) = [E (r,z,t)exp(−iω0τ + ik0z)+c.c.]/2, with frequency componentsE (r,z,ω). The laser pulse propagates in argon, featuredby its dispersion relationk(ω) [39]. Therefore, we write the propagation equations in theframe of the local timet ≡ τ − z/vg with τ the time in the frame of the laboratory andvg = (k′0)

−1 ≡ (dk/dω)|−1ω0

the group velocity of the pulse. The propagation of the envelopeis governed by:

∂ E

∂z=

i2K (ω)

[

∆⊥ +k2(ω)−K 2(ω)]

E +µ0ω

2K (ω)(iωPKerr−J ) (1)


where the frequency dependent operatorK (ω) ≡ k0 + k′0(ω − ω0), and PKerr(r,z,ω),J (r,z,ω) denote the Fourier transformed nonlinear polarization and current envelopes, re-spectively. The nonlinear polarization describes the optical Kerr response

PKerr(r,z,t) = ε02n0n2|E (r,z,t)|2E (r,z,t), (2)

wheren2|E |2 is the nonlinear (dimensionless) Kerr index, withn2 = 1.74×10−19 cm2/W for ar-gon at 1 atm [37,38]. The current envelope comprises two terms:J (r,z,t) = JPlasma(r,z,t)+JNLL (r,z,t), whereJPlasma accounts for plasma induced defocusing and absorption, andJNLL for nonlinear losses associated with optical field ionization. The coupling with the elec-tron plasma of densityNe(r,z,t), generated by optical field ionization, is described by evolutionequations forJPlasmaand for the density of argon atomsNAr(r,z,t), from which the electrondensityNe(r,z,t) is obtained by charge conservationNe(r,z,t) = N0−NAr(r,z,t):

∂JPlasma

∂ t=

e2

meNeE − JPlasma

τc(3)

wheree andme denote the electron charge and mass,τc = 190/p fs is the collision time inargon, and

∂NAr

∂ t= −W(|E |)NAr . (4)

The current for nonlinear losses reads as

JNLL =W(|E |)|E |2 NArUiE (5)

whereUi denotes the ionization potential of argon andW(|E |) the intensity dependent ioniza-tion rate. We finally model the field dependent ionization rateW(|E |) according to the Keldyshformulation [40] later revisited by Perelomov et al. [41], Ammosov et al. [42] and Ilkov etal. [43] with detailed formulas given in Ref. [18].

The propagation equation is solved by calculating the nonlinear polarization and currents inthe temporal domain at each step along the propagation axis. These sources are then Fourier-transformed to the frequency domain and used to propagate the frequency components of thelaser pulse to the next step, before the laser field is Fourier-transformed back into the temporaldomain.

Our calculations start with a 30 fs (FWHM), 780 nm Gaussian laser pulse. The input pulseenvelope thus reads

E (r,z= zin, t) = E0exp

[

− r2

w20

− ik0r2

2 f0

]

exp

[

− t2

τ2p

]

(6)

where the peak electric fieldE0 corresponds to an initial energy of 0.78 mJ andτp =

τFWHM/√

2ln2. The focusing conditions reproduce the experimental values withzin = 154cm, f0 = 150cm andw0 = 1.21 mm. The origin of the propagation axisz= 0 coincides withthe position of the curved mirror CM as in the experiments.

4. Results

4.1. Results: spectral analysis

In our experiment, a filament is initiated in the SIGC and is scanned with respect to the exitpinhole. Pulses are extracted at different positions along the propagation axis and are spectrally


−30

−20

−10

0

400 500 600 700 800 900

sp

ectr

al p

ow

er

de

nsity (

dB

)

wavelength (nm)

227 cm

240 cm

253 cm

Fig. 2. (Color online) Measured pulse spectra at three different positions behind the focus-ing optic extracted from the filament at 1000 mbar.

