1. Introduction

In strong-field atomic, molecular and optical physics, high-order harmonics generation (HHG) is an issue of wide interests [1–3 ]. A simple physical picture for the HHG is provided by a semiclassical three-step model [4]. In this model, the bound electron first tunnels out from the barrier formed by the Coulomb potential and the laser field. As it escapes from the potential, the electron behaves similarly to a classical particle and is accelerated by the laser field. When the laser field changes its direction, the electron can return to the vicinity of the nucleus and recombine with the core with falling into the initial bound state. During the recombination, the electron will reduce its energy and emit a high-energy photon ω. This process is termed as the HHG. If the electron returns with a kinetic energy E_p, the energy of the photon will be ω = E_p + I_p (atomic units of $\overline{h}=e=m_e=1$ are used throughout this paper unless mentioned otherwise) with I_p being the ionization potential of the initial state. Considering the electronic kinetic energy E_p ≥ 0, the photon energy ω predicted from the semiclassical model is always near to or larger than I_p.

In experiments and numerical simulations, however, the emitted photon energy ω can be smaller than I_p, which has been termed as below-threshold harmonics (BTHs). The origin of BTHs has attracted great theoretical and experimental attentions in recent years [5–10 ]. As it is natural to consider that the BTHs arise from multiphoton effects, experimental studies showed that the quantum paths [11] which are closely associated with tunneling influence significantly on the BTHs [12, 13]. The different contributions of long and short quantum trajectories to the harmonics near to the threshold have been identified in experiments [14] and described by a generalized three-step model where the Coulomb and the excitation effects are included [14, 15]. Recent studies show that besides quantum trajectories, the BTHs can also be importantly influenced by the resonance effect [16]. As present studies on BTHs mainly focus on atoms, the studies of BTHs from molecules are relatively less. For molecules, due to more degrees of freedom, the molecular response in strong laser fields shows some more complex effects such as the orientation effect [17–19 ], two-center interference [20–22 ], etc. Because of the orientation effect, the electron can emit harmonics not only parallel to the laser polarization (i.e., the parallel harmonics) but also perpendicular to the laser polarization (i.e., the perpendicular harmonics). As a result, the molecular HHG can show high ellipticity [23, 24]. This ellipticity phenomenon is another hot issue in strong-field-molecule interaction in recent years. Because it is related to not only the harmonic intensity but also the phase of the harmonic and therefore provides deep insights into the mechanism of HHG from aligned molecules [25–29 ]. When present studies on this ellipticity mainly focus on harmonics above the ionization threshold, the ellipticity of BTHs, which can provide other insights into the mechanism of low-order harmonic generation, is desired.

In this paper, we study the polarization properties of BTHs from aligned molecules in strong linearly-polarized laser fields with varied laser wavelengthes and orientation angles θ (θ, the angle between the molecular axis and the laser polarization). We concentrate on lower-order harmonics (LOHs) below the threshold. Our simulations show high ellipticity of below-threshold LOHs at some harmonic orders which presents strong angle dependence. Our analyses reveal that this high ellipticity is closely related to the resonance effect between the ground state and several lower excited states of the molecule, where the axis symmetry of the molecule plays an important role. Because the resonance effect is sensitive to the molecular orientation, the ellipticity of LOHs also shows the strong angle dependence. The results are expected to give suggestions on the complex origin of BTHs from aligned molecules.

