- Email: [email protected]
The isotopic dependence and influence of driving laser intensity on the harmonic yields of diatomic molecule ions X2+
Chemical Physics Letters xxx (xxxx) xxxx
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Research paper
The isotopic dependence and influence of driving laser intensity on the harmonic yields of diatomic molecule ions X2+ ⁎
Hang Liu, Yi Li, Liqiang Feng
Laboratory of Molecular Reaction Dynamics, School of Chemical and Environmental Engineering, Liaoning University of Technology, Jinzhou 121001, China
H I GH L IG H T S
of harmonic yields from X . • Modulation motion effect on modulation of harmonic yields from X • Nuclear duration effect on harmonic yields from X . • Pulse intensity effect on harmonic yields from X . • Laser • Chirp effect on harmonic yields from X . + 2
2
+
.
+ 2 + 2
+ 2
A R T I C LE I N FO
A B S T R A C T
Keywords: High-order harmonic generation Harmonic yields from X2+ Diatomic molecule X2+ Isotope effect
The harmonic yields from X2+ (including H2+, D2+ and T2+) driven by 800 nm laser with different laser intensities and pulse durations have been studied. At shorter pulse duration, the harmonic yields of X2+ follows as H2+ > D2+ > T2+, and the intensity difference is decreased as laser intensity increases. At longer pulse duration, the harmonic yields of X2+ follows as T2+ > D2+ > H2+, and the intensity difference is increased as laser intensity increases. Furthermore, the harmonic yield from X2+ can also be modulated by laser chirps. Finally, the physical mechanism is displayed clearly in ionization probability, internuclear distance and timeprofile of harmonic emission.
Keywords: 42.65.Ky 42.65.Re 32.80.Fb
1. Introduction High-order harmonic generation (HHG), which is produced when matters are under the intense laser field, as an efficient tool in ultrafast electron dynamics and attosecond science has been widely investigated over 30 years [1–5]. For now, the HHG can be found from atoms, molecules and solids [6–8]. Generally, for atomic system, the HHG can be explained by the ionization, acceleration and recombination model, which is called three-step model [9,10]. While, for the molecular system, some novel and interesting phenomena can be found due to the additional degree of freedom. For instance, for the simplest diatomic molecule H2+ and its isotopes D2+ and T2+, Zuo et al. [11] found the effect of charge-resonance-enhanced-ionization (CREI) on the HHG of H2+. With the assistance of CREI, Wang et al. [12] and Feng et al. [13] theoretically propose a two-color pump-probe scheme to enhance the harmonic yield
⁎
of H2+ and its isotopes. Further, Liu et al. [14,15] analyzed and reported that both CREI and dissociative-state ionization are contributed to HHG from H2+ and its isotopes. Moreover, due to the effect of nuclear motion, the amplitude [16–20] and frequency [21–25] modulations of HHG from X2+ (including H2+, D2+ and T2+) can be found. Furthermore, using the different modulation of HHG from X2+, He et al. [26] monitored the ultrafast vibrational dynamics of H2+ and D2+. Furthermore, in recent 10 years, Lu, Zhang and Han developed a program, named 'LZH-DICP' code [27]. By using this code, they found many interesting phenomena of HHG in atomic, molecular and solids' systems [27–32]. Although the harmonic yields from X2+ has been investigated for some given laser conditions, the changes of laser conditions for the harmonic yields from X2+ have not been reported. Thus, in this paper, we investigate the harmonic yields from X2+ driven by a 800 nm laser field with different laser intensities, pulse durations and laser chirps.
Corresponding author. E-mail address: [email protected] (L. Feng).
https://doi.org/10.1016/j.cplett.2019.136965 Received 17 October 2019; Received in revised form 13 November 2019; Accepted 14 November 2019 0009-2614/ © 2019 Elsevier B.V. All rights reserved.
Please cite this article as: Hang Liu, Yi Li and Liqiang Feng, Chemical Physics Letters, https://doi.org/10.1016/j.cplett.2019.136965
