Extension of Harmonic Cutoff and Generation of Isolated Sub-30 as Pulse in a Two-Color Chirped Laser Field

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1 Commun. Theor. Phys. 58 (2012) Vol. 58, No. 4, October 15, 2012 Extension of Harmonic Cutoff and Generation of Isolated Sub-30 as Pulse in a Two-Color Chirped Laser Field ZHANG Gang-Tai ( ), 1, BAI Ting-Ting (ÜÜÜ), 2 and ZHANG Mei-Guang (ß½) 1 1 Department of Physics and Information Technology, Baoji University of Arts and Sciences, Baoji , China 2 Department of Mathematics, Baoji University of Arts and Sciences, Baoji , China (Received February 27, 2012; revised manuscript received June 25, 2012) Abstract We theoretically investigate high-order harmonic and isolated attosecond pulse generation in a two-color chirped laser field, which is synthesized by a 9 fs/800 nm fundamental chirped pulse and a 9 fs/1600 nm controlling chirped pulse. Our numerical results show that, by using this method, not only is the harmonic cutoff significantly extended to the 948th order harmonic, but also the bandwidth of the supercontinuum spectrum is effectively broadened to about 1342 ev. In addition, due to the introduction of the chirp, the long quantum path is suppressed and only the short one is selected, and then an isolated 28 as pulse with a bandwidth of 155 ev is obtained directly. PACS numbers: Ky, Rm, Re Key words: high-order harmonic, attosecond pulse generation, supercontinuum, two-color chirped field 1 Introduction The emergence and development of attosecond (as) extreme ultraviolet pulses pave a way for studying and operating basic ultrafast electronic processes in atoms and molecules with unprecedented accuracy and resolution. [1 3] Presently, the high-order harmonic generation (HHG) is a unique way to produce attosecond pulses in experiments. The mechanism of the HHG can be well understood by the well-known semiclassical threestep model: [4] first an electron tunnels through the barrier formed by the atomic potential and the laser field, then it oscillates almost freely in the laser field, and finally it may return the ground state by recombining with the parent ion. During the recombination, a photon with energy equal to the ionization potential plus the kinetic energy of the recombining electron is emitted. It has been investigated theoretically [5] and experimentally [6] that an attosecond pulse train with a periodicity of half an optical cycle of the laser field is generated from HHG. However, for practical application, the straightforward attosecond metrology prefers an isolated attosecond pulse, so much effort has been paid out to obtain an isolated attosecond pulse. It has been shown that an isolated attosecond pulse can be obtained by using a few-cycle laser pulse [1 2,4] or polarization gating technique. [3,7] Currently, an isolated 80 as pulse has been achieved experimentally, [8] which is also the shortest attosecond pulse in experiment. However, the pulse of isolated attosecond pulse is still significantly longer than the time scale of electron motion in atoms., i.e., 24 as. In attosecond science, it has been suggested that the bandwidth of the attosecond pulse is more important than the duration. [9] Therefore, it is an intense desire to broaden the bandwidth of the attosecond pulse and push the duration to much shorter time. Recently, it has been proposed that the waveformcontrolled two-color field can generate a broad supercontinuum, which can support the generation of a broadband isolated attosecond pulse. By using the controlling technique, some authors have successfully synthesized a broadband isolated attosecond pulse. [10 12] Very recently, we also note that a variety of schemes with a chirp form are used to investigate the wavepacket behavior [13] and the attosecond pulse generation based on the HHG. [14 19] Here, we pay more attention to the latter aspect. Carrera et al. [14] extended the HHG cutoff and created an isolated 108 as pulse via coherent control of an intense few-cycle chirped laser pulse. Xiang et al. [15] showed that the extension of the harmonic cutoff and the generation of the ultrabroad continuum spectrum can be observed through the combination of a chirped few-cycle laser and static electric field, and then a clean isolated 10 as pulse can be produced with phase compensation. Feng et al. [16] theoretically investigated the combined chirp effects on the HHG and isolated attosecond pulse generation in the combination of an intense 5 fs, 800 nm fundamental chirped pulse combined with a weak 10 fs, 1200 nm controlling chirped pulse. Moreover, they also investigated the nuclear signatures effects on the generations of the molecular high- Supported by the Science Foundation of Baoji University of Arts and Sciences of China under Grant Nos. Zk10122, ZK11061, ZK11135, ZK11060, and ZK1032 Corresponding author, gtzhang79@163.com c 2011 Chinese Physical Society and IOP Publishing Ltd

