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A single isolated sub-50 attosecond pulse generation with a two-color laser field by a frequency-chirping technique
Chemical Physics Letters 511 (2011) 166–171
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Chemical Physics Letters journal homepage: www.elsevier.com/locate/cplett
A single isolated sub-50 attosecond pulse generation with a two-color laser field by a frequency-chirping technique Kun Zhao b,1, Tianshu Chu a,b,⇑ a Institute for Computational Sciences and Engineering, Laboratory of New Fiber Materials and Modern Textile, the Growing Base for State Key Laboratory, Qingdao University, Qingdao 266071, China b State Key Laboratory of Molecular Reaction Dynamics, Dalian Institute of Chemical Physics, Chinese Academy of Sciences, Dalian 116023, China
a r t i c l e
i n f o
Article history: Received 13 March 2011 In final form 1 June 2011 Available online 6 June 2011
a b s t r a c t We discuss the possibility of using the frequency-chirping technique to shorten the duration of the generated single attosecond pulse (SAP) by a two-color laser field of 800 and 1600 nm with few-cycle pulses. By adopting various combinations of the two frequency-chirped laser fields in our numerical simulation of ionizing He atom, we demonstrate that the best possible condition to obtain the shortest SAP is using the same chirping in both the fundamental and the half-harmonic laser fields without any phase effect and any delay time. There is a maximum increment of about 40 eV in the bandwidth of the XUV super-continuum in the cutoff (the second plateau) region. A single isolated attosecond pulse of 48 as can be generated that is further reduced to 9.7 as by phase compensation. Ó 2011 Elsevier B.V. All rights reserved.
1. Introduction The generation of attosecond (as, 1018 s) extreme ultraviolet (XUV) pulses has attracted much attention because of their unprecedented time resolution to investigate and manipulate basic ultrafast electronic processes, such as control of molecular motion, trace of bound electron motion, inner-shell electron excitation, above-threshold ionization and complex photochemical processes [1–6]. Compared with the train of attosecond pulses, the single isolated attosecond pulse is much more preferred for detecting and controlling the electronic dynamics [7]. Currently, the production of isolated attosecond pulses by means of high-order harmonic generation (HHG) in gas, can be experimentally achieved either with the technique of high-order harmonic generation (HHG) from a few-cycle driving pulse [8,9] or with the technique of temporal confinement of the HHG by polarization gating [10]. Moreover, the proposal of attosecond generation using multi-cycle intense driving laser were also presented recently. Usually, the driving pulses used in most experiments to date to obtain the single attosecond pulse last from 5 to 50 femtosecond (fs, 1015 s). However, in order to further shorten the duration of the derived attosecond pulse, the driving pulse needs to shorten to sub-5 fs that still meets technical challenges in current experimental conditions. ⇑ Corresponding author at: Institute for Computational Sciences and Engineering, Laboratory of New Fiber Materials and Modern Textile, the Growing Base for State Key Laboratory, Qingdao University, Qingdao 266071, China. E-mail addresses: [email protected], [email protected] (T. Chu). 1 Present address: Department of Physics, Tsinghua University, Beijing 100084, China. 0009-2614/$ - see front matter Ó 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.cplett.2011.06.005
An alternative novel method for producing attosecond pulse has been turn to the use of two- or three-color laser field that allowed for using of longer driving laser and thus can avoid the stringent requirement for an available ultrafast laser source (sub-5 fs) in the single field. With the two- or three-color method, the attosecond pulse duration can also be decreased in analogy to the ideal sub-5 fs driving pulse [11–13]. The physics behind this phenomenon is that the HHG is a highly nonlinear optical process, which is sensitive to a slight change in the driving laser field [14]. According to the classical three-step model [15], the broadband XUV supercontinuum and the derived attosecond pulse are resulting from the significantly varying driving electric field from one half-cycle to the next, leading to the difference of kinetic energy of an electron returning to its parent ion for each half-cycle. Frequency-chirping, i.e., a time-dependent change in the laser frequency during the pulse provides a new and easy way to create and control a focused wavepacket of the laser field [16–18]. It is usually easier to induce a chirping into a femtosecond pulse than to obtain a specific pulse shape. Besides, the change of wavepacket of the driving laser pulse with chirping may also affect the highly nonlinear HHG. Thus, the frequency-chirping technique may have potential to broaden the attainable XUV super-continuum in the HHG spectrum, so as to shorten the duration of the derived attosecond pulse to a certain extent. In the experimental sides, using the chirped pulses to generate isolated attosecond pulse from multi-cycle two-color driving lasers has recently been proposed and realized by Altucci et al. [19,20], utilizing two orthogonally polarized components properly chirped and delayed. In their new experimental scheme with group delay dispersion, the generation of a single attosecond pulse is attributed
