Abstract

A theory of high-order harmonic generation by heteronuclear diatomic molecules exposed to an elliptically polarized laser field is presented. This theory is based on the molecular strong-field approximation, which takes into account the dressing of the initial and final molecular bound states caused by the laser field. The laser-field-caused Stark shift of the effective ionization potential of the highest occupied molecular orbital changes the structure of the harmonic spectra. Both the even and odd harmonics appear in the harmonic spectra due to the broken inversion symmetry of the heteronuclear diatomic molecule. An interference minima condition that is valid for arbitrary heteronuclear diatomic molecules is derived. Our results are illustrated using examples of CO and NO molecules.

© 2012 Optical Society of America

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    [CrossRef]
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  39. L. Holmegaard, J. L. Hansen, L. Kalhjøj, S. L. Kragh, H. Stapelfeldt, F. Filsinger, J. Küpper, G. Meijer, D. Dimitrovski, M. Abu-samha, C. P. J. Martiny, and L. B. Madsen, “Photoelectron angular distributions from strong-field ionization of oriented molecules,” Nat. Phys. 6, 428–432 (2010).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  43. W. M. Huo, “Electronic structure of CO and BF,” J. Chem. Phys. 43, 624–647 (1965).
    [CrossRef]
  44. P. E. Cade and W. M. Huo, “Hartree-Fock-Roothaan wavefunctions for diatomic molecules,” At. Data Nucl. Data Tables 15, 1–39 (1975).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  49. C. C. Chirilă and M. Lein, “Explanation for the smoothness of the phase in molecular high-order harmonic generation,” Phys. Rev. A 80, 013405 (2009).
    [CrossRef]
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    [CrossRef]
  51. B. Borca, D. B. Milošević, A. F. Starace, A. V. Flegel, M. V. Frolov, and N. L. Manakov, “Anisotropy-induced polarization effects in harmonic generation by an absorptive medium,” in Super-Intense Laser-Atom Physics, B. Piraux and K. Rzążewski, eds., Vol. 12 of NATO Science Series II: Mathematics, Physics and Chemistry (Kluwer, 2001), pp. 249–258.
  52. H. Li, D. Ray, S. De, I. Znakovskaya, W. Cao, G. Laurent, Z. Wang, M. F. Kling, A. T. Le, and C. L. Cocke, “Orientation dependence of the ionization of CO and NO in an intense femtosecond two-color laser field,” Phys. Rev. A 84, 043429 (2011).
    [CrossRef]
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  54. S. Odžak and D. B. Milošević, “Molecular high-order harmonic generation: analysis of a destructive interference condition,” J. Phys. B 42, 071001(F) (2009).
    [CrossRef]

2011

B. B. Augstein and C. Figueira de Morisson Faria, “Multielectron corrections in molecular high-order harmonic generation for different formulations of the strong-field approximation,” J. Mod. Opt. 58, 1173–1187 (2011).
[CrossRef]

B. B. Augstein and C. Figueira de Morisson Faria, “Influence of asymmetry and nodal structures on high-harmonic generation in heteronuclear molecules,” J. Phys. B 44, 055601 (2011).
[CrossRef]

A. Etches, M. B. Gaarde, and L. B. Madsen, “Two-center minima in harmonic spectra from aligned polar molecules,” Phys. Rev. A 84, 023418 (2011).
[CrossRef]

E. Hasović, M. Busuladžić, W. Becker, and D. B. Milošević, “Dressed-bound-state molecular strong-field approximation: Application to above-threshold ionization of heteronuclear diatomic molecules,” Phys. Rev. A 84, 063418 (2011).
[CrossRef]

S. Odžak and D. B. Milošević, “Ellipticity and the offset angle of high harmonics generated by homonuclear diatomic molecules,” J. Phys. B 44, 125602 (2011).
[CrossRef]

