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Strong field physics with long wavelength lasers

To cite this Article: , 'Strong field physics with long wavelength lasers', Journal ofModern Optics, 54:7, 1075 - 1085To link to this article: DOI: 10.1080/09500340601135458URL: http://dx.doi.org/10.1080/09500340601135458

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© Taylor and Francis 2007

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Journal of Modern OpticsVol. 54, No. 7, 10 May 2007, 1075–1085

Strong field physics with long wavelength lasers

K. D. SCHULTZ*y, C. I. BLAGAz, R. CHIRLAx, P. COLOSIMOz,J. CRYANx, A. M. MARCHz, C. ROEDIGx, E. SISTRUNKx,

J. TATEx, J. WHEELERx, P. AGOSTINIx and L. F. DIMAUROx

yDepartment of Physics and Astronomy, Austin Peay State University,PO Box 4608, Clarksville, TN 37044, USA

zSUNY Stony Brook, Department of Physics and Astronomy, Stony Brook NY,11794-3800, USA

xThe Ohio State University, Department of Physics, 191 W. Woodruff Ave.,Columbus, OH 43210, USA

(Received 21 July 2006; in final form 17 November 2006)

The generation of short, intense, mid-infrared laser pulses allows for theexploration of atom–laser interactions deep in the tunnelling regime as well asproviding the ability to explore scaled interactions. In this paper we present recentexperimental and theoretical results for this largely unexplored parameter space.

1. Introduction

In the last decade or so, multi-kilohertz, chirped-pulse amplified (CPA) lasers haverevolutionized our understanding of an isolated atom interacting with an intenseelectromagnetic field. Such discoveries as above-threshold ionization (ATI) [1], high-harmonic generation (HHG) [2, 3], multiple ionization [4], and attosecond pulsegeneration [5, 6] have benefited from such lasers. Concurrent with these experimentaladvances, there has been a great deal of progress theoretically [7–9]. From theseadvances it was found that the physics underlying these discoveries reveals simplescaling laws such that the laser–atom interaction scales with wavelengthand intensity. Until recently, suitable lasers for such studies have beenTitanium–Sapphire (Ti:S) operating at a central wavelength near 0.8 mm andNd:YAG, YLF and glass lasers operating near 1.0 mm.

A quasi-classical model describing how a single electron bound to an atomresponds to an intense laser field is the rescattering or three-step model [10, 11] andhas been found to give good agreement with both experimental and theoreticalfindings. In this model, the electron is promoted to the continuum via tunnellingionization, which places the electron far from the ionic core with little or no initialkinetic energy. The electron then propagates under the combined influence of the ionpotential and the strong laser field where it escapes or recollides with the parent ion

*Corresponding author. Email: schultzk@apsu.edu

Journal of Modern Optics

ISSN 0950–0340 print/ISSN 1362–3044 online � 2007 Taylor & Francis

http://www.tandf.co.uk/journals

DOI: 10.1080/09500340601135458

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approximately one half of an optical period later. While in the laser field the electroncan gain from the field a maximum energy of 3.17Up, where Up is the ponderomotiveenergy, or the cycle-averaged kinetic energy of a free electron. The dynamics andenergy of the electron are determined by the phase of the field at the time ofionization. Upon returning to the ion the electron can be recaptured leading toHHG, rescatter elastically off the core and gain up to 10Up of energy, or it canliberate additional electrons through inelastic scattering. When experiment andtheory have been carefully compared for helium [12], the rescattering picture hasbeen shown to quantitatively capture the important physics. The condition for thissemi-classical description to be applicable is that the interaction energy be strongenough that the tunnelling or quasi-static approximation be valid.

Keldysh’s theory of ionization is a useful framework for determining by whatprocesses ionization takes place [13]. According to this theory the ratio of the time ittakes for the electron to tunnel through the barrier to the optical period of the lightfield parametrizes the interaction. If the field changes directions before the electronhas a chance to tunnel through the barrier, then ionization, if it occurs, takes placevia multi-photon ionization (MPI). Otherwise the electron can escape over or tunnelthrough the potential barrier. This so-called Keldysh parameter can be written as� ¼ ðIp=2UpÞ

1=2, where Ip is the ionization potential of the atom and Up is theponderomotive energy. In atomic units, Up ¼ I=4!2, where ! and I are the frequencyand intensity of the laser light respectively. For the majority of ionization studies todate the dominant experimental parameter has been the intensity. As anexperimental knob the intensity leaves much to be desired. It is easy to turn downthe intensity to explore the multi-photon regime (until you run out of dynamic rangein the experiment), but even with the highest intensities possible from a Terawatttabletop system a bare atom can only experience a relatively low intensity. Heliumhas the largest Ip of any neutral atom and saturates at an intensity of approximately1PWcm�2. Ions can survive higher intensities and important experiments measuringionization rates have been done [14], but to measure the electron energiescorresponding to a particular charge state would require ion trapping, ion beammeasurements, or low duty-cycle coincidence experiments, none of which are withouttheir difficulties.

