Optimal quasi-phase-matching for high-order harmonic generation in gases and plasma

Abstract

High-order harmonic generation (HHG) [1], if phase matched, may provide an important and convenient (in principle, table-top) source of short-wavelength coherent radiation. So far, however, HHG phase matching remains poor. The only experimental way to improve HHG phase matching has been weaker focusing of the pumping beam. An inherent disadvantage of this method is that it lowers the incident intensity thus decreasing overall power conversion efficiency. Moreover, this technique can have only a limited success in approaching the optimum phase matching which corresponds to large negative values of the parameter bΔk (b is the laser confocal parameter, and Δk is the dispersive phase mismatch of the medium), since in gases or, to much larger extent, in plasma, Δk is positive. Recently, J. M. Rax and N. J. Fish [2] have suggested plasma density modulation as a method to phase match relativistic third-harmonic generation in plasma. Their idea is essentially a ramification of the well known in nonlinear optics method of quasi-phase-matching (QPM). Almost all the effort in QPM has been concentrated on the second-harmonic generation in solids since obviously much more convenient ways to optimize higher-order harmonic generation (until recently -- almost exclusively, third-harmonic generation) in gases exist. As a result, to the best of our knowledge, no general consideration of quasi-phase-matching in gases and plasma, in particular for HHG, has been published yet. In the present paper, we consider quasi-phase-matching of HHG by a focused beam in plasma or a gas whose nonlinear succeptibility and refractive index are spatially modulated, in particular through the medium density modulation, and demonstrate that QPM is feasible with the available laser and plasma technology. In accordance with the HHG experimental conditions, we assume that harmonics are generated by bound electrons. To obtain analytic results, we rely on the perturbation expressions for the induced nonlinear polarization; our results remain valid beyond perturbation limits if some general assumptions hold regarding nonlinear polarization induced by strong laser field [3]. Accurate quantitative estimates of the improvement in HHG due to QPM requires nonperturbative calculations of HHG in homogeneous media with large negative bΔk; to our knowledge, no such calculations have been published yet. On the basis of a nonperturbative model introduced in Ref. [4], however, one might expect QPM-optimized harmonic intensity to be up to two order of magnitude larger, with available depth of plasma density modulation, than the output intensity attainable under currently employed loose focusing and poor phase matching. Even more important a consequence of QPM is the opportunity to use tight focusing so far deleterious for HHG; resulting much higher intensities would greatly increase power conversion efficiency as compared to the loose-focusing regime.