H. Li, D. Ray, S. De, I. Znakovskaya, W. Cao, G. Laurent, Z. Wang, M. F. Kling, A. T. Le, and C. L. Cocke, “Orientation dependence of the ionization of CO and NO in an intense femtosecond two-color laser field,” Phys. Rev. A 84, 043429 (2011).
[CrossRef]

X. S. Zhu, Q. B. Zhang, W. Y. Hong, P. X. Lu, and Z. Z. Xu, “Molecular orbital imaging via above-threshold ionization with circularly polarized pulses,” Opt. Express 19, 13722–13731 (2011).
[CrossRef]

2010

S. Odžak and D. B. Milošević, “Role of ellipticity in high-order harmonic generation by homonuclear diatomic molecules,” Phys. Rev. A 82, 023412 (2010).
[CrossRef]

L. Holmegaard, J. L. Hansen, L. Kalhjøj, S. L. Kragh, H. Stapelfeldt, F. Filsinger, J. Küpper, G. Meijer, D. Dimitrovski, M. Abu-samha, C. P. J. Martiny, and L. B. Madsen, “Photoelectron angular distributions from strong-field ionization of oriented molecules,” Nat. Phys. 6, 428–432 (2010).
[CrossRef]

D. Dimitrovski, C. P. J. Martiny, and L. B. Madsen, “Strong-field ionization of polar molecules: Stark-shift-corrected strong-field approximation,” Phys. Rev. A 82, 053404 (2010).
[CrossRef]

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]

2009

D. A. Telnov and S. I. Chu, “Effects of multiple electronic shells on strong-field multiphoton ionization and high-order harmonic generation of diatomic molecules with arbitrary orientation: An all-electron time-dependent density-functional approach,” Phys. Rev. A 80, 043412 (2009).
[CrossRef]

S.-K. Son and S. I. Chu, “Multielectron effects on the orientation dependence and photoelectron angular distribution of multiphoton ionization of CO2 in strong laser fields,” Phys. Rev. A 80, 011403(R) (2009).
[CrossRef]

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

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, 013401 (2009).
[CrossRef]

A. Abdurrouf and F. H. M. Faisal, “Theory of intense-field dynamic alignment and high-order harmonic generation from coherently rotating molecules and interpretation of intense-field ultrafast pump-probe experiments,” Phys. Rev. A 79, 023405 (2009).
[CrossRef]

S. Odžak and D. B. Milošević, “Interference effects in high-order harmonic generation by homonuclear diatomic molecules,” Phys. Rev. A 79, 023414 (2009).
[CrossRef]

K. Yoshii, G. Miyaji, and K. Miyazaki, “Measurement of molecular rotational temperature in a supersonic gas jet with high-order harmonic generation,” Opt. Lett. 34, 1651–1653 (2009).
[CrossRef]

S. Odžak and D. B. Milošević, “Molecular high-order harmonic generation: analysis of a destructive interference condition,” J. Phys. B 42, 071001(F) (2009).
[CrossRef]

C. C. Chirilă and M. Lein, “Explanation for the smoothness of the phase in molecular high-order harmonic generation,” Phys. Rev. A 80, 013405 (2009).
[CrossRef]

2008

G. N. Gibson and J. Biegert, “Influence of orbital symmetry on high-order-harmonic generation and quantum tomography,” Phys. Rev. A 78, 033423 (2008).
[CrossRef]

W. Boutu, S. Haessler, H. Merdji, P. Breger, G. Waters, M. Stankiewicz, L. J. Frasinski, R. Taïeb, J. Caillat, A. Maquet, P. Monchicourt, B. Carré, and P. Salières, “Coherent control of attosecond emission from aligned molecules,” Nat. Phys. 4, 545–549 (2008).
[CrossRef]

2007

C. B. Madsen, A. S. Mouritzen, T. K. Kjeldsen, and L. B. Madsen, “Effects of orientation and alignment in high-order harmonic generation and above-threshold ionization,” Phys. Rev. A 76, 035401 (2007).
[CrossRef]

