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R-dependent molecular harmonic generation from H2+
Physics Letters A 381 (2017) 859–864
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Physics Letters A www.elsevier.com/locate/pla
Discussion
R-dependent molecular harmonic generation from H+ 2 Liqiang Feng a,b , Hang Liu a,c,∗ a b c
Laboratory of Modern Physics, College of Science, Liaoning University of Technology, Jinzhou 121001, China State Key Laboratory of Molecular Reaction Dynamics, Dalian Institute of Chemical Physics, Chinese Academy of Sciences, Dalian 116023, China School of Chemical and Environmental Engineering, Liaoning University of Technology, Jinzhou 121001, China
a r t i c l e
i n f o
Article history: Received 28 November 2016 Received in revised form 4 January 2017 Accepted 8 January 2017 Available online 11 January 2017 Communicated by V.A. Markel Keywords: R-dependent molecular harmonic generation Non-Bohn–Oppenheimer time-dependent Schrödinger equation Charge-resonance-enhanced-ionization Dissociative ionization
a b s t r a c t R-dependent high-order harmonic spectra (R is the nuclear distance) from H+ 2 have been investigated through solving the Non-Bohn–Oppenheimer time-dependent Schrödinger equation. We found that (i) for the case of the few-cycle pulse, the harmonic emission mainly occurs from R = 3.7 to R = 6, caused by the charge-resonance-enhanced-ionization (CREI) process. (ii) For the case of the multi-cycle pulse, the harmonic emission can be separated into two parts, that is the charge-resonance-enhancedionization region from R = 3.7 to R = 8; and the dissociative ionization region when R > 10. (iii) Isotopic investigation showed that the R-dependent harmonic emission process can be moved towards the + smaller-R region as the masses of the nuclei are increased (D+ 2 and T2 ). (iv) Multi-minima on the harmonic spectra can be obtained, which is attributed to the two-center interference and the electron– nuclear coupling during the generation of the harmonics. The R-dependent ionization probabilities, the time-dependent nuclear motions and the time–frequency analyses of the harmonic spectra have been shown to explain the R-dependent molecular harmonic emission process. © 2017 Elsevier B.V. All rights reserved.
1. Introduction High-order harmonic generation (HHG) caused by the interactions of the intense pulses with the atoms and the molecules attracts lots of attentions in ultrafast optics and strong-field physics [1–7]. Generally, the harmonic emission process from atoms and molecules can be described by the ‘ionization-acceleration-recombination’ model [8], which is denoted as the three-step model. For atomic system, the harmonic emission process can be well understood through analyzing the three-step mode. However, for the case of the molecular system (e.g. the simplest molecule H+ 2 ion), the electron can recombine not only with its parent nucleus but also with its neighbor nucleus. Thus, when the harmonic spectra are generated, one wants to know (Q1) which nucleus (z-dependent harmonic emission spectra) and (Q2) which nuclear distance (R-dependent harmonic emission spectra) present the main role in harmonic emission process. To answer the first question, Lein et al. [9,10] first investigate the harmonic generation from the two-center of H+ 2 and the min-
*
Corresponding author at: Laboratory of Modern Physics, College of Science, Liaoning University of Technology, Jinzhou 121001, China. E-mail address: [email protected] (H. Liu). http://dx.doi.org/10.1016/j.physleta.2017.01.011 0375-9601/© 2017 Elsevier B.V. All rights reserved.
ima on the HHG spectra caused by the two-center interference has been observed. Han et al. [11] and Chelkowski et al. [12] find that the population of the excited state also plays an important role in the generation of the minima on the HHG spectra. Zhang et al. [13–16] theoretically investigate a series of the spatial distributions (z-dependent harmonic emission spectra) of the molecular high-order harmonic generation (MHHG) spectra by using the Thz, the static controlling pulse and the two-color field, and they find that the MHHG from two-H nuclei can be controlled when changing the laser profile of the combined field. Moreover, recently, we also investigate nuclear signature effect on spatial distribution of the MHHG spectra [17]. The results show that the contribution from the negative-H to the MHHG is higher than that from the positive-H and the intensity differences between the two H nuclei can be decreased with the increase of the mass of the nucleus. For the second question, Zuo et al. [18,19] first discussed charge-resonance-enhanced-ionization (CREI) phenomenon and its relation to the HHG. Recently, through analyzing the R-dependent MHHG spectra, Silva et al. [20] report the even harmonic generation process in H+ 2 and its isotopes. In this paper, to better understand the R-dependent spatial distribution of the MHHG, we further investigate the MHHG spec+ + tra from H+ 2 and its isotopes (D2 and T2 ). For the case of the few-cycle pulse, the harmonic emission is mainly caused by the CREI. For the case of the multi-cycle pulse, the harmonic emis-
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Fig. 1. R-dependent MHHG spectra from H+ 2 driven by the (a) few-cycle 5 fs/800 nm pulse and (b) multi-cycle 20 fs/800 nm pulse. The pulse intensity is I = 4.0 × 1014 W/cm2 . Total MHHG spectra and MHHG spectra without interferences (c) few-cycle pulse and (d) multi-cycle pulse. Total MHHG spectra and MHHG spectra without interferences from 20ω1 to 40ω1 orders (e) few-cycle pulse and (f) multi-cycle pulse.
sion is caused by the CREI and the dissociative ionization. Due to the slower motion of the heavy nucleus, the R-dependent spatial distribution of the MHHG can be moved towards the smaller-R region. Moreover, the multi-minima on the MHHG spectra can be obtained, which is caused by the two-center interference and the electron–nuclear coupling during the harmonic emission process.
