175 Laser Physics, Vol. 9, No. 1, 1999, pp. 175–183. Original Text Copyright © 1999 by Astro, Ltd. Copyright © 1999 by åÄàä “ç‡Û͇ /Interperiodica” (Russia). 1. INTRODUCTION Two recent experiments on high-order harmonic generation in helium [1, 2] have demonstrated that exposing a gas of atoms to an ultrashort laser pulse of a few optical cycles provides an efficient way to extend harmonic emission up to frequencies well into the so- called water window. This opens the route to the devel- opment of methods of interferometry in the XUV regime, leading, for instance, to important applications in biology. Using laser pulses of a few optical cycles has two main implications on the process of high-order har- monic generation. The first one concerns the saturation intensity, which may be much higher. In other words, the atoms may be exposed to higher intensities before being ionized, leading to the emission of higher-order harmonics. The second implication is the fact that the use of ultrashort laser pulses provides a natural and effi- cient scheme to produce subfemtosecond harmonic pulses. When an atom interacts with a laser pulse of a few optical cycles, the field intensity varies significantly on a time scale of one optical cycle, leading to new inter- esting effects [3]. In particular, the role of the phase of the laser field defined here as the phase of the electric field at the maximum of its envelope becomes critical. The purpose of the present contribution is to examine in detail how this phase affects the high-order harmonic spectra. From the theoretical point of view, the ultrashort duration of the laser pulses considered here requires a nonadiabatic treatment. In other words, the so-called slowly varying envelope approximation is not valid any more. However, this does not imply that the semiclassi- cal description [4] of the harmonic emission provides a wrong or inadequate picture. On the contrary, its pre- dictions are still, at least qualitatively, in very good agreement with those obtained by solving the corre- sponding full 3D time-dependent Schrödinger equa- tion. According to the semiclassical picture, the elec- tron first escapes from the core by tunneling through the Coulomb barrier, which is lowered by the strong laser field. Once free, it oscillates freely, driven by the external laser field. If it reencounters the residual ion, it may either recombine into the ground state, emitting a harmonic photon, or be scattered. It is important to stress that the main part of the dynamics of the har- monic emission process is directly determined by the phase accumulated by the electron on its trajectory in the continuum [5]. When the laser field intensity does not vary significantly on a time scale of one optical cycle, the maximum kinetic energy gained by the elec- tron in the continuum is given by 3.17U p , where U p is the quiver energy of free electrons in an oscillating field. It defines the energy of the highest harmonic in the spectrum given by I p + 3.17U p , where I p is the ion- ization potential of the atom. The paper is divided in three sections. The first one is devoted to some remarks about the description of an ultrashort laser pulse. In particular, we clarify the con- cept of pulse duration, which is ambiguous for pulses of a few optical cycles. In the second section, we describe briefly our method for solving the full 3D time-depen- dent Schrödinger equation and indicate how we per- form the time–frequency analysis of the atomic dipole acceleration. In the last section, we present and discuss in detail our results. We show that they may be inter- Phase Sensitivity of Harmonic Emission with Ultrashort Laser Pulses A. de Bohan*, Ph. Antoine*, D. B. Milo evi **, G. L. Kamta*, and B. Piraux* * Laboratoire de Physique Atomique et Moléculaire, Université Catholique de Louvain, Chemin du Cyclotron 2, Louvain-la-Neuve, B-1348 Belgium e-mail: piraux@fyam.ucl.ac.be ** Department of Physics and Astronomy, The University of Nebraska, Lincoln, Nebraska, 68588-0111 USA Received August 27, 1998 Abstract—We consider harmonic generation by atoms exposed to an intense laser pulse of a few femtoseconds. Our results, obtained by solving numerically the corresponding 3-dimensional time-dependent Schrödinger equation, demonstrate that the harmonic spectra are extremely sensitive to the phase of the laser field. Depend- ing on this phase, the harmonics in the cutoff are resolved or not resolved. The position of the cutoff itself varies with the phase and the so-called “plateau” region exhibits two well-distinct parts: a series of well-defined har- monics followed in the high-frequency region by a series of broad peaks which are not separated any more by twice the laser field frequency. These results are explained in terms of both quantum and classical dynamics. We also show that this phase sensitivity may be exploited in order to probe the phase of the electric field of an ultrashort laser pulse in a single shot experiment. Our discussion about this new method of diagnosis takes into account propagation effects. s ˇ c´ STRONG FIELD PHENOMENA