Chemical Physics Letters xxx (xxxx) xxxx
H. Liu, et al.
We find that the harmonic yields of X2+ are very sensitive to the laser conditions and the nuclear motion is the key point to lead to the different harmonic yields from diatomic molecule X2+. This work provides the information of harmonic yields from X2+ when different lasers are adopted, which gives a reference to the future experimental and theoretical works. For instance, the different harmonic yields from X2+ can provide guidance for monitoring the dynamics of isotopic molecules. 2. Method The time-dependent Schrödinger equation of X2+ driven by intense ⎡TR + Te + V (z , R)+⎤ ∂ψ (z , R, t ) =⎢ laser is given by [27,32–36] i ∂t ⎥ ψ (z , R, t ) . 1 1 + 2m + 1 zE (t ) ⎥ ⎢ i ⎣ ⎦ Here, R (0 < R < 30 a.u.) and z (−100 a.u. < z < 100 a.u.) are the internuclear distance and electronic coordinate with the spatial steps of 2m + 1 ∂2 1 ∂2 ΔR = 0.1 a.u. and Δz = 0.02 a.u.. TR = − m 2 , Te = − 4im 2 and
(
)
i ∂R
i
∂z
V (z , R) = 1/ R − 1/ (z − R/2)2 + 1
are the nuclear, electronic kinetic − 1/ (z + R/2)2 + 1 energy operators and potential energy, respectively. mi (i = H, D and T) is the mass of H, D and T nucleus. The time-dependent wave function, Ψ(z, R, t), can be propagated using the standard second-order splitoperator method with time space of dt = 0.1 [27]. E (t ) = E exp[−4 ln(2) t 2/ τ 2] cos(ω1 t + δ (t )) is the driven laser field with
Fig. 1. The harmonic yields from X2+ driven by a 800 nm pulse with different pulse durations. The laser intensities are (a) I = 5.0 × 1014 W/cm2 and (b) I = 7.0 × 1014 W/cm2.
Fig. 2. The laser profiles, ionization probabilities, time-dependent internuclear distances and time-frequency analyses of HHG from X2+ driven by 10 fs-800 nm pulse with (a)–(d) I = 5.0 × 1014 W/cm2 and (e)–(h) I = 7.0 × 1014 W/cm2. 2
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Fig. 3. The laser profiles, ionization probabilities, time-dependent internuclear distances and time-frequency analyses of HHG from X2+ driven by 20 fs-800 nm with (a)–(d) I = 5.0 × 1014 W/cm2 and (e)-(h) I = 7.0 × 1014 W/cm2.
E, ω1, τ and δ(t) being the laser amplitude, laser frequency, pulse duration and chirp form of the laser field. In this paper, the laser frequency is fixed at 0.056 a.u. (800 nm pulse) and the other laser parameters are changed. 1 2π
The HHG spectrum can be expressed asS (ω) =
2
∫ a (t ) e−iωtdt ,
where, a (t ) = −〈ψ (z , R, t )|[∇V + E (t )]|ψ (z , R, t )〉. The harmonic yield comes from the average intensity of the last 20 harmonics. t
The ionization probability is given byIP (t ) =
Rs
∫ dt ′ ∫ dRj (R, z, t ) , 0
0
∂
where, j (R, z , t ) = Im[ψ ∗ δ (z − z 0) ∂z ψ],and Rs = 25 a.u. is the absorbing position [27]. The internuclear distance < R(t) > can be expressed as < R (t ) > = 〈ψ|R|ψ〉2 / 〈ψ|ψ〉2 .The time–frequency analyses of HHG is given by A (t , ω) = ∫ a (t ′) ω W (ω (t ′ − t )) dt ′, where
W (x ) = ⎛ ⎝
1 ξ
⎞ eix e−x ⎠
2
/2ξ 2 and
ξ = 18 a.u. [37].
3. Results and discussion Fig. 1(a) and (b) show the harmonic yields from X2+ driven by a 800 nm laser field with different laser intensities and pulse durations. Here, we define the laser intensities of 5.0 × 1014 W/cm2 and 7.0 × 1014 W/cm2 as lower and higher laser intensities, respectively. Also, the 10 fs and 20 fs laser pulses are defined as shorter and longer pulse durations, respectively. Clearly, at shorter pulse duration, the
Fig. 4. The harmonic yields from X2+ as a function of pulse duration of (a) down-chirped pulse (i.e. β = −0.002) and (b) up-chirped pulse (i.e. β = +0.002). The laser intensity is 5.0 × 1014 W/cm2. 3
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Fig. 5. The laser profiles, ionization probabilities, time-dependent internuclear distances and time-frequency analyses of HHG from X2+ driven by 10 fs-800 nm down-chirped pulse and (e)-(h) 20 fs-800 nm down-chirped pulse.