Figure 1 shows a global survey of the experiments that have been done to probeboth the MPI and tunnelling regimes. Inclusion in the plot required that theexperiments either recorded ions and electrons or were state-resolved, thusexperiments that only looked at ionization rates of highly ionized atoms are notincluded. In no way is this plot meant to include all of the experiments that have beendone to date, but rather to be representative of what has been done. In figure 1(a),electric field amplitude is plotted versus frequency in atomic units. The light used inthese experiments ranges from the near-infrared (Ti:S and Nd:YAG, YLF, or glass)to the ultraviolet (excimer lasers); in addition the proposed parameter space forexperiments using mid-infrared lasers (MIR) is shown. The field amplitudes varyfrom 10�3 to 0.2 au but it is important to realize that the upper-limit of theseexperiments is not limited by the laser technology, but the atoms themselves. Thediagonal dashed line in figure 1(a) corresponds to �=1 for ground-state hydrogenand is meant to be a guide to delineate between MPI and tunnelling. The systems

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surveyed include ground-state hydrogen [15], helium [12, 16–20], and the other inertgases [16, 21–34]. Between all of these experiments 0.3<�<10, however, only twoneutral atoms, helium and neon, access the tunnelling regime. Looking at figure 1(a)it is seen that by using MIR lasers at intensities below saturation it is possible to getfurther into the tunnelling regime than was possible with previous experiments.

Looking at the equations for � and Up it is found that both scale more stronglywith wavelength than intensity. For example, He saturates in a femtosecond, 0.8 mmlaser field at 1PWcm�2, corresponding to Up�60 ev. Up for 4 mm light at the sameintensity is 25 times bigger or 1.5 keV. To obtain these energies at 0.8mm wouldrequire an intensity of 25PWcm�2, which no neutral atom could ever experience.Additionally, to get well into the tunnelling regime requires �� 1, therefore fora given wavelength higher intensities are needed. Since the majority of short pulseexperiments have taken place at 0.8mm, only He and Ne can experience intensitieshigh enough to ensure tunnelling ionization. Using longer wavelength light will helpcircumvent these problems. Using the example of helium, it is seen that for 0.8mmlight �¼ 0.45, while for 4 mm light �¼ 0.09. So with MIR light it is possible to probefurther into the tunnelling regime providing more stringent tests for the rescatteringmodel which assumes the quasi-static approximation. Not only is it possible toinvestigate further into the tunnelling regime with He, but at these wavelengths andintensities all the noble gas atoms tunnel ionize. Table 1 summarizes these results forhelium and xenon at intensities for which over the barrier ionization (IOTB) occurs.

Further implications for the rescattering model at long wavelengths are thoserelating to the electron wavepacket as it is propagating in the laser field. Once the

Figure 1. Experimental parameter space plotted as frequency versus (a) field amplitude and(b) Keldysh parameter in atomic units. The dashed and dotted lines are for �¼1 for thehydrogen atom with n¼1 and n¼100, respectively. (The colour version of this figure isincluded in the online version of the journal.)

Table 1. Scaled parameters for helium at IOTB¼ 1.5PWcm�2 and xenon

at IOTB¼ 0.09PWcm�2.

He Xe

0.8mm 2 mm 4mm 2mm 4 mm

� 0.4 0.15 0.07 0.45 0.22Up, keV 0.08 0.56 2.2 0.03 0.13Ecut off, keV 0.3 1.8 7.0 0.11 0.42�cut off, nm 4 0.7 0.18 11.2 2.9

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wavepacket is released into the continuum, the width of the wavepacket transverse to

the propagation is freely spreading and is given by ½�2o þ ð2T=�oÞ

2�1=2. Where T/ � is

the propagation time of the electron and �o is the initial width of the wavepacket. It

is therefore evident that for longer wavelengths, the electron spends more time in the

continuum and therefore experiences more spreading. For processes like inelastic

(e, 2e) scattering and HHG, where the wavepacket needs to overlap with the core, the

wavepacket spread will cause the yield to drop. Furthermore as Up increases at

longer wavelengths, the electron velocity increases to the point that the relativistic

regime is approached and the effect of the B-field of the light on the wavepacket

dynamics can no longer be ignored [35], nor can relativistic motion [36].It is possible to exploit the Keldysh picture further. Since � depends only on Ip

and Up, it is conceivable that there are appropriate atoms, intensities and

wavelengths such that their Keldysh parameter is the same. Then according to the

Keldysh theory the underlying dynamics will be the same between two different

atoms irradiated with different colour light. For example, Xe exposed to 50TWcm�2

light at 0.8 mm has the same Keldysh parameter as K atoms exposed to 1TWcm�2

light at 3.6mm. Figure 2 shows electron spectra for Xe and K. These spectra both

Electron energy (eV)

