Orientation dependence of high-order harmonic generation in molecules M. Lein, 1, * P. P. Corso, 2 J. P. Marangos, 1 and P. L. Knight 1 1 Blackett Laboratory, Imperial College of Science, Technology and Medicine, London SW7 2BW, United Kingdom 2 Dipartimento di Scienze Fisiche ed Astronomiche dell’Universita ` and Istituto Nazionale di Fisica della Materia, Via Archirafi 36, 90123 Palermo, Italy Received 21 October 2002; published 28 February 2003 We present two- and three-dimensional model calculations of high-order harmonic generation in H 2 + . The harmonic spectra exhibit clear signatures of intramolecular interference. An interference minimum appears at a harmonic order that depends on the molecular orientation. Harmonic generation in three-center molecules is studied on the basis of two-dimensional calculations for a H 3 2+ model system. From analytical considerations, the orientation dependence of the harmonic intensities in three-center molecules exhibits a double minimum due to intramolecular interference. In the numerical results, the double minimum is broadened into a single wide minimum. The effect of nonzero laser ellipticity on harmonic generation is investigated by means of two-dimensional simulations for H 2 + . We find that harmonic generation with elliptical polarization is gov- erned by interference effects similar to linear polarization. DOI: 10.1103/PhysRevA.67.023819 PACS numbers: 42.65.Ky, 33.80.Rv I. INTRODUCTION High-order harmonic generation HHG1–4 is the pro- cess in which a laser-driven system converts many incoming laser photons into a single high-energy photon. Using HHG as an efficient source of high-frequency coherent radiation is one of the main goals of research in this field 5–7. HHG has been studied in many different systems, but most of the experimental and theoretical work has been focused on at- oms. The recollision picture 8,9 explains HHG as a sequence of tunnel ionization, laser-driven motion of the free electron, and recombination with the core. HHG with small molecules resembles HHG with atoms because the wave packet associ- ated with a recolliding electron is typically much larger than the internuclear distance. However, since molecules have more degrees of freedom than atoms, their behavior in strong fields is richer and lends itself to targeted control by the experimenter. For example, HHG can be enhanced by pre- aligning the molecules in the interaction region 10,11. Fur- thermore, some molecules tolerate unusually high laser in- tensities 12–14. Therefore, one may hope that higher harmonic yields and higher photon energies can be reached with molecules. Several earlier theoretical studies have shown that har- monic generation with linearly polarized light is sensitive to the molecular orientation 15–19. The most dramatic orien- tation effect appears to be the interference between the con- tributions from the different atoms within the molecule 18,19, which can lead to a complete suppression of har- monics. The conditions for constructive and destructive in- terference were found to be rather simple and independent of the laser parameters. By varying the orientation of the mol- ecule, a certain harmonic can be maximized or minimized. For diatomic molecules, it was found that the harmonic order of an interference extremum depends only upon the projec- tion of the internuclear separation onto the polarization axis. The conclusions of Refs. 18,19 were largely based on two-dimensional 2D model calculations for two-center molecules in linearly polarized lasers. In this paper, we com- pare those with the results of 3D calculations, which require a much larger amount of CPU time. We confirm from the comparison that the interference effects are independent of the dimensionality of the system. Furthermore, our analysis is extended to 2D calculations for three-center molecules. Here, we also find pronounced interference structures. The main motivation for the study of three-center systems is the experiment of Refs. 10,11 in which laser-induced align- ment was demonstrated most clearly for CS 2 molecules. Fi- nally, we investigate HHG in elliptically polarized lasers. Again, clear signatures of interference are found. However, the simple model of intramolecular interference given in Ref. 19 does not apply to elliptical polarization because the model implies that the impact velocity of recolliding elec- trons is parallel to the polarization axis. Accordingly, the interference pattern becomes more complicated for nonzero ellipticity. This paper is organized as follows. Section II describes our numerical method. Sec. III gives a comparison between 2D and 3D results for H 2 + . In Sec. IV, we report on the 2D treatment of the three-center system H 3 2 + . Section V de- scribes HHG with elliptical laser polarization, based on 2D calculations for H 2 + . Finally, Sec. VI contains a short sum- mary and our conclusions. II. METHOD In our numerical approach, we solve the time-dependent Schro ¨ dinger equation for a molecule in a strong laser pulse with electric field E( t ), i   t  r, t  =  p 2 2 +p• A t  +V  r   r, t  , 1 *Present address: Max Planck Institute for the Physics of Com- plex Systems, No ¨thnitzer Strasse 38, D-01187 Dresden, Germany. PHYSICAL REVIEW A 67, 023819 2003 1050-2947/2003/672/0238196/$20.00 ©2003 The American Physical Society 67 023819-1