journal of modern optics, 2003, vol. 50, no. 3/4, 561–577
Investigations of electron wave-packet dynamics and highorder harmonic generation in laser-aligned molecules N. HAY1 , M. LEIN1 , R. VELOTTA1 , R. DE NALDA1 , E. HEESEL1 , M. CASTILLEJO2 , P. L. KNIGHT1 and J. P. MARANGOS1 1
The Blackett Laboratory, Imperial College of Science, Technology and Medicine, London SW7 2BW, UK; e-mail:
[email protected] 2 Instituto de Quı´ mica-Fı´ sica Rocasolano, CSIC, Serrano 119, 28006 Madrid, Spain (Received 19 April 2002) Abstract. We review recent work examining electron dynamics and highorder harmonic generation (HHG) in molecules exposed to high intensity (1013 –1015 W cm2 ) ultrashort pulse laser fields. The use of adiabatic laser alignment techniques, using relatively long duration (300 ps) laser pulses of moderately high intensity (1011 –1012 W cm2 ) enabled us to perform the first experiments investigating the role of the angle between the laser polarization vector and the molecular axis. We have shown that alignment of a sample can enhance the harmonic yield by a small but significant factor. HHG in model H2 and Hþ 2 molecules was treated theoretically by numerical integration of the Schro¨dinger equation. The unique influence of the dipole phase in molecular HHG is clearly demonstrated and is shown to offer a consistent explanation for our experimental observations. The electron wavepacket dynamics exhibit strong interference effects that result in the reduction or enhancement of harmonic yield at certain alignment angles. Experiments to further investigate these effects and to increase the modulation of the harmonics by molecular alignment are discussed.
1.
Introduction There has been extensive work in recent years studying high-order harmonic generation (HHG) resulting from the interaction of intense laser light with atoms. This process provides a unique source of coherent extreme ultraviolet (XUV) radiation [1]. Several workers have observed the similarity of HHG in atoms and small molecules [2, 3] and recently it was shown that if sufficiently short pulses are used ( 1016 cm3 ) that there is a detectable yield from the nonlinear process. Controllable alignment of molecules has been investigated previously because of the critical influence of the alignment and orientation on the dynamics of chemical reactions. Various orientation-dependent phenomena (steric effects) have been studied, such as the head vs tail asymmetry of the reaction probability and the branching ratios of various products [12]. We distinguish between alignment, where the molecular axis is preferentially aligned at some angle relative to a space-fixed axis (e.g. an electric field line) and orientation where the direction in which the molecule is pointing along the alignment axis is also controlled. For the rotationally symmetric
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molecules considered here, orientation is not possible. Note that an aligned distribution has inversion symmetry, while an oriented distribution does not. Early experiments probing the production of oriented molecules made use of the electrostatic hexapole technique [13, 14]. For a review, see Stolte [15]. In this technique, a molecular beam is directed along the centre of a set of charged rods such that the field strength is zero at the centre but increases closer to the rods. Thus, a molecule with a positive Stark shift will have a higher energy further from the centre. This energy derivative will exert a force on the molecule, and the molecules will be focused by the field. By adjusting the voltage on the rods, only molecules in a particular rotational state can be focused onto a desired point. This achieves rotational state selection. The best focusing is achieved if the energy increases as the square of the distance from the centre of the rods. The typical beam density one can obtain with this technique is very low and in practice the method is restricted to polar symmetric-top molecules (e.g. CH3I) and to polar linear molecules that behave as such [16]. The ‘brute force’ method of orienting polar molecules in a strong homogeneous electric field, as proposed by Loesch and Remscheid in 1990 [17] for molecular beams of rotationally cold molecules, does not have these restrictions, although the attainable orientation is in general much smaller. Brute force orientation comes about by mixing of rotational states owing to the interaction of the electric field with the molecular dipole moment. It is based on applying a strong uniform electric field so that orientation by Stark mixing of different rotational states occurs. The experimental feasibility of brute force orientation is subject to several restrictions, such as the high values required for the electric