2. Ellipticity phenomenon

2.1. Numerical methods

We assume that the molecular axis is along the x axis and the electric field E(t) is located in the xy plane. The Hamiltonian of the symmetrical molecule ${\mathrm{H}}_2^{+}$ studied here is H(t) = p ²/2 + V(r) + r·E(t) with the soft-Coulomb potential $V(\mathbf{r})=-Z/\sqrt{\xi +{\mathbf{r}}_1^2}-Z/\sqrt{\xi +{\mathbf{r}}_2^2}$. Here Z = 1 is the effective charge, ξ = 0.5 is the smoothing parameter. r ₁ = r − R ₁ and r ₂ = r − R ₂ with R ₁ and R ₂ being the positions of the nuclei that have the coordinates (R/2,0) and (-R/2,0) in the xy plane. R = 2 a.u. is the internuclear separation. The linearly-polarized electric field used here is $\mathbf{E}(t)=\overrightarrow{\mathbf{e}}f(t)E$ sinω ₀ t. $\overrightarrow{\mathbf{e}}$ is unit vector along the laser polarization. f (t) is the envelope function. E and ω ₀ are the amplitude and the frequency of the laser field. In our calculations, we use a ten-cycle laser pulse which is linearly ramped up for three optical cycles and then keeps at a constant intensity for seven additional cycles. The time-dependent Schrödinger equation (TDSE) $i\dot{\mathrm{\Psi }}(t)=H(t)\mathrm{\Psi }(t)$ is solved numerically by the spectral method [30]. In two-dimensional (2D) cases, we use a grid size of 409.6×409.6 a.u. for the x and y axes, which allows us to explore a wide laser-parameter region. We also check our main results in three-dimensional (3D) cases where a grid size of 204.8 × 204.8×51.2 a.u. for the x, y, and z axes is used. The laser intensity used in our calculations is I ₀ = 5 × 10¹⁴ W/cm².

In our 2D simulations, the ground state |0〉 with 1sσ_g symmetry (the 2D equivalent of the 1sσ_g symmetry), the first excited state |1〉 with 1sσ_u symmetry and the second excited state |2〉 with 2pπ_u symmetry have the ionization potential of I_{p 0} = |E ₀| = 1.11 a.u., I_{p 1} = |E ₁| =0.69 a.u., and I_{p 2}=|E ₂| = 0.55 a.u. respectively. Here, E_n is the eigenvalue of the eigenstate |n〉 of the field-free Hamiltonian H ₀ = p ²/2 +V(r). The corresponding 3D eigenvalues are similar to the 2D ones. With the TDSE wave function Ψ(t), the coherent part of the parallel or perpendicular spectrum can be evaluated using

To explore the roles of the excited states in the ellipticity of harmonics, the following expressions are also used to approximately evaluate the harmonic spectra:

The ellipticity of harmonics can be evaluated using

We mention that in Eq.(1)–Eq.(5), the HHG spectra are evaluated using the dipole acceleration. It has been shown that the dipole-acceleration spectra are somewhat different from those obtained using the dipole moment in TDSE simulations, especially as the ionization of the system is strong [31]. In our cases, the ionization isn’t very strong. For the lower-order harmonics relating to the resonance effect and mainly discussed in the paper, our extended simulations show that both dipole-acceleration and dipole-moment calculations give similar results.

2.2. Numerical results

To study the polarization properties of BTHs from aligned ${\mathrm{H}}_2^{+}$, a wide parameter region of laser wavelength ranging from λ = 400 nm to λ = 900 nm and molecular orientation ranging from θ = 10⁰ to θ = 80⁰ has been explored. Some typical results are presented in Fig. 1.

 

Fig. 1 The ellipticity of below-threshold harmonics from 2D and 3D ${\mathrm{H}}_2^{+}$ at λ = 400 nm [(a) and (c)] and λ = 760 nm [(b) and (d)] for different angles θ, obtained using Eq. (1).

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In Fig. 1(a) of λ = 400 nm, high ellipticity is observed for harmonic 5 (H5) which shows higher ellipticity at smaller angles θ. In addition, harmonics 7 and 9 (H7 and H9) which are near to the ionization threshold show small ellipticity here. For the longer wavelength of λ = 760 nm, high ellipticity appears at H9, as shown in Fig. 1(b). The ellipticity of H9 decreases as the angle increases. By comparison, the neighboring one of H7 shows smaller ellipticity which increases with the increase of the angle θ. For harmonics 13 to 17 near the threshold, higher ellipticity can also be observed here. The 3D results shown in Figs. 1(c) and 1(d) are similar to the 2D ones, especially for LOHs. Below, we will show that the high ellipticity of LOHs arises from the resonance effect and focus our discussions on 2D cases for simplicity.

