Spectrochimica Acta Part A: Molecular and Biomolecular Spectroscopy 130 (2014) 19–23
Contents lists available at ScienceDirect
Spectrochimica Acta Part A: Molecular and Biomolecular Spectroscopy journal homepage: www.elsevier.com/locate/saa
Theoretical exploration of control factors for the high-order harmonic generation (HHG) spectrum in two-color field Xinting Huang, Dapeng Yang, Li Yao ⇑ Institute of Computational Physics, Department of Physics, Dalian Maritime University, Dalian 116026, PR China Physics Laboratory, North China University of Water Resources and Electric Power, Zhengzhou 450011, PR China
h i g h l i g h t s
g r a p h i c a l a b s t r a c t
The laser-parameter effects on HHG
spectrum were studied. Different schemes were applied to
discuss the function of intensity and CEP. An isolated 16as pulse was obtained by an ultrabroad XUV continuum of 208 eV.
a r t i c l e
i n f o
Article history: Received 24 May 2013 Received in revised form 20 March 2014 Accepted 23 March 2014 Available online 2 April 2014 Keywords: Time-dependent wave packet Attosecond High-order harmonic generation Carrier-envelope phase
a b s t r a c t In this work, the laser-parameter effects on the high-order harmonic generation (HHG) spectrum and attosecond trains by mixing two-color laser field, a visible light field of 800 nm and a mid-infrared (mid-IR) laser pulses of 2400 nm, are theoretically demonstrated for the first time. Different schemes are applied to discuss the function of intensity, carrier-envelope phase (CEP) and pulse duration on the generation of an isolated attosecond pulse. As a consequence, an isolated 16as pulse is obtained by Fourier transforming an ultrabroad XUV continuum of 208 eV with the fundamental field of duration of 6 fs, the duration of 12 fs, the CEPs of the two driving pulses of p and the rel9 1014 W/cm2 of intensity, pffiffiffi ative strength ratio R ¼ 0:2. Ó 2014 Elsevier B.V. All rights reserved.
Introduction The creation of the attosecond pulses opens up a novel way of studying unprecedented resolution and it has been extensively studied over last decades since the first time attosecond pulse reported [1]. The attosecond science becomes an important discipline for the study on the temporal scale of electronic states in ⇑ Corresponding author at: Institute of Computational Physics, Department of Physics, Dalian Maritime University, Dalian 116026, PR China. Tel.: +86 41184705321. E-mail address: [email protected] (L. Yao). http://dx.doi.org/10.1016/j.saa.2014.03.098 1386-1425/Ó 2014 Elsevier B.V. All rights reserved.
atoms and molecules [2–6]. Great efforts have been taken to generate shorter isolated attosecond pulses (IAPs). The shorter IAP obtained, the better for application. The technology of the generation of IAPs had been developed in recent years such as Fourier synthesis of Raman sidebands [7] or by the process of high-order harmonic generation (HHG) [8]. The three-step model (TSM) of the HHG is well-known [9]. Firstly, electrons tunnel transit through the barrier which is formed by the coulomb potential and the ultra strong electric field. Secondly, these free electrons will be accelerated by the electric field and obtained a large amount of kinetic energy. Thirdly, the electrons will be driven back by the electric field through the half-cycle
20
X. Huang et al. / Spectrochimica Acta Part A: Molecular and Biomolecular Spectroscopy 130 (2014) 19–23
Fig. 1. Here is the electric field by synthesizing the fundamental pulse with the intensity 5 1014 W/cm2, the optical cycles ofp800 ffiffiffi nm and the duration of 6 fs and the control pulse 2400 nm, duration of 12 fs, R ¼ 0, phase equal to 0.
changing, which recombine with the ions. And the process of the recombination will emit harmonic photons at frequency hx ¼ Ip þ 3:17U p , where Ip is the ionization potential of the target atom and U p / Ik2 is the electron ponderomotive energy in the laser field, respectively [10]. This process makes photons energy, which is equal to the sum of ionization potential and electrons’ kinetic energy. Many efforts have been made for obtaining shorter attosecond pulse trains, which can probe and measure the electron dynamics deep inside atoms and molecules during the later years. Two techniques in the experiment were used currently for HHG [11–17]: a few-cycle driving pulse and the temporal confinement of HHG using polarization gating. The polarization-gating technique is convenient to be realized in the laboratory. However it
is difficult to generate attosecond pulses with durations less than 100as because of the limited duration of the driving pulse. Hence, a two-color laser field is used for the generation of the IAP. According to this scheme, a driving fundamental field (FP) is shaped by mixing the second pulse, control field (CP). Therefore, the generation of the IAP receives contributions from the HHG process. Recently, a parallel quantum electron and nuclei wave-packet dynamics program had been developed by Han and co-workers for investigating the laser-atom-molecule interaction in the nonperturbative regime with attosecond resolution by numerically solving the time-dependent Schrödinger equation [22–30]. Zhao et al. used a few-cycle field mixing a static electric field and make the spectrum increase to 191 eV [10]. In addition, Shao et al. used the two-color scheme and obtained an ultrabroad XUV continuum of 300 eV with a 12as attosecond pulse [18]. In this paper, we use a laser field at 800 nm combining with a mid-IR field of 2400 nm to get the IAP. In principle, the given high-energy of the mid-IR field [19–21] combining with the driving field can result in a larger extension of the cutoff of the XUV. The cycle of the mid-IR is longer than that of IR. The major advantage of mixing a mid-IR wavelength is to improve the efficiency of the HHG process, because mid-IR can make the electrons have more kinetic energy according to the second step of the TSM. Due to the huge energy, more XUV is emitted. Therefore, the cutoff could be larger and get better attosecond trains. Theoretical methods In the numerical simulations, we will solve the time-dependent Schrödinger equation to obtain both the HHG spectrum and the isolated attosecond pulse. All these work can be done via the one-dimensional parallel quantum wave-packet computer code LZH-DICP [22–30].
