1. Introduction

For the past two decades, great progress has been made in the study of high-order harmonic generation (HHG) [1–3]. The HHG has wide applications in attosecond science [4–6]. It can be well described by a three-step model [7, 8], where harmonics are emitted through the steps of tunneling, propagation, and recombination of the active electron with the parent ion.

A lot of work has been done to investigate the HHG from atoms and symmetric molecules like ${\text{H}}_2^{+}$ [9–17] and N₂ [18–21] with the central symmetry potential. Recently, much attention has been paid to asymmetric molecules such as HeH²⁺ [22–28], CO [29–42], HCl [43], BF [44] and OCS [45] in theoretical [22–30,32–35,38,39,41,42,44] and experimental [31,36,37,40,43,45] studies. The wave functions of these asymmetric molecules usually do not have a definite parity. According to the strong-field approximation (SFA) [8], the emission of harmonics is closely related to the highest occupied molecular orbital (HOMO) [5,38]. Due to the asymmetry of the molecular orbital, the HHG of the asymmetric molecule differs remarkably from the symmetric one. For example, the HHG spectra from symmetric molecules show a striking minimum [9]. This minimum has been identified as arising from effects of two-center interference as the rescattering electron recombines with the molecular potential with two-center characteristic. It can be used to probe the bond length of the molecule and has attracted broad experimental [46–48] and theoretical [49–52] interest in recent years. For the HHG spectra of asymmetric molecules, however, the simulations based on the numerical solution of the time-dependent Schrödinger equation (TDSE) show that the striking minimum disappear in some cases [53,54].

For atoms and symmetric molecules, the HHG spectra usually include only odd-order components. Very recently, it is shown that the interplay between electronic and nuclear degrees of freedom can induce the generation of even harmonics in symmetric molecules [55,56]. For asymmetric molecules, however, both odd and even harmonics are emitted, even though the nuclear motion is not considered. The emission of odd-even harmonics is one of the important characteristics of the HHG from asymmetric molecules. Present studies showed that odd and even harmonics from asymmetric molecules have different spectral properties [23,33,44] and carry different information of the target [36–38,40,41,45]. They therefore need to be studied separately [24]. The TDSE simulations further suggested that there are different HHG routes contributing to the emission of one odd or even harmonic, and the routes are subject to different intramolecular interference effects in the recombination process [53, 54, 57, 58]. These HHG routes are related to the rescattering electrons which ionize from and return to gerade and/or ungerade components of the asymmetric orbital. However, in TDSE simulations, all of the contributions of the different HHG routes are coupled together. So the role of individual routes remains unclear.

The molecular orbital tomography procedure with HHG provides unprecedented access to the inner workings of molecules [7]. The procedure is first proposed for symmetric molecules such as N₂ [7,59] and CO₂ [60] in experimental studies. Then it is generalized to asymmetric molecules such as HeH²⁺ [22] and CO [24,35] in theoretical works. In comparison with symmetric cases, the orbital tomography of asymmetric molecules with HHG needs a high degree of orientation of the target [37, 61] and therefore is more difficult to achieve in experiments. Recently, great efforts have been devoted to increasing the degree of orientation in experiments [36, 40]. In the asymmetric orbital tomography procedure through decoding odd-even high harmonics [24], it has been assumed that there are some specific emission routes which contribute mostly to odd-even HHG, as other routes play a small role. For the potential application of the procedure in experiments, a detailed theoretical study on the applicability of this assumption is also needed.

In this paper, we study the roles of different HHG routes in the emission of one odd or even harmonic from model HeH²⁺ molecule with 1σ valence orbital and model CO molecule with 5σ valence orbital. Our simulations are performed with a SFA model, which allows us to resolve the contribution of each route to odd-even HHG. Our results show that those routes, associated with the rescattering electrons ionized from the gerade component of the asymmetric orbital, play a dominating role in the emission of odd or even harmonics, in agreement with previous assumptions. Our analyses, based on the SFA calculations of the ionization probabilities for the gerade versus ungerade components of the asymmetric orbital, reveal that generally, there is very little electron ionized through tunneling from the ungerade component. The potential mechanism is associated with effects of two-center interference in tunneling, which occurs as the bound electron of the asymmetric molecule escapes through the laser-Coulomb-formed barrier. As the electron escaping from the gerade component is always subject to the constructive interference, the ungerade one is subject to the destructive one. We further reconstruct the asymmetric orbitals of HeH²⁺ and CO using route-resolved odd-even HHG spectra. Our simulations show that the routes associated with electrons ionized from the ungerade component play a small role in this reconstruction.

