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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 19, Iss. 2 — Jan. 17, 2011
  • pp: 436–447
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Two-center interference in high-order harmonic generation from heteronuclear diatomic molecules

Xiaosong Zhu, Qingbin Zhang, Weiyi Hong, Pengfei Lan, and Peixiang Lu  »View Author Affiliations


Optics Express, Vol. 19, Issue 2, pp. 436-447 (2011)
http://dx.doi.org/10.1364/OE.19.000436


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Abstract

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

1. 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

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

]. 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

2. W. Cao, P. Lu, P. Lan, X. Wang, and Y. Li, “Control of the launch of attosecond pulses,” Phys. Rev. A 75, 063423 (2007). [CrossRef]

6

6. S. Gilbertson, S. D. Khan, Y. Wu, M. Chini, and Z. Chang, “Isolated attosecond pulse generation without the need to stabilize the carrier-envelope phase of driving lasers,” Phys. Rev. Lett. 105, 093902 (2010). [CrossRef] [PubMed]

]. 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

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

], attosecond time-resolved electronic and nuclear dynamics probing in molecules [8

8. W. Li, X. Zhou, R. Lock, S. Patchkovskii, A. Stolow, H. C. Kapteyn, and M. M. Murnane, “Time-resolved dynamics in N2O4 probed using high harmonic generation,” Science 322, 1207–1211 (2008). [CrossRef] [PubMed]

, 9

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

], polarization manipulation of high harmonics and attosecond pulses [10

10. 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, 073902 (2009). [CrossRef] [PubMed]

, 11

11. O. Smirnova, S. Patchkovskii, Y. Mairesse, N. Dudovich, D. Villeneuve, P. Corkum, and M. Yu. Ivanov, “Attosecond circular dichroism spectroscopy of polyatomic molecules,” Phys. Rev. Lett. 102, 063601 (2009). [CrossRef] [PubMed]

].

A very important phenomenon for molecular HHG is the “minimum” in the harmonic spectrum. Lein et al have predicted the minima of H2 and H2+ theoretically by numerically solving the time-dependent Schrödinger equation (TDSE) [12

12. M. Lein, N. Hay, R. Velotta, J. P. Marangos, and P. L. Knight, “Role of the intramolecular phase in high-harmonic generation,” Phys. Rev. Lett. 88, 183903 (2002). [CrossRef] [PubMed]

, 13

13. M. Lein, N. Hay, R. Velotta, J. P. Marangos, and P. L. Knight, “Interference effects in high-order harmonic generation with molecules,” Phys. Rev. A 66, 023805 (2002). [CrossRef]

]. 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 Rcosθ=(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

14. G. L. Kamta and A. D. Bandrauk, “Three-dimensional time-profile analysis of high-order harmonic generation in molecules: nuclear interferences in H2+,” Phys. Rev. A 71, 053407 (2005). [CrossRef]

].

The destructive interference has been experimentally observed by Kanai et al [15

15. T. Kanai, S. Minemoto, and H. Sakai, “Quantum interference during high-order harmonic generation from aligned molecules,” Nature 435, 470–474 (2005). [CrossRef] [PubMed]

], where the HHG from CO2 has been investigated. The highest occupied molecular orbital (HOMO) of CO2 is mainly contributed by the two oxygen atoms, so CO2 can be treated as a stretched O2 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

16. C. Vozzi, F. Calegari, E. Benedetti, J. P. Caumes, G. Sansone, S. Stagira, M. Nisoli, R. Torres, E. Heesel, N. Kajumba, J. P. Marangos, C. Altucci, and R. Velotta, “Controlling two-center interference in molecular high harmonic generation,” Phys. Rev. Lett. 95, 153902 (2005). [CrossRef] [PubMed]

], and the result was also consistent with the two-center model supposing the recombining electron kinetic energy Ek=h̄ω-Ip. Moreover, Smirnova et al have found intensity dependent minima in their experimental work [9

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

]. 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

13. M. Lein, N. Hay, R. Velotta, J. P. Marangos, and P. L. Knight, “Interference effects in high-order harmonic generation with molecules,” Phys. Rev. A 66, 023805 (2002). [CrossRef]

