Ellipticity dependence of the near-threshold harmonics of H<sub>2</sub> in an elliptical strong laser field

Hua Yang; Peng Liu; Ruxin Li; Zhizhan Xu

doi:10.1364/OE.21.028676

1. Introduction

High-order harmonic generation (HHG) occurs when atoms or molecules are exposed to strong laser fields, emitting a spectrum from the extreme ultraviolet to the keV region [1]. HHG can be understood by the three-step model [2] which includes tunneling ionization, acceleration of electrons, and recombination. A quantum presentation of this model under strong-field approximation (SFA) [3] has also been developed. When the intense laser field is elliptically polarized, the orthogonal polarized component of the electric field can push the free electrons away from the parent ions, thus suppress the recombination and HHG yield. It has been demonstrated that the ellipticity dependence of HHG can be used to probe the wavepacket of the recombining free electrons both in experiments [4–6] and by calculations [7, 8].

However, for the harmonics with the photon energy near the ionization potential, i.e., the near-threshold (NT) harmonics, anomalous ellipticity dependence has been observed in atoms [9–11] as well as aligned molecules [12]: As the ellipticity increases, the intensity of some of the NT harmonics shows an anomalous maximum (AM) at a non-zero ellipticity. Comparing with the harmonic emission driven by the linearly polarized intense laser field, an enhancement of nearly one order in magnitude for the AM has been observed [9, 12]. More properties of the AMs such as alignment dependence have also been investigated and reported in the experiment of O₂ [12].

It is known that SFA is only effective for the harmonics well above the ionization threshold, and the origin of the NT harmonics goes beyond the three-step model [13]. For the generation of NT harmonics, the bound state may play an important role in their ellipticity dependence, which has been proposed in the previous discussions on multiphoton resonance [12], multi-wave-mixing progress [11], and Coulomb effects [10]. Numerically, through solving the time-dependent Schrödinger equation (TDSE), calculations on the interaction of elliptically polarized with atoms were carried out but only focusing on the high harmonics [14–16]. However, a complete theoretical method that can explain the AM of the threshold harmonics is still desirable. Difficulties include the imprecision of the soft-core potential and single-electron approximation, which neglect the electron-electron effects and hinder the exact calculation of bound states, especially for molecules.

In this paper we perform an ab initio calculation on the HHG of aligned H₂ molecules by means of time-dependent density functional theory (TDDFT), where the ionic potential and the electron-electron effects are attentively treated. AMs exist in the NT harmonics, which is consistent with the reported experimental observations on atoms [9–11] and O₂ molecules [12]. The AMs can be attributed to the system-dependent multiphoton effects of the orthogonally polarized component of the elliptical driving laser field. Our calculation shows that the structure of the bound-state, such as molecular alignment and bond length, can be sensitively reflected on the ellipticity dependence of the near-threshold harmonics. Thus information of the multiphoton process as well as the dynamical features of the bound states in strong laser fields can be got from study of the near-threshold harmonics, which can also offer better understanding of the HHG mechanisms and knowledge of extending the HHG spectroscopy to the threshold region.

2. Theoretical method

The three-dimensional TDDFT calculation is performed by propagating the molecular wavefunctions under the time-dependent Kohn-Sham (KS) equation,

i \frac{\partial}{\partial t} ψ_{i} (r, t) = [- \frac{1}{2} \nabla^{2} + v_{eff} (r, t)] ψ_{i} (r, t), i = 1, 2, ..., N .

where N is the total number of the KS orbitals. As the KS orbital of H₂ is double-occupied, we do not specify the spin-orbitals throughout the propagation. Since N = 1 for H₂, no unoccupied orbital is introduced, and no radiation from the transition of excited states is included. v_eff is the time-dependent effective potential, which can be written in the general form:

v_{eff} (r, t) = \int \frac{ρ (r', t)}{| r - r' |} d r' + v_{xc} (r, t) + v_{ps} + E (t) \cdot r,

