1. Introduction

When atoms or molecules are placed in a strong laser field, high-order harmonics can be generated in a highly non-perturbative fashion [1]. They enable the generation of highly coherent extreme ultraviolet (XUV) or soft X-ray light sources, as well as the sources of single attosecond pulses or attosecond pulse trains [2–4]. Since the high-order harmonic generation (HHG) provides effective means to investigate the time-resolved processes that occur on the attosecond timescale, it thus opens up a new field for attosecond science [1]. The rich physics has been embodied in the molecular HHG when the molecules are impulsively aligned or orientated in an ultrashort laser pulse [5–9], and it has been used to retrieve the electronic structure information from the measured harmonic spectra [10,11] and to detect structural evolution on sub-femtosecond time scales in dynamic molecular systems [12,13]. Its applications can also be found in molecular orbital reconstruction [14–16], controlling and imaging of chemical reactions [14–16], charge migration [17], and so on. Meanwhile, the aligned or orientated molecules can be applied for studying other strong-field phenomena, for example, the multiphoton ionization [18,19] and ultrashort pulse compression [20,21]. However, it is still challenging to precisely measure the intensity of aligning laser pulse or gas temperature, which are the key parameters to determine the degree of alignment or orientation [22–24].

Recently, some attempts have been demonstrated both experimentally and theoretically to solve this issue via the probe-laser induced molecular HHG. Taking advantage of the sensitive dependence of arising times of local minima and maxima at the rotational revivals in the time-resolved HHG spectra, He et al. accurately measured the rotational temperature and pump intensity [25]. This method has been verified by using aligned N₂ and CO₂ molecules. Later on, Jin et al. found that the positions of deep minima in the HHG spectrum of CO₂ molecules are very sensitive to the low degree of alignment, which can be used to retrieve the intensity of aligning laser pulse or gas temperature [26]. The two methods have their own limitations, and alternative approaches are needed. As well known, the aligned molecules driven by a linearly polarized laser can generate harmonics in both parallel and perpendicular polarization directions [27], therefore, the generated harmonics are elliptically polarized. Such measurement has been performed by Zhou et al. in an accurate polarization measurement of HHG from aligned molecules [27]. They found that harmonic emission from N₂ molecules can be strongly elliptically polarized and the ratio between perpendicular and parallel of HHG yields is changed dramatically with the pump-probe angle. This unique characteristic of molecular HHG, as we will show in the current investigation, can be useful for retrieving the information of both aligning laser and gaseous molecules. Our investigation is based on the development of a complete theory for quantitatively simulating previously reported experimental results.

To fully simulate the experimentally measured high harmonics of aligned or orientated molecules, we need to consider both single-molecule response and macroscopic response. The former part considers the induced dipole by the driving laser field, which can be obtained by solving the time-dependent Schrödinger equation (TDSE), or can be obtained by its equivalents, such as the strong-field approximation (SFA) [28–30] and quantitative rescattering (QRS) theory [31–33]. Since the first step of harmonic emission in the single-molecule response is tunnel ionization, at lower intensities, ionization and recombination would only occur for electrons in the highest occupied molecular orbital (HOMO). With the increase of driving laser intensity, electrons in the inner orbital, such as HOMO-1 and HOMO-2, may be ionized, which would contribute to the harmonic emission as well [34–36]. For the latter part, it can be calculated by solving the Maxwell’s wave equation of high-harmonic field in an ionizing gaseous medium [37]. With the QRS theory, Le et al. [38] have simulated the measurements in Zhou et al. [27] based on the single-molecule response, i.e., the propagation effects are ignored, and have only considered the contribution from the HOMO orbital. The discrepancy between the simulation and measurement remains.

The first goal of this paper is to resolve such discrepancy. We apply a full theory for simulating both perpendicular and parallel components of harmonic emission from aligned N₂ molecules in Zhou et al. [27]. The multiple-orbital contribution to the HHG in the single-molecule response and the macroscopic propagation of high-harmonic field in the medium are both included in the simulations. Second, we investigate how the intensity ratio between the harmonic perpendicular and parallel components depends on the probe-laser intensity and the alignment degree, which could be used to identify the multiple-orbital contribution to the HHG and to retrieve the parameters of aligning laser and gas medium, respectively. Finally, we calculate the polarization features of HHG by using a mid-infrared, 1600-nm laser aiming at extending the HHG into the higher photon energies.

2. Theoretical methods

2.1 Quantitative rescattering theory for a multiple-orbital molecular system with non-zero ionization phase

For a single orbital molecular system, in the QRS theory, each of the two components of the complex laser-induced dipole moment in the frequency domain is given by [31–33]

Here $\theta ^{\prime}$ and $\varphi ^{\prime}$ are polar and azimuthal angles of the molecular axis in the frame attached to the probe field, which refers to a right-handed coordinate system with the polarization direction of the probe field as the z’-axis and its propagation direction as the y’-axis of the coordinate system, $N({\theta^{\prime}} )$ is the alignment-dependent ionization probability, $W(\omega )$ is the recombining electron wave packet, which is the same for both components, and ${d^{||, \bot }}({\omega ,\theta^{\prime},\varphi^{\prime}} )$ is the parallel (or perpendicular) component of the photo-recombination (PR) transition dipole (complex in general). Equation (1) is also valid if the induced dipole is calculated by the TDSE or the SFA. For convenience, $W(\omega )$ is usually calculated as

In the pump-probe scheme, the molecules are initially isotropically distributed and then partially aligned by a weak and loosely focused pump laser beam, so the induced dipole ${D^{\textrm{||,} \bot }}({\omega ,\theta^{\prime},\varphi^{\prime}} )$ are coherently added with the molecular alignment distribution [31,43]. Assuming that the pump and probe laser pulses propagate collinearly and $\alpha$ is the angle between their polarization directions, we obtain the averaged induced dipole of aligned molecules [31,43]

Since $W(\omega )$ doesn’t depend on the alignment angle $\theta ^{\prime}$ and $\varphi ^{\prime}$, similar to Eq. (1), ${D^{\textrm{||,} \bot }}({\omega ,\theta^{\prime},\varphi^{\prime}} )$ can be written in an alternative formulation

In the above procedure, only a single molecular orbital is taken into account to calculate the induced dipole. For N₂ molecules, we include the first three orbitals, i.e., HOMO (1π_g), HOMO-1 (1π_u), and HOMO-2 (3σ_u). Following the procedure in Ref. [45], the induced dipole generated by each molecular orbital is coherently summed as

2.2 Macroscopic propagation of high harmonics in the medium

In this paper, we consider the experimental conditions of low laser intensities or low gas pressures. Therefore, the fundamental laser field can be described by a Gaussian beam analytically [48]. For high harmonics, we only include the nonlinear polarization induced by the driving laser. The three-dimensional propagation equation of the high-harmonic field in the moving frame ($z = z^{\prime}$ and $t^{\prime} = t - z/c$) is described as [49–52]

Here ${n_e}({t^{\prime},\theta^{\prime}} )$ is the alignment-dependent free-electron density, which is obtained from

After the macroscopic propagation in the gaseous medium, we obtain the parallel and perpendicular component of near-field harmonics on the exit face of the gas ($z^{\prime} = {z_{out}}$). Due to symmetry, the perpendicular component of harmonic vanishes for isotropically distributed molecules and partially aligned molecules with $\alpha = {0^ \circ }$ or ${90^ \circ }$. For partially aligned molecules with other angles, the perpendicular component would appear, but is smaller than the parallel component in general [38].