To illuminate this resonance-related-ellipticity mechanism, in Fig. 2, we show the wavelength dependence of ellipticity of harmonics at different angles for several typical cases of BTHs, calculated using Eq. (1) of full simulations. First, for the case of H5 in Fig. 2(a), in the wavelength region of 400 nm to 500 nm, higher ellipticity is observed at smaller orientation angles. Here, a striking ellipticity peak appears in the curve of θ = 20⁰ and the peak is located at λ = 440 nm. The latter is related to five-photon resonance between the 1sσ_g and 2pπ_u states. In the wavelength region of 500 nm to 700 nm, harmonic 5 in Fig. 2(a) also shows a smaller ellipticity peak, which appears in the curve of the large angle of θ = 80⁰ with the peak position of λ = 560 nm (relating to five-photon resonance between the 1sσ_g and 1sσ_u states). Similarly, for the case of H7 in Fig. 2(b), the striking ellipticity peak of θ = 20⁰ appears in the wavelength region of 500 nm to 700 nm with the peak position of λ = 610 nm (relating to seven-photon resonance between the 1sσ_g and 2pπ_u states), as the ellipticity peak of θ = 80⁰ is located in the wavelength region of 700 nm to 900 nm with the peak position of λ = 780 nm (relating to seven-photon resonance between the 1sσ_g and 1sσ_u states). For the case of H9 in Fig. 2(c), the ellipticity peak of θ = 20⁰ shifts to the region of 700 nm to 900 nm with the peak position of λ = 760 nm (relating to nine-photon resonance between the 1sσ_g and 2pπ_u states). The temporal evolution of the population of the first two excited states in some resonance cases of laser wavelength, mentioned above, is presented in Fig. 3. For comparison, results obtained at the neighboring wavelengthes are also shown. One can see that the excited states have the larger population in resonance cases than others, near to the peak of the laser field.

 

Fig. 2 The ellipticity of harmonic 5 (a), harmonic 7 (b), harmonic 9 (c) and harmonic 15 (d) from 2D ${\mathrm{H}}_2^{+}$ as a function of the laser wavelength at different angles θ.

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Fig. 3 The temporal evolution of the population of the first two excited states for the laser wavelength in resonant cases, compared with other wavelengthes: (a) five-photon resonance between the 1sσ_g and 1sσ_u states for λ = 560 nm; (b) seven-photon resonance between the 1sσ_g and 2pπ_u states for λ = 610 nm; (c) seven-photon resonance between the 1sσ_g and 1sσ_u states for λ = 780 nm; (d) nine-photon resonance between the 1sσ_g and 2pπ_u states for λ = 760 nm. The corresponding orientation angles θ are as shown.

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For higher-order harmonics, however, the situation is different. As shown in Fig. 2(d), the ellipticity curves of harmonic 15 show the strong oscillation in all wavelength cases and the striking ellipticity peak discussed above disappears here. These results imply that the ellipticity of higher-order harmonics arises from other mechanisms, which deserves a detailed study in the future.

Recently, the polarization properties of HHG from aligned ${\mathrm{H}}_2^{+}$ have been studied for a lower laser intensity in [32], where a strong dependence of the polarization parameters on the orientation angle is also observed. To check our results, we have extended our simulations to other laser intensities. On the whole, the wavelength-dependent peaks of BTHs observed in Fig. 2 are more striking for higher laser intensities. Next, we will explore the physical mechanisms behind these above phenomena.

3. Analysis on the ellipticity mechanism

3.1. Numerical analysis

For simplicity, we choose the typical case of 760 nm to analyze the ellipticity mechanism of below-threshold LOHs. The wavelength of 760 nm with ω ₀ = 0.06 a.u. corresponds to seven-photon resonance between the ground state |0〉 and the first excited state |1〉 and nine-photon resonance between |0〉 and the second excited state |2〉. At this wavelength, for the case of the small angle of θ = 20⁰, the ellipticity of LOHs is remarkable as shown in Fig. 4(c).

