PHYSICAL REVIEW A 87, 013825 (2013)

Broadband spectral-phase control in high-order harmonic generation

Carles Serrat*

UPC, Universitat Politecnica de Catalunya, Colom 11, 08222 Terrassa (Barcelona), Spain(Received 23 July 2012; published 22 January 2013)

I make evident a method for broadband spectral-phase control of light pulses produced in high-order harmonicgeneration (HHG). Using a feedback loop scheme, I demonstrate that a broad spectral region of the HHGoutput field can be coherently added to the intense infrared driving pulse with optimal attenuation and time delayparameters to control the phase of the HHG field generated in a second interaction region. I provide computationalevidence of the control scheme by considering an optimal flattening of the spectral phase for the production ofisolated attosecond pulses. This is proven by considering different spectral widths and for several central photonenergies, such as 36 eV, 70 eV, and 120 eV. An iterative procedure of the method allows one to obtain singlepulses with less than 1.2 cycles at a central photon energy of 36 eV. This control scheme is a fundamental toolthat can be implemented for amplitude and phase shaping of any suitable spectral region in HHG.

DOI: 10.1103/PhysRevA.87.013825 PACS number(s): 42.65.Ky, 32.80.Fb, 37.10.Jk

In this paper I demonstrate a method for controlling thespectral phase of broadband coherent light by consideringa feedback loop scheme [1] that takes advantage from thecoherent emission in high-order harmonic generation (HHG).Controlling the phase of a broad spectral region in HHGrepresents a formidable challenge of general implication tomany areas of science [2]. In particular, the shaping ofthe phase in a broadband is an essential ingredient for thereconstruction of the time profile of attosecond pulses. Thebroadband emission generated by HHG is a source for coherentattosecond light pulses, which can span photon energies fromthe ultraviolet to the soft-x-ray region [3]. Controlling theamplitude and phase of the harmonic radiation is crucial toproducing pulses with a duration as short as the bandwidth ofthe generated radiation in principle allows and to isolate themby compressing the energy generally delivered as pulse trainsfrom the HHG process. Indeed, isolated attosecond pulses havedemonstrated a great potential in the investigation of ultrafastelectronic processes, and they can be produced by spectralselection of the few harmonics in the cutoff region that are mostin phase [4–6]. Several other techniques have been investigatedto generate isolated and twin pulses [7–20]. These importantefforts are able or promising to produce isolated attosecondpulses by considering a particular spectral bandwidth of theharmonics spectrum. The feedback loop control presented inthis paper allows one to directly control the HHG processsuch that the light is emitted with the desired shape from anychosen spectral bandwidth of the harmonics spectrum. This inparticular results in a general and robust tool for the productionof isolated attosecond pulses.

In order to demonstrate the capabilities of the controlmethod, I show how the spectral phase of the HHG emissioncan be controlled in a broad region of the spectrum byconsidering the goal of confining its energy in the time domainas a significant example. As will be shown, by flattening thespectral phase in a large bandwidth of the harmonics spectrumthe production of very short isolated attosecond pulses isachieved. The technique for phase shaping makes use of afiltered XUV bandwidth from the HHG process to slightly

*carles.serrat-jurado@upc.edu

reshape the intense infrared (IR) driving pulse, by coherentlycombining a weak XUV pulse with the IR field. Althoughcombining attosecond XUV pulses with the driving IR fieldhas been studied for HHG [21–26], the possibility to shapethe phase of a broadband spectrum by combining a weakXUV pulse with the intense IR driving pulse has not yet beenconsidered. The present results show that a broadband spectralregion can be directly manipulated to obtain the desired shapeand that the technique can be implemented, in principle, for anyregion of the harmonics spectrum. The feedback loop schemebasically includes a variable intensity attenuator, a variabledelay line and a spectral phase analyzer, which are managedby a computer. The experimental realization will thereforebe limited by the existence of the optical elements at thedifferent bandwidths and photon energies that are of interest.The present work attempts to demonstrate the feasibility ofthe control method by analyzing different examples, since theobserved effect cannot be addressed in general due to theintrinsic characteristics of a coherent control tool based onfeedback loop schemes. Otherwise, the observed results arebased on a conceptually clear physical mechanism: as one-photon ionization is almost negligible for the short pulsedurations and low XUV intensities that are considered, themanipulation of the spectral phase is basically due to the smallIR amplitude reshape caused by the weak XUV fields, whichmodifies the electron wave-packet trajectories generatingharmonics in a broad region of the spectrum.