, 17

17. M. Lein, “Molecular imaging using recolliding electrons,” J. Phys. B 39, R135–R173 (2007). [CrossRef]

]. The two-center interference also plays an important role in the study of the attosecond nuclear motion [18

18. S. Baker, J. S. Robinson, M. Lein, C. C. Chirilǎ, R. Torres, H. C. Bandulet, D. Comtois, J. C. Kieffer, D. M. Villeneuve, J. W. G. Tisch, and J. P. Marangos, “Dynamic two-center interference in high-order harmonic generation from molecules with attosecond nuclear motion,” Phys. Rev. Lett. 101, 053901 (2008). [CrossRef] [PubMed]

] and the imaging of molecular orbital. But till now, researches on the two-center interference mainly focus on homonuclear diatomic molecule (HoDM), such as H2, N2, O2, or some other simple molecules with symmetric HOMOs like CO2 [19

19. Y. J. Chen, J. Liu, and B. Hu, “Reading molecular messages from high-order harmonic spectra at different orientation angles,” J. Chem. Phys. 130, 044311 (2009). [CrossRef] [PubMed]

23

23. R. M. Lock, X. Zhou, W. Li, M. M. Murnane, and H. C. Kapteyn, “Measuring the intensity and phase of high-order harmonic emission from aligned molecules,” Chem. Phys. 36622–32 (2009). [CrossRef]

]. 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 N2O and CO2 at parallel alignment, which is determined by the same internuclear distance of 2.3Å. Note that although N2O is an asymmetric molecule, it has a very similar HOMO to that of CO2 with a πg symmetry [22

22. R. Torres, T. Siegel, L. Brugnera, I. Procino, J. G. Underwood, C. Altucci, R. Velotta, E. Springate, C. Froud, I. C. E. Turcu, M. Yu. Ivanov, O. Smirnova, and J. P. Marangos, “Extension of high harmonic spectroscopy in molecules by a 1300 nm laser field,” Opt. Express 18, 3174–3180 (2010). [CrossRef] [PubMed]

,23

23. R. M. Lock, X. Zhou, W. Li, M. M. Murnane, and H. C. Kapteyn, “Measuring the intensity and phase of high-order harmonic emission from aligned molecules,” Chem. Phys. 36622–32 (2009). [CrossRef]

].

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.

2. 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 |R⃗C| and |R⃗O| are 0.65 Å and 0.48 Å respectively, so the internuclear distance R=|R⃗O-R⃗C|=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 p⃗ (i.e. particular de Broglie wavelength λ).

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. |R⃗C| and |R⃗O| are 0.65 Å and 0.48 Å respectively.

The HOMO of CO is σ-type, schematic of which can be seen in Fig. 1. According to the LCAO approximation,
ΨCO(r)=μcμψμ(rRμ)=ΨC(rRC)+ΨO(rRO),
(1)
where ΨCO is the HOMO of model CO constructed by the linear combination of atomic orbitals ψμ. 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

19. Y. J. Chen, J. Liu, and B. Hu, “Reading molecular messages from high-order harmonic spectra at different orientation angles,” J. Chem. Phys. 130, 044311 (2009). [CrossRef] [PubMed]

, 20

20. C. Figueira de Morisson Faria and B. B. Augstein, “Molecular high-order harmonic generation with more than one active orbital: quantum interference effects,” Phys. Rev. A 81, 043409 (2010). [CrossRef]

]:
ΨC(rRC)=ucuψu(rRC),
(2)
ΨO(rRO)=vcvψv(rRO),
(3)
where ΨC and ΨO represent the sum of orbitals contributed from atoms C and O respectively. The coefficients cμ (cu, cv) for the weights of contributions are obtained with the Gaussian 03 ab initio code [24