+∞ ∂ V ( z, R ) d( R , t ) = ψ ∗ ( z, R , t ) − ∂z −∞ 1 + 1+ E (t ) ψ( z, R , t )dz 2m p + 1
2. Method
The R-dependent MHHG spectra can be obtained through the Fourier transformation of the R-dependent time-dependent dipole acceleration,
The Non-Bohn–Oppenheimer time-dependent Schrödinger equation (NBO-TDSE) of H+ 2 and its isotopes can be expressed as [21–25] (atomic units are used throughout this paper unless stated otherwise),
i
∂ψ( z, R , t ) 1 ∂2 1 ∂2 = − − + V ( z, R ) 2 ∂t 2u N ∂ R 2u e ∂ z2 1 + 1+ zE (t ) ψ( z, R , t ). (2m p + 1)
(3)
0
3. Results and discussion
(1)
u N = m p /2, u e = 2m p /(2m p + 1) are the reduced masses of the nucleus and the electron, where m p are the masses of the H, D, T. Soft Coulomb potential of H+ 2 and its isotopes is V ( z, R ) = 1/ R −
2 1 Ttotal −i ωt P ( R , ω) = √ d( R , t )e dt . 2π
(2)
1/ ( z − R /2)2 + 1 − 1/ ( z + R /2)2 + 1. Here, the isotopic variants cannot change the form of the potential energy surfaces, thus, we use the same Coulomb potential to describe H+ 2 and its isotopes. This is a very useful approximation in the strong-field physics dynamics [12,22] and the chemical reaction dynamics [26,27]. R and z denote the nuclear (nuclear distance) and the electronic coordinates, respectively, which are defined by −100 < z < 100 with z = 0.2 and 0 < R < 30 with R = 0.1. The time step is chosen to be t = 0.1 Laser pulse is E (t ) = E exp[−4 ln(2)t 2 /τ 2 ] sin(ω1 t ), where E, ω1 , and τ are the amplitude, the frequency and the time of the full width at half maximum (FWHM) of the laser pulse. The R-dependent ionization probabilities (IPs) can be calculated t by P iR (t ) = 0 dt j ( R , z s , t ), where j = u1 Im[ψ ∗ δ( z − z0 ) ∂∂z ψ], and e z s = ±25. The nuclear motion can be expressed as +∞time-dependent P R ( R , t ) = −∞ |ψ( z, R , t )|2 dz. The R-dependent time-dependent dipole acceleration is given by [28]
Figs. 1(a) and 1(b) show the R-dependent MHHG spectra from H+ 2 driven by the few-cycle 5 fs/800 nm and multi-cycle 20 fs/800 nm pulses, respectively. The pulse intensity is I = 4.0 × 1014 W/cm2 . As seen, in the presence of the few-cycle pulse (Fig. 1(a)), the probability of the MHHG from the equilibrium internuclear distance R = 2.0 is very small and the MHHG mainly occurs from R = 3.7 to R = 6. Particularly, the below-threshold (lower than 19H) MHHG can be found from R = 3.7 to R = 6, and the above-threshold (higher than 19H) MHHG only happens around R = 4 For the case of the multi-cycle pulse (Fig. 1(b)), the MHHG can be separated into two parts, for instance: the highintensity harmonic emission process from R = 3.7 to R = 8; and the low-intensity harmonic emission process when R > 10. Particularly, the below-threshold MHHG can be found in the whole region, while the above-threshold MHHG can be obtained only around R = 3.7 ∼ 6. As we know the harmonic emission is caused by the processes of ionization–acceleration–recombination, and the ionization process of H+ 2 can be separate into two parts [18,19, 29,30], that is (i) the direct ionization/CREI region from R < 10, | z| > 25 and (ii) the dissociative ionization region from R > 10, | z| > 25. Through analyzing the R-dependence of the MHHG spectra shown in Figs. 1(a) and 1(b), we see that the harmonic emission is only contributed from the CREI region for the case of the few-cycle pulse. With the increase of the pulse duration, the contribution of the dissociative ionization to the MHHG spectrum is
L. Feng, H. Liu / Physics Letters A 381 (2017) 859–864
Table 1 Minima of the harmonics at different internuclear distances. R
2
3
4
5
6
7
m=0 m=1 m=2 m=3
22ω1 88ω1 – –
9ω1 39ω1 88ω1 –
5ω1 22ω1 49ω1 88ω1
3ω1 14ω1 31ω1 56ω1
2ω1 9ω1 22ω1 39ω1
1ω1 7ω1 16ω1 28ω1
enhanced, but the CREI also plays the main role in harmonic emission process. It is well known that MHHG presents the fingerprints of interferences [9,10]. Thus, to see the interference phenomena on the MHHG spectra, the total MHHG spectra (which can be approximately written as P total (ω) = | S 1 (ω)|2 + | S 2 (ω)|2 + 2 Re[ S 1 (ω) S 2∗ (ω)]) and the MHHG spectra | S 1 (ω)|2 + | S 2 (ω)|2 without the interferences, where | S j (ω)|2 , j = 1, 2 can be interpreted as the harmonic spectrum only originating from the nucleus j, have been shown in Figs. 1(c) and 1(d) for the cases of the few-cycle and the multi-cycle pulses, respectively. As seen, the two spectra are on top of each other in most parts of the harmonics, implying that the interference term is typically negligible. However, there are still some minima in the harmonic spectra. Since these minima are not present in the | S 1 (ω)|2 + | S 2 (ω)|2 result, it is caused by interference term. To clearly see these minima, part of the harmonic spectra from 20ω1 to 40ω1 orders have been shown in Figs. 1(e) and 1(f) for the above two cases. Clearly, three (21ω1 , 23ω1 and 37ω1 ) and six (21ω1 , 23ω1 , 28ω1 , 32ω1 , 34ω1 and 36ω1 ) visible minima on the MHHG spectra can be found for the cases of the few-cycle and the multi-cycle pulses. As we know the minima on the MHHG spectra are caused by the two-center interference [9,10] and the positions of the minima are relevant to the internuclear distance (R) and the angle between the laser polarization and the molecular axis (θ ), which can be expressed as R cos θ = (2m + 1)λ/2, m = 0, 1, 2... . In the present linearly polarized laser pulse, the positions of the minima are only decided by the internuclear distance. Here, if the fixed nuclei model is used and according to the above equation, there should be only one minimum from 20ω1 to 40ω1 harmonic orders, as seen in Table 1. However, the nuclei are not fixed in the present electron–nuclear
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dynamics model, and the minima can be formed in every different internuclear distance. Since the total harmonic signal is the sum of the harmonics from the different internuclear distances, thus leading to the multi-minima on the harmonic spectra. Particularly, for the case of the few-cycle pulse, due to the harmonics from 20ω1 to 40ω1 orders are produced from R = 3.7 to R = 4.6 (see Fig. 1(a)). Thus, according to the above minima equation, the three minima on the MHHG spectra come from R = 3.9 ∼ 4.1 (21ω1 and 23ω1 ) and R = 4.5 ∼ 4.6 (37ω1 ), respectively. For the case of the multi-cycle pulse, due to the extended contributions from the larger R, the harmonics from 20ω1 to 40ω1 orders are produced from R = 3.7 to R = 7 (see Fig. 1(b)). As a result, the six minima on the MHHG spectra come from R = 3.9 ∼ 4.1 (21ω1 and 23ω1 ), R = 5.2 ∼ 5.4, R = 7 (28ω1 ) and R = 4.5 ∼ 5.0 (32ω1 , 34ω1 and 36ω1 ) respectively. From the above analyses, we see that the multi-minima on the MHHG spectra are caused by the two-center interference and the electron–nuclear coupling during the harmonic emission process, and the CREI region plays the main role in the generations of the minima. To better understand the R-dependent MHHG from the direct ionization/CREI and the dissociative ionization, in Figs. 2(a) and 2(b), we present the R-dependent IPs [31,32] for the cases of the few-cycle and multi-cycle pulse, respectively. For the case of the few-cycle pulse (Fig. 2(a)), the IPs can be enhanced from R = 3 to R = 6, which is well defined as the CREI. And the maximum IP is achieved at R = 4, which is responsible for the dominated harmonic emission from R = 3.7 to R = 6 and the maximum harmonic emission around R = 4. With the increase of the nuclear distance, the IPs decrease very fast, which means the probability of the dissociative ionization is very small and is responsible for the low-intensity harmonic emission in this region. For the case of the multi-cycle pulse (Fig. 2(b)), an enhancement of the IPs can also be found from R = 3 to R = 8 with the maximum value of 0.07 at R = 5, thus leading to the maximum harmonic cutoff extension in this region. As the nuclear distance increased, although the IPs are decreased, the declining rate is much slower in comparison with the few-cycle pulse case. Thus, a visible contribution from the dissociative ionization to the MHHG can be obtained in the multicycle pulse case. To understand the different phenomenon of the
+ Fig. 2. R-dependent IPs of H+ 2 for the cases of (a) few-cycle pulse and (b) multi-cycle pulse. Time-dependent nuclear motions of H2 for the case of (c) few-cycle pulse and (d) multi-cycle pulse. T is the optical cycle of 800 nm pulse.