3. Results and discussion

3.1 Comparison of simulations and measurements

First, we simulate the intensity ratio between perpendicular and parallel components of harmonic fields from aligned N₂ molecules with the complete theory. We use a 120-fs pump laser pulse with an intensity of 3.0×10¹³ W/cm², and a 30-fs probe laser pulse with an intensity of 2×10¹⁴ W/cm². Both pump and probe lasers are with the wavelength of 800 nm and the electric fields of both the pump and probe laser pulse are taken to have a Gaussian form, given by

 

Fig. 1. Experimental (a) and theoretical (b) (c) intensity ratios between perpendicular and parallel components of harmonic yields, as a function of pump-probe angle for aligned N₂ molecules. Experimental results in (a) are taken from Zhou et al. [27], and theoretical results in (c) are taken from Le et al. [38]. The simulated results in (b) are calculated by taking into account of the macroscopic propagation effects and multiple-orbital interference with the ionization phase difference of 1.2 π and π between HOMO and HOMO-1 and between HOMO and HOMO-2, respectively.

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We comment that the remaining discrepancies between theory and experiment may come from the neglect of multiple electron correlation effects in the single-molecule theory, or the absence of nonlinear optical effects on the driving laser pulse. To reveal the role of each individual factor in the simulation, the results are shown in Fig. 2 by removing some factors. Figure 2(a) shows the single-molecule simulation results by only including the single HOMO orbital. If the macroscopic propagation is taken into account in Fig. 2(b), the intensity ratios are reduced, but they are still not distinguished for each harmonic order. Figure 2(c) gives the single-molecule results by including the three molecular orbitals with the relative phases indicated in the figure. Clearly the ratios for H23 are overestimated and the ratios for H19 and H21 are underestimated. If the relative phases between two orbitals are chosen differently, as indicated in Fig. 2(d), even with the inclusion of propagation effects, the simulated results cannot compare with the experimental ones.

 

Fig. 2. Intensity ratios between perpendicular and parallel components of harmonic yields, as a function of pump-probe angle for aligned N₂ with each individual factor taken into account. The single-molecule simulation results by only including the single HOMO orbital are given in (a), the macroscopic propagation is taken into account in (b), the single-molecule simulation results by including the three molecular orbitals with the relative phases are given in (c), the simulation results with propagation effects and zero relative phases between two orbitals are given in (d). The laser parameters used are the same as in Fig. 1(b).

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In our simulations, we include the multiple-orbital interference with ionization phase in the single-molecule response, which is absent in Le et al. simulations [38]. There has been a debate whether the relative ionization phase between different molecular orbitals is non zero [46,47]. Our results indicate that the non-zero relative ionization phase indeed exists and should be included in the theory. Note that the measurement of intensity ratio between harmonic perpendicular and parallel component with involving a polarizer is relatively easy and accurate. With such measurements available for an extended spectral region and for other molecular targets, the relative ionization phase in a multiple-orbital molecular system can be further checked. Le et al. concluded that the macroscopic propagation effect (or phase-matching effect) affects both harmonic components in the same way, thus it cannot change the intensity ratio between two components [38]. We have checked that this conclusion is only valid if the shape of “averaged” transition dipole in Eq. (5) doesn’t depend on the intensity of probe laser. In reality, the angular distribution of ionization probability $N({\theta^{\prime}} )$ in Eq. (5) is slightly changed with the laser intensity, so does the “averaged” transition dipole, leading to the slight change in the intensity ratio between two harmonic components by including the propagation effect.

In the experiment, the phase difference between two harmonic components can also be measured, and the harmonic field is generally elliptically polarized. The geometry of the polarization ellipse is described by two parameters, orientation angle $\phi$ and ellipticity $\varepsilon$. The orientation angle $\phi$ of the harmonic field ellipse is defined as the angle between the major axis of the ellipse and the polarization direction of probe field [27]. Ellipticity $\varepsilon$ is defined as the ratio of the minor axis to the major axis of the harmonic field ellipse. With the intensity ratio and phase difference between two harmonic components, $\phi$ and $\varepsilon$ are given by [27]

 

Fig. 3. Phase difference between the two polarization components of aligned N₂ molecules for different pump-probe angles with r=10.6 µm (a) and r=17.3 µm (b) at the exit of gas jet. The experimental results in Zhou et al. [27] and the theoretical results in Le et al. [38] are shown in (c) and (d), respectively.

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Fig. 4. Harmonic ellipticity from aligned N₂ molecules for pump-probe angles $\alpha = {40^ \circ },{50^ \circ }$ and 60^○ with r=10.6 µm (a) and r=17.3 µm (b) at the exit of gas jet. Zhou et al. experimental results [27] and Le et al. theoretical results [38] are given in (c) and (d), respectively.

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3.2 Analysis of the contributions from inner molecular orbitals

Next, we try to understand how the inner molecular orbitals contribute to the intensity ratio between two harmonic components. As shown in Fig. 5, we first check the alignment-dependent ionization probability $N(\theta ^{\prime})$ of individual molecular orbital with the increase of laser intensity, which are quite different because each orbital has different symmetry and ionization potential. For example, the ionization probability peaks at ${90^ \circ }$ for the HOMO-1, however, it peaks at 0^◦ for both HOMO and HOMO-2. The relative ionization probabilities between two orbitals are strongly dependent on laser intensity. In Fig. 5(b), when the laser intensity is 2.0×10¹⁴ W/cm² (For convenience, we set I₀ equal to 10¹⁴ W/cm², then this intensity can be written as 2.0I₀), the ionization probability from the HOMO-1 (HOMO-2) is a factor of 5(20) smaller than that of the HOMO near ${90^ \circ }$ (${0^ \circ }$). With the decrease (or increase) of laser intensity, these factors become bigger (or smaller) in Fig. 5(a) [or in Fig. 5(c)].

 

Fig. 5. Ionization probability from HOMO, HOMO-1 and HOMO-2 at laser intensities of 1.5×10¹⁴ W/cm² (1.5 I₀) (a), 2.0×10¹⁴ W/cm² (2.0 I₀) (b) and 2.5×10¹⁴ W/cm² (2.5 I₀) (c). The data of HOMO-1 and HOMO-2 have been multiplied by some factors indicated in the figures. The calculation is performed by using the MO-ADK theory under an 800-nm, 30-fs laser.

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As shown in Fig. 6, at one selected pump-probe angle of ${50^ \circ }$, the inner orbitals have the non-neglectable contributions to both harmonic components at given laser intensities. Compared to the harmonics obtained with the HOMO only, their perpendicular components are not changed much by adding two inner orbitals, however, their parallel components are greatly modified. And such modification changes with the harmonic order due to the photon-energy dependence of the photo-recombination cross section. One can see that the smaller orders (H13 and H19) have bigger changes than larger orders (H21 and H29), which is about the same for all laser intensities. Therefore, from Fig. 6(b) one can explain why the peak value of intensity ratio between perpendicular and parallel components is increased with the harmonic order in Fig. 1(b).

 

Fig. 6. Comparison of the macroscopic high-harmonic spectra (envelope only) with HOMO only and with three molecular orbitals. The pump-probe angle is 50 degrees. Laser intensities in the center of gas jet are 1.5×10¹⁴ W/cm², 2.0×10¹⁴ W/cm², and 2.5×10¹⁴ W/cm², respectively. Other macroscopic conditions are the same as those in Fig. 1(b). I₀= 10¹⁴ W/cm².