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to the interplay between polarization, ionization gating, and trajectory selection operated by suitable phase-matching condition [20], which represents both the microscopic and macroscopic [21] aspects of the attosecond pulse generation. By using their proposed method of mixing an infrared assistant pulse with a Ti:sapphire main pulse, Midorikawa et al. showed that an isolated attosecond pulse can be produced using a multi-cycle two-color pulse with a duration longer than 30 fs [22,23], they also discussed the effects of the carrier-envelope phase (CEP) and the relative intensity on the generation of isolated attosecond pulse. Further, a better wavelength scaling of harmonic yield in a two-color infrared field has been theoretically demonstrated [24]. It should be mentioned that the double optical gating (DOG) technique combining two powerful optical gating methods of polarization gating and two-color gating [25–27] provides useful tool for generating isolated attosecond pulses with multi-cycle driving pulse. Here, in this theoretical work, we investigated in the few-cycle regime the effect of frequency-chirping on the broadband XUV continuum and the generation of single isolated attosecond pulse, using a two-color laser field with few-cycle driving pulses and interacting with helium (He). To the best of our knowledge, this is the first theoretical investigation aiming on exploiting frequency-chirping technique to shorten the single isolated attosecond pulse. 2. Methods In our wavepacket simulation, a 6 fs/800 nm few-cycle laser pulse and its half-harmonic pulse, a 12 fs/1600 nm few-cycle laser pulse, are combined to serve as the driving pulse for generating the single isolated attosecond pulse. The intensities of the fundamental and the half-harmonic pulses are chosen to be 6.0 1014 and 2.0 1014 W cm2, respectively. The model atom of interaction in this simulation is helium (He). With the GAUSSIAN pulse envelopes, the synthesized electric field can be expressed as
167
3. Results and discussion First of all, the HHG power spectra with different combinations of frequency-chirping are simulated in order to find the best condition for the generation of isolated attosecond pulse, as shown in Figure 1. The frequency-chirping hardly changes the high-order cutoff points [34] of XUV super-continuum spectra (the second plateau region) because of the same maximal peak intensities of the driving pulses with different chirping. However, the frequency-chirping can obviously change the low-order cutoff points [34] of XUV super-continuum spectra (the second plateau region), owing to the very different intensities and positions of the second maximal peaks. Thus, with a frequency-chirping technique, we can change and control the bandwidth of the XUV super-continuum spectrum in the cutoff region, so as to control the duration of generated single isolated attosecond pulse. With a negative chirping of the halfharmonic laser pulse, in spite of the type of the fundamental laser pulse, the bandwidths of the XUV continuum spectra in the cutoff region are obviously reduced. Thus, the frequency-chirping of the half-harmonic laser pulse plays a more important role than that of the fundamental laser pulse in this model. Only with a positive chirping of both the fundamental laser pulse and the half-harmonic laser pulse simultaneously, the bandwidth of the XUV continuum spectrum can increase to the maximal extent. Then, with a positive chirping of both the fundamental and halfharmonic laser pulses simultaneously, the effect of relative phase u and the initial time delay t0 between this two laser pulses are investigated respectively, and the corresponding HHG power spectra are shown in Figure 2. With a relative phase, the bandwidth of the XUV continuum spectrum is obviously reduced. Taking the
EðtÞ ¼ E1 exp½4lnð2Þt 2 =s21 cosðx1 t þ c1 t2 Þ þ E2 exp½4lnð2Þ ðt t 0 Þ2 =s22 cos½x2 ðt t0 Þ þ c2 ðt t 0 Þ2 þ u
ð1Þ
Here, E1 and E2 are the amplitudes of the electric fields of the fundamental and the half-harmonic laser pulse, respectively; x1 and x2 are the frequencies; s1 and s2 are the corresponding pulse durations (full width at half maximum, FWHM); and c1 and c2 are the frequency-chirping parameters. If the frequency is increasing with time during the pulse, i.e., the sign of chirping parameter is positive, we call it as a positive-chirping; otherwise we call it a negative-chirping. u is the relative phase and t0 is the initial time delay between this two laser pulses. To investigate the HHG spectrum and the attosecond pulse generation, we numerically solved the time-dependent Schrödinger equation (TDSE) by means of the sine discrete variable representation (DVR) and split-operator method [28–30]. All the calculations were performed with the developed parallel quantum wavepacket computer code LZH-DICP [31]. For our simulation, the soft-core potential [32]
pffiffiffiffiffiffiffiffiffiffiffiffiffi VðrÞ ¼ 1= a þ r 2
ð2Þ
with the parameter a of 0.484 was chosen, so that the ionization energy Ip of 24.6 eV for the ground state of He is close to the experimental value. The initial wavepacket was constructed by integrating the TDSE in imaginary time space in the absence of the driving field with an initial guess. The harmonic spectrum is then obtained by taking the modulus squared of its Fourier transform of the time-dependent atomic dipole acceleration from the Ehrenfest theorem [33]. Besides, the generation of ultrafast isolated attosecond pulse is through superposing several harmonics by the inverse Fourier transformation.
Figure 1. The HHG power spectra with different combinations of frequencychirping (type: ct2-like). The relative phase u and the initial time delay t0 between this two laser pulses are set to zero, and the absolute values of chirping parameters of the fundamental laser pulse c1 and half-harmonic laser pulse c2 are 0.0372 fs2. The positive sign of chirping parameter stands for a positive chirping, and the negative sign for a negative chirping. If the laser pulse has no frequency-chirping, we use zero to denote. The harmonic intensities have been multiplied by factors of 1, 103, 106, and so on form bottom to top for the purpose of clarity.
