Two-center interference in high-order harmonic generation from heteronuclear diatomic molecules

Two-center interference for heteronuclear diatomic molecules (HeDM) is investigated. The minimum in the high-order harmonic spectrum, as a consequence of the destructive interference, is shifted to lower harmonic orders compared with that in a homonuclear case. This phenomenon is explained by performing phase analysis. It is found that, for an HeDM, the high harmonic spectrum contains information not only on the internuclear separation but also on the properties of the two separate centers, which implies the potential application of estimating the asymmetry of molecules and judging the linear combination of atomic orbitals (LCAO) for the highest occupied molecular orbital (HOMO). Moreover, the possibility to monitor the evolution of HOMO itself in molecular dynamics is also promised. © 2011 Optical Society of America OCIS codes: (190.7110) Ultrafast nonlinear optics; (190.4160) Multiharmonic generation; (300.6560) Spectroscope, x-ray. References and links 1. M. Lewenstein, Ph. 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Introduction
When atoms and molecules are exposed to intense laser irradiation, high-order harmonics are generated.The high-order harmonic generation (HHG) has been an attractive topic for the passed two decades.The three-step model can help us understand its physical mechanism: (i) an atom or molecule emits its electron to continuous state, (ii) the electron is accelerated in the laser field, (iii) the accelerated electron recombines with the parent ion and high energy photon is radiated [1].HHG from atoms is first concerned for the potential applications to serve as a source of coherent radiation in the EUV band and to produce attosecond pulses [2][3][4][5][6].When investigations of HHG are turned to molecules, new challenges and opportunities come forth.Many novel applications are promised, such as molecular orbital tomographic imaging [7], attosecond time-resolved electronic and nuclear dynamics probing in molecules [8,9], polarization manipulation of high harmonics and attosecond pulses [10,11].
A very important phenomenon for molecular HHG is the "minimum" in the harmonic spectrum.Lein et al have predicted the minima of H 2 and H + 2 theoretically by numerically solving the time-dependent Schrödinger equation (TDSE) [12,13].They demonstrate that this kind of intensity independent minimum, which is named structural minimum, is a consequence of the interference between the emissions from the two centers, when the de Broglie wavelength of the returning electron, the internuclear distance R and alignment angle θ satisfy the relationship Rcos θ =(2m+1)λ /2.This model indicates that the destructive interference requires a phase difference of π which originates from the recombination processes of the returning wave packet to separate centers.The two-center interference discussed here can be interpreted as reproduction of Young's two-slit experiment in the microscopic world.A further insight into this phenomenon has then been gained theoretically by Kamta et al [14].
The destructive interference has been experimentally observed by Kanai et al [15], where the HHG from CO 2 has been investigated .The highest occupied molecular orbital (HOMO) of CO 2 is mainly contributed by the two oxygen atoms, so CO 2 can be treated as a stretched O 2 molecule with two emitting center.It was found that their experimental data agreed well with the theoretical prediction given by Lein's model.Almost at the same time, Vozzi et al also observed this phenomenon in their experiment [16], and the result was also consistent with the two-center model supposing the recombining electron kinetic energy E k =hω-I p .Moreover, Smirnova et al have found intensity dependent minima in their experimental work [9].They explain that this kind of minimum, which is named dynamical minimum, results from the interference of multiple channels.
A structural minimum contains very important information, from which structure of objective molecule can be probed and symmetry of the bond can be judged [13,17].The two-center interference also plays an important role in the study of the attosecond nuclear motion [18] and the imaging of molecular orbital.But till now, researches on the two-center interference mainly focus on homonuclear diatomic molecule (HoDM), such as H 2 , N 2 , O 2 , or some other simple molecules with symmetric HOMOs like CO 2 [19][20][21][22][23]. Minima from those symmetric molecules are just dependent on internuclear distances and alignment angles, while little information on the combining atoms is included.For example, minima were observed at the same position around 55eV in both harmonic spectra from N 2 O and CO 2 at parallel alignment, which is determined by the same internuclear distance of 2.3 Å.Note that although N 2 O is an asymmetric molecule, it has a very similar HOMO to that of CO2 with a π g symmetry [22,23].
In this paper, two-center interference for heteronuclear diatomic molecule (HeDM), which has seldom been studied to the best of our knowledge, is investigated.We focus on the structural minimum caused by this two-center interference and find that the minimum shows new characteristics.The phase difference between the two emissions in an HeDM consists of two parts: the spatial phase difference and the additional recombining phase difference.The latter part arises from the unequal recombining processes, which will induce a shift of the interference minimum in the spectrum.This implies that information not only on the internuclear separation but also on the properties of the two separate centers is packed in the harmonic spectrum.Our study indicates that the two-center interference offers opportunities to estimate the asymmetry of the objective HeDM and to judge the linear combination of diatomic orbital (LCAO) for molecular orbitals.Moreover, the possibility to monitor the evolution of the HOMO itself in molecular dynamics is promised.