field and the dipole moment and the low values needed for the rotational constant and the temperature of the beam. The beam density obtainable with this technique is much higher than that arising from the hexapole technique, mainly due to the different distance between the nozzle and scattering centre (a few centimetres for the brute force method against hundreds of centimetres for the hexapole selector). Nonetheless, the presence of a strong electric field (typically tens of kV cm1 ) prevents the use of brute force alignment in conjunction with the densities needed for HHG as a strong discharge would occur. A general method to align molecules is the use of collisions in an anisotropic velocity distribution. When a velocity gradient exists between two molecular species, as in a free expansion, collisions along the direction of the flow will preferentially align angular momenta to be perpendicular to the flow. One of the species, having larger velocity, is the ‘diluent’ (typically helium) whereas the molecular species to be aligned represents the ‘seed’. The molecules with angular momentum parallel to the flow present a larger cross-sectional area for collisional reorientation, while molecules with angular momentum perpendicular to the flow are less likely to be hit. As modelled by Zare and co-workers [18], even if all such collisions simply randomize the direction of the angular momenta, this will lead to a net alignment of angular momenta perpendicular to the gas flow. This kind of alignment is usually referred as ‘negative alignment’; in fact, if the rotor molecules (the seed) are faster than the diluent a preferential alignment of the angular momenta parallel to the velocity is expected (positive alignment). The collisional alignment, formerly used for linear molecules, has recently been applied to the planar benzene molecule [19].
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The use of an optical (oscillating) field offers several possibilities to align molecules. Among them, single-photon excitation has received much interest since Zare developed the theory for aligning linear and symmetric top molecules [20]. The photoexcitation method achieves a measure of spatial control over a molecular axis by aligning the molecule in the excited state [21]. If the internuclear axis is parallel to the polarization of the laser, the molecule is more likely to be excited than if the axis is perpendicular. Hence, the excited molecules end up with the internuclear axis directed predominantly along the polarization vector. With this method, there is no physical separation between aligned and non-aligned molecules. It is the subsequent analysis of the rotational state of resulting products that allows one to evince the role of steric effects in the chemical reaction [22]. More recently, the use of a strong off-resonant optical field for adiabatic alignment has been developed. Alignment has been demonstrated at low densities ( 1010 cm3 ) for molecules having no permanent dipole moment [23, 24] where the electric field of a laser pulse induces a molecular dipole moment and simultaneously exerts a torque upon it [25]. Under the conditions discussed below, this torque leads to an ensemble of molecules having an angular distribution of molecular axes that is peaked about the direction of the electric field. From the quantum mechanical point-of-view, the eigenstates of aligned molecules in the adiabatic limit are pendular states and can be labelled by the quantum numbers J~ and M of the field-free rotational state [26]. The degree of alignment is quantified by the expectation value of cos2 , hcos2 i, where is the polar angle between the molecular axis and the electric field vector. It ranges from 1/3, corresponding to an isotropic distribution, to 1 when the molecule is in a perfectly aligned state along the electric field vector. For an ensemble of molecules in thermal equilibrium, the average of the Boltzmann distribution of the rotational states has also to be considered; thus we have [23, 26]: hcos2 i ¼
X j~
wj~
M¼þ X j~
hcos2 ij~;M :
ð1Þ
M¼j~
In equation (1), hcos2 ij~;M is the expectation value of cos2 over the rotational state characterized by the quantum numbers J~ and M and wJ~ ¼ exp½J~ðJ~ þ 1Þ= =Qr with Qr the rotational partition function and ¼ kT=B the reduced rotational temperature (B is the rotational constant of the molecule). In [25], asymptotic expressions for hcos2 ij~;M are reported as a function of the anisotropy parameter c ¼ ð!jj !? Þ1=2 where !jj;? ¼ jj;? E02 =4B with jj;? the parallel and normal polarizability of the molecule and E0 the electric field amplitude. In earlier work [23, 27–29], a 3:5 ns duration laser pulse was used to align several species of neutral molecule in samples of density