2 558 Communications in Theoretical Physics Vol. 58 order harmonics and the attosecond pulses, and obtained an isolated 57 as (16 as) pulse without (with) phase compensation through optimizing the laser parameters under the optimal initial vibrational state. [17] Li et al. [18] directly obtained an isolated 26 as pulse with a bandwidth of ev from the supercontinuum around the cutoff of HHG by an intense few-cycle chirped laser and its high-order harmonic pulses. Wu et al. [19] theoretically demonstrated both a 1670 ev broadband supercontinuum and an intense isolated 38 as pulse could be generated by combining a chirped 5 fs/800 nm fundamental laser with a 12 fs/1600 nm subharmonic laser field. However, in these above schemes with a chirp form, the requirement for the driving pulse is rather stringent: a few-cycle laser pulse with a stabilized carrier-envelope phase (CEP) is required, which can only be achieved with a state-of-the-art laser system. Thereby, a great amount of effort has been focused on producing a single attosecond pulse by using a multicycle laser pulse. It has been suggested that the requirement for single attosecond generation with a fewcycle driving pulse has been released in a two-color field scheme with a multicycle driving laser pulse. [20 25] Pfeifer et al. used the multicycle 800 nm pulse and a weak second harmonic field [20] or a weak subharmonic field [21] to generate single attosecond pulses. Cao et al. [22] theoretically presented a method for efficient isolated attosecond pulse generation using the combination of a fundamental and a weak second harmonic field in the multicycle regime. Zeng et al. [23] demonstrated that, with a moderate laser intensity ( W cm 2 ), an isolated attosecond pulse of 220 as can be generated with the two-color laser pulse consisting of a 35 fs/800 nm pulse and a 46 fs/1150 nm pulse. Tang et al. [24] proposed a scheme to generate isolated sub-100-as pulses from a helium atom with 30 fs lasers. Chen et al. [25] obtained an intense 38 as isolated pulse from a coherent superposition state by quantum path control in a multicycle 800 nm laser pulse in combination with a controlling 1600 nm laser pulse. However, the duration of the shortest isolated attosecond pulse achievable is further limited by the intrinsic chirp of the harmonic emission, which arises from the laser-intensitydependent atomic dipole phase. Therefore, the chirp compensation of attosecond harmonic pulses is very crucial for the generation of transform-limited attosecond pulses. To realize the goal, some works have been made to obtain the transform-limited attosecond pulses. [26 28] Recently, Kohler et al. [29] proposed a method to acquire a complete control of the chirp by shaping a laser pulse and employing XUV light or X rays for ionization, and showed that this method allows for the formation of attosecond pulses with arbitrary chirp, including the possibility of attochirp-free HHG and bandwidth-limited attosecond pulses. Though isolated attosecond pulse generations from the single chirped pulse, [14,17,30] the two-color laser pulse, the fundamental chirped pulse combined with a controlling chirp-free pulse, [18 19,31 32] and the two-color chirped pulse [16] have been investigated, respectively, little research has been reported in the two-color chirped pulse with a multicycle driving pulse. It is because of this, in this paper, we theoretically investigate the HHG and the isolated attosecond pulse generation in a two-color chirped laser field synthesized by a multicycle fundamental chirped pulse and a controlling chirped pulse. Our goal lies in extending the harmonic cutoff, broadening the supercontinuum, and generating an ultrashort isolated attosecond pulse. It is shown that, with this method, the harmonic cutoff is extended significantly and the bandwidth of the supercontinuum spectrum is broadened obviously. In addition, due to the introduction of the chirp, the electron trajectories for the HHG are significantly modulated, as a result, the long quantum path is suppressed, and only the short quantum path is picked out to contribute to the supercontinuum generation. By superposing the harmonics from 400th to 500th order, an isolated 28 as pulse with a bandwidth of 155 ev is directly obtained. To demonstrate these results, we perform the classical trajectory simulation by the semiclassical three-step model and the quantum time-frequency analysis in terms of the wavelet transformation of the time-dependent induced dipole acceleration. 