The first term is the classical Hartree potential that describes the electron-electron interaction and ρ is the electron density. The second term is the exchange-correlation potential including all non-trivial many body effects. The Leeuwen-Baerends functional [17], whose accuracy has been extensively benchmarked in many calculations of HHG [18–21], is employed in the second term. The last two terms account for the potential due to the interaction of electrons with nuclei and external laser field, respectively. The nuclear distance of the molecule is fixed during the propagation. v_ps gives the ionic potential modeled by the norm-conserving, nonlocal Troullier-Martines pseudopotential [22]. The potential of the laser field is expressed using the dipole approximation and the length gauge. The calculation of the ground state and the propagation of the KS orbital are carried out using the real-space code Octopus [23].

The harmonic spectrum in each polarization direction is obtained from Fourier transformation of the dipole acceleration calculated using Ehrenfest’s theorem [24], which has been suggested as the most accurate method for calculating HHG when dealing with molecular systems [25]:

H_{p} (ω) \propto {| a_{p} (ω) |}^{2} = {| \int {\ddot{d}}_{p} (t) e^{i ω t} d t |}^{2},

where p presents the polarization direction along x or z in our calculation.

The elliptically polarized laser field is described as

E (t) = E_{0} f (t) {(\frac{1}{1 + ε^{2}})}^{1 / 2} [e_{z} \sin (ω t) + e_{x} ε \sin (ω t - π / 2)],

where E₀ stands for the amplitude of the electric field, f (t) is the laser envelope, and e_x and e_z are unit vectors. We choose the peak intensity of 1.5 × 10¹⁴ W/cm², the wavelength of 800 nm, the time-step of dt = 0.02 a.u. (atomic units) and the ellipticity of 0 ≤ ε ≤ 0.5. A 9-cycle, trapezoidal envelope, which stays constant for 6 cycles between a 1.5-cycle linear ramp and a 1.5-cycle linear decay, as shown in Fig. 1, is used. The middle cycles correspond to most of the HHG yields, and prevent the amplitude of the orthogonal laser component from cycle-to-cycle change.

Fig. 1 Amplitude of the z- and x- polarization component of the elliptical laser field at ε = 0.2.

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The molecule is placed in a three-dimensional rectangular grid cell of |x_max| = 20 a.u. (1 a.u. = 0.0529 nm) and |y_max| = 15 a.u.. The size of the |z_max| is chosen according to the electron excursion distance. Classically the shortest excursion distance of the electrons for the cut-off harmonic is z_cut = 1.13 E_z /ω² ~23 a.u., while the maximum excursion distance of the long trajectory is z_long = 2E_z /ω² ~40 a.u.. Since the long trajectory can affect the ellipticity dependence of the near-threshold harmonics and no AM was observed in experiments for the long path [12], we take |z_max| = 29 a.u., which is larger than z_cut but shorter than z_long, thus can roughly block out the long trajectory [15]. The electron wavepackets originally in long trajectories may return and reach the nucleus area, by which the molecules are prematurely ionized. However, the influence to the bound state is still small, because of the lateral diffusion of electron wavepackets during the long excursion. Thus the premature ionization is not important for the bound state. The grid space of dx = dy = dz = 0.4 a.u. is used in the ground state calculation and the time-propagation. For H₂ whose equilibrium bond length is ~1.4 a.u., the calculated ionization energy is 15.01 eV, which gives a small difference compared with the experimental result of 15.37 eV [26]. A bonus imaginary potential of Δr = 6 a.u. and V₀ = 0.6 a.u. outside the grid cell in each dimension is appended to absorb the reflecting wavefunction in the boundary region [27]:

V (r) = {\begin{cases} 0 for 0< r < R, \\ - i V_{0} \sin^{1 / 8} [\frac{π (r - R)}{2 Δ r}] for R < r < R + Δ r . \end{cases}

