The generation and utilisation of half-cycle cut-offs in high harmonic spectra

Laser Photonics Rev. 4, No. 6, 697–719 (2010) / DOI 10.1002/lpor.200900028 697

Abstract High-order harmonic spectra are composed of a co-

herent sum of half-cycle emissions, the cut-off energy of which

depend sensitively on different sub-cycle portions of the driving

laser field. By selecting the correct focal geometry the half-cycle

cut-off emissions can be preferentially selected over the lower

energy plateau emissions through phase matching, such that

they form macroscopic half-cycle cut-off features in the far-field

spectrum. The energy of these macroscopic half-cycle cut-offs

can then be used to retrieve the waveform of the driving laser

field. The processes through which these macroscopic half-cycle

cut-offs are formed and their applications, both for measuring

the laser waveform and the generation of wavelength tunable

isolated attosecond pulses, are reviewed in detail.

A wavelet transform of the simulated on-axis harmonic field

generated by an atomic gas jet driven by an intense few-cycle

laser pulse. The focal geometry has been selected to only phase

match the half-cycle cut-offs. The energies of these half-cycle

cut-offs form a “fingerprint” of the laser field, which can be used

to determine important properties of the field waveform.

© 2010 by WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

The generation and utilization of half-cycle cut-offsin high harmonic spectra

Luke E. Chipperfield1,*, Joseph S. Robinson2, Peter L. Knight1, Jonathan P. Marangos1, and John W. G. Tisch1

1 Imperial College London, London, SW7 2BW, UK2 Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA

Received: 23 May 2009, Revised: 19 September 2009, Accepted: 5 October 2009

Published online: 9 December 2009

Key words: High-order harmonic generation, carrier-envelope phase, phase matching, few-cycle pulses, attosecond, HHG, CEP.

PACS: 32.80.Rm, 42.65.Ky, 42.65.Re

1. Introduction

Progress in attosecond technology over the last decade hasled to the development of techniques capable of revealingand even controlling electron dynamics within atomic andmolecular systems, see e. g. [1–7]. These dynamics typi-cally evolve on attosecond timescales, so their illuminationrequires a probe that can be controlled with comparabletemporal resolution. Technologies such as chirped pulse am-plification [8] and hollow-core fibre pulse compression [9]provide a route through which intense few-cycle pulses of

optical to mid-infra-red frequencies can now be reliablygenerated [10, 11], corresponding to pulses of only a fewfemtoseconds in duration. These few-cycle pulses have rev-olutionised the field of femtosecond chemistry by allowingnuclear dynamics within molecules to be resolved. How-ever, in the majority of cases the electron dynamics evolveon a much faster timescale than the envelope of these few-cycle pulses. Fortunately, the sub-cycle variation in theirelectric field can, instead, provide a suitable probe. Thefield waveform provides a force that can be controlled withsub-femtosecond precision and is also strong enough to

* Corresponding author: e-mail: [email protected]

© 2010 by WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

698 L. E. Chipperfield, J. S. Robinson, et al.: Half-cycle cut-offs in high harmonic spectra

compete with the potential the electron is bound within.The use of the oscillating field, as opposed to the muchslower variation in the pulse envelope, in this way has beentermed “lightwave electronics” [12].

Clearly, an essential component of any technique usingsub-cycle variations in the applied laser field is the precisedetermination of the laser waveform. The first completecharacterisation of the waveform within a pulse of lightwas made by Goulielmakis et al. [13] using the attosecondstreak camera technique [14, 15]. However, although im-pressive, this is a complex technique requiring the priorproduction of a sub-femtosecond XUV pulse and a largenumber of measurements (typically over a thousand, witheach measurement averaged over a thousand shots). Alter-natively, the amplitude and phase of the pulse’s spectral en-velope can be retrieved using techniques such as frequencyresolved optical gating (FROG) [16] (see e. g. [17] for areview) and spectral phase interferometry for direct electric-field reconstruction (SPIDER) [18]. However, the valuesof the spectral phase found through such techniques arerelative, leaving the absolute phase ambiguous. The abso-lute phase of the spectrum determines the carrier-envelopephase (CEP), which is the phase of the carrier field at thepeak of the pulse envelope. Therefore, SPIDER and FROGtechniques are blind to the CEP of the laser pulse.

A linearly polarised laser field, with a central frequencyof ��, CEP of � and peak field amplitude of �� correspond-ing to a peak intensity of � � �

��

�

�, can be represented

in the time domain as

��

��

�� (1)

where �� is the complex laser envelope function with themaximum of its modulus at � � � where �� . For apulse with a CEP of � � � (a cosine pulse) the peak of theenvelope coincides with a maximum of the field waveform,but for a CEP of � � � � (a sine pulse) the waveform iszero at this point, so that �� will never reach the value of��. For a long (narrowband) pulse, in which there is littlevariation in the value of �� over an optical cycle, theinfluence of the CEP on the interaction dynamics will be in-significant. However, it is now possible to fabricate pulsesof light with only a few oscillations of the electric fieldwithin the full width at half maximum intensity (FWHM)of the envelope, for which ��

�. Pulses shorter

than three cycles at FWHM are now routinely producedand, recently, even the production of a sub-1.5 cycle pulsehas been achieved [19]. For such pulses �� varies signif-icantly within one half-cycle of the field, producing largeamplitude changes from one field peak to the next. There-fore, the CEP is a very significant parameter in determiningthe temporal structure of few-cycle laser waveforms andcan have a very significant impact on the interaction dy-namics [20, 21]; making the determination and control ofthe CEP essential. Due to the differing group and phasevelocities within the cavities of few-cycle laser systems,the CEP varies from pulse-to-pulse. Fortunately, technolo-gies enabling the pulse-to-pulse CEP of few-cycle laser

systems to be stabilised are now well developed [22, 23],along with the ability to adjust the CEP. However, the valueto which the CEP of the laser system is locked is unknownand, although small, there remains a pulse-to-pulse jitterin the CEP about this fixed value; for which values of 50–280 mrad have been reported [20, 24].

At intensities of the order of 1014 Wcm�� the electricfield is great enough to significantly distort the binding po-tential experienced by electrons bound within atomic andmolecular systems. Along the laser polarisation axis thepotential is suppressed, forming a potential barrier throughwhich the bound electron can tunnel through to the freestates of the continuum beyond. The highly non-linear de-pendence of the tunnel ionisation rate on the field amplitudelimits ionisation to times close to the half-cycle peaks ofthe field waveform. After ionisation the laser field can drivethe freed electron far away from the parent ion, so that itsfurther evolution is dominated by the laser field, which canpromote it to very high energies. For electrons that do notinteract again with the core, most of this energy is lost asthe electron travels out of the focus of the laser field. How-ever, when the laser field reverses direction the electron isaccelerated back towards its parent ion with which it mayrecollide. The recolliding continuum wavepacket interfereswith the remaining bound state, resulting in high frequencyoscillations in the electron’s dipole moment, radiation fromwhich leads to high-order harmonic generation (HHG). Al-ternatively, the free electron may be rescattered into thecontinuum and be further accelerated by the laser field, pro-ducing a high energy contribution to the high-order abovethreshold ionisation (HATI) spectrum. Due to the influenceof the laser field on the electron during the time it is free inthe continuum, details of the laser waveform are mappedonto the structure of the HHG spectrum and the momentumspectrum of the ionised electrons. The process is repeatedevery half-cycle of the field, with the energy distributionof each half-cycle burst of radiation and ionised electronsdependent on a different fraction of the field sampled bythe electron while it was in the continuum.

Several techniques have been proposed for measur-ing the absolute CEP of a few-cycle laser field, includ-ing measuring the photoelectron emission from metalsurfaces [25–27] and terahertz emission from plasmasin air [28, 29]. However, the most established absoluteCEP measurement techniques rely on the sensitivity ofthe strong-field recollision process to the laser waveform.“Stereo ATI” detects the CEP of few-cycle pulses throughthe asymmetry in the ATI electrons ejected in oppositedirections along the polarisation axis of the laser [30].By measuring the ATI electrons in this way the electronsionised by the odd half-cycles are separated from thosegenerated by the even half-cycles, as they are ejected inopposite directions. For few-cycles pulses, the differencebetween successive half-cycles of the field envelope is greatenough to generate an asymmetry between these two ATIspectra that is significant enough for it to be used to mea-sure the CEP. Although the direct photo-ionised electronscan be used for this [31], the asymmetry in the higher en-

© 2010 by WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.lpr-journal.org

Laser Photonics Rev. 4, No. 6 (2010) 699

ergy rescattered electrons has a far greater sensitivity tothe CEP [32], which was the parameter used by Pauluset al. [33] to make the first accurate determination of theCEP from a few-cycle pulse. Recently, Wittmann et al. [24]increased the sensitivity of the stereo ATI technique byseparating the ATI spectra into a high and a low energyrange, enabling the first single-shot CEP measurement tobe made.