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Fig. 3. Time–frequency analyses of the MHHG for the cases of (a) few-cycle pulse with R < 10; (b) few-cycle pulse with R > 10; (c) multi-cycle pulse with R < 10; (d) multi-cycle pulse with R > 10.
+ Fig. 4. R-dependent MHHG spectra from D+ 2 driven by the (a) few-cycle pulse; (b) multi-cycle pulse and from T2 driven by the (c) few-cycle pulse; (d) multi-cycle pulse.
dissociative ionization from the few-cycle and multi-cycle pulses, the time-dependent nuclear motions for the above two pulses have been shown in Figs. 2(c) and 2(d), respectively. For the case of the few-cycle pulse (Fig. 2(c)), we see that at the end of the laser pulse, the maximum nuclear distance is around R = 6, corresponding to the direct ionization/CREI region, which means in this short pulse, most of the electron has already been direct ionized before the dissociation process. As a result, we only observe the MHHG from the CREI region in the few-cycle pulse. For the case of the multicycle pulse (Fig. 2(d)), the nuclear distance begins to increase at t = 10T (T is the optical cycle of 800 nm pulse), and it gets to the dissociation requirement of R = 10 at t = 12T. We see that in the multi-cycle pulse, the molecule has enough to dissociate, thus leading to the enhancement of the dissociative ionization and is
responsible for the visible contribution to the MHHG from the dissociative ionization shown in Fig. 1(b). Fig. 3 shows the time–frequency analyses of the MHHG by using the wavelet transformation [33]. For the case of the few-cycle pulse with R < 10 (Fig. 3(a)), four harmonic radiation peaks can be found during the harmonic emission process, and the recombination energy from the maximum harmonic radiation peak (from t = 2.25T to t = 2.75T) is in agreement with the calculated result shown in Fig. 1(c). For the case of the few-cycle pulse with R > 10 (Fig. 3(b)), there are only two harmonic radiation peaks during the harmonic emission process, and the harmonic radiation time (i.e. from t = 3.25T to t = 3.75T and from t = 3.75T to t = 4.25T) is at the later of the laser pulse, where is the nuclear distance beginning to increase (see Fig. 2(c)). It notes that although two harmonic
L. Feng, H. Liu / Physics Letters A 381 (2017) 859–864
Fig. 5. Total MHHG spectra and MHHG spectra without interferences from 20ω1 to 40ω1 orders for the cases of D+ 2 driven by the (a) few-cycle pulse; (b) multi-cycle pulse and T+ 2 driven by the (c) few-cycle pulse; (d) multi-cycle pulse.
radiation peaks can be obtained in the dissociative ionization region, the intensities of the harmonic radiation peaks are very weak compared with the CREI region (see the color bar). Thus, their contributions to the total MHHG can be ignored in comparison with the harmonic radiation peaks from the CREI region. For the case of the multi-cycle pulses with R < 10 (Fig. 3(c)), due to the increased pulse duration, there are many harmonic radiation peaks during the harmonic emission process, and the intensities of the harmonic radiation peaks from t < 10T are larger than those from t > 10T. This is because that when t < 10T, the nuclear distance is small and there is only the direct ionization/CREI responsible for the ionization probability and the harmonic emission process. While when t > 10T, the nuclear distance is remarkably increased. As a result, both direct ionization/CREI and dissociative ionization are
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responsible for the ionization probability and the harmonic emission process (see Fig. 2(d)). For the case of the multi-cycle pulse with R > 10 (Fig. 3(d)), the harmonic radiation peaks can only be observed when t > 12T. This is attributing to the dissociation process only happening at the latter of the laser pulse with t > 12T (see Fig. 2(d)). Although the intensities of the harmonic radiation peaks from R > 10 (dissociative ionization) are smaller than those from the R < 10 (direct ionization/CREI), their intensities are much larger than those from the few-cycle pulse case. Thus, we can observe their contributions to the total MHHG shown in Fig. 1(b). Figs. 4(a)–4(b) and 4(c)–4(d) show the R-dependent MHHG + spectra from D+ 2 and T2 driven by the above few-cycle and multicycle pulses, respectively. Clearly, similar R-dependent MHHG spectra from the CREI and the dissociative ionization can be found + in D+ 2 and T2 but with a smaller-R shift. For example, (i) for the case of D+ driven by the few-cycle pulse (Fig. 4(a)), the harmonic 2 emission from the CREI mainly occurs from R = 3.7 to R = 5; (ii) for the case of D+ 2 driven by the multi-cycle pulse (Fig. 4(b)), the harmonic emission from the CREI and the dissociative ionization mainly occurs from R = 3.7 to R = 7 and from R = 10 to R = 15 (here we only consider the dissociative ionization from R = 10 to R = 15 due to the IPs from R > 15 are very small), respectively; (iii) for the case of T+ 2 driven by the few-cycle pulse (Fig. 4(c)), the harmonic emission from the CREI mainly occurs from R = 3.7 to R = 4; (iv) for the case of T+ 2 driven by the multicycle pulse (Fig. 4(d)), the harmonic emission from the CREI and the dissociative ionization occurs from R = 3.7 to R = 6 and from R = 10 to R = 13, respectively. + Figs. 5(a)–5(d) show the harmonic spectra of D+ 2 and T2 from 20ω1 to 40ω1 orders. As seen, the multi-minima on the MHHG spectra can also be observed. However, due to the compression of the R-dependence of the MHHG, the positions of the minima are shift. For instance, (i) the minimum in the 25ω1 harmonic order from D+ 2 driven by the few-cycle pulse come from R = 3.7; (ii) the minima in the 25ω1 , the 28ω1 the 32ω1 and the 38ω1 from D+ 2 driven by the multi-cycle pulse come from R = 3.7, R = 5.2 ∼ 5.4, R = 4.9 ∼ 5.1 and R = 4.5 ∼ 4.6; (iii) the minimum in 24ω1
+ Fig. 6. Time-dependent nuclear motions of D+ 2 for the case of (a) few-cycle pulse; (b) multi-cycle pulse and of T2 for the case of (c) few-cycle pulse; (d) multi-cycle pulse.