harmonic yields of X2+ follows as H2+ > D2+ > T2+, and the harmonic difference will be decreased as laser intensity increases. At longer pulse duration, the harmonic yields of X2+ nearly follows as T2+ (D2+) > D2+ (T2+) > H2+ (note here D2+ > T2+ > H2+ for the case of lower laser intensity and longer pulse duration), and the harmonic difference is increased as laser intensity increases. To understand the harmonic yields from X2+ driven by different laser fields, in Figs. 2 and 3, we give the laser profiles, ionization probabilities (IPs), time-dependent internuclear distances < R (t) > and time-frequency analyses of HHG driven by the above mentioned laser fields. Generally, the harmonic emission occurs at every half cycle of laser field. Thus, there are many harmonic emission peaks (HEPs) during the laser-X2+ driven time. For the cases of shorter pulse durations (Fig. 2), the intensities of HEPs on falling part of laser field are much higher than those on rising part of laser field. Thus, the harmonic yields of X2+ are mainly coming from the HEPs of P1~3, as shown in Fig. 2. As we know that the harmonic yield is sensitive to the ionization probability. Here, due to the larger ionization probability from light nucleus, the intensities of P1~3 from light nucleus (i.e. H2+) are much higher than those from heavy nucleus (i.e. D2+ and T2+), thus leading to the harmonic yields of X2+ following as H2+ > D2+ > T2+. What is the reason behind the different ionization probabilities from X2+. According to the investigations from Zuo et al. [11] and Liu et al. [14,15], we know that the ionization of X2+ (H2+,
D2+ and T2+) can be separated into three parts, such as the direct ionization at smaller internuclear distance; the CREI from R(t) = 4.0 a.u. to R(t) = 12.0 a.u.; and the dissociating ionization at larger internuclear distance. Moreover, the ionization probability can be remarkably enhanced in the CREI region. Thus, through analyzing the time-dependent internuclear distance of X2+ shown in Fig. 2(b)–(d), we see that due to the faster motion of light nucleus (i.e. H2+), the internuclear distance is larger than 4.0 a.u. on the falling part of laser field, where is in the CREI region. That is to say the larger ionization probability can be found in this region. On the contrary, the slower motion of heavy nucleus only leads to the smaller internuclear distance on the falling part of laser field (i.e. the internuclear distance is around 4.0 a.u. for D2+ and even smaller than 4.0 a.u. for T2+), which results in the smaller ionization probability in this region. As laser intensity increases, the internuclear distances of X2+ are all extended and enter into the CREI region, as shown in Fig. 2(f)–(h). That is to say the ionization probabilities of X2+ can all be enhanced. Moreover, the differences of ionization probabilities of X2+ are decreased in this region compared with those from lower laser intensity case. This is the reason behind the smaller intensity difference of harmonic yield from X2+. Therefore, we see that different nuclear motion is the key point for different ionizations and harmonic yields from X2+. For the case of longer pulse duration with lower laser intensity [Fig. 3(a)–(d)], the harmonic yields from X2+ are coming from the 4
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Fig. 6. The laser profiles, ionization probabilities, time-dependent internuclear distances and time-frequency analyses of HHG from X2+ driven by 10 fs-800 nm upchirped pulse and (e)–(h) 20 fs-800 nm up-chirped pulse.
HEPs of P1~4. The intensities of P1 and P2 are similar for H2+, D2+ and T2+. However, the intensities of P3 and P4 follow as D2+ > T2+ > H2+, which leads to the harmonic yields of X2+ following as D2+ > T2+ > H2+. Theoretical analyses show that, the HEPs of P1~4 are coming from t = 10 T to t = 12 T, where the internuclear distances of H2+, D2+ and T2+ are all in the CREI region. However, due to the faster nuclear motion of H2+, the ionization probability is advanced, which leads to the enhancement and reduction of harmonic emission efficiency on the rising and falling (i.e. the decrease of P3 and P4) parts of laser field, respectively, as shown in Fig. 3(b). For the cases of D2+ and T2+, due to the slower nuclear motion, the electron mainly ionizes around the middle part of laser field, and the larger ionization probability on the falling part of laser field can still lead to the high-intensity HEPs in this region, which is responsible for the higher intensities of P3 and P4 shown in Fig. 3(c) and (d). However, the much slower nuclear motion of T2+ leads to the smaller ionization probability and weaker HEPs. This is the main reason behind the different harmonic yields from X2+. At higher laser intensity [Fig. 3(e)–(h)], the harmonic yields are coming from P-3, P-2 and P1 for H2+; from P-3, P-2, P1 and P2 for D2+; and from P-3, P-2 and P1~3 for T2+, respectively, which results in the harmonic yield of X2+ following as T2+ > D2+ > H2+. Through analyzing ionization probabilities and time-dependent internuclear distances of X2+, we see that the faster nuclear motion of H2+ leads to the advance of ionization process