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We then choose harmonic orders H17, H19 and H21, and plot intensity ratio as a function of the pump-probe angle at different laser intensities in Fig. 7. In Figs. 7(a)–7(c), we consider the contribution of both the HOMO and two inner orbitals in the single-molecule response, while in Figs. 7(d)–7(f) only the HOMO orbital contribution is included. One can see that with only the HOMO, the change of intensity ratio with the pump-probe angle at a fixed laser intensity almost cannot be distinguished for different harmonic orders, which is similar to Le et al. simulations in the single-molecule level [38] as shown in Fig. 1(c). With adding the interference from inner orbitals, the value of intensity ratio becomes smaller or remains the same depending on the laser intensity. It is clearly shown that when the inner orbitals contribute to the harmonic generation, the intensity ratio sensitively depends on the harmonic order and laser intensity. Therefore, by tuning the input laser power we may detect the multiple-orbital contribution to the HHG [53] by examining the intensity ratio between the perpendicular and parallel components of high harmonics.

 

Fig. 7. The intensity ratio between the perpendicular and parallel components of macroscopic harmonics for H17, H19 and H21 as a function of the pump-probe angle at different laser intensities (black: 1.5×10¹⁴ W/cm², red: 2×10¹⁴ W/cm², and blue: 2.5×10¹⁴ W/cm²). In the single-molecule response, the multiple-orbital contribution is included in (a)–(c), and only the HOMO orbital contribution is taken into account in (d)–(f). I₀= 10¹⁴ W/cm².

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3.3 Retrieval of molecular alignment by using the polarization characteristics of high harmonics

Then we discuss the possibility to retrieve the molecular alignment by using the intensity ratio and the phase difference between the perpendicular and parallel harmonic components. Figures 8(a) and 8(b) show the intensity ratio as a function of maximum alignment degree and pump-probe angle for H17 and H21. For each fixed pump-probe angle, the intensity ratio increases as the maximum alignment degree increases. Once we know the measured intensity ratio at one pump-probe angle, for example, at 50 degrees, and the maximum alignment degree expressed in terms of < cos²θ> at the first half-revival can be determined as shown in Fig. 8(c). And then this value can be used to retrieve the intensity of pump laser at a known gas temperature or to retrieve the gas temperature at a known pump-laser intensity as shown in Fig. 8(d). This figure is obtained by assuming that the degree of molecular alignment can be calculated with the rigid-rotor model [6,54]. The above procedure can be repeated for other pump-probe angles to reduce the error introduced in the experiment.

 

Fig. 8. (a)–(b): Intensity ratio as a function of maximum alignment degree and pump-probe angle for H17 (a) and H21 (b). (c) Maximum alignment degree as a function of intensity ratio for H17 (same for H21) for the pump-probe angle of 50°. (d) Maximum alignment degree expressed in terms of < cos²θ> at the first half-revival as a function of laser intensity when the gas temperature is fixed (as shown in the figure). The pump laser has a wavelength of 800 nm, an intensity of 2.0 ×10¹⁴ W/cm², and full width at half maximum duration of 120 fs. (I₁,T₁), (I₂,T₂), (I₃,T₃) are three different sets of the pump intensity and gas temperature reconstructed according to an intensity ratio.

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In the pump-laser frame, defined as a right-handed coordinate system with the polarization direction of the pump field as the z-axis and its propagation direction as the y-axis, the alignment distribution $\rho (\theta )$ can be retrieved as well by using the measured intensity ratio and phase difference as a function of pump-probe angle. We propose a retrieval method based on the single-molecule QRS model with including the HOMO contribution to HHG, in which one assumes that the angle-dependent ionization rate and photo-recombination cross sections for fixed-in-space molecules are known from the theory. We put the details of this method in the appendix A and discuss the retrieved results here. Figure 9 compares with the real (used to calculate the macroscopic HHG) and the retrieved alignment distribution. In the retrieval of alignment distribution, we use the intensity of 1.8×10¹⁴ W/cm² to calculate the ionization probability, which is a little bit lower than the peak intensity at the center of gas medium for the macroscopic HHG. With H17 and H21, retrieved results are quite similar at low or high alignment degree. For the lower degree of alignment, the angle distribution can be perfectly retrieved, see Figs. 9(a) and 9(b). For the higher degree, the retrieved alignment distribution has some discrepancies with the real one, especially at small angles as shown in Figs. 9(c) and 9(d). Due to the sinθ term in the integral element, the contribution of small angles in the alignment distribution to HHG has been greatly decreased, thus bigger error showing up in the small-angle range. On the other hand, the retrieved alignment degree < cos²θ> which includes the sinθ term in the integration is very close to the real value, and the relative error does not exceed 2%.

 

Fig. 9. Real and retrieved alignment distribution $\rho (\theta )$ in the pump frame, with real alignment degree < cos²θ>=0.54 in (a), (b) and < cos²θ>=0.70 in (c), (d). H17 [(a) and (c)] and H21 [(b) and (d)] are used in the retrieval procedure. The intensity of probe laser used for retrieval is 1.8×10¹⁴ W/cm², and n_max = 3 is used in the retrieval.

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Since the intensity of probe laser in the experiment is usually not measured precisely, we check how our retrieval method works if there are some uncertainties in the probe intensity. We fix the target alignment same as those in Figs. 10(a) and 10(b), and use two intensities of 1.6×10¹⁴ W/cm² and 2.0×10¹⁴ W/cm² as probe ones in the retrieval procedure, which are about 10% lower or higher than the one used before. The retrieved alignment distributions are shown in Fig. 10. One can see that only for angles smaller than 30 degrees, retrieved distributions are different from the real ones. If one compares the retrieved degree of alignment < cos²θ>, the relative error is about within 6%.

 

Fig. 10. Real and retrieved alignment distribution $\rho (\theta )$ in the pump frame, with real alignment degree < cos²θ>=0.54. In the retrieval, probe intensities of 1.6×10¹⁴ W/cm² and 2.0×10¹⁴ W/cm² are used in (a), (b) and (c), (d), respectively, and H17 and H21 are used in (a), (c) and (b), (d), respectively. n_max = 2 is used in the retrieval.

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We also check whether our retrieval method is eligible when the contribution of inner orbitals to the HHG becomes important. As shown in Fig. 11, when the macroscopic HHG used for retrieval includes the multiple-orbital interference, our single-orbital based retrieval method does not perform as well as when only a single-orbital contributes. Nevertheless, it can still provide some qualitative information on alignment distribution and provides with a reference value of alignment degree < cos²θ>. Since the contributions of the inner orbitals are different with varying the harmonic order, as shown in Figs. 6 and 7, the retrieved alignment distributions with different orders are also different as shown in Fig. 11.

 

Fig. 11. Real and retrieved alignment distribution in the pump frame. The intensity of probe laser used in the retrieval is 1.5×10¹⁴ W/cm². The contribution of the inner orbitals is included for macroscopic HHG.

