SUPPLEMENTARY INFORMATION

Transcription

1 SUPPLEMENTARY INFORMATION DOI: /NPHYS2029 Generalized Molecular Orbital Tomography Supplementary Information C. Vozzi, M. Negro, F. Calegari, G. Sansone, M. Nisoli, S. De Silvestri, and S. Stagira I. RETRIEVAL OF SINGLE-MOLECULE RESPONSE The expression for the XUV spectral intensity I(ω, τ) (see Methods): I(ω, τ) S(ω, τ) 2 π/2 2 = F (θ, τ) Ẽ(ω, θ) dθ 0 (1) was approximated by the matrix equation F(τ i,θ j ) Ẽ(θ j,ω k )= S(τ i,ω k ) (2) where is the matrix product and the matrices F, Ẽ, S were obtained from the corresponding continuous functions F, Ẽ, S at the sampling points τ i,θ j,ω k along the delay, angular and spectral dimensions respectively. The experimental measurement I(ω, τ) was acquired with a 0.1-eV step on the spectral axis and with a 75-fs step on the temporal axis. The angular distribution F was calculated according to the experimental conditions by approximating the CO 2 molecules as rigid rotors [3] with a thermal energy distribution at 75 K, using a 1-deg. step on the θ axis and a 10-fs step on the temporal axis. The experimental measurement was interpolated and resampled to 0.35-eV and 10-fs steps on the spectral and temporal axes respectively before exploiting it in the retrieval algorithm. The amplitude of the macroscopic response A = S was determined from measurements. In order to operate on a smooth data set, the measured spectra were filtered in the Fourier domain along the spectral dimension, in order to select only the DC component and cancel the harmonic modulation; this operation is allowed as far as one is not interested in the temporal structure of the XUV emission but only in the molecular dipole phase. An iterative retrieval algorithm based on the Kaczmarz method with relaxation [4] was applied to Eq. (2). We seeded the algorithm with the guess Ẽ0 = E th exp[iφ] where E th is NATURE PHYSICS 1

2 the XUV emission predicted by the SFA and Φ is a random phase. Setting S 0 = A, the algorithm at the n-th iteration operates in the following way: Ẽ n = K{ S n 1, F, Ẽn 1} B n = F Ẽn (3) S n = A exp[i B n ] where K{ } represents the result of the Kaczmarz algorithm applied to Eq. (2), which seeks for a solution of the equation in a least-square sense [4], and B n represents the phase of B n. The iterations are stopped when the sum ( B n A) 2 performed over all the matrix elements converges to a suitably small value; the retrieved single-molecule response is then given by Ẽn. It is worth noting that, since the method is based on the interference of XUV emission from different molecular orientations, it can retrieve the phase behaviour as a function of θ at fixed XUV photon energy, but it cannot directly retrieve the phase relationship between contributions at neighbouring frequencies. Indeed the retrieved single-molecule XUV contribution Ẽ(θ j, ω k ) departs from the actual emitted field Ẽa(θ j, ω k ) by a (random) phase recalibration term P = exp[if(ω k )] which is a function of ω only: Ẽ a (θ j, ω k ) = Ẽ(θ j, ω k ) exp[if(ω k )]. (4) Hence the retrieved macroscopic XUV emission S will be related to the actual XUV macroscopic emission S a by the relation: S a (τ i, ω k ) = F(τ i, θ j ) Ẽa(θ j, ω k ) = S(τ i, ω k ) exp[if(ω k )] (5) For any chosen delay τ i, the phase recalibration term can then be simply obtained by: f(ω k ) = S a (τ i, ω k ) S(τ i, ω k ). (6) Hence the complete reconstruction of the single-molecule emission can be achieved if the actual spectral phase of the macroscopic XUV emission S a is known at a given delay τ i. In simple molecules, this recalibration issue can be easily solved if theoretical predictions about the spectral phase of the harmonic emission are available. This is the case of carbon dioxide; in particular, it has been theoretically shown [5] (and experimentally confirmed [2]) that the dipole contribution in CO 2 has an almost flat spectral phase for molecules oriented at angles 2