and the larger enhancement of ionization probability can be found from t = 8.5 T to t = 10.75 T [see Fig. 3(e)], where is the harmonic emission region of P-3, P-2 and P1. Thus, the intensities of P-3, P-2 and P1 are higher than the others for H2+. While, due to the slower and much slower nuclear motions of D2+ and T2+, the larger enhancement of ionization probability can be found from t = 8.5 T to t = 11.25 T for D2+ and from t = 8.5 T to t = 11.75 T for T2+ [see Fig. 3(e)], where are the harmonic emission regions of P-3, P-2, P1, P2 (for D2+) or P-3, P-2, P1~3 (for T2+), respectively. Thus, the intensities of P-3, P-2, P1, P2 for D2+or P-3, P-2, P1~3 for T2+are higher than the others, respectively. As can be seen, the slower nuclear motion in longer pulse duration leads to slower enhancement of ionization probability, which leads to more intense HEPs with higher photon energy and is the reason behind the harmonic yield of X2+ following as T2+ > D2+ > H2+. Next, we investigate the chirp effect of laser field on the harmonic yields from X2+. The chirp form is chosen to be δ(t) = βω1t2, where β is the chirp parameter of the laser field. It should be illustrated that the unit of the chirp parameter β is rad/s2. However, for convenience, the unit of chirp parameter is omitted in the following discussion. Here, we choose β = −0.002 and β = +0.002 as the down-chirped and upchirped pulses, respectively. The laser intensity is 5.0 × 1014 W/cm2. Fig. 4(a) and (b) show the harmonic yields from X2+ driven by the down-chirped and up-chirped pulses with different pulse durations. As can be seen, when the down-chirped pulse is used, the similar changes
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of harmonic yields from X2+ can be found, that is, H2+ > D2+ > T2+ for shorter pulse duration and D2+ > T2+ > H2+ for longer pulse duration. While, as up-chirped pulse is used, the harmonic yields from X2+ are nearly the same for different diatomic molecule X2+. Theoretical analyses show that as down chirp introduces, the decrease of instantaneous frequency on the falling part of laser field can lead to the extension of emission photon energy from P2 and P3 [38,39], as shown in Fig. 5. Thus, the harmonic cutoff region is coming from P2 and P3. Through analyzing the laser profile, we see that the intensities of P2 and P3 are very sensitive to the ionizations around t = 10.5 T and t = 11 T (near the middle part of laser field). However, the down chirp can only modulate the instantaneous frequency on the falling part of laser field (i.e. t > 11 T). That is to say these two points are nearly the same as chirp-free pulse case. Thus, harmonic yields from X2+ driven by down-chirped pulse will still follow as H2+ > D2+ > T2+ for shorter pulse duration and D2+ > T2+ > H2+ for longer pulse duration. As up chirp introduces, the decrease of instantaneous frequency on the rising part of laser field can lead to the extension of HEPs on rising part of laser field [38,39]. However, the smaller ionization probability on the rising part of laser field only leads to the very weak intensities of these HEPs, which can be ignored on the harmonic spectrum. Thus, the real harmonic cutoff region is contributed by P1 for the up-chirped pulse case. We see that although different nuclear motions of X2+ can lead to different ionization probabilities and HEPs. However, these differences are all on the falling part of laser field, and they are almost useless to P1. Therefore, the intensity of P1 is nearly the same for H2+, D2+ and T2+, which is responsible for the similar harmonic yield of X2+ when the up-chirped pulse is adopted (see Fig. 6).
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4. Conclusion In conclusion, we theoretically investigate the harmonic yields from H2+, D2+ and T2+ driven by different laser conditions. We find that the harmonic yield of X2+ is very sensitive to laser conditions. Through theoretical analyses, we see that the nuclear motion is the key point to result in the modulation of HHG from different diatomic molecule X2+.The results can be summarized into 4 parts. Firstly, for the cases of shorter pulse durations, the harmonic yields of X2+ follows as H2+ > D2+ > T2+, and the harmonic difference is decreased as laser intensity increases. Secondly, for the cases of longer pulse durations, the harmonic yields of X2+ follows as T2+ (or D2+) > D2+ (or T2+) > H2+, and the intensity difference is increased as laser intensity increases. Thirdly, when the down-chirped pulse is used, the similar changes of harmonic yields from X2+ can be found as those in chirp-free pulse case. Finally, when the up-chirped pulse is used, the harmonic yields from X2+ are nearly the same for different diatomic molecule X2+. This work provides the information of harmonic yield from X2+ when the different lasers are adopted, which will provide the reference to the future experimental and theoretical works.
Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgements This work was supported by the National Natural Science Foundation of China (Grants No. 11504151), Basic research project of Liaoning Provincial Education Department (Grants No. JJL201915405) and Liaoning Natural Science Foundation (Grants No. 2019-MS-167). 6
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H. Liu, et al. harmonic generation from H2+ in nonhomogeneous laser field, Opt. Exp. 24 (2016) 19736. [33] Y. Li, R.L.Q. Feng, Y. Qiao, Improvement of harmonic spectra from superposition of initial state driven by homogeneous and inhomogeneous combined field, Can. J. Phys. (2019), https://doi.org/10.1139/cjp-2019-0002. [34] L.Q. Feng, Molecular harmonic extension and enhancement from H2+ ions in the presence of spatially inhomogeneous fields, Phys. Rev. A 92 (2015) 053832. [35] L. Li, M. Zheng, R.L.Q. Feng, Y. Qiao, Waveform control in generations of intense water window attosecond pulses via multi-color combined field, Int. J. Mod. Phys. B 33 (2019) 1950130.
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Research paper
The isotopic dependence and influence of driving laser intensity on the harmonic yields of diatomic molecule ions X2+ ⁎
Hang Liu, Yi Li, Liqiang Feng
Laboratory of Molecular Reaction Dynamics, School of Chemical and Environmental Engineering, Liaoning University of Technology, Jinzhou 121001, China
H I GH L IG H T S
of harmonic yields from X . • Modulation motion effect on modulation of harmonic yields from X • Nuclear duration effect on harmonic yields from X . • Pulse intensity effect on harmonic yields from X . • Laser • Chirp effect on harmonic yields from X . + 2
2
+
.
+ 2 + 2
+ 2
A R T I C LE I N FO
A B S T R A C T
Keywords: High-order harmonic generation Harmonic yields from X2+ Diatomic molecule X2+ Isotope effect
The harmonic yields from X2+ (including H2+, D2+ and T2+) driven by 800 nm laser with different laser intensities and pulse durations have been studied. At shorter pulse duration, the harmonic yields of X2+ follows as H2+ > D2+ > T2+, and the intensity difference is decreased as laser intensity increases. At longer pulse duration, the harmonic yields of X2+ follows as T2+ > D2+ > H2+, and the intensity difference is increased as laser intensity increases. Furthermore, the harmonic yield from X2+ can also be modulated by laser chirps. Finally, the physical mechanism is displayed clearly in ionization probability, internuclear distance and timeprofile of harmonic emission.
Keywords: 42.65.Ky 42.65.Re 32.80.Fb
1. Introduction High-order harmonic generation (HHG), which is produced when matters are under the intense laser field, as an efficient tool in ultrafast electron dynamics and attosecond science has been widely investigated over 30 years [1–5]. For now, the HHG can be found from atoms, molecules and solids [6–8]. Generally, for atomic system, the HHG can be explained by the ionization, acceleration and recombination model, which is called three-step model [9,10]. While, for the molecular system, some novel and interesting phenomena can be found due to the additional degree of freedom. For instance, for the simplest diatomic molecule H2+ and its isotopes D2+ and T2+, Zuo et al. [11] found the effect of charge-resonance-enhanced-ionization (CREI) on the HHG of H2+. With the assistance of CREI, Wang et al. [12] and Feng et al. [13] theoretically propose a two-color pump-probe scheme to enhance the harmonic yield
⁎
of H2+ and its isotopes. Further, Liu et al. [14,15] analyzed and reported that both CREI and dissociative-state ionization are contributed to HHG from H2+ and its isotopes. Moreover, due to the effect of nuclear motion, the amplitude [16–20] and frequency [21–25] modulations of HHG from X2+ (including H2+, D2+ and T2+) can be found. Furthermore, using the different modulation of HHG from X2+, He et al. [26] monitored the ultrafast vibrational dynamics of H2+ and D2+. Furthermore, in recent 10 years, Lu, Zhang and Han developed a program, named 'LZH-DICP' code [27]. By using this code, they found many interesting phenomena of HHG in atomic, molecular and solids' systems [27–32]. Although the harmonic yields from X2+ has been investigated for some given laser conditions, the changes of laser conditions for the harmonic yields from X2+ have not been reported. Thus, in this paper, we investigate the harmonic yields from X2+ driven by a 800 nm laser field with different laser intensities, pulse durations and laser chirps.
Corresponding author. E-mail address: [email protected] (L. Feng).
https://doi.org/10.1016/j.cplett.2019.136965 Received 17 October 2019; Received in revised form 13 November 2019; Accepted 14 November 2019 0009-2614/ © 2019 Elsevier B.V. All rights reserved.