Theoretical model
We apply CO as the target molecule, because effective destructive interference could be observed due to the comparable contributions to HOMO from atoms C and O.As shown in Fig. 1, (X,Y,Z) is the molecule frame, and (x,y,z) is the laboratory frame.The orientation angle θ is defined as the angle between the directions of the intrinsic dipole moment of CO (Z axis in the molecule frame) and the electric field (at the ionization time).The length | R C | and | R O | are 0.65 Å and 0.48 Å respectively, so the internuclear distance R=| R O -R C |=1.13 Å. Assume that the incident pulse is linearly polarized along x axis, and the accelerated electron returns in the direction of polarization with particular momentum p (i.e.particular de Broglie wavelength λ ).The HOMO of CO is σ -type, schematic of which can be seen in Fig. 1.According to the LCAO approximation, where Ψ CO is the HOMO of model CO constructed by the linear combination of atomic orbitals ψ μ .These atomic orbitals are approximated by Gaussian basis set.As the recombination at respective center is concerned, the linear summation is divided into 2 groups according to different contributing atoms [19,20]:  We employ the Lewenstein model [1] to calculate the harmonic radiation.The timedependent dipole moment is given by In this equation, E(t) is the electric field of the laser pulse, A(t) is the corresponding vector potential, ε is a positive regularization constant.p st and S st are the stationary momentum and quasiclassical action, which are given by where I p =14.014 eV is the ionization energy.The harmonic spectrum is then obtained by Fourier transforming the time-dependent dipole acceleration a(t) [25]: where a(t) = ẍ(t), ω L is the frequency of the driving pulse and q corresponds to the harmonic order.
The transition dipole moment between the ground and the continuum state is calculated by The linear combination for HOMO of CO has been divided into two groups, so the transition dipole moment can be written as In the above equation, the terms growing linearly with length R C and R O are artifacts coming from the lack of orthogonality between the Volkov states and the field-free bound states, and actually they have little influence on the result [19,20].We neglect these two terms and get where êc and êo are unit vectors of d C and d O respectively.The dipole moments are obtained by where two terms are included.The former term, called the recombining phase difference, represents the phase difference from the unequal recombination processes; while the latter one, called the spatial phase difference, represents the phase difference that originates when the wave packet of returning electron travels through the projection of the internuclear distance on the laser polarization axis.For a bonding HoDM, Eq. ( 12) will reduce to [18][19][20]26] and only the spatial term in Eq. ( 14) remains.An antibonding HoDM can be seen as a special kind of HeDM, for which an intrinsic phase difference of π exists.Unless stated otherwise, HoDM refers to the bonding kind in this paper.
For the recombination, a modification should be applied, that is, d rec ( p)= d CO ( p k ) and effective momentum p k = p 2 + 2I p p/| p| [19,26].This modification implies that the recombination occurs within the potential well and the acceleration effect of the binding potential is considered in the modified Lewenstein model.As discussed in previous works, the returning electron is accelerated and the corresponding wavelength decreases when the electron enters the potential well before recombination near the nuclei [13,14].The interference is determined by the wavelength where the electron is faster in the core region instead of the wavelength corresponding to the asymptotic energy far away [17].The results with and without this modification have been compared: if the modification is not applied, the minimum position estimated is far away from the exact numerical result [27].
To confirm this modified model, we compare our result obtained by this model with Lein's obtained by solving TDSE numerically.As shown in Fig. 3, the two results agree well considering the positions of minima [12,13].Consistent results with Kamta's are also obtained in various angles [14], which confirms that this modified model is appropriate.