As schematically illustrated in Fig. 1, a 5 fs laser pulse withGaussian temporal profile, carrier-envelope phase CEP = 0◦,central wavelength of 800 nm, and peak intensity of 5 ×1014 W/cm2 interacts with a gas medium—helium in thepresent simulations (ionization potential 24.59 eV), producinghigh-order harmonics with a photon energy cutoff at ≈127 eV.The feedback loop scheme evolves as follows: a first spectralfilter selects the desired bandwidth of the generated radiationin the XUV region that is to be optimized. The energy of thisXUV1 pulse is modified by a variable light attenuator (A1)that scans for optimal intensities, and then it passes througha variable delay line (D1) that scans for the position withrespect to the IR pulse where the XUV1 pulse is to be added.Coherently combined, the resulting IR + XUV1 pulse is thensent to a second helium gas target that produces a second

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CARLES SERRAT PHYSICAL REVIEW A 87, 013825 (2013)

FIG. 1. (Color online) Schematic illustration of the numericalexperiment. An intense IR laser pulse interacts with a first gas target—helium in this case. The output is filtered in a particular spectralregion—and occasionally split in several pulses, in order to modify theXUV intensity, which is achieved by using variable light attenuatorsAi . The attenuated XUV pulses are combined with the original IRpulse with a given delay provided by variable delay lines Di . Thecombination of the IR + XUV pulses generate harmonics in a secondgas target. The output from the second interaction region is analyzed,in this case by a phase analyzer (e.g., RABBIT, FROG CRAB), foroptimization in a iterative loop procedure managed by a computer.

high-order harmonics signal. The output from the second gastarget is filtered to select the same bandwidth as in the firstinteraction region, and the spectral phase of this last generatedXUV pulse is reconstructed by using a spectral-phase analyzer.The information of the spectral amplitude is then examined bya numerical algorithm that performs a linear regression fromthe data of the amplitude versus photon energy and calculatesits χ2 coefficient, such that the flattest spectral amplitude canbe selected. In other words, by performing the phase analysisat each cycle of the loop, i.e., for each intensity and positionthat is scanned, an optimal value of the attenuation and theposition with respect to the driving field is achieved, whichgive a combination of IR and XUV1 pulses that produce theoptimal flattening of the spectral phase. Additional loops canbe implemented by splitting the XUV pulse from the firstgas target (XUV1, XUV2, . . . ,XUVn; see Fig. 1), so that anoptimal driving pulse shape can be obtained iteratively toachieve the desired shape of the spectral phase.

The simulations have been performed by numericallysolving the atomic response in the single-atom nonadiabaticstrong-field approximation [27] in order to demonstrate thecontrol mechanism in the simplest case. Propagation effectscan, however, be important, i.e., the shown results applyfor low-pressure HHG processes and for short propagationlengths. I show first the results obtained by considering a singlefeedback loop, which is relatively simple to implement andalready results in isolated short attosecond pulses that mightbe sufficient for most applications. In order to demonstratethat the XUV pulse from a chosen region of the harmonicsspectrum—when properly attenuated and added at the correctposition—is a valid field to optimally reshape the IR pulse, thesimulations have been performed considering different centralphoton energies and for different bandwidths of the harmonicsspectrum. In Fig. 2 the optimization has been obtained byfiltering the harmonics spectrum in a spectral region centered

FIG. 2. (Color online) Optimization of the HHG spectral region100–140 eV obtained by considering a single feedback loop. In (a) thespectral intensity of the original and optimized spectra are shown, asindicated. The spectral phase of the original and optimized spectra areshown in (b). (c) and (d) show the temporal evolution of the intensitiesof the XUV pulses, before and after optimization, respectively, whichare centered at a photon energy of 120 eV. Note that the FWHM ofthe subpulse with a highest peak intensity is shortened by 91 as dueto the spectral-phase optimization.