Please cite this article as: Hang Liu, Yi Li and Liqiang Feng, Chemical Physics Letters, https://doi.org/10.1016/j.cplett.2019.136965
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We find that the harmonic yields of X2+ are very sensitive to the laser conditions and the nuclear motion is the key point to lead to the different harmonic yields from diatomic molecule X2+. This work provides the information of harmonic yields from X2+ when different lasers are adopted, which gives a reference to the future experimental and theoretical works. For instance, the different harmonic yields from X2+ can provide guidance for monitoring the dynamics of isotopic molecules. 2. Method The time-dependent Schrödinger equation of X2+ driven by intense ⎡TR + Te + V (z , R)+⎤ ∂ψ (z , R, t ) =⎢ laser is given by [27,32–36] i ∂t ⎥ ψ (z , R, t ) . 1 1 + 2m + 1 zE (t ) ⎥ ⎢ i ⎣ ⎦ Here, R (0 < R < 30 a.u.) and z (−100 a.u. < z < 100 a.u.) are the internuclear distance and electronic coordinate with the spatial steps of 2m + 1 ∂2 1 ∂2 ΔR = 0.1 a.u. and Δz = 0.02 a.u.. TR = − m 2 , Te = − 4im 2 and
(
)
i ∂R
i
∂z
V (z , R) = 1/ R − 1/ (z − R/2)2 + 1
are the nuclear, electronic kinetic − 1/ (z + R/2)2 + 1 energy operators and potential energy, respectively. mi (i = H, D and T) is the mass of H, D and T nucleus. The time-dependent wave function, Ψ(z, R, t), can be propagated using the standard second-order splitoperator method with time space of dt = 0.1 [27]. E (t ) = E exp[−4 ln(2) t 2/ τ 2] cos(ω1 t + δ (t )) is the driven laser field with
Fig. 1. The harmonic yields from X2+ driven by a 800 nm pulse with different pulse durations. The laser intensities are (a) I = 5.0 × 1014 W/cm2 and (b) I = 7.0 × 1014 W/cm2.
Fig. 2. The laser profiles, ionization probabilities, time-dependent internuclear distances and time-frequency analyses of HHG from X2+ driven by 10 fs-800 nm pulse with (a)–(d) I = 5.0 × 1014 W/cm2 and (e)–(h) I = 7.0 × 1014 W/cm2. 2
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Fig. 3. The laser profiles, ionization probabilities, time-dependent internuclear distances and time-frequency analyses of HHG from X2+ driven by 20 fs-800 nm with (a)–(d) I = 5.0 × 1014 W/cm2 and (e)-(h) I = 7.0 × 1014 W/cm2.
E, ω1, τ and δ(t) being the laser amplitude, laser frequency, pulse duration and chirp form of the laser field. In this paper, the laser frequency is fixed at 0.056 a.u. (800 nm pulse) and the other laser parameters are changed. 1 2π
The HHG spectrum can be expressed asS (ω) =
2
∫ a (t ) e−iωtdt ,
where, a (t ) = −〈ψ (z , R, t )|[∇V + E (t )]|ψ (z , R, t )〉. The harmonic yield comes from the average intensity of the last 20 harmonics. t
The ionization probability is given byIP (t ) =
Rs
∫ dt ′ ∫ dRj (R, z, t ) , 0
0
∂
where, j (R, z , t ) = Im[ψ ∗ δ (z − z 0) ∂z ψ],and Rs = 25 a.u. is the absorbing position [27]. The internuclear distance < R(t) > can be expressed as < R (t ) > = 〈ψ|R|ψ〉2 / 〈ψ|ψ〉2 .The time–frequency analyses of HHG is given by A (t , ω) = ∫ a (t ′) ω W (ω (t ′ − t )) dt ′, where
W (x ) = ⎛ ⎝
1 ξ
⎞ eix e−x ⎠
2
/2ξ 2 and
ξ = 18 a.u. [37].
3. Results and discussion Fig. 1(a) and (b) show the harmonic yields from X2+ driven by a 800 nm laser field with different laser intensities and pulse durations. Here, we define the laser intensities of 5.0 × 1014 W/cm2 and 7.0 × 1014 W/cm2 as lower and higher laser intensities, respectively. Also, the 10 fs and 20 fs laser pulses are defined as shorter and longer pulse durations, respectively. Clearly, at shorter pulse duration, the
Fig. 4. The harmonic yields from X2+ as a function of pulse duration of (a) down-chirped pulse (i.e. β = −0.002) and (b) up-chirped pulse (i.e. β = +0.002). The laser intensity is 5.0 × 1014 W/cm2. 3
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Fig. 5. The laser profiles, ionization probabilities, time-dependent internuclear distances and time-frequency analyses of HHG from X2+ driven by 10 fs-800 nm down-chirped pulse and (e)-(h) 20 fs-800 nm down-chirped pulse.