Electron energy (eV)

Up

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(a)

(b)

Figure 2. Photoelectron spectra for �¼ 1.2 ionizing (a) xenon with 70TWcm�2 of 0.8 mmlight and (b) potassium with 1.2TWcm�2 of 3.6mm light.

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have ATI peaks with spacings equal to the appropriate photon energy. Similarly, bychoosing the proper conditions it is possible to produce HHG in alkali atoms at longwavelengths [37, 38]. An advantage of using alkali atoms exposed to MIR light is theopportunity to easily prepare excited states, thereby ‘tuning’ the HHG spectrum [39].

2. Experimental apparatus

2.1 Laser systems

Experiments were performed using two separate laser systems. The first has beendescribed elsewhere [37, 38], but recent modifications merit a brief description here.Difference frequency generation (DFG) from Ti:S and Nd:YLF (Yttrium LithiumFluoride) mode-locked lasers is used to produce 3–4 mm light. The Ti:S producespulses centred at 815 nm with a pulse width of 100 fs and 2.6mJ of energy and issynchronized to the Nd:YLF laser which produces 15 ps, 550 mJ pulses centred at1053 nm. The pump (Ti:S) and the signal (Nd:YLF) are mixed in a 5mm long KTPcrystal. Nominally the phase matching is collinear, but in practice the pump andsignal are slightly non-collinear to facilitate separation of the idler. For theexperiments discussed in this paper the crystal is tuned for 3.6 mm and producespulses 100 fs long with 160 mJ of energy.

The second system uses the idler from DFG from a commercial opticalparametric amplifier [40]. The pump is an amplified Ti:S laser with 4.5mJ of energyand a pulse width of 50 fs. This signal is generated from superfluorescence in a�-BaB2O4 (BBO) crystal pumped with a small amount of the Ti:S beam. The signal isthen amplified in this BBO before passing through a second BBO where it is mixedwith the majority of the pump beam. This second amplifying stage uses a nominallycollinear geometry for phase matching, however, as in the previous laser system aslight detuning away from collinearity is used to facilitate separation of the pump,signal and idler. The idler is tunable from 1.6 to 2.6 mm. For these experiments theidler puts out �600 mJ of energy at 2 mm with a pulse width of 50 fs providing peakintensities in excess of 1PWcm�2, which is near the saturation intensity of helium.

2.2 Photoelectron spectrometers

Two different time-of-flight (TOF) spectrometers are used in these studies tomeasure photoelectron spectra, each capable of running at multi-kHz rates. The firstTOF spectrometer is used for study of the alkali and alkaline atoms as well as theinert gases. It is capable of running in electron or ion collection mode, but notsimultaneously. The flight tube is kept grounded for electron spectroscopy so thatthe electrons are allowed to freely propagate to the multi-channel plate (MCP)detector. For ion collection there are three field plates used in the Wiley–McClarenconfiguration [41] which are tuned to maximize the resolution for the ion underinvestigation. The flight tube and interaction region have mu-metal shielding toreduce the effects of any stray magnetic fields. The flight tube is 40 cm long and the

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timing resolution of the electronics is 1 ns, giving an energy resolution of better than5% at energies below 300 eV. This spectrometer is currently being used to study therare gases and a precision leak valve is used to backfill the chamber to the correcttarget density. To minimize space-charge effects and pulse pile-up the pressure isadjusted such that the count rate is approximately 0.5 events/shot or lower.

The second spectrometer is a coincidence spectrometer and has been discussed indepth elsewhere [20, 34] although in the results presented here it has not been used incoincidence mode. Briefly, the spectrometer is a pulsed-plate dual-sided TOF designwhich measures the electron energy or ion m/q distribution. The resolution of theelectron energy analyser and mass spectrometer is 5% and 1/300, respectively. Usingthe spectrometer in coincidence mode allows the determination of the efficiencies ofthe two detectors [42] and these have been measured to be 30% and 1% for ions andelectrons respectively. The background pressure is 10�12 bar and like the previousspectrometer a precision leak valve is used to control the atomic density to maintaina count rate of 0.5 events/shot or lower.

