High harmonic spectra contributed by HOMO-1 orbital of aligned CO<sub>2</sub> molecules

Jiawei Li; Peng Liu; Hua Yang; Liwei Song; Shitong Zhao; Hui Lu; Ruxin Li; Zhizhan Xu

doi:10.1364/OE.21.007599

1. Introduction

High-order harmonic generation (HHG) has been applied in producing ultrafast coherent extreme ultraviolet (XUV) radiation and attosecond pulses, and it has become a tool of investigating molecular structure in an ultrafast time scale. HHG is well explained by the three-step model: electrons tunnel into the continuum from the ground state in the intense laser fields; the electrons are then accelerated by a strong oscillating laser field; ultimately the electrons have a probability of being driven back to recombine with the parent ion and emit high-energy harmonic photons [1]. The information of molecular orbital and/or structure can be retrieved from the recombination probability obtained from the harmonic spectra of pre-aligned molecules [2–5]. In ultrafast intense laser fields, molecules are “kicked” by the laser field which induces Stark force and creates a rotational wave packet. The wave packet dephases quickly after the laser pulse is over and re-phases periodically at the intervals of the rotational period of the molecules (revivals). At the revivals, alignment degree $< \cos^{2} θ >$ experiences characteristic modulations indicating the molecules are aligned parallel (alignment) or perpendicular (anti-alignment) to the laser polarization direction. Therefore the corresponding temporal modulation of harmonic generation reveals structural dynamics and rotational dynamics of the molecules [6–8].

For the linear polyatomic CO₂ molecules, both Kanai et al. [9] and Vozzi et al. [10] observed that the time evolved harmonic emission exhibit inverted modulation versus the alignment parameter $< \cos^{2} θ >$ . The modulation inversion can be attributed to the interference of recombining electrons originated from the two oxygen atoms in the CO₂ molecule. Further studies indicate the time evolved harmonic spectra can reflect the angular dependence and the phase variation of HHG at different orders [11–14]. Most recently, the high-order revivals of aligned CO₂ molecules have been revealed to extract the informations of continuum electron dynamics from high harmonic emission [8]. Among these studies, only the HOMO orbital has been taken into consideration by assuming that the inner orbital of CO₂ molecule does not play dominant role in harmonic emission.

The contribution of inner orbital to harmonic generation has been pointed out even though they have higher ionization potential. From temporal modulation of HHG from aligned N₂ molecules, the HOMO-1 orbital of N₂ has been shown to contribute to the harmonic spectrum when the molecules are aligned perpendicular to the laser polarization [15–19]. Using the high harmonic interferometry, the HOMO-2 orbital of CO₂ was found to be pronounced when the molecules aligned parallel to the laser polarization [3,20–22]. However, it is interesting to note that the experimental observation of HOMO-1 orbital in the temporal evolved harmonic emission from aligned CO₂ molecules has not been reported so far. Recently Wu et al. reported the fluorescence emission of $C O_{2}^{+} (A^{2} Π_{u} \to X^{2} Π_{g})$ and $C O_{2}^{+} (B^{2} Σ_{u} \to X^{2} Π_{g})$ in neutral CO₂ molecules irradiated by intense femtosecond laser pulses, which provides direct evidence that the HOMO-1 orbital is simultaneously involved in the process of tunneling ionization [23]. It is therefore desirable to identify the effect of inner orbital HOMO-1 of CO₂ in the harmonic spectra of aligned molecules, because the determination of multi-orbital effect in HHG is a crucial step for the accurate tomography imaging of molecular orbital.

In this work we distinguish the contribution of HOMO-1 orbital in the time evolved harmonic spectra of field-free aligned CO₂ molecules through experiments and numerical calculations. The temporal yield of the 25th harmonics (noted as H25) at the 1/2 revival show a peak when the molecules are best aligned along the laser polarization. The peak has been attributed to contribution of HOMO-2 orbital [3] and it appears till H31 in our experiments. However, we found that the harmonic emission at anti-alignment exists for H33 in the cut-off region. This feature suggests the contribution of HOMO-1 orbital of CO₂, instead of HOMO-2 or the two-center interference effect of HOMO. The angular distributions of tunneling ionization from the three orbitals of CO₂ have been revealed respectively.