M. Lein, “Molecular imaging using recolliding electrons,” J. Phys. B 40, R135–R173 (2007).
[CrossRef]

2006

J. Levesque and P. B. Corkum, “Attosecond science and technology,” Can. J. Phys. 84, 1–18 (2006).
[CrossRef]

A. Scrinzi, M. Yu. Ivanov, R. Kienberger, and D. M. Villeneuve, “Attosecond physics,” J. Phys. B 39, R1–R39 (2006).
[CrossRef]

C. B. Madsen and L. B. Madsen, “High-order harmonic generation from arbitrarily oriented diatomic molecules including nuclear motion and field-free alignment,” Phys. Rev. A 74, 023403 (2006).
[CrossRef]

D. B. Milošević, “Strong-field approximation for ionization of a diatomic molecule by a strong laser field,” Phys. Rev. A 74, 063404 (2006).
[CrossRef]

2005

M. F. Guest, I. J. Bush, H. J. J. van Dam, P. Sherwood, J. M. H. Thomas, J. H. Van Lenthe, R. W. A. Havenith, and J. Kendrick, “The GAMESS-UK electronic structure package: algorithms, developments and applications,” Mol. Phys. 103, 719–747 (2005).
[CrossRef]

G. Lagmago 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]

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]

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

2004

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]

G. Lagmago Kamta and A. D. Bandrauk, “High-order harmonic generation from two-center molecules: time-profile analysis of nuclear contributions,” Phys. Rev. A 70, 011404(R) (2004).

M. Spanner, O. Smirnova, P. B. Corkum, and M. Y. Ivanov, “Reading diffraction images in strong field ionization of diatomic molecules,” J. Phys. B 37, L243–L250 (2004).
[CrossRef]

X. Chu and S. I. Chu, “Role of the electronic structure and multielectron responses in ionization mechanisms of diatomic molecules in intense short-pulse lasers: an all-electron ab initio study,” Phys. Rev. A 70, 061402 (2004).
[CrossRef]

2003

D. B. Milošević and F. Ehlotzky, “Scattering and reaction processes in powerful laser fields,” Adv. At. Mol. Opt. Phys. 49, 373–532 (2003).
[CrossRef]

2002

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

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

2000

B. Borca, A. V. Flegel, M. V. Frolov, N. L. Manakov, D. B. Milošević, and A. F. Starace, “Static-electric-field-induced polarization effects in harmonic generation,” Phys. Rev. Lett. 85, 732–735 (2000).
[CrossRef]

1999

P. Salières, A. L’Huillier, P. Antoine, and M. Lewenstein, “Study of the spatial and temporal coherence of high-order harmonics,” Adv. At. Mol. Opt. Phys. 41, 83–142 (1999).
[CrossRef]

1998

R. Kopold, W. Becker, and M. Kleber, “Model calculations of high-harmonic generation in molecular ions,” Phys. Rev. A 58, 4022–4038 (1998).
[CrossRef]

1996

D. B. Milošević and B. Piraux, “High-order harmonic generation in a bichromatic elliptically polarized laser field,” Phys. Rev. A 54, 1522–1531 (1996).
[CrossRef]

1994

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]

1993

P. B. Corkum, “Plasma perspective on strong field multiphoton ionization,” Phys. Rev. Lett. 71, 1994–1997 (1993).
[CrossRef]

M. W. Schmidt, K. K. Baldridge, J. A. Boatz, S. T. Elbert, M. S. Gordon, J. H. Jensen, S. Koseki, N. Matsunaga, K. A. Nguyen, S. J. Su, T. L. Windus, M. Dupuis, and J. A. Montgomery, “General atomic and molecular electronic structure system,” J. Comput. Chem. 14, 1347–1363 (1993).
[CrossRef]

1988

M. Ferray, A. L’Huillier, F. X. Li, L. A. Lompré, G. Mainfray, and C. Manus, “Multiple-harmonic conversion of 1064  nm radiation in rare-gases,” J. Phys. B 21, L31–L35 (1988).
[CrossRef]

1987

1976

B. A. Zon and E. I. Sholokhov, Zh. Eksp. Teor. Fiz. 70, 887–898 (1976) [“Quasienergy spectra of a dipolar molecule and of the hydrogen atom,” Sov. Phys. JETP 43, 461–466 (1976)].