 

Fig. 4 Parallel (a) and perpendicular (b) spectra, the ellipticity (c) and the phase difference δ ₁ (d) of below-threshold harmonics from 2D ${\mathrm{H}}_2^{+}$ with λ = 760 nm at θ = 20⁰. Here, the black-square curves indicate the results of full simulations of Eq. (1), the red-circle curves indicate that obtained using Eq. (3) where the transition from the first excited state to the ground state is excluded. The blue-triangle curves indicate that obtained using Eq. (4) where besides the first excited state, the transition from the second excited state to the ground state is also excluded.

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As we expect that this high ellipticity observed in Fig. 4(c) is related to the resonance effect, we further compare the spectra of full simulations to those of Eq. (3) and Eq. (4) where we exclude the bound-bound transitions of |1〉 to |0〉 and (|1〉 + |2〉) to |0〉, respectively. Here, we will focus on H7 and H9 which are mostly possible to relate to the resonance. We mention that for the low orders, the results of Eq. (2) agree with the full simulations of Eq. (1). So we do not show the spectra of Eq. (2) here.

For the parallel case in Fig. 4(a), in comparison with the full simulations of the black-square curve, a striking fall is observed in the red-circle curve of Eq. (3) for H7. The blue-triangle curve of Eq. (4) is similar to that of Eq. (3) here. These results imply that the first excited state dominates in the emission of the parallel harmonic of H7. For the perpendicular case in Fig. 4(b), this striking fall is observed in the blue-triangle curve of Eq. (4) for H9 as the red-circle curve of Eq. (3) agrees with the full simulations at this order. One thus can expect that the perpendicular harmonic of H9 is contributed mainly by the second excited state. Furthermore, a careful analysis tells that the first excited state contributes mostly to the perpendicular harmonic of H7, but the parallel harmonic of H9 is contributed comparably by both these two excited states. In other words, harmonics 7 and 9 are influenced differently by the two excited states. As the main contributions to both the parallel and perpendicular harmonics of H7 come from the first excited state, the parallel harmonic of H9 is influenced by both these two excited states. This influence induces a phase difference of δ1 ∼π/2 for H9, as shown in Fig. 4(d) and accordingly high ellipticity of H9, as shown in Fig. 4(c). The phase difference δ ₁ for H7 is near to π and accordingly harmonic 7 of full simulations in Fig. 4(c) shows low ellipticity.

It should be mentioned that in strong laser fields such as I ₀ = 5×10¹⁴ W/cm² used here, the bound electron of ${\mathrm{H}}_2^{+}$ is easily subject to the Stark effect, which can induce the shifts of the electronic energy levels and mix the electronic states with different symmetries [33, 34]. Generally, the Stark effect is stronger for higher excited states which are located farther from the nuclei [35]. With assuming a dc field which has an amplitude of E = 0.12 a.u., we have evaluated the Stark shifts of electronic states of ${\mathrm{H}}_2^{+}$ numerically. At small angles such as θ = 20⁰, the Stark-shifted ionization potential of the 1sσ_g state is 1.15 a.u., that of the 1sσ_u state is 0.72 a.u., and it is 0.62 a.u. for the 2pπ_u state. The energy differences between these field-dressed states are similar to the corresponding field-free ones. We thus expect that the analyses in Fig. 4 based on the field-free states are also applicable as the Stark effect is considered.

As the orientation angle changes, the influence of these two excited states on LOHs also changes. For the case of the large angle of θ = 80⁰, as shown in Figs. 5(a) and 5(b), when the parallel and perpendicular harmonics of H9 at θ = 80⁰ both are contributed mainly by the second excited state, the parallel harmonic of H7 is influenced mainly by the second excited state and the perpendicular one is influenced mainly by the first excited state. These different influences also result in a phase difference of δ ₁ ∼ 0.14π for H7 and accordingly a higher ellipticity of this order, as seen in Figs. 5(c) and 5(d). Because the phase difference of δ ₁ ~ 0.14π for H7 in Fig. 5(d) is smaller than δ ₁ ~ π/2 for H9 in Fig. 4(d), the ellipticity of H7 in Fig. 5(c) is also lower than that of H9 in Fig. 4(c). These numerical analyses shed light on why the ellipticity peak of θ = 80⁰ in Fig. 2(b) is lower than that of θ = 20⁰ in Fig. 2(c).