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Figure 2. The HHG power spectra with (a) different relative phases, (b) different initial time delays, under the condition of the positive chirping of both the fundamental and half-harmonic laser pulses simultaneously. The harmonic intensities have been multiplied by factors of 1, 103, 106, and so on form bottom to top for the purpose of clarity.
initial time delay into account, the spectrum bandwidth also decreases with the absolute value of initial time delay increasing. Therefore, without relative phase and the initial time delay, the bandwidth of the XUV continuum spectrum can achieve its maximum, propitious to generate the ultrafast attosecond pulse. Under this maximal spectrum bandwidth condition, the values of frequency-chirping parameters are also changed in order to derive broader XUV super-continuum spectra, and the corresponding HHG power spectra are shown in Figure 3. With only increasing chirping parameter of half-harmonic laser pulse c2 and keeping the chirping parameter of the fundamental laser pulse c1 constant, the bandwidths of the XUV super-continuum spectra in the cutoff region increase first but then decrease obviously, giving the maximal point at c1 = c2 = 0.0372 fs2. With only increasing chirping parameter c1 and keep c2 constant, the bandwidths of the XUV super-continuum spectra only slightly change, but also show similar behaviors, giving the maximal value at the same chirping condition. Besides, we can see that the frequency-chirping of the half-harmonic laser pulse also plays a more important role, in agree with the phenomenon discussed previously. From the discussion above, we find that the frequency-chirping technique provides an effective new way to broaden the bandwidth of the XUV super-continuum spectrum. With the best frequency-chirping condition of c1 = c2 = 0.0372 fs2, the bandwidth of the XUV super-continuum spectrum in the cutoff region can increase by about 40 eV to the maximum. Then, under this best
Figure 3. The HHG power spectra with different values of frequency-chirping parameters: (a) only changing the chirping parameter of the half-harmonic laser pulse c2 and keep the chirping parameter of the fundamental laser pulse c1 constant of 0.0372 fs2, (b) only changing chirping parameters c1 and keep c2 constant of 0.0372 fs2. The fundamental laser pulse has a positive chirping, as well as the halfharmonic laser pulse simultaneously, and these two pulses have no relative phase and initial time delay. The harmonic intensities have been multiplied by factors of 1, 103, 106, and so on form bottom to top for the purpose of clarity.
chirping condition, we further reveal the physics behind the HHG phenomenon and investigate the generation of single isolated attosecond pulse in the following sections. The electric field of the driving laser pulse and the dependence of the kinetic energy on the ionization and recombination times are studied and shown in Figure 4. According to the classical three-step model [15], it is obvious that the electrons ionize from point A, then accelerate and gain kinetic energies by point B, and finally recombine to the parent ion core between points B and C, releasing its high kinetic energies by emitting high order harmonics. Only the third step is direct related with the durations of the
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Figure 4. (a) The electric fields of the driving laser pulse without the frequencychirping (shown in dashed black line) and with the frequency-chirping (shown in solid red line). (b) The dependence of the kinetic energy on the ionization and recombination times without the frequency-chirping (shown in dashed black and blue lines) and with the frequency-chirping (shown in solid cyan and red lines). (For interpretation of references to color in this figure legend, the reader is referred to the web version of this article.)
generated attosecond pulse. As can be seen in Figure 4, with a positive frequency-chirping, the frequency of the electric field after the main peak is increased, so as to increase the frequency of recombination process. Besides, with the positive frequency-chirping, the main peak of the recombination is slightly increased and the second peak slightly closes up to the main peak, which is in agree with the behavior of electric field above. Though the positive frequency-chirping can also decrease the frequency of the electric field before the main peak in this model, the slightly change of ionization has no effect on the durations of generated attosecond pulse. We also applied a negative frequency-chirping before the main peak and a positive frequency-chirping after the main peak simultaneously, and found its HHG power spectra have no obvious change (not shown here). It can also be seen that the harmonics higher than Ip + 90 eV can be filtered out to generate the attosecond pulse with the frequency-chirping. This continuum with a 345 eV bandwidth can support an isolated attosecond pulse of approximately 9 as in the Fourier transform limit. Figure 5a shows the temporal profile of the generated single isolated attosecond pulse with the frequency-chirping technique. By directly superposing the XUV spectral continuum with a bandwidth of 77.7 eV, from 155th to 205th order, a single isolated attosecond pulse of 48 as can be generated without phase compensation. However, if the phase compensation for the chirp of XUV spectral continuum is introduced and assume the phases of harmonics are the same, the duration of attosecond pulse can be clearly reduced. Then, by utilizing the phase compensation and superposing the
Figure 5. The temporal profiles of the generated single isolated attosecond pulses (a) with a frequency-chirping technique, (b) without a frequency-chirping technique. The profile without phase compensation is shown in red line and the profile with phase compensation is shown in black line. Here for a clear show, the scaling of the red and the black lines is artificially adjusted. (For interpretation of references to color in this figure legend, the reader is referred to the web version of this article.)
