1. Introduction

High-order harmonic generation (HHG) through the laser-matter interaction has been a topic of great interest over the last two decades [1] for its potential applications as a tabletop source of XUV and soft x-ray emission. The underlying physics of HHG can be understood by a semiclassical three-step model [2,3]. The maximal energy of the emitted harmonic photon obeys the cutoff law: I_p + 3.17I₀λ²/4, where I_p is the ionization potential of the atom, I₀ and λ are the intensity and the wavelength of the laser pulse, respectively. Nowadays, HHG has become an important approach to generate attosecond pulses [4–9] and also provides a novel tool to real-time probe electron dynamics in atoms and molecules [10–13] and to tomographically image molecular orbital [14–19].

However, these potential applications of HHG is still limited by two factors: bandwidth and yield of the harmonic spectrum. Since a broader harmonic spectrum could benefit the generation of shorter attosecond pulses and also provide an opportunity for detecting core-to-valence transitions [20], considerable effort has been paid to extend the cutoff energy. According to the cutoff law, one straightforward way for cutoff extension is increasing the intensity or wavelength of the laser pulse [21–23]. However, increasing the intensity may cause the ground state depletion (i.e., the saturation of the ionization yield) and the phase-mismatch induced by the dispersion of free electrons, which will suppress the HHG yield. While increasing the wavelength is demonstrated to accompany with a dramatic decrease of HHG yield [22]. Consequently, some other schemes for cutoff extension are studied, such as the use of two-color, multi-color fields [24–26] and plasmon-enhanced fields [27]. On the other hand, increasing harmonic yield is helpful to improve the signal-to-noise ratio(S/N) and then to achieve better spatio-temporal resolution of ultrafast measurement. In order to increase the high-harmonic yield, some elaborate schemes have also been proposed, e.g., using the ionization gating [28], UV-assisted fields [29] and Rydberg states [30].

Recently, molecular high-order harmonic generation (MHHG) has attracted much attention. Comparing to HHG in atoms, some new phenomena are discovered in MHHG. An important example is the structural and dynamical minimums due to the multi-center structure [31–33] and multi-orbital dynamics [34–37]. Another example is the modulation in harmonic amplitude [38–40] and frequency [41] resulting from the nuclear motion. Zuo and Bandrauk demonstrated that the charge-resonance-enhanced ionization (CREI) will occur at a large internuclear distance [42,43] and then lead to efficient harmonic emission [44–46]. However, Lein et al. reported that the harmonic cutoff of ${\mathrm{H}}_2^{+}$ will be reduced due to the nuclear motion [38]. To overcome this issue, Lara-Astiaso et al. proposed to enhance the harmonic cutoff in ${\mathrm{H}}_2^{+}$ by using a negatively-chirped laser pulse [47]. Yet, in their scheme, the Fourier-limited duration of the chirped laser pulse is less than one optical cycle, which is extremely difficult to obtain experimentally.

In this work, we propose a two-pulse scheme to overcome the suppression of HHG yield near the cutoff region. In our scheme, a multi-cycle 800-nm pump pulse is employed to steer the ${\mathrm{H}}_2^{+}$ to a larger internuclear distance where the ionization is enhanced. Then a sequential 1600-nm probe field irradiates the ${\mathrm{H}}_2^{+}$ to generate harmonics. By solving the time-dependent Schrödinger equation (TDSE), we find that in our scheme the harmonic cutoff is extended and the harmonic yield is enhanced by about 2–3 orders of magnitude compared to that in the probe pulse alone. The harmonic yield can be further enhanced by increasing the intensity of the pump pulse. In addition, our scheme is demonstrated to work well in a wide range of the time delay between these two sequential pulses. By using a probe laser pulse with longer wavelength, our scheme can be used for efficient harmonic emission in the water window region.