 

Fig. 5 Same as Fig. 4 but θ = 80⁰.

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3.2. Analytical study

The different angle dependence of ellipticity of H7 and H9 observed in Fig. 4 and Fig. 5 (also see Fig. 1) can also be understood as follows. Considering that the main contributions to H7 and H9 come from the bound-bound transitions of these two excited states 1sσ_u and 2pπ_u to the ground state 1sσ_g, we can denote the coherent parts of harmonic emissions parallel and perpendicular to the laser polarization simply using

Here, d ₁ = 〈0|x|1〉, and d ₂ = 〈0|y|2〉. For convenience, we denote the terms in Eq. (7) using

We mention that at θ = 0⁰ and 90⁰, Eq. (7) predicts the disappearance of the perpendicular component F_⊥. This is in agreement with our TDSE simulations. Next, we explain the ellipticity of H7 and H9 revealed in Fig. 4 and Fig. 5 using these above expressions. Our analyses show that the expressions give an applicable description of the angle dependence of these harmonics.

As mentioned above, the ellipticity requires that the yields of perpendicular harmonics are comparable to the parallel ones and there is a phase difference of δ ₁ ~π/2 between them. For the case of H7 at θ = 20⁰ in Fig. 4, the seven-photon resonance between |0〉 and |1〉 plays an important role in generating this harmonic. In this case, one can expect that |a ₁ (ω)| has an amplitude far larger than |a ₂(ω)|. At the same time, the value of cos² θ is nearly one order of magnitude larger than sin² θ at θ = 20⁰. Combining Eq. (7) with Eq. (8), we have $F_{\Vert }~d_{\Vert }^1$ and $F_{\perp }~d_{\perp }^1$. This expression implies that the parallel and perpendicular harmonics of H7 are mainly contributed by the first excited state and therefore the phase difference between them is small, in agreement with our previous numerical analyses. Since cosθ > sin θ at θ = 20⁰, one also arrives at $d_{\Vert }^1>d_{\perp }^1$ which implies that the yields of the parallel harmonic are larger than the perpendicular one at this order. As a result, harmonic 7 doesn’t show ellipticity basically, as observed in Fig. 4(c).

For H9 in Fig. 4 of θ = 20⁰, the nine-photon resonance between |0〉 and |2〉 is mainly responsible for the emission of this harmonic. Similar to the analyses for H7, we have |a ₂(ω)| ≫|a ₁ (ω)|. Considering cos² ≫ θ sin² θ at θ = 20⁰, we also have $F_{\Vert }~d_{\Vert }^1+d_{\Vert }^2$ and $F_{\perp }~d_{\perp }^2$. This expression implies that as the main contribution to the perpendicular harmonic of H9 comes from the second excited state, the parallel one is contributed comparably by both the first and the second excited states. The interference of the two different excited-state routes differentiates the phase of the parallel harmonic from the perpendicular one. On the other hand, the yields of parallel and perpendicular harmonics are comparable since all contributions to these two components are suppressed by a small factor. For example, for the contribution of $d_{\Vert }^1$, that is |a ₁(ω)| and for $d_{\Vert }^2$, that is sin² θ. For $d_{\perp }^2$, it is sinθ. Accordingly, high ellipticity can be expected, as discussed in Fig. 4.