at 120 eV and considering a bandwidth of 40 eV. The resultingXUV pulse is combined with the IR intense pulse and sent tothe second interaction region by scanning its intensity fromα = 1010 to α = 1011, in steps of �α = 1010, where α is anattenuation coefficient that has been defined with respect tothe peak intensity of the IR pulse as IXUV = IIR/α, with IXUV

and IIR being the peak intensities of the XUV and the IRpulses, respectively. The position at which the XUV pulse iscombined with the IR pulse has been scanned from θ = −180◦to θ = 180◦, in steps of �θ = 10◦, where θ is the angularlocation defined with respect to the IR cycle. From Fig. 2, it canbe observed that the spectral phase in the region 100–140 eVis considerably flattened already by considering a singlefeedback loop. Figure 2(a) shows the spectral intensity fromthe first interaction region (black full line) and the optimizedspectrum (red dotted line), which has been obtained for theoptimal values α = 1010 and θ = 10◦. As commented above,the amplitude of the harmonics is not substantially changed inthis case, which is a consequence of using a XUV pulse with apeak intensity of 10 orders of magnitude less than the one of theIR pulse. Figure 2(b) shows the spectral phase of the harmon-ics. It is clear from this picture that the region of the spectralphase that has been filtered, which is shown by the gray dashedline in Fig. 2(b), is flattened considerably when the optimalXUV1 pulse is added. As shown by Figs. 2(c) and 2(d), thisflattening of the spectral phase results in a substantial compres-sion of the initial XUV radiation, which in this case consists ofbasically two attosecond bursts [Fig. 2(c)], and being reshapedto an isolated attosecond pulse of 155 as [Fig. 2(d)], which isan ≈4.5 cycles pulse with a central photon energy of 120 eV.

As a second example, the optimization of the spectralphase has been performed for a region of the spectrum wellin the plateau region. The filtered spectral region in thiscase is centered at 70 eV and has a bandwidth of 30 eV.

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FIG. 3. (Color online) Optimization of the HHG spectral region55–85 eV obtained by considering a single feedback loop. In (a) thespectral intensity of the original and optimized spectra are shown, asindicated. The spectral phase of the original and optimized spectra areshown in (b). (c) and (d) show the temporal evolution of the intensitiesof the XUV pulses, before and after optimization, respectively, whichare centered at a photon energy of 70 eV. In this case, the subpulsewith a highest peak intensity is shortened by 8 as, while the satellitesubpulses almost disappear due to the spectral-phase optimization.

The numerical algorithm considers only a single feedbackloop and has scanned again from α = 1010 to α = 1011, insteps of �α = 1010, and from θ = −180◦ to θ = 180◦, insteps of �θ = 10◦. The optimal values in this case resultin α = 5 × 1010 and θ = −120◦. Figure 3(a) shows howthe spectral intensity is only slightly modified, while theoptimized phase is considerably flattened [Fig. 3(b)]—i.e.,the χ2 coefficient of the corresponding linear regression isconsiderably reduced, as it is required by the target definedin the optimization algorithm. As a result, the XUV radiationfrom the first gas target [Fig. 3(c)] is compressed to an isolatedattosecond pulse of only 119 as [Fig. 3(d)], which correspondsto a basically two-cycles pulse at 70 eV.

These results can in general be further improved by addi-tional optimization loops. Specifically, once the optimal pulseXUV1 has been obtained from the first loop, a second XUV2

pulse from the first interaction region is used to reshape theIR + XUV1 field, so that a new optimal attenuation parameterand position angle is obtained by the computer algorithmto achieve the optimal IR + XUV1 + XUV2 pulse, and thisprocedure can be continued iteratively by adding furtherfeedback loops. In order to be able to compare the ability of themethod, the next simulations have been performed by usingparameter values close to the ones used in the experimentsreported in [8]. In this case the peak intensity of the drivingpulse has been lowered to 2 × 1014 W/cm2, which gives thephoton energy cutoff at ≈70 eV. Figure 4 shows the resultsfrom the optimization performed by using two consecutiveloops in a region of the spectrum centered at 36 eV andconsidering a bandwidth of 30 eV. The numerical algorithmscans in this case from α = 1010 to α = 2 × 1011, in stepsof �α = 1010, and from θ = −360◦ to θ = 360◦, in steps of�θ = 10◦. The optimal values from the first loop are α = 1011