2. Theoretical description

2.1. Odd-even HHG with SFA

We study the HHG with the SFA at the fixed-nuclear approximation here. The time-dependent dipole moment is given by [8]

2.2. Routes for odd vs even harmonics

As discussed in [8], the selection rule for the emission of one odd or even harmonic is associated with the product of these two bound-continuum transition dipoles L = [e⃗_θ · d_i(p)][e⃗_θ · d_r(p)] in Eq. (2).

For atoms and symmetric molecules with the initial state |0〉 having a definite parity, only odd harmonics are emitted. In this case, the route of odd HHG can be denoted simply using $L_{00}^o=〈\mathbf{p}\left|{\overrightarrow{\mathbf{e}}}_{\theta }\cdot \mathbf{r}\right|0〉〈0\left|{\overrightarrow{\mathbf{e}}}_{\theta }\cdot \mathbf{r}\right|\mathbf{p}〉$. The expression indicates that the electron ionizes from and recombines with the same initial state |0〉 with an odd or even parity. Note that the values of the matrix elements in $L_{00}^o$ are not zero only when the initial state |0〉 and the continuum state |p〉 have different parities.

For asymmetric molecules such as CO or HeH²⁺ with the initial state |0〉, which is composed of the gerade component |0_g〉 and the ungerade one |0_u〉 with different weights, both odd and even harmonics will be emitted. In this case, the product L can be written as $L=\left[{\overrightarrow{\mathbf{e}}}_{\theta }\cdot {\mathbf{d}}_i(\mathbf{p})\right]\left[{\overrightarrow{\mathbf{e}}}_{\theta }\cdot {\mathbf{d}}_r(\mathbf{p})\right]=L_{gg}^o+L_{uu}^o+L_{gu}^e+L_{ug}^e$, which includes two odd (even) HHG routes $L_{gg}^o$ and $L_{uu}^o$ ($L_{gu}^e$ and $L_{ug}^e$) contributing to the emission of one odd (even) harmonic [38]. Specifically, these two odd routes are

With the above discussions, for molecules with an asymmetric initial state |0〉, we can divide Eq. (2) into the following parts:

2.3. Relations between spectra and dipoles

As discussed in [5], because the laser field distorts the Coulomb potential remarkably in the ionization process, the ionization step of HHG is not usually sensitive to the molecular structure but the recombination step does so. As a result, the HHG spectra and recombination dipoles of symmetric molecules show a close relation [14,51]. In the single-molecule case, that is [5]

For asymmetric molecules, the situation is somewhat more complex. As discussed above, there are two different HHG routes contributing to the emission of one odd or even harmonic. If we assume that the main odd-even routes dominate in odd-even HHG, a close relation between odd (even) spectra and relevant odd (even) recombination dipoles can also be obtained. That is [38]

One of the aims of the present work is to check the applicability of the above expression of Eq. (12), which has been used in imaging the asymmetric orbital [38], tracing the electron dynamics [53, 54, 57] and probing the nuclear dynamics [58] of asymmetric molecules in theoretical studies. With considering the experimental factors of the orientation effect and the alignment distribution, a developed version of Eq. (12) has also been used in calibrating the degree of orientation of top molecules [61]. In particular, similar expressions to the developed version of Eq. (12) have been used in probing the shape resonance [40] and the charge migration [62] within asymmetric molecules in experiments.

Furthermore, in [14], a simple HHG model based on Eq. (11) has been developed which provides deep insights into the mechanism of orientation dependence of HHG from symmetric molecules [43]. The extension of relevant studies to asymmetric molecules associated with Eq. (12) is another aim of the present work.