The structure of the field waveform is also encodedwithin the HHG spectrum. Therefore, if the phase and am-plitude of the HHG spectrum could be measured, it wouldbe reasonably straightforward to extract the different half-cycle bursts and use them to reconstruct the laser waveform.However, in an experiment it is the power spectrum thatis measured, with the loss of the phase information, andit is the macroscopic response of an enormous numberof emitters in a gas jet that is measured, rather than theemissions from a single atom. This prevents the simpleresolution of the half-cycle bursts. As the emission fromeach half-cycle is in the propagation direction of the laserfield the HHG spectrum also lacks the asymmetry used bystereo ATI to distinguish two different sets of half-cycleemissions. Despite this, the interference pattern in the HHGspectrum generated by a few-cycle laser pulse has longbeen known to be sensitive to the CEP [34]. Yakovlev etal. [35] proposed a route by which the interference be-tween the successive bursts of radiation could be used, viatime-frequency analysis, to determine the most energetichalf-cycle cut-off (HCO) energies, from which the pulseCEP could be retrieved. However, this technique has notyet been realised experimentally.

In this paper we review a technique that we have devel-oped which makes use of macroscopic propagation effectsto preferentially select, through phase matching, the singleatom HCO emissions from a small region of the interactionvolume, such that they form Macroscopic HCO (MHCO)features in the final HHG spectrum. The energies of theMHCOs can be used to reconstruct the sub-cycle struc-ture of the field waveform. This procedure was proposed byChipperfield et al. [36] and experimentally demonstrated byHaworth et al. [37], who used it to retrieve the CEP of a 3.2cycle (8.5 fs) pulse. The main advantage of this techniqueover stereo ATI is the use of the harmonic emission, whichis generally considered easier to detect compared to theATI electrons. The second advantage is that lower energyHCOs, which are generated in the tails of the pulse, canbe detected. As the gradient of the envelope is greatest forthe half-cycles in the head and tails of the pulse their HCOemissions have the greatest sensitivity to the pulse CEP.Therefore, these HCO emissions can be used to retrieve theCEP of pulses of three cycles and above. In comparison,stereo ATI has, so far, been shown to be more suited topulses shorter than two cycles, for which the higher energyATI emissions, generated close to the peak of the pulseenvelope, have sufficient sensitivity to diagnose the CEP.There are two main drawbacks of the HCO technique, com-pared to stereo ATI. The first is that the measurement is notmade at the laser focus, which complicates the diagnostic

technique. The second drawback is that for the emissionsfrom two HCO bursts to be distinguishable, within an HHGpower spectrum, the difference in their energies must belarge compared to the central frequency of the driving laserfield. Therefore, this requires the driving laser field to havea relatively high intensity or long wavelength.

This review will begin by investigating HHG in thesingle atom limit and how the structure of the field wave-form is encoded into each half-cycle burst of radiation. Aspreviously stated, it is not possible to extract this informa-tion directly from a power spectrum, due to the loss of thephase information. However, the review will then discusshow, by using the correct focal geometry, phase matchingcan be used to preferentially select the half-cycle cut-offemissions within a small region of the interaction volume.This leads to the manifestation of MHCOs in the measuredpower spectrum, the energies of which can be extracted. Atechnique that can use these measured MHCO energies toretrieve the pulse waveform is then presented. Finally, thepossibility of using MHCOs for the production of wave-length tunable isolated attosecond pulses from driving laserpulses over three cycles is discussed.

2. Half-cycle cut-offs in single atomHHG spectra

The most intuitive picture of HHG is provided througha semi-classical description known as the three-stepmodel [38, 39]. In this model the electron’s evolution ispartitioned into two regimes, one in which the atomic bind-ing potential dominates and the other, during its evolution inthe continuum, in which it is the laser field that dominates.This partitioning is the widely used strong field approxi-mation (SFA) [40–42]. This reduces the process of highharmonic generation into three steps: (1) tunnel ionisationof the atom, which ejects the highest lying electron into thecontinuum; (2) laser driven evolution of the freed electronin the continuum; and (3) recollision of the electron withthe core with the conversion of its excess energy into ahigh frequency photon. Within the three-step picture thedynamics of the freed electron are treated classically anddriven purely by the laser field. Upon ionisation, at time��, the electron is assumed to appear in the continuum at�� , with zero initial velocity, �� . The ac-celeration �, velocity � and position � of the electron aregiven by

��

�� (2)

��

�

��

��

��

�� (3)

��

�

��

��

��

�(4)

where ��

��

�� is the vector potential, � is

the fundamental charge and � is the mass of an electron.

www.lpr-journal.org © 2010 by WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim


Figure 1 (online color at: www.lpr-journal.org) The classical elec-

tron trajectories generated by a 3-cycle laser field are plotted in

pink to blue, with blue indicating higher recollision energies. The

laser field, which is plotted by the red line, has a peak intensity of

� � �� Wcm�� and a CEP � � �. The vector potential, scaled

by ��, is plotted by the dashed black line. The electron trajectory

with the highest recollision energy is plotted by the thick blue line.

The electron recollision energies are shown in the lower plot by the

blue lines.

The electron is defined to recollide with the parent ionwhen it returns to the origin, requiring that the recollidingtrajectories satisfy �� , where �� is the recollisiontime. The recollision energy of these trajectories is given by

��

��

�� (5)

The periodicity of the laser field generates groups of rec-ollision trajectories every half-cycle with ionisation timesclose to the half-cycle electric field extremum. The recolli-sion energy of these trajectories loosely follows the valueof ��

�, first increasing with recollision time (and theduration of the trajectory, � � �� ), peaking just beforethe maximum of ��

� and then decreasing, see Fig. 1.This generates bursts of high frequency radiation close toevery peak in the vector potential. The recollision trajec-tories generated each half-cycle can be split into the shorttrajectories, for which the recollision energy, � , increaseswith the trajectory duration � , and the long trajectories, forwhich � decreases with � .

To distinguish between the half-cycle groups of trajecto-ries we will label them according to the timing of the peakin the vector potential around which they recollide. The vec-tor potential peaks occur when the phase, � �� , ofthe carrier field in Eq. (1) is given by � � �� whereN is an integer. Therefore, the half-cycle groups of trajecto-ries will be labelled as �� , see Figs. 2 and 4. The trajecto-ries with the greatest recollision energy within a half-cyclegroup have ionisation and recollision times correspondingto phases of � � �� and � � �� , re-spectively [35]. These HCO trajectories, therefore, sampleoverlapping sub-cycle portions of the laser field ��

in duration, which, within the validity of the SFA, fullydetermines their recollision energy.

Close to the envelope maximum of a long, many cycle,pulse, the HCO recollision energy is the same for eachhalf-cycle and is given by [38, 39]

�max � ��

�

��

��

��

� (6)

where �� is the ponderomotive potential of the laser field,which is the average energy of a free electron oscillatingin the field. For a few-cycle laser pulse the magnitude ofthe field envelope varies from one half-cycle to the next.Therefore, the value of �� is not applicable for calculatingthe HCO recollision energies, other than for approximat-ing the HCO recollision energy closest to the envelopepeak. Instead, if the variation of the envelope over thistime is not too large, then the recollision amplitude can beapproximated with a half-cycle dependent ponderomotivepotential that depends on the envelope magnitude local tothe half-cycle [37, 43]. More specifically, the HCO recol-lision energies can be approximated by the magnitude ofthe vector potential envelope, ��, at the time �� ,which is close to the peak of the electric field immediatelypreceding the recollision time,

�HCO��

�� (7)

where

��

��

�

�� (8)

��

��

��

�� (9)



Figure 2 (online color at: www.lpr-journal.org) (a) Classically

calculated recollision energies for a 2-cycle laser pulse with � �� Wcm�� and � � �. In (b) a linear frequency chirp of

+50 fs� has been applied to the spectrum of the driving laser field

used for (a). The red crosses mark the HCO recollision energies,

in (a) they are labelled �� according to the scheme discussed in

the text. The solid black lines, given by �� ,

approximate the HCO recollision energies.

as shown in Fig. 2. Here and in the rest of the review adriving laser with a central wavelength of 750 nm (�� rad s��) is used. A change in the CEP of ��shifts the laser carrier wave beneath the envelope by atime of ��. Therefore, by measuring the HCO ener-gies generated by an unknown laser waveform for a rangeof CEP values, from Eq. (7), the envelope of the vectorpotential, and hence of the electric field, can be recon-structed [43, 44] so long as the central frequency �� isknown, as illustrated in Fig. 3. As the HCO energies arealso sensitive to changes in the central frequency, lower fre-quencies generate higher energy recollisions, the effect ofintroducing a frequency chirp on the pulse is also detectable.In Fig. 2b the effect of introducing a linear frequency chirp,�, on the HCO energies is shown, where the laser pulse

Figure 3 (online color at: www.lpr-journal.org) The classically

calculated HCO recollision energies, left y-axis, plotted in grey

against CEP, top x-axis, for an 8 fs driver pulse. The Gaussian

envelope of the driving laser pulse, plotted in red against time,

shows excellent agreement with the calculated HCO energies.