Fig. 2. (a and c) Classical returning-kinetic energy maps in the left column is displayed. (b and d) Quantum time–frequency analyses is shown for the HHG in p the ffiffiffi right column. Panel (a and b) are corresponding to the fundamental laser pulse, 800 nm, 6 fs, 5 1014 W/cm2 and control pulse, 2400 nm, 12 fs, the relative strength ratio R ¼ 0:2, 14 2 phase equal to 0. Panel (c and d) are corresponding to the fundamental laser pulse, 800 nm, 6 fs, 9 10 W/cm and control pulse, 2400 nm, 12 fs, the relative strength ratio pffiffiffi R ¼ 0:2, phase = 0.
X. Huang et al. / Spectrochimica Acta Part A: Molecular and Biomolecular Spectroscopy 130 (2014) 19–23
21
In the simulation, helium, as a typical noble gas medium, which can be approximated as a one-electron system, is chosen for providing the HHG spectrum pffiffiffiffiffiffiffiffiffiffiffiffiffi for the results. A soft-core potential of helium VðrÞ ¼ 1= a þ r2 is applied. The parameter a is set to be 0.484 and the ionization potential Ip is 24.6 eV, which can be matched by the experimental information. The fundamental pulse (FP) alone can be exhibited:
EðtÞ ¼ E0 exp 4 lnð2Þt 2 =s20 cosðw0 tÞ
ð5Þ
And the synthesized two-color electric field by the fundamental pulse and the control pulse (CP) can be expressed as:
EðtÞ ¼ E0 exp 2 lnð2Þt2 =s20 cosðx0 tÞ h i o pffiffiffi þ R exp 2 lnð2Þðt þ rtÞ2 =s1 cos½x1 ðt þ rtÞ
ð6Þ
pffiffiffi here, phase ¼ 2p TD1t ¼ x1 Dt, and the relative strength ratio R of the control pulse can be depicted:
pffiffiffi E1 R¼ E0
ð7Þ
In the equations above, x0, x1 are the frequencies of the FP and the CP, respectively; E0, E1 depict the different electric field amplitudes; s0, s1 are the corresponding pulse durations (Full Width Half Maximum – FWHM). Results and discussion Pulse intensity effect
Fig. 3. (a) The profiles of the laser field synthesizing by two-color fields is corresponding to the HHG spectra (b) with the only changing of intensity of the FP 14 2 from 5 1014 W/cm2 to 9 p10 ffiffiffi W/cm and the other parameters are 800 nm/6 fs of FP, 2400 nm/12 fs of CP, R ¼ 0:2, phase = 0.
Here is the Schrödinger equation:
i
h i @ b Wðr; tÞ ¼ H b0 þ V b ðr; tÞ Wðr; tÞ wðr; tÞ ¼ H @t
ð1Þ
b 0 is the field-free Hamiltonian, H b 0 ¼ T þ V ¼ 1 r2 þ V C ; where H 2 VC is the Coulomb potential; V(r, t) = r E(t), is defined as the interaction potential between the atom and the strong electric field. The coordinate r is one-dimensional and Schrödinger equation is solved for the function of quantum wave-packet. The advancement of the wave-packet can be well depicted by the standard second-order split-operator method:
wðt þ dtÞ ¼ expðiTdt=2Þ expðiVdtÞ expðiTdt=2ÞwðtÞ þ oðdt3 Þ ð2Þ Here, T depicts the kinetic-energy operator and V depicts the interaction potential. The one-dimension time-dependent induced dipole acceleration will be calculated by using Ehrenfest theorem:
Vðr; tÞ Wðr; tÞ r VC þ EðtÞWðr; tÞ ¼ Wðr; tÞ r
dA ðtÞ ¼
Wðr; tÞ
dx ðtÞ ¼ ð3Þ
And the obtained harmonic spectra will be shown by Fourier transforming the time-dependent dipole acceleration:
2 Z 1 dðtÞeixt dt PA ðxÞ ¼ pffiffiffiffiffiffiffi 2p
The generation of the IAPs is studied under the two-color scheme and the intensity of the pulse is discussed firstly and the HHG process is analyzed according to the TSM. Fig. 1a is the plot of the electric field which shows five peaks labeled as A, B, C, D and E, respectively. Firstly, from the TSM, the atoms are ionized and the electrons are free around the peak A. Then the electrons are accelerated at point B with the changing of the electric fields’ direction in the second place. During this process, the electrons will gain a large amount of energy. Finally, the electrons with great energy will recombine with the ions between the second peak B and the third peak C and the harmonic spectrum will be obtained at the same time. The whole process happens during a period of half optic cycle of the pulse. Similar to the process from A to B and C, the other two independent periodical processes are from B to C to D and C to D to E. There will be three cutoffs by the process A–B–C–D–E. According to the above analyses, the ionizing process determines the harmonic efficiency, while the accelerating process determines the cutoff frequency on the HHG spectra. The continuum, which is the key to get better IAPs, is defined as a spread of harmonics with the frequency between the largest cutoff of energy and the second largest cutoff energy. And the larger continuum we take, the better IAPs we will get. Therefore, we focus on making the largest cutoff higher to obtain larger continuum. The physical meaning of the time–frequency analyses are performed to transform the dipole response of the He atom [31–36]. Herein, the induced dipole acceleration is obtained by the Fourier transform. The expression is taken as:
ð4Þ
Z
dðtÞxt0 ;x ðtÞdt
ð8Þ
The Morlet wavelet [37] is chosen as the kernel of the mother wavelet.
pffiffiffiffiffi
xt0 ;x ðtÞ ¼ xWðtÞ½xðt t0 Þ WðtÞ ¼ ð1=
pffiffiffi it t2 =2s2 sÞe e
ð9Þ
22
X. Huang et al. / Spectrochimica Acta Part A: Molecular and Biomolecular Spectroscopy 130 (2014) 19–23
Fig. 4. (a and b) Shows a variation with the CEPs changing from p to p in each 1/5p (first scheme). Panel (c) shows a variation with the CEPs changing from p to p in each 1/5p (second scheme). Panel (d) shows a variation with the CEPs changing from p to p and its effects on the continuum. Here the other parameters are 800 nm/6 fs, pffiffiffi 9 1014 W/cm2 of FP, 2400 nm/12 fs of CP, R ¼ 0:2, phase = 0.
