2D semiclassical model for high harmonic generation from gas

Vol. 43 No. I1SCIENCE| N CHINAS¨ Series a￡0November 20002D semiclassical model for high harmonic generation from gasCHEN Liming"3AAeA÷6UWe" Oa caYcE ZHANG Jief¨0AwU妞CHEN Zhaoyang"3A NOze& JIANG Wenmiarf12-lAAace1. Laboratory of Optical Physicsf-tnstitute of Physicsf-Chinese Academy of Sciencesf Beijing 100080f-Chinaf2. Laboratory for Laser Fusionf institute of Nuclear Physics and Chemistry Mianyang 621900f-Chinaz3. Shanghai Institute of Optics and Fine Mechanics -Chinese Academy of Sciencesf-Shanghai 201800f-ChinaCorrespondence should be addressed to Chen Liming"email imchen aphy iphy. ac cncReceived December 62-1999Abstract The electron behavior in laser field is described in detail. Based on the 1dmodelf-a 2D semiclassical model is proposed analytically using 3D DC-tunneling ionization theory. Lotsof harmonic features are explained by this model -including the analytical demonstration of the maximum electron energy 3. 17 U. Finally -some experimental phenomena such as the increase of thecutoff harmonic energy with the decrease of pulse duration and theanomalousi luctuations in the cutoff region are explained by this modKeywordsf high harmonic generation -tunneling ionization ultrashort laser pulseHigh harmonics generation HHGECs an intriguing and experimentally well-confirmed phemenon which results from ionization of gas atom by laser field hhg has been studied in thesimple but illustrative models including classical modelsuiE-semiclassical models 2e and quantummodel5u354eY The semiclassical model put forward by Corkum and Kulander et al. explains theemission of high harmonic radiation in terms of three discrete stepsf Firste the electron tunnelsthrough the barrier formed by the atomic Coulomb potential and the laser fieldE>the quasi-freeelectron subsequently acquires kinetic energy from the laser fieldE> and half an optical cycle laterit returns to its parent ion and emits a photon. But the Corkum model has not considered the ef-fect of magnetic field of the laser and the laser pulse duration which is essential in ultrashortlaser-atom interactions. Sof it is necessary to adjust this model in ultrashort laser-atom interactions. This paper is the extended 2D model which can be used in the larger range1 Theoretical modelDuring the laser-atom interactions the atom will ionize in the laser field. When the Keldyshparameter y>1fwthe ionization is multiphoton f Whereas y< 1 corresponds to the tunnelinregime. The key difference between them is the coer transition time andhe laser optical cycle while the tunneling time is中国煤化工For atom tunneling ionization ratel-there areCNMHGE checked the experimental results and theoretical predictions - and found them fitted well with each other in the inten-sity of 10 W/cm2 except the Szoke s equations. Sof we can use the three-dimensional DC-tunheling theory which is not time-averaged when we want to calculate the behavior of electron inlaser fields捕No. ll2D SEMICLASSICAL MODEL FOR HIGH HARMONIC GENERATION FROM GAS1203eZ￡nE=4W1//y/-2e3/2where Wa=4 10s is the atomic frequency unitE>Ei= Ea/ EH is the ratio of the ionizationpotential of the atom we studied and hydrogen. Ef ZEpfc E" nEg Ea= 1. 16x 102"Tb/AEGf nfSea is the atomic unit of field. Rh is the Bohr radius. n is the laser wavelength. E nECsthe well-known solution of one-dimentional wave equation. Here the field is statice and no atomicshell dependence is includedThe evolution of electron density n is given by;g= Pnowhere no is the initial gas densityAfter ionizationf-the ion remains its position for its large mass. The motion of the ionizeelectron in a linearly polarized laser field A=&"n 2@ is governed by Lagrangian equation￡"420In non-relativistic limite we haveu/c- a= al u/c￡"6￡0Cwhere ai and a, are the constants to be determined by initial conditions. Since the electron haselocity when itda2=0￡whotrength at ionization and then we getu1/c=a-a1￡u2/c=1-√1￡￡"720=C￡"8￡07The trajectory of the electron is then fully determinedIn generale u free electron quivering in laser field has very low probalility of collision withions. For examplef-in an ionized gas with Z=502wno=10cm End temperature Te= 100eⅤ￡ the electron- ion collision rate is merely y/≈4.8×10-5. However￡ since the electronsare ionized at rest e their behavior in the initial stage is somewhat special. We will show that inthe first half optical cycle after ionization an electron ionized bv a linearly polarized light hasrather good chance to be recombined. This inter中国煤化工 vant to the high harmonic generation from rare gasCNMHGLet us first consider a planar laser of infinite durationz-ti ytc aosif"kn2e-where kG/c. The trajectory isx= a EUcost;-cost￡"t-tE所ntE方数据4-4+sim22t+3si¨2t￡0cost(2sint￡¨10￡cSCIENCE IN CHINAC Series azoVol. 