The time x-axis scale has been set to match the CEP scale given

� � �� (so that its scale is reversed). Reproduced, with

permission, from [43].

after chirping, ��, is given by

��

��

�

��

(10)

��

��

��

�� (11)

If the power spectrum of the laser pulse is known, whichis usually the case in an experiment, a CEP scan would beunnecessary, as the laser waveform could be reconstructedfrom the set of HCO energies generated by one pulse. Thiscan be achieved by fitting the set of generated HCO energiesto a library of calculated ones, enabling the determination ofthe intensity, CEP and chirp of the laser field to be retrievedfrom a single measurement [37]. This process, and thatof [43, 44], will be discussed in more detail in Sect. 5.1.

It is not possible to measure the HCO recollision ener-gies directly. However, as discussed in Sect. 1, upon recol-lision the electron may generate a photon with an energygiven by [45]

��

� ��

� �� (12)

where � is the ionisation potential of the ground state fromwhich it was ionised (the factor of 1.3 in front of the � isderived from a quantum mechanical correction to the classi-cal recollision process, see [45]). Alternatively, the electronmay scatter off the ground state and be further acceleratedby the laser field through the process of high-order abovethreshold ionisation (HATI) [46]. Therefore, the HCO ener-gies are contained within both the HHG spectrum and theHATI electron spectrum.



The pedagogical picture provided by the classical three-step model is supported by the Feynman path integral ap-proach taken by Lewenstein et al. [45]. In this model theSFA is taken so that the state of the electron is restrictedto either the atomic ground state, unperturbed by the laserfield, ��

�� , or a Volkov state, which isdriven solely by the laser field,

�� (13)

� ��

��

�

��

� �

��

��

where � is the electron’s asymptotic (or drift) momentum,�� is a plane wave state of momentum � and �� with �� a unit vector pointing in the direction ofthe laser polarisation. The electron is initially in the groundstate and can be excited by the laser field into the contin-uum as a Volkov state, with asymptotic momentum �, atsome time ��. The spontaneous emission of a photon withfrequency can then occur via the transition from theVolkov state back to the ground state, at a time ��.

Within the SFA, the amplitude of the frequency com-ponent of the oscillating electron dipole moment, ��, isgiven by the coherent integration over all possible paths,which are defined by the parameters ��, � � �� and �,leading to such dipole oscillations:

��

��

��

��

�

��

��

��

��

�

��

��

��

(14)

The amplitude �� is dependent on the ground stateamplitude and the transition matrix elements between theground and Volkov states. The action evaluated along thesystem’s path in Eq. (14) is given by

��

��

� ��

��

��

(15)

The probability rate for the spontaneous emission of a pho-ton of angular frequency is given by [47]

��

��

�� (16)

Therefore, the power spectrum of the radiation from a singleatom is given by � �� .

Although Eq. (14) is an integration over all possiblepaths, it is dominated by the contributions from pathsfor which the action is stationary � � � �. Therefore,Eq. (14) can be simplified through the application of thesaddle-point approximation (SPA). The action is stationarywith respect to � for the values of the asymptotic momen-tum which satisfy the three-step model requirement that the

electron must return to the point of its origin. Therefore,the three-dimensional integral over � in Eq. (14) can beeliminated by selecting the saddle-point momenta,

��

�

� ��

��

�� (17)

This step is generally taken, and is done so in this review,when calculating the HHG spectra using the SFA. The SPAcan be further applied to eliminate the integrals over �� and� [45]. This reduces Eq. (14) to a coherent sum over a finitenumber of complex quantum orbits [46, 48–52]

��

�� (18)

��

��

� ��

�� (19)

where �� is the recollision time, �� the trajectory duration

and �� the dipole moment for the � quantum orbit (QO).The complex amplitude �� includes the transitionmatrix elements along with factors arising from the SPA,see e. g. [36]. The SPA fails close to the cut-off energy foreach QO. However, the uniform approximation [51, 53, 54]provides a more rigorous treatment that does not have thisfailing, but it does not distinguish between the long andshort trajectories. Therefore, the uniform approximation isused in this review unless the long and short trajectories aretreated separately.

The real values of the complex QOs closely resemblethe classical trajectories and so can be grouped into pairsof long and short orbits that recollide every half-cycle ofthe laser field, in the same way as the classical trajectories.Using the QO model the harmonic spectrum generated byeach half-cycle pair of quantum orbits can be calculatedseparately. The single atom HHG spectrum is constructedfrom these half-cycle spectra, as shown in Fig. 4b. Althoughquantum effects have replaced the sharp, classical, cut-offenergy with a smooth but rapid fall-off [55], the relationshipbetween the HCO cut-off energy (the energy above whichthe fall-off begins) and the electric-field during the elec-tron’s short sojourn in the continuum, as given by Eq. (5), isstill valid. Therefore, the waveform of the laser field can bereconstructed by retrieving these HCO energies. However,when combined in the total single atom spectrum, as shownby the single atom HHG spectrum in Fig. 4b, the individualHCO energies are lost and only the most energetic HCOcan be easily determined.

In Fig. 5 a wavelet transform, see e. g. [56], has beenperformed upon the single atom dipole acceleration, cal-culated by numerically solving the one-dimensional time-dependent Schrodinger equation (TDSE), see e. g. [57], toreveal the half-cycle bursts of radiation from which thespectrum is composed. The wavelet transform has beenoverlaid with the values generated by the QO model toillustrate the good agreement between the two methods,despite the QO model taking the SFA. Such time-frequency



Figure 4 (online color at: www.lpr-journal.org) Classically calculated HCO trajectories are plotted in (a) alongside the driving laser

pulse, which is a 2-cycle pulse with � � � and � � � � �� Wcm��. In (b) the individual spectra generated from each half-cycle,

calculated using the QO model, are plotted in colours that match the relevant HCO trajectories in (a). These half-cycle spectra coherently

combine to form the HHG spectrum plotted in black. The most intense half-cycle spectrum is the first one generated after the peak,

as expected, because the electron trajectories which generate this radiation are ionised at the peak of the envelope. The classical HCO

energies for each half-cycle, calculated by Eq. (12), are indicated by circles.

Figure 5 (online color at: www.lpr-journal.org)

A pseudo colour plot of the wavelet transformed

dipole acceleration of a neon atom driven by a

2-cycle laser pulse with � � � � �� Wcm��

and � � �, calculated by numerically solving the

TDSE. The black overlaid lines are the recollision

times of the quantum orbits for the same system

parameters.

analysis enables the retrieval of the HCO energies from thecombined spectrum. However, in an experiment the detec-tor used to measure the harmonic spectrum is only sensitiveto intensity, losing the phase information and renderingsuch a transform impossible. Nevertheless, the interferencebetween the half-cycle spectra can still be used to deter-mine their relative separation, as proposed by Yakovlev etal. [35], see Fig. 6. Alternatively, the HCO energies canbe found by using phase matching within the interactionregion to isolate individual HCO emissions [36, 37]. Thisapproach will be discussed in the next section.

3. The generation of macroscopic half-cyclecut-offs

In practice, single atom HHG spectra are not measured.Instead, the measured HHG spectrum is a result of the col-lective emissions from an ensemble of atoms. Although theradiation is spontaneous, the atoms radiate cooperatively be-cause of their mutual interaction with the spatially coherentdriving laser field, so that the harmonic field energy growsquadratically with the number of interacting atoms [58].However, variations in the laser waveform within the in-teraction region can inhibit the coherent build up of theharmonic field. Therefore, phase matching plays a pivotalrole in determining the structure of the final HHG spectrum.

The generating medium usually consists of a gas jetlocated close to the focus of a laser beam. The electric field




Same as Fig. 5, but with the wavelet transform per-

formed on the power spectrum of the dipole ac-

celeration. The removal of the phase information

in the signal prevents the resolution of individual

half-cycle bursts. However, the interferences in the

power spectrum contain information about the rel-

ative separation of the bursts, which is revealed in

the wavelet transform. This enables the presence

of two half-cycle spectra with cut-offs at approxi-

mately the 80th harmonic to be inferred from the

peaks in the wavelet spectrum at separation times

of �� and � . See ref [35] for a full description

of this technique.

of a laser pulse in the fundamental Gaussian mode that ispropagating in a vacuum along the z-axis, is given by

�� (20)

� ��

��

�

��

��

where � is the radial coordinate, � ��, �� is the pulseenvelope in the frequency domain, ��

�

�� is the

Rayleigh range and �� is the lowest order Laguerre-Gaussian solution given by [59]

��

��

��

��

� � ��

��

��

��

(21)

where �� and �� is the radius of the beam at thefocus. �� defines the amplitude of �� at the focus, � � �,so that as a function of �, the on-axis, � � �, peak envelopeamplitude is given by

��

�

� (22)

where �� . The peak focal intensity of the laserpulse is given by � � ��

�

��.As it propagates through the focus the laser pulse under-

goes a � phase shift, relative to a plane wave, known as theGouy phase, � � �� . As well as causing a CEP slipof �, the variation of the Gouy phase with frequency alsogenerates a shift in the time of the pulse peak, ��, relativeto �� for broad bandwidth pulses. The on-axis laserintensity, CEP and pulse peak time through the focus areplotted in Fig. 7a and the derivatives of these values withrespect to the propagation direction are plotted in Fig. 7b.