where s = 30 in the calculations. The intensity of the photons is shown by the color1 scale in the Fig. 2b and d. We focus on the five peaks labeled from A1 to A5 which indicate the maximum of the returning electrons and recombination with the ions. By comparing Fig. 2b and d, the height of the largest peak, A3, is significantly increased and the increasing of the neighbor two peaks is almost invariable in the meantime. Consequently, the enlarged energy between the first largest peak and the second largest peak induces the extension of the continuum for the HHG emission. The laser parameters of the synthesized field we use are 800 nm/6 fs and 2400/12 fs. Here the CEPs and phase, are all set to be zero. Then the harmonic spectrum will be showed within the intensity of the FP varying from 5 1014 W/cm2 to 9 1014 W/cm2 and the strength ratio is kept with 0.2. As it is shown by Fig. 3a and b, we can see that the second-largest cutoff increases slightly by the change of the intensity, while this is not the case with the largest cutoff which increases linearly with the intensity of the pulse. The continuum obtained are 104 eV, 125 eV, 149 eV, 171 eV and 195 eV which are corresponding to the field EFP = 5 1014 W/cm2, EFP = 6 1014 W/cm2, EFP = 7 1014 W/cm2, EFP = 8 1014 W/cm2 and EFP = 9 1014 W/cm2, respectively. The largest continuum will be emerged by which the intensity of the FP is fixed to 9 1014 W/cm2. As it is shown in Fig. 2b and c, the classical returning kinetic-energy and quantum maps can be observed. The relative strength is kept invariable while 1 For interpretation of color in Fig. 2, the reader is referred to the web version of this article.
the intensity of the FP is changed in the previous work. At this time, pffiffiffi the relative strength ratio R related with intensities of the FP and CP is fixed to investigate the effects on the HHG process. Meanwhile, pffiffiffi the strength ratio R varies from 1/5 to 1 in each 1/5. Then we find that the bandwidth of the harmonic is deteriorated except the pffiffiffi situation that R equals to 1/5. Therefore, the spectra do not contribute to the generation of an isolated attosecond pffiffiffi pulse. Consequently, the parameters EFP = 9 1014 W/cm2 and R ¼ 0:2, are shifted for the later discussion. CEP effects The HHG spectra are plotted as a function of the CEP of the two electronic fields by three schemes. In the first scheme, we investigate the situation by changing one of the CEPs of the driving fields, while the other is kept zero pffiffiffiat each step. We fix the intensity of the FP to be 9 1014 W/cm2, R to be 0.2 and make the CEP of CP to be kept zero. Then the CEP of the FP varies from p to p in each 1/5p rad. But the HHG spectra obtained in the Fig. 4a are nearly changeless in the above process. This time, the CEP of the CP is the only parameter which is changed for the discussion. As it is shown in Fig. 4b, the largest cutoffs are invariable, but the second largest cutoffs are changed obviously. A 204 eV continuum is obtained when the CEP of the CP is made to p. And the value of the CEPs of the CP are made increased but the HHG spectra get bad which cannot contribute more for the generation of IAPs. Then, we will take second scheme to discuss the effect on the generation of the
X. Huang et al. / Spectrochimica Acta Part A: Molecular and Biomolecular Spectroscopy 130 (2014) 19–23
Fig. 5. The temporal profiles of the attosecond train generated by using a two-color scheme with the parameters of fundamental pulse 800 nm/6 p fsffiffiffiEFP = 9 1014 W/ cm2, control pulse 2400 nm/12 fs, the relative strength ratio R ¼ 0:2, phase = 0 and CEPCP = 0.2p.
IAPs of the CEP. It is that we make the CEP of the CP equal to that of FP and the HHG spectra have been shown in Fig. 4c, which are almost same as the Fig. 2b. It indicates that the CEP of the CP plays an important role in the process of HHG. On this occasion, we take the third scheme for a larger continuum in the following work. At this time, the CEP of the CP is equal to the minus of that of the FP. Such as the CEP of the control laser pulse is equal to p, while the one of the fundamental laser pulse is equal to p. The CEP of FP varies p/5 from p to p at each manipulation, then the HHG spectrum is obtained. We plot the spectra to show the effects of CEP on the processing of the HHG and it is depicted in Fig. 4d. In the figure, we can see that the largest cutoff gets the slightly extended when the CEPs are 1/5p and 1/5p, while the second largest cutoff decreases. The width of the continuum has not got a big amplification. According to the above discussion, the effect of the CEP is studied to make the IAP shorter for the application on probing electron motion in atoms and molecules (see Fig. 5). Pulse duration effect The duration of the fundamental pulse is kept as 6 fs firstly because it is extremely difficult for producing a pulse within a shorter duration. Then we make the duration of the control pulse equal to 6 fs 20 fs 24 fs 30 fs 32 fs, respectively. Nevertheless, we can see little change of the continuum by the different duration of the control pulse. In the meantime, the HHG spectra