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3.4 Extension to the higher photon-energy region

With the advance of laser technology, the mid-infrared laser pulses have become available in the lab, which can be used to extend the spectral region of HHG. Here we investigate the HHG polarization properties of aligned N₂ molecules with a mid-infrared laser. A 1600-nm laser (photon energy of 0.77eV) with a duration of 60 fs and an intensity of 2.0×10¹⁴ W/cm² is used for simulations, the parameters used for the pump laser are the same as those in Section 3.1. The calculated intensity ratio, phase difference, and ellipticity are shown in Fig. 12 for selected harmonic orders with consideration of the macroscopic propagation effects and the multiple-orbital interference. The cutoff of HHG spectrum is near H217 (or photon energy of 167 eV). In Fig. 12(a), the behavior of intensity ratio with the increase of pump-probe angle is similar for different orders, however, the peak value of intensity ratio doesn’t monotonically decrease with the increase of harmonic order, which is different from that in Ref. [55]. In Ref. [55], the intensity ratio was calculated in the single-molecule level with including the HOMO only. The discrepancy between two simulations is mostly due to the interference from inner orbitals. It is necessary to perform experiments for the HHG in the extended photon-energy region to reveal the possible multiple-orbital contribution by using longer-wavelength lasers. The phase difference and ellipticity in Figs. 12(b) and 12(c) for selected pump-probe angles show similar values for higher-order harmonics in comparison with lower-order ones.

 

Fig. 12. The intensity ratio (a) between two components of high harmonics with a 1600-nm laser, and the corresponding phase difference (b) and ellipticity (c) for selected pump-probe angles.

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4. Conclusion

In conclusion, we have simulated parallel and perpendicular components from harmonic emissions of transiently aligned N₂ molecules with a complete theory by including both single-molecule and macroscopic responses. And the multiple-orbital interference has been considered in the single-molecule QRS theory. Our simulations agree very well with the measurements in Zhou et al. experiment [27]. We have studied the dependence of the intensity ratio between two harmonic components on the intensity of probe laser, from which we may analyze whether the inner molecular orbitals contribute to the HHG. Then we have investigated how the intensity ratio changes with the alignment degree. With the theory of rotational wave packet, we can retrieve the intensity of pump laser or the gas temperature by using the value of alignment degree obtained from measured intensity ratio. Furthermore, we have developed a retrieval method to get the alignment distribution with the measured intensity ratio and phase difference between two harmonic components. This method is based on the QRS theory in the single-molecule level by including the HOMO contribution only. In the region of higher photon energies accessed by using the probe laser with 1600 nm wavelength, we have found that the intensity ratio does not monotonically decrease with the increase of the harmonic order, which is different from the previous results [55] with a single orbital of HOMO. This requires some experimental efforts with mid-infrared lasers to resolve the discrepancy in the theories and to reveal the contribution to the HHG from inner orbitals.

Appendix A

In the QRS theory, each of the two components of the complex induced dipole is given by [31–33]

(17)$${D^{\textrm{||,} \bot }}({\omega ,\theta^{\prime},\varphi^{\prime}} )= N{({\theta^{\prime}} )^{1/2}}W(\omega ){d^{\textrm{||}, \bot }}({\omega ,\theta^{\prime},\varphi^{\prime}} ),$$

Here $\theta ^{\prime}$ and $\varphi ^{\prime}$ are polar and azimuthal angles of the molecular axis in the frame attached to the probe field, which refers to a right-handed coordinate system with the polarization direction of the probe field as the z’-axis and its propagation direction as the y’-axis of the coordinate system, $N({\theta^{\prime}} )$ is the alignment-dependent ionization probability, $W(\omega )$ is the recombining electron wave packet, which is the same for both components, and ${d^{\textrm{||}, \bot }}({\omega ,\theta^{\prime},\varphi^{\prime}} )$ is the parallel (or perpendicular) component of the photo-recombination (PR) transition dipole.

Let $\theta$ and $\varphi$ be the polar and azimuthal angles of the molecular axis in the frame attached to the pump laser field, and the angle between the polarization directions of pump and probe laser fields is $\alpha$. The relationship between $\theta$ and $(\theta ^{\prime},\varphi ^{\prime})$ is [31,43]

(18)$$\textrm{cos}\theta = \cos \theta ^{\prime}\cos \alpha + \sin \theta ^{\prime}\sin \alpha \cos \varphi ^{\prime},$$

The alignment distribution in the probe-laser frame can be expressed as

(19)$$\rho ({\alpha ,\theta^{\prime},\varphi^{\prime}} )= \rho ({\theta ({\alpha ,\theta^{\prime},\varphi^{\prime}} )} ).$$

Here the alignment distribution in the pump-laser frame does not depend on the azimuthal angle $\varphi$, which is true for linear molecules. In the probe-laser frame, for linear molecules, we have

(20)$${D^{\textrm{||}}}({\omega ,\theta^{\prime},\varphi^{\prime}} )= {D^{\textrm{||}}}({\omega ,\theta^{\prime}} ),$$

and

(21)$${D^ \bot }({\omega ,\theta^{\prime},\varphi^{\prime}} )= {D^ \bot }({\omega ,\theta^{\prime},\varphi^{\prime} = 0} )\textrm{cos}\varphi ^{\prime}.$$

Therefore, the component of alignment averaged induced dipole parallel to the polarization direction of probe laser is

(22)$${\bar{D}^{\textrm{||}}}({\omega ,\alpha } )= \int_\textrm{0}^\pi {\int_\textrm{0}^{\textrm{2}\pi } {{D^{\textrm{||}}}({\omega ,\theta^{\prime}} )\rho ({\alpha ,\theta^{\prime},\varphi^{\prime}} )\textrm{sin}\theta ^{\prime}d\theta ^{\prime}d\varphi ^{\prime},} } $$

and its perpendicular component can be calculated as

(23)$${\bar{D}^ \bot }({\omega ,\alpha } )= \int_0^\pi {\int_0^{2\pi } {{D^ \bot }({\omega ,\theta^{\prime}} )\rho ({\alpha ,\theta^{\prime},\varphi^{\prime}} )\textrm{sin}\theta ^{\prime}\cos \varphi ^{\prime}d\theta ^{\prime}d\varphi ^{\prime}.} } $$

For linear molecules, the alignment distribution $\rho (\theta )$ does not depend on azimuthal angle $\varphi$ and is an even function, that is

(24)$$\rho (\theta )= \rho ({\pi \textrm{ - }\theta } ).$$

Taking Legendre polynomial ${P_\textrm{n}}({\cos \theta } )$ as the basis of the expansion of the generalized Fourier series, we can always expand $\rho (\theta )$ as

(25)$$\rho (\theta )= \sum\limits_{\textrm{n = }0}^\infty {{p_{2n}}{P_{2n}}({\cos \theta } ),} $$

where ${p_{2n}}$ is the expansion coefficient. Since $\rho (\theta )$ is an even function of $\cos \theta $, Eq. (25) only contains even-order Legendre polynomials ${P_{2n}}({\cos \theta } )$, which is a polynomial of ${\cos ^2}\theta $ . Therefore, Eq. (25) can be written as

(26)$$\rho (\theta )= {a_0} + \sum\limits_{n = 1}^\infty {{a_n}{{\cos }^{2n}}\theta } .$$

Truncate the infinite series expansion by setting a large n_max, we have

(27)$$\rho (\theta )= {a_0} + \sum\limits_{n = 1}^{{n_{\max }}} {{a_n}{{\cos }^{2n}}\theta } .$$

Substituting Eq. (27) into Eq. (22) and Eq. (23), we can get

(28)$${\bar{D}^{\textrm{||}}}({\omega ,\alpha } )= {a_0}I_{\textrm{||}}^0(\alpha )+ \sum\limits_{n = 1}^{{n_{\max }}} {{a_n}I_{\textrm{||}}^n(\alpha )} ,$$

and

(29)$${\bar{D}^ \bot }({\omega ,\alpha } )= \sum\limits_{n = 1}^{{n_{\max }}} {{a_n}I_ \bot ^n(\alpha )} ,$$

with

(30)$$I_{\textrm{||}}^n(\alpha )= \int_0^\pi {\int_0^{2\pi } {{D^{\textrm{||}}}({\omega ,\theta^{\prime}} ){{\cos }^{2n}}\theta ({\alpha ,\theta^{\prime},\varphi^{\prime}} )\sin \theta ^{\prime}d\theta ^{\prime}d\varphi ^{\prime},} } $$