analyzed. Figure 2 shows recorded pulse spectra integrated over 50 ms at three positions withinthe filament in 1000 mbar argon. The distances are measured with respect to the focusing mir-ror. All recorded spectra are corrected by the spectrometer sensitivity which is high in the bluespectral region and decreases in the IR. We observe that the spectral bandwidth of the pulsesgrows with increasing propagation length. The spectrum at extraction positionz= 227 cm hasa full-width ∆λ of ∆λ = 120 nm at− 20 dB, i.e. at 1 % of the maximum. Afterz= 253 cm ofpropagation the spectrum spans over more than one octave (∆λ = 434 nm). The shape of thespectrum extracted at the shortest propagation distance (z= 227 cm) exhibits the typical sym-metrical structure from SPM. With increasing propagation length, the spectra become moreand more asymmetric and shift into the blue spectral region due to the formation of free elec-trons [44]. The full spectral evolution is shown in Figs. 3(a)–3(e). For five different pressures,a spectral map is recorded by extracting pulse spectra in steps of 4 mm along the propagationaxis with an integration time of 50 ms. In all cases, the spectral bandwidth increases duringthe propagation in the filament as discussed above. The spectral broadening, which is initiallycaused by SPM, becomes asymmetric with increasing propagation distance. Higher pressuresresult in stronger nonlinearities, and the critical power for self-focusing decreases. Thus, the fil-amentation process and spectral broadening begin at shorter distances from the focusing opticwhich can be observed by comparing the maps at different pressures.

From the measured spectra, we extract a Fourier-limited pulse duration (FWHM) as a func-tion of propagation distance, as shown in Fig. 4(a). The Fourier limit decreases as the propa-gation distance increases and reaches below 3 fs in the 1000 mbar case. The different offsetsof the Fourier-limited pulse durations at positionz= 226 cm confirm the earlier onset of thefilamentation process at different pressures. In order to investigate the Fourier-limited pulse du-ration after the full filamentation stage without passing the pinholes, a 1.5-m-long cell withoutvacuum transition was installed (not shown in Fig. 1). The Fourier-limited pulse duration of thegenerated white-light after 4 m of propagation is shown by the filled circles in Fig. 4(a). Fig-ure 4(b) shows the center wavelength versus propagation length. The asymmetry of the spectrachanges during the propagation so that the central wavelength shifts into the IR and then intothe blue spectral region. This behavior can be explained by simulations of the time structurealong the filament, presented in section B. After 4 m of propagation the central wavelengthreaches the same value for all pressures.

The beam profile at two different extraction positions is measured for a filament generated in800 mbar argon, see Fig. 5. For this end, a charge-coupled device camera (WinCamD, DataRayInc.) is placed about 1.5 m behind the truncating pinhole. The profiles demonstrate a Gaussian-shaped intensity distribution which corresponds to the white-light core containing most of the


a

a b c

d e f

spectral power density (dB)

Fig. 3. (Color online) Spectral evolution of propagating pulses inside a filament on a log-arithmiccolor (gray) scale. (a)-(e) Normalized integrated pulse spectra versus the positionof extraction at five different argon pressures. (f) Numerically calculated map of single shotpulse spectra at 1000 mbar argon.

energy. A small ellipticity at position 226 cm results from diffraction at the pinhole.

4.2. Comparison to simulation

The experimental results are compared to numerical simulations at conditions comparable to theexperiment. Figure 3(f) shows simulated pulse spectra at 1000 mbar argon over the propagationlength inside the filament. An increasing spectral broadening is observed for pulses propagatinglonger in the filament matching well with the corresponding measurements. In the simulations, aconsiderable spectral amount is generated in the IR spectral region which could not be measuredin the experiment due to a low sensitivity of the spectrometer. Thus, only the wavelengthsup to 950 nm are considered for the comparison. The Fourier-limited pulse duration of thesimulated pulse spectra is shown as dashed line in Fig. 4(a). The spectral broadening results ina decrease of the Fourier limit, reaching similar values as in the experiment. We interpret thedifference in the Fourier limited pulse duration in the early stage of the filament as resultingfrom a comparison of multiple shot spectra with a single shot simulation. As observed in theexperiment, the central wavelength shifts into the IR, and subsequently into the blue spectralregion, see Fig. 4(b).