get worse with the increasing of the duration. Then the duration of the fundamental pulse is fixed to 12 fs and the same steps will be taken as the former work. However, the HHG spectrum is distorted in this work. Conclusions We theoretically demonstrate the laser-parameter effects on high-order harmonic generation by carrying out time-dependent wave packet calculation in a two-color pulse scheme by choosing the noble gas. In the work, the two-color scheme, mixing a laser field at 800 nm and a mid-IR field at 2400 nm, is used for exploring the effects of the intensity of the pulse on the process of HHG. There are two schemes that the former is to keep the relative strength ratio at 0.2 and change the intensity of the FP from 3 1014 W/cm2 to 9 1014 W/cm2, while the latter one is to keep
23
the intensity invariable and change the relative strength ratio with the CEP equal to zero. We find that the continuum increases linearly as the intensity of the FP increases. It shows that we can make the intensity up to extend the continuum for getting better HHG. And the intensity of the fundamental field plays an important role for get shorter attosecond. The CEP effect is explored in three ways, in which one of the FP and the CP is kept invariable while the other are changed, two CEPs are equivalent and one of the CEPs equals to the other. In this part, we can see the CEPs of the control pulse make the continuum longer. Different CEPs values of the two pulse can make the HHG spectrum different and the CEPs of the control pulse plays the major role. And we will focus on the study of the CEPs on the control of the pulse. It is also a significant parameter on the experiment. Then we discuss the duration of the two fields. Only a series of duration can make practicable HHG spectra and the HHG spectra in our work benefits from the shorter duration. Finally, we make a EFP = 9 1014 W/cm2 800 nm fundamental pulse mix a control pulse when relative strength ratio is 0.2 with the compensation that CEPs of the two driving pulses are equal to p to get the IAP. It has been demonstrated that a 208 eV continuum supporting an isolated 16as pulse can be obtained and it can be applied for probing ultrafast electron motion with an unprecedented time resolution. Acknowledgements This work was supported by the Natural Science Foundation of Liaoning Province (Grant Nos. L2010055, L2010057, 201102016). The Fundamental Research Funds for the Central Universities (Grant No. 3132013102). References [1] M. Hentschel, R. Kienberger, C. Spielmann, G.A. Reider, N. Milosevic, T. Brabec, P. Corkum, U. Heinzmann, M. Drescher, F. Krausz, Nature 414 (2001) 509. [2] P.B. Corkum, F. Krausz, Nat. Phys. 3 (2007) 381. [3] J. Hu, K.L. Han, G.Z. He, Phys. Rev. Lett. 95 (2005) 123001. [4] W. Xu, G. Zhao, Cent. Eur. J. Phys. 10 (2012) 253. [5] C.X. Yao, T.J. Shao, G.J. Zhao, J. Mod. Opt. 58 (2011) 954. [6] M. Drescher, M. Hentschel, R. Kienberger, M. Uiberacher, V. Yakovlev, A. Scrinizi, T. Westerwalbesloh, U. Kleineberg, U. Heinzmann, F. Krausz, Nature 803 (2002) 419. [7] T. Brabec, F. Krausz, Rev. Mod. Phys. 72 (2000) 545. [8] I.J. Sola, Nat. Phys. 2 (2006) 319. [9] P.B. Corkum, Phys. Rev. Lett. 71 (1993) 1994. [10] G.J. Zhao, X.L. Guo, T.J. Shao, K. Xue, New J. Phys. 13 (2011) 093035. [11] G. Sansone, Science 314 (2006) 443. [12] G. Sansone, Phys. Rev. A 79 (2009) 053410. [13] G. Sansone, Phys. Rev. A 80 (2009) 063837. [14] Y.L. Yu, Phys. Rev. A 80 (2009) 053423. [15] Z.H. Chang, Phys. Rev. A 76 (2007) 051403. [16] T.J. Shao, G.J. Zhao, C.X. Yao, B. Wen, Laser Phys. 23 (2013) 025301. [17] X.M. Feng, S. Gilbertson, H. Mashiko, H. Wang, S.D. Khan, M. Chini, Y. Wu, K. Zhao, Z.H. Chang, Phys. Rev. Lett. 103 (2009) 183901. [18] T.J. Shao, G.J. Zhao, B. Wen, H. Yang, Phys. Rev. A 82 (2010) 063838. [19] R. Torres, Phys. Rev. A81 (2010) 051802. [20] C.L. Cheng, G.J. Zhao, T.J. Shao, H.B. Zhu, Eur. Phys. J. D 64 (2011) 171. [21] M. Negro, C. Vozzi, K. Kovacs, C. Altucci, R. Velotta, F. Frassetto, L. Poletto, P. Villoresi, S. De Silvestri, V. Tosa, S. Stagira, Laser Phys. Lett. 8 (12) (2011) 875– 879. [22] Y.H. Guo, R.F. Lu, K.L. Han, G.Z. He, Int. J. Quantum Chem. 109 (2009) 3410. [23] L.Q. Feng, T.S. Chu, Phys. Lett. A 375 (2011) 3641. [24] L. Feng, T. Chu, Phys. Rev. A 84 (2011) 053853. [25] T.S. Chu, Y. Zhang, K.L. Han, Int. Rev. Phys. Chem. 25 (2006) 201. [26] K.L. Han, G.Z. He, N.Q. Lou, J. Chem. Phys. 105 (1996) 8699. [27] T.S. Chu, K.L. Han, Phys. Chem. Chem. Phys. 10 (2008) 2431. [28] K.L. Han, G.Z. He, J. Photochem. Photobiol. C: Photochem. Rev. 8 (2007) 55. [29] J. Hu, K.L. Han, G.Z. He, Phys. Rev. Lett. 95 (2005) 123001–123007. [30] J. Hu, M.S. Wang, K.L. Han, G.Z. He, Phys. Rev. A 74 (2006) 063417. [31] J.J. Carrera, X.M. Tong, S.I. Chu, Phys. Rev. A 74 (2006) 023404. [32] X.M. Tong, S.I. Chu, Phys. Rev. A 61 (2000) 021802. [33] X. Chu, S.I. Chu, Phys. Rev. A 64 (2001) 063404. [34] G.W.F. Drake, M.M. Cassar, R.A. Nistor, Phys. Rev. A 65 (2002) 054501. [35] T. Zhang, G.W.F. Drake, Phys. Rev. A 54 (1996) 4882. [36] R. El-Wazni, G.W.F. Drake, Phys. Rev. A 80 (2009) 064501. [37] P. Antoine, B. Piraux, A. Maquet, Phys. Rev. A 51 (1995) R1750.