and θ = −320◦. Once the optimal XUV1 is found, a second

FIG. 4. (Color online) Optimization of the region 21–51 eVobtained by considering two consecutive feedback loops. In (a)the spectral intensity from the original spectrum (black full line),the optimized spectrum from the first loop (blue dotted line), andthe optimal spectrum from the second loop (red dashed line) areshown. The spectral phase for the same cases are shown in (b), asindicated. (c) and (d) show the temporal evolution of the intensities ofthe XUV pulses before and after the optimization, respectively. Thespectral-phase optimization by using two iterative loops is clearly ableto strongly compress the energy of the initial XUV field, with almostnegligible satellite subpulses in the final output. The inset in (d) showsthe electric field of the optimized XUV pulse, which consists of a lessthan 1.2 cycles pulse at a central photon energy of 36 eV.

feedback loop is implemented by using a second XUV2 pulse,with α = 5 × 1010 and θ = −20◦ as the optimal intensity andposition for XUV2. Figure 4(a) shows the spectral intensityof the original spectrum (black line), the one of the optimizedspectrum from the first loop (dotted blue line), and the finaloptimal spectrum from the second feedback loop (dashed redline). As it can be observed from Fig. 4(a), there is someintensity enhancement in this case, which is comparable to theXUV-induced enhancement that has been reported previously[21–26,28]. As it will be shown next, however, the reshapingof the intensity profile of the harmonics is not an impedimentfor the spectral phase to be optimally flattened. Figure 4(b)shows the spectral phase of the harmonics in the case of theoriginal spectrum (black line), the optimal spectrum from thefirst loop (dotted blue line), and the final optimal spectrumobtained from the second feedback loop (dashed red line). Thefirst loop is already able to flatten considerably the spectralphase in the broad spectral region that has been considered (21–51 eV), which is revealed by a reduction of the correspondinglinear regression χ2 coefficient in the optimal case. The XUV2

added as the result of the second feedback loop further flattensthe spectral phase, which results in a short isolated attosecondpulse of ≈130 as, as shown in Fig. 4(d). In the inset of Fig. 4(d)the electric field of this final attosecond pulse has been plotted.By using two feedback loops, hence an almost 1.2 cyclespulse at 36 eV is produced, which is comparable to isolatedattosecond pulses obtained by other methods [8].

In conclusion, a feedback loop scheme has been demon-strated for shaping the spectral phase in broad spectral regionsof HHG. The tool is based on the combination of the IR

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intense driving pulse with the weak signal produced by theHHG process in a selected spectral bandwidth and can beapplied in an iterative procedure. The success of the methodis determined by the resolution of the scan in the intensityof the XUV pulse and the delay with respect to the IR pulsethat is performed by the optimization loop. For the simulationsreported in the present study the optimization is found to bestrongly robust for a resolution of �θ = 10◦ (i.e., for a spatialresolution of ≈22 nm, or a temporal delay resolution of ≈74 as,with a driving pulse of 800 nm) and considering variations ofthe optimal XUV pulse intensity of 10%. Also, the numericalalgorithm that has been used in the present study representsthe simplest numerical configuration, which has been chosenin order to demonstrate the capabilities of the method inthe simplest case, although it can certainly be improved toseek for the final solution by using some learning-type loopschemes. It has been thus demonstrated numerically that theradiation produced by HHG can be successfully used in afeedback loop scheme for weakly perturbing the IR intensedriving pulse so that particular shapes of the spectral phasecan be achieved in a broad region of the spectrum. Using asingle loop, isolated attosecond pulses have been obtained at

different central photon energies, which shows the reliability ofthe method in a relatively simple experimental setup. The useof additional feedback loops allows one to further optimizethe final solution. In the present study, the target definedin the numerical algorithm has been accurately obtained byusing two consecutive feedback loops. Although in the presentsimulations the flattening of the spectral phase, in particularXUV regions, has been considered as the optimization targetfor attosecond pulse compression, the method can readily beapplied to obtain different spectral-phase shapes and at otherphoton energies and pulse durations, e.g., for the coherentcontrol of femtosecond and attosecond pulses in the x-rayregion, which paves the way for a conceptually generalcoherent x-ray pulse shaper.

