Phase-matched relativistic harmonic generation in plasmas is observed for the first time. Thirdharmonic light is detected and discriminated spectrally and angularly from the harmonics generated from competing processes. Its angular pattern is a narrow forward-directed cone, which is consistent with phase matching of a high-order transverse mode in a plasma. The signal level is found to be on the same order of magnitude for a circularly polarized pump pulse as for a linearly polarized pump

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... PACS numbers: 52.40.Nk, 42.65.Ky It had been predicted theoretically [1,2] and recently verified experimentally that the free electrons in the focus of a high-intensity laser [3] will oscillate relativistically and thus emit harmonics with unique angular patterns [4] . This process, known as relativistic nonlinear Thomson scattering, scatters light into a broad range of angles (ϳ180 ± ) with each lobe having a .30 ± angular width for the low-order harmonics. In this paper we report the experimental observation of the third harmonic emitted into a narrow, hollow cone pointing in the direction of laser propagation. To explain these results, a new theory is presented that describes how phase matching in a plasma can result provided that the harmonic radiation has a highorder transverse structure. The signal level of the harmonic for a circularly polarized pump pulse was found to be on the same order of magnitude as that for a linearly polarized pump pulse, which is characteristically different from harmonic generation from electrons bound to atoms. The increased efficiency due to phase matching might eventually lead to interesting coherent, ultrashort-duration and short-wavelength light sources, in which there is much current interest [5] . Theory for coherent emission in the direction of propagation of the laser beam, referred to as relativistic harmonic generation, has been derived [6, 7] . It indicates that-because of the mismatch between the phase velocities of the laser pulse and the generated harmonics and because of the collective response of the plasma-the conversion efficiency should be low unless a means for phase matching, or quasi phase matching [7], is implemented. Experimentally, Liu et al. [8] tried to measure the third-harmonic light produced from relativistic harmonic generation but ultimately could give only an upper limit on the conversion efficiency. Meyer and Zhu [9] claimed to have observed the second relativistic harmonic generated under the condition of beam filamentation. However, such second-harmonic light has been later identified by several groups [10,11] to be associated with the transverse plasma-density depression driven by laser self-channeling or filamentation [12] , as is also evident from its broad angular width caused by the plasma density gradient [9] . In this experiment, we used a Ti:sapphire-Nd:glass laser system based on chirped-pulse amplification that produces #2-J, 400-fs pulses at 1.053 mm wavelength. The 50-mmdiam laser beam was beam split into two beams. The first beam was sent to a type-II KDP (potassium dihydrogen phosphate) crystal to produce a second-harmonic pulse which was then focused by an f͞3 off-axis parabolic mirror into the center of a gas jet (hydrogen or helium, with a flat top of 750 mm in length and a gradient of 250 mm) to preionize the gas into a plasma (i.e., used as a prepulse). The second beam, the pump pulse, was sent through a delay line to vary its temporal delay and then through a half-wave wave plate and a thin-film polarizer to vary its energy. After that, it was focused by an f͞3.3 lens onto the same gas jet with its focus overlapping with, and its direction collinearly counterpropagating with respect to, that of the second-harmonic prepulse. The prepulse has a focal spot of 12 mm FWHM (full width at half maximum), and the pump pulse has a focal spot of 14 mm. In this experiment, the prepulse energy was fixed at ϳ90 mJ, corresponding to a peak vacuum intensity of 2 3 10 17 W͞cm 2 , which is 3 orders of magnitude higher than the ionization threshold [13] of hydrogen, enough to produce a plasma column of .50 mm diameter. The pump-pulse intensity was varied by adjusting the half-wave wave plate and the maximum was about 2 3 10 17 W͞cm 2 , which corresponds to a normalized vector potential a 0 8.5 3 10 210 l͓mm͔I 1͞2 ͓W͞cm 2 ͔ 0.4. The polarization of the prepulse was set to be linearly polarized, and that of the pump pulse was controlled to be either linearly polarized or circularly polarized by use of a quarter-wave wave plate located before the focusing lens. For the data presented below, the pump pulse was linearly polarized. The third-harmonic light generated by the pump pulse and emitted in the forward direction was collected by the focusing parabolic mirror of the prepulse and was 5528 0031-9007͞00͞84(24)͞5528(4)$15.00