43Figpresentstralectonesof electrons being ionized at different timelot:=-20 and 5021where a0=0.02 and zi =0. In order to get a better viewf un amplified scale is used for the axisof z. The electrons entering the areac Hike that being ionized at t:=50f are considered to returnto the vicinity of its parent iorf defined as bohr radius rb=3. 3 x 10 fC It is clear that theseelectrons will be recombined with a photon of energy Ey=mu/2+ Ea emitted. Let us firstconsider the displacement in z direction after an optical cycle isTagUo. 5 +sin t fY= Ta/2￡"112⑩日0.0-0.02」0.0Time of ionization/()Fig. 1. The trajectory of electron driven by the sameFig. 2. The relation between electron energy and ionia-laser field with different ionization limnlion tumeElectron will return to the vicinity of ion in the condition of z. 2RB. We can get three importantosiNe.IE ifCElectrons failed to return in the first half optical cycle have little chance to do so in theirfuture flights. After the initial stage an ionized electron behaves more and more like a quiveringfree electron. The probability of collision with ions is then described by yei/aE" liECBecause the effect of magnetic field was considered a maximum laser intensity shouldbe put forward in order to ensure that some electrons return into Bohr radius in an optical cycleWhen laser intensity exceeds this valuee mo harmonic will be generated because the displacementin z axis is greater than the Bohr radius and the electron escapes from the parent ion. For exam-plef this intensity is 6. x 104Wi%m I ao=0. 021ECfor hydrogen in 800-nm wavelength ifcondition was satisfied. This intensity together with saturation intensity decide the maximum valufor hhgE iiECNot all the electrons will return to parent ion and emit photon in an optical cycle asfig. I shows. From eq f 10ECas x=Of we can get the relation between ionization time and therecombination time. Electrons will go back to ions if they are ionized at 0170.Another featureis that the recombination time is inversly proportional to the ionization timef. e. the earlier theelectrons are ionized the longer they stay in the laser fieldFig. 2 presents the kinetic energy of returmed electron Ek= mu/2 as a function of ionization timef where z;=0fw0=0.01. Since the field is periodical this figure represents the elec-tron behavior in each opticle cycle. Depending on中国煤化工 onization￡ some electrons will go back with different kinetic energy andCNMH Gons ionized at t=17will have the maximum kinetic energy on their returnt and the maximum energy happens to beEL/U3. 17. These are in agreement with Corkum i s prediction It is well known that themaximum kinetic energy is the physical origin of the hymax =3.17Up+ Ea law for the high har-monic radiation cutoff. This important cutoff law can be proved to be true for all laser strengthswhen the而高 nation time 2" t010 -from eq￡"8 BCwe can getNo. ll2D SEMICLASSICAL MODEL FOR HIGH HARMONIC GENERATION FROM GAS1205Ct-E=cost+rsi"t-￡国￡¨12Ecwhere t=t-ti fand thenco¨t-z/2o=d"z￡则U¨z￡4￡BE￡¨3￡where&rfe sis"T/2fc cs"r/2f6-6t2G sih"t/22CSof -the generated photon energy isE1uPAz￡GEEdeFor d0 while the electrons have the maximum energy one can get t=4.12o one canget [=4.42 from eq.29fCand then Emax =3. 17 Up. The corresponding ionized time is t:=tthe When electrons are ionized by circularly polarized laserf-the velocity and displacement ofelectron in z axis are￡"15E⑩a fa t-t Costi - sint sint f￡"16￡cThe displacement in the z axis is the same as that in the x axis. One can then get the expressionof the displacement of electron from parent iongsht+t:￡y￡¨170It is always greater than the bohr radius. One can also get the recombination electron energyEk=8 U, sirfat-t, 02fY It is greater than the electrons heated by linearly polarized lassersSof one can conclude that the electrons heated by circularly polarized laser never return tothe parent ion and emit photons. That is fit for experimental resultThen we consider a Gaussian pulse of duration tf&"n2O- aosif"ky2explEn-L/2￡lwhere L= CT. The motion of ionized electrons can still be described by egs f@ 9f@ The recombination of electrons is associated with photon radiation. And the power of radiation is given2 u+ Ea￡"19￡0Note that time delay must be accounted the radiation power is of recombination time. Using Larmor formula we get the power radiated per unit solid angledPdn-8sin-Apf7￡¨20￡cwhere 0 is measured from the direction of polarization. At 0=T/2f-dPd.Pe yh中国煤化工￡"21EcThe foansform o fCf tfciCNMHGdo2 Results and discussionWe taru3t-this 2D semiclassical model to explain the effect of high harmonic with pulse du-SCIENCE IN CHINAC Series azoVol. 