To explain this shift and to understand the connection with the filamentation dynamics, wehave followed simulated FROG (Frequency-resolved optical gating) traces of the axial pulseenvelope along the propagation distance [45]. FROG traces are defined as the spectrograms ofthe envelopeE (r = 0,t,z)

F (t,ω) =

∣

∣

∣

∣

12π

∫ +∞

−∞exp(−iωt ′)E (t ′)g(t ′− t)dt′

∣

∣

∣

∣

2

(7)

whereg(t ′− t) is the gate function at time delayt andω is the angular frequency. We chosea Gaussian gate functiong(t ′ − t) with 5 fs FWHM. FROG traces give an indication of thefrequency content and of the local chirp carried by a pulse. A positive (resp. negative) slopeof the FROG trace in(t,ω) variables corresponds to a positive (resp. negative) pulse chirpthat will lead to temporal broadening (resp. compression) by further propagation in a normallydispersive medium.


0

5

10

15

20

25

Fourier limited pulse duration (fs)

-

700

720

740

760

780

800

230 240 250

position of extraction (cm)

600 mbar

central wavelength (nm)

800 mbar1000 mbar1000 mbar (simulated)

400

(a)

(b)

Fig. 4. (Color online) The evolution of (a) the Fourier-limited pulse duration (FWHM), and(b) the central wavelength versus propagation at three different pressures. The dashed linesshow the simulation results for an argon pressure of 1000 mbar. The filled circles in (a) and(b) indicate the converged Fourier limit and central wavelength for the fully propagatedfilament.

a b

Fig. 5. (Color online) Measured beam profiles in the far field from the truncatedfilament226 cm (a) and 256 cm (b) after the focusing optic.

-20 10 20

time (fs)

0.1

-0.1

0.5

1.5

1

-20-20-20 -10-10-10 0000 10101010

time (fs)time (fs)time (fs)

0

0

-0.1

0.1

0.5

0

0.5

1

1.5x10

14

2

1

0

x1017

Fig. 6. (Color online) Simulated FROG traceF (t,ω) (first line), intensityI(t, r) (secondline) and electron densityρ(t, r) (third line) distributions at four propagation distanceswithin the gas cell.


160 180 200 220 240 260 280100

101

102

z (cm)

∆t FWHM(fs)

20 µm

50 µm

Fig. 7. (Color online) Simulated pulse duration (FWHM) of the radially averaged pulseintensityover a radius of 20 or 50µm, as functions of the propagation distance.

Figure 6 shows the FROG traces of the pulse as it propagates in the gas cell and under-goes filamentation. For an understanding of the newly generated frequencies we simultane-ously monitored the evolution of the pulse intensity and electron density distributions. Dur-ing the initial focusing/self-focusing stage, SPM broadens the spectrum symmetrically and thepulse undergoes a slow temporal compression. Around and immediately after the nonlinear fo-cus (numerically obtained atz = 239 cm), the pulse dynamics exhibits asymmetry along thetime axis, mainly resulting from plasma defocusing of its trailing part (positive times, Fig. 6,z= 242 cm). The most intense part of the pulse moves toward negative times where the lowfrequencies of the spectrum are the most amplified (z= 246 cm). At positive times, the trail-ing part of the pulse undergoes a refocusing stage which results in a trailing intensity peakthat enhances the plasma density. As shown in the FROG trace atz= 252 cm, this results ina dominant redshifted leading peak and a subdominant but growing blueshifted trailing peak,both carrying positive chirps. It is a standard in the filamentation dynamics that the leadingpeak experiences nonlinear losses and thus disappears earlier than the rest of the pulse [46].Correspondingly, the weight of the redshifted leading part in the center of mass of the spectrumis progressively replaced by the weight of this growing trailing component. Overall, this leadsto a blueshift of the spectrum central wavelength, as observed in Fig. 4(b). This dynamics isrecurrent along the propagation distance, leading to new refocused pulses that are shifted tem-porally toward positive times (delayed) and spectrally towards shorter wavelengths. The FROGtrace atz= 284 cm is taken at the position of the second refocusing cycle and shows that thechirp carried by this blueshifted pulse is still positive and becomes stronger. In spite of thispositive chirp, few-cycle pulses of duration supported by the spectral broadening are still gen-erated by further propagation as these mainly result from the space-time refocusing dynamicswhich is much stronger than the temporal broadening that would be induced by group velocitydispersion [45]. We finally note the importance of the weight of the refocusing pulse in thedetermination of the central wavelength of the spectrum: When a pulse refocusing occurs inthe central part (t ∼ 0), a leading peak and a previously focused trailing peak simultaneouslyundergo the effect of nonlinear absorption, hence the weight of the long- and short-wavelengthcomponents in the spectrum, featuring respectively the leading and trailing peaks, decreases tothe benefit of the central wavelength components of the refocused pulse. These may be in turnblueshifted if the refocused peak propagates over a sufficiently long distance. This explains thefinal increase of the central wavelength after the filament shown in Fig. 4(b).