Whether or not a harmonic emission survives propaga-tion depends on whether the single atom contributions arephase matched. The variations in the intensity, CEP and

time of the generating laser field as it propagates throughthe interaction region all affect the phase of the single atomemissions. Therefore, for a particular harmonic to be phasematched on-axis, the effect of these variations on the har-monic’s phase must match the variation in the phase of theharmonic field with propagation distance, which is givenby � . For perfect phase matching the condition,

��

��

�

��

��

�

�� (23)

must be satisfied, where � is the wavenumber of the laser’scentral frequency, �� is the phase gradient of the QOdipole, �, with respect to intensity [60, 61] and �� isthe phase gradient with respect to the laser CEP. The valuesof �� and �� are defined as

��

� ��

�

�

� �� (24)

��

��

�

�

�� (25)

These functions are plotted in Fig. 8 and the gradients� ��, �� and �� are plotted in Fig. 7b.

The intensity dependence of the harmonic emissions,��, is approximately proportional to the trajectory dura-tion. This is because the term in the action �� thatis most sensitive to changes in the laser intensity is thatwhich is gained while the electron is in the continuum,

��

��

� ��

��

�� (26)

Both �� and �� increase proportional to ��and the values �� and �� increase proportional to ��. There-fore,

��

�

��

�

�� (27)



Figure 7 (online color at: www.lpr-journal.org) Properties of a 3-cycle FWHM laser pulse with �� Wcm��, �� μm,

�� mm, �� nm and � � � as it propagates, in a vacuum, through the focus.

Figure 8 (online color at: www.lpr-

journal.org) (a) Intensity dependence, ��,and (b) CEP dependence, ��, of the �half-cycle, calculated using the QO model,

generated by a 3-cycle laser pulse with an

intensity of � � � � �� Wcm�� and CEP

of � � �. The red lines correspond to the

short quantum orbits and the blue to the long.

The magenta circles correspond to the HCO

quantum orbit, calculated by the uniform

approximation. The SPA, which is used to

calculate the long and short QO contributions

separately, is invalid close to the HCO energy.

This causes the “kink” in the values of �at the HCO energy, which should be ignored.

The dominant linear �� component of

the CEP dependence, ��, has been removed

from (b) for clarity.

where �� is the normalised action in the continuumalong the quantum path, �, if the values �� and �� ofthe laser field are set to unity, so that � � � �� and�� . The value of �� is approximately pro-portional to � �. Therefore, the longer duration quantumpaths have a greater sensitivity to changes in the intensityof the driving laser pulse. The values of ��, for the�� long and short QOs are plotted in Fig. 8a against har-monic order. As shown in Fig. 8a by the black arrow, themagnitude of �� increases with the trajectory duration.

The effect of varying the CEP of the driving laser fieldon the half-cycle emissions can be decomposed into twomain parts. The primary effect is a simple linear time shift,given by��, due to the carrier shifting under the laser en-velope. The second is an intensity shift, experienced by thehalf-cycle as it is shifted towards or away from the envelopepeak. The first effect dominates over the second, particu-larly for half-cycle emissions close to the envelope peak, cf.the function �� with the values of �� plot-

ted in Fig. 8b. For long trajectory emissions generated attimes for which the laser envelope has the greatest intensitygradient the second effect does become more significant.However, to simplify the arguments used in this sectionit will be assumed that the phase of the half-cycle emis-sions varies linearly with CEP. If it is also assumed that � � ��, then Eq. (23) simplifies to

��

��

��

��

� �� (28)

where �� is the phase mismatch vector. For a fixed valueof �� the radiation will sum constructively over the co-herence length, ��, which is

��

�� (29)

At the focus �� and the magnitude of �� ismaximal. Therefore, on-axis phase matching is not possible



Figure 9 (online color at: www.lpr-journal.org) The intensity dependence of the �� to �� QOs are plotted at three different positions,

on-axis, within the focus of a 3-cycle laser pulse with �� nm, �� μm (�� mm) and an intensity at the focus of

� �� Wcm��. The blue lines indicate perfect phase matching, the QOs within the two red dashed lines have coherence lengths of

�2 mm, and those within the two green dot-dashed line have coherence lengths of �1 mm. If the gas jet width is such that is selects

only QOs with �� mm then for � � �mm, (a), a sub-set of each of the half-cycle long QO solutions are phase matched. At

� � ��mm, (b), the solutions close to the HCO frequency are phase matched but neither the long or the short QO solutions are well

phase matched. Downstream of the focus, with � � �mm (c), the whole set of short QO solutions are well phase matched. However, if

the gas jet selects QOs with �� mm (those within the green dashed lines) then the HCOs are never phase matched in isolation. The

magenta circles give the HCO intensity dependence, calculated using the uniform approximation; for clarity, the SPA calculated values

close to the HCO frequency have not been plotted. In (b) the plots have been labelled �� according to scheme discussed in the text.

for any of the quantum orbit solutions. Similarly, beforethe focus, � � �, the intensity gradient is positive and�� , so that on-axis phase matching is not pos-sible. Note that this is not the case off-axis, see e. g. [62].Downstream from the focus, � � �, the intensity gradientis negative, so that phase matching is possible. The blueline in Fig. 9 indicates the values of �� which, givenEq. (28), result in perfect phase matching, i. e. � �.

The width of the gas jet relative to the Rayleigh rangeof the laser beam determines how close to zero mustbe for the emission to survive propagation. For a laser fieldwith �� mm a thin gas jet, with a FWHM density of�mm, will select the QOs within the green lines in Fig. 9.In this configuration there are two phase matching regimes.Close to the focus the high Gouy phase gradient selectsQOs with high intensity dependencies. Therefore, only thelong trajectories are phase matched, as indicated in Fig. 9a.Within the blue coloured regions in Fig. 10, the �� longtrajectory emissions have �� mm. This figure revealsthat the long trajectories are phase matched within � � � ��mm on-axis. However, the sensitivity of the long QOsto the laser field intensity reduces as the harmonic order�� increases, which is the opposite dependence givenby ��. Therefore, only a sub-set of the long QOs arephase matched for any gas jet position within this region.Further downstream of the focus the Gouy phase gradientdrops to a value low enough for it to balanced by the lower

intensity dependencies of the short QOs, as indicated inFig. 9c. The range over which the short trajectories arephase matched is illustrated by the red coloured region inFig. 10. As the sensitivity of the short trajectories to thelaser intensity increases with harmonic order, almost allthe frequencies (below the cut-off) of the short trajectoryemissions are phase matched within the region 4 mm �� 7 mm.

The long and short QOs associated with each half-cycleconverge at the HCO frequency. Close to the HCO fre-quency the QOs can not be accurately separated into longand short paths and are instead described using the uniformapproximation. The value of �� given by the uniformapproximation at each HCO frequency is indicated by themagenta circles in Figs. 8a, 9, and 11. The QOs close tothe HCO energy have intensity dependencies which aresignificantly different to the majority of the long and shortQOs, as can be seen in Fig. 8a and Fig. 9. It is this propertythat enables the HCO QOs to be selected through phasematching. In Fig. 9b the QOs within the two dashed redlines have �� mm. As the figure indicates, a rela-tively long interaction region centred at a suitable positionin the focus will phase match the HCO radiation but not thehalf-cycle plateau (long and short) radiation.

Changes in the intensity or wavelength of the drivinglaser do not alter the shape of electric field waveform thatleads to the generation of the �� HCO, only its magni-



Figure 10 (online color at: www.lpr-journal.org) A plot of the

on-axis phase matching conditions for the �� half-cycle emissions,

calculated using the QO model, indicating the harmonic orders

that are phase matched at different propagation distances. The

properties of the generating pulse are �� Wcm��,

� � �, and �� μm and the field waveform is calculated us-

ing Eq. (20) at each point in �. The phase matching conditions for

the long trajectory emissions are plotted in blue and those for the

short trajectories are plotted in red. Within the regions of light blue

(red) the long (short) trajectory emissions have �� mm,

corresponding to the �� QOs that fall within the green dot-dashed

lines in Fig. 9. The overlying regions of dark blue (red) indicate

where the long (short) trajectory emissions have �� mm,

corresponding to the �� QOs within the red dashed lines in Fig. 9.