2 Model and Methods In our numerical calculations, the HHG spectrum and the attosecond pulse generation can be studied by numerically solving the 1D time-dependent Schrödinger equation with the splitting operator method. [33] The simulation method is based on a single-active-electron approximation and has widely been used for HHG simulation. The harmonic spectrum is obtained by Fourier transforming the time-dependent dipole acceleration for a model atom. The temporal profiles of the attosecond pulses can be generated by simply performing inverse Fourier transformations of the XUV supercontinua in different spectra regions. In our simulation, we choose a soft-core Coulomb potential model V (x) = z/ x 2 + a 2 and set z = 1 and a = corresponding to the ionization energy of ev for the ground state of an Ne atom. The two-color chirped laser pulse is synthesized by a 9 fs/800 nm fundamental chirped pulse and a 9 fs/1600 nm controlling chirped pulse. The intensities of the 800 nm fundamental chirped pulse and 1600 nm controlling chirped pulse are W/cm 2 and W/cm 2, respectively. The electric field of the synthesized laser pulse can be expressed as E(t) = E 1 f 1 (t)cos[ω 1 t + δ 1 (t)]

3 No. 4 Communications in Theoretical Physics E 2 f 2 (t)cos[ω 2 t + δ 2 (t)], (1) where E i and ω i (i = 1, 2) are the amplitudes and frequencies of the 800 nm fundamental chirped pulse and 1600 nm controlling chirped pulse, respectively. f i (t) = exp[ 2 ln2(t/τ i ) 2 ] (i = 1, 2) present the envelopes of the two chirped pulses and τ i (i = 1, 2) are the corresponding pulse durations at full width at half maximum (FWHM). δ i (t) (i = 1, 2) are the time profiles of the carrier-envelope phases (CEPs), which has the time-varying hyperbolic tangent form δ i (t) = β i tanh[(t t i )/T i ] (i = 1, 2). The chirp form is controlled by adjusting the two parameters β i and T i (i = 1, 2). t i (i = 1, 2) are used to adjust the sweep range. In the present work, T i and t i (i = 1, 2) are chosen to be 200 a.u. and 0, respectively. Due to the recent advancements in comb laser technology, it is highly likely that such a time-varying CEP can be achieved in the near future. [14,34 35] Moreover, the form of the frequencychirped pulse has been extensively adopted in the study of HHG and attosecond pulse generation. [14 16,31 32] 3 Results and Discussion In Fig. 1(a), we present the harmonic spectra of the Ne atom in the fundamental chirped pulse (β 1 = 6.25), in the fundamental chirped pulse in combination with a 9 fs/1600 controlling chirp-free pulse (β 1 = 6.25 and β 2 = 0), and in the two-color chirped laser pulse (β 1 = 6.25 and β 2 = 4.4), respectively. For the case of single fundamental chirped pulse, the harmonic spectrum shows a double-plateau structure, as shown by the solid gray curve in Fig. 1(a). The spectrum cutoff is only at the 241st order harmonic, and the spectral structure above 128ω 1 (ω 1 is the central frequency of the fundamental chirped pulse) is continuous. For the case of the fundamental chirped pulse in combination with a 9 fs/1600 controlling chirp-free pulse, the harmonic spectrum reveals a double-plateau structure, which is similar to that of the single fundamental chirped pulse, as shown by the dashed dark-gray curve in Fig. 1(a). However, the spectrum cutoff is extended to 483ω 1, and the harmonic spectrum above 282ω 1 is continuous. Thus the bandwidth of the supercontinuum spectrum for adding the controlling chirp-free pulse is much broader than that in the fundamental chirped pulse alone. Moreover, for the above two optical fields, the supercontinuum shows a strongly modulated structure, which is due to the interference of the long and short quantum paths. As is well known, the strongly modulated supercontinuum with two quantum paths is not beneficial to generate an isolated attosecond pulse. Simultaneously, these results imply that both the selection of single quantum path and the generation of isolated attosecond pulse can not be achieved in the above two cases. In order to obtain a single attosecond pulse, the less modulated supercontinuum with single quantum path contribution should be achieved. This aim can be realized by a two-color chirped pulse. Here, by adding a 9 fs/1600 nm controlling chirped pulse (β 2 = 4.4) to the fundamental chirped pulse (β 1 = 6.25), the situation is completely different from the above two cases. To substantiate this fact, the HHG spectrum