The different spectral weights associated with each intensity peak also affect the calculatedpulse duration. Figure 7 shows FWHM pulse durations for the radially averaged intensity pro-files as functions of the propagation distance. At each distance, only the most intense peak ofthe averaged profile is monitored. Minima of the pulse duration corresponding to few-cyclepulses between 4 and 5 fs are obtained after each refocusing stage averaging over a transmittedbeam radius of 50µm. Closer around the axis (see averaging radius of 20µm), pulse durationsbetween 1 and 3 fs are reached. The shortest intensity peaks in the temporal profile, however,


A B

Fig. 8. (Color online) High-order harmonics generated directly in the filament extracted atdifferent positions after the focusing optic.

B

A

D = 5.5 mm

D = 7.5 mm

spike position z

, z (cm)

AB

230

240

250

−1000 0 1000

chirp (fs2)

(b)

B

A

D = 5.5 mm

D = 7.5 mm

spike position z

, z (cm)

AB

(a)

230

240

250

600 700 800 900 1000

argon pressure (mbar)

Fig. 9. (Color online) (a) Position of the intensity spikes A and B versus pressure of thenonlinearmedium for the experiment (box, cross) and theory (lines) for two different aper-tures (D= 5.5 mm and 7.5 mm). (b) Position of intensity spikes versus chirp for 1000 mbarargon. The symbols denote the same as in (a).

are mainly obtained with a blueshifted spectral content and a positive chirp.

4.3. High-order harmonic generation

Due to the complex temporal structure of the pulses formed during the propagation in the fila-ment the temporal characterization with for example SPIDER is difficult. To give an estimationof the pulse duration we analyzed the high-order harmonic radiation generated inside the fila-ment. The measurements are consistent with the theoretical prediction that HHG in a filamentoccurs at the refocusing stage in the filament in which an ultrashort sub-pulse is formed on-axiswith an intensity which can be higher than the clamping intensity. We attribute these sub-pulsesto intensity spikes referred to Ref. [32,33]. For the filament in 1000 mbar argon, we record theharmonic intensity generated along the filament, see Fig. 8, and identify two confined regionswhere harmonics are generated while the whole filament extends over 30 cm. One region Aat z≈ 236 cm with a discrete harmonic spectrum and another region B atz≈ 242 cm showsa continuous spectral shape. From the discrete harmonic structure in region A, we concludethat the generating pulse consists of multiple cycles. The continuous XUV spectrum in regionB indicates the occurrence of an intense spike with a near single-cycle time structure [33].


Table 1. Offset Values Determined by a Fit to the Data Shown (a) in Fig. 9(a), and (b) inFig. 9(b), Respectively (see text for details)

D (mm) LA (cm) LB (cm)5.5 18±1 27±27.5 10±1 21±1

(a)

D (mm) LA (cm) LB (cm)5.5 15±2 22±17.5 9±1 20±2

(b)

The position of both regions A and B can hardly be seen in the linear diagnostics of radiallyaveraged IR spectra (cf. 3(e)), because the intensity spikes appear only on-axis.