3 FIG. 1: (a) Normalized experimental XUV intensity spectra; (b) normalized retrieved spectral intensity; (c) retrieved spectral phase. θ > 50. Such molecular distribution is in practice obtained in correspondence of the delay τ = ps (molecular anti-alignment). We took advantage of this feature imposing, at each intermediate step of the algorithm, a recalibration of the spectral phase of S n in order to have at anti-alignment a constant phase of the macroscopic response in all the spectral range. The result of the retrieval procedure is illustrated in Figure 1; in particular, panel (a) shows the experimental XUV intensity as a function of the photon energy and of the delay τ after elimination of the harmonic modulation by Fourier filtering. The macroscopic XUV intensity retrieved by the reconstruction algorithm is reported in Fig. 1(b) and appears in very good agreement with the experimental data. Figure 1(c) shows the retrieved spectral phase of the macroscopic XUV emission, with a peculiar jump of about 2.2 rad located between 50 and 65 ev. It worth noting that, in those molecular species for which a theoretical modelling is lacking, the phase recalibration required by the retrieval procedure can still be determined with a completely experimental approach, thus avoiding any knowledge of the molecular properties. Indeed, the phase between neighboring harmonics emitted by the macroscopic molecular 3

4 FIG. 2: (a) Average phase and amplitude, with corresponding error bars, of the retrieved harmonic field at a molecular orientation of 30 degrees extracted from 200 simulation runs. (b) Error evolution (calculated as the departure between the experimental and the retrieved macroscopic response) as a function of simulation steps for different gas temperatures and aligning pulse intensities. ensemble at a given delay (hence S a ) can be determined with a standard experimental technique like RABBIT [2]. In order to prove the reliability of the retrieval procedure, we show in fig. 2(a) the average phase and amplitude (with corresponding error bars) of the retrieved harmonic field at a molecular orientation of 30 degrees, extracted from 200 simulation runs. Each run was seeded with a different random initial guess. One can see that the calculated standard deviation is almost negligible apart from the region around 50 ev, in correspondence of the phase jump; nevertheless in this region the retrieved amplitude is very low owing to the structural interference minimum, hence the error on the retrieved signal is higher. Figure 2(b) shows the error evolution (i.e. the discrepancy between the experimental and the retrieved macroscopic emission) as a function of simulation steps for different gas temperatures and aligning pulse intensities. This plot shows that the retrieval procedure is very sensitive to a small departure of the input data from the experimental ones. In particular a departure of about 20 K in gas rotational temperature, or a change of about 10% in the aligning pulse intensity clearly slow down the convergence and worsen the agreement between experimental and retrieved macroscopic response, thus confirming the reliability of 4

5 the retrieved results. II. ROUTES TOWARDS THE EXTENSION OF THE RETRIEVAL TECHNIQUE TO COMPLEX MOLECULAR SPECIES The reconstruction of the single-molecule XUV emission can be in principle extended to complex molecules. From the experimental point of view, this task requires the tridimensional (3D) alignment of the investigated molecules, which consists in fixing the three axes of the molecular frame in the laboratory frame; in polar species, also the orientation of the head-tail direction of the molecular dipole moment would be required for a reliable reconstruction. Both 3D alignment [6] and orientation [7] can be performed nowadays with all-optical techniques. The generalized retrieval algorithm is based on a generalization of Eq. 1 in order to take into account the relevant angular degrees of freedom of the molecule; it is worth noting that the evolution of the molecular angular distribution F (which is required by the retrieval procedure) as a function of time and angular coordinates can be determined even in complex species [3], although at the expense of a much larger computational cost with respect to linear molecules. The discretization of the generalized equation corresponds to a (larger) set of linear equations in the unknown Ẽ, which can again be solved by the Kaczmarz approach. In principle, all the components of the XUV field vector emitted by each molecule are involved in the macroscopic response and can be determined by the retrieval algorithm. Since the amount of unknowns is larger with respect to the case of linear molecules, a larger set of experimental data, for instance several temporal scans of the harmonic emission corresponding to different driving polarization directions with respect to the 3D alignment frame, is required for a reliable retrieval. It must be stressed that the analysis of more complex species could require the introduction of additional steps; in particular, the presence of harmonics with elliptical polarization could introduce further difficulties since only a partial interference among emissions from molecules at different orientation would occur in this case. Nevertheless by filtering the harmonic emission with a reflective linear polarizer before the detector, the regime of full interference among XUV molecular contributions (which is at the base of the retrieval procedure) could be restored. In the CO 2 case here considered, such polarization filtering was not required since for a driving field parallel to the alignment direction, the macroscopic 5

