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Transcript
274 / QELS 2001 / FRIDAY MORNING tions are in good agreement with the experimental results. In conclusion, we have performed a detailed analysis of the harmonic generation process in the few-optical-cycle regime. In particular the spectral profile of the harmonics with respect to the order, their intensity and relative spectral shift, and their divergence properties were measured as a function of the position of the gas jet. We have developed a nonadiabatic three-dimensional propagation model, whose results are in good agreement with the experiments. - c .c v) 3 4m v .0 Photon energy (ev) C v) W - . I - C 80 100 120 140 160 Photon energy (eV) QFF5 Fig. 1. Harmonic spectrum in helium generated by 7-fs excitation pulses. The inset shows on a logarithmic scale an enlarged portion of the cutoff spectral region shows discrete and well resolved harmonics up to 160 eV. The number of photons per pump pulse in the 50-150 eV spectral range is -7 x lo6, which corresponds to a conversion efficiency of -3 x As shown in the inset of Fig. 1, which displays an enlarged portion of the harmonic spectrum on a logarithmic scale, harmonic radiation at energies as high as 190 eV can be obtained at lower efficiency. Using the spectrograph focal-plane images we have measured the far field divergence of the harmonic beam. Figure 2 shows a three-dimensional harmonic spectrum, from which we can obtain the conversion efficiency and the divergence of the single harmonics. As displayed in Fig. 3, which shows the harmonic divergence as a function of the order, the spatial width of the beam broadens with order from 1.5 mrad to 3.2 mrad ranging from 231dharmonic up to 6Sthharmonic, and then narrows in the cutoff region. This analysis was performed for various positions of the gas jet with respect to the laser beam waist. The harmonic divergence is remarkably affected by the jet position. The experimental results have been analyzed in terms of a nonadiabatic three-dimensional numerical model3 The single-atom response is calculated using the strong field model, generalized References 1. T. Brabec and E Krausz, “Intense few-cycle laser fields: frontiers of nonlinear optics,” Rev. Mod. Phys. 72,545-591 (2000). 2. M. Nisoli, S. De Silvestri, 0. Svelto, “Generation of high energy 10-fs pulses by a new pulse compression technique,” Appl. Phys. Lett. 68,2793-2795 (1996). 3. P. Villoresi, P. Ceccherini, L. Poletto, G. Tondello, C. Altucci, R. Bruzzese, C. de Lisio, M. Nisoli, S. Stagira, G. Cerullo, S. De Silvestri, 0.Svelto, “Spectral features and modeling of high-order harmonics generated by sub-10fs pulses,” Phys. Rev. Lett. 85, 2494-2497 (2000). QFF6 Harmonic M e r QFF5 Fig. 3. Divergence of the harmonic beam vs harmonic order l L 4 5 am Theory of Relativistic Harmonic Generation Q-Han Park, Physics Department, Kyunghee University, Seoul 130-701, Korea Robert W. Boyd, Institute ofOptics, University of Rochester, Rochester, NY 14627 USA to account for nonadiabatic effects. Ionization ef- fects and ground state depletion are included using the Ammosov-Delone-Krainovtheory The single atom response is inserted into a three-dimensional propagation model, which takes into account both temporal plasma-induced phase modulation, and spatial plasma lensing effects on the fundamental beam. The computer simula- QFF5 Fig. 2. Three-dimensional harmonic spectrum John E. Sipe, Department ofphysics, University of Toronto, Toronto, Canada M5S lA7 Alexander L. Gaeta, School ofApplied and Engineering Physics, Cornell University, Ithaca, NY 14853 USA In a recent publication, Chen et al.’ describe their observation of phase matched third-harmonic generation (THG) in a highly ionized gas under laser intensities sufficiently large (2 x 1017 Wlcm’,) that relativistic effects are expected to be important. The harmonic radiation is found to be emitted in a narrow cone (of radius 5.6 degrees) in the near forward direction, with a reasonably large (2 x lo-*) conversion efficiency. Perhaps most surprisingly they found that the intensity of the THG was comparable for linear and circular polarization of their incident laser beam. In traditional nonlinear optics the process of THG is known to vanish identically for an isotropic medium for a circularly polarized fundamental laser beam? Nonlinear optical interactions are often described in terms of a nonlinear optical susceptibility, which for