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harmonic order from T+ 2 driven by the few-cycle pulse come from R = 3.8; (iv) the minima in the 22ω1 , the 24ω1 the 28ω1 , the 34ω1 and the 36ω1 from T+ 2 driven by the multi-cycle pulse come from R = 3.8 ∼ 4.1, R = 5.2 ∼ 5.4 and R = 4.6 ∼ 4.8. To explain the isotope effect on the R-dependent MHHG spectra, in Fig. 6, we present the time-dependent nuclear motions for + the cases of D+ 2 and T2 . As seen, with the increase of the nuclear + mass, the maximum nuclear distances for D+ 2 and T2 driven by the few-cycle pulse are around R = 5 and R = 4, respectively, as shown in Figs. 6(a) and 6(c), which means nuclear motion will be slowdown with the increase of the nuclear mass and is responsible for the smaller R shift of the R-dependent MHHG in the heavy nucleus. Similar results for the cases of the multi-cycle pulse, as + shown in Figs. 6(b) and 6(d) for D+ 2 and T2 , the nuclear distances + begin to increase at t = 12T (for D2 ) and t = 14T (for T+ 2 ), and they get to the dissociation requirement at t = 14T (for D+ 2 ) and t = 16T (for T+ ). At the end of the laser pulse, the nuclear dis2 tances from dissociative ionization are from R = 10 to R = 15 and + from R = 10 to R = 13 for D+ 2 and T2 , respectively, thus leading to the smaller R shift of the R-dependent MHHG shown in Fig. 4. 4. Conclusions In conclusion, we theoretically investigated the R-dependent + + spatial distribution of the MHHG from H+ 2 , D2 and T2 . For the case of the few-cycle pulse, the harmonic emission is mainly caused by the CREI. For the case of the multi-cycle pulse, the harmonic emission is caused by the CREI and the dissociative ionization. Due to the slower motion of the heavy nucleus, the R-dependent spatial distribution of the MHHG present a smaller R shift with the increase of the nuclear mass. Moreover, the multi-minima on the MHHG spectra can be found, which is caused by the two-center interference and the electron–nuclear coupling during the harmonic emission process, and the CREI region plays the main role in the generations of the minima. Acknowledgements This work was supported by National Natural Science Foundation of China (No. 11504151), the Doctoral Scientific Research
Foundation of Liaoning province (No. 201501123), and the Scientific Research Foundation of Liaoning Provincial Education Department (No. JL201615405). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17]
F. Krausz, M. Ivanov, Rev. Mod. Phys. 81 (2009) 163. E. Goulielmakis, M. Schultze, M. Hofstetter, et al., Science 320 (2008) 1614. X.B. Bian, A.D. Bandrauk, Phys. Rev. Lett. 105 (2010) 093903. K.J. Yuan, A.D. Bandrauk, Phys. Rev. Lett. 110 (2013) 023003. P.C. Li, X.X. Zhou, G.L. Wang, Z.X. Zhao, Phys. Rev. A 80 (2009) 053825. R.F. Lu, H.X. He, Y.H. Guo, K.L. Han, J. Phys. B, At. Mol. Opt. Phys. 42 (2009) 225601. L.Q. Feng, T.S. Chu, Phys. Lett. A 376 (2012) 1523. P.B. Corkum, Phys. Rev. Lett. 71 (1993) 1994. M. Lein, N. Hay, R. Velotta, J.P. Marangos, P.L. Knight, Phys. Rev. Lett. 88 (2002) 183903. M. Lein, N. Hay, R. Velotta, J.P. Marangos, P.L. Knight, Phys. Rev. A 66 (2002) 023805. Y.C. Han, L.B. Madsen, J. Phys. B, At. Mol. Opt. Phys. 43 (2010) 225601. S. Chelkowski, T. Bredtmann, A.D. Bandrauk, Phys. Rev. A 88 (2013) 033423. J. Zhang, X.L. Ge, T. Wang, T.T. Xu, J. Guo, X.S. Liu, Phys. Rev. A 92 (2015) 013418. J. Zhang, X.F. Pan, C.L. Xia, D. Du, T.T. Xu, J. Guo, X.S. Liu, Laser Phys. Lett. 13 (2016) 075302. J. Zhang, H.F. Liu, X.F. Pan, H. Du, J. Guo, X.S. Liu, Chin. Phys. B 25 (2016) 053202. J. Zhang, X.F. Pan, H. Du, T.T. Xu, J. Guo, X.S. Liu, Opt. Commun. 382 (2017) 495. L.Q. Feng, W.L. Li, Laser Phys. 27 (2017) 016002.
[18] T. Zuo, S. Chelkowski, A.D. Bandrauk, Phys. Rev. A 48 (1993) 3837. [19] T. Zuo, A.D. Bandrauk, Phys. Rev. A 52 (1995) R2511. [20] R.E.F. Silva, P. Rivière, F. Morales, O. Smirnova, M. Ivanov, F. Martín, Sci. Rep. 6 (2016) 32653. [21] R.F. Lu, P.Y. Zhang, K.L. Han, Phys. Rev. E 77 (2008) 066701. [22] L.Q. Feng, T.S. Chu, J. Chem. Phys. 136 (2012) 054102. [23] L.Q. Feng, T.S. Chu, Phys. Rev. A 84 (2011) 053853. [24] L.Q. Feng, Phys. Rev. A 92 (2015) 053832. [25] J. Hu, K.L. Han, G.Z. He, Phys. Rev. Lett. 95 (2005) 123001. [26] M.D. Chen, K.L. Han, N.Q. Lou, Chem. Phys. Lett. 357 (2002) 483. [27] T.S. Chu, Y. Zhang, K.L. Han, Int. Rev. Phys. Chem. 25 (2006) 201. [28] K. Burnett, V.C. Reed, J. Cooper, P.L. Knight, Phys. Rev. A 45 (1992) 3347. [29] Y.W. Duan, W.K. Liu, J.M. Yuan, Phys. Rev. A 61 (2000) 053403. [30] S. Chelkowski, C. Foisy, A.D. Bandrauk, Phys. Rev. A 57 (1998) 1176. [31] M. Mohebbi, S. Batebi, Opt. Commun. 296 (2013) 113. [32] R.F. Lu, C.Y. Xiao, K.M. Deng, H.P. Wu, Chem. Phys. 382 (2011) 88. [33] P. Antoine, B. Piraux, A. Maquet, Phys. Rev. A 51 (1995) R1750.