2. Experimental setup and result

The experiments were performed using a Ti:sapphire based chirped pulse amplification laser system (Legend-USP-HE followed by a cryogenic multi-pass amplifier, Coherent Inc.), which produces 40 fs laser pulses of 1k Hz at the center wavelength of 800 nm with the maximum pulse energy of 10 mJ. In experiments, about 6 mJ of the output energy were used to be split into two beams: one beam used as the pump pulse (for aligning molecules) and the other as the probe pulse (for driving HHG from molecules) whose polarization was the same as the pump beam. The two beams were collinearly focused with a lens (f = 300 mm) onto a pulsed supersonic molecular beam located in a high vacuum chamber. The laser focus was about 1 mm downstream of a 0.5 mm diameter nozzle orifice and about 2mm before it, where only the short trajectory was in the phase-matching condition. Stagnation pressure of CO₂ gas (99.998%) was about 2 bar, leading to a rotational temperature of 80 Kelvin (K) according to the estimation based on the parameters of the supersonic molecular beam. The spot size of pump laser beam crossing with molecules was measured to be 150 µm (FWHM) and the laser field intensity was estimated to be 6.0 × 10¹³ Wcm⁻². The probe laser energy was adjustable by using a half-wave plate and a high extinction film polarizer. The HHG spectra were detected by a home-made flat-field grating spectrometer equipped with a soft-x-ray CCD camera (Princeton Instruments, PI: SX 400).

Alignment is important for one to see the contribution of the inner orbitals which are only pronounced at certain angles. We optimized the alignment condition by fine adjusting the intensity of pump pulses, which cannot be very high for avoiding ionization. When CO₂ molecules are irradiated by the pump pulses whose duration τ_on = 40 fs are much shorter than the rotational period T_rot = 42.7 ps, nonadiabatic field-free alignment is achieved by instantaneous excitation of a rotational wave packet $ψ (t) = Σ_{J, M} A_{J, M} (t) | J, M 〉$ [24–26]. The time evolution of the wave packet can be calculated by solving the time dependent Schrödinger equation (TDSE). The temporal evolved alignment parameter $〈 \cos^{2} θ 〉 (t)$ is defined as,

〈 \cos^{2} θ 〉 (t) = \sum_{J} ρ_{J} \sum_{J M}^{} {〈 ψ (t) | \cos^{2} θ | ψ (t) 〉}_{J M} (t) .

where ρ_J represents the initial Boltzmann distribution of the molecules over all rotational levels

| J, M 〉

. The harmonic emissions are then obtained by introducing the probe pulses at the varied time delays for comparing with the modulation of alignment degree.

In the experimental result shown in Fig. 1 , we focus on the temporal variations of molecular alignment and harmonic emission at the 1/2 revival. As the probe pulse energy is about 1.8 mJ, the peak intensity is estimated to be ~1.9 × 10¹⁴ Wcm⁻². We can see from Fig. 1 that, for H25-H31, on-axis peaks appear at the delay time of 21.2 ps when the molecules are most aligned along the pump laser polarization. The on-axis emission peak increases to the strongest at H29 and disappears at H33.

Fig. 1 Experimental result of harmonic yields from aligned CO₂ molecules for probe laser intensities of ~1.9 × 10¹⁴ Wcm⁻², as a function of the pump-probe delay time.

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The on-axis peak can be due to the emission from the HOMO-2 orbital which favors ionization at parallel alignment [3,20–22]. Because of the higher ionization potential, contribution from inner orbital is pronounced near the cut-off. Using the method of high harmonic interferometry, O. Smirnova et al. had observed the contribution of HOMO-2 orbital to harmonic yield at parallel alignment near cut-off from aligned CO₂ molecules [3]. Therefore HOMO-2 orbital can make contribution to the on-axis emission peaks for H25-H31, but it is interesting to see that the on-axis peak disappears at H33 while there is still harmonic emission at the anti-alignment condition.