As the orientation angle increases, the situation changes. For the case of θ = 80⁰ in Fig. 5, cosθ ≪ sinθ with cos² θ ∼ 0. For H7 with |a ₁(ω)| ≫|a ₂(ω)|, considering cos² θ~0, we $F_{\Vert }~d_{\Vert }^2$ and $F_{\perp }~d_{\perp }^1$. For H9 with |a ₁(ω)| ≪ |a ₂(ω)|, we have $F_{\Vert }~d_{\Vert }^2$ and $F_{\perp }~d_{\perp }^2$. In this case, the parallel and perpendicular harmonics of H7 are influenced by different excited-state routes resulting in a larger phase difference and accordingly higher ellipticity of this harmonic order. For H9, since both contributions to parallel and perpendicular harmonics come from the second excited state 2pπ_u. No phase difference is expected and the harmonic order does not show ellipticity basically here. It is interesting to note that for the case of the large angle, the perpendicular harmonic of H7 is somewhat stronger than the parallel one.

Besides H7 and H9, the two excited states also have some influences on other harmonic orders such as harmonic 11. As one can see from Fig. 4 and Fig. 5 that the 2pπ_u excited state contributes more to higher orders than the 1sσ_u excited state. This is easy to understand since the 2pπ_u state is Stark shifted more strongly than the 1sσ_u state. However, all of the influences of the two states on higher orders are not so remarkable as they do for H7 and H9.

From the above analyses, we can conclude that high ellipticity of LOHs generally appears at the harmonic order at which several different excited states all play an important role due to the Stark effect. The interference of the different excited-state channels can induce a phase difference of δ ₁ ∼ π/2 between parallel and perpendicular harmonics and thus lead to high ellipticity of the corresponding LOH. In our cases, this occurs at the orders n agreeing with n ≈ (I_{p 0} − I_{p 1})/ω ₀ for 1sσ_g-1sσ_u resonance or n ≈ (I_{p 0} −I_{p 2})/ω ₀ for 1sσ_g-2pπ_u resonance. For the former case of 1sσ_g-1sσ_u resonance relating to the symmetry of the orbital with respect to the x axis, the ellipticity increases as the angle increases and for the latter case of 1sσ_g-2pπ_u resonance relating to the y-axis symmetry, the opposite trend of this angle dependence of ellipticity is observed and the ellipticity is usually higher than the former case.

With these discussions, we return to Fig. 2 and explain the wavelength dependence of ellipticity observed there. For the wavelength region of 400 nm to 500 nm, the 2pπ_u excited state plays an important role in emitting H5, so higher ellipticity appears in H5 at smaller angles. For 500 nm to 700 nm, the 2pπ_u excited state plays an important role in generating H7, and the 1sσ_u excited state plays an important role in the emission of H5, so higher ellipticity is observed for H7 at smaller angles, and harmonic 5 shows higher ellipticity at larger angles. Similarly, for the region of 700 nm to 900 nm, higher ellipticity appears at H9 when the angle is smaller, and it occurs at H7 for larger angles.

4. Conclusions

In summary, we have studied the ellipticity of below-threshold harmonics from aligned molecules exposed to strong and linearly-polarized laser fields. We pay attention to lower-order harmonics below the threshold. The ellipticity of lower-order harmonics shows the strong dependence on the laser wavelength and the orientation angle. Our analyses reveal that this ellipticity of lower-order harmonics is closely related to the resonance effect. The interference between different excited-state emission channels arising from the resonance effect is mainly responsible for the appearance of high ellipticity. As the resonance effect shows the strong dependence on the laser wavelength and the orientation angle, so do this ellipticity. A simple model is proposed to describe the angle dependence of ellipticity of lower-order harmonics.

This angle dependence of ellipticity of lower-order harmonics is observed for the simple molecule ${\mathrm{H}}_2^{+}$. However, we expect that it will also appear for other molecules with complex symmetries such as N₂ [36–38 ] and CO₂ [39–41 ], for which multielectron effects influence importantly on the HHG. As this angle dependence of ellipticity is closely associated with the symmetry of the excited state involved in the resonance, it is possible to probe the symmetries of the excited states, which play important roles in the evolution of the system, through the measurement of this angle-dependent ellipticity. In particular, due to the polarization properties, these lower-order harmonics from aligned molecules can also be used to produce pulsed VUV elliptically polarized radiation with high efficiency.

We note that recent experiments [42] have observed effects of a (shape) resonance on the ellipticity of harmonics generated from a molecule.