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Discussion
R-dependent molecular harmonic generation from H+ 2 Liqiang Feng a,b , Hang Liu a,c,∗ a b c
Laboratory of Modern Physics, College of Science, Liaoning University of Technology, Jinzhou 121001, China State Key Laboratory of Molecular Reaction Dynamics, Dalian Institute of Chemical Physics, Chinese Academy of Sciences, Dalian 116023, China School of Chemical and Environmental Engineering, Liaoning University of Technology, Jinzhou 121001, China
a r t i c l e
i n f o
Article history: Received 28 November 2016 Received in revised form 4 January 2017 Accepted 8 January 2017 Available online 11 January 2017 Communicated by V.A. Markel Keywords: R-dependent molecular harmonic generation Non-Bohn–Oppenheimer time-dependent Schrödinger equation Charge-resonance-enhanced-ionization Dissociative ionization
a b s t r a c t R-dependent high-order harmonic spectra (R is the nuclear distance) from H+ 2 have been investigated through solving the Non-Bohn–Oppenheimer time-dependent Schrödinger equation. We found that (i) for the case of the few-cycle pulse, the harmonic emission mainly occurs from R = 3.7 to R = 6, caused by the charge-resonance-enhanced-ionization (CREI) process. (ii) For the case of the multi-cycle pulse, the harmonic emission can be separated into two parts, that is the charge-resonance-enhancedionization region from R = 3.7 to R = 8; and the dissociative ionization region when R > 10. (iii) Isotopic investigation showed that the R-dependent harmonic emission process can be moved towards the + smaller-R region as the masses of the nuclei are increased (D+ 2 and T2 ). (iv) Multi-minima on the harmonic spectra can be obtained, which is attributed to the two-center interference and the electron– nuclear coupling during the generation of the harmonics. The R-dependent ionization probabilities, the time-dependent nuclear motions and the time–frequency analyses of the harmonic spectra have been shown to explain the R-dependent molecular harmonic emission process. © 2017 Elsevier B.V. All rights reserved.
1. Introduction High-order harmonic generation (HHG) caused by the interactions of the intense pulses with the atoms and the molecules attracts lots of attentions in ultrafast optics and strong-field physics [1–7]. Generally, the harmonic emission process from atoms and molecules can be described by the ‘ionization-acceleration-recombination’ model [8], which is denoted as the three-step model. For atomic system, the harmonic emission process can be well understood through analyzing the three-step mode. However, for the case of the molecular system (e.g. the simplest molecule H+ 2 ion), the electron can recombine not only with its parent nucleus but also with its neighbor nucleus. Thus, when the harmonic spectra are generated, one wants to know (Q1) which nucleus (z-dependent harmonic emission spectra) and (Q2) which nuclear distance (R-dependent harmonic emission spectra) present the main role in harmonic emission process. To answer the first question, Lein et al. [9,10] first investigate the harmonic generation from the two-center of H+ 2 and the min-
*
Corresponding author at: Laboratory of Modern Physics, College of Science, Liaoning University of Technology, Jinzhou 121001, China. E-mail address: [email protected] (H. Liu). http://dx.doi.org/10.1016/j.physleta.2017.01.011 0375-9601/© 2017 Elsevier B.V. All rights reserved.
ima on the HHG spectra caused by the two-center interference has been observed. Han et al. [11] and Chelkowski et al. [12] find that the population of the excited state also plays an important role in the generation of the minima on the HHG spectra. Zhang et al. [13–16] theoretically investigate a series of the spatial distributions (z-dependent harmonic emission spectra) of the molecular high-order harmonic generation (MHHG) spectra by using the Thz, the static controlling pulse and the two-color field, and they find that the MHHG from two-H nuclei can be controlled when changing the laser profile of the combined field. Moreover, recently, we also investigate nuclear signature effect on spatial distribution of the MHHG spectra [17]. The results show that the contribution from the negative-H to the MHHG is higher than that from the positive-H and the intensity differences between the two H nuclei can be decreased with the increase of the mass of the nucleus. For the second question, Zuo et al. [18,19] first discussed charge-resonance-enhanced-ionization (CREI) phenomenon and its relation to the HHG. Recently, through analyzing the R-dependent MHHG spectra, Silva et al. [20] report the even harmonic generation process in H+ 2 and its isotopes. In this paper, to better understand the R-dependent spatial distribution of the MHHG, we further investigate the MHHG spec+ + tra from H+ 2 and its isotopes (D2 and T2 ). For the case of the few-cycle pulse, the harmonic emission is mainly caused by the CREI. For the case of the multi-cycle pulse, the harmonic emis-
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Fig. 1. R-dependent MHHG spectra from H+ 2 driven by the (a) few-cycle 5 fs/800 nm pulse and (b) multi-cycle 20 fs/800 nm pulse. The pulse intensity is I = 4.0 × 1014 W/cm2 . Total MHHG spectra and MHHG spectra without interferences (c) few-cycle pulse and (d) multi-cycle pulse. Total MHHG spectra and MHHG spectra without interferences from 20ω1 to 40ω1 orders (e) few-cycle pulse and (f) multi-cycle pulse.
sion is caused by the CREI and the dissociative ionization. Due to the slower motion of the heavy nucleus, the R-dependent spatial distribution of the MHHG can be moved towards the smaller-R region. Moreover, the multi-minima on the MHHG spectra can be obtained, which is caused by the two-center interference and the electron–nuclear coupling during the harmonic emission process.
