Control the high-order harmonics cutoff through the. combination of chirped laser and static electric field

Transcription

1 Control the high-order harmonics cutoff through the combination of chired laser and static electric field Yang Xiang,, Yueing iu Shangqing Gong State Key Laboratory of High Field Laser Physics, Shanghai Institute of Otics and Fine echanics, Chinese cademy of Science, Shanghai 8, China Graduate School of Chinese cademy of Sciences, eijing 49 China College of Comuter Science and Technology, Henan Polytechnic University, Jiaozuo 454 China bstract:the high harmonic generation from atoms in the combination of chired laser ulse and static field is theoretically investigated. For the first time, we exlore a further hysical mechanism of the significant extension of high harmonic generation cutoff based on three-ste model. It is shown that the cutoff is substantially extended due to the asymmetry of the combined field. If aroriate arameters are chosen, the cutoff of high harmonic generation can reach I + 4U. Furthermore, an ultrabroad suer-continuum sectrum can be generated. When the hases are roerly comensated for, an isolated 9 attosecond ulse can be obtained. PCS numbers: 4.65.Ky It is well known that high harmonics are generated when atoms or molecules are irradiated by intense laser field. In recent years, high harmonic generation (HHG) has become a very interesting toic of laser-atom interaction [-] because of a very imortant alication of attosecond (as) ulses generation. Usually, there are two techniques to generate attosecond ulses [4]. One is to use few-cycle laser ulse, then the cutoff is continuous and by sectrally selecting it a single attosecond ulse can be obtained [5]. The other is to use multi-cycle intense infrared ulse, then select many harmonics in the lateau and an attosecond ulse train can be generated [6]. any control ways have been studied in order to gain an isolated attosecond ulse [7-9]. In the resence of an intense static electric field, one can get a ulse of attosecond []. Using a chired few-cycle laser field, a ulse of 8 attosecond can be generated []. From the works mentioned above, one may want to know what would haen if the combination of chired laser field and a static electric field is adoted in the HHG rocess. ased on this idea, we now investigate the HHG sectrum of atoms in this combined field. We use one-dimensional time-deendent Schrödinger equation (TDSE) to describe the interaction between an atom and the combination of chired laser field and the static electric field in the single active electron and diole aroximations (The atom units are used in all equations in this aer, unless otherwise mentioned.): ( xt), i = + V x xe t x ψ ( ) ( t) ψ ( x, t), ()

2 where the Coulomb otential V(x) is reresented by the soft-core otential which can be V x exressed as: ( ) = / ρ + x.here, ρ is the softening arameter. We choose ρ =.6957 which corresonds to the ionization otential ( I ) of 4.6 ev for the ground state of the helium atom. The combined field is described as E() t = F f ( t) cos( ωt+ δ ( t) ) α, () where F, () f t,ω and () t δ is the amlitude, enveloe, angular frequency and time rofile of the carrier enveloe hase (CEP) of the laser field, and α is the ratio between the amlitude of the static and laser field. If symmetry will be destroyed if α = and δ ( t) =, ( ) α or ( t) E t is of symmetry. However the δ. In all of our calculations, F is 5. 4 W/cm, ω is.57 a.u. (corresonds to wavelength λ =8 nm) and α is equal to.4 which is aroximate to that in Ref. []. lthough such a high static field can hardly be exerimentally achieved nowadays, a low-frequency laser field (such as CO lasers) can be used instead []. The enveloe of the laser field f () t has a tye of Gaussian with 5 femtosecond full width at half maximum. The CEP considered in this aer is δ ( t) = β tanh ( t t ) / τ []. The ( ) arameters β, t and τ are used to control the chir form. Due to the recent advancement of comb laser technology, it is highly likely that such a time-varying CEP can be achieved in near future [,, 4]. In our work, τ is chosen to be a.u.. In order to find how the combined field imacts on the HHG, we consider the HHG of helium atom in four field cases: (a) chir-free laser field; (b) combination of chir-free laser field and static electric field; (c) chired laser field; (d) combination of chired laser field and static electric field. The arameters β and t for the chired laser field in (c) and (d) are 6.5 and τ /7.. These fields are shown in Fig.. The HHG sectrums of the four cases are shown in Fig.. The cutoff of the sectrums for (b), (c) and (d) is about 7st, 4st and 55st-order harmonic resectively, and each of them is much higher than the well-known value of I +.7U (about 77th order harmonic, just as shown in Fig.(a)), where U ( = F /(4 ω ))is the onderomotive energy. To our surrise, the cutoff of HHG in (d) is much higher than that in (b) or (c). Furthermore, an ultra-broad suer-continuum sectrum which covers about 45 orders harmonics aears in the case of combination of chired laser field and static electric field. Generally, the hysical origin of the HHG from linearly olarized laser field can be qualitatively understood in the framework of semi-classical model which is called three-ste-method (ST) [5]: first, the electron tunnels through the barrier formed by the Coulomb otential and the laser field; next, it oscillates almost freely in the laser field; finally, it may return back and recombine with the arent ion. During the recombination, a hoton is emitted. The returning kinetic energies of the electron in different laser fields are shown in Fig.. For