3. High harmonic generation

HHG at long wavelengths is an interesting area which is being vigorously pursued bythis group [37]. It has been shown experimentally and theoretically that HHG cutsoff at photon energies of Ipþ 3.17Up, and therefore it should be possible to generatehigher energy harmonics with MIR than has been possible to date. To helpdetermine the validity of the scaling of electron energies and consequently harmonicenergies, results obtained from the numerical solutions of the time-dependentSchrodinger equation (TDSE) within the single active electron (SAE) approximationfor wavelengths ranging from to 2.0mm are presented. The code used is based on thecode of Muller [43] and provides qualitative agreement with the quasi-classicalresults expected for �<1 [44].

For these calculations argon is used as the model atom, since it is a prototypicalatom studied in laboratories and a computationally efficient model potential forargon, with spin–orbit coupling neglected, and has been developed for TDSEcalculations [43]. The atom is subjected to an N-cycle flat-top laser pulse describedby the vector potential AðtÞ ¼ Aoz cos!t, with a half-cycle turn-on and turn-off. Thecode outputs both angle-resolved photoelectron energy spectra (PES) and theelectron dipole moment as a function of time. However, at intensities where there issubstantial ionization of the ground state there is also a substantial backgroundobscuring the harmonics due to the non-zero dipole moment at the end of thepulse [45]. The electron acceleration, on the other hand, is always zero at the end ofthe pulse and is the more physically relevant quantity. The acceleration wascalculated using Ehrenfest’s theorem, aðtÞ ¼ h�ðtÞj�@zVðrÞ þ EðtÞj�ðtÞi, where E(t) isthe electric field of the laser. This term just adds an oscillation at the fundamentaland is suppressed. The HHG spectra presented here were obtained by taking theFourier transform of the time-dependent acceleration.

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Figure 3(a) shows the angle integrated PES in scaled energy units for argonexposed to 8-cycle long 0.8 and 2.0 mm light at an intensity of 0.16PWcm�2. Suchlight fields correspond to Up of 9.5 and 59.7 eV for 0.8 and 2.0mm, respectively.Figure 3(b) shows the HHG spectra for the same laser parameters. The contrastbetween the two wavelengths is striking in both the PES and HHG spectra. In thePES, a clear break near 2Up as well as a long plateau ending in a sharp cut-off at10Up for the 2.0mm case is seen. This is what one would expect from the quasi-classical rescattering model. However, there are no such indications in the results at0.8mm. Here it is clear that there is a MPI contribution causing a monotonic declinein the yield extending beyond 10Up in contrast to what one expects if the ionizationwere purely tunnelling. The results are just as striking in the HHG spectra. For the0.8mm light the photon signal slowly falls off with an indistinct cut-off beyond theexpected classical cut-off of 45 eV. This behaviour is again attributed to MPI. Thelonger wavelength has a distinct cut-off at the expected Emax� 200 eV. To achievesuch an energy in argon with 0.8 mm light requires intensities well beyond argon’ssaturation intensity.

More energetic electrons are not the only advantage of working with the inertgases at long wavelengths. Attosecond pulses are created by summing severalharmonics together; the more harmonics that are mixed the shorter the pulse that canbe obtained if the relative phase between the harmonics is fixed. Unfortunately, anarbitrary number of harmonics cannot be mixed, because differing harmonicsoriginate from electron wave packets that have differing propagation times. Thisdifference in propagation times corresponds to a differing phase between successiveharmonics, therefore limiting the number of harmonics to be summed to someoptimal number [46], unless this attochirp is compensated for in the experiment

Scaled energy (E/UP)

Energy (eV)

HH

inte

nsity

Ele

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(a)

(b)

Figure 3. TDSE calculations of (a) PES of argon exposed to 0.8mm (dashed) and 2 mm(solid) light at an intensity of 0.16PWcm�2. (b) HHG spectra for the same conditions.(The colour version of this figure is included in the online version of the journal.)

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[47, 48]. The chirp on the harmonics can be written as �/T/Up/ 1/I�, where T is the

propagation time of the electron. Once again the advantage of working at long

wavelengths is evident. Not only is it possible to generate higher energy harmonics,

but by being able to sum more of them together it is possible to obtain shorter

attosecond pulses than was previously possible at 0.8 mm.Figure 4 shows numerical results that confirm the ��1 scaling of the harmonics.