(31)$$I_ \bot ^n(\alpha )= \int_0^\pi {\int_0^{2\pi } {{D^ \bot }({\omega ,\theta^{\prime}} ){{\cos }^{2n}}\theta ({\alpha ,\theta^{\prime},\varphi^{\prime}} )} \sin \theta ^{\prime}\cos \varphi ^{\prime}d\theta ^{\prime}d\varphi ^{\prime}.} $$

The coefficients ${a_n}$ can be related to < cos^2kθ> ($k = 0,1,\ldots {n_{\max }}$) in the following,

(32)$$\left\langle {\textrm{co}{\textrm{s}^{2k}}\theta } \right\rangle = \int_0^\pi {{{\cos }^{2k}}\theta \rho (\theta )\sin \theta d\theta } = \frac{{2{a_0}}}{{2k + 1}} + 2\sum\limits_{n = 1}^{{n_{\max }}} {\frac{{{a_n}}}{{2({k + n} )+ 1}}} ,$$

and < cos^2kθ> with k = 0 corresponding to the normalization condition of $\rho (\theta )$,

$$2{a_0} + 2\sum\limits_{n = 1}^{{n_{\max }}} {\frac{{{a_n}}}{{2n + 1}}} = 1,$$

which gives

(33)$${a_0} = \frac{1}{2} - \sum\limits_{n = 1}^{{n_{\max }}} {\frac{{{a_n}}}{{2n + 1}}} .$$

Substituting Eq. (33) into Eq. (32), we have

(34)$$\left\langle {\textrm{co}{\textrm{s}^{2k}}\theta } \right\rangle - \frac{1}{{2k + 1}} = 2\sum\limits_{n = 1}^{{n_{\max }}} {{a_n}\left[ {\frac{1}{{2({k + n} )+ 1}} - \frac{1}{{({2k + 1} )({2n + 1} )}}} \right]} .$$

Let

$$\begin{array}{l} A = [{{a_1},\ldots {a_{{n_{\max }}}}} ],\\ Cs = \left[ {\left\langle {{{\cos }^2}\theta } \right\rangle - \frac{1}{3},\ldots ,\left\langle {{{\cos }^{2{n_{\max }}}}\theta } \right\rangle - \frac{1}{{2{n_{\max }} + 1}}} \right]. \end{array}$$

we can rewrite Eq. (34) as

(35)$$C\textrm{s} = EA,$$

with

(36)$${E_{ij}} = \frac{2}{{2({i + j} )+ 1}} - \frac{2}{{({2i + 1} )({2j + 1} )}},1 \le i,j \le {n_{\max }}.$$

The coefficients array A is given by

(37)$$A = {E^{ - 1}}C\textrm{s} = BCs,$$

with

(38)$$B = {E^{ - 1}}.$$

Substituting Eq. (36) into Eq. (28), we obtain

(39)$${\bar{D}^ \bot }({\omega ,\alpha } )= \sum\limits_{m = 1}^{{n_{\max }}} {l_ \bot ^m(\alpha )} \left[ {\left\langle {{{\cos }^{2m}}\theta } \right\rangle - \frac{1}{{2m + 1}}} \right],$$

with

(40)$$l_ \bot ^m(\alpha )= \sum\limits_{n = 1}^{{n_{\max }}} {I_ \bot ^n(\alpha ){B_{nm}}} .$$

Substitute Eq. (36) into Eq. (29), we get

(41)$${\bar{D}^{\textrm{||}}}({\omega ,\alpha } )= l_{\textrm{||}}^0(\alpha )+ \sum\limits_{m = 1}^{{n_{\max }}} {l_{\textrm{||}}^m(\alpha )\left[ {\left\langle {{{\cos }^{2m}}\theta } \right\rangle - \frac{1}{{2m + 1}}} \right],} $$

with

(42)$$l_{\textrm{||}}^0(\alpha )= \frac{{I_{\textrm{||}}^0(\alpha )}}{2},l_{\textrm{||}}^m(\alpha )= \sum\limits_{n = 1}^{{n_{\max }}} {\left[ {I_{\textrm{||}}^n(\alpha )- \frac{{I_{\textrm{||}}^0(\alpha )}}{{2n + 1}}} \right]} {B_{nm}}.$$

The ratio of the perpendicular and parallel component of the alignment averaged induced dipoles is

(43)$$\sqrt {R(\alpha )} {e^{i\delta (\alpha )}} = \frac{{{{\bar{D}}^ \bot }({\omega ,\alpha } )}}{{{{\bar{D}}^{\textrm{||}}}({\omega ,\alpha } )}} = \frac{{\sum\nolimits_{m = 1}^{{n_{\max }}} {l_ \bot ^m(\alpha )\left[ {\left\langle {{{\cos }^{2m}}\theta } \right\rangle - \frac{1}{{2m + 1}}} \right]} }}{{l_{\textrm{||}}^0(\alpha )+ \sum\nolimits_{m = 1}^{{n_{\max }}} {l_{\textrm{||}}^m(\alpha )\left[ {\left\langle {{{\cos }^{2m}}\theta } \right\rangle - \frac{1}{{2m + 1}}} \right]} }},$$

where $R(\alpha )$ is the intensity ratio and $\delta (\alpha )$ is the phase different for pump-probe angle $\alpha$. From Eq. (43), we obtain

(44)$$\sqrt {R(\alpha )} {e^{i\delta (\alpha )}}l_{\textrm{||}}^0(\alpha )+ \sum\limits_{m = 1}^{{n_{\max }}} {\left[ {\sqrt {R(\alpha )} {e^{i\delta (\alpha )}}l_{\textrm{||}}^m(\alpha )- l_ \bot^m(\alpha )} \right]\left[ {\left\langle {{{\cos }^{2m}}\theta } \right\rangle - \frac{1}{{2m + 1}}} \right]} = 0.$$

Setting

$$\begin{array}{l} y(\alpha )={-} \sqrt {R(\alpha )} {e^{i\delta (\alpha )}}l_{||}^0(\alpha ),\\ {x_m}(\alpha )= \left[ {\sqrt {R(\alpha )} {e^{i\delta (\alpha )}}l_{||}^m(\alpha )- l_ \bot^m(\alpha )} \right],\\ {c_m} = \left\langle {{{\cos }^{2m}}\theta } \right\rangle - \frac{1}{{2m + \textrm{1}}}, \end{array}$$

then Eq. (44) becomes

(45)$$\textrm{y}(\alpha )= \sum\limits_{m = 1}^{{n_{\max }}} {{c_m}{x_m}(\alpha )} .$$

By selecting N(N $\ge $ n_max) pump-probe angles ${\alpha _{\textrm{i},}}1 \le i \le N,$ we can get the coefficients ${c_m}$ by fitting Eq. (45). Then substituting < cos^2mθ>=c_m+1/(2m+1) into Eq. (33) and Eq. (37), we get $[{{a_0},{a_1},\ldots ,{a_{{n_{\max }}}}} ]$ and $\rho (\theta )$. When calculating $l_{\textrm{||}}^m(\alpha )$ and $l_ \bot ^m(\alpha )$, one assumes that the parameters of probe laser are known, such as intensity, wavelength, and duration, and ionization rate and photo-recombination transition dipoles for fixed-in-space molecules from theory are precise.