Result and discussion
Parallel alignment with the orientation angle θ =0 • is investigated first.According to the twocenter model, and destructive interference takes place when the electron recombines with energy 29.54 eV or 265.87 eV for an HoDM with internuclear distance R=1.13 Å.To extend the cutoff to higher harmonic orders with relative low laser intensity, mid-infrared driving pulse at 1300 nm is used [22], which implies the first and second interference minimum should locate around the 31st and 279th harmonic orders.Figure 4 shows the harmonic spectrum of CO at θ =0 • .Calculation is performed for laser intensity I=3×10 14 W/cm 2 and a duration of 30 fs full width at half maximum, which can be achieved by the optical parametric amplifier (OPA) technique nowadays [28].The sine squared envelope is employed as f(t)=sin 2 ( πt T p ), where T p determines the pulse duration.A minimum is observed around the 131st harmonic, which is far away from the 31st or 279th harmonic in the HoDM case.
To explain this phenomenon and understand the underlying mechanism, phase analysis is performed.The recombining phases for the two centers versus returning electron energy are shown in Fig. 5(a), as indicated by the yellow and blue curves respectively.Then the recombining phase difference is obtained, which is indicated by the green curve in Fig. 5(a).In Fig. 5(b), both the recombining phase difference and the spatial phase difference (blue curve) are presented, the sum of which turns to be the total phase difference (red curve) according to Eq. (14).All the phases are shifted into the interval from -π to π.
The two-center model indicates that the interference takes place when the total phase difference equals π (or -π).As depicted by Eq. ( 14), in an HoDM with the same internuclear distance, the total phase difference is dominated by the spatial phase difference, so the minima will locate where the blue curve in Fig. 5 Harmonic order Harmonic intensity (arb.units) Fig. 4. Harmonic spectrum of CO at θ =0 • , with laser intensity I=3×10 14 W/cm 2 , wavelength λ =1300 nm and a duration of 30 fs.
the recombining phase difference is added (green curve).This induces a shift of the position where the total phase difference reaches π to 14.99 eV and 121.6 eV (red curve), which imply the 15th and the 127th harmonics for 1300 nm.The latter well explains the observed minimum in Fig. (4).But there is no minimum around the 15th harmonics.The discrepancy arises from the inaccuracy at low orders caused by the strong-field approximation (SFA) in Lewenstein model, and from the fact that the amplitude of d C and d O are not comparable enough to form efficient destructive interference there.The result illustrates that minima in HeDMs are not dependent on the internuclear separation alone.Shift of the position of minimum, which originates from the difference between the two recombination processes, is an important measurement for the asymmetry of the molecule.Note that the asymmetry is probed by the different phase properties of each recombination process, and the asymmetry of the phase property is probed actually.Further, this phase asymmetry is a consequence of the asymmetric LCAO for HOMO, since the phase of each recombination process is affected by the combination of atomic orbitals for each center.So, the asymmetry of the HOMO of objective molecule is probed finally.An antibonding HoDM is the opposite case of bonding HoDM, where a constant phase shift of π (or -π) exists in full energy region.The amplitude difference will not shift the minimum, but the degree of destructive interference will be reduced.
According to the molecular orbital theory, only atomic orbitals with close energy levels can efficiently combine to molecular orbitals.For an HoDM, pairs of identical atomic orbitals overlap to form molecular orbitals with equal contributions.But the case is more complex in HeDMs.It is not that easy to say how an HOMO is formed in an HeDM.The HOMO is combined by some atomic orbitals with various coefficients, or some HOMOs are non-bonding orbitals for lone pairs only from one atom.Our investigation has offered an opportunity to identify the molecular orbitals, with the help of the relationship between the minimum and the orbital combination.Since the position of minimum is dependent on the orbital combination, the LCAO can be known by comparing the observation of the position of minimum with the theoretical predictions [29].