+∞ ∂ V ( z, R ) d( R , t ) = ψ ∗ ( z, R , t ) − ∂z −∞ 1 + 1+ E (t ) ψ( z, R , t )dz 2m p + 1
2. Method
The R-dependent MHHG spectra can be obtained through the Fourier transformation of the R-dependent time-dependent dipole acceleration,
The Non-Bohn–Oppenheimer time-dependent Schrödinger equation (NBO-TDSE) of H+ 2 and its isotopes can be expressed as [21–25] (atomic units are used throughout this paper unless stated otherwise),
i
∂ψ( z, R , t ) 1 ∂2 1 ∂2 = − − + V ( z, R ) 2 ∂t 2u N ∂ R 2u e ∂ z2 1 + 1+ zE (t ) ψ( z, R , t ). (2m p + 1)
(3)
0
3. Results and discussion
(1)
u N = m p /2, u e = 2m p /(2m p + 1) are the reduced masses of the nucleus and the electron, where m p are the masses of the H, D, T. Soft Coulomb potential of H+ 2 and its isotopes is V ( z, R ) = 1/ R −
2 1 Ttotal −i ωt P ( R , ω) = √ d( R , t )e dt . 2π
(2)
1/ ( z − R /2)2 + 1 − 1/ ( z + R /2)2 + 1. Here, the isotopic variants cannot change the form of the potential energy surfaces, thus, we use the same Coulomb potential to describe H+ 2 and its isotopes. This is a very useful approximation in the strong-field physics dynamics [12,22] and the chemical reaction dynamics [26,27]. R and z denote the nuclear (nuclear distance) and the electronic coordinates, respectively, which are defined by −100 < z < 100 with z = 0.2 and 0 < R < 30 with R = 0.1. The time step is chosen to be t = 0.1 Laser pulse is E (t ) = E exp[−4 ln(2)t 2 /τ 2 ] sin(ω1 t ), where E, ω1 , and τ are the amplitude, the frequency and the time of the full width at half maximum (FWHM) of the laser pulse. The R-dependent ionization probabilities (IPs) can be calculated t by P iR (t ) = 0 dt j ( R , z s , t ), where j = u1 Im[ψ ∗ δ( z − z0 ) ∂∂z ψ], and e z s = ±25. The nuclear motion can be expressed as +∞time-dependent P R ( R , t ) = −∞ |ψ( z, R , t )|2 dz. The R-dependent time-dependent dipole acceleration is given by [28]
Figs. 1(a) and 1(b) show the R-dependent MHHG spectra from H+ 2 driven by the few-cycle 5 fs/800 nm and multi-cycle 20 fs/800 nm pulses, respectively. The pulse intensity is I = 4.0 × 1014 W/cm2 . As seen, in the presence of the few-cycle pulse (Fig. 1(a)), the probability of the MHHG from the equilibrium internuclear distance R = 2.0 is very small and the MHHG mainly occurs from R = 3.7 to R = 6. Particularly, the below-threshold (lower than 19H) MHHG can be found from R = 3.7 to R = 6, and the above-threshold (higher than 19H) MHHG only happens around R = 4 For the case of the multi-cycle pulse (Fig. 1(b)), the MHHG can be separated into two parts, for instance: the highintensity harmonic emission process from R = 3.7 to R = 8; and the low-intensity harmonic emission process when R > 10. Particularly, the below-threshold MHHG can be found in the whole region, while the above-threshold MHHG can be obtained only around R = 3.7 ∼ 6. As we know the harmonic emission is caused by the processes of ionization–acceleration–recombination, and the ionization process of H+ 2 can be separate into two parts [18,19, 29,30], that is (i) the direct ionization/CREI region from R < 10, | z| > 25 and (ii) the dissociative ionization region from R > 10, | z| > 25. Through analyzing the R-dependence of the MHHG spectra shown in Figs. 1(a) and 1(b), we see that the harmonic emission is only contributed from the CREI region for the case of the few-cycle pulse. With the increase of the pulse duration, the contribution of the dissociative ionization to the MHHG spectrum is
L. Feng, H. Liu / Physics Letters A 381 (2017) 859–864
Table 1 Minima of the harmonics at different internuclear distances. R
2
3
4
5
6
7
m=0 m=1 m=2 m=3
22ω1 88ω1 – –
9ω1 39ω1 88ω1 –
5ω1 22ω1 49ω1 88ω1
3ω1 14ω1 31ω1 56ω1
2ω1 9ω1 22ω1 39ω1
1ω1 7ω1 16ω1 28ω1
enhanced, but the CREI also plays the main role in harmonic emission process. It is well known that MHHG presents the fingerprints of interferences [9,10]. Thus, to see the interference phenomena on the MHHG spectra, the total MHHG spectra (which can be approximately written as P total (ω) = | S 1 (ω)|2 + | S 2 (ω)|2 + 2 Re[ S 1 (ω) S 2∗ (ω)]) and the MHHG spectra | S 1 (ω)|2 + | S 2 (ω)|2 without the interferences, where | S j (ω)|2 , j = 1, 2 can be interpreted as the harmonic spectrum only originating from the nucleus j, have been shown in Figs. 1(c) and 1(d) for the cases of the few-cycle and the multi-cycle pulses, respectively. As seen, the two spectra are on top of each other in most parts of the harmonics, implying that the interference term is typically negligible. However, there are still some minima in the harmonic spectra. Since these minima are not present in the | S 1 (ω)|2 + | S 2 (ω)|2 result, it is caused by interference term. To clearly see these minima, part of the harmonic spectra from 20ω1 to 40ω1 orders have been shown in Figs. 1(e) and 1(f) for the above two cases. Clearly, three (21ω1 , 23ω1 and 37ω1 ) and six (21ω1 , 23ω1 , 28ω1 , 32ω1 , 34ω1 and 36ω1 ) visible minima on the MHHG spectra can be found for the cases of the few-cycle and the multi-cycle pulses. As we know the minima on the MHHG spectra are caused by the two-center interference [9,10] and the positions of the minima are relevant to the internuclear distance (R) and the angle between the laser polarization and the molecular axis (θ ), which can be expressed as R cos θ = (2m + 1)λ/2, m = 0, 1, 2... . In the present linearly polarized laser pulse, the positions of the minima are only decided by the internuclear distance. Here, if the fixed nuclei model is used and according to the above equation, there should be only one minimum from 20ω1 to 40ω1 harmonic orders, as seen in Table 1. However, the nuclei are not fixed in the present electron–nuclear
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dynamics model, and the minima can be formed in every different internuclear distance. Since the total harmonic signal is the sum of the harmonics from the different internuclear distances, thus leading to the multi-minima on the harmonic spectra. Particularly, for the case of the few-cycle pulse, due to the harmonics from 20ω1 to 40ω1 orders are produced from R = 3.7 to R = 4.6 (see Fig. 1(a)). Thus, according to the above minima equation, the three minima on the MHHG spectra come from R = 3.9 ∼ 4.1 (21ω1 and 23ω1 ) and R = 4.5 ∼ 4.6 (37ω1 ), respectively. For the case of the multi-cycle pulse, due to the extended contributions from the larger R, the harmonics from 20ω1 to 40ω1 orders are produced from R = 3.7 to R = 7 (see Fig. 1(b)). As a result, the six minima on the MHHG spectra come from R = 3.9 ∼ 4.1 (21ω1 and 23ω1 ), R = 5.2 ∼ 5.4, R = 7 (28ω1 ) and R = 4.5 ∼ 5.0 (32ω1 , 34ω1 and 36ω1 ) respectively. From the above analyses, we see that the multi-minima on the MHHG spectra are caused by the two-center interference and the electron–nuclear coupling during the harmonic emission process, and the CREI region plays the main role in the generations of the minima. To better understand the R-dependent MHHG from the direct ionization/CREI and the dissociative ionization, in Figs. 2(a) and 2(b), we present the R-dependent IPs [31,32] for the cases of the few-cycle and multi-cycle pulse, respectively. For the case of the few-cycle pulse (Fig. 2(a)), the IPs can be enhanced from R = 3 to R = 6, which is well defined as the CREI. And the maximum IP is achieved at R = 4, which is responsible for the dominated harmonic emission from R = 3.7 to R = 6 and the maximum harmonic emission around R = 4. With the increase of the nuclear distance, the IPs decrease very fast, which means the probability of the dissociative ionization is very small and is responsible for the low-intensity harmonic emission in this region. For the case of the multi-cycle pulse (Fig. 2(b)), an enhancement of the IPs can also be found from R = 3 to R = 8 with the maximum value of 0.07 at R = 5, thus leading to the maximum harmonic cutoff extension in this region. As the nuclear distance increased, although the IPs are decreased, the declining rate is much slower in comparison with the few-cycle pulse case. Thus, a visible contribution from the dissociative ionization to the MHHG can be obtained in the multicycle pulse case. To understand the different phenomenon of the
+ Fig. 2. R-dependent IPs of H+ 2 for the cases of (a) few-cycle pulse and (b) multi-cycle pulse. Time-dependent nuclear motions of H2 for the case of (c) few-cycle pulse and (d) multi-cycle pulse. T is the optical cycle of 800 nm pulse.