2.4. Analytical expressions of odd vs even dipoles

Next, we explore the analytical expressions of the dipoles $D_g(\mathbf{p},\theta )={\overrightarrow{\mathbf{e}}}_{\theta }\cdot {\mathbf{d}}_r^g(\mathbf{p})=〈0_g\left|{\overrightarrow{\mathbf{e}}}_{\theta }\cdot \mathbf{r}\right|\mathbf{p}〉$ and $D_u(\mathbf{p},\theta )={\overrightarrow{\mathbf{e}}}_{\theta }\cdot {\mathbf{d}}_r^u(\mathbf{p})=〈0_u\left|{\overrightarrow{\mathbf{e}}}_{\theta }\cdot \mathbf{r}\right|\mathbf{p}〉$, for model molecules HeH²⁺ and CO which we will explore in the paper.

Cases of model HeH²⁺—For the asymmetric initial state |0〉 = |0_g〉 + |0_u〉, in the linear combination of atomic orbital-molecular orbital (LCAO-MO) approximation [24], the 1σ initial-state wave function ${\psi }_{1\sigma }(\mathbf{r})={\psi }_{1\sigma }^g(\mathbf{r})+{\psi }_{1\sigma }^u(\mathbf{r})$ has the form of ψ_1σ(r) = 〈r|0〉 ≈ N_f(a₁e^−κr_a + a₂e^−κr_b) with ${\psi }_{1\sigma }^g(\mathbf{r})=〈\mathbf{r}|0_g〉=\left[{\psi }_{1\sigma }(\mathbf{r})+{\psi }_{1\sigma }(-\mathbf{r})\right]/2$ and ${\psi }_{1\sigma }^u(\mathbf{r})=〈\mathbf{r}|0_u〉=\left[{\psi }_{1\sigma }(\mathbf{r})-{\psi }_{1\sigma }(-\mathbf{r})\right]/2$. Here, a₁ = Z₁/B, a₂ = Z₂/B, $B={\left(Z_1^2+Z_2^2\right)}^{1/2}$, r_a = |r − R_a|, r_b = |r − R_b|, and $\kappa =\sqrt{2I_p}$. R_a and R_b are the positions of these two nuclei to the origin with R_a = |R_a| = [Z₂/(Z₁ + Z₂)]R, and R_b = |R_b| = [Z₁/(Z₁ + Z₂)]R. R is the internuclear distance and I_p is the ionization potential. N_f is the normalization factor. Z₁ and Z₂ are the effective changes of the He core and the H core for model HeH²⁺, respectively. The above expressions imply that we have chosen the center of charge associated with these effective changes as the coordinate origin. The asymmetry of the 1σ wave function ψ_1σ(r) depends on the parameters of Z₁, Z₂, R and I_p.

Note, for real HeH²⁺ with Z₁ = 2 and Z₂ = 1, the parameters κ associated with the He core and the H core in the above expressions should be different. As discussed in [27], when the use of the same κ for He and H cores gives a rough evaluation of the exact odd-even dipoles D_g and D_u, the use of different κ remarkably improves the comparison between analytical and exact odd-even dipoles. Here, for simplicity, we use the same κ for He and H cores, with which compact expressions of the analytical dipoles can be obtained and a analytical deduction of the roles of different odd-even HHG routes can be performed conveniently. In addition, such approach may be applicable for molecules (like CO) for which the effective charge ratio Z₁/Z₂ is close to one.

With the plane wave approximation for the continuum electron |p〉 and considering the electronic momenta p parallel to the laser polarization e⃗_θ, the dipoles D_g(u) ≡ D_g(u)(p, θ) associated with 1σ orbital can be written as

In the expression of D_g(u), the part G_g(u) in the first term represents the interference of the electronic wave between these two atomic centers of the molecule. It plays an important role in the behavior of the dipole D_g(u) [14]. Meanwhile, the amplitude of the second term associated with G′_g(u) is usually smaller than the first term and it gives a rectification to the interference patterns induced by the term G_g(u).