24. M. J. Frisch, G. W. Trucks, H. B. Schlegel, G. E. Scuseria, M. A. Robb, J. R. Cheeseman, J. A. Montgomery Jr., T. Vreven, K. N. Kudin, J. C. Burant, J. M. Millam, S. S. Iyengar, J. Tomasi, V. Barone, B. Mennucci, M. Cossi, G. Scalmani, N. Rega, G. A. Petersson, H. Nakatsuji, M. Hada, M. Ehara, K. Toyota, R. Fukuda, J. Hasegawa, M. Ishida, T. Nakajima, Y. Honda, O. Kitao, H. Nakai, M. Klene, X. Li, J. E. Knox, H. P. Hratchian, J. B. Cross, V. Bakken, C. Adamo, J. Jaramillo, R. Gomperts, R. E. Stratmann, O. Yazyev, A. J. Austin, R. Cammi, C. Pomelli, J. W. Ochterski, P. Y. Ayala, K. Morokuma, G. A. Voth, P. Salvador, J. J. Dannenberg, V. G. Zakrzewski, S. Dapprich, A. D. Daniels, M. C. Strain, O. Farkas, D. K. Malick, A. D. Rabuck, K. Raghavachari, J. B. Foresman, J. V. Ortiz, Q. Cui, A. G. Baboul, S. Clifford, J. Cioslowski, B. B. Stefanov, G. Liu, A. Liashenko, P. Piskorz, I. Komaromi, R. L. Martin, D. J. Fox, T. Keith, M. A. Al-Laham, C. Y. Peng, A. Nanayakkara, M. Challacombe, P. M. W. Gill, B. Johnson, W. Chen, M. W. Wong, C. Gonzalez, and J. A. Pople, “Gaussian 03, Revision C.02,” Gaussian Inc., Wallingford, CT (2010).

]. Here ΨC and ΨO 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 ΨC and ΨO are also shown in Fig. 2(a) and Fig. 2(c). We can see that ΨC is mainly an combination of 2s and 2pz orbitals, while ΨO is purely 2pz approximately.

Fig. 2 Sectional view of (a)ΨC, (b)ΨCO, (c)ΨO. The black dots represent the nuclear positions.

We employ the Lewenstein model [1

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

] to calculate the harmonic radiation. The time-dependent dipole moment is given by
x(t)=itdt[πɛ+i(tt)/2]3/2×drec[pst(t,t)A(t)]dion[pst(t,t)A(t)]×exp[iSst(t,t)]E(t)+c.c..
(4)
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. pst and Sst are the stationary momentum and quasiclassical action, which are given by
pst(t,t)=1ttttA(t)dt,
(5)
Sst(t,t)=(tt)Ip12pst2(t,t)(tt)+12ttA2(t)dt,
(6)
where Ip=14.014 eV is the ionization energy.

The harmonic spectrum is then obtained by Fourier transforming the time-dependent dipole acceleration a(t) [25

25. A. D. Bandrauk, S. Chelkowski, D. J. Diestler, J. Manz, and K.-J. Yuan, “Quantum simulation of high-order harmonic spectra of the hydrogen atom,” Phys. Rev. A 79, 023403 (2009). [CrossRef]

]:
Sq|a(t)exp(iqωLt)|2,
(7)
where a(t) = (t), ωL 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
dCO(p)=ΨCO(r)|r|eipr.
(8)
The linear combination for HOMO of CO has been divided into two groups, so the transition dipole moment can be written as
dCO(p)=ΨC(rRC)|r|eipr+ΨO(rRO)|r|eipr
(9)
=ΨC(r1)|r1+RC|eipr1eipRC+ΨO(r2)|r2+RO|eipr2eipRO.
(10)
In the above equation, the terms growing linearly with length RC and RO 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

19. Y. J. Chen, J. Liu, and B. Hu, “Reading molecular messages from high-order harmonic spectra at different orientation angles,” J. Chem. Phys. 130, 044311 (2009). [CrossRef] [PubMed]

, 20

20. C. Figueira de Morisson Faria and B. B. Augstein, “Molecular high-order harmonic generation with more than one active orbital: quantum interference effects,” Phys. Rev. A 81, 043409 (2010). [CrossRef]

]. We neglect these two terms and get
dCO(p)=dC(p)eipRC+dO(p)eipRO
(11)
=|dC(p)|eiϕCipRCe^c+|dO(p)|eiϕOipROe^o,
(12)
where êc and êo are unit vectors of d⃗C and d⃗O respectively. The dipole moments are obtained by d⃗C = 〈ΨC(r⃗)|r⃗|e−ip⃗·r⃗〉 and d⃗O = 〈ΨO(r⃗)|r⃗|eip⃗·r⃗〉 respectively, where the integrations are performed numerically. The two terms in Eq. (12) describe the recombination process of each center respectively. -p⃗·· R⃗C and -p⃗·· R⃗O represent phases gained by the returning electrons before recombination. The phases ϕC and ϕO, which arise from the recombination, are obtained by the phase angles of the transition dipole moments d⃗C and d⃗O respectively. The total phase difference between the emissions from the two centers is defined as
Δϕ=[ϕC+(pRC)][ϕO+(pRO)]
(13)
=(ϕCϕO)+pRcosθ,
(14)
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