The green line represents the classically calculated �� HCO fre-

quency. For harmonic orders close to the HCO frequency the uni-

form approximation has been used to calculate the phase matching

conditions for the HCO emissions. Within the magenta coloured

region, these HCO emissions have �� mm. Between the

two black, dot-dashed, lines the HCO emissions (magenta) are

well phase matched but the lower frequency emissions from the

long and short trajectories (blue and red) are not. A 2 mm FWHM

gas jet placed within this region will preferentially select the sin-

gle atom HCO emissions such that they form a MHCO feature in

the far-field HHG spectrum, cf. Fig. 12.

tude and duration. Alternatively, changes in the pulse CEPdo alter the waveform shape (for few-cycle pulses). TheHCO energy, �� , generated by the �� half-cycle is givenby Eq. (12). The dominant component of Eq. (12) is pro-portional to ��

�, which is proportionalto ��

�� . Therefore, the �� HCO energy can be approxi-

mated by

��

�� (30)

where �� is dependent only on the shape of the laserwaveform, not on the laser’s wavelength or pulse energy;see [63] for an investigation into optimising �� . The inten-sity dependence, � , of the QO that generates the HCO�� can be approximated as � � �� , giving

the relation

� ��

��

� (31)

Given that �� , where � is a con-stant, then setting the phase matching condition �� in Eq. (28) implies the approximate relationship for theHCO QOs

�� (32)

� ��

��

�

�� (33)

where � �� is the position in the focus at which the HCOemissions are phase matched. As Eq. (32) is not dependenton ��, �� or �� it is applicable to any focal geometry andany pulse energy and wavelength. In Fig. 9 the HCO QOenergies �� , marked by the magenta circles, vary ap-proximately linearly with the HCO frequency; indicatingthat variations in �� with � are approximately givenby linear variations in the amplitude of the generating half-cycle. Therefore, together with Eq. (32), this implies thatall the HCOs are phase matched at approximately the samepoint in the focus and, from Fig. 9b, this point is given by

� ��

� � (34)

which is independent of the values of ��, �� and ��.Simarly, the coherence length required to select only theHCO emissions is

��

� � (35)

These relationships are illustrated in Fig. 11. In Fig. 11a thebeam waist has been reduced and in Fig. 11b the drivinglaser wavelength has been increased. In both cases therelationships given in Eq. (34) and Eq. (35) hold.

In Fig. 10 the long (short) trajectory emissions fromthe �� half-cycle with �� mm fall within the darkblue (red) region. Simarly, the HCO emissions from the�� half-cycle with �� mm fall within the magentaregion. If a gas jet with an interaction length of � �mmis centred at � �mm, for the focal geometry used forFig. 10, then the HCO emissions within the black dashedlines in Fig. 10 will be phase matched but the plateau emis-sions will not. Therefore, these atomic HCO emissions willsurvive propagation to form a peak in the far-field HHGspectrum, we designate this a macroscopic HCO (MHCO),to distinguish them from the atomic HCOs.

Calculating the generation of MHCOs requires a fullpropagation model. To calculate the response of a macro-scopic ensemble of atoms to a focused, intense, few-cyclelaser pulse, the approach of [64–66] is taken. If the polar-isation of the medium does not change significantly overdistances shorter than the driving wavelength, which is truein the usual cases of HHG from gas jets, in which veryhigh free-electron densities are avoided, then the slowlyevolving wave approximation (SEWA), see [11, 67], can




journal.org) The intensity dependence for

the �� to �� QOs for the same conditions

as Fig. 9 but setting the beam waist of the

generating laser field to 30 μm in (a) and

the central wavelength to 1000 nm in (b).

The gas jet position in each case has been

chosen to optimise the phase matching of

the HCO QOs, � � ��mm in (a) and

� � �mm in (b), which give approximately

the same �� ratios as in Fig. 9b. In (a)

the values within the red dashed lines have

coherence lengths of � ��mm while in (b)

they are � ��mm.

be applied to the Maxwell wave equation. Therefore, ina co-moving coordinate frame, where �� , thesystem can be described by

��

��

�

�

��

��

��

��

�� (36)

where ��

��

��

��

��

��

�is the transversal

Laplace operator, � is the radial coordinate and �� is the polarisation field. As the density of atoms can beassumed to be constant along the direction perpendicular tothe laser propagation direction, a cylindrically symmetriccoordinate system has been used.

The radiation field is decomposed into a component de-scribing the driving laser field, ��

��, and one describ-ing the generated XUV field, ��

��. Such a decom-position is beneficial due to the different approximationsthat can be applied to the two fields. At the relatively lowatom densities considered here (� �� cm�� correspond-ing to �� bar) the dispersion at IR wavelengths causedby the neutral neon atoms, see e. g. [68], is too low to havea significant effect over the short interaction lengths in-volved (�1 cm). Therefore, only the dominant polarisationresponse of the free-electrons, ��

��, is included.Alternatively, for the higher frequencies that make up theXUV radiation field, the effect of the free-electrons canbe ignored. The dominant effect is, instead, the high fre-quency radiation generated by the non-linear response ofthe atoms, as discussed in Sect. 2. The higher frequenciesare also more susceptible to absorption by the neutral atoms,which is included by using the XUV absorption coefficients �� given by [69]. Therefore, after Fourier transformingEq. (36) the driving laser field is determined by solving�

�

��

��

��

�

��

��

�� (37)

where �� is given by the Fourier transform of��

��. Similarly, the generated XUV field is deter-mined by solving

��

��

��

��

�

��

(38)where �� is the amplitude, per unit �, of the electricfield with polarisation �� radiated by the atoms driven by thelaser field �� , given by Eq. (37), from the surfacerepresented by �� with area �� , where�� is the radial step size.

A Gaussian density profile is used to represent theatomic gas jet so that the density of atoms, ��, is givenby

��

��

�

��

�� (39)

where �� is the peak atomic density, �� is the position ofthe centre of the gas jet and � is the gas jet FWHM. Theatomic density is assumed to be invariant with �, which en-ables the fields to be propagated with cylindrical symmetry,this is valid for a few millimetre wide gas jet in the focusof a laser field with a beam waist of a few micrometers.

Across the surface �� the laser field waveform can beassumed to be constant, so long as �� is sufficiently small.Therefore, the atoms in the volume �� experi-ence identical laser fields, the �� phase component isremoved in the co-moving coordinate frame, and so ra-diate coherently. The power spectrum of the cooperativeemission from � � �� atoms is given by

��

��

�� (40)

The intensity of the field propagating through the area ��is �� . Therefore, the harmonic field, with




Pseudo colour plot of the build-up of the on-

axis harmonic spectrum generated by the �� half-

cycle emissions, showing the generation of the

�� MHCO through the phase match selection of

the single atom HCOs generated within the region

��mm � � � ��mm. The blue dashed line

is the classical HCO cut-off energy, calculated at

each point in �, and the green dashed line is of the

normalised gas jet density ��.

polarisation ��, generated per unit � is given by

��

��

��

��

�� (41)

The frequency components of the single atom dipole

moment, ��, in Eq. (41) are calculated using the methodsdiscussed in Sect. 2. If the full spectrum is required, the

SFA model is used, so that �� is given by Eq. (14), aftertaking the SPA for the momentum integral. If it is onlythe emissions from a single half-cycle of the field that areneeded, then the QO model, Eq. (19), using the uniformapproximation, is used to calculate the contributions fromthe pair of QOs associated with the half-cycle.

The electric fields are propagated by taking small stepsin � through the interaction region. Given the fields in theplane at �, the Crank-Nicholson finite-difference method isused to solve Eq. (37) and Eq. (38) to find the fields in theplane at � ��. The XUV field can be propagated purelyin the frequency domain, with �� calculated ateach point in � for each step in �. However, to calculatethe effects of free-electron dispersion and the non-linearresponse of the atoms for Eq. (38), the IR field must betransformed into the time-domain each step. After evolvingthe XUV field through the interaction region it is propa-gated analytically, through vacuum, to the far-field, usingHuygens integral in cylindrical coordinates [59]. For anelectric field at the plane �� the electric field at the plane�� is given by

��

��

��

�

� ��

��

��

��

��

��

��

��

�� (42)

which is in the form of a Hankel transform.

The effects of free-electron dispersion on the propaga-tion of the fields will be discussed in Sect. 4. For the restof this section it will be assumed that the density of atomsis such that the free electron density never reaches levelshigh enough to have a significant affect on propagation.

Therefore, �� and Eq. (20) can be used in-stead of Eq. (37) to analytically calculate the properties ofthe laser field through the focus. This is to simplify thefollowing discussion and because free electron dispersioneffects should be minimised if the MHCOs are to be usedfor determining the laser waveform, as will be discussed inSect. 4 and Sect. 5.

The generation of MHCOs is now simulated usinga three-cycle laser pulse with �� nm, �� , � � � and �� μm (�� mm).A neon gas jet is used for the interaction, with �� cm�, �� and �� mm. Theon-axis properties of the laser field, given by Eq. (20), at� � �� mm are � � � �� Wcm�� and � � � ��.In Fig. 12 the on-axis build-up of the �� half-cycle har-monic amplitude has been plotted, for which the QO modelhas been used to calculate the atomic dipole moment. Theselection of the HCO emissions at � � �mm can be clearlyseen. The horizontal dashed line in Fig. 12 is at the peakof the generated MHCO and the vertical dashed line indi-cates the location of the steepest rate of harmonic build-up.The HCO cut-off energy of the �� emissions in Fig. 12 at� � �mm is greater than in Fig. 4 because the CEP of thelaser field at � � �mm has increased to � � � ��. There-fore, the region of the electric field that generates the ��emissions has moved closer to the peak of the envelope,where the field magnitude is greater, so generating a highercut-off energy.