3. Results and discussions

In our calculation the nuclei of H₂ are fixed with the equilibrium distance of 1.4 a.u. as well as the elongated distance of 2.2 and 3.0 a.u.. Different interatomic distances are considered thus different bound-state structures and ionization potenticals are employed. For the case of such elongated H₂, the Leeuwen-Baerends functional used in our calculation has been demonstrated to give accurate ionization energies and HHG yields [21]. We first discuss the results for the bond length of 2.2 a.u., which is close to the bond length of O₂ (~2.28 a.u.) and gives representative results. The harmonics are also distinguished under perpendicular and parallel alignment of molecules, where the polarization direction of the major laser component (E_z) is perpendicular or parallel to the molecular axis, respectively. The ionization energy of the KS orbital calculated at the bond length of 2.2 a.u. is ~12.9 eV, corresponding to the harmonic order of 8. Harmonic spectrum at zero ellipticity for H₂ at the bond length of 2.2 a.u. under both alignments is given in Fig. 2. It is noteworthy that the harmonics beyond cutoff (~H33) may be generated from the imperfect absorption of the absorbing boundaries [28], which gives little influence to the lower harmonics in our calculations.

Fig. 2 Harmonic spectrum at zero ellipticity for H₂ at the bond length of 2.2 a.u. under both alignments.

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Figure 3 shows the calculated intensity of the generated harmonics (H_x + H_z) from the perpendicularly and parallelly aligned H₂ molecules, where the polarization direction of E_z is perpendicular or parallel to the molecular axis, respectively. Each harmonic is normalized to its intensity at ε = 0. The AMs can be clearly distinguished in the NT harmonics. For the perpendicular alignment [Fig. 3(a)], H5 as well as H7 reaches to the maximum intensity at ε ~0.2. For the parallel alignment [Fig. 3(b)], an AM appears at ε ~0.1 for H11. No AM is found in the harmonics well above the threshold. Instead, they exhibit a near-Gaussian ellipticity dependence with a constant width due to quantum effects of the tunnelling process, as reported previously [29]

Fig. 3 Normalized harmonic intensity as a function of ellipticity for perpendicularly (left) and parallelly (right) aligned H₂ at the bond length of 2.2 a.u.. Each harmonic is normalized to its intensity at zero ellipticity.

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Figure 4 plots the z-component of the harmonics (H_z) under perpendicular and parallel alignments, respectively. Similar ellipticity dependence for all harmonics for both alignment conditions is observed and no AM can be found. As one can see, the lower harmonics are more insensitive to the ellipticity than the near-cutoff harmonics. This phenomenon has been observed in experiments using the laser intensities not only in tunnelling regime [4, 9, 30] but also in multiphoton-ionization regime [31, 32]. In the latter case, the deviation of the ellipticity dependence was found to be small from the prediction of perturbation theory, especially for the lowest harmonics. The ellipticity sensitivity is predicted to increase with the harmonic order, which is the case in our calculation result for these near-threshold harmonics.

Fig. 4 z-component of the harmonics (H_z) as a function of ellipticity for perpendicularly (left) and parallelly (right) aligned H₂ at the bond length of 2.2 a.u.. Each harmonic is normalized to its intensity at zero ellipticity.

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Since the H_z of the NT harmonics indicates a smooth variation, the x-component of the harmonics must be essential to the AMs observed in the total harmonics shown in Fig. 3. In Fig. 5 the spectra of the x-component of the harmonics (H_x) as a function of ellipticity under both alignments are given. In contrary to H_z, the maximum intensity appears at a non-zero ellipticity for each H_x. Meanwhile, the intensity of each H_x is zero at ε = 0, regardless of the alignment. The non-monotonic ellipticity dependence of H_x has been experimentally observed in the near-threshold harmonics of Ne [9] by respective measurements of H_x and H_z. At ε = 0 H_x was observed to have a minimum while H_z exhibited a maximum, which is consistent with our calculation. When it comes to the above-threshold harmonics, the three-step model regime begins to take effect, and the intensity of H_x shows a rapid decay as the harmonic order increases. The decay is more significant under perpendicular alignment.