Cases of model CO—We continue to discuss the case of model CO molecule with 5σ valence orbital that is a valence orbital within a single-particle approximation. In the LCAO-MO approximation, the 5σ ground state wave function has the form of ψ_5σ(r) ≈ 〈r|0〉 = N_f(a₁z_ae^−κr_a − a₂z_be^−κr_b) with z_a = z − R_a, and z_b = z + R_b. The definitions of the parameters in the above expression are similar as for model HeH²⁺. Then the dipoles D_g(u) associated with 5σ orbital have the following forms:

2.5. Routes for ionization from polar molecules

The ionization is the first step of the HHG process. Theoretical studies on strong-field tunneling ionization from molecules have been performed by Tong et. al. [64, 65]. To understand the mechanism of odd-even HHG from polar molecules, it is meaningful for studying the strong-field ionization routes of polar molecules. In the frame of the SFA [63], the ionization probability of molecules at the angle θ can be written as N(θ) = ∫ dv|c(v, θ)|², where the amplitude $c(\mathbf{v},\theta )=-i{\int }_0^{T_p}d{t}^{\prime }\mathbf{E}(t^{\prime })\cdot {\mathbf{d}}_i\left(\mathbf{v}+\mathbf{A}(t^{\prime })\right)e^{iS(\mathbf{v},t^{\prime })}$. Here, v is the drift velocity of the continuum electron, $S(\mathbf{v},t^{\prime })={\int }^{t^{\prime }}\left[{(\mathbf{v}+\mathbf{A}(t^{″}))}^2/2+I_p\right]d{t}^{″}$ is the quasiclassical action and T_p is the length of the total pulse. For asymmetric molecules, as discussed above, the main (minor) HHG routes are associated with the electrons ionizing from the gerade (ungerade) components of the asymmetric orbital. Accordingly, we can define the main (minor) ionization routes. The ionization amplitudes for relevant routes can be written as

2.6. Saddle-point analyses of different routes

Cases of HHG—According to the quantum-orbit theory [8,66–68], the saddle-point equations for HHG are

The saddle-point equations of Eq. (19) and Eq. (20) tell that the instantaneous momentum p_st(t′_s, t_s) + A(t′_s) = e⃗_θ · (p_st (t′_s, t_s) + A(t′_s)) in the ionization step is imaginary with $p_{\mathit{st}}\left(t_s^{\prime },t_s\right)+A\left(t_s^{\prime }\right)=\pm i\sqrt{2I_p}$ and the momentum p_st(t′_s, t_s) + A(t_s) = e⃗_θ · (p_st(t′_s, t_s) + A(t_s)) is real in the recombination process with $p_{\mathit{st}}\left(t_s^{\prime },t_s\right)+A\left(t_s\right)=\pm \sqrt{2(\omega -I_p)}$. As a result, the behaviors of the dipoles ${\mathbf{d}}_r^{g(u)}$ versus ${\mathbf{d}}_i^{g(u)}$ in Eq. (21) and Eq. (22) differ significantly. One of the important differences between the ionization and recombination dipoles comes from the interference terms G_g and G_u, which are introduced in Eq. (13) to Eq. (16) and reflect the molecular structure. For simplicity, as in [14], we omit the terms associated with G′_g and G′_u, which usually have smaller values, then Eq. (21) and Eq. (22) can be rewritten as

Cases of ionization—Similarly, the saddle-point equation for ionization is [63,69–72],

Discussions—On the whole, the interference terms G_g and G_u are functions of cos or sin and depend on the amplitudes of a₁ and a₂, the internuclear distance R, the ionization potential I_p, the orientation angle θ, and the symmetry of the molecular orbital. In the HHG recombination process with real electronic momenta, the interference terms will induce cos-type and sin-type interference patterns, which differ remarkably from each other [24].

In the ionization process with imaginary electronic momenta, the influences of the cos-type and sin-type interference terms are also different. This can be seen with analyzing the properties of cos and sin functions with imaginary arguments ix [14], which have been shown to contribute to the remarkable increase of ionization for symmetric molecules, in comparison with atoms with similar I_p [72,73]. For example, as the value of |cos(ix)| = |e^x + e^−x|/2 is always larger than unity, i.e., |cos(ix)| ≥ 1, the value of | sin(ix)| = |e^x − e^−x|/2 can be smaller than unity. The value of |cos(ix)| is always larger than |sin(ix)|, i.e., |cos(ix)| > |sin(ix)|, etc.. The properties differentiate the influences of cos-type interference terms from sin-type ones in the ionization process, especially for those associated with the main ionization routes. In Sec. 3, we will discuss different influences of the interference terms in detail, with the cases of HeH²⁺ (cos-type interference terms for the main ionization route) versus CO (sin-type one).