The harmonic radiation constitutes an ideal tool for tracking the filament. We analyzed theharmonic spectral maps (e.g. Fig. 8) at different pressures and diameters of the first apertureregarding the center locationzA ,zB of the individual regions (A, B). Figure 9(a) shows theposition of the two regions versus pressure at two diameters of the aperture (D= 5.5 mmand 7.5 mm). Below a pressure of 700 mbar, region B disappears which can be account to asmaller filament length at a low pressure, associated with an earlier termination [47]. We canexplain the shift of region A and B by the pressure dependence of the nonlinear refractiveindexn2(p) ∝ p [48]. Increasing the pressure leads to an stronger self-focusing due to highernonlinearities. The position of the beam collapse can be calculated for a collimated input beamby [36]

Lc =0.367zR

√

(√

PPeak/Pcr−0.852)2−0.0219(8)

with the Rayleigh rangezR and the peak powerPPeak. Using a focusing optic, Eq. (8) is amendedby the focal lengthf and the beam collapses at a smaller distance given by [18,49]

Lc, f = (L−1c + f−1)−1. (9)

To consider the different diameters of the aperture, we use forD = 5.5 mm a beam waist of2.8 mm at the focusing optic and a transmitted pulse energy of 0.75 mJ, corresponding to a peakpower of≈ 20 GW. ForD = 7.5 mm, we use a beam waist of 3.9 mm and a transmitted pulseenergy of 1 mJ, corresponding to a peak power of≈ 27 GW. ConsideringLc, f as the beginningof the filament and defining the location of the spikes by the relative offsetLA andLB in thefilament, respectively, their absolute positions are given by

zi = Lc, f (p)+Li , i = A, B. (10)

As shown by the curves in Fig. 9, this model matches the experimental data for values ofLA

and LB listed in Table 1(a). We observe a good agreement between the spike positions andthe model by considering a single offset-pairLA , LB for each input beam width, i.e aperturediameter D. Spike A appears a few cm beyond the nonlinear focus whereas spike B appearsat a the position where the trailing peak is refocused. The simulation results of Fig. 6 showthat a refocusing cycles requires a propagation distance of 10 - 12 cm in good agreement withthe offset-differenceLA −LB. This confirms that spike B is associated with a refocusing eventand with the shortest pulse durations [32, 33]. We investigate also the dependence of beamcollapse on the entrance pulse duration for the two different aperture sizes at a fixed pressure of1000 mbar, see Fig. 9(b). The chirp is controlled by the separation of the compressor gratings inthe amplifier system. We can stretch the pulses by a positive or negative dispersion of± 1000 fs2

to a duration of approximately 87 fs before the second spike vanishes. The first spike can be


observed at an even wider range of chirp values. We use Eqs. (8) and (9) and assume a changein the peak power of the pulse according to

Ppeak(β ) =Ppeak(0)

√

τ2p +(2·β/τp)2

. (11)

for an applied group delay dispersionβ and a Fourier-limited pulse durationτp = 35 fs. We finda good agreement of the model for both spikes, again for a single pair of relative offsets listedin Table 1(b). This shows that the nonlinear diagnostic using high-order harmonics directlyfrom the filament allows to track the refocusing stages inside the femtosecond light channel.In particular, we are able to verify a basic filamentation model predicting the position of thenonlinear focus.

5. Conclusion

In summary, we presented a study on the nonlinear light evolution inside a filament using twodifferent diagnostics. The spectral reshaping of pulses within a filament is experimentally ob-served in a detailed investigation by extracting pulses at different positions directly from thehigh intensity light channel. Numerical simulations are in excellent agreement with the exper-imental data supporting the interpretation of the physics of these highly nonlinear and dynam-ical effects. The peak intensity of the pulses generated within the filament is high enough toproduce high-order harmonics directly in the filament [33]. The spectral structure of these har-monics carries information about the duration of the generating pulse and can be tracked alongthe propagation axis. The beginning of the filament and the occurring refocusing cycles canbe monitored by the position of the intensity spikes self-focusing dependence. Our experimen-tal method opens the way for better understanding of the dynamics during the filamentationprocess.

Acknowledgments

This work was funded by Deutsche Forschungsgemeinschaft within the Cluster of ExcellenceQUEST, Centre for Quantum Engineering and Space-Time Research. M. B. Gaarde was sup-ported by the National Science Foundation under Grant No. PHY-1019071.


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Tracking spectral shapes and temporal dynamics along a femtosecond filament