for the case is presented, as shown by the dotted black curve in Fig. 1(a). Clearly, one can see that for the case of the two-color chirped pulse the harmonic spectrum shows a similar structure to the cases of the above two optical fields, whereas the first cutoff of the spectrum is reduced to 83ω 1 and the second one of the spectrum is remarkably extended to 948ω 1, thus the harmonic spectrum above the 83rd becomes continuous, which corresponds to an ultrabroad supercontinuum with a bandwidth of about Intensity/arb.units (a) (b) A chirped 800 nm chirped 800 nm+chirp free 1600 nm chirped 800 nm+chirped 1600 nm T10-3 T Harmonic order (ω/ω 1 ) Harmonic order (ω/ω 1 ) Fig. 1 (a) Harmonic spectra of the Ne atom in the fundamental chirped pulse (solid gray curve), in the fundamental chirped pulse in combination with a 9 fs/1600 nm controlling chirp-free pulse with a peak intensity of W/cm 2 (dashed dark-gray curve), and in the two-color chirped laser pulse synthesized by the fundamental chirped pulse and a 9 fs/1600 nm controlling chirped pulse (dotted black curve). For the purpose of clarity, the harmonic intensities of the fundamental chirped pulse and the fundamental chirped pulse in combination with a 9 fs/1600 nm controlling chirped pulse are multiplied by factors of 10 3 and 10 6, respectively. (b) Time-frequency distribution of the HHG spectrum corresponding to the dotted black curve in Fig. 1(a). B

4 560 Communications in Theoretical Physics Vol ev. Compared with the cases of using the single fundamental chirped pulse and the fundamental chirped pulse combined with the controlling chirp-free pulse, the bandwidth of the supercontinuum in the two-color chirped field becomes much broader. Furthermore, the modulation in the supercontinuum is largely removed, and the spectrum is much smoother for the harmonics in the second plateau, implying one quantum path has been selected. To well understand the spectral structure shown by the dotted black curve in Fig. 1(a), we investigate the emission time of the harmonics for the case in terms of the time-frequency analysis method. [36] The result is shown in Fig. 1(b). As shown in this figure, there are two main peaks contributing to the harmonics, which is marked as A, and B, respectively. The maximal harmonic orders for A and B are about 83, and 948, respectively. The intensity of A is stronger than that of B, in other words, the harmonic yield of A are higher than that of B, resulting in a double-plateau structure in the spectrum shown by the dotted black curve in Fig. 1(a). Furthermore, the harmonics above the 83rd order only originate from the contribution of B, forming an ultrabroad xuv supercontinuum with a bandwidth of about 1342 ev. For the harmonics below the 83rd order, there are at least two quantum paths contributing to the same harmonic, the interference between these quantum paths leads to the irregular spectrum structure. Particularly, for the peak B, the long quantum path is suppressed and the short one is retained, which is responsible for a broadband smooth supercontinuum in the second plateau and at the same time indicates that the selection of the single short path is achieved successfully. Considering all the results above, we can conclude that the broadband supercontinuum with single quantum path can be generated in the two-color chirped field, which can support the generation of a favorable isolated attosecond pulse. In the following, we consider the attosecond pulse generation in the two-color chirped field, the calculation results are shown in Fig. 2. For comparison, the inset in Fig. 2(a) shows the temporal profile of the attosecond pulse in the fundamental chirped pulse in combination with the controlling chirp-free pulse. In Fig. 2(a), we superpose the harmonics from 400th to 500th order, an isolated 28 as pulse with a bandwidth of 155 ev is directly obtained without any phase compensation. This is in sharp contrast to the case of the inset in Fig. 2(a), where two radiation pulses are synchronically emitted in every optical cycle due to this fact that there are two quantum paths with different emission times corresponding to the same harmonic. In addition, the different ultrashort isolated attosecond pulses can be obtained by superposing different harmonic orders in the supercontinuum region. Figure 2(b) presents the temporal profile of the attosecond pulse generated by superposing the harmonics from 440th to 505th order, an isolated 39 as pulse with high signal-to-noise ratio is produced. Figure 2(c) presents the temporal profile of the generated isolated attosecond pulse by superposing