6 harmonic emission is linearly polarized. It is important to stress that, by exploiting the general approach we mentioned above, one could in principle retrieve the XUV vector field and not only one component of such field; in the CO 2 case, the retrieval of the projection of the XUV field on the alignment axis was anyway enough for the tomographic reconstruction. III. ION-FIELD CORRECTION TO MOLECULAR TOMOGRAPHY The XUV field contribution of a target molecule can be calculated as the dipole acceleration in the direction ˆx of the diriving laser field, according to the relation (written in the velocity form): Ẽ(ω, θ) ωf t { Ψ(r, t, θ) ˆp Ψ(r, t, θ) } (7) where ˆp = i x, F t { } is the Fourier transform operator from the temporal to the spectral domain and Ψ(r, t, θ) is the time-dependent molecular wavefunction; hereafter the molecular orientation at an angle θ with respect to the ˆx direction will be explicitly indicated for the sake of clarity. Neglecting bound-bound and continuum-continuum transitions, Eq. (7) can be transformed into the relation: Ẽ(ω, θ) iω(k)a(k) ψ 0 (r, θ) x k θ with ψ 0 (r, θ) the HOMO wavefunction, k θ the continuum scattering state with asymptotic momentum k and a(k) the amplitude of the recombining wavepacket in the k-space. The dispersion relation ω = k 2 /2 + I p is used in the previous expression, being I p the ionization potential of the molecule. It is worth noting that in general the scattering state k θ cannot be represented exactly by a plane wave, owing to the electrostatic potential U(r, θ) of the molecular ion. Expressing k θ in terms of the potential U with a first-order perturbation approach [8], one obtains: [ k θ e ikx 1 i k ] U(r, θ)dx this relation reduces to the plane wave expression for negligible potential U. It is worth noting that the function C(r, θ) = U(r, θ)dx is integrated along the asymptotic propagation direction ˆx of the wavepacket, that corresponds to the laser polarization direction. Hence, (8) (9) 6

7 for different molecular orientations the function C changes, as expected from a non-spherical potential well. Inserting Eq. (9) into Eq. (8), one gets: Ẽ(ω, θ) [( ω(k)a(k) e ikx ψ 0 (r, θ) k kf k {ψ 0 } 1 k F k {Uψ 0 } if k {Cψ 0 } ) U(r, θ) i k where F k { } represents the Fourier operator in the spatial domain. ] U(r, θ)dx d 3 r = (10) It is worth noting that the bound wavefunction ψ 0 and the potential U are assumed to be real functions. In this work the wavepacket amplitude a(k) can be considered real as well, since the retrieval procedure we exploited is not sensitive to the attochirp of the XUV emission. Moreover, since the wavefunction ψ 0 shows inversion symmetry, its Fourier transform is real. Recalling that the bound wavefunction ψ 0 obeys the Schrödinger equation (in atomic units) 2 2 ψ 0 + Uψ 0 = I p ψ 0 (11) it is easy to show that, exploiting the properties of the Fourier transform in k space, ) F k {Uψ 0 } = (I p + k2 F k {ψ 0 } (12) 2 hence We can then write: ( Ẽ(ω, θ) 3k ω(k)a(k) 2 + I ) p F k {ψ 0 } if k {Cψ 0 } (13) k ] Re [Ẽ(ω, θ) ω(k)a(k) ( 3k 2 + I ) p F k {ψ 0 } + Im [F k {Cψ 0 }] (14) k Since the function Cψ 0 has not a well defined symmetry, one cannot make general assumptions on the second term in the right-hand side of Eq. contribution to the overall term, we show in Fig. 3(a) the quantity (14); in order to determine its Re[Ẽ(ω, θ)] as determined from the experimental data, whereas we display in Fig. 3(b) the first term in the ( ) 3k right-hand side of equation (14), namely the quantity ω(k)a(k) + Ip F 2 k k {ψ 0 } as calculated from the theoretical HOMO wavefunction ψ 0. As can be seen, the major difference between the two figures is for low alignment angles 0 θ 40 and in the energy region between 35 and 55 ev. This difference is attributed to both a residual influence of the ion field on the XUV emission and on the approximations we made in deriving Eq. (14); those 7