The selection of the HCO radiation through phasematching is repeated for all half-cycles, generating a se-ries of MHCOs in the far-field HHG spectrum, see Fig. 13.However, the interferences between the half-cycle emis-



Figure 13 (online color at: www.lpr-journal.org) On-axis

HHG spectra generated by a 3-cycle laser pulse, with ��

� � �� Wcm�� and � � � at the focus and a beam waist

of �� μm in a neon gas jet with �� mm

and �� mm, after propagation to the far-field. The top plot,

in solid black, is of the full spectrum, with the atomic response

calculated using the SFA model. The lower plots, in solid blue,

are of the component half-cycle spectra calculated using the QO

model, with each plot labelled by the HC number �� . The dashed

green line in the top plot is of the HHG spectrum after convolution

with a Gaussian mask to filter out the high frequency interference

modulations filtered out. This filtering reveals the peaks due to

three of the MHCOs, generated by the ��, �� and �� half-cycles,

from which it is composed.

sions can make it hard to distinguish the MHCO peaks,see the complete HHG spectrum plotted in black in Fig. 13.Therefore, the HHG spectrum should be convolved with aGaussian mask to eliminate these interferences and enablethe MHCO peaks to be distinguished, see the dashed greenline in Fig. 13. A wavelet transform of the on-axis harmonicradiation after propagation, plotted in Fig. 14a, illustrateshow the plateau radiation has not survived propagation,leaving only the HCO bursts of which the far-field HHGspectrum is composed. Increasing the wavelength of thedriving laser field increases the spectral and temporal iso-lation of the MHCOs, as shown by the wavelet transform,plotted in Fig. 14b, of the harmonic signal produced by aneon gas jet driven by a laser pulse with the parametersgiven in Fig. 11b.

In Fig. 14 the wavelet transforms of the far-field, on-axis, HHG spectra are overlaid with the classical HCO

energies plotted against their recollision time. The classicalHCO energies are found by calculating the half-cycle max-ima of Eq. (12) using the electric field given by Eq. (20) at� � ��. These figures illustrate the strong connectionbetween the MHCO energies and the microscopic HCOenergies generated by the laser field at a defined point inthe interaction region.

4. The influence of free-electrons on theMHCO generation process

The generation of free electrons is an unavoidable by-product of the HHG process. To simplify the discussionon the phase matching process leading to the formation ofMHCOs the effect of free-electrons was not included inSect. 3. However, as they are intrinsic to any HHG experi-ment, the effect of free-electrons in the interaction regionon the phase matching of the XUV field will be discussedin this section.

The polarisation response due to the oscillatory motionof the free electrons in the laser field is given by

��

��

�� (43)

where �� is the plasma frequency, given by

��

��

�� (44)

and

��

��

�

��

��

��

(45)

is the free electron density. The ionisation rate, �� , is calculated analytically using ADK the-ory [70]. The effect of the free electrons on the refractiveindex is given by [71]

��

��

��

�

��

� (46)

As the free electrons are generated through tunnel ioni-sation their density is very sensitive to the intensity of theIR field. This leads to large temporal and radial variationsin the free electron density and commensurate variations inthe refractive index [72]. The time-dependence in the refrac-tive index leads to spectral blue-shifting of the driving laserpulse and the variation in intensity across the beam profileproduces a transverse variation in the refractive index thatacts as a defocusing lens, causing “plasma defocusing”. Thespectral blue-shifting effects are concentrated around thetime of the pulse peak, as the ionisation rate , and, hence,the temporal gradient in the free electron density, is great-est (unless pulse intensity is such that the ground state is



Figure 14 (online color at: www.lpr-journal.org) Wavelet power spectra of the on-axis harmonic radiation after propagation to the

far-field: (a) generated by a laser field with the same properties as Fig. 13 in a 2 mm wide gas jet placed 4 mm after the laser focus; (b)

generated with the same conditions as (a) but using a laser pulse with a central wavelength of �� nm and a 0.8 mm wide gas jet placed

3 mm after the laser focus; (c) and (d) are generated using the same conditions as (a) but with the inclusion of free-electron dispersion

effects, with a peak atom density of �� cm�� in (c) and ��

�� cm�� in (d). For each figure the wavelet plots are overlaid

with the classical HCO energies, plotted against recollision time, generated by the electric field given by Eq. (20) at � � ��.

depleted earlier in the pulse). Similarly, plasma defocusingeffects are most significant at the peak and “tail” (or trailingedge) of the pulse, for which the radial gradient in the freeelectron density is greatest. Therefore, the “head” (leadingedge) of the pulse is least affected by free-electron disper-sion. The on-axis intensity waveforms of the laser pulseafter propagating to the centre a gas jet with �� mm,�� mm and peak atom densities of ��

�� cm��,��

�� cm��, and �� cm�� are plotted in

Fig. 15 and compared to the waveform given by Eq. (20).In the centre of the gas jet, at the end of the pulse, thetotal ionisation fraction is �� . As shown inthe figures, a peak density of ��

�� cm�� has littleeffect on the laser waveform. However, at the much higherdensity of ��

�� cm�� the peak and tail of the laserpulse have been significantly blue-shifted and defocused;resulting in significant changes to the HHG process. For adetailed review of the effects that free electron dispersioncan have on the HHG process see e. g. [66, 73]

There is no significant difference between the harmonicfield generated by propagating through the gas jet with��

�� cm�� and that in which free electron effectsare neglected, as plotted in Fig. 14a. However, at densitiesof ��

�� cm�� the free electron dispersion ef-fects begin to become significant. Therefore, to illustratethe effects that free electrons have on the MHCO genera-tion process, the simulation used to generate Fig. 14a wasrepeated, using Eq. (37), for gas jets with ��

�� cm��,which is plotted in Fig. 14c, and ��

�� cm��, whichis plotted in Fig. 14d. Comparing Fig. 14a with Fig. 14cand (d) reveals that increasing the atom density does notprevent the formation of MHCO features. However, it caneffect the phase matching to such an extent that the ener-gies of the MHCOs are radically changed, cf. Fig. 14a withFig. 14d. The classical HCO energies, calculated for theanalytic laser waveform at � � ��, that are plotted inFig. 14a are also plotted in Figs. 14(c) and (d). Comparisonwith these classical HCO energies reveals that the presence



Figure 15 (online color at: www.lpr-journal.org) Intensity wave-

forms of the laser pulse after propagating to the centre of a neon

gas jet with �� mm and �� mm. The solid blue is

the waveform the laser pulse would have if it were propagat-

ing through vacuum, i. e. that given by Eq. (20). The three other

waveforms are calculated using Eq. (37), which includes free elec-

tron dispersion effects, with peak gas jet densities of: �� cm��,

dotted magenta; �� cm��, dot-dashed green; and �� cm��,

dashed red.

of significant free electron dispersion effects can preventthe simple connection between the MHCO energies andthose calculated for the analytical laser field. Therefore,although MHCOs can be generated in conditions in whichfree electron dispersion effects are significant, such condi-tions are unsuitable for using them as a diagnostic of theunperturbed laser waveform, the process for which will bediscussed in Sect. 5.

5. Utilising the MHCOs

5.1. Reconstructing the laser field

As previously discussed, the energy of each MHCO peakis directly related to the microscopic HCO energies of theatomic emissions generated within a confined, on-axis re-gion of good phase matching. Therefore, the energies of theMHCOs provide multiple sub-cycle snapshots of the laserwaveform. Using the semi-classical calculation of their en-ergy given by Eq. (12), the MHCO energies can, therefore,be used to retrieve properties of the driving laser waveformat the point in the focus in which the atomic HCOs werephase matched.

As the MHCOs are not generated at the focus of thelaser pulse an important consideration is the location ofthis generation region. In Fig. 16 the on-axis gradient ofthe MHCO harmonic amplitude along the propagation di-rection is plotted for the �� to �� half-cycles for the sameconditions as in Fig. 13. This is to reveal the point in thefocus at which the bulk of the harmonic field was gener-ated. Although there is some variation in the position at

which the MHCOs are generated the values are all within��mm. In particular the three MHCOs distinguishable inthe far-field HHG spectrum (��, �� and ��) are all gener-ated within ��mm of each other. Importantly, the positionof the MHCO generation regions are all close to the valueof �� mm, which was found in Sect. 3.Therefore, if it is valid to describe the propagation of thelaser field using Eq. (20), the measured MHCO energiescan be used to determine properties of the laser waveformat �� and Eq. (20) can be used to determinethe laser properties at the focus (or any other position). Tosummarise, the ability to use the MHCO energies to directlymeasure the laser waveform requires that:

1. The MHCO energies correspond to the classically calcu-lated single atom HCO energies at the point in the focusin which the MHCOs were generated.

2. The laser waveform is not significantly affected by prop-agation through the interaction region.

3. The MHCOs are all generated in approximately the samepoint in the focus.

4. The point at which the MHCOs are generated can becalculated analytically.

If the focal geometry is setup to preferentially phase matchthe HCO emissions, then point 1 is satisfied. Point 2 re-quires the density of atoms in the gas jet to be low enoughfor the affect the atoms have on the laser field to be insignif-icant. If point 2 is valid, then points 3 and 4 also hold, asillustrated in Fig. 14 and Fig. 16.