Fig. 5 Spectra of the x-component of the harmonics as a function of ellipticity under perpendicular (left) and parallel (right) alignments at the bond length of 2.2 a.u.. The maximum of each harmonic is marked by the square.

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It is also noted in Fig. 5 that the ellipticity, at which the H_x peaks, is dependent on the harmonic orders. For the parallel alignment [Fig. 5(b)], as the harmonic order increases, the ellipticity of the peak H_x changes rapidly from above 0.5 to 0.3 in H3 ~H7, then slowly shifts to ~0.2 in the next 4 harmonic orders. The shifting maximum indicates the observed shift of AMs in experiments. For aligned O₂ [12] under parallel alignment, H9 reached a maximum at ~0.2, larger than that of ~0.15 for the higher harmonic of H11. Similar tendency can also be found in Ne [11], but less pronounced. The shift is also alignment-dependent. Comparing Fig. 5(a) with Fig. 5(b), it is noteworthy that the maximum tends to appear at a smaller ellipticity for the parallel alignment than for the perpendicular alignment.

For the near-threshold harmonics, when the x-component is comparable with or larger than the z-component, the AM can appear in the total harmonic emission. The x- and z- components of the NT harmonics with AMs are retrieved and plotted in Fig. 6. For H5 under perpendicular alignment [Fig. 6(a)], the x-component (H5_x) is significant compared with the z-component (H5_z) thus an AM can appear. While H5_x maximizes at the ellipticity of ~0.35, the total intensity of H5 peaks at the ellipticity of ~0.2. The adjacent H7 [Fig. 6(b)] shows similar curves but a stronger AM, because of a larger ratio of H7_x over H7_z. H11 under the parallel alignment [Fig. 6(c)], whose energy is above-threshold, obeys the same rule. As H11_x is relatively weak compared with H11_z, totally H11 has an AM but less pronounced than H5 and H7. For other harmonics, H_x is weak compared with H_z, therefore no AM exists.

Fig. 6 Intensities of different polarization components of H5 (a) and H7 (b) under perpendicular alignment and H11 (c) under parallel alignment at the bond length of 2.2 a.u..

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To gain a better understanding of the HHG mechanisms in different energy regimes, we perform the time-frequency analysis by wavelet transformation of the dipole acceleration [33], where a Gaussian filter centred at a chosen frequency is interposed:

a_{ω} (t) = \int a (t') \exp [- \frac{ω^{2} {(t - t')}^{2}}{2 τ^{2}}] \exp [- i ω (t - t')] d t'

The temporal width of the Morlet windows is modulated by τ/ω, thus the relative bandwidth of ∆ω/ω is fixed for all harmonics [34]. We chose the parameter τ = 22, and the analyzing function (in FWHM) comprises about seven oscillations, similar to previous reports [15, 35]. Figure 7 shows the typical time profile of different harmonics at zero-ellipticity. For the NT harmonic of H5, the time profile is a smooth function of time without periodicity, which indicates that the multiphoton mechanism dominates the harmonic generation. The overall time profile has a slope may due to the ground state depletion, which makes the electrons more tightly bounded thus lower the radiations after the first few cycles. For higher order harmonics of H13-H29, the time profile contains two peaks per optical cycle, indicating that the three-step process begins to play a role. Also the time profile of H5 in the first optical cycle is not zero, but correlates with the trapezoidal laser envelope, which is different from the high harmonics dominated by three-step process. Evidently the multiphoton process can be qualitatively distinguished from the three-step process as the harmonic orders are near the threshold [20, 33].

Fig. 7 Time profile of the dipole accelerations for different harmonics of the perpendicular aligned H₂ at the bond length of 2.2 a.u..

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As shown in our calculation, the AMs of near-threshold harmonics can mainly be attributed to the non-monotonous ellipticity dependence of the H_x, while the H_z shows the monotonous one. This suggests that the H_z and the H_x are generated from different mechanisms. It is notable that the intensity of the orthogonal laser component is always much smaller than that of the z-component in our calculation. They may have different performance on the generation of the near-threshold harmonics, although can be both regarded as playing a part in the multiphoton mechanism in board terms.

