Few-cycle nonlinear optics of multicomponent media Hervé Leblond, 1, * S. V. Sazonov, 2,† I. V. Mel’nikov, 3,4,‡ D. Mihalache, 5 and F. Sanchez 1 1 Laboratoire P.O.M.A., U.M.R.-C.N.R.S. 6136, Université d’Angers, 2 Boulevard Lavoisier, 49045 Angers Cedex 01, France 2 Russian Research Centre “Kurchatov Institute,” Kurchatov Square, 123182, Moscow, Russian Federation 3 High Q Laboratories Inc., 2 Gledhill Crescent, Hamilton, Ontario, Canada L9C 6H4 4 Optolink Ltd., bldg. 5, proezd 4806, Zelenograd, 124498 Moscow, Russian Federation 5 Department of Theoretical Physics, Horia Hulubei National Institute for Physics and Nuclear Engineering (IFIN-HH), 407 Atomistilor, Magurele-Bucharest 077125, Romania Received 18 July 2006; published 14 December 2006 Using Maxwell-Bloch equations, we analyze the response of a two-component medium of two-level atoms driven by a two-cycle optical pulse beyond the traditional approach of slowly varying amplitudes and phases. We show that the notions of carrier, envelope, phase, and group velocities can be generalized to this situation. For optical pulses of a given duration, we show that the optical field can form a temporal soliton. DOI: 10.1103/PhysRevA.74.063815 PACS numbers: 42.65.Re, 05.45.Yv, 42.65.Tg, 42.81.Dp I. INTRODUCTION The recent success in solid-state mode-locked lasers has resulted in the generation of two-cycle optical pulses 1,2. These pulses have been immediately exploited to generate both unipolar single-cycle electromagnetic pulses and even shorter, attosecond pulses in extreme UV in a variety of non- linear media 3,4 where their wide spectra and intense elec- tric fields raised concerns of an adequate description of a few-cycle pulse FCP laser-matter interaction within the slowly varying envelope approximation SVEA operating with a quasimonochromatic field 5–12. Considerable progress has been made in obtaining a description of both resonant 5,6,9,17 and nonresonant 7,8,11–23 spatiotem- poral dynamics of the FCP beyond the SVEA. In the Maxwell-Bloch formulation 8–11,16,17, the dy- namic response of a medium is modeled by truncating the density-matrix equation either using a long- or short-wave approximation. This enables us to arrive at an integrable non- linear evolution equation and build up a two-soliton solution that can be treated as a correct envelopeless representation of a single-cycle optical pulse 11,12,17. More recent numeri- cal solutions of its natural extension onto the 2+1-D propagation 14,18 have exhibited noticeable departures from the Brabec-Krausz results 10, which were obtained within the SVEA. However all this research in the FCP phe- nomenology has been elaborated for a single-component me- dium that can be either a two-level resonant system or non- resonant nonlinear matrix alone. In this paper, we consider a more general case of the two-component medium where we can derive a nonlinear evolution equation, and its respective two-cycle solution. In the integrable case of the FCP plane-wave propagation, an adequate interpretation of the breather solution allows the demonstration of the physically meaningful quantities of car- rier frequency, envelope, and phase and group velocity, which emerge self-consistently outside the limitations of the SVEA. In the nonintegrable case, we show that the FCP dynamics are extremely sensitive to the relative strength of the two qualitatively different optical nonlinearities and third-order dispersion. Moreover, when analyzing the impact of the coherent absorption and cubic nonlinearity on a two- cycle optical pulse a remarkable feature is distinguished: a stabilization of carrier-envelope phase. This article is organized as follows. In Sec. II, we recall the nonlinear evolution equation for the electric field and its derivation from the Maxwell-Bloch equations for the two- component medium. In Sec. III, the reinterpretation of the breather solution is presented whereas in Sec. IV we show numerically that the propagation of the two-cycle soliton and envelope-phase stabilization also occurs in the nonintegrable case. II. TWO-CYCLE OPTICAL PULSE IN TWO-COMPONENT NONLINEAR MEDIUM—A MODEL Citing the examples of ions embedded in a crystal host, multiple bands in semiconductors, and defects generated in a guiding structure, we note that optically nonlinear condensed matter often contains more than one polarizable component, even though only one may be of primary interest. Here we formulate the evolution beyond the SVEA approximation for optically nonlinear materials, which have more than one po- larizable component. The equation governing the evolution of an optical FCP in a two-component medium was first derived in Ref. 21. We recall briefly the derivation below. We consider the time-dependent propagation of a two- dimensional femtosecond pulse through a two-component medium. The response of the medium upon interaction with the femtosecond electromagnetic field Ex , z , t is described by the total macroscopic polarization P obtained by sum- ming over all components, P =Re j=1 2 N j d j R j , d j being the dipole transition matrix element and N j the atomic density of the j th component. The time dependence of the off-diagonal density-matrix elements  21  j  R j is given by the Maxwell- Bloch equations, *FAX: 33 241735216. Email address: herve.leblond@univ- angers.fr † Email address: sazonov.sergey@gmail.com ‡ FAX: 416 9713020. Email address: ivm@highqlabs.com PHYSICAL REVIEW A 74, 063815 2006 1050-2947/2006/746/0638158 ©2006 The American Physical Society 063815-1