Despite these disagreements arising from different interference terms, the contrast of the main versus minor ionization routes for CO and HeH²⁺ with different symmetries is similar. This is one of the important properties of tunnel ionization and is significant in the mechanism of odd-even HHG from asymmetric molecules. Let us discuss this point in more detail. For HeH²⁺ with 1σ valence orbital, the interference terms |G_g(u) (± iκ, θ)| associated with the main (minor) ionization route are

We mention that for model HeH²⁺ with 2σ valence orbital, the interference term associated with the main (minor) ionization route is $G_u^{2\sigma }$ ($G_g^{2\sigma }$), which has a form as $G_g^{5\sigma }$ ($G_u^{5\sigma }$). Similarly, for model CO with 6σ valence orbital, the interference term associated with the main (minor) ionization route is $G_u^{6\sigma }$ ($G_g^{6\sigma }$), which has a form as $G_g^{1\sigma }$ ($G_u^{1\sigma }$). In these cases, the main ionization route is associated with the ungerade component of the asymmetric orbital. However, the contrast of the main versus minor ionization routes is similar to the cases of Eq. (30) and Eq. (31) discussed above.

We can define a parameter

3. Results and discussions

3.1. Comparison of HHG of different routes

In the following, we study the roles of different HHG routes and verify our discussions in Sec. 2 with numerical solution of SFA for model HeH²⁺ and CO.

Cases of model HeH²⁺—In Fig. 1, we plot odd-even harmonics emitted by different routes and relevant dipoles of asymmetric molecules HeH²⁺ at θ = 0° and θ = 40°. For the case of θ = 0° in Fig. 1(a), one can observe that the full odd spectrum (thin-solid-black curve), obtained by results of Eq. (7) plus Eq. (8), basically agrees with the main-route odd spectrum (dash-black) of Eq. (7), as the result of Eq. (8) (short-dash-magenta), associated with the minor odd route, differs significantly from the full one. Similar phenomena are also observed for the cases of even harmonics, where the main even route of Eq. (9) still plays a dominating role.

 

Fig. 1 Harmonic spectra (a,b) and relevant dipoles (c,d) for model HeH²⁺ with I_p = 1.11 a.u. at θ = 0° (a,c) and θ = 40° (b,d). In (a) and (b), we plot the full odd spectra of Eq. (7) plus Eq. (8) (thin-solid-black), the main-route odd spectra of Eq. (7) (dash-black), the minor-route odd spectra of Eq. (8) (short-dash-magenta), full even spectra of Eq. (9) plus Eq. (10) (bold-solid-red), main-route even spectra of Eq. (9) (short-dot-red) and minor-route even spectra of Eq. (10) (dash-dot-blue). In (c) and (d), we plot the main-route odd dipole |${\left|D_g^{1\sigma }(\mathbf{p},\theta )\right|}^2$ of Eq. (13) (solid-black) and the main-route even dipole ${\left|D_u^{1\sigma }(\mathbf{p},\theta )\right|}^2$ of Eq. (14) (dash-red) with p²/2 = ω − I_p. Here, the vertical arrows indicate the intersections of odd-even HHG spectra and relevant odd-even dipoles. The horizontal arrows indicate the interference-induced hollows in spectra and relevant dipoles. The molecular parameters used are as shown. The laser parameters used are I = 5 × 10¹⁴ W/cm ² and λ = 800 nm.

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The main characteristics of the full odd versus even spectra also agree with the predictions of the odd versus even dipoles of Eqs. (13) and (14), associated with the main routes shown in Fig. 1(c). For example, the odd and even spectra show a striking intersection, consistent with the behaviors of odd versus even dipoles, as indicated by the vertical arrows. In addition, the full odd spectrum also shows a striking hollow, which is also in agreement with the prediction of the odd dipole, as indicated by the horizontal arrows. These comparisons verify the applicability of Eq. (12), which stands for the asymmetric orbital imaging procedure with odd-even HHG.

For the case of θ = 40° in Figs. 1(b) and 1(d), the situation is similar to θ = 0°. One can observe that the minor odd-even routes contribute little to odd-even HHG and the full odd versus even spectra show a close relation with the odd versus even dipoles associated with the main routes. The above phenomena also hold for other molecular parameters of Z₁, Z₂ and R in our extended simulations.