We take ${n_{\max }} = 2$ as an example to explicitly show the retrieval procedure. The alignment distribution can be written as

(46)$$\rho (\theta )= {a_0} + {a_1}{\cos ^2}\theta + {a_2}{\cos ^4}\theta .$$

E is

(47)$$E = \textrm{8}\left( {\begin{array}{{cc}} {\frac{\textrm{1}}{{\textrm{45}}}}&{\frac{\textrm{2}}{{\textrm{105}}}}\\ {\frac{\textrm{2}}{{\textrm{105}}}}&{\frac{\textrm{4}}{{\textrm{225}}}} \end{array}} \right),$$

with $B = {E^{ - 1}}.$

The coefficients in array A are related to the elements of B by

(48)$${a_1} = {B_{11}}\left( {\left\langle {{{\cos }^2}\theta } \right\rangle - \frac{1}{3}} \right) + {B_{12}}\left( {\left\langle {{{\cos }^4}\theta } \right\rangle - \frac{1}{5}} \right),$$

(49)$${a_2} = {B_{21}}\left( {\left\langle {{{\cos }^2}\theta } \right\rangle - \frac{1}{3}} \right) + {B_{22}}\left( {\left\langle {{{\cos }^4}\theta } \right\rangle - \frac{1}{5}} \right).$$

${a_0}$ can be calculated by using Eq. (33). The alignment averaged induced dipoles are

(50)$${\bar{D}_ \bot }({\omega ,\alpha } )= l_ \bot ^1\left( {\left\langle {{{\cos }^2}\theta } \right\rangle - \frac{1}{3}} \right) + l_ \bot ^2\left( {\left\langle {{{\cos }^4}\theta } \right\rangle - \frac{1}{5}} \right),$$

with

(51)$$l_ \bot ^1 = I_ \bot ^1{B_{11}} + I_ \bot ^2{B_{21,}}l_ \bot ^2 = I_ \bot ^1{B_{12}} + I_ \bot ^2{B_{22}}.$$

And

(52)$${\bar{D}^{\textrm{||}}}({\omega ,\alpha } )= l_{\textrm{||}}^0 + l_{\textrm{||}}^1\left( {\left\langle {{{\cos }^2}\theta } \right\rangle - \frac{1}{3}} \right) + l_{\textrm{||}}^2\left( {\left\langle {{{\cos }^4}\theta } \right\rangle - \frac{1}{5}} \right),$$

with

(53)$$\begin{array}{l} l_{||}^0 = \frac{1}{2}I_{||}^0,\\ l_{||}^1 = \left( {I_{||}^1 - \frac{1}{3}I_{||}^0} \right){B_{11}} + \left( {I_{||}^2 - \frac{1}{5}I_{||}^0} \right){B_{21}},\\ l_{||}^2 = \left( {I_{||}^1 - \frac{1}{3}I_{||}^0} \right){B_{21}} + \left( {I_{||}^2 - \frac{1}{5}I_{||}^0} \right){B_{22}}. \end{array}$$

Thus we have

(54)$$\begin{aligned} \sqrt {R(\alpha )} {e^{i\delta (\alpha )}} &= \frac{{{{\bar{D}}^ \bot }({\omega ,\alpha } )}}{{{{\bar{D}}^{\textrm{||}}}({\omega ,\alpha } )}}\\ &= \frac{{l_ \bot ^1\left( {\left\langle {{{\cos }^2}\theta } \right\rangle - \frac{1}{3}} \right) + l_ \bot ^2\left( {\left\langle {{{\cos }^4}\theta } \right\rangle - \frac{1}{5}} \right)}}{{l_{\textrm{||}}^0 + l_{\textrm{||}}^1\left( {\left\langle {{{\cos }^2}\theta } \right\rangle - \frac{1}{3}} \right) + l_{\textrm{||}}^2\left( {\left\langle {{{\cos }^4}\theta } \right\rangle - \frac{1}{5}} \right)}}, \end{aligned}$$

and with

(55)$$\begin{array}{l} y(\alpha )={-} \sqrt {R(\alpha )} {e^{i\delta (\alpha )}}l_{\textrm{||}}^0(\alpha ),\\ {x_1}(\alpha )= \left[ {\sqrt {R(\alpha )} {e^{i\delta (\alpha )}}l_{\textrm{||}}^1(\alpha )- l_ \bot^1(\alpha )} \right],\\ {x_2}(\alpha )= \left[ {\sqrt {R(\alpha )} {e^{i\delta (\alpha )}}l_{\textrm{||}}^2(\alpha )- l_ \bot^2(\alpha )} \right],\\ {c_1} = \left\langle {{{\cos }^2}\theta } \right\rangle - \frac{1}{3},\\ {c_2} = \left\langle {{{\cos }^4}\theta } \right\rangle - \frac{1}{5}, \end{array}$$

We finally get

(56)$$y(\alpha )= {c_1}{x_1}(\alpha )+ {c_2}{x_2}(\alpha ).$$

In Fig. 13, we check our retrieval method by using the high harmonics of aligned N₂ molecules calculated with the HOMO in the single-molecule level. If we choose n_max=1, i.e., only one constant and a cos²θ term are included in the alignment distribution, the retrieved distribution has a big discrepancy compared to the one (or real one) used for HHG simulations. If we increase n_max to 2, the retrieved distribution almost can reproduce the real one except for very small angles. It doesn’t improve the retrieved result by further increasing n_max to 3. In our retrieval procedure, we thus choose n_max as 2 or 3.

 

Fig. 13. Comparison of the retrieved alignment distributions by setting n_max=1, 2 and 3. The single-molecule high harmonics simulated by only including the HOMO are used to test the retrieval method. Intensity of probe laser is 2.0×10¹⁴ W/cm², maximum alignment degree is < cos²θ>=0.54, and high harmonics are obtained for pump-probe angles of 50°.

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Funding

National Key Research and Development Program of China (2018YFB0504400); National Natural Science Foundation of China (11774175, 11834004, 91850209, 91950102); Science and Technology Commission of Shanghai Municipality (18DZ1112700); Strategic Priority Research Program of Chinese Academy of Sciences (XDB16030300); Key Research Program of Frontier Sciences of Chinese Academy of Sciences (QYZDJ-SSW-SLH010).

Disclosures

The authors declare no conflicts of interest.