2. Theoretical model

We investigate the electronic motion inside ${\mathrm{H}}_2^{+}$ by numerically solving the time-dependent Schrödinger equation, which consists of one-dimensional motion of nuclei and one-dimensional motion of electron [48, 49]. In our model, the ${\mathrm{H}}_2^{+}$ molecular ion is assumed to be aligned along the polarization direction of the linearly polarized laser pulses. Rotational motion of the molecules and transversal spreading of electronic wavefunction are not included. The TDSE is expressed as (Hartree atomic units are used throughout):

Here, R is the internuclear distance, z is the electron position measured from the center-of-mass of two protons, μ = m_p/2 and μ_e = 2m_p/(2m_p + 1) are the reduced masses (m_p is the mass of the proton). α = 1 and β = 0.03 are the soft-core parameters. In the dipole approximation, the length-gauged laser-molecule interaction is V (t) = [1 + 1/(2m_p + 1)]zE(t). E(t) is the electric field of two sequential pulses, which is given by

Here, E₀, E₁ are the electric field amplitudes, τ, τ₁ are the pulse durations, ω, ω₁ are the central frequencies of the probe pulse and the pump pulse, respectively. Δt is the time delay between them and the initial phase ϕ₀ is set to π. We solve Eq. (1) by using the Crank-Nicolson method. To eliminate artificial reflections from boundaries, the wave function is multiplied by a sin^1/6− masking functions at each time step. The initial state is the ground state 1sσ_g of ${\mathrm{H}}_2^{+}$, which is obtained by using imaginary time propagation method of the field-free TDSE. Its energy is −0.78 a.u. and the equilibrium internuclear distance is $R_e=\frac{〈{\psi }_0|R|{\psi }_0〉}{〈{\psi }_0|{\psi }_0〉}=2.6\mathrm{a}.\mathrm{u}.$ (ψ₀ is the initial wavefunction). Harmonic spectrum is obtained by the Fourier transformation of the dipole acceleration $a(t)=〈\psi (t)|-\frac{\partial V_0}{\partial z}+E(t)|\psi (t)〉$. For comparison, we also solve Eq. (1) with the internuclear distance fixed at R_e, which is called the static case in the following.

3. Result and discussion

We first investigate the role of nuclear motion in HHG from ${\mathrm{H}}_2^{+}$ by using a 5-cycle, 1600-nm laser pulse with the peak intensity of 2.5×10¹⁴W/cm² [shown in Fig. 1(a)]. Figure 1(b) presents the harmonic spectrum (solid line) obtained by solving the TDSE in non-BO treatment (moving case). For comparison, the result in the fixed-nuclei approximation (static case) is also plotted as dashed line in Fig. 1(b). One can see that the harmonic yield below 117th order (H1-H117) for the moving ${\mathrm{H}}_2^{+}$ is greater than that for the static ${\mathrm{H}}_2^{+}$, which is in agreement with previous result [41]. This phenomenon can be well understood by the ionization enhancement induced by the nuclear motion. In detail, in the static case, the nuclear wave packet is restricted to the Franck-Condon (FC) region (i.e., at the equilibrium internuclear distance), where the ionization rate is inappreciable due to the large ionization threshold [see Procedure 1 in Fig. 2]. While in the moving case, the nuclear wave packet forced by the laser field could go beyond the FC region and move to larger internuclear distances, where the ionization rate is higher than that at the equilibrium internuclear distance due to the smaller ionization potential [see Procedure 2 in the Fig. 2]. This can be confirmed by the internuclear distance of the moving ${\mathrm{H}}_2^{+}$ in Fig. 1(c) (solid line), which has reached a peak of 3.6 a.u. at 4T₀ (T₀ is the optical cycle of the 1600-nm probe pulse). Apart from the CREI effect (bond-length dependent ionization rate), the two-center interference effect also plays an important role in the HHG yield of ${\mathrm{H}}_2^{+}$. According to the two-center interference model [31–33], one can expect the 0- and 1-order destructive interferences for H25-H53 and H229-H259, as well as the 1-order constructive interference for H103-H131 in the static case. While in the moving case, the 0-order interference minimum will be significantly smeared off due to the rapid change of the internuclear distance. One can only observe the 1-order constructive and destructive interferences at H53-H81 and H119-H149 when the internuclear distance is already close to the maximum value R=3.6 a.u. These expected extrema can be clearly seen in the harmonic spectra shown in Fig. 1(b). It must be emphasized that the two-center interference can only influence the HHG yield in the vicinity of interference extrema and will not affect that far from the extrema. However, as presented in Fig. 1(b), the whole harmonic spectrum in the moving ${\mathrm{H}}_2^{+}$ is enhanced by 1–2 orders of magnitude when compared to that in the static case. This result indicates that the enhancement of HHG in our work is mainly due to the CREI effect.