harmonic yields of X2+ follows as H2+ > D2+ > T2+, and the harmonic difference will be decreased as laser intensity increases. At longer pulse duration, the harmonic yields of X2+ nearly follows as T2+ (D2+) > D2+ (T2+) > H2+ (note here D2+ > T2+ > H2+ for the case of lower laser intensity and longer pulse duration), and the harmonic difference is increased as laser intensity increases. To understand the harmonic yields from X2+ driven by different laser fields, in Figs. 2 and 3, we give the laser profiles, ionization probabilities (IPs), time-dependent internuclear distances < R (t) > and time-frequency analyses of HHG driven by the above mentioned laser fields. Generally, the harmonic emission occurs at every half cycle of laser field. Thus, there are many harmonic emission peaks (HEPs) during the laser-X2+ driven time. For the cases of shorter pulse durations (Fig. 2), the intensities of HEPs on falling part of laser field are much higher than those on rising part of laser field. Thus, the harmonic yields of X2+ are mainly coming from the HEPs of P1~3, as shown in Fig. 2. As we know that the harmonic yield is sensitive to the ionization probability. Here, due to the larger ionization probability from light nucleus, the intensities of P1~3 from light nucleus (i.e. H2+) are much higher than those from heavy nucleus (i.e. D2+ and T2+), thus leading to the harmonic yields of X2+ following as H2+ > D2+ > T2+. What is the reason behind the different ionization probabilities from X2+. According to the investigations from Zuo et al. [11] and Liu et al. [14,15], we know that the ionization of X2+ (H2+,
D2+ and T2+) can be separated into three parts, such as the direct ionization at smaller internuclear distance; the CREI from R(t) = 4.0 a.u. to R(t) = 12.0 a.u.; and the dissociating ionization at larger internuclear distance. Moreover, the ionization probability can be remarkably enhanced in the CREI region. Thus, through analyzing the time-dependent internuclear distance of X2+ shown in Fig. 2(b)–(d), we see that due to the faster motion of light nucleus (i.e. H2+), the internuclear distance is larger than 4.0 a.u. on the falling part of laser field, where is in the CREI region. That is to say the larger ionization probability can be found in this region. On the contrary, the slower motion of heavy nucleus only leads to the smaller internuclear distance on the falling part of laser field (i.e. the internuclear distance is around 4.0 a.u. for D2+ and even smaller than 4.0 a.u. for T2+), which results in the smaller ionization probability in this region. As laser intensity increases, the internuclear distances of X2+ are all extended and enter into the CREI region, as shown in Fig. 2(f)–(h). That is to say the ionization probabilities of X2+ can all be enhanced. Moreover, the differences of ionization probabilities of X2+ are decreased in this region compared with those from lower laser intensity case. This is the reason behind the smaller intensity difference of harmonic yield from X2+. Therefore, we see that different nuclear motion is the key point for different ionizations and harmonic yields from X2+. For the case of longer pulse duration with lower laser intensity [Fig. 3(a)–(d)], the harmonic yields from X2+ are coming from the 4
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Fig. 6. The laser profiles, ionization probabilities, time-dependent internuclear distances and time-frequency analyses of HHG from X2+ driven by 10 fs-800 nm upchirped pulse and (e)–(h) 20 fs-800 nm up-chirped pulse.
HEPs of P1~4. The intensities of P1 and P2 are similar for H2+, D2+ and T2+. However, the intensities of P3 and P4 follow as D2+ > T2+ > H2+, which leads to the harmonic yields of X2+ following as D2+ > T2+ > H2+. Theoretical analyses show that, the HEPs of P1~4 are coming from t = 10 T to t = 12 T, where the internuclear distances of H2+, D2+ and T2+ are all in the CREI region. However, due to the faster nuclear motion of H2+, the ionization probability is advanced, which leads to the enhancement and reduction of harmonic emission efficiency on the rising and falling (i.e. the decrease of P3 and P4) parts of laser field, respectively, as shown in Fig. 3(b). For the cases of D2+ and T2+, due to the slower nuclear motion, the electron mainly ionizes around the middle part of laser field, and the larger ionization probability on the falling part of laser field can still lead to the high-intensity HEPs in this region, which is responsible for the higher intensities of P3 and P4 shown in Fig. 3(c) and (d). However, the much slower nuclear motion of T2+ leads to the smaller ionization probability and weaker HEPs. This is the main reason behind the different harmonic yields from X2+. At higher laser intensity [Fig. 3(e)–(h)], the harmonic yields are coming from P-3, P-2 and P1 for H2+; from P-3, P-2, P1 and P2 for D2+; and from P-3, P-2 and P1~3 for T2+, respectively, which results in the harmonic yield of X2+ following as T2+ > D2+ > H2+. Through analyzing ionization probabilities and time-dependent internuclear distances of X2+, we see that the faster nuclear motion of H2+ leads to the advance of ionization process