Contents lists available at ScienceDirect
Spectrochimica Acta Part A: Molecular and Biomolecular Spectroscopy journal homepage: www.elsevier.com/locate/saa
Theoretical exploration of control factors for the high-order harmonic generation (HHG) spectrum in two-color field Xinting Huang, Dapeng Yang, Li Yao ⇑ Institute of Computational Physics, Department of Physics, Dalian Maritime University, Dalian 116026, PR China Physics Laboratory, North China University of Water Resources and Electric Power, Zhengzhou 450011, PR China
h i g h l i g h t s
g r a p h i c a l a b s t r a c t
The laser-parameter effects on HHG
spectrum were studied. Different schemes were applied to
discuss the function of intensity and CEP. An isolated 16as pulse was obtained by an ultrabroad XUV continuum of 208 eV.
a r t i c l e
i n f o
Article history: Received 24 May 2013 Received in revised form 20 March 2014 Accepted 23 March 2014 Available online 2 April 2014 Keywords: Time-dependent wave packet Attosecond High-order harmonic generation Carrier-envelope phase
a b s t r a c t In this work, the laser-parameter effects on the high-order harmonic generation (HHG) spectrum and attosecond trains by mixing two-color laser field, a visible light field of 800 nm and a mid-infrared (mid-IR) laser pulses of 2400 nm, are theoretically demonstrated for the first time. Different schemes are applied to discuss the function of intensity, carrier-envelope phase (CEP) and pulse duration on the generation of an isolated attosecond pulse. As a consequence, an isolated 16as pulse is obtained by Fourier transforming an ultrabroad XUV continuum of 208 eV with the fundamental field of duration of 6 fs, the duration of 12 fs, the CEPs of the two driving pulses of p and the rel9 1014 W/cm2 of intensity, pffiffiffi ative strength ratio R ¼ 0:2. Ó 2014 Elsevier B.V. All rights reserved.
Introduction The creation of the attosecond pulses opens up a novel way of studying unprecedented resolution and it has been extensively studied over last decades since the first time attosecond pulse reported [1]. The attosecond science becomes an important discipline for the study on the temporal scale of electronic states in ⇑ Corresponding author at: Institute of Computational Physics, Department of Physics, Dalian Maritime University, Dalian 116026, PR China. Tel.: +86 41184705321. E-mail address: [email protected] (L. Yao). http://dx.doi.org/10.1016/j.saa.2014.03.098 1386-1425/Ó 2014 Elsevier B.V. All rights reserved.
atoms and molecules [2–6]. Great efforts have been taken to generate shorter isolated attosecond pulses (IAPs). The shorter IAP obtained, the better for application. The technology of the generation of IAPs had been developed in recent years such as Fourier synthesis of Raman sidebands [7] or by the process of high-order harmonic generation (HHG) [8]. The three-step model (TSM) of the HHG is well-known [9]. Firstly, electrons tunnel transit through the barrier which is formed by the coulomb potential and the ultra strong electric field. Secondly, these free electrons will be accelerated by the electric field and obtained a large amount of kinetic energy. Thirdly, the electrons will be driven back by the electric field through the half-cycle
20
X. Huang et al. / Spectrochimica Acta Part A: Molecular and Biomolecular Spectroscopy 130 (2014) 19–23
Fig. 1. Here is the electric field by synthesizing the fundamental pulse with the intensity 5 1014 W/cm2, the optical cycles ofp800 ffiffiffi nm and the duration of 6 fs and the control pulse 2400 nm, duration of 12 fs, R ¼ 0, phase equal to 0.
changing, which recombine with the ions. And the process of the recombination will emit harmonic photons at frequency hx ¼ Ip þ 3:17U p , where Ip is the ionization potential of the target atom and U p / Ik2 is the electron ponderomotive energy in the laser field, respectively [10]. This process makes photons energy, which is equal to the sum of ionization potential and electrons’ kinetic energy. Many efforts have been made for obtaining shorter attosecond pulse trains, which can probe and measure the electron dynamics deep inside atoms and molecules during the later years. Two techniques in the experiment were used currently for HHG [11–17]: a few-cycle driving pulse and the temporal confinement of HHG using polarization gating. The polarization-gating technique is convenient to be realized in the laboratory. However it
is difficult to generate attosecond pulses with durations less than 100as because of the limited duration of the driving pulse. Hence, a two-color laser field is used for the generation of the IAP. According to this scheme, a driving fundamental field (FP) is shaped by mixing the second pulse, control field (CP). Therefore, the generation of the IAP receives contributions from the HHG process. Recently, a parallel quantum electron and nuclei wave-packet dynamics program had been developed by Han and co-workers for investigating the laser-atom-molecule interaction in the nonperturbative regime with attosecond resolution by numerically solving the time-dependent Schrödinger equation [22–30]. Zhao et al. used a few-cycle field mixing a static electric field and make the spectrum increase to 191 eV [10]. In addition, Shao et al. used the two-color scheme and obtained an ultrabroad XUV continuum of 300 eV with a 12as attosecond pulse [18]. In this paper, we use a laser field at 800 nm combining with a mid-IR field of 2400 nm to get the IAP. In principle, the given high-energy of the mid-IR field [19–21] combining with the driving field can result in a larger extension of the cutoff of the XUV. The cycle of the mid-IR is longer than that of IR. The major advantage of mixing a mid-IR wavelength is to improve the efficiency of the HHG process, because mid-IR can make the electrons have more kinetic energy according to the second step of the TSM. Due to the huge energy, more XUV is emitted. Therefore, the cutoff could be larger and get better attosecond trains. Theoretical methods In the numerical simulations, we will solve the time-dependent Schrödinger equation to obtain both the HHG spectrum and the isolated attosecond pulse. All these work can be done via the one-dimensional parallel quantum wave-packet computer code LZH-DICP [22–30].