Figure 4 was generated by calculating the return times of given harmonics for the

short trajectories. To do this a plot of electron return times as a function of return

energy was generated and is shown in figure 5. To generate this plot, a portion of the

full HHG spectra was selected and inverse Fourier transformed to produce the time

dependence of the harmonics. This procedure is done many times to produce

Kin

etic

ene

rgy

(E/U

P)

Time (t /T)

Kin

etic

ene

rgy

(E/U

P)

1 2

3

4

5

(a)

(b)

Figure 5. TDSE return times as a function of harmonic energy for driving fields of (a) 2 mmand (b) 0.8mm with an intensity of 0.16PWcm�2. The numeric labels refer to the �n trajectory.The top trace plots the driving electric field.

Figure 4. TDSE calculations of harmonic chirp in argon as a function of wavelength for anintensity of 0.16PWcm�2, obtained by determining the slope of �1 in figure 5. The solid line isthe expected ��1 classical scaling.

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the contour plot shown in figure 5. There are many interesting features in this plotand many of these are discussed elsewhere [44], but briefly it is seen that at 2 mm theelectron makes many returns to the core unlike the electron dynamics at 0.8 mm.The importance of these higher order returns is not clear at the moment and is asubject of interest. The other important result is the classical-like dynamics thatthe electron has when driven by a MIR pulse. The black lines in figure 5 are theexpected classical trajectories. In figure 5(b) it is obvious that while thereis coalescing of the electron around the short classical trajectory, the electronwave-packet rapidly diffuses and the role of higher-order returns is masked.However, the electron dynamics in the 2 mm light show much more classicalbehaviour. Interferences do not occur until much later in the laser pulse, and sohigher-order returns become much more evident and strictly follow the expectedclassical dynamics. It is not clear what the role of these returns has in the generationof attosecond pulses, but is clear that some focusing effect is happening. From thesecalculations it appears possible to create an attosecond pulse train with pulse widthsof 100 as.

4. Photoelectron spectra

Figure 6(a) and (b) show preliminary photoelectron spectra of xenon, scaled suchthat each of the curves has unit area, at wavelengths of 0.8, 2.0 and 3.6 mm inlaboratory units and scaled units, respectively. The data taken at the MIR

Nor

mal

ized

ele

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/100

meV

Electron energy (eV) Electron energy (Up)

(a) (b)

Figure 6. (a) Comparison of xenon PES at different wavelengths and intensities. Blacksolid line: 80TWcm�2

¼ Isat, 0.8 mm light, � ¼ 1.2. Grey solid line: 40TWcm�2, 2.0 mmlight, �¼ 0.63. Dotted line 40TWcm�2, 3.6 mm light, �¼ 0.35. (b) Same as above, but plottedversus Up.

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wavelengths are at the same intensity of 40TWcm�2 and the spectrum taken at0.8mm is at saturation. Figure 6(a) clearly shows that the 3.6 mm light belowsaturation is producing electrons that are four times more energetic than thoseproduced from 0.8mm light at saturation. Scaling the electron energy (figure 6(b))clearly shows the dynamics changing from MPI at the shortest wavelength totunnelling ionization at the longest. For the data taken at saturation �¼1.2 and clearATI peaks are visible as well as photoelectron energies in excess of 10Up, which aresignatures of MPI. At 2.0mm �¼ 0.63 for this intensity. ATI peaks are not visible forthe scale used in this plot, but there is faint ATI structure spaced a photon energyapart (0.62 ev). There is also what appears to be a strong 2Up break indicative oftunnelling ionization becoming important. Finally, for 40TWcm�2, 3.6mm lightinteracting with xenon �¼ 0.35. It is expected for this value of the Keldyshparameter that ionization dominantly occurs via tunnelling. The data here supportsuch a view. There are no visible ATI peaks, there is a clear 2Up break, and a cut-offnear 10Up, and at these higher energies the spectra are limited by the spectrometer’sfixed temporal resolution. It is evident for MIR light that the main mechanismfor ionization in xenon appears to be tunnelling, even below xenon’ssaturation intensity. Furthermore, the electron yields in the plateau seem to supportthe notion that a wavepacket spread at longer wavelengths causes rescatteringto turn off.

5. Conclusions

In summary, we have shown that by using longer wavelength light, it is possible forthe experimentalist to delve deeper into the tunnelling regime than previouslypossible. By doing so we are capable of producing more energetic electrons andhigher energy photons from HHG. The dynamics of an electron in MIR fieldsappears to allow for shorter attosecond pulses, enabling experiments to approachthe 100 as barrier. We have presented calculations supporting these expectations andpreliminary experimental data showing that at these wavelengths, even xenonappears to tunnel ionize and therefore generates much more energetic electrons thanhas been seen previously.

Acknowledgements

This work was performed with support from US DOE/BES under contract DE-FG-02-04ER15614 and NSF under contract GRT962706.

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