Another possible reason for the on-axis peaks is the two-center interference effect of the HOMO orbital. The previous studies have revealed that for the CO₂ molecules with a HOMO orbital of anti-bonding π_g symmetry a phase jump of π appears at the harmonic orders of maximum destructive two-center interference [27–30], which can be calculated by

R \cos (θ) = n λ, n = 1, 2, 3, ...

where R represents the distance of two center emitters, θ is the angle between molecular axis and probe (HHG) laser polarization, λ is the de Broglie wavelength of a free electron. At the pump field intensity of 6 × 10¹³ Wcm⁻², calculation of the angular probability function

{| ψ (θ, t) |}^{2} \sin θ

at the maximum alignment degree

< \cos^{2} θ >

indicates that most of the molecules are aligned at θ = 30° of the laser polarization. At this angle, Eq. (2) predicts that the maximum destructive interference takes place at H23. After H25, harmonic emission from the HOMO orbital around parallel alignment begins to increase, because the θ for the destructive interference shifts to lager angle as the λ decreases and the yields at small angles will not be depressed any more. This is consistent with our observation shown in Fig. 1 that the on-axis emission peak starts to appear at H25. So the HOMO can also make contribution to the on-axis peaks.

However, the existence of off-axis peaks (harmonics at anti-alignment) at H33 when on-axis peaks disappear cannot be explained by the two-center interference of HOMO. Because the π phase shift is supposed to exist till the next destructive interference position which is determined by Rcos(θ) = 2λ, where the harmonic energy is more than 156 eV, over 100 orders, and beyond our experimental observation. Therefore other origin that has pronounced harmonic yield at the anti-alignment should be responsible for the feature at H33. As is known that the HOMO-1 orbital of CO₂ favors ionization at perpendicular alignment and its contributions to harmonic signal are pronounced near the cut-off. So we suggest the contributions from HOMO-1 are the reason for the feature at H33.

3. Theory simulation and angular distributions of HHG

In order to clarify the effects of inner orbital, numerical calculations are performed using the extended Lewenstein strong field approximation model [31–34]. In this model, the time-dependent dipole moment for a molecule aligned along the z axis is calculated by using:

\begin{matrix} r (t) = i {\int_{0}^{\infty} d τ (\frac{π}{ε + i τ / 2})}^{3 / 2} [\cos θ d_{z}^{*} (t) + \sin θ d_{y}^{*} (t)] \\ \times [\cos θ d_{z}^{} (t - τ) + \sin θ d_{y}^{} (t - τ)] \\ \times E (t - τ) \exp [- i S_{s t} (t, τ)] a^{*} (θ, t) a (θ, t - τ) + c . c .. \end{matrix}

where d_z and d_y are the z and y components of the transition dipole moment from the initial state to the continuum state approximated by plane wave [34]:

d_{y} = \frac{1}{{(2 π)}^{3 / 2}} \int_{- \infty}^{\infty} e^{- i (p_{x} x + p_{y} y + p_{z} z)} y ϕ (x, y, z) d x d y d z,

and the d_z has a similar equation. E(t) is the laser filed linearly polarized on the y - z plane with an angle θ versus the molecular axis. When we consider the contribution to the HHG from more than one orbital, the dipoles are added up coherently. The wavefunction φ(x,y,z) of each molecular orbital is calculated with the GAMESS code using Hartree-Fock method with the 6-311 + + G** basis set [35]. Without considering the Coulomb distortion, the SFA is questionable at lower harmonic orders. However, the experimental result we discuss is near the cut-off where the SFA can give an acceptable prediction [12].