Fig. 2. (a and c) Classical returning-kinetic energy maps in the left column is displayed. (b and d) Quantum time–frequency analyses is shown for the HHG in p the ffiffiffi right column. Panel (a and b) are corresponding to the fundamental laser pulse, 800 nm, 6 fs, 5 1014 W/cm2 and control pulse, 2400 nm, 12 fs, the relative strength ratio R ¼ 0:2, 14 2 phase equal to 0. Panel (c and d) are corresponding to the fundamental laser pulse, 800 nm, 6 fs, 9 10 W/cm and control pulse, 2400 nm, 12 fs, the relative strength ratio pffiffiffi R ¼ 0:2, phase = 0.
X. Huang et al. / Spectrochimica Acta Part A: Molecular and Biomolecular Spectroscopy 130 (2014) 19–23
21
In the simulation, helium, as a typical noble gas medium, which can be approximated as a one-electron system, is chosen for providing the HHG spectrum pffiffiffiffiffiffiffiffiffiffiffiffiffi for the results. A soft-core potential of helium VðrÞ ¼ 1= a þ r2 is applied. The parameter a is set to be 0.484 and the ionization potential Ip is 24.6 eV, which can be matched by the experimental information. The fundamental pulse (FP) alone can be exhibited:
EðtÞ ¼ E0 exp 4 lnð2Þt 2 =s20 cosðw0 tÞ
ð5Þ
And the synthesized two-color electric field by the fundamental pulse and the control pulse (CP) can be expressed as:
EðtÞ ¼ E0 exp 2 lnð2Þt2 =s20 cosðx0 tÞ h i o pffiffiffi þ R exp 2 lnð2Þðt þ rtÞ2 =s1 cos½x1 ðt þ rtÞ
ð6Þ
pffiffiffi here, phase ¼ 2p TD1t ¼ x1 Dt, and the relative strength ratio R of the control pulse can be depicted:
pffiffiffi E1 R¼ E0
ð7Þ
In the equations above, x0, x1 are the frequencies of the FP and the CP, respectively; E0, E1 depict the different electric field amplitudes; s0, s1 are the corresponding pulse durations (Full Width Half Maximum – FWHM). Results and discussion Pulse intensity effect
Fig. 3. (a) The profiles of the laser field synthesizing by two-color fields is corresponding to the HHG spectra (b) with the only changing of intensity of the FP 14 2 from 5 1014 W/cm2 to 9 p10 ffiffiffi W/cm and the other parameters are 800 nm/6 fs of FP, 2400 nm/12 fs of CP, R ¼ 0:2, phase = 0.
Here is the Schrödinger equation:
i
h i @ b Wðr; tÞ ¼ H b0 þ V b ðr; tÞ Wðr; tÞ wðr; tÞ ¼ H @t
ð1Þ
b 0 is the field-free Hamiltonian, H b 0 ¼ T þ V ¼ 1 r2 þ V C ; where H 2 VC is the Coulomb potential; V(r, t) = r E(t), is defined as the interaction potential between the atom and the strong electric field. The coordinate r is one-dimensional and Schrödinger equation is solved for the function of quantum wave-packet. The advancement of the wave-packet can be well depicted by the standard second-order split-operator method:
wðt þ dtÞ ¼ expðiTdt=2Þ expðiVdtÞ expðiTdt=2ÞwðtÞ þ oðdt3 Þ ð2Þ Here, T depicts the kinetic-energy operator and V depicts the interaction potential. The one-dimension time-dependent induced dipole acceleration will be calculated by using Ehrenfest theorem:
Vðr; tÞ Wðr; tÞ r VC þ EðtÞWðr; tÞ ¼ Wðr; tÞ r
dA ðtÞ ¼
Wðr; tÞ
dx ðtÞ ¼ ð3Þ
And the obtained harmonic spectra will be shown by Fourier transforming the time-dependent dipole acceleration:
2 Z 1 dðtÞeixt dt PA ðxÞ ¼ pffiffiffiffiffiffiffi 2p
The generation of the IAPs is studied under the two-color scheme and the intensity of the pulse is discussed firstly and the HHG process is analyzed according to the TSM. Fig. 1a is the plot of the electric field which shows five peaks labeled as A, B, C, D and E, respectively. Firstly, from the TSM, the atoms are ionized and the electrons are free around the peak A. Then the electrons are accelerated at point B with the changing of the electric fields’ direction in the second place. During this process, the electrons will gain a large amount of energy. Finally, the electrons with great energy will recombine with the ions between the second peak B and the third peak C and the harmonic spectrum will be obtained at the same time. The whole process happens during a period of half optic cycle of the pulse. Similar to the process from A to B and C, the other two independent periodical processes are from B to C to D and C to D to E. There will be three cutoffs by the process A–B–C–D–E. According to the above analyses, the ionizing process determines the harmonic efficiency, while the accelerating process determines the cutoff frequency on the HHG spectra. The continuum, which is the key to get better IAPs, is defined as a spread of harmonics with the frequency between the largest cutoff of energy and the second largest cutoff energy. And the larger continuum we take, the better IAPs we will get. Therefore, we focus on making the largest cutoff higher to obtain larger continuum. The physical meaning of the time–frequency analyses are performed to transform the dipole response of the He atom [31–36]. Herein, the induced dipole acceleration is obtained by the Fourier transform. The expression is taken as:
ð4Þ
Z
dðtÞxt0 ;x ðtÞdt
ð8Þ
The Morlet wavelet [37] is chosen as the kernel of the mother wavelet.
pffiffiffiffiffi
xt0 ;x ðtÞ ¼ xWðtÞ½xðt t0 Þ WðtÞ ¼ ð1=
pffiffiffi it t2 =2s2 sÞe e
ð9Þ
22
X. Huang et al. / Spectrochimica Acta Part A: Molecular and Biomolecular Spectroscopy 130 (2014) 19–23
Fig. 4. (a and b) Shows a variation with the CEPs changing from p to p in each 1/5p (first scheme). Panel (c) shows a variation with the CEPs changing from p to p in each 1/5p (second scheme). Panel (d) shows a variation with the CEPs changing from p to p and its effects on the continuum. Here the other parameters are 800 nm/6 fs, pffiffiffi 9 1014 W/cm2 of FP, 2400 nm/12 fs of CP, R ¼ 0:2, phase = 0.