For example, we assume a linear combination for CO: We then calculate the harmonic spectrum of this artificial CO, which predicts a minimum around the 119th order.As shown in Fig. 6, harmonic spectra from the real and artificial HOMO are both presented.To obtain a clearer observation, thick curves are plotted to represent the smoothed harmonic spectra where fine structure has been eliminated by convolution with a Gaussian of appropriate width [13]: where σ = 5ω L .The minima of the two curves locate at different frequencies in the spectra.By observing the minimum position, we will be able to judge which combination the HOMO is really combined of.Moreover, once the relationship between the harmonic spectra and the LCAO is already known, it will be possible to monitor molecular dynamics.How the combination of the orbitals changes or how the HOMO evolves can be deduced by reading the harmonic spectra.The electric field is alternating in an laser pulse, and the asymmetric molecule will experience opposite electric field in adjacent half optical cycle.This means that harmonics at θ and π − θ orientations will be generated in one shot.The HHG and phase analysis for CO oriented at θ =180 • have been investigated, shown in Fig. 7.The applied laser parameters and the representations of each curve are the same as in Fig. 4   It is shown in Fig. 7 that the reversal of electric field does not change the shape of the harmonic spectra, and all the phases are just reversed.According to the definition of total phase difference in Eq. ( 14), in the reversed electric field The total phase difference has just changed its sign, but the position where it reaches π orπ does not change.So the minimum position will not change and the shape of the spectrum remains the same.Note that we are emphasizing the "shape" of the spectrum.The two-center interference is insensitive to the reversal of electric field.But the ionization rate might be increased or decreased in opposite electric field.The symmetry between half-cycles in HHG is broken due to the preferred direction of ionization or Column focusing effect [30].This system-induced gating can be utilized to manipulate the HHG to obtain a wide supercontinuum [31].Recently, a research has been done for the HHG of CO as a polar molecule [32].The author indicates that the effect of system-induced gating of CO will be weakened by the Stark shift.As we are focusing on the two-center interference effect in this paper and the ionization-induced asymmetry is limited by the stark shift, the assumption that HHG of CO is insensitive to the reversal of electric field is employed in this paper.
Next, we extend our investigation to all the orientation angles.Phase analysis for spatial phase difference alone is first performed, shown in Fig. 8.The boundaries of the red and blue colors represent where the phase differences reach π (or −π).For an HoDM which has the same internuclear distance as CO, the minima will locate in harmonic spectra along the boundary lines in Fig. 8.The two pairs of boundary lines also match with the prediction by Eq. ( 16).
Phase analysis for total phase difference is shown in Fig. 9.The boundaries of the red and blue colors represent where the phase differences reach π (or −π) too.Compare Fig. 9 with Fig. 8, the boundary lines, which predict the interference minima, shift to lower energy.This phenomenon is due to the added recombining phase difference as discussed above.The conclusion that all the phases just change the signs in opposite electric field can also be extended to all orientation angles.
Moreover, new characteristic is observed.For an HoDM, there is no minimum at angles close to 90 • , i.e., perpendicular alignment (Fig. 8).The position of the minimum tends to infinity  according to the calculation of Eq. ( 16).But for CO, the boundary lines bend to 90 • .It is because although the spatial phase difference decreases rapidly as the angle tends to 90 • , the recombining phase difference remains.At these angles, minima are mainly affected by the recombining phase difference, which reflect the properties of the atoms.
Harmonic spectra at all orientation angles are obtained in Fig. 10.To observe the entire variation of minima, a stronger intensity I=6×10 14 W/cm 2 is applied to locate most of the minima in the plateau region.The other parameters are kept the same as in Fig. 4. Two pairs of "minimum grooves" are observed, which is consistent to the prediction of phase analysis in Fig. 9.The symmetry with respect to 90 • is observed, which has been discussed in the parallel condition.The reason for the blurring and disappearing of the first minimum grooves at small angles and angles tending to 180 • has been discussed above too.The HHG efficiency decreases as θ tends to 90 • due to the nodal structure of the HOMO [33].As an extension of the investigation for parallel alignment, it is possible to judge the LCAO information about the HOMO and monitor molecular dynamics by reading the shift and the bend of the minima curves rather than by measuring the shift of one minimum only.For a highly asymmetric HeDM like HF, whose non-bonding HOMO is dominantly contributed by the fluorine atom, angle independent result is obtained.Fig. 10.Harmonic spectra at all orientation angles in arbitrary unit.I=6×10 14 W/cm 2 , the other parameters are the same as in Fig. 4.