In Eq. (3), S_st is the quasiclassical action at stationary point for the electron propagating in the laser field [31]:

S_{s t} (t, τ) = \int_{t - τ}^{t} ({[p_{s t} (t, τ) - A (t')]}^{2} / 2 + I_{p}) d t',

where

p_{s t} (t, τ) = \int_{t - τ}^{t} A (t') d t' / τ

is the canonical momentum at the stationary point, with A the vector potential. And the I_p is the ionization potential which is different for each molecular orbital. The depletion of the neutral molecules is included by

a (θ, t) = \exp [- \int_{- \infty}^{t} W (θ, t') / 2 d t']

. The alignment-dependent ionization probability for each orbital is

N (θ) = 1 - \exp [- \int W (θ, t) d t]

,

W (θ, t)

is the ionization rate for an instantaneous field strength at ionization time, which is calculated using the method of Murray et al. [36,37].

The time evolution of HHG yield for each orbital can be written as

f_{n} (t) \propto \int N (θ) S_{n} (θ) {| ψ (θ, t) |}^{2} \sin (θ) d θ,

where

S_{n} (θ)

is obtained through the Fourier transform of the dipole moment, for the (2n + 1)th harmonic from CO₂ molecules aligned at a fixed angle θ, and t is the delay time.

{| ψ (θ, t) |}^{2} \sin (θ)

is the weighted angular distribution of the molecules, in which

ψ (θ, t)

is obtained by solving the TDSE of rotational wave packet.

In the first subplot at Fig. 2(a) , the calculated alignment degree $〈 \cos^{2} θ 〉 (t)$ based on the experimental conditions is shown in green dotted dash line, indicating an increase of molecular alignment followed by a decrease. The experimental intensities of H23-H33 are plotted in blue solid lines respectively. One can see that the temporal modulation of 23rd order harmonic emission is inverted according to the molecular alignment. This inversion is consistent with the previous observations and explained by the two-center interference of the recombining electron wave packets including the ground state depletion effect [11,38]. The on-axis peak appears at H25-H31, disappears at H33. The calculation is consistent with the experimental result. According to our simulation, the on-axis peaks are due to the contributions from HOMO-2 and HOMO orbital. Because of its symmetry, the tunneling ionization and HHG of HOMO-2 is pronounced at parallel alignment, so it mainly contributes to the “on-axis peaks” of H25-H31. According to [36,37], the HOMO also has a considerable ionization rate below 45° and it has a relatively lower ionization potential, so HOMO can also make contribution to the “on-axis peaks”, as HOMO-2 does. On the other hand, HOMO-1 has preferable ionization and HHG at larger alignment angles because of its π_u symmetry. The harmonic yields of H33 at the anti-alignment mostly come from the HOMO-1 orbital. It should be noted that HOMO-1 also contributes to H25-H31 at anti-alignment in our calculation.

Fig. 2 (a) The comparison of measured (blue solid line) and calculated (red dash line) H23-H33 yield from aligned CO₂ molecules for probe laser intensities of 1.9 × 10¹⁴ Wcm⁻², as functions of the pump-probe delay time. Error bars are labeled on 50% of data for clearance. Calculated alignment degree $〈 \cos^{2} θ 〉 (t)$ is shown in green dotted dash line in the first subplot. (b) The calculated result without the contribution from the HOMO-1 orbital.

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4. Result at lower probe laser intensity

In experiments we decrease the energy of probe pulses to 1.4 mJ (estimated intensity of 1.2 × 10¹⁴ Wcm⁻²), and the evolved harmonic emissions are given in Fig. 3 . As the intensity of driving field becomes lower, the drop of the cut-off energy decreases the observed harmonic order down to H29. We can see that harmonic emission exhibits inverted modulation versus molecular alignment at H23, and becomes consistent with molecular alignment after H25. HOMO orbital are mainly responsible for the yield of H27 and H29 at maximum alignment and the HOMO-2 may contribute to the harmonic yield a little. This is consistent with our previous conclusion that the two-center interference makes the emission from HOMO becomes more important around the parallel alignment than that around the anti-alignment after H25. The observation that harmonic signal at anti-alignment is weak near the cut-off which suggests that there is little contribution from HOMO-1 at low probe laser intensity.