 

Fig. 1 (a) Electric field of the 1600-nm probe laser pulse. The intensity and duration are 2.5×10¹⁴W/cm² and 5 optical cycles, respectively. (b) Harmonic spectra of ${\mathrm{H}}_2^{+}$ obtained by solving the TDSE in non-BO treatment (red solid line) and in fixed-nuclei approximation (blue dashed line). (c) Time-frequency analysis of the harmonic spectrum in the moving ${\mathrm{H}}_2^{+}$ [red solid line in Fig. 1(b)] and the time-dependent internuclear distance 〈R〉 of ${\mathrm{H}}_2^{+}$. (d) Time-frequency analysis of spectrum in the static ${\mathrm{H}}_2^{+}$ [blue dashed line in Fig. 1(b)].

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Fig. 2 Mechanism of HHG in ${\mathrm{H}}_2^{+}$. Procedure 1: Ionization at the equilibrium internuclear distance (in the FC region). Procedure 2: Evolution of the nuclear wave packet and the ionization forced by the 1600nm probe pulse alone. Procedure 3′-3″: Evolution of the nuclear wave packet driven by a multi-cycle 800-nm pump pulse (procedure 3′) and the ionization driven by a sequential 1600-nm probe pulse (procedure 3″).

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On the other hand, one can see that the harmonic cutoff for the moving ${\mathrm{H}}_2^{+}$ is considerably reduced compared to that for the static ${\mathrm{H}}_2^{+}$. According to the cutoff law, the harmonic cutoff energy in our simulation should be 220 eV (corresponding to 285th harmonic), which is in good agreement with the static ${\mathrm{H}}_2^{+}$ and far larger than that for the moving case. To clarify the physics underlying this difference, we have performed the time-frequency analysis for the harmonic spectra and the results are shown in Figs. 1(c) and 1(d). One can see that, for the moving ${\mathrm{H}}_2^{+}$ [Fig. 1(c)], the dominant harmonics are emitted at 3.2–4.0T₀, where the internuclear distance reaches its maximum (3.6 a.u.). In this range, the harmonics are generated by the electrons accelerated at the second peak of the probe pulse on the falling part [marked by P₂ in Fig. 1(a)]. Its effective amplitude is about 0.0557 a.u., which corresponds to the harmonic cutoff (133rd harmonic). While for the static ${\mathrm{H}}_2^{+}$, the harmonics with cutoff energy is mainly emitted at 2.4–2.9T₀, where the electrons are accelerated by the highest peak of the probe field [P₁ in Fig. 1(a)]. Therefore, the harmonic cutoff energy of the static ${\mathrm{H}}_2^{+}$ is higher compared to the moving case.

In the following, we demonstrate a method to extend the harmonic cutoff of the moving ${\mathrm{H}}_2^{+}$ by using two sequential laser pulses to manipulate the nuclear motion. The basic idea of our scheme is sketched by procedures 3′ and 3″ in Fig. 2. In this scheme, we first employ a multi-cycle pump pulse to steer the molecular nuclei dissociation and prepare the ${\mathrm{H}}_2^{+}$ with the internuclear distance well outside the FC region (procedure 3′). Sequentially, a long-wavelength probe pulse interacts with the dissociative ${\mathrm{H}}_2^{+}$ to produce high harmonics (procedure 3″). In this way, the induced CREI is helpful for generating an efficient broadband harmonic spectrum in the sequential probe pulse. Our scheme is demonstrated by using a 15-cycle, 800-nm pump pulse in combination with a 5-cycle, 1600-nm probe pulse. The laser intensities of these two pulses are 2.0×10¹⁴W/cm² and 2.5×10¹⁴W/cm², respectively. The time delay between the two pulses is chosen as Δt = 4.75T₀. The synthesized laser field is presented in Fig. 3(a). Figure 3(b) shows the harmonic spectrum (solid line) generated in these two sequential laser pulses. For comparison, the harmonic spectrum generated with the 1600-nm probe pulse alone is also presented (dashed line). One can see that in our two-pulse scheme, the harmonic cutoff is extended to 200.9 eV (259th harmonic), and the harmonic intensity is about 2–3 orders of magnitude higher than that in the 1600-nm probe pulse alone.

