Nonadiabatic three-dimensional model of high-order harmonic generation in the few-optical-cycle regime E. Priori, G. Cerullo,* M. Nisoli, S. Stagira, and S. De Silvestri Istituto Nazionale per la Fisica della Materia, Centro di Elettronica Quantistica e Strumentazione Elettronica del Consiglio Nazionale della Ricerche, Dipartimento di Fisica, Politecnico di Milano, Piazza Leonardo Da Vinci 32, 20133 Milano, Italy P. Villoresi, L. Poletto, and P. Ceccherini Istituto Nazionale per la Fisica della Materia, Laboratorio di Elettronica Quantistica, Dipartimento di Elettronica e Informazione, Universita ` di Padova, Padova, Italy C. Altucci Istituto Nazionale per la Fisica della Materia, Dipartimento di Chimica, Universita ` della Basilicata, Potenza, Italy R. Bruzzese and C. de Lisio Istituto Nazionale per la Fisica della Materia, Dipartimento di Scienze Fisiche, Universita ` di Napoli Federico II, Napoli, Italy Received 30 December 1999; published 5 May 2000 A numerical model to calculate the high-order harmonics spectrum of a macroscopic gas target irradiated by a few-optical-cycle laser pulse is presented. The single-atom response, calculated within the nonadiabatic strong-field approximation, is the source term of a three-dimensional propagation code. The simulation results show remarkably good agreement with experiments performed in neon using laser pulses with durations of 30 and 7 fs. Both simulations and experiments show discrete and well-resolved harmonics even for the shortest driving pulses. PACS numbers: 42.65.Ky, 42.65.Re, 32.80-t I. INTRODUCTION High-order harmonic generation HHG1,2 is a rapidly developing topic in the field of laser-atom interaction. Be- sides its fundamental interest, it represents an attractive tech- nique for the production of ultrafast coherent radiation in the UV and soft-x-ray regions of the spectrum using a tabletop system. Recent advances in ultrashort-pulse laser technology have allowed the generation of light pulses with 20–30-fs duration and mJ-level energy; these pulses can be further shortened to sub-10-fs duration by hollow fiber compression 3. A number of theoretical studies have shown the advan- tages of using such short pulses for HHG 4–6; since atoms can be exposed to higher electric fields before a significant fraction is ionized, the conversion efficiency increases and higher energy photoemission can be achieved. In addition, the use of short driving pulses should allow one to combine the fields of several harmonics and generate, under suitable conditions, soft-x-ray pulses with attosecond duration 7. Recently, several HHG experiments, performed using both 20–30-fs pulses 8–11 and sub-10-fs pulses 12–15, showed a strong dependence of the spectral characteristics of the observed harmonics on the experimental conditions driving pulse duration and intensity and laser-gas interaction geometry. To understand these results, a detailed theoretical model for the calculation of HHG spectra in the few-optical- cycle regime of the driving pulse is required. Typical HHG calculations consist of two parts: first the microscopic single-atom response to the driving field is calculated, then it is inserted as a source term in the propa- gation equations of the harmonic field, to obtain the macro- scopic response of the excited nonlinear medium. The most accurate way to obtain the single-atom response is the nu- merical solution of the three-dimensional time-dependent Schro ¨ dinger equation 16,17 for the atom interacting with the laser field. Since this approach is quite time consuming, analytical models based on the approximate solution of the Schro ¨ dinger equation have been proposed; a particularly suc- cessful one, developed by Lewenstein and co-workers 18,19, is known as the strong-field approximation SFA, and shows good qualitative agreement with the exact solu- tion. When the single-atom response is used to study propa- gation effects, two approaches are followed. A first set of studies uses the adiabatic approximation, assuming that the atomic response is determined by the instantaneous laser in- tensity. Using this approach three-dimensional propagation models, calculating the single-atom response either with the Schro ¨ dinger equation 20–23 or with the SFA 24–26, have been developed. Other studies use a nonadiabatic ap- proach, calculating for each atom of the nonlinear medium the full response to the electric field of the driving pulse, again using either Schro ¨ dinger equation 27–29 or the SFA 30,31. However, to our knowledge, in the nonadiabatic case, only one-dimensional plane wave geometries were considered. For few-optical-cycle pulses the adiabatic approach is clearly not valid, because the electric field varies consider- ably during one optical cycle, and the atomic response to the entire pulse not to an instantaneous intensity needs to be calculated. On the other hand, in many experimental condi- *Electronic address: giulio.cerullo@polimi.it PHYSICAL REVIEW A, VOLUME 61, 063801 1050-2947/2000/616/0638018/$15.00 ©2000 The American Physical Society 61 063801-1