Cases of model CO—Next, we turn to model CO with 5σ valence orbital. Relevant results are shown in Fig. 2, where we have used the parameters of Z₁/Z₂ = 1.5, R = 2.13 a.u. and I_p = 0.515 a.u.. For both cases of θ = 0° and θ = 40°, one can observe that the main-route odd spectrum (dash-black) matches well with the full odd spectrum (thin-solid-black), and they are several orders of magnitude higher than the minor-route odd spectrum (short-dash-magenta), implying that the minor odd route has a small influence here. By comparison, the minor-route even spectrum (dash-dot-blue) is only several times lower than the full even spectrum (bold-solid-red), which also basically agrees with the main-route even spectrum (short-dot-red), indicating that the minor even route plays a more important role here.

 

Fig. 2 Same as Fig. 1, but for model CO with I_p = 0.515 a.u.. The dipoles in (c) and (d) are calculated with Eq. (15) (main odd, solid-black) and Eq. (16) (main even, dash-red). The molecular parameters used are as shown. The laser parameters used are I = 3 × 10¹⁴ W/cm² and λ = 800 nm.

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However, as we increase the internuclear distance R or the ionization potential I_p, the influences of the minor even route decrease remarkably, as shown in Fig. 3, where we plot the cases of R = 4 a.u. with I_p = 0.515 a.u. (the left column) and I_p = 0.98 a.u. with R = 2.08 a.u. (right). The latter case is associated with the parameters of CO⁺. In Figs. 3(a) and 3(b), the yields of even harmonics associated with the minor even route of Eq. (10) become remarkably lower than the full even ones, with the prevailing contributions of the main even route of Eq. (9). When comparing the full odd and even spectra to relevant dipoles in Fig. 2 and Fig. 3, it can be seen that the odd versus even dipoles of the main route, still give a good prediction of the relative yields of odd versus even harmonics. From the results in Fig. 1 and Fig. 3 for model HeH²⁺ and CO with different R and I_p, we can conclude that on the whole, the minor emission routes have a small role in odd-even HHG, and this role is influenced by the symmetry of the molecular orbital and the molecular parameters, in agreement with our discussions in Sec. 2. This role seems more remarkable for CO than for HeH²⁺. Next, we analyze the potential mechanism with studying the ionization dynamics of different routes for polar molecules.

 

Fig. 3 Same as Fig. 2, but results obtained for I_p = 0.515 a.u. and R = 4 a.u. (the left column) and for I_p = 0.98 a.u. and R = 2.08 a.u. (right) at θ = 0°.

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3.2. Comparison of ionization of different routes

As discussed in Sec. 2.6, there are two ionization routes contributing to the ionization yields of polar molecules, i.e., the main one and the minor one. They are subject to different interference effects and the main route with the constructive interference is expected to play a dominating role.

In Fig. 4, we plot the ratio γ(θ) of Eq. (32) for HeH²⁺. In Fig. 4(a), we fix the molecular parameters with changing the laser intensity and in Fig. 4(b), we fix the laser intensity with changing the molecular parameters. From Fig. 4(a), one can observe that this ratio is of the order of 10² and it increases significantly as the orientation angle increases. This ratio is not very sensitive to the laser intensity. However, this ratio depends strongly on the ionization potential I_p and the internuclear distance R. As shown in Fig. 4(b), with fixing the ionization potential and the laser parameters, this ratio increases remarkably as the internuclear distance decreases. In addition, this ratio is larger for smaller I_p. The results basically agree with our saddle-point analyses below Eq. (32).

 

Fig. 4 Angle dependence of ratio γ(θ) of Eq. (32) for ionization probability of main versus minor routes of HeH²⁺ at different laser and molecular parameters, with Z₁/Z₂ = 1.54 and λ = 800 nm. In (a), the molecular parameters are as in Fig. 1, the laser intensities are as shown. In (b), the laser intensities are as in Fig. 1, the molecular parameters are as shown. The log₁₀ scale is used here.