1975

P. E. Cade and W. M. Huo, “Hartree-Fock-Roothaan wavefunctions for diatomic molecules,” At. Data Nucl. Data Tables 15, 1–39 (1975).
[CrossRef]

1965

W. M. Huo, “Electronic structure of CO and BF,” J. Chem. Phys. 43, 624–647 (1965).
[CrossRef]

Abdurrouf, A.

A. Abdurrouf and F. H. M. Faisal, “Theory of intense-field dynamic alignment and high-order harmonic generation from coherently rotating molecules and interpretation of intense-field ultrafast pump-probe experiments,” Phys. Rev. A 79, 023405 (2009).
[CrossRef]

Abu-samha, M.

L. Holmegaard, J. L. Hansen, L. Kalhjøj, S. L. Kragh, H. Stapelfeldt, F. Filsinger, J. Küpper, G. Meijer, D. Dimitrovski, M. Abu-samha, C. P. J. Martiny, and L. B. Madsen, “Photoelectron angular distributions from strong-field ionization of oriented molecules,” Nat. Phys. 6, 428–432 (2010).
[CrossRef]

Altucci, C.

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]

Antoine, P.

P. Salières, A. L’Huillier, P. Antoine, and M. Lewenstein, “Study of the spatial and temporal coherence of high-order harmonics,” Adv. At. Mol. Opt. Phys. 41, 83–142 (1999).
[CrossRef]

Atkins, P. W.

P. W. Atkins and R. S. Friedman, Molecular Quantum Mechanics, 3rd ed. (Oxford University, 2001).

Augstein, B. B.

B. B. Augstein and C. Figueira de Morisson Faria, “Influence of asymmetry and nodal structures on high-harmonic generation in heteronuclear molecules,” J. Phys. B 44, 055601 (2011).
[CrossRef]

B. B. Augstein and C. Figueira de Morisson Faria, “Multielectron corrections in molecular high-order harmonic generation for different formulations of the strong-field approximation,” J. Mod. Opt. 58, 1173–1187 (2011).
[CrossRef]

Balcou, Ph.

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]

Baldridge, K. K.

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Figures (6)

Fig. 1.
Fig. 1.

Coordinate space (first row) and momentum space (second row) wave functions of the 5σ HOMO of CO (first column) and the 2π HOMO of NO (second column), presented in the zx and pzpx planes, respectively. The wave functions are calculated using the method described in [43,44] and references therein.

Fig. 2.
Fig. 2.

High-order harmonic spectra of the CO molecule, obtained using a linearly polarized laser field having the intensity 4×1014W/cm2 and photon energy ω=1.55eV. The angle between the laser-field polarization axis and the molecular axis is plotted along the abscissa, while the harmonic order is along the ordinate. Panels in the first row correspond to the undressed case: μ=α=α=0. For the remaining panels, the dipole is 1.0546 a.u. Panels in the second row correspond to the case α=α=0, while, for the panels in the third row, we have α=5.3673a.u. and α=5.49a.u. The results shown in the first (second) column are obtained using only the component Tnϵ^ (Tnk^×ϵ^). The results in the third column represent the sum of both contributions: Eq. (14) with |TK|2=|TKϵ^|2+|TKk^×ϵ^|2. The curves nmin(θL), which express the interference minima condition as the solution of the transcendental equation defined by Eq. (22), are presented by white curves in the second column.

Fig. 3.
Fig. 3.

Same as in Fig. 2, but for the NO molecule and the intensity 2×1014W/cm2.

Fig. 4.
Fig. 4.

Same as in the lower row of Fig. 2, but for the case when the electron was ionized at the carbon atom.