As no unoccupied orbital is introduced in the calculation, the multiphoton mechanism may derive from a multiphoton resonance of the electrons interacting with the laser fields. A trajectorial description without involving the tunnelling process was proposed to give a qualitative interpretation for the generation of the below-threshold harmonics [12]. It was suggested that the intermediate resonance of the bound charge could contribute to the generation of these low harmonics. Intuitively speaking, unlike the semi-classical process which dominates the generation of the near-cutoff harmonics, the orthogonal laser field with a low intensity may not prohibit but contribute to H_x. However, if the ellipticity is large and the x-component of the laser field is strong enough to incur an over-excitation, the intensity of H_x may be suppressed.

Our calculation suggests that the resonance depends on the alignment, especially for H_x of the lowest harmonics. It can be seen from Fig. 5 that the absolute intensity of H_x is stronger for the case of perpendicular alignment, where the orthogonal laser component is in fact parallel to the elongating direction of the electron orbital. Thus the resonance amplitude may be enhanced due to a large electron polarization, and consequently the AMs tend to be more dominant. However, for the aligned O₂ [12], the AMs are more pronounced for the parallel alignment. Besides the unknown detecting efficiency for each polarization component in experiments, the reason may also lies in the fact that the outmost occupied orbital of O₂ has a π_g symmetry where the near-core electron tends to spread outwards because of the anti-bonding nature.

Ionization potential of the bound state may play a similar role. The electron with a lower ionization potential is more loosely bounded, thus is expected to exhibit stronger AMs. We perform the calculations at different bond lengths of 3.0 and 1.4 a.u., using the same calculating parameters. The calculated ionization potentials are 11.8 and 15.01 eV, respectively. Generally the near-threshold harmonics exhibit similar features as in Figs. 3–5, and the AMs are also found. The normalized near-threshold harmonics under different alignments are plotted in Fig. 8. For the large bond length of 3.0 a.u., the enhancement of the AM by a factor of ~8 can be found. In contrary, for the bond length of 1.4 a.u., only a weak AM can be found under perpendicular alignment.

Fig. 8 Calculated harmonics under perpendicular (left) and parallel (right) alignments at the bond length of 3.0 a.u. (top) and 1.4 a.u. (bottom). Each harmonic is normalized to its intensity at zero ellipticity.

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4. Conclusion

Ellipticity dependence of the near-threshold harmonics are calculated by means of TDDFT. For the simple molecular system of H₂, anomalous maximums can be found, as those observed in experiments of atoms and aligned molecules. H_x is found to dominant the appearance of the AM, as long as its intensity is significant compared with H_z. For these low harmonics, multiphoton effects play a dominate role rather than the three-step process, but may still be different between H_x and H_z. Our calculation suggests that the ellipticity dependence of the low harmonics can be sensitive to both alignment and bond length, especially for H_x. Thus the near-threshold harmonic can offer an effective tool for the study of the multiphoton process as well as the dynamical features of the bound states in strong laser fields. The calculation also has the potential to be applied to more complex systems, due to the multi-electron features of TDDFT. Our calculation indicates that the polarization direction of the NT harmonics can be strongly deviated. If there were conditions that the radiation was highly elliptical, it would be very appealing for future HHG studies since it would provide a way to create elliptical harmonics without losing efficiency. However, we still lack knowledge of their phase relationship. The phase difference of the two components can’t be easily determined from the time-dependent dipole or its frequency spectrum via Fourier transformation in our calculation, which still needs further investigation.

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Ellipticity dependence of the near-threshold harmonics of H₂ in an elliptical strong laser field

Abstract

1. Introduction

2. Theoretical method

3. Results and discussions

4. Conclusion

References and links

Cited By

Figures (8)

Equations (6)

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