the harmonics from 495th to 560th order, an isolated 40 as pulse with a clean temporal profile is obtained. Figure 2(d) shows the temporal profile of the generated isolated attosecond pulse via superposing 71 order harmonics from 420 to 490. It can be clearly seen that a clean isolated 43 as pulse is generated. To explore to the underlying physics of the harmonic cutoff extension, the supercontinuum broadening and the isolated attosecond pulse generation, we perform the classical trajectory simulation in terms of the semiclassic three-step model. For comparison, the result of adding the controlling chirp-free pulse is also given. Figure 3(a) presents the electric field of the fundamental chirped pulse in combination with the controlling chirp-free pulse. Figure 3(b) shows the corresponding dependence of the kinetic energy on the ionization (gray diamonds) and emission times (black circles). As shown in Fig. 3(b), the electrons are mainly ionized near the peaks of the laser field, which are marked as A, B, and C, respectively. The highest kinetic energy of the electrons ionized near the peak B is 727 ev corresponding to the cutoff position in the second plateau (i.e., 483rd order), and the second-highest kinetic energy of the electrons ionized near the peak A or peak C is about 416 ev corresponding to the cutoff position in the first plateau (i.e., 282nd order), which results in a broadband supercontinuum with a bandwidth of 311 ev. Furthermore, for the harmonic photons with energy higher than I p +416 ev, there are two classes of trajectories corresponding to the same kinetic energy, which are the so-called long and short trajectories. A trajectory with earlier ionization but later emission times is called the long trajectory, and a trajectory with later ionization but earlier emission times is called the short trajectory. As the kinetic energy increases, the emission time for the short trajectory increases, that for the long trajectory decreases, and finally the emission times of two trajectories are equal for the highest kinetic energy. This result indicates the harmonics in the second plateau are not emitted at the same time. Therefore, the superposition of several harmonics will lead to two radiation pulses in every one optical cycle, as shown by the inset in Fig. 2(a). By introducing a chirp to the above optical field shown in Fig. 3(a), i.e., in the two-color chirped field case, the electron trajectories of the HHG can be significantly modulated. Figure 3(c) shows the electric field of the two-color chirped field. Figure 3(d) shows the corresponding dependence of the kinetic energy on the ionization (gray diamonds) and emission times (black circles).

5 No. 4 Communications in Theoretical Physics 561 T10-6 T10-6 Intensity/arb.units Intensity/arb.units T10-6 Intensity/arb.units Intensity/arb.units T10-6 Fig. 2 Temporal profile of the attosecond pulse in the two-color chirped field: (a) by superposing the harmonics from 400th to 500th order, (b) by superposing the harmonics from 440th to 505th order, (c) by superposing the harmonics from 495th to 560th order, (d) by superposing the harmonics from 420th to 490th order. The inset in Fig. 2(a) shows the temporal profile of the attosecond pulse in the fundamental chirped pulse in combination with the controlling chirp-free pulse. All parameters are the same as those in Fig. 1. Kinetic energy/ev Electric field/a.u. Kinetic energy/ev Electric field/a.u. Fig. 3 (a) Electric field of the fundamental chirped pulse in combination with the controlling chirp-free pulse. (b) Dependence of the kinetic energy on the ionization (gray diamonds) and emission times (black circles) in the field shown in (a). (c) Electric field of the two-color chirped field. (d) Dependence of the kinetic energy on the ionization (gray diamonds) and emission times (black circles) in the field shown in (c). As shown in Fig. 3(c), there are three dominant ionization peaks (marked as A, B, and C) corresponding to the harmonic generation. The highest kinetic energy of the peak B is 1448 ev, corresponding to the second cutoff energy at I p ev (i.e., 948ω 1 ). Since the highest kinetic energy of the electron gained in the two-color chirped