where s = 30 in the calculations. The intensity of the photons is shown by the color1 scale in the Fig. 2b and d. We focus on the five peaks labeled from A1 to A5 which indicate the maximum of the returning electrons and recombination with the ions. By comparing Fig. 2b and d, the height of the largest peak, A3, is significantly increased and the increasing of the neighbor two peaks is almost invariable in the meantime. Consequently, the enlarged energy between the first largest peak and the second largest peak induces the extension of the continuum for the HHG emission. The laser parameters of the synthesized field we use are 800 nm/6 fs and 2400/12 fs. Here the CEPs and phase, are all set to be zero. Then the harmonic spectrum will be showed within the intensity of the FP varying from 5 1014 W/cm2 to 9 1014 W/cm2 and the strength ratio is kept with 0.2. As it is shown by Fig. 3a and b, we can see that the second-largest cutoff increases slightly by the change of the intensity, while this is not the case with the largest cutoff which increases linearly with the intensity of the pulse. The continuum obtained are 104 eV, 125 eV, 149 eV, 171 eV and 195 eV which are corresponding to the field EFP = 5 1014 W/cm2, EFP = 6 1014 W/cm2, EFP = 7 1014 W/cm2, EFP = 8 1014 W/cm2 and EFP = 9 1014 W/cm2, respectively. The largest continuum will be emerged by which the intensity of the FP is fixed to 9 1014 W/cm2. As it is shown in Fig. 2b and c, the classical returning kinetic-energy and quantum maps can be observed. The relative strength is kept invariable while 1 For interpretation of color in Fig. 2, the reader is referred to the web version of this article.
the intensity of the FP is changed in the previous work. At this time, pffiffiffi the relative strength ratio R related with intensities of the FP and CP is fixed to investigate the effects on the HHG process. Meanwhile, pffiffiffi the strength ratio R varies from 1/5 to 1 in each 1/5. Then we find that the bandwidth of the harmonic is deteriorated except the pffiffiffi situation that R equals to 1/5. Therefore, the spectra do not contribute to the generation of an isolated attosecond pffiffiffi pulse. Consequently, the parameters EFP = 9 1014 W/cm2 and R ¼ 0:2, are shifted for the later discussion. CEP effects The HHG spectra are plotted as a function of the CEP of the two electronic fields by three schemes. In the first scheme, we investigate the situation by changing one of the CEPs of the driving fields, while the other is kept zero pffiffiffiat each step. We fix the intensity of the FP to be 9 1014 W/cm2, R to be 0.2 and make the CEP of CP to be kept zero. Then the CEP of the FP varies from p to p in each 1/5p rad. But the HHG spectra obtained in the Fig. 4a are nearly changeless in the above process. This time, the CEP of the CP is the only parameter which is changed for the discussion. As it is shown in Fig. 4b, the largest cutoffs are invariable, but the second largest cutoffs are changed obviously. A 204 eV continuum is obtained when the CEP of the CP is made to p. And the value of the CEPs of the CP are made increased but the HHG spectra get bad which cannot contribute more for the generation of IAPs. Then, we will take second scheme to discuss the effect on the generation of the
X. Huang et al. / Spectrochimica Acta Part A: Molecular and Biomolecular Spectroscopy 130 (2014) 19–23
Fig. 5. The temporal profiles of the attosecond train generated by using a two-color scheme with the parameters of fundamental pulse 800 nm/6 p fsffiffiffiEFP = 9 1014 W/ cm2, control pulse 2400 nm/12 fs, the relative strength ratio R ¼ 0:2, phase = 0 and CEPCP = 0.2p.
IAPs of the CEP. It is that we make the CEP of the CP equal to that of FP and the HHG spectra have been shown in Fig. 4c, which are almost same as the Fig. 2b. It indicates that the CEP of the CP plays an important role in the process of HHG. On this occasion, we take the third scheme for a larger continuum in the following work. At this time, the CEP of the CP is equal to the minus of that of the FP. Such as the CEP of the control laser pulse is equal to p, while the one of the fundamental laser pulse is equal to p. The CEP of FP varies p/5 from p to p at each manipulation, then the HHG spectrum is obtained. We plot the spectra to show the effects of CEP on the processing of the HHG and it is depicted in Fig. 4d. In the figure, we can see that the largest cutoff gets the slightly extended when the CEPs are 1/5p and 1/5p, while the second largest cutoff decreases. The width of the continuum has not got a big amplification. According to the above discussion, the effect of the CEP is studied to make the IAP shorter for the application on probing electron motion in atoms and molecules (see Fig. 5). Pulse duration effect The duration of the fundamental pulse is kept as 6 fs firstly because it is extremely difficult for producing a pulse within a shorter duration. Then we make the duration of the control pulse equal to 6 fs 20 fs 24 fs 30 fs 32 fs, respectively. Nevertheless, we can see little change of the continuum by the different duration of the control pulse. In the meantime, the HHG spectra