XUV spectral continuum with a full bandwidth of about 340 eV, from 80th to 300th order, a single isolated attosecond pulse of 9.7 as with a clean profile can be derived, which is very close to the Fourier transform limit. To further reinforce the impression on the chirp effects, in Figure 5b, we show the corresponding temporal profile of the generated attosecond pulse for the zero-chirp driving field case where the single isolated attosecond pulses of 80 and 14.2 as (with and without phase compensation) are both wider than those in the frequency-chirping case. Comparison with the previously reported two-color scheme generating an isolated attosecond pulse of 85 as without phase compensation [12] revealed also that there indeed is an improvement in the pulse width with the introducing of frequency-chirping techniques. Moreover, a recent work has reported an interesting phenomenon: chirp of fundamental field of the two-color scheme can control modulation of high-order harmonic fields [35]. In principle, two pulses should show up in the time domain due to that two quantum paths, long and short, can interfere and contribute to the HHG spectrum. As known, the long quantum path
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has earlier ionization time and later emission time while the short path has later ionization time and earlier emission time. Hence, the electron with long path travels a longer time in the continuous state, resulting in a lower harmonic efficiency due to the quantum diffusion [36]. This in turn implies an existing possibility that high order harmonic is solely contributed by the short quantum path leading to only one pulse showing up in the time-domain, which is just the present case of Figure 5a with the optimal condition. Meanwhile we also investigated another type of frequencychirping expressed by,
EðtÞ ¼ E1 exp½4lnð2Þt 2 =s21 cosðx1 t þ c1 t3 Þ þ E2 exp½4lnð2Þ ðt t 0 Þ2 =s22 cos½x2 ðt t0 Þ þ c2 ðt t 0 Þ3 þ u
ð3Þ
The HHG power spectra using this type of frequency-chirping are shown in Figure 6a, and the best condition of frequencychirping is found to be the negative chirping in both the fundamental laser pulse and the half-harmonic laser pulse simultaneously, with the chirping parameters of 0.0372 fs3. Then, with this best chirping, we analyze the dependence of the kinetic energy on the ionization and recombination times, the corresponding results are shown in Figure 6b. Similarly, with this frequency-chirping technique, we can also generate a single isolated sub-50 attosecond pulse, which is shown in Figure 6c. It is interesting to learn the combined chirp effects by putting together the above two types of c2t2-like and c3t3-like as follows:
EðtÞ ¼ E1 exp½4lnð2Þt 2 =s21 cosðx1 t þ c2 t2 Þ þ E2 exp½4lnð2Þt2 =s22 cosðx2 t þ c3 t 3 Þ
ð4Þ
Figure 7 shows the HHG power spectra for all possible combinations among the zero-chirp, positive chirp and negative chirp of the two electronic fields. In this case, we see that for both positive and negative chirps in the E2 field, using the fundamental chirp of c2t2 has almost no effect on the second plateau region. This is understandable because the c3t3 chirp modulates the electronic field more rapidly than the c2t2 chirp, thus, if we put the two chirps together, significant influence on the low-order cut off point of the supper continuum band obviously comes from the c3t3 chirp, and a similar situation can be deduced for the vice versa in Eq. (4) (i.e., exchange the two chips). Thus, the useful information we can provide for experimentalists includes: first, the chirp type should be paid attention to; second, the same positive chirp in the type of ct2-like and the same negative chirp in the type of c0 t3-like are beneficial for two-color attosecond generation; third, the chirp parameter value of 0.0372 is the most appropriate one; and forth, the relative phase and the carrier-envelope phase should be kept to zero for two-color attosecond generation. We note that all above points are drawn with the relative intensity of 3:1 which is the optimal condition for (6 fs/800 nm + 12 fs/1600 nm) two-color attosecond generation without chirp effects. Finally, we clarified one thing about our theoretical investigation, that is, we considered the microscopic control of single attosecond pulse generation, due to that what we have done here is by carrying out the TDSE analysis within a single-atom scope and thus mainly describing the microscopic atom-laser interaction. However, generation of single atosecond pulse bears much more than just a radiation type from single atom. Actually, it is a coherence sum of the radiation from all the atoms in the gas, which is commonly referred as ‘phase matching’ of the radiation [21]. Thus, for a complete understanding of the generation of an isolated attosecond pulse, both the microscopic and the macroscopic [21] aspects of this generation should be combined together. In connection to the latter, the Maxwell wave equation can be used to describe the macroscopic propagation and phase matching of
Figure 6. (a) The HHG power spectra with different combinations of frequencychirping (type: c0 t3-like). The harmonic intensities have been multiplied by factors of 1, 103, 106, and so on form bottom to top for the purpose of clarity. (b) The dependence of the kinetic energy on the ionization and recombination times. The ionization is shown in black square points and the recombination is shown in red round points. (c) The temporal profiles of the generated single isolated attosecond pulses. The profile without phase compensation is shown in red line and the profile with phase compensation is shown in black line. (For interpretation of references to color in this figure legend, the reader is referred to the web version of this article.)
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change the durations of generated isolated attosecond pulse. Under the best chirping condition, the bandwidth of the XUV super-continuum spectrum in the cutoff region can increase by about 40 eV to the maximum. A single isolated attosecond pulse of 48 as can be generated without phase compensation, and can be further reduced to 9.7 as by phase compensation. The frequency-chirping technique is found to be capable of shortening the duration of the isolated attosecond pulse from two-color few-cycle pulses with suitable condition. Acknowledgments This work was supported by the National Natural Science Foundation of China (Grant No. 10874096 and 20633070) and Qingdao University Research Fund (Grant No. 063-06300510). References
Figure 7. The HHG power spectra with different combinations of frequencychirping (type: combination of c2t2-like and c3t3-like). The harmonic intensities have been multiplied by factors of 1, 103, 106, and so on form bottom to top for the purpose of clarity.