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Fig. 3. Time–frequency analyses of the MHHG for the cases of (a) few-cycle pulse with R < 10; (b) few-cycle pulse with R > 10; (c) multi-cycle pulse with R < 10; (d) multi-cycle pulse with R > 10.
+ Fig. 4. R-dependent MHHG spectra from D+ 2 driven by the (a) few-cycle pulse; (b) multi-cycle pulse and from T2 driven by the (c) few-cycle pulse; (d) multi-cycle pulse.
dissociative ionization from the few-cycle and multi-cycle pulses, the time-dependent nuclear motions for the above two pulses have been shown in Figs. 2(c) and 2(d), respectively. For the case of the few-cycle pulse (Fig. 2(c)), we see that at the end of the laser pulse, the maximum nuclear distance is around R = 6, corresponding to the direct ionization/CREI region, which means in this short pulse, most of the electron has already been direct ionized before the dissociation process. As a result, we only observe the MHHG from the CREI region in the few-cycle pulse. For the case of the multicycle pulse (Fig. 2(d)), the nuclear distance begins to increase at t = 10T (T is the optical cycle of 800 nm pulse), and it gets to the dissociation requirement of R = 10 at t = 12T. We see that in the multi-cycle pulse, the molecule has enough to dissociate, thus leading to the enhancement of the dissociative ionization and is
responsible for the visible contribution to the MHHG from the dissociative ionization shown in Fig. 1(b). Fig. 3 shows the time–frequency analyses of the MHHG by using the wavelet transformation [33]. For the case of the few-cycle pulse with R < 10 (Fig. 3(a)), four harmonic radiation peaks can be found during the harmonic emission process, and the recombination energy from the maximum harmonic radiation peak (from t = 2.25T to t = 2.75T) is in agreement with the calculated result shown in Fig. 1(c). For the case of the few-cycle pulse with R > 10 (Fig. 3(b)), there are only two harmonic radiation peaks during the harmonic emission process, and the harmonic radiation time (i.e. from t = 3.25T to t = 3.75T and from t = 3.75T to t = 4.25T) is at the later of the laser pulse, where is the nuclear distance beginning to increase (see Fig. 2(c)). It notes that although two harmonic
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Fig. 5. Total MHHG spectra and MHHG spectra without interferences from 20ω1 to 40ω1 orders for the cases of D+ 2 driven by the (a) few-cycle pulse; (b) multi-cycle pulse and T+ 2 driven by the (c) few-cycle pulse; (d) multi-cycle pulse.
radiation peaks can be obtained in the dissociative ionization region, the intensities of the harmonic radiation peaks are very weak compared with the CREI region (see the color bar). Thus, their contributions to the total MHHG can be ignored in comparison with the harmonic radiation peaks from the CREI region. For the case of the multi-cycle pulses with R < 10 (Fig. 3(c)), due to the increased pulse duration, there are many harmonic radiation peaks during the harmonic emission process, and the intensities of the harmonic radiation peaks from t < 10T are larger than those from t > 10T. This is because that when t < 10T, the nuclear distance is small and there is only the direct ionization/CREI responsible for the ionization probability and the harmonic emission process. While when t > 10T, the nuclear distance is remarkably increased. As a result, both direct ionization/CREI and dissociative ionization are
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responsible for the ionization probability and the harmonic emission process (see Fig. 2(d)). For the case of the multi-cycle pulse with R > 10 (Fig. 3(d)), the harmonic radiation peaks can only be observed when t > 12T. This is attributing to the dissociation process only happening at the latter of the laser pulse with t > 12T (see Fig. 2(d)). Although the intensities of the harmonic radiation peaks from R > 10 (dissociative ionization) are smaller than those from the R < 10 (direct ionization/CREI), their intensities are much larger than those from the few-cycle pulse case. Thus, we can observe their contributions to the total MHHG shown in Fig. 1(b). Figs. 4(a)–4(b) and 4(c)–4(d) show the R-dependent MHHG + spectra from D+ 2 and T2 driven by the above few-cycle and multicycle pulses, respectively. Clearly, similar R-dependent MHHG spectra from the CREI and the dissociative ionization can be found + in D+ 2 and T2 but with a smaller-R shift. For example, (i) for the case of D+ driven by the few-cycle pulse (Fig. 4(a)), the harmonic 2 emission from the CREI mainly occurs from R = 3.7 to R = 5; (ii) for the case of D+ 2 driven by the multi-cycle pulse (Fig. 4(b)), the harmonic emission from the CREI and the dissociative ionization mainly occurs from R = 3.7 to R = 7 and from R = 10 to R = 15 (here we only consider the dissociative ionization from R = 10 to R = 15 due to the IPs from R > 15 are very small), respectively; (iii) for the case of T+ 2 driven by the few-cycle pulse (Fig. 4(c)), the harmonic emission from the CREI mainly occurs from R = 3.7 to R = 4; (iv) for the case of T+ 2 driven by the multicycle pulse (Fig. 4(d)), the harmonic emission from the CREI and the dissociative ionization occurs from R = 3.7 to R = 6 and from R = 10 to R = 13, respectively. + Figs. 5(a)–5(d) show the harmonic spectra of D+ 2 and T2 from 20ω1 to 40ω1 orders. As seen, the multi-minima on the MHHG spectra can also be observed. However, due to the compression of the R-dependence of the MHHG, the positions of the minima are shift. For instance, (i) the minimum in the 25ω1 harmonic order from D+ 2 driven by the few-cycle pulse come from R = 3.7; (ii) the minima in the 25ω1 , the 28ω1 the 32ω1 and the 38ω1 from D+ 2 driven by the multi-cycle pulse come from R = 3.7, R = 5.2 ∼ 5.4, R = 4.9 ∼ 5.1 and R = 4.5 ∼ 4.6; (iii) the minimum in 24ω1
+ Fig. 6. Time-dependent nuclear motions of D+ 2 for the case of (a) few-cycle pulse; (b) multi-cycle pulse and of T2 for the case of (c) few-cycle pulse; (d) multi-cycle pulse.