8 FIG. 3: (a) Real component of the measured Ẽ(ω,θ); (b) first term in the right-hand side of equation (14), see text. contributions cannot be easily accounted for in the HOMO reconstruction. Nevertheless the overall agreement between the two figures is good, hence we assume for the sake of the HOMO retrieval that the role of this quantity is minor with respect to the first term in Eq. (14). Though not rigorous, this assumption is supported by the good agreement between the HOMO reconstruction and the theoretical expectation, as reported in the main text. Hence we will write: ] Re [Ẽ(ω, θ) F k {ψ 0 } ( ) ( k 2 3k a(k) 2 + I p 2 + I ) (15) p k In the framework of the mentioned approximations, Eq. (15) provides the ion-field correction to the usual relationship between the XUV harmonic emission and the HOMO structure. It is worth noting that this correction is not reliable for very low k values [8]. On the other hand, for very large k values (thus, very high electron kinetic energies), the usual SFA relationship is retrieved again apart from a constant amplitude factor. Since the spatial frequencies k sampled by the HHG experiment are limited to a bidimensional domain, one can retrieve directly only the bidimensional projection of the molecular orbital. In order to invert Eq. (15) and retrieve this projection, one has to calculate a(k). 8

9 This quantity is usually determined by calibration procedures, exploiting the harmonic spectrum in a companion atom (Krypton, in the CO 2 case) associated with the SFA prediction of the transition dipole term [1]. Nevertheless this approach is sensitive to the Coulomb field of the atomic ion, which is not taken into account by SFA and could alter the retrieved orbital. Hence a(k) was determined directly from the SFA applied to a hydrogenoid atom with the same ionization potential of CO 2 [5]. Once the reconstruction of the Fourier transform of the HOMO is obtained as a function of k and θ, it must be resampled in the k x, k y domain before inversion. This operation requires to take into account the symmetry of the wavefunction, thus imposing the relationships F k {ψ 0 } = F k {ψ 0 } and F kx, k y {ψ 0 } = F kx,k y {ψ 0 }. It is worth noting that the conversion of the Fourier transform of the HOMO from the k, θ domain to the whole k x, k y plane is in general not possible in those molecules lacking any symmetry of the HOMO wavefunction. Indeed the XUV spectra generated by multicycle laser pulses contain contributions from electrons colliding onto the parent molecular ion from both the two sides; it is not possible to disentagle the contributions related to electron momenta with opposite directions, unless the HOMO possesses a (un)gerade symmetry. Nevertheless it has been theoretically shown [9] that the exploitation of single-cycle laser pulses with stable CEP would overcome this issue, since a single electron recollision onto the oriented molecular ion (with a given momentum direction) would be obtained in that case; hence the extension of the measurements from the k to the -k momentum half space would be obtained by changing the pulse CEP by π, without any assumption about the HOMO inversion symmetry. [1] Itatani, J. et al. Tomographic imaging of molecular orbitals. Nature 432, (2004). [2] Boutu, W. et al. Coherent control of attosecond emission from aligned molecules. Nature Physics 4, (2008). [3] Stapelfeldt, H. & Seideman, T. Colloquium: Aligning molecules with strong laser pulses. Reviews of Modern Physics 75, (2003). [4] Popa, C. & Zdunek, R. Kaczmarz extended algorithm for tomographic image reconstruction 9

10 from limited-data. Mathematics and Computers in Simulation 65, (2004). [5] Le, A., Lucchese, R. R., Tonzani, S., Morishita, T. & Lin, C. D. Quantitative rescattering theory for high-order harmonic generation from molecules. Physical Review A 80, (2009). [6] Viftrup S. S. et al., Holding and Spinning Molecules in Space. Physical Review Letters 99, (2007). [7] De S. et al., Field-Free Orientation of CO Molecules by Femtosecond Two-Color Laser Fields. Physical Review Letters 103, (2009). [8] Landau, L. D. & Lifshitz, E. M. Quantum mechanics, Non-relativistic Theory (Pergamon Press, Oxford, 1991). [9] van der Zwan E. V. et al., Molecular orbital tomography using short laser pulses. Physical Review A 78, (2008). 10