the radiation. Just as shown in the recent studies [19–21], the macroscopic propagation of the fundamental and the harmonic laser fields through gas atoms plays a major role in increasing the final attosecond signal level by orders of magnitude. It is also known that the improved signal-to-noise ratio poses another critical factor for obtaining a usable single isolated attosecond pulse. Thus, besides the present issue on shortening XUV pulse duration, how to increase the magnitude of the attosecond signal level shown in this study presents an equally important issue, which requires subsequent works focusing on the macroscopic propagation of laser field through real gas target by solving the propagation equations starting from the Maxwell wave equation. 4. Conclusion In conclusion, we present a study of using frequency-chirping technique to shorten the duration of isolated attosecond pulse generated by a two-color laser field with few-cycle pulses, and theoretically investigate the effect of frequency-chirping on the broadband XUV super-continuum and the generation of single isolated attosecond pulse. We show that the frequency-chirping can obviously control the frequency of the recombination process, so as to
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Contents lists available at ScienceDirect
Chemical Physics Letters journal homepage: www.elsevier.com/locate/cplett
A single isolated sub-50 attosecond pulse generation with a two-color laser field by a frequency-chirping technique Kun Zhao b,1, Tianshu Chu a,b,⇑ a Institute for Computational Sciences and Engineering, Laboratory of New Fiber Materials and Modern Textile, the Growing Base for State Key Laboratory, Qingdao University, Qingdao 266071, China b State Key Laboratory of Molecular Reaction Dynamics, Dalian Institute of Chemical Physics, Chinese Academy of Sciences, Dalian 116023, China
a r t i c l e
i n f o
Article history: Received 13 March 2011 In final form 1 June 2011 Available online 6 June 2011
a b s t r a c t We discuss the possibility of using the frequency-chirping technique to shorten the duration of the generated single attosecond pulse (SAP) by a two-color laser field of 800 and 1600 nm with few-cycle pulses. By adopting various combinations of the two frequency-chirped laser fields in our numerical simulation of ionizing He atom, we demonstrate that the best possible condition to obtain the shortest SAP is using the same chirping in both the fundamental and the half-harmonic laser fields without any phase effect and any delay time. There is a maximum increment of about 40 eV in the bandwidth of the XUV super-continuum in the cutoff (the second plateau) region. A single isolated attosecond pulse of 48 as can be generated that is further reduced to 9.7 as by phase compensation. Ó 2011 Elsevier B.V. All rights reserved.
1. Introduction The generation of attosecond (as, 1018 s) extreme ultraviolet (XUV) pulses has attracted much attention because of their unprecedented time resolution to investigate and manipulate basic ultrafast electronic processes, such as control of molecular motion, trace of bound electron motion, inner-shell electron excitation, above-threshold ionization and complex photochemical processes [1–6]. Compared with the train of attosecond pulses, the single isolated attosecond pulse is much more preferred for detecting and controlling the electronic dynamics [7]. Currently, the production of isolated attosecond pulses by means of high-order harmonic generation (HHG) in gas, can be experimentally achieved either with the technique of high-order harmonic generation (HHG) from a few-cycle driving pulse [8,9] or with the technique of temporal confinement of the HHG by polarization gating [10]. Moreover, the proposal of attosecond generation using multi-cycle intense driving laser were also presented recently. Usually, the driving pulses used in most experiments to date to obtain the single attosecond pulse last from 5 to 50 femtosecond (fs, 1015 s). However, in order to further shorten the duration of the derived attosecond pulse, the driving pulse needs to shorten to sub-5 fs that still meets technical challenges in current experimental conditions. ⇑ Corresponding author at: Institute for Computational Sciences and Engineering, Laboratory of New Fiber Materials and Modern Textile, the Growing Base for State Key Laboratory, Qingdao University, Qingdao 266071, China. E-mail addresses: [email protected], [email protected] (T. Chu). 1 Present address: Department of Physics, Tsinghua University, Beijing 100084, China. 0009-2614/$ - see front matter Ó 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.cplett.2011.06.005
An alternative novel method for producing attosecond pulse has been turn to the use of two- or three-color laser field that allowed for using of longer driving laser and thus can avoid the stringent requirement for an available ultrafast laser source (sub-5 fs) in the single field. With the two- or three-color method, the attosecond pulse duration can also be decreased in analogy to the ideal sub-5 fs driving pulse [11–13]. The physics behind this phenomenon is that the HHG is a highly nonlinear optical process, which is sensitive to a slight change in the driving laser field [14]. According to the classical three-step model [15], the broadband XUV supercontinuum and the derived attosecond pulse are resulting from the significantly varying driving electric field from one half-cycle to the next, leading to the difference of kinetic energy of an electron returning to its parent ion for each half-cycle. Frequency-chirping, i.e., a time-dependent change in the laser frequency during the pulse provides a new and easy way to create and control a focused wavepacket of the laser field [16–18]. It is usually easier to induce a chirping into a femtosecond pulse than to obtain a specific pulse shape. Besides, the change of wavepacket of the driving laser pulse with chirping may also affect the highly nonlinear HHG. Thus, the frequency-chirping technique may have potential to broaden the attainable XUV super-continuum in the HHG spectrum, so as to shorten the duration of the derived attosecond pulse to a certain extent. In the experimental sides, using the chirped pulses to generate isolated attosecond pulse from multi-cycle two-color driving lasers has recently been proposed and realized by Altucci et al. [19,20], utilizing two orthogonally polarized components properly chirped and delayed. In their new experimental scheme with group delay dispersion, the generation of a single attosecond pulse is attributed