Fig. 5.
Fig. 5.

Same as in Fig. 3, but for the case when the electron was ionized at the nitrogen atom.

Fig. 6.
Fig. 6.

Parameters of elliptic dichroism, presented in false color, as a function of the angle θL and the harmonic order n. The laser-field ellipticity is ε=0.1. The results presented in the first, second, and third columns correspond to the cases δnϵ^(ε), δnk^×ϵ^(ε), and δn(ε), respectively. The results presented in the first and second row correspond to the parameters of Fig. 2 (CO) and Fig. 3 (NO), respectively.

Equations (24)

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E(t)=EL[e^Lzsin(ωt+φ1)+e^Lxεsin(ωt+φ2)],EL=E01+ε2,
e^Lz=z^cosθL+x^sinθL,e^Lx=z^sinθL+x^cosθL.
E(t)=EL[cosθLsin(ωt+φ1)εsinθLsin(ωt+φ2)],E(t)=EL[sinθLsin(ωt+φ1)+εcosθLsin(ωt+φ2)].
|ψ˜b(τ)=exp{i[τμ·E(t)dt+12j,k=x,y,zαjkτEj(t)Ek(t)dt]}|ψb(τ).
Ipd(t)=Ip+μ·E(t)+12j,k=x,y,zαjkEj(t)Ek(t).
12j,k=x,y,zαjkEj(t)Ek(t)12[αE2(t)+αE2(t)].
ΔS=EL28{(α+α)(ε2+1)+(αα)[(ε21)cos(2θL)+2εcos(φ1φ2)sin(2θL)]}.
μS(τ)=τμ·E(t)dt=μz^τE(t)dt=μEL[cosθLcos(ωτ+φ1)εsinθLcos(ωτ+φ2)]/ω,
αS(τ)=EL216ω{[α+α+(αα)cos(2θL)]sin(2ωτ+2φ1)2ε(αα)sin(2ωτ+φ1+φ2)sin(2θL)+ε2[α+α+(αα)cos(2θL)]sin(2ωτ+2φ2)}.
|ψ˜j(τ)=|ψjexp{i[(IpΔS)τ+μS(τ)+αS(τ)]},
ϕei(r,t;R)=J=A,BacJaψa(0)(rJ)eiEei(R)t.
rA=r+1λ2Rr+1,rB=r+1λ2Rr1.
r+s2Rrs=r+sλ2R,i.e.,ssλ.
wK=12π(ωKc)3|TK|2.
TKqq=i0TdtTtdtd3kei[S(k;t,t)+ωKqqt]×s=±1ei(sλ)[k+A(t)]·R0/2eK*·Ψ˜s(0)q(t)|r|k+A(t)×s=±1ei(sλ)[k+A(t)]·R0/2k+A(t)|r·E(t)|Ψ˜s(0)q(t).
|Ψ˜s(0)q(t)=gq(t)acsa|ψa(0)
gq(t)={1,q=uei[μS(t)ΔSt+αS(t)],q=d
ωKqq={nω(q=q=uorq=q=d)nω+ΔS(q=d,q=u)nωΔS(q=u,q=d).
S(k;t,t)=ttdt[k+A(t)]2/2Ip(tt),
TK=TKeK=TKϵ^ϵ^+TKk^×ϵ^k^×ϵ^,
mq(t)=gq(t)*eiλ[kst+A(t)]·R0/2s=±1acsaeis[kst+A(t)]·R0/2eK*·ma(t),
s=±1acsaeis[kst+A(t)]·R0/2eK*·ma(t)=0.
δn(ε)=|Tn(+|ε|)|2|Tn(|ε|)|2|Tn(+|ε|)|2+|Tn(|ε|)|2,
δnj(ε)=|Tnj(+|ε|)|2|Tnj(|ε|)|2|Tnj(+|ε|)|2+|Tnj(|ε|)|2,j=ϵ^,k^×ϵ^.

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