Macroscopic Effects in Noncollinear High-Order Harmonic Generation

Abstract

We study two-color high-order harmonic generation using an intense driving field and its weak second harmonic, crossed under a small angle in the focus. Employing sum-and difference-frequency generation processes, such a noncollinear scheme can be used to measure and control macroscopic phase matching effects by utilizing a geometrical phase mismatch component, which depends on the noncollinear angle. We further show how spatial phase effects in the generation volume are mapped out into the far field allowing a direct analogy with temporal carrier envelope effects in attosecond pulse generation. DOI: 10.1103/PhysRevLett.112.143902 PACS numbers: 42.65.Ky, 42.65.Re High-order harmonic generation (HHG) in gases using multicolor optical fields is becoming a common tool in attosecond science. Multicolor HHG has been implemented by mixing the fundamental field with waves at low harmonic [1][2][3] and at incommensurate [4,5] frequencies. Other schemes employ attosecond pulse trains mixed with the fundamental field [6] as well as synthesized light transients [7] for driving the HHG process. The control and flexibility brought about by using two or more driving fields is used, for example, to improve the efficiency of the generation process [3,8], to monitor the single atom response [9,10], and to implement gating techniques [11]. High-order wave mixing can further be beneficial for achieving phase matching [1,[12][13][14][15].Using multiple optical driving fields also allows for noncollinear geometries. This concept, used extensively in low-order frequency conversion processes such as optical parametric amplification, has barely been applied to attosecond science. In an early work, Birulin and co-workers [16] addressed theoretically the general concept of HHG in a noncollinear geometry. By mixing the fundamental and a weak second harmonic at an angle, Bertrand and co-workers [17] demonstrated the spatial separation of multiple extreme ultraviolet (XUV) beams corresponding to different sets of absorbed photon numbers from the two fields. A similar scheme was recently used [18] for reconstructing the spatiotemporal characteristics of the generated attosecond pulses. Other authors [19][20][21] concentrated on noncollinear sum-frequency generation (SFG) processes driven by two identical laser fields. Such a scheme allows for separation of the harmonics from the fundamental but suffers from phase matching problems.In this Letter, we theoretically and experimentally examine the influence of noncollinear geometries in HHG, analyzing in detail the macroscopic aspects. We show for the first time, to our knowledge, how a weak noncollinear field can be used as a probe to monitor and control phase matching. We show that difference-frequency generation (DFG), where a photon (or several photons) is emitted in the conversion process, is in general more favorable than SFG [22], which becomes very inefficient with increasing noncollinear angle. This leads to the counterintuitive result that the XUV radiation is dominantly emitted ou...

Phase matching of noncollinear sum and difference frequency high harmonic generation above and below the critical ionization level

“…Previous investigations of noncollinear HHG have shown that phase matching effects can determine the angle of preferential harmonic emission and used this angular dependence to infer the difference in the magnitude of the harmonic and driving laser wavevectors [43]. In this work, we experimentally and theoretically investigate the macroscopic physics of phasematched noncollinear HHG at high pressures.…”

“…Due to the angular separation between the driving laser beams, phase-matching in noncollinear HHG [41,43] is fundamentally different from either single-beam HHG or collinear HHG [3,24,[44][45][46][47] and presents both new challenges and new opportunities. Previous investigations of noncollinear HHG have shown that phase matching effects can determine the angle of preferential harmonic emission and used this angular dependence to infer the difference in the magnitude of the harmonic and driving laser wavevectors [43].…”

“…However, in a noncollinear geometry we must consider the projection of the driving laser wave vectors along the direction of the harmonic emission due to the angular differences between them (Fig. 1e-f) [28,43,[48][49][50]. This projection results in a modified phase-matching equation:…”

“…Typically, critical ionization places an upper limit on the laser intensities that can be used to drive HHG because at ionization levels above η c , the free-electron contribution to the index dominates, resulting in a driving laser phase velocity much greater than c [51,52]. Exceptions to this rule include quasi phase matching [53] or UV-driven HHG, where bright HHG beams can be produced through effective phase matching in multiply-ionized plasmas [43].…”

“…1i). Note that regardless of the specific conditions, the noncollinear phase mismatch varies as the cosine of the angular separation between the emitted harmonics and the driving lasers, which behaves quadratic in the small angle approximation [43] (see Appendix B, Fig. 6).…”