Note added in proof. Recently, the reported control schemehas been successfully applied by considering the amplitudeof the harmonics as the optimization parameter, which ispublished in [29].

Support from the Spanish Ministry of Economy andCompetitiveness through “Plan Nacional” (FIS2011-30465-C02-02) is acknowledged.

[1] H. Rabitz et al., Science 288, 824 (2000).[2] P. Agostini and L. F. DiMauro, Rep. Prog. Phys. 67, 813 (2004).[3] A. Scrinzi et al., J. Phys. B 39, R1 (2006).[4] M. Hentschel et al., Nature (London) 414, 509 (2001).[5] R. Kienberger et al., Science 297, 1144 (2002).[6] R. Kienberger et al., Nature (London) 427, 817 (2004).[7] D. G. Lappas and A. L’Huillier, Phys. Rev. A 58, 4140 (1998).[8] G. Sansone et al., Science 314, 443 (2006).[9] T. Pfeifer et al., Opt. Lett. 31, 975 (2006).

[10] T. Pfeifer, L. Gallmann, M. J. Abel, P. M. Nagel, D. M. Neumark,and S. R. Leone, Phys. Rev. Lett. 97, 163901 (2006).

[11] Z. Chang, Phys. Rev. A 71, 023813 (2005).[12] Z. Chang, Phys. Rev. A 76, 051403 (2007).[13] R. Lopez-Martens et al., Phys. Rev. Lett. 94, 033001 (2005).[14] M. J. Abel, Chem. Phys. 366, 9 (2009).[15] X. Feng, S. Gilbertson, H. Mashiko, H. Wang, S. D. Khan,

M. Chini, Y. Wu, K. Zhao, and Z. Chang, Phys. Rev. Lett. 103,183901 (2009).

[16] N. M. Naumova, J. A. Nees, I. V. Sokolov, B. Hou, and G. A.Mourou, Phys. Rev. Lett. 92, 063902 (2004).

[17] P. Raith et al., Opt. Lett. 36, 283 (2011).[18] C. Winterfeldt et al., Rev. Mod. Phys. 80, 117 (2008).[19] M. C. Kohler et al., Adv. At. Mol. Opt. Phys. 61, 159 (2012).[20] A. B. H. Yedder, C. Le Bris, O. Atabek, S. Chelkowski, and

A. D. Bandrauk, Phys. Rev. A 69, 041802(R) (2004).[21] K. Ishikawa, Phys. Rev. Lett. 91, 043002 (2003).[22] K. J. Schafer, M. B. Gaarde, A. Heinrich, J. Biegert, and

U. Keller, Phys. Rev. Lett. 92, 023003 (2004).[23] M. B. Gaarde, K. J. Schafer, A. Heinrich, J. Biegert, and

U. Keller, Phys. Rev. A 72, 013411 (2005).[24] E. J. Takahashi, T. Kanai, K. L. Ishikawa, Y. Nabekawa, and

K. Midorikawa, Phys. Rev. Lett. 99, 053904 (2007).[25] C. Figueira de Morisson Faria and P. Salieres, Laser Phys. 17,

390 (2007).[26] A. Fleischer and N. Moiseyev, Phys. Rev. A 77, 010102

(2008).[27] M. Lewenstein, P. Balcou, M. Y. Ivanov, A. L’Huillier, and P. B.

Corkum, Phys. Rev. A 49, 2117 (1994).[28] J. Seres et al., Nature Phys. 3, 878 (2007).[29] C. Serrat, Appl. Sci. 2, 816 (2012).

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