get worse with the increasing of the duration. Then the duration of the fundamental pulse is fixed to 12 fs and the same steps will be taken as the former work. However, the HHG spectrum is distorted in this work. Conclusions We theoretically demonstrate the laser-parameter effects on high-order harmonic generation by carrying out time-dependent wave packet calculation in a two-color pulse scheme by choosing the noble gas. In the work, the two-color scheme, mixing a laser field at 800 nm and a mid-IR field at 2400 nm, is used for exploring the effects of the intensity of the pulse on the process of HHG. There are two schemes that the former is to keep the relative strength ratio at 0.2 and change the intensity of the FP from 3 1014 W/cm2 to 9 1014 W/cm2, while the latter one is to keep
23
the intensity invariable and change the relative strength ratio with the CEP equal to zero. We find that the continuum increases linearly as the intensity of the FP increases. It shows that we can make the intensity up to extend the continuum for getting better HHG. And the intensity of the fundamental field plays an important role for get shorter attosecond. The CEP effect is explored in three ways, in which one of the FP and the CP is kept invariable while the other are changed, two CEPs are equivalent and one of the CEPs equals to the other. In this part, we can see the CEPs of the control pulse make the continuum longer. Different CEPs values of the two pulse can make the HHG spectrum different and the CEPs of the control pulse plays the major role. And we will focus on the study of the CEPs on the control of the pulse. It is also a significant parameter on the experiment. Then we discuss the duration of the two fields. Only a series of duration can make practicable HHG spectra and the HHG spectra in our work benefits from the shorter duration. Finally, we make a EFP = 9 1014 W/cm2 800 nm fundamental pulse mix a control pulse when relative strength ratio is 0.2 with the compensation that CEPs of the two driving pulses are equal to p to get the IAP. It has been demonstrated that a 208 eV continuum supporting an isolated 16as pulse can be obtained and it can be applied for probing ultrafast electron motion with an unprecedented time resolution. Acknowledgements This work was supported by the Natural Science Foundation of Liaoning Province (Grant Nos. L2010055, L2010057, 201102016). The Fundamental Research Funds for the Central Universities (Grant No. 3132013102). References [1] M. Hentschel, R. Kienberger, C. Spielmann, G.A. Reider, N. Milosevic, T. Brabec, P. Corkum, U. Heinzmann, M. Drescher, F. Krausz, Nature 414 (2001) 509. [2] P.B. Corkum, F. Krausz, Nat. Phys. 3 (2007) 381. [3] J. Hu, K.L. Han, G.Z. He, Phys. Rev. Lett. 95 (2005) 123001. [4] W. Xu, G. Zhao, Cent. Eur. J. Phys. 10 (2012) 253. [5] C.X. Yao, T.J. Shao, G.J. Zhao, J. Mod. Opt. 58 (2011) 954. [6] M. Drescher, M. Hentschel, R. Kienberger, M. Uiberacher, V. Yakovlev, A. Scrinizi, T. Westerwalbesloh, U. Kleineberg, U. Heinzmann, F. Krausz, Nature 803 (2002) 419. [7] T. Brabec, F. Krausz, Rev. Mod. Phys. 72 (2000) 545. [8] I.J. Sola, Nat. Phys. 2 (2006) 319. [9] P.B. Corkum, Phys. Rev. Lett. 71 (1993) 1994. [10] G.J. Zhao, X.L. Guo, T.J. Shao, K. Xue, New J. Phys. 13 (2011) 093035. [11] G. Sansone, Science 314 (2006) 443. [12] G. Sansone, Phys. Rev. A 79 (2009) 053410. [13] G. Sansone, Phys. Rev. A 80 (2009) 063837. [14] Y.L. Yu, Phys. Rev. A 80 (2009) 053423. [15] Z.H. Chang, Phys. Rev. A 76 (2007) 051403. [16] T.J. Shao, G.J. Zhao, C.X. Yao, B. Wen, Laser Phys. 23 (2013) 025301. [17] X.M. Feng, S. Gilbertson, H. Mashiko, H. Wang, S.D. Khan, M. Chini, Y. Wu, K. Zhao, Z.H. Chang, Phys. Rev. Lett. 103 (2009) 183901. [18] T.J. Shao, G.J. Zhao, B. Wen, H. Yang, Phys. Rev. A 82 (2010) 063838. [19] R. Torres, Phys. Rev. A81 (2010) 051802. [20] C.L. Cheng, G.J. Zhao, T.J. Shao, H.B. Zhu, Eur. Phys. J. D 64 (2011) 171. [21] M. Negro, C. Vozzi, K. Kovacs, C. Altucci, R. Velotta, F. Frassetto, L. Poletto, P. Villoresi, S. De Silvestri, V. Tosa, S. Stagira, Laser Phys. Lett. 8 (12) (2011) 875– 879. [22] Y.H. Guo, R.F. Lu, K.L. Han, G.Z. He, Int. J. Quantum Chem. 109 (2009) 3410. [23] L.Q. Feng, T.S. Chu, Phys. Lett. A 375 (2011) 3641. [24] L. Feng, T. Chu, Phys. Rev. A 84 (2011) 053853. [25] T.S. Chu, Y. Zhang, K.L. Han, Int. Rev. Phys. Chem. 25 (2006) 201. [26] K.L. Han, G.Z. He, N.Q. Lou, J. Chem. Phys. 105 (1996) 8699. [27] T.S. Chu, K.L. Han, Phys. Chem. Chem. Phys. 10 (2008) 2431. [28] K.L. Han, G.Z. He, J. Photochem. Photobiol. C: Photochem. Rev. 8 (2007) 55. [29] J. Hu, K.L. Han, G.Z. He, Phys. Rev. Lett. 95 (2005) 123001–123007. [30] J. Hu, M.S. Wang, K.L. Han, G.Z. He, Phys. Rev. A 74 (2006) 063417. [31] J.J. Carrera, X.M. Tong, S.I. Chu, Phys. Rev. A 74 (2006) 023404. [32] X.M. Tong, S.I. Chu, Phys. Rev. A 61 (2000) 021802. [33] X. Chu, S.I. Chu, Phys. Rev. A 64 (2001) 063404. [34] G.W.F. Drake, M.M. Cassar, R.A. Nistor, Phys. Rev. A 65 (2002) 054501. [35] T. Zhang, G.W.F. Drake, Phys. Rev. A 54 (1996) 4882. [36] R. El-Wazni, G.W.F. Drake, Phys. Rev. A 80 (2009) 064501. [37] P. Antoine, B. Piraux, A. Maquet, Phys. Rev. A 51 (1995) R1750.