 

Fig. 3 (a) Electric field (red solid line) synthesized by the 800-nm pump pulse (black dash-dotted line) and the sequential 1600-nm probe pulse (blue dashed line). (b) Harmonic spectra of ${\mathrm{H}}_2^{+}$ in the synthesized laser field (red solid line) and in the 1600-nm probe pulse alone (blue dashed line).

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To demonstrate the role of the 800-nm pump pulse in our two-pulse scheme, we have plotted in Fig. 4(a) the time-frequency analysis of the harmonic spectrum as well as the time-dependent internuclear distance 〈R〉 (solid line) driven by the two sequential laser pulses. The time-dependent internuclear distance 〈R〉 for 1600-nm probe pulse alone is also presented as the dashed line for comparison. It’s obvious that the highest harmonic in the two sequential laser pulses is generated at 7.5T₀, which is accelerated at the highest peak of the 1600-nm probe field. The highest harmonic in the 1600-nm probe pulse alone is accelerated at the second highest peak of the laser field on the falling part [P2 in Fig. 1(a)]. Consequently, the cutoff energy in the two sequential laser pulses is much higher than in the 1600-nm probe pulse alone. On the other hand, due to the excitation of the 800-nm pump pulse, the internuclear distance of ${\mathrm{H}}_2^{+}$ is effectively stretched to 3.7 a.u. (far beyond the FC region) at the beginning of the 1600-nm probe pulse (t = 6T₀). Within the duration of the 1600-nm probe pulse (6–9.75T₀), the internuclear distance 〈R〉 is maintained above 3.5 a.u., which is larger than that in the 1600-nm pulse alone. Hence, the ionization rate as well as the harmonic yield are substantially increased in our scheme as shown in Fig. 3(b). For further check, we have calculated the time evolution of the electron probability density ρ(z; t) = ∫ ψ(R, z; t)^*ψ(R, z; t)dR in the two sequential laser pulses in Fig. 4(b). One can see that the ionization is negligible for t < 4T₀. This is because the ${\mathrm{H}}_2^{+}$ is still in the FC region, where the ionization rate is low due to relatively large ionization potential. The ionization of electron mainly occurs at 6.15–8.25T₀, i.e., within the duration of the 1600-nm probe pulse, where the internuclear distance is large and therefore the ionization rate is enhanced due to the CREI mechanism. This result is in good agreement with the time-frequency analysis in Fig. 3(a). Here, it’s worth noting that at the tail of the 800-nm pump pulse (4.0–6.0T₀), the internuclear distance has gone beyond the FC region and partial harmonics are generated. However, in this range, the maximal harmonic energy is only 78.3 eV (about 101st harmonic), which is far smaller than that driven by the sequential 1600-nm probe pulse.

 

Fig. 4 (a) Time-frequency profile of harmonic spectrum in the synthesized two-pulse field, and the time-dependent internuclear distance 〈R〉 in the synthesized field (purple solid line) and in the 1600-nm probe field alone (black dashed line). (b) Time evolution of the electron probability density in the synthesized two-pulse field (black solid line).