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to the interplay between polarization, ionization gating, and trajectory selection operated by suitable phase-matching condition [20], which represents both the microscopic and macroscopic [21] aspects of the attosecond pulse generation. By using their proposed method of mixing an infrared assistant pulse with a Ti:sapphire main pulse, Midorikawa et al. showed that an isolated attosecond pulse can be produced using a multi-cycle two-color pulse with a duration longer than 30 fs [22,23], they also discussed the effects of the carrier-envelope phase (CEP) and the relative intensity on the generation of isolated attosecond pulse. Further, a better wavelength scaling of harmonic yield in a two-color infrared field has been theoretically demonstrated [24]. It should be mentioned that the double optical gating (DOG) technique combining two powerful optical gating methods of polarization gating and two-color gating [25–27] provides useful tool for generating isolated attosecond pulses with multi-cycle driving pulse. Here, in this theoretical work, we investigated in the few-cycle regime the effect of frequency-chirping on the broadband XUV continuum and the generation of single isolated attosecond pulse, using a two-color laser field with few-cycle driving pulses and interacting with helium (He). To the best of our knowledge, this is the first theoretical investigation aiming on exploiting frequency-chirping technique to shorten the single isolated attosecond pulse. 2. Methods In our wavepacket simulation, a 6 fs/800 nm few-cycle laser pulse and its half-harmonic pulse, a 12 fs/1600 nm few-cycle laser pulse, are combined to serve as the driving pulse for generating the single isolated attosecond pulse. The intensities of the fundamental and the half-harmonic pulses are chosen to be 6.0 1014 and 2.0 1014 W cm2, respectively. The model atom of interaction in this simulation is helium (He). With the GAUSSIAN pulse envelopes, the synthesized electric field can be expressed as
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3. Results and discussion First of all, the HHG power spectra with different combinations of frequency-chirping are simulated in order to find the best condition for the generation of isolated attosecond pulse, as shown in Figure 1. The frequency-chirping hardly changes the high-order cutoff points [34] of XUV super-continuum spectra (the second plateau region) because of the same maximal peak intensities of the driving pulses with different chirping. However, the frequency-chirping can obviously change the low-order cutoff points [34] of XUV super-continuum spectra (the second plateau region), owing to the very different intensities and positions of the second maximal peaks. Thus, with a frequency-chirping technique, we can change and control the bandwidth of the XUV super-continuum spectrum in the cutoff region, so as to control the duration of generated single isolated attosecond pulse. With a negative chirping of the halfharmonic laser pulse, in spite of the type of the fundamental laser pulse, the bandwidths of the XUV continuum spectra in the cutoff region are obviously reduced. Thus, the frequency-chirping of the half-harmonic laser pulse plays a more important role than that of the fundamental laser pulse in this model. Only with a positive chirping of both the fundamental laser pulse and the half-harmonic laser pulse simultaneously, the bandwidth of the XUV continuum spectrum can increase to the maximal extent. Then, with a positive chirping of both the fundamental and halfharmonic laser pulses simultaneously, the effect of relative phase u and the initial time delay t0 between this two laser pulses are investigated respectively, and the corresponding HHG power spectra are shown in Figure 2. With a relative phase, the bandwidth of the XUV continuum spectrum is obviously reduced. Taking the
EðtÞ ¼ E1 exp½4lnð2Þt 2 =s21 cosðx1 t þ c1 t2 Þ þ E2 exp½4lnð2Þ ðt t 0 Þ2 =s22 cos½x2 ðt t0 Þ þ c2 ðt t 0 Þ2 þ u
ð1Þ
Here, E1 and E2 are the amplitudes of the electric fields of the fundamental and the half-harmonic laser pulse, respectively; x1 and x2 are the frequencies; s1 and s2 are the corresponding pulse durations (full width at half maximum, FWHM); and c1 and c2 are the frequency-chirping parameters. If the frequency is increasing with time during the pulse, i.e., the sign of chirping parameter is positive, we call it as a positive-chirping; otherwise we call it a negative-chirping. u is the relative phase and t0 is the initial time delay between this two laser pulses. To investigate the HHG spectrum and the attosecond pulse generation, we numerically solved the time-dependent Schrödinger equation (TDSE) by means of the sine discrete variable representation (DVR) and split-operator method [28–30]. All the calculations were performed with the developed parallel quantum wavepacket computer code LZH-DICP [31]. For our simulation, the soft-core potential [32]
pffiffiffiffiffiffiffiffiffiffiffiffiffi VðrÞ ¼ 1= a þ r 2
ð2Þ
with the parameter a of 0.484 was chosen, so that the ionization energy Ip of 24.6 eV for the ground state of He is close to the experimental value. The initial wavepacket was constructed by integrating the TDSE in imaginary time space in the absence of the driving field with an initial guess. The harmonic spectrum is then obtained by taking the modulus squared of its Fourier transform of the time-dependent atomic dipole acceleration from the Ehrenfest theorem [33]. Besides, the generation of ultrafast isolated attosecond pulse is through superposing several harmonics by the inverse Fourier transformation.
Figure 1. The HHG power spectra with different combinations of frequencychirping (type: ct2-like). The relative phase u and the initial time delay t0 between this two laser pulses are set to zero, and the absolute values of chirping parameters of the fundamental laser pulse c1 and half-harmonic laser pulse c2 are 0.0372 fs2. The positive sign of chirping parameter stands for a positive chirping, and the negative sign for a negative chirping. If the laser pulse has no frequency-chirping, we use zero to denote. The harmonic intensities have been multiplied by factors of 1, 103, 106, and so on form bottom to top for the purpose of clarity.