Conclusion
Two-center interference for a typical HeDM CO is investigated.Harmonic spectra are obtained using modified Lewenstein model.The positions of minima in the spectra are found to be shifted to lower harmonic orders compared with that from an HoDM with the same internuclear distance.Phase analysis is performed to explain this phenomenon.It shows that the shift results from the additional recombining phase difference, which pushes the position where the value of total phase difference equals π to lower harmonic orders.It is found that the result of such interference is not influenced by the reversal of the electric field.The investigation is also extended to all orientation angles.In addition to the shift of minima, new phenomenon that the angle-dependent minimum curves bend to 90 • is observed.No minimum will be found at angles close to perpendicular alignment in the harmonic spectra from an HoDM, but in the HeDM case minimum appears at these angles where the recombining phase difference dominates the interference.Our investigation indicates that information on not only the molecular structure but also the atomic orbital combination for HOMO at the two centers is packed in the harmonic spectra of an HeDM, which can be judged by probing the HHG signals.Moreover, our study also implies the possibility to monitor the evolution of the HOMOs in molecular dynamics.Finally, besides diatomic molecules, our analysis method for the two-center interference is applicable to any asymmetric molecule whose HOMO is mainly contributed by two centers.

Fig. 1 .
Fig. 1.The schematic for the process of recombination.(X,Y,Z) is the molecule frame, and (x,y,z) is the laboratory frame.θ is defined as the angle between the directions of the intrinsic dipole moment of CO and the electric field.| R C | and | R O | are 0.65 Å and 0.48 Å respectively.
where Ψ C and Ψ O represent the sum of orbitals contributed from atoms C and O respectively.The coefficients c μ (c u , c v ) for the weights of contributions are obtained with the Gaussian 03 ab initio code[24].Here Ψ C and Ψ O are two portions of the HOMO of CO rather than independent atomic orbitals of C and O.Sectional view of the constructed HOMO is shown in Fig.2(b), the portions Ψ C and Ψ O are also shown in Fig.2(a) and Fig.2(c).We can see that Ψ C is mainly an combination of 2s and 2pz orbitals, while Ψ O is purely 2pz approximately.

Fig. 3 .
Fig. 3. Harmonic spectra of H 2 obtained using the Lewenstein model with the modified effective momentum p k .780nm laser is used with intensity I=5×10 14 W/cm 2 and a duration of 26 fs.Internuclear distance of H 2 is 1.4 a.u. and the alignment angle is (a) 0 o , and (b) 30 o respectively.
(b) equals π.The positions are right 29.54 eV and 265.87 eV, just the same as the prediction from two-center model.While for the CO molecule,

Fig. 5 .
Fig. 5. Phase analysis for the recombination of CO.

Fig. 6 .
Fig.6.Comparison of HHG from two probable HOMO of CO, a real one (blue) and an artificial one with orbital combination described in the text (red).The thick curves represent the smoothed harmonic spectra.

Fig. 7 .
Fig. 7.The HHG and phase analysis for CO oriented at θ =180 • .The applied laser parameters and the representations of each curve are the same as in Fig. 4 and Fig 5.

Fig. 8 .
Fig. 8. Phase analysis for spatial phase difference.The boundaries of the red and blue colors represent where the phase differences reach π (or −π).All the phases are shifted in the interval from -1 to 1 (in unit of π).

Fig. 9 .
Fig. 9. Phase analysis for total phase difference.The boundaries of the red and blue colors represent where the phase differences reach π (or −π).All the phases are shifted in the interval from -1 to 1 (in unit of π).
r respectively, where the integrations are performed numerically.The two terms in Eq. (12) describe the recombination process of each center respectively.-p • R C and -p • R O represent phases gained by the returning electrons before recombination.The phases φ C and φ O , which arise from the recombination, are obtained by the phase angles of the transition dipole moments d C and d O respectively.The total phase difference between the emissions from the two centers is defined as