Molecular high-order harmonic generation: analysis of a destructive interference condition

S Odžak¹ and D B Milošević^1,2

¹ Faculty of Science, University of Sarajevo, Zmaja od Bosne 35, 71000 Sarajevo, Bosnia and Herzegovina
² Max-Born-Institut, Max-Born-Strasse 2a, 12489 Berlin, Germany

E-mail: milo@bih.net.ba

Received 11 February 2009
Published 23 March 2009

High-order harmonic generation (HHG) by molecular gases has recently attracted a lot of attention in the context of molecular imaging [1] and attophysics [2–9]. In particular, by tomographic reconstruction of molecular orbitals, Itatani et al [10] have reconstructed the highest occupied molecular orbital (HOMO) of a N₂ molecule. In this experiment, the HHG spectra of aligned N₂ molecules were measured for various orientations of the molecular axis with respect to the laser polarization axis.

Physically, HHG is usually described as a three-step process [11]: first, the molecule is ionized, second, the ionized electron propagates in the laser field and returns to the parent ion and third, this electron recombines with the parent ion and a high-order harmonic is emitted. In the final state, the molecule is in its ground state and we have a photon of energy nω, where n is the harmonic order and ω is the laser photon energy (we use atomic units). Since molecules are multi-centre systems, one can expect the multiple-point emitter or multiple-slit type interference, which manifests as minima and maxima in the HHG spectra. The position of these minima and maxima has been the subject of many theoretical and experimental investigations. A simple model, supported by numerical results for the simplest molecular ion H ⁺ ₂, was suggested in [12, 13]. However, this model is not applicable to more complex diatomic molecules such as N₂ or O₂. Recent measurements of HHG spectra from aligned N₂, O₂ and CO₂ molecules are controversial and have motivated further theoretical investigations [1, 14, 15].

Ab initio calculations of the molecular HHG spectra are presently possible only for the simplest molecules. Therefore, approximate theories have to be developed. Furthermore, from such theories one should be able to withdraw the interference conditions in an analytical form. This is the aim of our present paper. We have recently introduced molecular strong-field approximation (MSFA) [16]. It should be mentioned that various versions of the MSFA exist (we have reviewed this in [16]). Our version is based on the length gauge, the dressed initial state and the undressed final molecular state. This choice is in accordance with the findings of [17] (length gauge), [18–21] (dressed initial state) and [16] (undressed final state). At the instant of ionization the laser field should be strong, while the electron returns to the parent ion when the field is weak. This also justifies the laser-field dressing (undressing) of the initial (final) molecular state. The results obtained in [16] are in agreement with the ab initio results for the H₂ molecule. In the present paper, using the MSFA theory of [16], we generalize the destructive interference condition of [1, 12] to arbitrary diatomic molecules. We analyse this condition in the general case and we illustrate it using the example of an Ar₂ molecule, having 5σ_u HOMO, for various internuclear distances.

We use the notation of [16] and denote the relative electron and nuclear coordinate by r and R, respectively. For the ground-state electronic wavefunctions of the neutral homonuclear diatomic molecules we use the Hartree–Fock–Roothaan wavefunctions [18]

where ψ_a(r) are the Slater-type orbitals and the sum over a denotes the sum over atomic orbitals. The real coefficients c_sa at the different atomic centres (denoted by the index s  =  ±1) are equal up to a sign, i.e., c_–1a  =  s_aλ c_1a, where [16]

Here m_a is the magnetic quantum number of the atomic orbital a and m_λ  =  |m_a| is the value of the projection of orbital angular momentum on the internuclear axis (for example, for σ states it is m_λ  =  0, and m_λ  =  1 for π states). The factor comes from the inversion of the z coordinate of the second centre. For the Ar₂ molecule having the 5σ_u HOMO it is . We choose ten atomic orbitals: a  =  1s, 2s, 2s ', 3s, 3s ', 2p, 2p ', 3p, 3p ', 3d. We will present results for different values of the internuclear distance: between R  =  3 au and R  =  7.2 au (this is the equilibrium distance). The coefficients c_sa, the exponents of the Slater-type orbitals and the corresponding ionization energies I_P of Ar₂ are taken from [22].