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Figure 2. The HHG power spectra with (a) different relative phases, (b) different initial time delays, under the condition of the positive chirping of both the fundamental and half-harmonic laser pulses simultaneously. The harmonic intensities have been multiplied by factors of 1, 103, 106, and so on form bottom to top for the purpose of clarity.
initial time delay into account, the spectrum bandwidth also decreases with the absolute value of initial time delay increasing. Therefore, without relative phase and the initial time delay, the bandwidth of the XUV continuum spectrum can achieve its maximum, propitious to generate the ultrafast attosecond pulse. Under this maximal spectrum bandwidth condition, the values of frequency-chirping parameters are also changed in order to derive broader XUV super-continuum spectra, and the corresponding HHG power spectra are shown in Figure 3. With only increasing chirping parameter of half-harmonic laser pulse c2 and keeping the chirping parameter of the fundamental laser pulse c1 constant, the bandwidths of the XUV super-continuum spectra in the cutoff region increase first but then decrease obviously, giving the maximal point at c1 = c2 = 0.0372 fs2. With only increasing chirping parameter c1 and keep c2 constant, the bandwidths of the XUV super-continuum spectra only slightly change, but also show similar behaviors, giving the maximal value at the same chirping condition. Besides, we can see that the frequency-chirping of the half-harmonic laser pulse also plays a more important role, in agree with the phenomenon discussed previously. From the discussion above, we find that the frequency-chirping technique provides an effective new way to broaden the bandwidth of the XUV super-continuum spectrum. With the best frequency-chirping condition of c1 = c2 = 0.0372 fs2, the bandwidth of the XUV super-continuum spectrum in the cutoff region can increase by about 40 eV to the maximum. Then, under this best
Figure 3. The HHG power spectra with different values of frequency-chirping parameters: (a) only changing the chirping parameter of the half-harmonic laser pulse c2 and keep the chirping parameter of the fundamental laser pulse c1 constant of 0.0372 fs2, (b) only changing chirping parameters c1 and keep c2 constant of 0.0372 fs2. The fundamental laser pulse has a positive chirping, as well as the halfharmonic laser pulse simultaneously, and these two pulses have no relative phase and initial time delay. The harmonic intensities have been multiplied by factors of 1, 103, 106, and so on form bottom to top for the purpose of clarity.
chirping condition, we further reveal the physics behind the HHG phenomenon and investigate the generation of single isolated attosecond pulse in the following sections. The electric field of the driving laser pulse and the dependence of the kinetic energy on the ionization and recombination times are studied and shown in Figure 4. According to the classical three-step model [15], it is obvious that the electrons ionize from point A, then accelerate and gain kinetic energies by point B, and finally recombine to the parent ion core between points B and C, releasing its high kinetic energies by emitting high order harmonics. Only the third step is direct related with the durations of the
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Figure 4. (a) The electric fields of the driving laser pulse without the frequencychirping (shown in dashed black line) and with the frequency-chirping (shown in solid red line). (b) The dependence of the kinetic energy on the ionization and recombination times without the frequency-chirping (shown in dashed black and blue lines) and with the frequency-chirping (shown in solid cyan and red lines). (For interpretation of references to color in this figure legend, the reader is referred to the web version of this article.)
generated attosecond pulse. As can be seen in Figure 4, with a positive frequency-chirping, the frequency of the electric field after the main peak is increased, so as to increase the frequency of recombination process. Besides, with the positive frequency-chirping, the main peak of the recombination is slightly increased and the second peak slightly closes up to the main peak, which is in agree with the behavior of electric field above. Though the positive frequency-chirping can also decrease the frequency of the electric field before the main peak in this model, the slightly change of ionization has no effect on the durations of generated attosecond pulse. We also applied a negative frequency-chirping before the main peak and a positive frequency-chirping after the main peak simultaneously, and found its HHG power spectra have no obvious change (not shown here). It can also be seen that the harmonics higher than Ip + 90 eV can be filtered out to generate the attosecond pulse with the frequency-chirping. This continuum with a 345 eV bandwidth can support an isolated attosecond pulse of approximately 9 as in the Fourier transform limit. Figure 5a shows the temporal profile of the generated single isolated attosecond pulse with the frequency-chirping technique. By directly superposing the XUV spectral continuum with a bandwidth of 77.7 eV, from 155th to 205th order, a single isolated attosecond pulse of 48 as can be generated without phase compensation. However, if the phase compensation for the chirp of XUV spectral continuum is introduced and assume the phases of harmonics are the same, the duration of attosecond pulse can be clearly reduced. Then, by utilizing the phase compensation and superposing the
Figure 5. The temporal profiles of the generated single isolated attosecond pulses (a) with a frequency-chirping technique, (b) without a frequency-chirping technique. The profile without phase compensation is shown in red line and the profile with phase compensation is shown in black line. Here for a clear show, the scaling of the red and the black lines is artificially adjusted. (For interpretation of references to color in this figure legend, the reader is referred to the web version of this article.)