3 comarison, the ionization otential is added. It is aarent that the trajectories of the electron in these field cases are quite different. Usually, there are two dominant quantum aths with different emission times contributing to each harmonic in each half otical cycle (O.C.) []: the ositive sloe section (short ath) and negative sloe section (long ath) in returning kinetic energy ma. In Fig. (b), due to the effect of static electric field, only the short aths contribute to the harmonics above I +.6U, and the maximum returning kinetic energy (RKE) is u to8. U (about 54 harmonics). That is to say, the static electric field can eliminate some long aths of the electron. In Fig. (c), the long aths of the electron are not eliminated as aarently as those in Fig.(b), but the RKE is u to.7u (about 5 harmonics) because of the chired laser field. However, when the combination of the chired laser and static electric field is alied, the electron returns within less than.5 otical cycles, and for harmonics above I +.U, only the short ath contributes to the harmonic generation, just as Fig. (d) dislays. The RKE is u to 7.5U (about 5 harmonics) which is much higher than the summation of that in Fig. (b) and Fig. (c). If we mark the maximum eak (cutoff-energy) and the eak just below it with and resectively, we can see clearly that it is the difference between the eak and that contributes to the continuum sectrum, as is mentioned in Ref. [8]. Just as reviewed in the introduction, the continuous sectrum is beneficial to generate single attosecond ulse. This will be discussed later (a) 5 (c) (b).. 5 (d).. 5 Figure (a) chir-free laser field; (b) combination of chir-free laser field and static electric field; (c) chired laser field; (d) combination of chired laser field and static electric field. In order to exlore the extremely extended cutoff, we look into two rocesses in TS: (i) once the electron tunnels, it accelerates in the electric field; (ii) when the electric field inverses, the electron then slows down and turn back to the arent ion. If the imulse in rocess (i) is weak, the kinetic energy the electron gains is small, as well as the returning kinetic energy. However, if the imulse in rocess (i) is strong, the imulse in rocess (ii) should be much stronger than the imulse in rocess (i), otherwise, the electron can not return to the arent ion. So, in order to gain

4 a large returning kinetic energy, both the imulse in rocess (i) and the difference between the two imulses should be large. ased on this, one can easily find the two imulses which contribute to the RKE. These imulses are reresented by the shadow areas which marked with and resectively in each field of Fig.. Comared with the returning energy ma in Fig., we find that the electron of the maximum energy just returns in area. From Fig., we can see that there is not so much difference of imulse in the four fields. evertheless, due to the static electric field or the chir, the symmetry of the field is broken. Thus, the difference of imulse and imulse enlarges, so the cutoff of case (b) or (c) is extended, comared with that of case (a). For the combination of the chired laser and the static field, the symmetry is broken further, and hence there is a greater difference between imulse and imulse. s a result, the cutoff is extended significantly. From the analysis above, we conclude that roer asymmetry of the field can lead to the extension of the HHG cutoff extremely. log(p(ω)) Return log(p(ω)) Return (a) 5 5 (b) log(p(ω)) log(p(ω)) Return Return (c) (d) Figure Comarison of the returning kinetic energy of the electron and HHG sectrums in different field conditions: (a) chir-free laser field; (b) combination of chir free laser and static electric field; (c) chired laser field; d) combination of chired laser and static electric field. The arameters are same to those in Fig.. Since the cutoff of HHG can be extremely extended in the resence of chired laser and static electric field, it is instructive to find out how the chir arameters imact on the HHG cutoff. The 4