18. S. Baker, J. S. Robinson, M. Lein, C. C. Chirilǎ, R. Torres, H. C. Bandulet, D. Comtois, J. C. Kieffer, D. M. Villeneuve, J. W. G. Tisch, and J. P. Marangos, “Dynamic two-center interference in high-order harmonic generation from molecules with attosecond nuclear motion,” Phys. Rev. Lett. 101, 053901 (2008). [CrossRef] [PubMed]

20

20. C. Figueira de Morisson Faria and B. B. Augstein, “Molecular high-order harmonic generation with more than one active orbital: quantum interference effects,” Phys. Rev. A 81, 043409 (2010). [CrossRef]

, 26

26. T. Kanai, S. Minemoto, and H. Sakai, “Ellipticity dependence of high-order harmonic generation from aligned molecules,” Phys. Rev. Lett. 98, 053002 (2007). [CrossRef] [PubMed]

]
dHoDM(p)=2datom(p)cos(pR/2),
(15)
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.

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

12. M. Lein, N. Hay, R. Velotta, J. P. Marangos, and P. L. Knight, “Role of the intramolecular phase in high-harmonic generation,” Phys. Rev. Lett. 88, 183903 (2002). [CrossRef] [PubMed]

, 13

13. M. Lein, N. Hay, R. Velotta, J. P. Marangos, and P. L. Knight, “Interference effects in high-order harmonic generation with molecules,” Phys. Rev. A 66, 023805 (2002). [CrossRef]

]. Consistent results with Kamta’s are also obtained in various angles [14

14. G. L. Kamta and A. D. Bandrauk, “Three-dimensional time-profile analysis of high-order harmonic generation in molecules: nuclear interferences in H2+,” Phys. Rev. A 71, 053407 (2005). [CrossRef]

], which confirms that this modified model is appropriate.

Fig. 3 Harmonic spectra of H2 obtained using the Lewenstein model with the modified effective momentum p⃗k. 780nm laser is used with intensity I=5×1014 W/cm2 and a duration of 26 fs. Internuclear distance of H2 is 1.4 a.u. and the alignment angle is (a) 0°, and (b) 30° respectively.

3. Result and discussion

Parallel alignment with the orientation angle θ=0° is investigated first. According to the two-center model,
R|cosθ|=(2m+1)λ/2,m=0,1,
(16)
and
λ=h/p,
(17)
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

22. R. Torres, T. Siegel, L. Brugnera, I. Procino, J. G. Underwood, C. Altucci, R. Velotta, E. Springate, C. Froud, I. C. E. Turcu, M. Yu. Ivanov, O. Smirnova, and J. P. Marangos, “Extension of high harmonic spectroscopy in molecules by a 1300 nm laser field,” Opt. Express 18, 3174–3180 (2010). [CrossRef] [PubMed]

], 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=3×1014 W/cm2 and a duration of 30 fs full width at half maximum, which can be achieved by the optical parametric amplifier (OPA) technique nowadays [28

28. E. J. Takahashi, T. Kanai, Y. Nabekawa, and K. Midorikawa, “10 mJ class femtosecond optical parametric amplifier for generating soft x-ray harmonics,” Appl. Phys. Lett. 93, 041111 (2008). [CrossRef]

]. The sine squared envelope is employed as f(t)=sin2(πtTp), where Tp 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.

Fig. 4 Harmonic spectrum of CO at θ=0°, with laser intensity I=3×1014 W/cm2, wavelength λ=1300 nm and a duration of 30 fs.
Fig. 5 Phase analysis for the recombination of CO.

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(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, 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 dC and dO 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

29. D. Shafir, Y. Mairesse, D. M. Villeneuve, P. B. Corkum, and N. Dudovich, “Atomic wavefunctions probed through strong-field light-matter interaction,” Nature Phys. 5, 412–416 (2009). [CrossRef]

].