XUV spectral continuum with a full bandwidth of about 340 eV, from 80th to 300th order, a single isolated attosecond pulse of 9.7 as with a clean profile can be derived, which is very close to the Fourier transform limit. To further reinforce the impression on the chirp effects, in Figure 5b, we show the corresponding temporal profile of the generated attosecond pulse for the zero-chirp driving field case where the single isolated attosecond pulses of 80 and 14.2 as (with and without phase compensation) are both wider than those in the frequency-chirping case. Comparison with the previously reported two-color scheme generating an isolated attosecond pulse of 85 as without phase compensation [12] revealed also that there indeed is an improvement in the pulse width with the introducing of frequency-chirping techniques. Moreover, a recent work has reported an interesting phenomenon: chirp of fundamental field of the two-color scheme can control modulation of high-order harmonic fields [35]. In principle, two pulses should show up in the time domain due to that two quantum paths, long and short, can interfere and contribute to the HHG spectrum. As known, the long quantum path
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has earlier ionization time and later emission time while the short path has later ionization time and earlier emission time. Hence, the electron with long path travels a longer time in the continuous state, resulting in a lower harmonic efficiency due to the quantum diffusion [36]. This in turn implies an existing possibility that high order harmonic is solely contributed by the short quantum path leading to only one pulse showing up in the time-domain, which is just the present case of Figure 5a with the optimal condition. Meanwhile we also investigated another type of frequencychirping expressed by,
EðtÞ ¼ E1 exp½4lnð2Þt 2 =s21 cosðx1 t þ c1 t3 Þ þ E2 exp½4lnð2Þ ðt t 0 Þ2 =s22 cos½x2 ðt t0 Þ þ c2 ðt t 0 Þ3 þ u
ð3Þ
The HHG power spectra using this type of frequency-chirping are shown in Figure 6a, and the best condition of frequencychirping is found to be the negative chirping in both the fundamental laser pulse and the half-harmonic laser pulse simultaneously, with the chirping parameters of 0.0372 fs3. Then, with this best chirping, we analyze the dependence of the kinetic energy on the ionization and recombination times, the corresponding results are shown in Figure 6b. Similarly, with this frequency-chirping technique, we can also generate a single isolated sub-50 attosecond pulse, which is shown in Figure 6c. It is interesting to learn the combined chirp effects by putting together the above two types of c2t2-like and c3t3-like as follows:
EðtÞ ¼ E1 exp½4lnð2Þt 2 =s21 cosðx1 t þ c2 t2 Þ þ E2 exp½4lnð2Þt2 =s22 cosðx2 t þ c3 t 3 Þ
ð4Þ
Figure 7 shows the HHG power spectra for all possible combinations among the zero-chirp, positive chirp and negative chirp of the two electronic fields. In this case, we see that for both positive and negative chirps in the E2 field, using the fundamental chirp of c2t2 has almost no effect on the second plateau region. This is understandable because the c3t3 chirp modulates the electronic field more rapidly than the c2t2 chirp, thus, if we put the two chirps together, significant influence on the low-order cut off point of the supper continuum band obviously comes from the c3t3 chirp, and a similar situation can be deduced for the vice versa in Eq. (4) (i.e., exchange the two chips). Thus, the useful information we can provide for experimentalists includes: first, the chirp type should be paid attention to; second, the same positive chirp in the type of ct2-like and the same negative chirp in the type of c0 t3-like are beneficial for two-color attosecond generation; third, the chirp parameter value of 0.0372 is the most appropriate one; and forth, the relative phase and the carrier-envelope phase should be kept to zero for two-color attosecond generation. We note that all above points are drawn with the relative intensity of 3:1 which is the optimal condition for (6 fs/800 nm + 12 fs/1600 nm) two-color attosecond generation without chirp effects. Finally, we clarified one thing about our theoretical investigation, that is, we considered the microscopic control of single attosecond pulse generation, due to that what we have done here is by carrying out the TDSE analysis within a single-atom scope and thus mainly describing the microscopic atom-laser interaction. However, generation of single atosecond pulse bears much more than just a radiation type from single atom. Actually, it is a coherence sum of the radiation from all the atoms in the gas, which is commonly referred as ‘phase matching’ of the radiation [21]. Thus, for a complete understanding of the generation of an isolated attosecond pulse, both the microscopic and the macroscopic [21] aspects of this generation should be combined together. In connection to the latter, the Maxwell wave equation can be used to describe the macroscopic propagation and phase matching of
Figure 6. (a) The HHG power spectra with different combinations of frequencychirping (type: c0 t3-like). The harmonic intensities have been multiplied by factors of 1, 103, 106, and so on form bottom to top for the purpose of clarity. (b) The dependence of the kinetic energy on the ionization and recombination times. The ionization is shown in black square points and the recombination is shown in red round points. (c) The temporal profiles of the generated single isolated attosecond pulses. The profile without phase compensation is shown in red line and the profile with phase compensation is shown in black line. (For interpretation of references to color in this figure legend, the reader is referred to the web version of this article.)
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change the durations of generated isolated attosecond pulse. Under the best chirping condition, the bandwidth of the XUV super-continuum spectrum in the cutoff region can increase by about 40 eV to the maximum. A single isolated attosecond pulse of 48 as can be generated without phase compensation, and can be further reduced to 9.7 as by phase compensation. The frequency-chirping technique is found to be capable of shortening the duration of the isolated attosecond pulse from two-color few-cycle pulses with suitable condition. Acknowledgments This work was supported by the National Natural Science Foundation of China (Grant No. 10874096 and 20633070) and Qingdao University Research Fund (Grant No. 063-06300510). References
Figure 7. The HHG power spectra with different combinations of frequencychirping (type: combination of c2t2-like and c3t3-like). The harmonic intensities have been multiplied by factors of 1, 103, 106, and so on form bottom to top for the purpose of clarity.
the radiation. Just as shown in the recent studies [19–21], the macroscopic propagation of the fundamental and the harmonic laser fields through gas atoms plays a major role in increasing the final attosecond signal level by orders of magnitude. It is also known that the improved signal-to-noise ratio poses another critical factor for obtaining a usable single isolated attosecond pulse. Thus, besides the present issue on shortening XUV pulse duration, how to increase the magnitude of the attosecond signal level shown in this study presents an equally important issue, which requires subsequent works focusing on the macroscopic propagation of laser field through real gas target by solving the propagation equations starting from the Maxwell wave equation. 4. Conclusion In conclusion, we present a study of using frequency-chirping technique to shorten the duration of isolated attosecond pulse generated by a two-color laser field with few-cycle pulses, and theoretically investigate the effect of frequency-chirping on the broadband XUV super-continuum and the generation of single isolated attosecond pulse. We show that the frequency-chirping can obviously control the frequency of the recombination process, so as to
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