43ration. Fig. 3 shows the emitted photon frequency spectrum produced by 50 fs laser pulse interacted with He gas atomi where a =0.01. One can get from the figure that it is odd times wave freenciesf and the maximum order is 53] choosing the magnitude drop to 4 orders as the cutoffecIn order to increase the cutoff energy of harmonicsf-higher laser intensity mustincrease the electrons energy and then increase the emitted photon energy under the same conditions. But the laser intensity cannot increase to the saturation intensity>. e. there is a saturationvalue for a certain gas density f a certain laser pulse duration and a certain species gas atom. Be-yond this value most of the gas atoms ionized and the intensity of harmonics will minimize to ze-ro. From eq.f3fB-one can get the time-dependence ionization rate curvet as fig. 4 showsfCOne can conclude that the saturate intensity increases with the decrease of laser duration. We repeat the calculation and get the time spectrum and frequency spectrum of He with 5 fs durationand a=0.022 as fig. 5 shows. One can get that the harmonics order increases rapidly over 100fTthe cutoff order is beyond 250Harmonic order(o,Fig. 3. The frequency spectrum of harmonics producedFig. 4. The ionization rate of the atom generated byby laser with 50 fs pulse duration. The highest harmoniultrashort intense pulseorder is5￡ao=0,.010三4.0204060801001Harmonic order(oig. 5. The time spectrun arcand frequency spectrin bECof high harmonics with 5 fs pulse duration interactedwith He The harmonic order is over loe a=0. 0220中国煤化工For the harmonic cutoff energy is proportionatensity and ionizationpotential the cutoff energy will drop if weTYHCNMHGsaturate intensity andionization potential. Fig. 6 shows the frequency spectrum of Xef -where a =0.02 and at 5 fspulse duration just as fig. 5. Compared with the frequency spectrum of He gasf-the harmonic or-ders of Xe gas decrease rapidlyFinalbyfsugcalculation can explain the large fluctuations observed experimentallyusein theNo. ll2D SEMICLASSICAL MODEL FOR HIGH HARMONIC GENERATION FROM GAS1207X-ray intensity at wavelength 10 nm in spite of the relatively small energy fluctuation of the 5 fspuls&5%EC Fig. 7 shows the extreme ultraviolef XUVECharmonic emission at the highest energies emitted for the same parameters as fig. 5 for initial laser phase of 0 and 90. The strongphase deof the XUV yield near cutoff region translates into corresponding fluctuation ofthe XUV emission produced by non-phase-stabilized pulses. For the laser initial phase is not con-trolled and hence cannot be kept constant in our laser pulses currently -so this anomalous fluctua-tion in the cutoff region is unavoidable until the carrier phase can be controlled三33>10-21304050200210220230240250260270Harmonic order (()Harmonic order(OJ)Fig. 6. The frequency spectrum of harmonics producedFig. 7. XUV hamonic emission spectra at the highby 5 fs pulse interacted with Xe. The harmonics orderes in sIn￡" dotted linel￡ Cand cosin" solid linedabout 59rier in the same laser intensity with 5 fs pulse durationThe electron behavior in laser field is described in detail in this papemiclassical modelf-a 2D semiclassical model is proposed analytically using 3D DC-tunnelinionization theory. Lots of harmonic features are explained by this modele including the analyticaldemonstration of the maximum electron energy 3. 17 Up. Finally some experimental phenomenasuch as the increase of cutoff energy with the decrease of laser duration and thei anomalous i fluc-tuations in the cutoff region can be explained by this modelAcknowledgements This work was supported by the National Natural Science Foundation of China" Grant No. 19854001EC-State Key Laboratory of High Temperature and High Density Foundation of CAEIE Grant No. 9804cC1. Krausef. L. f-SchaferfK. J. f-Kulanderf-K. C. f-High-ordler harmonic generation from atoms and ions in the high intensityregimef-Phys, Rev. Lett. f-1992f-6824c8935Corkumf-P. B E Plasma perspective on strong-field multiphoton ionizationf-Phys. Rev. Lett. f-1993E7f"132E919943. Lewensteinf-Mf-Balcouf-P f-lvanovf-M. Y, et al. f-Theory of high-harmonic generation by low-frequency laser fields-Phvs.Rev.AE1994249”321174. Protopapas￡M￡ Lappas.G.￡ Keitel.H.etal.ERC attosecond pulses from intenselaser fieldsE-Phys. Rev. AE-19952-58529-R2933中国煤化工5. Changfv E-RundquistEnA E-Wangf-H. et al. E-GenerationRev,le.-997v916￡92967C NMH GSIng high hanmonicsf-Phys6. Zhouf.f-Peatrossf-yf-Murnanef-M. M. et al. f-Enhanced high-harmonic generation using 25 fs laser pulses -Phys. RevLet.199￡-765￡7527. Augstf6 f-Meyerhoferf-D. D. fStricklandf-D. et al. f-Laser ionization of noble gases by coulomb-barrier suppression]Opt. Soc. Am. BE-1991f-842C 858发攻请E-Sartaniaf-S. et al. f-Generation of coherent X-rays in the waterwindow using 5-femtoseconde1997E278"279961