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harmonic order from T+ 2 driven by the few-cycle pulse come from R = 3.8; (iv) the minima in the 22ω1 , the 24ω1 the 28ω1 , the 34ω1 and the 36ω1 from T+ 2 driven by the multi-cycle pulse come from R = 3.8 ∼ 4.1, R = 5.2 ∼ 5.4 and R = 4.6 ∼ 4.8. To explain the isotope effect on the R-dependent MHHG spectra, in Fig. 6, we present the time-dependent nuclear motions for + the cases of D+ 2 and T2 . As seen, with the increase of the nuclear + mass, the maximum nuclear distances for D+ 2 and T2 driven by the few-cycle pulse are around R = 5 and R = 4, respectively, as shown in Figs. 6(a) and 6(c), which means nuclear motion will be slowdown with the increase of the nuclear mass and is responsible for the smaller R shift of the R-dependent MHHG in the heavy nucleus. Similar results for the cases of the multi-cycle pulse, as + shown in Figs. 6(b) and 6(d) for D+ 2 and T2 , the nuclear distances + begin to increase at t = 12T (for D2 ) and t = 14T (for T+ 2 ), and they get to the dissociation requirement at t = 14T (for D+ 2 ) and t = 16T (for T+ ). At the end of the laser pulse, the nuclear dis2 tances from dissociative ionization are from R = 10 to R = 15 and + from R = 10 to R = 13 for D+ 2 and T2 , respectively, thus leading to the smaller R shift of the R-dependent MHHG shown in Fig. 4. 4. Conclusions In conclusion, we theoretically investigated the R-dependent + + spatial distribution of the MHHG from H+ 2 , D2 and T2 . For the case of the few-cycle pulse, the harmonic emission is mainly caused by the CREI. For the case of the multi-cycle pulse, the harmonic emission is caused by the CREI and the dissociative ionization. Due to the slower motion of the heavy nucleus, the R-dependent spatial distribution of the MHHG present a smaller R shift with the increase of the nuclear mass. Moreover, the multi-minima on the MHHG spectra can be found, which is caused by the two-center interference and the electron–nuclear coupling during the harmonic emission process, and the CREI region plays the main role in the generations of the minima. Acknowledgements This work was supported by National Natural Science Foundation of China (No. 11504151), the Doctoral Scientific Research
Foundation of Liaoning province (No. 201501123), and the Scientific Research Foundation of Liaoning Provincial Education Department (No. JL201615405). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17]
F. Krausz, M. Ivanov, Rev. Mod. Phys. 81 (2009) 163. E. Goulielmakis, M. Schultze, M. Hofstetter, et al., Science 320 (2008) 1614. X.B. Bian, A.D. Bandrauk, Phys. Rev. Lett. 105 (2010) 093903. K.J. Yuan, A.D. Bandrauk, Phys. Rev. Lett. 110 (2013) 023003. P.C. Li, X.X. Zhou, G.L. Wang, Z.X. Zhao, Phys. Rev. A 80 (2009) 053825. R.F. Lu, H.X. He, Y.H. Guo, K.L. Han, J. Phys. B, At. Mol. Opt. Phys. 42 (2009) 225601. L.Q. Feng, T.S. Chu, Phys. Lett. A 376 (2012) 1523. P.B. Corkum, Phys. Rev. Lett. 71 (1993) 1994. M. Lein, N. Hay, R. Velotta, J.P. Marangos, P.L. Knight, Phys. Rev. Lett. 88 (2002) 183903. M. Lein, N. Hay, R. Velotta, J.P. Marangos, P.L. Knight, Phys. Rev. A 66 (2002) 023805. Y.C. Han, L.B. Madsen, J. Phys. B, At. Mol. Opt. Phys. 43 (2010) 225601. S. Chelkowski, T. Bredtmann, A.D. Bandrauk, Phys. Rev. A 88 (2013) 033423. J. Zhang, X.L. Ge, T. Wang, T.T. Xu, J. Guo, X.S. Liu, Phys. Rev. A 92 (2015) 013418. J. Zhang, X.F. Pan, C.L. Xia, D. Du, T.T. Xu, J. Guo, X.S. Liu, Laser Phys. Lett. 13 (2016) 075302. J. Zhang, H.F. Liu, X.F. Pan, H. Du, J. Guo, X.S. Liu, Chin. Phys. B 25 (2016) 053202. J. Zhang, X.F. Pan, H. Du, T.T. Xu, J. Guo, X.S. Liu, Opt. Commun. 382 (2017) 495. L.Q. Feng, W.L. Li, Laser Phys. 27 (2017) 016002.
[18] T. Zuo, S. Chelkowski, A.D. Bandrauk, Phys. Rev. A 48 (1993) 3837. [19] T. Zuo, A.D. Bandrauk, Phys. Rev. A 52 (1995) R2511. [20] R.E.F. Silva, P. Rivière, F. Morales, O. Smirnova, M. Ivanov, F. Martín, Sci. Rep. 6 (2016) 32653. [21] R.F. Lu, P.Y. Zhang, K.L. Han, Phys. Rev. E 77 (2008) 066701. [22] L.Q. Feng, T.S. Chu, J. Chem. Phys. 136 (2012) 054102. [23] L.Q. Feng, T.S. Chu, Phys. Rev. A 84 (2011) 053853. [24] L.Q. Feng, Phys. Rev. A 92 (2015) 053832. [25] J. Hu, K.L. Han, G.Z. He, Phys. Rev. Lett. 95 (2005) 123001. [26] M.D. Chen, K.L. Han, N.Q. Lou, Chem. Phys. Lett. 357 (2002) 483. [27] T.S. Chu, Y. Zhang, K.L. Han, Int. Rev. Phys. Chem. 25 (2006) 201. [28] K. Burnett, V.C. Reed, J. Cooper, P.L. Knight, Phys. Rev. A 45 (1992) 3347. [29] Y.W. Duan, W.K. Liu, J.M. Yuan, Phys. Rev. A 61 (2000) 053403. [30] S. Chelkowski, C. Foisy, A.D. Bandrauk, Phys. Rev. A 57 (1998) 1176. [31] M. Mohebbi, S. Batebi, Opt. Commun. 296 (2013) 113. [32] R.F. Lu, C.Y. Xiao, K.M. Deng, H.P. Wu, Chem. Phys. 382 (2011) 88. [33] P. Antoine, B. Piraux, A. Maquet, Phys. Rev. A 51 (1995) R1750.