6 562 Communications in Theoretical Physics Vol. 58 field is much more higher than that in the field shown in Fig. 3(a), the cutoff position of the HHG spectrum for the case is greatly extended. In addition, the harmonic cutoff extension can be interpreted from another point of view, namely after the electrons are ionized, the longer the time interval of the reversed electric field is, the higher kinetic energy the quasifreely electrons obtain from the driving laser field, as has been reported in Refs. [18, 31]. From Figs. 3(c) and 3(d), one can see that the electron at the peak B ionized at about 2.42 o.c. is accelerated by the reversed electric fields both from 1.97 to 1.39 o.c. and from 0.66 to 0.66 o.c., and obtains 1448 ev kinetic energy when returning to the parent ion at about 0.57 o.c. However, in Figs. 3(a) and 3(b), the electron at the peak B ionized at about 0.95 o.c. is accelerated by the reversed electric field from 0.52 to 0.52 o.c., and obtains 727 ev kinetic energy when returning to the parent ion at about 0.42 o.c. Therefore, the extension of the harmonic cutoff for the two-color chirped field case is due to that the electron experiences a longer time acceleration in the reversed electric field. For the peak C, the highest kinetic energy of the electron is 993 ev, which mainly contributes to the harmonics up to the 640th order. For the peak A, the highest kinetic energy of the electron is at about 130 ev, corresponding to the first cutoff energy at I p +130 ev (i.e., 83ω 1 ). Thus the cutoff energy difference between the two peaks (B and A) is much larger than that in Fig. 3(b). The enhanced cutoff energy difference explains why the supercontinuum spectrum generated by the two-color chirped field can be significantly broadened. Furthermore, for the harmonic photons with energy higher than I p +130 ev, the contribution of the long trajectory has been obviously suppressed, only two short trajectories are kept down. Note, though the ionization times of the peaks C and B are completely different, the emission times of the short trajectories in the two peaks are nearly equal. In other words, the harmonics with energies beyond I p +130 ev are emitted almost at the same time, i.e., are well phase-locked. Since the phase-locked harmonics cover an extremely broad bandwidth, which is beneficial for the generation of an ultrashort isolated attosecond pulse, as shown in Fig. 2. In this work, by introducing a properly chirped parameter to the laser field shown in Fig. 3(a), we realize the extension of both the harmonic cutoff and the supercontinuum. But one wants to know what the effect of the chirped parameter β 2 is. For this purpose, we investigate the HHG spectra in the two-color chirped field with three different chirped parameters: β 2 = 1, β 2 = 2, and β 2 = 4.4, respectively. Other parameters are the same as those in Fig T10 8 T Harmonic order (ω/ω 1 ) β 2 =1 β 2 =2 β 2 =4.4 Fig. 4 Harmonic spectra in the two-color chirped field with different chirped parameters: β 2 = 1, β 2 = 2, and β 2 = 4.4, respectively. For the purpose of clarity, the harmonic intensities for β 2 = 2 and β 2 = 4.4 are multiplied by factors of 10 3 and 10 8, respectively. Other parameters are the same as those in Fig. 1. Figure 4 gives our calculated results. At first glance, the harmonic spectra for these three cases present a similar two-plateau structure, but the spectral cutoff and the bandwidth of the supercontinuum are different. For the case of β 2 = 1, the spectrum cutoff is only at the 541st harmonic, and the harmonics higher than 276th order are regular and continuous, corresponding the supercontinuum with a bandwidth of about 411 ev. For the case of β 2 = 2, the spectrum cutoff is further extended to the 607th harmonic, and the supercontinuum ranges from 268th to 607th order, covering a 526 ev bandwidth, which is superior to that for the case of β 2 = 1. For the case of β 2 = 4.4, the cutoff of the harmonic spectrum is significantly expanded to the 948th harmonic, which is much larger than that for the cases of β 2 = 1 and β 2 = 2. In addition, the bandwidth of supercontinuum spectrum for the case is up to 1342 ev, ranging from 83rd to 948th, which is also much bigger than the above two cases. Most importantly, for β 2 = 4.4 case, the modulation in the supercontinuum is much reduced and the spectrum becomes smoother and more regular, compared with the above two cases. Based on the above analysis, we conclude that, with increasing the chirped parameter β 2, the harmonic cutoff and the bandwidth of the supercontinuum are all enhanced, and simultaneously the modulation on the supercontinuum is further eliminated, which is beneficial to the generation of a favorable isolated attosecond pulse. Note, though the spectrum cutoff and the bandwidth of the supercontinuum are the highest for the case of β 2 = 5.0 (Other parameters are the same as those in Fig. 1), the β 2 = 4.4 is the most optimal for directly synthesizing the isolated attosecond pulse according our numerical calculations.