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As the ionization is the first step of HHG and this ratio reflects the weight of the contributions of the main ionization route in ionization, one can expect that the main odd-even HHG routes associated with the main ionization route also contribute mostly to odd-even HHG, consistent with the results in Fig. 1. In addition, our extended HHG simulations with the molecular parameters in Fig. 4(b) also verify the increasing contributions of the main HHG routes with decreasing R or I_p, in agreement with the behaviors of the ionization curves in Fig. 4(b).

The situation is different for model CO, as shown in Fig. 5(a). Here, this ratio decreases with increasing the angle. In particular, at small angles, this ratio is of the order of 10, which is lower than the case of HeH²⁺ and implies that the minor HHG routes associated with the minor ionization route can play a more important role here. This ratio begins to increase remarkably for large R and large I_p, and becomes near to the order of 10², as shown in Fig. 5(b), implying that the contributions of the minor HHG routes begin to decrease remarkably in the cases. The ionization results explain relevant HHG results in Fig. 2 and Fig. 3. We note that in Fig. 5(a), this ratio for CO is not also very sensitive to the laser intensity, consistent with our saddle-point analyses.

 

Fig. 5 Same as Fig. 4 but for model CO with Z₁/Z₂ = 1.5 and λ = 800 nm. In (a), the molecular parameters are as in Fig. 2, the laser intensities are as shown. In (b), the laser intensities are as in Fig. 2, the molecular parameters are as shown.

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We further make simulations for model HeH²⁺ with 2σ valence orbital and model CO with 6σ valence orbital. Relevant results are presented in Fig. 6. Here, one can observe that this ratio of the main ionization route to the minor one is now of the order of 10 in Fig. 6(a) for HeH²⁺ with 2σ valence orbital and the order of 10² in Fig. 6(b) for CO with 6σ valence orbital, showing a contrary trend for results of 1σ versus 5σ. Our extended simulations for HHG also show that the role of the minor HHG routes becomes more remarkable for the case of 2σ than 5σ, in agreement with the ionization results.

 

Fig. 6 (a) Same as Fig. 4(b) but for model HeH²⁺ with 2σ valence orbital. (b) Same as Fig. 5(b) but for model CO with 6σ valence orbital.

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The main ionization routes of 1σ and 6σ are subject to the cos-style interference and those of 2σ and 5σ to the sin-style one. The situation reverses for the minor ones. We anticipate that the main routes associated with cos-style interference could play a more important role in ionization and HHG, in comparison with those related to sin-style interference. In addition, for cos-style main routes such as for 1σ and 6σ, this role increases for smaller R and I_p, and for sin-style main routes such as for 2σ and 5σ, the contrary trend holds. The discussions suggest the possible conditions for which the asymmetric orbital reconstruction procedure with odd-even HHG could be more applicable.

3.3. Orbital reconstructions with different routes

Here, we perform a direct comparison of asymmetric orbital reconstructions with odd-even HHG spectra obtained with different routes. We follow the procedure as in [38] with using full odd-even HHG spectra and spectra related to the main odd-even routes.

In the first column of Fig. 7, we show the exact LCAO-MO results for the gerade component with 1σ_g symmetry, the ungerade component with 1σ_u symmetry and the total wave function of the 1σ orbital of HeH²⁺. The corresponding reconstructions with the odd-even HHG spectra including all routes and only the main routes are shown in the second and the third columns of Fig. 7. All of the reconstructions basically reproduce the corresponding results. In particular, reconstructions with the main routes agree well with the full ones, implying that the minor odd-even HHG routes play a negligible role here.

 

Fig. 7 Comparison of the exact results (left column), the reconstructions of full routes (middle column), and the reconstructions of main route (right column) for the gerade component (first row), the ungerade component (second row), and the total component (third row) of the 1σ wave function ψ_1σ of model HeH²⁺, projected in the xoz plane. The laser and molecular parameters used are as in Fig. 1.