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We next investigate the influence of the intensity of the 800-nm pump pulse on HHG in this two-pulse scheme. Figure 5(a) shows the spectra with the intensity of the 800-nm pump pulse varying from 2.0 to 3.0×10¹⁴W/cm². Other parameters are the same as in Fig. 4. For comparison, the harmonic spectrum in the 1600-nm probe pulse alone is also presented. One can see that the harmonic spectrum is gradually enhanced as the intensity of the 800-nm pump pulse is increased. Because a higher pump intensity will force the nuclear wavepacket to larger internuclear distances [see Fig. 5(b)] and then lead to higher ionization during the 1600-nm probe pulse. However the HHG yield is saturated when the intensity of pump pulse reaches 3.0×10¹⁴W/cm². This is because in this case the internuclear distance R has gone beyond the critical distance (~7 a.u.) of CREI [42]. Here, it must be emphasized that to achive the enhancement of HHG, the intensity of the probe pulse in our scheme should be kept well below the ionization saturation level (even in the CREI case), since too strong ionization because of CREI will lead to serious ground-state depletion and phase mismatch, and then suppress the HHG yield. Moreover, the influence of the time delay between the two pulses on HHG in ${\mathrm{H}}_2^{+}$ is also investigated. As shown in Fig. 6, our two-pulse scheme is robust for the harmonic cutoff extension as well as the harmonic yield enhancement with the time delay changing from 3.75 to 5.75T₀.

 

Fig. 5 (a) Harmonic spectra of ${\mathrm{H}}_2^{+}$ in the two-pulse field with different intensities of 800-nm pump pulse. (b) Corresponding time-dependent internuclear distance 〈R〉 of ${\mathrm{H}}_2^{+}$.

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Fig. 6 Harmonic spectra of ${\mathrm{H}}_2^{+}$ in two-pulse field with different time delays (solid lines). For comparison, the spectrum in the 1600-nm probe pulse alone is also presented (dashed line).

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To further increase the photon energy of the harmonic cutoff, we consider employing a probe pulse with longer wavelength in our scheme. Figure 7(a) shows the harmonic spectrum (red solid line) obtained by using a 2400-nm probe pulse in combination with the 800-nm pump pulse. The intensity of the 2400-nm pulse is 2.5×10¹⁴W/cm² and the time delay Δt is set to be 2.5 ${\mathrm{T}}_0^{\prime }$ (${\mathrm{T}}_0^{\prime }$ is the optical cycle of the 2400-nm laser pulse). The result with the 2400-nm pulse alone is also presented (blue dashed line) for comparison. One can see that the harmonics in the sequential pulses is extend to 658th harmonic (338.7 eV), which has reached into the water window (284~530 eV). Moreover, the harmonic yield in the two-pulse field is significantly increased. This is due to the large internuclear distance 〈R〉 (above 3.7 a.u.) in the duration of the 2400-nm pulse [see solid line in Fig. 7(b)], which gives rise to the CREI and therefore enhances the HHG efficiency.

 

Fig. 7 (a) Harmonic spectrum of ${\mathrm{H}}_2^{+}$ in the two-pulse field synthesized by a 800-nm pump pulse and a sequential 2400-nm probe pulse (red solid line). For comparison, the harmonic spectrum in the 2400-nm pulse alone is also plotted (blue dashed line). (b) Time-frequency profile of the harmonic spectrum in the synthesized two-pulse field. Red solid and blue dashed lines are the time-dependent internuclear distance 〈R〉 of ${\mathrm{H}}_2^{+}$ in the synthesized two-pulse field and in the 2400-nm probe pulse alone, respectively.

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4. Conclusion

In conclusion, HHG of ${\mathrm{H}}_2^{+}$ in two sequential laser pulses is investigated. Under the control of a multi-cycle 800-nm pump pulse, the ${\mathrm{H}}_2^{+}$ is stretched well outside the FC region. Thus the induced CREI benefits the generation of an efficient broadband harmonic spectrum. Our simulations of the TDSE show that the harmonic cutoff is extended to 209.6 eV (270th harmonic) and the harmonic yield is enhanced by 2–3 orders of magnitude compared to that in the 1600-nm probe pulse alone. By increasing the intensity of the 800-nm pump pulse, the harmonic yield can be further increased. The effect of the time delay between the pump and probe pulses is also investigated. Our scheme is demonstrated to work well with the time delay within a broad range of 3.75–5.75T₀. Moreover, by using a longer wavelength of the probe pulse, efficient harmonic emission in the the water window region can be achieved. The proposed two-pulse scheme can effectively prevent the cutoff suppression of HHG in ${\mathrm{H}}_2^{+}$ and provide a promising way for generating a broadband harmonic spectrum with high efficiency.