7 No. 4 Communications in Theoretical Physics 563 I 1 =4.0T10 14 W/cm 2 I 1 =4.5T10 14 W/cm 2 I 1 =5.0T10 14 W/cm 2 T10 4 T10 2 I 2 =0.9T10 14 W/cm 2 I 2 =1.5T10 14 W/cm 2 I 2 =2.0T10 14 W/cm 2 T10 4 T10 2 Harmonic order (ω/ω 1 ) Harmonic order (ω/ω 1 ) T10-6 T10-7 Fig. 5 (a) Harmonic spectra with three different intensities of the fundamental chirped pulse: W/cm 2, W/cm 2, and W/cm 2, respectively. (b) Harmonic spectra with three different intensities of the controlling chirped pulse: W/cm 2, W/cm 2, and W/cm 2, respectively. For the purpose of clarity, the harmonic intensities for W/cm 2 ( W/cm 2 ) and W/cm 2 ( W/cm 2 ) are multiplied by factors of 10 2 and 10 4, respectively. (c) Temporal profiles of the attosecond pulses by superposing the harmonics from 520th to 630th (dashed curve) and from 520 to 640 (solid curve) for I 1 = W/cm 2. (d) Temporal profiles of the attosecond pulses by superposing the harmonics from 800th to 880th (dashed curve) and from 800th to 890th (solid curve) for I 2 = W/cm 2. Other parameters are the same as those in Fig. 1. Finally, we also investigate the influence of the intensities of the two chirped pulses on the harmonic cutoff and the supercontinumm. From our calculation, we find that increasing the intensity of either the fundamental chirped pulse or the controlling chirped pulse will result in the extension of the harmonic cutoff and the widening of the supcontinuum. Figure 5(a) is the harmonic spectra with three different intensities of the fundamental chirped pulse. As shown in this figure, with the increase of the intensity of the fundamental chirped pulse, the first cutoff of the spectrum has almost no change, whereas the second one of the spectrum is markedly enlarged, thus the frequency difference between the first and the second cutoffs, which also leads to the extension of the supercontinuum spectrum. This is because the supcontinuum origins from the harmonics in the second plateau in our two-color chirped scheme. The same is true if the intensity of the controlling chirped pulse is increased, as shown in Fig. 5(b). Just the harmonic cutoff and the bandwidth of the supercontinuum are greatly extended in comparison with that in Fig. 5(a). Furthermore, the modulation of the low-order harmonics for the supercontinuum is much removed with increasing the intensity of the controlling chirped pulse, and the continuous harmonics show a much smoother structure. This unique property above can be used for producing a broadband supercontinuum, which is propitious to the generation of an ultrashort isolated attosecond pulse. However, since high harmonics are not synchronized on attosecond time scale, selecting the entire available spectral range no longer provides the shortest possible pulses due to the time-phase dispersion, as has been reported in Ref. [37]. From our calculation, we find that there exists an optimal bandwidth for the generation of the shortest attosecond pulses in the given conditions of laser parameters, beyond which the pulse lengthens as dispersion dominates. [37] In order to clarify better, we present the temporal profiles of the generated attosecond pulses for the cases of I 1 = W/cm 2 and I 2 = W/cm 2, respectively. As is shown in Figs. 5(c) and 5(d), by superposing more harmonics, the attosecond pulse duration lengthens but rather shortens, which best support our point of view.