the case of third-harmonic generation relates the polarization of the medium to the third power of the electric field amplitude of the incident laser beam. Harmonic light is radiated in as a consequence of the timevarying polarization of the medium. This procedure presupposes the validity of the electric dipole approximation, both in the calucation of the nonlinear susceptibility and in terms of calculating how the response of the medium leads to new frequency components. Detailed examination of the tensor nature of the nonlinear susceptibility FRIDAY MORNING shows that the nonlinear response vanishes for a circularly polarized intput beam. In this paper, we show that the argument about the vanishing THG breaks down under relativistic conditions. We consider the motion of a free electron in an intense laser field of wavelen h h For field strengths of the order of . . for h = Ipm corresponds to an inmc lk which tensity of the order of 2 x l O I 7 Wlcm', the electron is accelerated to relavistic velocities in a single optical period, and thus the excursion of the electron from its equilibrium poistion occur over a distance of the order of h. Under such conditions, the radiation emitted by the electron cannot be described within the electric dipole approximation, and in fact cannot be well described by a multipole expansion for a finite number of terms. We instead calculate the intensity of the emitted radiation through use of the LenardWiechert potentials, as described below. The motion of a free electron in an intense electromagnetic field of frequency o is well known. For circularly polarized light the electon rotates at frequency w in a circle of radius ro = pclo where p = eEo/ywand $ = m2c? ie2E2,Io2. For linearly polarized light the electron moves in a figure-eight pattern of dimensions comparable to r,,. Even though the motion is strictly harmonic for the circularly polafized situation, and contains only frequencies o and 2 o in the linearly polarized case, the radiation field contains all harmonics of the fundamental frequency in both cases because of retardation effects. In particular, the vector potential of the n-th harmonic component of the radiated wave at distance R, from the center of the electron orbit is given by SHG-circular / QELS 2001 / 275 THG-circular P + where k,, = olc, k = nolc and k has magnitude k and points in the direction of the observation point, and i'gives the instantaneous position of the electron. The power per unit solid angle emitted by this harmonic component is then given by dIn c - IZxdnl2R2, dQ 811 We have evaluated d, analytically for the circularly polarized case and numerically for the linearly polarized case. Some of the predictions of this model are shown in Fig. 1-Fig. 3. Figure 1 shows radiation patterns in the polar coordinate for the sencond and the third harmonic generations in the case of the circularly polarized light. Figure 2 and figure 3 show radiation patterns for the linear case at various values of angle $. We find that this model predicts that linearly and circularly polarized light are roughly equally efficient at exciting harmonic generation. Also, we note that the model predicts the correct conversion efficiency, at least to order of magnitude. One important prediction of the present model is that second-harmonic generation (SHG) should also be emitted, with an efficiency no smaller than that of the third-harmonic. Although Chen et al.' make no mention of the observation of SHG in their initial publication, SHG has been observed in subsequent work.4The efficiency of the emission of the SHG is likely to depend upon the subtleties of the phase matching process, which wiU be discussed. QFF6 Fig. 1. Radiation patterns for the second and the third harmonic generations in the case of the circularlypolarized light. SHG-phi = 0 deg SHG-phi = 45 deg SHG-phi = 90 deg 0. 0.0 -0.0 -0. QFF6 Fig. 2. Radiation patterns for the second harmonic generations in the case of the linearly polarized light. THG-phi = 45 deg THG-phi=gO deg MG-phi = 0 deg -4 QFF6 Fig. 3. Radiation patterns for the third harmonic generations in the case of the linearly polarized light. Reference 3. 1. S.-Y. Chen, A. Maksimchuk, E. Esarey, and D. Umstadter, Phys. Rev. Lett., 84, 5528,2000. 2. J.F. Ward and G.H.C. New, Phys. Rev. 185,57, 4. 1969. R.W. Boyd, Nonlinear Optics, Academic Press, Boston, 1992. See especially section 4.2. D. Umstadter, personal correspondence.