It can be shown [16] that the T-matrix element for emission of the nth harmonic is proportional to a factor which comes from the recombination of the returned electron wave packet into the HOMO of the diatomic molecule represented by a linear combination of even (s_aλ  =   + 1) and odd (s_aλ  =  –1) atomic orbitals. We will use the notation a ₊  ≡ a(s_aλ  =   + 1) and a_– ≡ a(s_aλ  =  –1). The harmonic photon energy nω is determined by the energy-conserving condition at the recombination time t: nω  =  I_P  +  [k_st  +  A(t)]²/2 (this condition is obtained using the saddle-point method [16]). Here is the stationary electron momentum, where A(t)  =  A₀cos ωt is the amplitude of the vector potential of the linearly polarized laser field and t ' is the ionization time. For fixed R and , the above-mentioned factor is a function of the molecular orientation angle θ_L

where q  =  pR/2 and

with λ  =  |m_λ|  =  |m_a|. Here are the momentum-space Slater-type orbitals, are the real spherical harmonics, are the Legendre polynomials, and we used a result from [23]. For the Ar₂ molecule, the even orbitals are p orbitals while the odd orbitals are s and d orbitals. In this case, the functions and are real, so that the function F(θ_L) is real. In the general case, the function F(θ_L) is always real or purely imaginary. In the latter case, we can redefine the factor F(θ_L) → iF(θ_L), so that F(θ_L) is always real. Then the interference minima condition is a real nonlinear equation F(θ_L)  =  0 over the variable θ_L. Introducing new variable x  =  cos θ_L and denoting the corresponding sums in (3) by A_±, we can rewrite this condition as

or, for A_±(x) ≠ 0,

The functions A_±(x) and f(x) depend on the particular molecular orbital. Let us analyse the case of 5σ_u HOMO of the Ar₂ molecule for which λ  =  0. According to (5), if we take into account only the s states, then A ₊   =  0 and condition (5) takes the form sin(qx)  =  0, from which it follows that pRx  =  2(j  +  1)π, j  =  0, 1, 2,  ..., i.e.

where j_c is determined by the cutoff value of the harmonic order n (we neglect the case pRx  =  0 since then p  =  0, i.e., n_minω  =  I_P, or x  =  0 so that θ_L  =  π/2 for every n). Analogously, if only the p orbitals contribute then A_–  =  0 so that cos(qx)  =  0 and pRx  =  (2j  +  1)π, j  =  0, 1, 2,  ..., i.e., the interference minima condition takes the form

If both s and p orbitals contribute, then from (4) it follows that f(x) x is a straight line whose intersections with the curve tan(qx), (2j – 1)π/2 ≤ qx ≤ (2j  +  1)π/2, j  =  0, 1, 2,  ..., j_max give the solution of the nonlinear equation (6). The value of j_max depends on the value of q  =  pR/2 and is determined by the condition 0 ≤ x ≤ 1. In the general case, all orbitals s, p and d contribute and we have f(x) x/(α  +  βx²), so that the intersections of this curve with the curve tan(qx) give the required solutions.

In figure 1 we present the HHG spectra of the Ar₂ molecule, obtained using the MSFA method described in [16], for various internuclear distances: R  =  3 au, 3.8 au, 4.2 au, 4.6 au, 5 au and 7.2 au. Harmonic emission rates are presented in false colours as functions of the angle θ_L between the laser field polarization axis and the molecular axis (abscissa) and of the harmonic order n (ordinate). The obtained numerical results reproduce very well the interference minima which are obtained as the solutions of the nonlinear equation (5) and are presented by white lines. For the upper left panel (the shortest internuclear distance R  =  3 au) there is only one solution of (5) below the harmonic cutoff. With the increase of the internuclear distance (upper right panel, R  =  3.8 au), the second solution starts to appear and is noticeable only in the cutoff region. With a further increase of R (middle panels (R  =  4.2 au and 4.6 au) and the lower left panel (R  =  5 au)), this second solution moves deeper into the plateau region. For the largest internuclear distance (lower right panel, R  =  7.2 au), the third solution of (5) appears.

We have seen that the numerical solutions of (5) over n and θ_L, presented by white curves in figure 1, agree very well with the numerical results for the harmonic emission rate. Let us now analyse in more detail the interference condition (5) and its special cases (6)–(8). For this purpose, we introduce the de Broglie electron wavelength, which corresponds to the electron momentum p_min,

and we present the solutions of (5) for the interference minima condition in the (Rcos θ_L, λ_min) plane. For each value of 0 ≤ θ_L ≤ π/2 we find the corresponding solution for n_min and then, using (9), we present a point in this plane. The number of solutions below the harmonic cutoff depends on the value of R. For R  =  7.2 au only the p orbitals contribute to the HOMO of Ar₂ so that the interference minima condition reduces to (8), which can be rewritten as

Three solutions shown in the lower right panel of figure 1 (R  =  7.2 au) are presented in figure 2. It is evident that each of these solutions is a straight line which goes through the origin. They corresponds to (10) for j  =  0, 1, 2.