For example, we assume a linear combination for CO:
ΨCO(r)=33ψC,2s(rRC)+33ψC,2pz(rRC)33ψO,2pz(rRO).
(18)
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

13. M. Lein, N. Hay, R. Velotta, J. P. Marangos, and P. L. Knight, “Interference effects in high-order harmonic generation with molecules,” Phys. Rev. A 66, 023805 (2002). [CrossRef]

]:
Ssmooth(ω)=S(ω˜)exp((ω˜ω)2/σ2)dω˜,
(19)
where σ = 5ωL. 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 πθ 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 and Fig 5.

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 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.

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
Δϕrev=(ϕC+ϕO)+pRcos(πθ)=Δϕ.
(20)
The total phase difference has just changed its sign, but the position where it reaches π or −π 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

30. G. L. Kamta, A. D. Bandrauk, and P. B. Corkum, “Asymmetry in the harmonic generation from nonsymmetric molecules,” J. Phys. B 38, L339–L346 (2005). [CrossRef]

]. This system-induced gating can be utilized to manipulate the HHG to obtain a wide supercontinuum [31

31. Q. Zhang, P. Lu, W. Hong, Q. Liao, and S. Wang, “Control of high-order harmonic generation from molecules lacking inversion symmetry with a polarization gating method,” Phys. Rev. A 80, 033405 (2009). [CrossRef]

]. Recently, a research has been done for the HHG of CO as a polar molecule [32

32. A. Etches and L. B. Madsen, “Extending the strong-field approximation of high-order harmonic generation to polar molecules: gating mechanisms and extension of the harmonic cutoff,” J. Phys. B 43, 155602 (2010). [CrossRef]

]. 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).

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).

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.

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).

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=6×1014 W/cm2 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

33. B. K. McFarland, J. P. Farrell, P. H. Bucksbaum, and M. Gühr, “High harmonic generation from multiple orbitals in N2,” Science 322, 1232–1235 (2008). [CrossRef] [PubMed]

]. 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=6×1014 W/cm2, the other parameters are the same as in Fig. 4.

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.

Acknowledgment

This work was supported by the National Natural Science Foundation of China under Grants No. 60925021, 10904045 and the National Basic Research Program of China under Grant No. 2011CB808101.

References and links

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S. Gilbertson, S. D. Khan, Y. Wu, M. Chini, and Z. Chang, “Isolated attosecond pulse generation without the need to stabilize the carrier-envelope phase of driving lasers,” Phys. Rev. Lett. 105, 093902 (2010). [CrossRef] [PubMed]

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O. Smirnova, S. Patchkovskii, Y. Mairesse, N. Dudovich, D. Villeneuve, P. Corkum, and M. Yu. Ivanov, “Attosecond circular dichroism spectroscopy of polyatomic molecules,” Phys. Rev. Lett. 102, 063601 (2009). [CrossRef] [PubMed]

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M. Lein, N. Hay, R. Velotta, J. P. Marangos, and P. L. Knight, “Role of the intramolecular phase in high-harmonic generation,” Phys. Rev. Lett. 88, 183903 (2002). [CrossRef] [PubMed]

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M. Lein, N. Hay, R. Velotta, J. P. Marangos, and P. L. Knight, “Interference effects in high-order harmonic generation with molecules,” Phys. Rev. A 66, 023805 (2002). [CrossRef]

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G. L. Kamta and A. D. Bandrauk, “Three-dimensional time-profile analysis of high-order harmonic generation in molecules: nuclear interferences in H2+,” Phys. Rev. A 71, 053407 (2005). [CrossRef]

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M. Lein, “Molecular imaging using recolliding electrons,” J. Phys. B 39, R135–R173 (2007). [CrossRef]

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C. Vozzi, F. Calegari, E. Benedetti, R. Berlasso, G. Sansone, S. Stagira, M. Nisoli, C. Altucci, R. Velotta, R. Torres, E. Heesel, N. Kajumba, and J. P. Marangos, “Probing two-centre interference in molecular high harmonic generation,” J. Phys. B 39, S457–S466 (2006). [CrossRef]

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Y. J. Chen and B. Hu, “Strong-field approximation for diatomic molecules: comparison between the length gauge and the vlocity gauge,” Phys. Rev. A 80, 033408 (2009). [CrossRef]

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E. J. Takahashi, T. Kanai, Y. Nabekawa, and K. Midorikawa, “10 mJ class femtosecond optical parametric amplifier for generating soft x-ray harmonics,” Appl. Phys. Lett. 93, 041111 (2008). [CrossRef]

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D. Shafir, Y. Mairesse, D. M. Villeneuve, P. B. Corkum, and N. Dudovich, “Atomic wavefunctions probed through strong-field light-matter interaction,” Nature Phys. 5, 412–416 (2009). [CrossRef]

30.