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Relevant results for model CO with 5σ valence orbital are presented in Fig. 8. Again, all of the reconstructions of full routes and main routes reproduce the basic characteristics of the gerade component with 3σ_g symmetry, the ungerade component with 2σ_u symmetry and the whole wave function of 5σ valence orbital. As the reconstructions of full routes also agree with the main ones, a careful analysis with subtracting the results in Fig. 8(j) from those in Fig. 8(f) tells that the amplitude of the lobe around x = −3 a.u. in Fig. 8(f) is somewhat larger than the corresponding one in Fig. 8(j). Accordingly, the asymmetry of the 5σ valence orbital in Fig. 8(j) seems somewhat larger than that in Fig. 8(f). This difference reflects the contributions of the minor routes to HHG, which are omitted in the reconstruction simulations with main routes. As discussed in Fig. 2, in comparison with model HeH²⁺, the minor even route plays a more important role in the emission of even harmonics.

 

Fig. 8 Same as Fig. 7, but for model CO with 5σ valence orbital. The laser and molecular parameters used are as in Fig. 2.

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We mention that in Fig. 7 and Fig. 8, the appearance of the two central lobes near the x axis for reconstructions of the ungerade components of 1σ_u and 2σ_u arises from the orbital nodes of these components at z = 0, which will induce numerical singularities in the length-form tomography procedure [38,59]. Finally, the reconstruction results in Fig. 7 and Fig. 8 show the applicability of the present asymmetric orbital tomography procedure, based on the assumption that the main odd and even HHG routes contribute mostly to odd-even HHG. The minor odd and even HHG routes play a small role in this procedure.

3.4. Extended discussions on orientation effects, Coulomb effects, and nuclear motion

So far, our discussions are based on perfect alignment and orientation, the plane-wave approximation and the fixed nuclear approximation. The reconstruction procedure based on Eq. (12) can also be generalized for considering imperfect alignment and orientation and the Coulomb effect. With considering the orientation effect, the calculated odd-even HHG spectra in our extended simulations for model HeH²⁺ and CO still show the good agreement with the odd-even dipoles related to the main routes, similar to the TDSE results shown in [61]. The reconstructed orbitals with full and main routes are also in good agreement with each other, implying that the minor routes play a small role in this reconstruction. In particular, the asymmetry of the reconstructed orbital decreases as the degree of orientation decreases, similar to the results shown in the Supplemental Material in [38]. In this Supplemental Material, comparisons between results of plane-wave-based models and Coulomb-wave-based ones have also been shown. There are differences, but the main characteristics of the reconstructed orbitals are similar for plane-wave-based models and Coulomb-wave-based ones.

The situation is more complex as the nuclear motion is considered. In [58], it has been shown that for vibrating HeH⁺, the interaction between the laser field and the permanent dipole of the asymmetric molecule induces the rapid spreading of the initial nuclear wave packet, and the odd-even HHG occurs at a critical distance larger than the equilibrium separation. This critical distance depends on the molecular orientation. The phenomena go beyond the description of SFA. As a result, the present SFA-based reconstruction procedure, where it is assumed that the initial-state component of the target is not influenced by the laser field basically and the spectral amplitude is not sensitive to the molecular orientation, can not be directly applied to the vibrating asymmetric system. New asymmetric orbital reconstruction procedures which consider these nuclear-motion-related phenomena need to be developed.

4. Conclusion

In summary, we have studied odd-even HHG from asymmetric molecules with different symmetries using a route-resolved approach based on the SFA. We have analyzed the contributions of different ionization-recombination HHG routes, associated with the rescattering electron ionizing from and returning to the gerade and/or ungerade components of the asymmetric orbital, to the emission of odd or even harmonics. There are two HHG routes for each harmonic. These two routes can be divided into the main one and the minor one, depending on properties of tunneling ionization of polar molecules in strong laser fields.

Specifically, during tunneling as the bound electron passes through the barrier formed by the intense laser field and the asymmetric two-center potential, it can be subject to the constructive or destructive interference, associated with different ionization routes. As the constructive two-center interference in tunneling increases the ionization probability of relevant route remarkably, the destructive one suppresses the ionization yields of relevant route substantially, which differentiates the main ionization route (related to constructive interference) from the minor one (destructive interference). Because the ionization is the first step of HHG, the recombination routes associated with the main ionization route also contribute mainly to odd-even HHG. On the whole, the HHG routes related to the minor ionization route play a small role in the emission of odd-even harmonics, and this role is affected by the symmetry of the molecular orbital, the molecular parameters and the orientation angle, etc.. Further simulations show that the minor routes have less influence on the asymmetric orbital tomography procedure with odd-even HHG.