For smaller internuclear distances, the contributions of odd orbitals s and d become important. If only the s orbitals contribute, then, using (7), we obtain

In figure 3 we present, for various internuclear distances and for j  =  0, case (10) by red (gray) lines and case (11) by black lines. In addition to this, the solutions of (5) for the realistic Ar₂ HOMO, which includes all s, p and d orbitals, are presented by green dashed lines. For small angles θ_L and lower harmonic order n these solutions are positioned between the red (gray) (p) and the black (s) lines. The exception is the lower right panel of figure 3, where only the p orbitals contribute so that the green dashed and red (gray) curves overlap. With the increase of θ_L and n, the green dashed curves intersect the red (gray) curves and, for θ_L → π/2, they tend to different points on the abscissa.

In order to explain such behaviour of the interference minima curves, in figure 4 we analyse the interference minima condition for the internuclear distance R  =  3 au (upper left panel of figure 3). For a fixed harmonic order n, the transcendental equation (6) over the variable x  =  cos θ_L helps us to better understand this condition. In figure 4 we present, in a graphical form, the solutions of (6) for n  =  57, 69, 71 and 73. The function f(x)  =  –A ₊ (x)/A_–(x) is calculated using the known form of the momentum-space Slater-type orbitals [16] and is presented by a black line. Its intersection with the curve tan(qx), which is presented by a blue (gray) line, gives the solution of the mentioned transcendental equation. This solution is denoted by a green circle. For low harmonic orders (the example n  =  57 is shown in the upper left panel of figure 4) the behaviour of the function f(x) is regular and the point of its intersection with the curve tan(qx) corresponds to a point on the green dashed curve in figure 3, which lies between the red (gray) line and the black line. For some values of n and x the function A_–(x) is zero so that f(x)  =  –A ₊ (x)/A_–(x) goes to infinity. This case is shown in the remaining three panels of figure 4. With the increase of n, the value of x at which A_–(x)  =  0 moves to the right. For n  =  69 this point is left of the point of intersection of the curves f(x) and tan(qx). The corresponding interference minima solution still lies between the red (gray) line and the black line in the upper left panel of figure 3. With a further increase of n, the asymptotes of the functions f(x) and tan(qx)  =  sin(qx)/cos(qx) overlap (lower left panel of figure 4). At this point we have cos(qx)  =  0 and A_–(x)  =  0 and only the p orbitals contribute so that the green dashed curve in the upper left panel of figure 3 intersects the red (gray) curve (this happens exactly for n  =  70.4773 and θ_L  =  67.7°; we presented the physical case of the nearest odd harmonic n  =  71). With a further increase of n, the asymptote of f(x) moves to the right of that of tan(qx) so that f(x) intersects the positive branch of tan(qx) (lower right panel of figure 4). The solution which corresponds to this intersection point lies outside the region between the red (gray) and black lines in figure 3. The results presented in figure 4 can also be interpreted in terms of the behaviour of the function tan(qx) as a function of x. Namely, since , with the increase of n, the factor q in the argument of tan(qx) increases, so that the asymptote of tan(qx) moves to the left on the x-axis.

Let us now analyse what happens when our solutions approach the λ_min axis in figure 3. In this case we have Rcos θ_L x → 0. Since, according to (4) for Ar₂, the p orbital contribution is proportional to cos θ_L, the s orbital contribution does not depend on θ_L and the d orbital contribution is proportional to 3cos²θ_L – 1, relation (5) can be rewritten as f(p)xcos(qx)  +  [g(p)  +  x²]sin(qx)  =  0. For x → 0, keeping only the terms proportional to x, we have f(p)  +  g(p)q  =  0. The solution of this nonlinear equation over p gives a constant value for p and, according to (9), a constant value for λ_min. For x → 0, i.e. θ_L → π/2, we obtain λ_min → 1.849 23 au, 1.583 33 au and 1.390 19 au, for R  =  3 au, 4 au and 5 au, respectively. This agrees with the results presented in figure 3.

From figure 1 we see that, for larger values of R, there are more solutions for the interference minima condition. In figure 3 we have presented only the first such solution. Now, in figure 5, we present the second solution which corresponds to j  =  1 in (10) (red (gray) lines) and (11) (black lines). For the second solution, the above-analysed intersections of the green dashed curves (which correspond to the Ar₂ HOMO described by the linear combination of s, p and d orbitals) and the red (gray) curves (only the p orbitals) appear for smaller values of θ_L, which is in accordance with the results presented in figure 1. For θ_L → π/2 we obtain λ_min → 0.467 654 au, 0.439 224 au and 0.419 668 au, for R  =  3.8 au, 4.6 au and 5 au, respectively. For R  =  7.2 au only the p orbitals contribute to the HOMO of the Ar₂ molecule so that the green dashed and red (gray) curves overlap.