G. L. Kamta, A. D. Bandrauk, and P. B. Corkum, “Asymmetry in the harmonic generation from nonsymmetric molecules,” J. Phys. B 38, L339–L346 (2005). [CrossRef]

31.

Q. Zhang, P. Lu, W. Hong, Q. Liao, and S. Wang, “Control of high-order harmonic generation from molecules lacking inversion symmetry with a polarization gating method,” Phys. Rev. A 80, 033405 (2009). [CrossRef]

32.

A. Etches and L. B. Madsen, “Extending the strong-field approximation of high-order harmonic generation to polar molecules: gating mechanisms and extension of the harmonic cutoff,” J. Phys. B 43, 155602 (2010). [CrossRef]

33.

B. K. McFarland, J. P. Farrell, P. H. Bucksbaum, and M. Gühr, “High harmonic generation from multiple orbitals in N2,” Science 322, 1232–1235 (2008). [CrossRef] [PubMed]

OCIS Codes
(190.4160) Nonlinear optics : Multiharmonic generation
(190.7110) Nonlinear optics : Ultrafast nonlinear optics
(300.6560) Spectroscopy : Spectroscopy, x-ray

ToC Category:
Nonlinear Optics

History
Original Manuscript: November 8, 2010
Revised Manuscript: December 17, 2010
Manuscript Accepted: December 18, 2010
Published: January 3, 2011

Citation
Xiaosong Zhu, Qingbin Zhang, Weiyi Hong, Pengfei Lan, and Peixiang Lu, "Two-center interference in high-order harmonic generation from heteronuclear diatomic molecules," Opt. Express 19, 436-447 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-2-436


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References

  1. M. Lewenstein, Ph. Balcou, M. Yu. Ivanov, A. L’Huillier, and P. B. Corkum, “Theory of high-harmonic generation by low-frequency laser fields,” Phys. Rev. A 49, 2117–2132 (1994). [CrossRef] [PubMed]
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  6. S. Gilbertson, S. D. Khan, Y. Wu, M. Chini, and Z. Chang, “Isolated attosecond pulse generation without the need to stabilize the carrier-envelope phase of driving lasers,” Phys. Rev. Lett. 105, 093902 (2010). [CrossRef] [PubMed]
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  11. O. Smirnova, S. Patchkovskii, Y. Mairesse, N. Dudovich, D. Villeneuve, P. Corkum, and M. Yu. Ivanov, “Attosecond circular dichroism spectroscopy of polyatomic molecules,” Phys. Rev. Lett. 102, 063601 (2009). [CrossRef] [PubMed]
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  29. D. Shafir, Y. Mairesse, D. M. Villeneuve, P. B. Corkum, and N. Dudovich, “Atomic wavefunctions probed through strong-field light-matter interaction,” Nat. Phys. 5, 412–416 (2009). [CrossRef]
  30. G. L. Kamta, A. D. Bandrauk, and P. B. Corkum, “Asymmetry in the harmonic generation from nonsymmetric molecules,” J. Phys. B 38, L339–L346 (2005). [CrossRef]
  31. Q. Zhang, P. Lu, W. Hong, Q. Liao, and S. Wang, “Control of high-order harmonic generation from molecules lacking inversion symmetry with a polarization gating method,” Phys. Rev. A 80, 033405 (2009). [CrossRef]
  32. A. Etches, and L. B. Madsen, “Extending the strong-field approximation of high-order harmonic generation to polar molecules: gating mechanisms and extension of the harmonic cutoff,” J. Phys. B 43, 155602 (2010). [CrossRef]
  33. B. K. McFarland, J. P. Farrell, P. H. Bucksbaum, and M. Gühr, “High harmonic generation from multiple orbitals in N2,” Science 322, 1232–1235 (2008). [CrossRef] [PubMed]

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