and the larger enhancement of ionization probability can be found from t = 8.5 T to t = 10.75 T [see Fig. 3(e)], where is the harmonic emission region of P-3, P-2 and P1. Thus, the intensities of P-3, P-2 and P1 are higher than the others for H2+. While, due to the slower and much slower nuclear motions of D2+ and T2+, the larger enhancement of ionization probability can be found from t = 8.5 T to t = 11.25 T for D2+ and from t = 8.5 T to t = 11.75 T for T2+ [see Fig. 3(e)], where are the harmonic emission regions of P-3, P-2, P1, P2 (for D2+) or P-3, P-2, P1~3 (for T2+), respectively. Thus, the intensities of P-3, P-2, P1, P2 for D2+or P-3, P-2, P1~3 for T2+are higher than the others, respectively. As can be seen, the slower nuclear motion in longer pulse duration leads to slower enhancement of ionization probability, which leads to more intense HEPs with higher photon energy and is the reason behind the harmonic yield of X2+ following as T2+ > D2+ > H2+. Next, we investigate the chirp effect of laser field on the harmonic yields from X2+. The chirp form is chosen to be δ(t) = βω1t2, where β is the chirp parameter of the laser field. It should be illustrated that the unit of the chirp parameter β is rad/s2. However, for convenience, the unit of chirp parameter is omitted in the following discussion. Here, we choose β = −0.002 and β = +0.002 as the down-chirped and upchirped pulses, respectively. The laser intensity is 5.0 × 1014 W/cm2. Fig. 4(a) and (b) show the harmonic yields from X2+ driven by the down-chirped and up-chirped pulses with different pulse durations. As can be seen, when the down-chirped pulse is used, the similar changes
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of harmonic yields from X2+ can be found, that is, H2+ > D2+ > T2+ for shorter pulse duration and D2+ > T2+ > H2+ for longer pulse duration. While, as up-chirped pulse is used, the harmonic yields from X2+ are nearly the same for different diatomic molecule X2+. Theoretical analyses show that as down chirp introduces, the decrease of instantaneous frequency on the falling part of laser field can lead to the extension of emission photon energy from P2 and P3 [38,39], as shown in Fig. 5. Thus, the harmonic cutoff region is coming from P2 and P3. Through analyzing the laser profile, we see that the intensities of P2 and P3 are very sensitive to the ionizations around t = 10.5 T and t = 11 T (near the middle part of laser field). However, the down chirp can only modulate the instantaneous frequency on the falling part of laser field (i.e. t > 11 T). That is to say these two points are nearly the same as chirp-free pulse case. Thus, harmonic yields from X2+ driven by down-chirped pulse will still follow as H2+ > D2+ > T2+ for shorter pulse duration and D2+ > T2+ > H2+ for longer pulse duration. As up chirp introduces, the decrease of instantaneous frequency on the rising part of laser field can lead to the extension of HEPs on rising part of laser field [38,39]. However, the smaller ionization probability on the rising part of laser field only leads to the very weak intensities of these HEPs, which can be ignored on the harmonic spectrum. Thus, the real harmonic cutoff region is contributed by P1 for the up-chirped pulse case. We see that although different nuclear motions of X2+ can lead to different ionization probabilities and HEPs. However, these differences are all on the falling part of laser field, and they are almost useless to P1. Therefore, the intensity of P1 is nearly the same for H2+, D2+ and T2+, which is responsible for the similar harmonic yield of X2+ when the up-chirped pulse is adopted (see Fig. 6).
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4. Conclusion In conclusion, we theoretically investigate the harmonic yields from H2+, D2+ and T2+ driven by different laser conditions. We find that the harmonic yield of X2+ is very sensitive to laser conditions. Through theoretical analyses, we see that the nuclear motion is the key point to result in the modulation of HHG from different diatomic molecule X2+.The results can be summarized into 4 parts. Firstly, for the cases of shorter pulse durations, the harmonic yields of X2+ follows as H2+ > D2+ > T2+, and the harmonic difference is decreased as laser intensity increases. Secondly, for the cases of longer pulse durations, the harmonic yields of X2+ follows as T2+ (or D2+) > D2+ (or T2+) > H2+, and the intensity difference is increased as laser intensity increases. Thirdly, when the down-chirped pulse is used, the similar changes of harmonic yields from X2+ can be found as those in chirp-free pulse case. Finally, when the up-chirped pulse is used, the harmonic yields from X2+ are nearly the same for different diatomic molecule X2+. This work provides the information of harmonic yield from X2+ when the different lasers are adopted, which will provide the reference to the future experimental and theoretical works.
Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgements This work was supported by the National Natural Science Foundation of China (Grants No. 11504151), Basic research project of Liaoning Provincial Education Department (Grants No. JJL201915405) and Liaoning Natural Science Foundation (Grants No. 2019-MS-167). 6
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