8 564 Communications in Theoretical Physics Vol Conclusions In conclusion, we theoretically investigate high-order harmonic and isolated attosecond pulse generation in the synthesized field of a fundamental chirped pulse and a controlling chirped pulse. It is shown that the method not only can significantly extend the harmonic cutoff but also can evidently broaden the bandwidth of the supercontinuum spectrum. In addition, by the introduction of the chirp, the electron trajectories of the HHG can be significantly modulated, as a result, the long quantum path is suppressed and only the single short path is selected to contribute to the harmonic generation. Then by superposing some properly selected harmonic orders, an ultrashort isolated 28 as pulse with a bandwidth of 155 ev is obtained directly. Such an ultrashort attosecond pulse will allow one to detect and control the electronic dynamics inside atoms and molecules. It should be noted that our two-color chirped scheme presents several characteristics. First, the 800 nm fundamental chirped pulse adopted here is a multicycle one, which is currently much easier to achieve in experiment in comparison with the former studies (where the driving pulse is adopted to be a 5 fs pulse at 800 nm). [14 19] Second, by increasing the intensity of either the fundamental chirped pulse or the controlling chirped pulse, both the harmonic cutoff and the bandwidth of the supcontinuum can be remarkably extended, which ensures the supercontinuum with a broad bandwidth and supports the generation of the isolated ultrashort attosecond pulse. Third, our two-color chirped scheme with the same chirp form is favorable for the experimental realization of the ultrashort isolated pulse generation. References [1] M. Hentschel, et al., Nature (London) 414 (2001) 509. [2] R. Kienberger, et al., Nature (London) 427 (2004) 817. [3] G. Sansone, et al., Science 314 (2006) 443. [4] P.B. Corkum, Phys. Rev. Lett. 71 (1993) [5] P. Tzallas, et al., Nature (London) 426 (2003) 267. [6] I.P. Christov, M.M. Murnane, and H.C. Kapteyn, Phys. Rev. Lett. 78 (1997) [7] M. Ivanov, P.B. Corkum, T. Zuo, and A. Bandrauk, Phys. Rev. Lett. 74 (1995) [8] E. Goulielmakis, et al., Science 320 (2008) [9] G.L. Yudin, A.D. Bandrauk, and P.B. Corkum, Phys. Rev. Lett. 96 (2006) [10] Z.N. Zeng, Y. Cheng, X.H. Song, R.X. Li, and Z.Z. Xu, Phys. Rev. Lett. 98 (2007) [11] W.Y. Hong, P.X. Lu, P.F. Lan, Z.Y. Yang, Y.H. Li, and Q. Liao, Phys. Rev. A 77 (2008) [12] Z. Zhai, R.F. Yu, X.S. Liu, and Y.J. Yang, Phys. Rev. A 78 (2008) (R). [13] H. Zhang, K.L. Han, G.Z. He, and N.Q. Lou, Chem. Phys. Lett. 289 (1998) 494. [14] J.J. Carrera and S.I. Chu, Phys. Rev. A 75 (2007) [15] Y. Xiang, Y.P. Niu, and S.Q. Gong, Phys. Rev. A 79 (2009) [16] L.Q. Feng and T.S. Chu, Phys. Rev. A 84 (2011) [17] L.Q. Feng and T.S. Chu, J. Chem. Phys. 136 (2012) [18] P.C. Li, X.X. Zhou, G.L. Wang, and Z.X. Zhao, Phys. Rev. A 80 (2009) [19] J. Wu, G.T. Zhang, C.L. Xia, and X.S. Liu, Phys. Rev. A 82 (2010) [20] T. Pfeifer, et al., Opt. Lett. 31 (2006) 975. [21] T. Pfeifer, et al., Phys. Rev. Lett. 97 (2006) [22] W. Cao, P.X. Lu, P.F. Lan, X.L. Wang, and G. Yang, Opt. Express 15 (2007) 530. [23] Z.N. Zeng, Y.X. Leng, R.X. Li, and Z.Z. Xu, J. Phys. B 41 (2008) [24] S.S. Tang and X.F. Chen, Phys. Rev. A 82 (2010) [25] J.G. Chen, Y.J. Yang, S.L. Zeng, and H.Q. Liang, Phys. Rev. A 83 (2011) [26] Y.H. Zheng, Z.N. Zeng, P. Zou, L. Zhang, X.F. Li, P. Liu, R.X. Li, and Z.Z. Xu, Phys. Rev. Lett. 103 (2009) [27] P. Zou, Z.N. Zeng, Y.H. Zheng, Y.Y. Lu, P. Liu, R.X. Li, and Z.Z. Xu, Phys. Rev. A 81 (2010) [28] K.T. Kim, K.S. Kang, M.N. Park, T. Imran, G. Umesh, and C.H. Nam, Phys. Rev. Lett. 99 (2007) [29] M.C. Kohler, C.H. Keitel, and K.Z. Hatsagortsyan, Opt. Express 19 (2011) [30] Y.P. Niu, Y. Xiang, Y.H. Qi, and S.Q. Gong, Phys. Rev. A 80 (2009) [31] J.J. Xu, B. Zeng, and Y.L. Yu, Phys. Rev. A 82 (2010) [32] J.J. Xu, Phys. Rev. A 83 (2011) [33] M.D. Feit, J.A. Fleck, and A. Steiger, J. Comput. Phys. 47 (1982) 412. [34] T. Udem, R. Holzwarth, and T.W. Hansch, Nature (London) 416 (2002) 233. [35] S.T. Cundiff and J. Ye, Rev. Mod. Phys. 75 (2003) 325. [36] P. Antoine and B. Piraux, Phys. Rev. A 51 (1995) R1750. [37] Y. Mairesse, et al., Science 302 (2003) 1540.