Figure 16 (online color at: www.lpr-journal.org) Plots of the

derivatives, with respect to the propagation direction, �, of the

harmonic field of each of the four MHCOs plotted in Fig. 13. The

position of the peak in the derivative indicates the location in the

focus at which the majority of the harmonic field amplitude was

generated. Each MHCO is calculated separately using the QO

model for the same parameters as Fig. 13. Despite the relatively

large gas jet width of �mm the three MHCOs that are most visible

in the final far-field spectrum, ��, �� and ��, are all generated

within ��mm of each other.



As discussed in Sect. 2, the HCO energies can be ap-proximated by a time dependent ponderomotive potential,

��

�� (47)

where �� , as illustrated in Fig. 2. Thevalue of �� is approximately constant with � and can beestimated as � �� rad, where �� rad for an fs laser pulse [43]. Therefore, by measuring the HCOenergies for a range of CEP values, the time-dependentponderomotive potential can be determined over a portionof the laser pulse. Given the central frequency of the drivinglaser field, this can be used to determine the timescale ofthe portion measured and the envelope of the electric fieldduring this time. The complete field envelope can then befound by fitting it to these measured values, as reportedin [43, 44], see Fig. 17.

An alternative method to using the approximate relationbetween the HCO energies and the field envelope, givenby Eq. (47), is to calculate them by finding the half-cyclemaxima of the function given in Eq. (12), i. e.

��

��

��

��

��

��

��

��

��

given ��

��

��

��

� � �

�

and � � � � � � �

(48)

where �� is the vector potential of the driving fieldnormalised with respect to �� and ��. As the inverse ofEq. (48) is not know, determining the profile of the drivingfield requires the extraction of the HCO energies from thespectrum and then comparison of these values with a libraryof theoretical HCO energies, calculated using Eq. (48). Ifthe power spectrum of the driving laser field is known,which is usually the case, then this spectrum can be usedto generate the driving field used as the input to Eq. (48),as was done in [37]. The spectrum can be given a rangeof CEP and frequency chirp values, �, and the HCO ener-gies calculated can be scaled up for a range of intensities,producing three variables over which the theoretical pulsesare scanned over. Therefore, each value of CEP, � and in-tensity, produces a unique set of HCO energies which arecompared to the experimental ones, producing a matchingerror given by:

� ��

�

��

��

�� (49)

where �� are the energies of the experimental HCOs,labelled by � which increases from the highest energy HCOto the lowest, and � is the number of experimental HCOpeaks detected. �� are the intensity scaled theoreticalHCO energies which have been matched to the relevantexperimental HCO. This process is summarised in Fig. 19and a plot of the matching error � , given by Eq. (49),

Figure 17 (online color at: www.lpr-journal.org) (a) Measured

harmonic spectra as a function of relative CEP, with the harmonic

modulation filtered out. The white points mark the local maxima

in the spectra, which correspond to the HCO energies. (b) The

measured HCO energies from (a) are used to reconstruct the field

envelope for a portion of the laser field, plotted by the grey dots.

A � fs Gaussian envelope, plotted by the black dashed line, is

fitted to these values. The red dashed line plots the electric field

leading to the most energetically detected HCO. Reproduced, with

permission, from [43].

over CEP and � is shown in Fig. 20. The algorithm returnsthe pulse parameters that produce the best match (lowest� ). In [37] the technique, as described above, was used todetermine the properties of a 3.4-cycle laser pulse. Over thefull range of the experimental spectra shown in Fig. 18 therange of intensities retrieved was �� Wcm��

and the range of chirp, �, values was 1–6 fs�.The underlying philosophy behind this measurement

technique is the same as that for CEP detection using stereoATI. Stereo ATI works by comparing the cut-off energiesof the two photoelectron spectra generated by HATI butemitted in opposite directions. These cut-off energies inthe photoelectron HATI spectrum are equivalent to the half-cycle cut-offs in the HHG spectrum and are generated bythe two highest energy half-cycle recollisions. The fact thatthe two HATI spectra are emitted in opposite directions




Experimental and theoretical positions of half-cycle

cut-offs vs. CEP. Theoretical plot of HCO energies

(black curves). The vertical bands are spatially in-

tegrated experimental high harmonic spectra for

different values of CEP from the locking system.

The data is smoothed in the spectral direction to

remove the modulation from individual harmonics.

The colour-map shows relative harmonic intensity.

Reproduced with permission from [37].


This figure illustrates the procedure by which

the theoretical HCO energies can be matched to

the experimental ones. The colourmap is the raw

experimental image of the HHG spectrum. The

lower plots are of the smoothed (green) and un-

smoothed (blue) spectra spatially integrated be-

tween ��mrad, generated from the experimental

image. The black crosses show the experimental

HCOs detected numerically, while the red dashed

lines show the energies of the theoretical HCOs

which most closely matched the experimental ones.

The theoretical HCOs are labelled according to the

order in which they were emitted. The theoretical

HCOs labelled 2, 3 and 4 are used for the com-

parison with the experimental ones, as the HCO la-

belled 1 would be masked by the others. The red cir-

cles are to give an approximate guide to the relative

intensities of the theoretical HCOs, based purely

on the differences in the electric-field amplitudes

upon ionisation, and are not used by the algorithm.

The best match retrieved (which is the one shown)

was for a pulse with a CEP of � � �� with an

additional frequency chirp of �� fs�, correspond-

ing to a pulse duration of �� fs, and an intensity of

�� Wcm��. Reproduced from [74].

allows them to be detected independently of each other,allowing differences in the cut-off energies to be easily re-solved; this also provides an asymmetry that enables theresolution of differences of � in the CEP, which is not pos-sible when using HCOs extracted from the HHG spectrum.However, a method for accessing the lower energy cut-offenergies hidden within the photoelectron spectra has yetto be demonstrated. It is through utilising these lower en-ergy cut-offs, which are generated further into the temporalwings of the pulse, that provides the HCO measurementtechnique with its high degree of accuracy. In particular,this enables the HCO measurement technique to be appliedto pulses over three cycles in duration. The addition of

an XUV pulse has also been proposed, see [75, 76], as aroute through which the resolution of the HCO measure-ment technique could be improved and the � ambiguity inthe CEP removed. However, this technique has yet to beexperimentally demonstrated.

5.2. Wavelength tunable isolated attosecondpulses

Due to the spectral isolation of the MHCOs they are excel-lent candidates for generating isolated attosecond pulses.




journal.org) For each value of CEP

and group delay dispersion (GDD) (re-

ferred to as � in the rest of the pa-

per) the colour-map displays the best

match, using Eq. (49), of the theoretical

HCOs with those from the experimental

spectrum shown in Fig. 19lower is bet-

ter. Also shown are line outs giving the

best matches for the range of GDD val-

ues (above) and CEP values (to the right).

With knowledge of the pulse spectrum

the HCO energies detected within a sin-

gle HHG spectra measurement (averaged

over 1000 shots) are used to determine the

CEP and GDD of the driving laser field.

Reproduced from [74].


journal.org) The effect of spatio-spectral

filtering on the temporal profile of emitted

high harmonics generated within a HCO

phase matching regime. The figure shows

the calculated temporal profile of a fil-

tered HHG spectrum. The HHG spectrum

is generated with a CEP of � � ��. (a)

Temporal profile of a single MHCO, iso-

lated by 1 mrad half-angle on-axis spatial

filtering, as indicated by the black dashed

line. Under this condition an isolated at-

tosecond pulse is generated with an ex-

cellent contrast ratio of better than 10:1.

The central frequency of this pulse can be

controlled by varying the CEP of the driv-

ing laser. (b) Temporal-spatial profile of

the harmonics after spectral filtering. (c)

Temporal profile of the harmonic emis-

sion, after spectral filtering but no spa-

tial filtering, showing a train of attosec-

ond pulses with contributions from differ-

ent half-cycles of the pulse. Reproduced

from [37].

This has been shown to be theoretically possible throughspatio-spectral filtering (SSF) [37], see Fig. 21. Spatio fil-tering is used to select only the on-axis MHCOs. Spectralfiltering can then be used to select only the emissions fromone MHCO. The MHCOs generated in the tail of the pulsehave the greatest isolation in energy and, therefore, are thebest candidates for generating an isolated attosecond pulse.

The energy of the MHCOs generated in the pulse tail variesalmost linearly. Therefore, the central wavelength of anattosecond pulse generated in such a way can be tuned bycontrolling the CEP of the driving laser field.