Fig. 3 Comparison of measured (blue solid line) and calculated (red dash line) H23-H29 yield from aligned CO₂ molecules by the probe laser intensity of 1.2 × 10¹⁴ Wcm⁻², as a function of pump-probe delay. Error bars are labeled on 50% of data for clearance.

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In order to elucidate this effect, we calculate the angular distributions of the tunneling ionization from the three orbitals using the method proposed by Murray et al. [36,37]. As shown in Fig. 4 , the ionization probability of the HOMO-1 becomes higher with increasing the angle and the HOMO-2 shows an opposite trend. When the laser intensity is 1.2 × 10¹⁴ Wcm⁻², the tunneling ionization of inner orbitals are weaker than HOMO even at the angles where electronic density is maximum. When the laser intensity is 1.9 × 10¹⁴ Wcm⁻², the tunneling ionization rate of HOMO-1 increases and becomes much larger than that of HOMO near 90° and the tunneling ionization of HOMO-2 also increases. This result confirms that the lower laser intensity decreases the contributions of HOMO-1 especially at the anti-alignment angle. This suggests that HOMO-1 of CO₂ can only be distinguished at higher probe laser intensity, which is consistent with the results on N₂ [16].

Fig. 4 Calculated angular distributions of the tunneling ionization of HOMO (blue line), HOMO-1 (red line) and HOMO-2 (green line) at (a) 1.2 × 10¹⁴ Wcm⁻² and (b) 1.9 × 10¹⁴ Wcm⁻² respectively.

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In our calculation, we have also tried to use even higher driving laser field for the expectation of stronger effect of HOMO-1, however, the ground state depletion effect becomes significant, which suppresses both the contributions of HOMO and HOMO-1. As a result, no clear phenomenon is found from the contribution of HOMO-1.

5. Summary

In conclusion, we have investigated the influence of multiple orbital to the HHG from aligned CO₂ molecules as a function of pump-probe delay time. At the relatively high intensity of probe laser, harmonic yields present a small peak when the molecules are best aligned. The on-axis peak disappears near the cut-off which cannot be explained by the two-center interference of the HOMO orbital or the emission from the HOMO-2 orbital. Simulation reproduced the experimental observation, and it shows that the disappearance of the on-axis peak and the much stronger emission at anti-alignment of H33 indicate the harmonic yields around anti-alignment are from the HOMO-1 orbital. The HOMO-1 also contributes to the harmonic emission at the perpendicular alignment for H25-H31. The on-axis peaks at H25-H31 are due to the emission from HOMO-2 and HOMO orbitals. At the lower probe laser intensity, the angular distributions of tunneling ionization from HOMO and HOMO-1 show that contribution from HOMO-1 can be ignored. The experimental observation of the contributions from HOMO-1 to the evolving HHG from the aligned CO₂ is the first time. From this work we can learn that HOMO-1 orbital affect the time evolved harmonic spectra significantly when the laser intensity increases, especially at the orders near the cut-off. Our work also suggests that relative contributions from different orbital to harmonic emission are varied for different harmonic orders. We may control their ratio by fine-tuning the laser intensity at a particular order which is useful for the tomography imaging of molecular orbital.

Acknowledgments

This work is supported by National Natural Science Foundation of China (Grant Nos. 60978012, 11274326 and 11134010), the 973 Program of China (2011CB808103), and the State Key Laboratory of High Field Laser Physics of China.

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High harmonic spectra contributed by HOMO-1 orbital of aligned CO₂ molecules

Abstract

1. Introduction

2. Experimental setup and result

3. Theory simulation and angular distributions of HHG

4. Result at lower probe laser intensity

5. Summary

Acknowledgments

References and links

Cited By

Figures (4)

Equations (6)

Optics Express