References

1. F. Krausz and M. Ivanov, “Attosecond physics,” Rev. Mod. Phys. 81(1), 163–234 (2009). [CrossRef]  

2. M. Hentschel, R. Kienberger, C. Spielmann, G. A. Reider, N. Milosevic, T. Brabec, P. Corkum, U. Heinzmann, M. Drescher, and F. Krausz, “Attosecond metrology,” Nature 414(6863), 509–513 (2001). [CrossRef]  

3. G. Sansone, E. Benedetti, F. Calegari, C. Vozzi, L. Avaldi, R. Flammini, L. Poletto, P. Villoresi, C. Altucci, and R. Velotta, “Isolated single-cycle attosecond pulses,” Science 314(5798), 443–446 (2006). [CrossRef]  

4. T. Remetter, P. Johnsson, J. Mauritsson, K. Varjú, Y. Ni, F. Lépine, E. Gustafsson, M. Kling, J. Khan, R. López-Martens, K. J. Schafer, M. J. J. Vrakking, and A. L’Huillier, “Attosecond electron wave packet interferometry,” Nat. Phys. 2(5), 323–326 (2006). [CrossRef]  

5. L. Cai, J. Marango, and B. Friedrich, “Time-Dependent Alignment and Orientation of Molecules in Combined Electrostatic and Pulsed Nonresonant Laser Fields,” Phys. Rev. Lett. 86(5), 775–778 (2001). [CrossRef]  

6. H. Stapelfeldt and T. Seideman, “Colloquium: Aligning molecules with strong laser pulses,” Rev. Mod. Phys. 75(2), 543–557 (2003). [CrossRef]  

7. O. Ghafur, A. Rouzée, A. Gijsbertsen, W. K. Siu, S. Stolte, and M. J. J. Vrakking, “Impulsive orientation and alignment of quantum-state-selected NO molecules,” Nat. Phys. 5(4), 289–293 (2009). [CrossRef]  

8. Y. Ohshima and H. Hasegawa, “Coherent rotational excitation by intense nonresonant laser fields,” Int. Rev. Phys. Chem. 29(4), 619–663 (2010). [CrossRef]  

9. K. Lin, I. Tutunnikov, J. Qiang, J. Ma, Q. Song, Q. Ji, W. Zhang, H. Li, F. Sun, X. Gong, H. Li, P. Lu, H. Zeng, Y. Prior, I. S. Averbukh, and J. Wu, “All-optical field-free three-dimensional orientation of asymmetric-top molecules,” Nat. Commun. 9(1), 5134 (2018). [CrossRef]  

10. S. Minemoto, T. Umegaki, Y. Oguchi, T. Morishita, A.-T. Le, S. Watanabe, and H. Sakai, “Retrieving photorecombination cross sections of atoms from high-order harmonic spectra,” Phys. Rev. A 78(6), 061402 (2008). [CrossRef]  

11. P. M. Kraus, D. Baykusheva, and H. J. Wörner, “Two-pulse orientation dynamics and high-harmonic spectroscopy of strongly-oriented molecules,” J. Phys. B: At., Mol. Opt. Phys. 47(12), 124030 (2014). [CrossRef]  

12. R. Cireasa, A. E. Boguslavskiy, B. Pons, M. C. H. Wong, D. Descamps, S. Petit, H. Ruf, N. Thiré, A. Ferré, J. Suarez, J. Higuet, B. E. Schmidt, A. F. Alharbi, F. Légaré, V. Blanchet, B. Fabre, S. Patchkovskii, O. Smirnova, Y. Mairesse, and V. R. Bhardwaj, “Probing molecular chirality on a sub-femtosecond timescale,” Nat. Phys. 11(8), 654–658 (2015). [CrossRef]  

13. A. Ehn, J. Bood, Z. Li, E. Berrocal, M. Aldén, and E. Kristensson, “FRAME: femtosecond videography for atomic and molecular dynamics,” Light: Sci. Appl. 6(9), e17045 (2017). [CrossRef]  

14. J. Itatani, J. Levesque, D. Zeidler, H. Niikura, H. Pépin, J. C. Kieffffer, P. B. Corkum, and D. M. Villeneuve, “Tomographic imaging of molecular orbitals,” Nature 432(7019), 867–871 (2004). [CrossRef]  

15. C. Vozzi, M. Negro, F. Calegari, G. Sansone, M. Nisoli, S. De Silvestri, and S. Stagira, “Generalized molecular orbital tomography,” Nat. Phys. 7(10), 822–826 (2011). [CrossRef]  

16. C. Zhai, X. Zhang, X. Zhu, L. He, Y. Zhang, B. Wang, Q. Zhang, P. Lan, and P. Lu, “Single-shot molecular orbital tomography with orthogonal two-color fields,” Opt. Express 26(3), 2775–2784 (2018). [CrossRef]  

17. P. M. Kraus, B. Mignolet, D. Baykusheva, A. Rupenyan, L. Horný, E. F. Penka, G. Grassi, O. I. Tolstikhin, J. Schneider, F. Jensen, L. B. Madsen, A. D. Bandrauk, F. Remacle, and H. J. Wörner, “Measurement and laser control of attosecond charge migration in ionized iodoacetylene,” Science 350(6262), 790–795 (2015). [CrossRef]  

18. I. V. Litvinyuk, K. F. Lee, P. W. Dooley, D. M. Rayner, D. M. Villeneuve, and P. B. Corkum, “Alignment-dependent strong field ionization of molecules,” Phys. Rev. Lett. 90(23), 233003 (2003). [CrossRef]  

19. T. K. Kjeldsen, C. Z. Bisgaard, L. B. Madsen, and H. Stapelfeldt, “Role of symmetry in strong-field ionization of molecules,” Phys. Rev. A 68(6), 063407 (2003). [CrossRef]  

20. R. A. Bartels, T. C. Weinacht, N. Wagner, M. Baertschy, C. H. Greene, M. M. Murnane, and H. C. Kapteyn, “Phase modulation of ultrashort light pulses using molecular rotational wave packets,” Phys. Rev. Lett. 88(1), 013903 (2001). [CrossRef]  

21. J. Wu, H. Cai, H. Zeng, and A. Couairon, “Femtosecond filamentation and pulse compression in the wake of molecular alignment,” Opt. Lett. 33(22), 2593–2595 (2008). [CrossRef]  

22. A. S. Alnaser, X. M. Tong, T. Osipov, S. Voss, C. M. Maharjan, B. Shan, Z. Chang, and C. L. Cocke, “Laser-peak intensity calibration using recoil-ion momentum imaging,” Phys. Rev. A 70(2), 023413 (2004). [CrossRef]  

23. C. Smeenk, J. Z. Salvail, L. Arissian, P. B. Corkum, C. T. Hebeisen, and A. Staudte, “Precise in-situ measurement of laser pulse intensity using strong field ionization,” Opt. Express 19(10), 9336–9344 (2011). [CrossRef]  

24. S. Xu, X. Sun, B. Zeng, W. Chu, J. Zhao, W. Liu, Y. Cheng, Z. Xu, and S. L. Chin, “Simple method of measuring laser peak intensity inside femtosecond laser filament in air,” Opt. Express 20(1), 299–307 (2012). [CrossRef]  

25. Y. He, L. He, P. Wang, B. Wang, S. Sun, R. Liu, B. Wang, P. Lan, and P. Lu, “Measuring the rotational temperature and pump intensity in molecular alignment via high harmonic generation,” Opt. Express 28(14), 21182–21191 (2020). [CrossRef]  

26. C. Jin, S.-J. Wang, S.-F. Zhao, A.-T. Le, and C. D. Lin, “Robust control of minima of high-order harmonics by fine tuning the alignment of CO₂ molecules for shaping attosecond pulses and probing molecular alignment,” Phys. Rev. A 102(1), 013108 (2020). [CrossRef]  

27. X. Zhou, R. Lock, N. Wagner, W. Li, H. C. Kapteyn, and M. M. Murnane, “Elliptically Polarized High-Order Harmonic Emission from Molecules in Linearly Polarized Laser Fields,” Phys. Rev. Lett. 102(7), 073902 (2009). [CrossRef]  

28. M. Lewenstein, P. Balcou, M. Y. Ivanov, A. L’Huillier, and P. B. Corkum, “Theory of high-harmonic generation by low-frequency laser fields,” Phys. Rev. A 49(3), 2117–2132 (1994). [CrossRef]  