The appearance of the second solutions can also be explained by analysing, for a fixed harmonic order n, the transcendental equation (6) over the variable x  =  cos θ_L. In figure 6 we graphically present its solutions for n  =  51 and for two values of the internuclear distance R  =  3 au and 5 au of the Ar₂ molecule. The intersection of the function f(x) with the curve tan(qx), presented similarly as in figure 4, gives the solution of the mentioned transcendental equation. Only the solutions which satisfy the condition 0 ≤ x ≤ 1 are allowed. They are positioned left of the dashed red line and are denoted by the green circles. The forbidden solutions (right of the x  =  1 line) are denoted by the red circles. For R  =  3 au we have only one allowed solution (this corresponds to the upper left panel of figure 3), while for R  =  5 au we have two such solutions (the lower left panel of figures 3 and 5).

To summarize, we have shown that the HHG spectra of aligned homonuclear diatomic molecules exhibit clear minima for particular molecular orientation. The position and the number of these minima strongly depend on the internuclear distance, on the molecular symmetry and on the atomic orbitals of which a particular HOMO consists. We have analysed in detail these destructive interference minima using the Ar₂ molecule as an example. We have found interesting analytical formulae (5)–(8) which have particularly simple form (9)–(11) if they are expressed as functions of the parameter Rcos θ_L and the effective de Broglie wavelength of the recombining electron. These conditions can be useful in the investigation of the molecular structure and dynamics.

Acknowledgment

This work was supported by the Federal Ministry of Education and Science, Bosnia and Herzegovina.

[1] 

Lein M 2007 J. Phys. B: At. Mol. Opt. Phys. 40 R135 
IOPscience

[2] 

Levesque J and Corkum P B 2006 Can. J. Phys. 84 1 
CrossRef

[3] 

Scrinzi A, Ivanov M Yu, Kienberger R and Villeneuve D M 2006 J. Phys. B: At. Mol. Opt. Phys. 39 R1 
IOPscience

[4] 

Milošević D B, Paulus G G, Bauer D and Becker W 2006 J. Phys. B: At. Mol. Opt. Phys. 39 R203 
IOPscience

[5] 

Winterfeldt C, Spielmann C and Gerber G 2008 Rev. Mod. Phys. 80 117 
CrossRef

[6] 

Gaarde M B, Tate J L and Schafer K J 2008 J. Phys. B: At. Mol. Opt. Phys. 41 132001 
IOPscience

[7] 

Midorikawa K, Nabekawa Y and Suda A 2008 Prog. Quantum Electron. 32 43 
CrossRef

[8] 

Pfeifer T, Abel M J, Nagel P M, Jullien A, Loh Z H, Bell M J, Neumark D M and Leone S R 2008 Chem. Phys. Lett. 463 11 
CrossRef

[9] 

Nisoli M and Sansone G 2009 Prog. Quantum Electron. 33 17 
CrossRef

[10] 

Itatani J, Levesque J, Zeidler D, Niikura H, Pépin H, Kieffer J C, Corkum P B and Villeneuve D M 2004 Nature 432 867 
CrossRefPubMed

[11] 

Corkum P B 1993 Phys. Rev. Lett. 71 1994 
CrossRefPubMed

[12] 

Lein M, Hay N, Velotta R, Marangos J P and Knight P L 2002 Phys. Rev. Lett. 88 183903 
CrossRefPubMed

[13] 

Lein M, Hay N, Velotta R, Marangos J P and Knight P L 2002 Phys. Rev. A 66 023805 
CrossRef

[14] 

Mairesse Y, Levesque J, Dudovich N, Corkum P B and Villeneuve D M 2008 J. Mod. Opt. 55 2591 
CrossRef

[15] 

Le A T, Della Picca R, Fainstein P, Telnov D, Lein M and Lin C D 2008 J. Phys. B: At. Mol. Opt. Phys. 41 081002 
IOPscience

[16] 

Odžak S and Milošević D B 2009 Phys. Rev. A 79 023414 
CrossRef

[17] 

Bauer D, Milošević D B and Becker W 2005 Phys. Rev. A 72 023415 
CrossRef

[18] 

Milošević D B 2006 Phys. Rev. A 74 063404 
CrossRef

[19] 

Becker W, Chen J, Chen S G and Milošević D B 2007 Phys. Rev. A 76 033403 
CrossRef

[20] 

Busuladžić M, Gazibegović-Busuladžić A, Milošević D B and Becker W 2008 Phys. Rev. Lett. 100 203003 
CrossRefPubMed

[21] 

Busuladžić M, Gazibegović-Busuladžić A, Milošević D B and Becker W 2008 Phys. Rev. A 78 033412 
CrossRef

[22] 

Gilbert T L and Wahl A C 1967 J. Chem. Phys. 47 3425 
CrossRef

[23] 

Wahl A C, Cade P E and Roothaan C C J 1964 J. Chem. Phys. 41 2578 
CrossRef