5 variation of the RKE of electron with the chir arameters β and is obtained by TS, and the result is shown in Fig.. arently the otimal RKE of electron is u to 4 at β = 9. and t = τ /8.. The HHG sectrum for these arameters is resented in Fig.4. One can find that the continuum sectrum is broadened to 7 order harmonics. t U β=6.5 β=7. β=8. β=9. β=. Ek/U t/τ Figure The maximum returning energy of electron as a function of β and t. log(p(ω)) Figure 4 HHG sectrum for the combination of chired laser and static field. The chir arameters are: β = 9., t = τ /8.. ow, we consider the attosecond ulse generation from HHG of Fig.4. y adding a frequency window with a bandwidth of 5 order harmonics to the suer-continuum sectrum and making an inverse Fourier transformation, an isolated attosecond ulse is generated without any hase comensation. If we move this frequency window along harmonic order axis, the duration of the attosecond ulse changes a little, though the intensity decreases. That is to say, the ulse duration is not sensitive to the osition of the frequency window. The temoral rofiles of the attosecond ulses are shown in Fig.5 (a). The shortest ulse duration we show in this figure is 56 attosecond. One may think that if we enlarge the width of the frequency window, the attosecond ulse duration will shorten further. ctually, due to the hase mismatch, the ulse duration lengthens but rather shortens, as shown in Fig.5 (b) (comared with the red dot line in (a)). However, if the hase mismatch over such an ultra-broad continuum sectral can be roerly comensated for, an isolated 9 attosecond ulse with a clean temoral rofile (Fig.5(c)) could be theoretically obtained. 5

6 as.6.4 intensity (arb.units).5 57 as 56 as intensity (arb.units) as time (as) time (as) (a) (b).5 intensity (arb.units).5 9 as time (as) x 4 (c) Figure 5 The temoral rofiles of the attosecond ulses generated from the continuum sectrum without (a), (b) and with (c) hase comensation. The harmonics order used in (a) are: th- 7th (blue dash line), 4th- 9th (red dot line), 6th- th (green solid line).the harmonics order used in (b) is: 4th-th.The harmonics order used in (c) is: th-7th. In conclusion, the HHG of helium atom in the combination of chired laser field and static electric field has been investigated. We demonstrated that roer asymmetry of the field lead to the significant extension of the HHG cutoff. If aroriate arameters are used, an ultra-broad continuum sectrum which covers about 7 order harmonics is obtained. y imosing a band-ass filter with bandwidth of 5 order harmonics on the continuum sectrum, a sub- attosecond isolated ulse is generated without any hase comensation. If all the hase of the continuum sectrum is roerly comensated for, an isolated 9 attosecond ulse with clean rofile can be obtained. Considering the racticability in exeriment, we investigate the HHG sectrum of He atom in combination of chired laser field and a low-frequency laser field (CO laser, the intensity and wavelength is 8 w/cm, 9.6 μm resectively).still an ultra-broad continuum sectrum (covers about 6 order harmonics) is obtained and with roer hase comensation, an isolated 9 attosecond ulse can be generated, which have little differences with that discussed above. cknowledgements 6

7 The work is suorted by the ational asic Research Program of China (Grant o.6c94, 6788), the Project of cademic Leaders in Shanghai (Grant o.7xd4) and the Knowledge Innovation Program of the Chinese cademy of Sciences. References: [] Carsten Winterfeldt et al. Rev od. Phy. 8, 7 (8). [] Thomas Pfeifer et al. Phys. Rev. Lett. 97, 69 (6). [] Zhinan Zeng et al. Phys. Rev. 77, 46 (8). [4] Y. ariresse et al. Phys. Rev. Lett. 9, 69 (4). [5] R. Kienberger et al. ature (London), 47, 87 (4) [6] Y. ariresse et al. Science,, 54 (). [7] Wei Cao et al. Phys. Rev. 75, 64 (7). [8] Zhinan Zeng et al. Phys. Rev. Lett. 98, 9 (7) [9] Juan J. Carrera et al. Phys. Rev (6). [] Weiyi Hong et al. J. Phys. : t. ol. Ot. Phys. 4, (7). [] Juan J.Carrera and Shih-I Chu. Phys.Rev. 75, 87 (7). [] orca et al. Phys. Rev. Lett. 85, 7 (). [] S. T. Cundiff and J Ye, Rev. od. Phys () [4] Th. Udem et al. ature (London) 46, (). [5] P.. Corkum. Phys. Rev. lett. 7, 997 (99). 7