29. X. X. Zhou, X. M. Tong, Z. X. Zhao, and C. D. Lin, “Role of molecular orbital symmetry on the alignment dependence of high-order harmonic generation with molecules,” Phys. Rev. A 71(6), 061801 (2005). [CrossRef]  

30. X. X. Zhou, X. M. Tong, Z. X. Zhao, and C. D. Lin, “Alignment dependence of high-order harmonic generation from N₂ and O₂ molecules in intense laser fields,” Phys. Rev. A 72(3), 033412 (2005). [CrossRef]  

31. A.-T. Le, R. R. Lucchese, S. Tonzani, T. Morishita, and C. D. Lin, “Quantitative Rescattering Theory for high-order harmonic generation from molecules,” Phys. Rev. A 80(1), 013401 (2009). [CrossRef]  

32. C. D. Lin, A.-T. Le, C. Jin, and H. Wei, “Elements of the quantitative rescattering theory,” J. Phys. B: At., Mol. Opt. Phys. 51(10), 104001 (2018). [CrossRef]  

33. C. D. Lin, A.-T. Le, C. Jin, and H. Wei, Attosecond and Strong-Field Physics: Principles and Applications (Cambridge University, 2018), pp.209–213.

34. A.-T. Le, R. R. Lucchese, and C. D. Lin, “Uncovering multiple orbitals influence in high harmonic generation from aligned N₂,” J. Phys. B: At., Mol. Opt. Phys. 42(21), 211001 (2009). [CrossRef]  

35. J. Yao, G. Li, X. Jia, X. Hao, B. Zeng, C. Jing, W. Chu, J. Ni, H. Zhang, H. Xie, C. Zhang, Z. Zhao, J. Chen, X. Liu, Y. Cheng, and Z. Xu, “Alignment-Dependent Fluorescence Emission Induced by Tunnel Ionization of Carbon Dioxide from Lower-Lying Orbitals,” Phys. Rev. Lett. 111(13), 133001 (2013). [CrossRef]  

36. G. Li, H. Xie, J. Yao, W. Chu, Y. Cheng, X. Liu, J. Chen, and X. Xie, “Signature of multi-channel interference in high-order harmonic generation from N₂ driven by intense mid-infrared pulses,” Acta Phys. Sin-Ch Ed. 65(22), 224208 (2016). [CrossRef]  

37. C. Jin, A.-T. Le, and C. D. Lin, “Medium propagation effects in high-order harmonic generation of Ar and N₂,” Phys. Rev. A 83(2), 023411 (2011). [CrossRef]  

38. A.-T. Le, R. R. Lucchese, and C. D. Lin, “Polarization and ellipticity of high-order harmonics from aligned molecules generated by linearly polarized intense laser pulses,” Phys. Rev. A 82(2), 023814 (2010). [CrossRef]  

39. X. M. Tong, Z. X. Zhao, and C. D. Lin, “Theory of molecular tunneling ionization,” Phys. Rev. A 66(3), 033402 (2002). [CrossRef]  

40. S.-F. Zhao, C. Jin, A.-T. Le, T. F. Jiang, and C. D. Lin, “Determination of structure parameters in strong-field tunneling ionization theory of molecules,” Phys. Rev. A 81(3), 033423 (2010). [CrossRef]  

41. R. R. Lucchese and V. McKoy, “Studies of differential and total photoionization cross sections,” Phys. Rev. A 26(3), 1406–1418 (1982). [CrossRef]  

42. R. R. Lucchese, G. Raseev, and V. McKoy, “Studies of differential and total photoionization cross sections of molecular nitrogen,” Phys. Rev. A 25(5), 2572–2587 (1982). [CrossRef]  

43. M. Lein, R. De Nalda, E. Heesel, N. Hay, E. Springate, R. Velotta, M. Castillejo, P. L. Knight, and J. P. Marangos, “Signatures of molecular structure in the strong field response of aligned molecules,” J. Mod. Opt. 52(2-3), 465–478 (2005). [CrossRef]  

44. C. Jin, A.-T. Le, S.-F. Zhao, R. R. Lucchese, and C. D. Lin, “Theoretical study of photoelectron angular distributions in single-photon ionization of aligned N₂ and CO₂,” Phys. Rev. A 81(3), 033421 (2010). [CrossRef]  

45. C. Jin, A.-T. Le, and C. D. Lin, “Analysis of effects of macroscopic propagation and multiple molecular orbitals on the minimum in high-order harmonic generation of aligned CO₂,” Phys. Rev. A 83(5), 053409 (2011). [CrossRef]  

46. Y. Mairesse, J. Higuet, N. Dudovich, D. Shafir, B. Fabre, E. Mével, E. Constant, S. Patchkovskii, Z. Walters, M. Y. Ivanov, and O. Smirnova, “High Harmonic Spectroscopy of Multichannel Dynamics in Strong-Field Ionization,” Phys. Rev. Lett. 104(21), 213601 (2010). [CrossRef]  

47. O. Smirnova, Y. Mairesse, S. Patchkovskii, N. Dudovich, D. Villeneuve, P. Corkum, and M. Y. Ivanov, “High harmonic interferometry of multi-electron dynamics in molecules,” Nature 460(7258), 972–977 (2009). [CrossRef]  

48. C. Jin, A.-T. Le, and C. D. Lin, “Retrieval of target photorecombination cross sections from high-order harmonics generated in a macroscopic medium,” Phys. Rev. A 79(5), 053413 (2009). [CrossRef]  

49. M. Geissler, G. Tempea, A. Scrinzi, M. Schnürer, F. Krausz, and T. Brabec, “Light Propagation in Field-Ionizing Media: Extreme Nonlinear Optics,” Phys. Rev. Lett. 83(15), 2930–2933 (1999). [CrossRef]  

50. E. Priori, G. Cerullo, M. Nisoli, S. Stagira, S. De Silvestri, P. Villoresi, L. Poletto, P. Ceccherini, C. Altucci, R. Bruzzese, and C. de Lisio, “Nonadiabatic three-dimensional model of high-order harmonic generation in the few-optical-cycle regime,” Phys. Rev. A 61(6), 063801 (2000). [CrossRef]  

51. V. Tosa, H. T. Kim, I. J. Kim, and C. H. Nam, “High-order harmonic generation by chirped and self-guided femtosecond laser pulses. I. Spatial and spectral analysis,” Phys. Rev. A 71(6), 063807 (2005). [CrossRef]  

52. M. B. Gaarde, J. L. Tate, and K. J. Schafer, “Macroscopic aspects of attosecond pulse generation,” J. Phys. B: At., Mol. Opt. Phys. 41(13), 132001 (2008). [CrossRef]  

53. H.-J. Liang, X. Fan, S. Feng, L.-Y. Shan, Q.-H. Gao, B. Yan, R. Ma, and H.-F. Xu, “Quantum interference of multi-orbital effects in high-harmonic spectra from aligned carbon dioxide and nitrous oxide,” Chin. Phys. B 28(9), 094207 (2019). [CrossRef]  

54. J. Ortigoso, M. Rodriguez, M. Gupta, and B. Friedrich, “Time evolution of pendular states created by the interaction of molecular polarizability with a pulsed nonresonant laser field,” J. Chem. Phys. 110(8), 3870–3875 (1999). [CrossRef]  

55. A.-T. Le and C. D. Lin, “Polarisation states of high harmonic generation from aligned molecules,” J. Mod. Opt. 58(13), 1158–1165 (2011). [CrossRef]