Pfeifer et al. [43, 44] have combined the effect ofMHCO selection with the modification of the driving laserfield through high free electron densities to produce a



promising route to generate wavelength tunable isolatedattosecond pulses. On the rising edge of the field envelopethe free electron density builds up until it is high enough toinhibit the HHG process by preventing the phase matchingof harmonics generated from subsequent half-cycles [43].This enables MHCOs to be generated in the head, or lead-ing edge, of the pulse, where the intensity gradient of theenvelope is greatest, but not at later times; removing thehalf-cycle radiation generated in the tail of the pulse en-velope, which usually dominates the HHG spectrum. Theenergy of the leading edge MHCOs can be finely controlledby varying the pulse CEP. Lineouts of the harmonic spectragenerated in such a way are plotted against pulse CEP inFig. 17, which clearly shows the linear variations in the en-ergy of the MHCOs with CEP. After filtering out the lowerenergy emissions these, therefore, provide an ideal sourceof wavelength tunable isolated attosecond pulses.

6. Summary

The atomic HHG spectrum is composed of bursts of ra-diation generated each half-cycle of the laser field. Thesehalf-cycle spectra consist of radiation generated by bothlong and short quantum paths, with very different phasematching properties, which converge at the HCO energy.Usually a focal geometry is selected to optimise the radi-ation generated by either the short quantum paths or thelong. We have reviewed the process by which the radiationgenerated by the HCO quantum paths is phase matched butthe long and short plateau emissions are not. This resultsin the formation of MHCO features, at the energies of theHCOs, in the far-field HHG spectrum. The HCO radiationhas an intensity dependence, �� , which is approximatelythe median between the values of �� for the plateau ra-diation generated by the long and short quantum paths. Theselection of the HCOs through phase matching, therefore,simply requires the positioning of the centre of the generat-ing medium, �� , just before the region of optimum, on-axis,short path phase matching and just after the optimum posi-tion for on-axis long path phase matching. This conditionis satisfied for

�� (50)

The interaction length must also be long enough so thatthe required degree of phase matching is strict enoughto prevent the build up of plateau emissions. Ideally, theinteraction length, ��, should also not be so long that thelaser field waveform and the phase matching conditionschange significantly within the interaction region.

The MHCO features in the far-field HHG spectrum areformed from the atomic HCO radiation, generated from therecollision of electrons driven along paths in the continuumthat take the electron far away from the parent ion. Forintense, long wavelength, driving laser fields the evolutionof the electron along this path is approximately classical

and so the recollision energy can be suitably approximatedby the semi-classical model. It is possible to detect multipleMHCOs in an HHG spectrum, each of which has an energydetermined, in a way that can be simply calculated, on adifferent sub-cycle portion of the laser waveform. Conse-quently, the field waveform can be found by matching themeasured MHCO energies to those calculated classicallyfor a waveform that is a function of the unknown variablesof the field waveform, such as CEP, intensity and pulseduration. The main source of error in this technique is inlocating the position in the focus at which point the atomicHCO emissions were phase matched. However, there isonly a small range of values over which the phase match-ing can isolate the HCO emissions and the process is wellunderstood theoretically, so that minimising these errorsis not difficult. Therefore, the generation of MHCOs inHHG spectra, through phase match selection, provides aneffective and relatively simple route through which thewaveform of an intense few-cycle laser field can be recon-structed. As it utilises the half-cycle radiation generated inthe pulse tail, which are more sensitive to variations in thepulse CEP, it can be used for pulses of over three-cycles.If the spectrum of the driving laser field is known, whichis usually the case, then the waveform can be determinedfrom a single measurement of the MHCO energies.

As the XUV radiation propagates parallel to the drivinglaser field the use of MHCOs as an in-line pulse diagnosticis not practical unless the generated XUV radiation is tobe used downstream; in which case the diagnostic is insitu. However, the phase matching requirements for theselection of HCO radiation radiation are not limited by theRayleigh range of the laser pulse. Therefore, a small frac-tion of the main laser pulse energy could be removed to beused for the generation of the MHCOs separately from themain beam path. The high intensities required to generateMHCOs with large enough differences in their energies tobe distinguishable, could then be achieved through tightfocusing, reducing the portion of pulse energy consumed.

The MHCOs originating from the half-cycle emissionsin the head or tail of the laser pulse, where the envelopegradient is greatest, are spectrally isolated from the otherMHCOs. Therefore, they can be used to generate an isolatedattosecond pulse, either through spatio-spectral filtering, assuggested in Ref. [37], or through generating high free elec-tron densities to remove subsequent half-cycle emissions,as accomplished in Ref. [44]. The central frequency of theseMHCOs is sensitive, approximately linearly, to the CEPof the driving laser field. Therefore, once isolated throughfiltering, the laser CEP can be used to provide a fine levelof control over their central frequency.

Acknowledgements The authors gratefully acknowledge support

from the EPSRC (grants EP/E028063 and EP/F034601). The

simulations within this work were conducted on the Imperial

College High Performance Computing Service facilities.



Luke Chipperfield works as a post-doctoral research associate withinthe Quantum Optics and Laser Sci-ence (QOLS) group at Imperial Col-lege London. In 2003 he received anMPhys degree in Theoretical Physicsfrom Durham University. He went onto study for a PhD in the QOLS groupat Imperial College London, under

the supervision of Prof. Jon Marangos, which he com-pleted in 2007. The research for his PhD focused on sim-ulating high-order harmonic generation (HHG) in atoms,including both microscopic and macroscopic effects.His current research interests include the interaction ofatoms and molecules with intense multi-colour fields,the development of techniques for optimising the HHGyield and increasing the spectral cut-off and strong-fieldrecollision processes in extended media.

Joseph Robinson received his MSciin Physics from Imperial CollegeLondon in 2002, and stayed there tocontinue his PhD studies in the Quan-tum Optics and Laser Science Group,under the supervision of Prof. JohnTisch. His research included ultrafastlaser physics and high-intensity laser-matter interactions, receiving his PhD

in 2006. In 2008, he moved to the University of Califor-nia at Berkeley and subsequently the Material SciencesDivision of the Lawrence Berkeley National Laboratory,where he is currently a Postdoc. His present researchfocuses on the study of attosecond timescale dynamicsin condensed matter systems.

Professor Sir Peter Knight is SeniorPrincipal at Imperial College respon-sible for the College’s research strat-egy and deputy to the Rector of Im-perial. He is a member of the Im-perial College Management Boardand Council, and Professor of Quan-tum Optics. He was knighted in the

Queen’s Birthday Honours List in 2005 for his workin optical physics. He was until 2008 Principal of theFaculty of Natural Sciences at Imperial College London.He was Head of the Physics Department, Imperial Col-lege London from 2001 to 2005. Peter Knight is a Past-President of the Optical Society of America and was for7 years a member of their Board of Directors. He is aDirector of the OSA Foundation. He was coordinator ofthe SERC Nonlinear Optics Initiative, past-chair of theEPS Quantum Electronics and Optics Division and Edi-tor of the Journal of Modern Optics from 1987 to 2006.He is Editor of Contemporary Physics and serves on a

number of other Editorial Boards. He is a Thomson-ISI“Highly Cited Author.” His research centres on theoreti-cal quantum optics, strong field physics and especiallyon quantum information science.

Prof. Jon Marangos graduated witha BSc in Physics from ImperialCollege in 1982 followed by aPhD obtained in 1986 (supervisorProf. D. D. Burgess). He was ap-pointed an EPSRC Advanced Fellowin 1990, a Lecturer in Physics at Im-perial in 1995 and a Reader in 1g9.In 2002 he was made Professor of

Laser Physics. From 2003 to 2008 he was the Head ofthe Quantum Optics and Laser Science group and iscurrently the Director of the Blackett Laboratory LaserConsortium. His current interests are a) non-linear op-tical processess for generation of coherent VUV radia-tion; b) investigation of atomic and molecular coherenceeffects (e.g. EIT) and enhanced non-linear frequencymixing; c) High intensity laser- matter interactions espe-cially looking at interactions with molecules and clus-ters; d) development of experiments towards measure-ment of processes on the Attosecond time-scale; e) con-trolling the electron dynamics driven in complex sys-tems by strong laser fields; and f) developing high powersub-femtosecond light sources.

Professor John Tisch is an experimen-tal physicist working in the Quan-tum Optics and Laser Science Groupwithin the Blackett Laboratory LaserConsortium (BLLC) at Imperial Col-lege. He received his BSc Honoursdegree (First Class) from the Univer-sity of Tasmania and won the Aus-tralian Institute of Physics prize for

best student in the final year. He was awarded a Com-monwealth Scholarship to study for a PhD (graduatingin 1995) at Imperial College under the supervision ofProf Henry Hutchinson. He was awarded an EPSRC Ad-vanced Fellowship in 1998 to support his post-doctoralresearch in the BLLC. In 2001 he moved to the ETHZurich to work as Team Leader in the Ultrafast LaserPhysics group of Prof Ursula Keller. He was awardeda lectureship at Imperial College in 2003 and was pro-moted to Reader in Laser Science in 2005. In 2009 hewas made Professor of Laser Physics. His research inter-ests are ultrafast laser physics and high-intensity laser-matter interactions, especially the use of femtosecondlaser pulses to generate coherent x-ray pulses of attosec-ond duration and to observe and manipulate the motionof electrons in matter on the attosecond time-scale andon the Angstrøm length-scale.



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Documents

The generation and utilisation of half-cycle cut-offs in high harmonic spectra