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Physical Review A

Intensity dependence of the phase of harmonics in one- and two-frequency laser fields

W.-C. Liu, Charles W. Clark

Published May 1, 1996• 5 citations
Publisher

Abstract

The phase relationship between an induced atomic dipole moment and a driving laser field is an essential aspect of high harmonic generation (HHG) in a dense medium. A ``phase-matching'' criterion must be fulfilled to obtain coherent amplification of harmonic radiation, which is critical for practical HHG. We have performed numerical calculations on model atoms to survey the intensity dependence of the phase in the case of strong one- or two-frequency laser fields. The calculations show that multiphoton resonances significantly affect the yield of harmonic generation and that in the ``plateau'' regime, the phases exhibit only weak dependence upon the intensity of the driving field. \textcopyright{}1996 The American Physical Society.

DOI
10.1103/physreva.53.3582

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The phase relationship between an induced atomic dipole moment and a driving laser field is an essential aspect of high harmonic generation (HHG) in a dense medium. A phase-matching'' criterion must be fulfilled to obtain coherent amplification of harmonic radiation, which is critical for practical HHG. We have performed numerical calculations on model atoms to survey the intensity dependence of the phase in the case of strong one- or two-frequency laser fields. The calculations show that multiphoton resonances significantly affect the yield of harmonic generation and that in the plateau'' regime, the phases exhibit only weak dependence upon the intensity of the driving field. \textcopyright{}1996 The American Physical Society.
Harmonic generation is studied in an idealized model medium in which the laser field essentially interacts only with a single atomic transition and where ionization can be neglected. Such two-level models are known to generate high harmonics with a plateau and a cutoff. We put forward a quantitative rule for the cutoff frequency in the harmonic spectrum in this model which we relate to the ``IP{\mathrm{I}}_{\mathrm{P}}IP​+3UP{3\mathrm{U}}_{\mathrm{P}}3UP​ rule'' in real systems with ionization. We examine the phase properties of the emitted harmonics and show that their coherence and thus phase matching is more favorable in the cutoff regime than in the plateau of the harmonic spectrum. This behavior is known to occur in realistic models of neon, and that this occurs also for the simple idealized two-level model strongly suggests that the sudden change of phase coherence from the cutoff to the plateau regime is as intrinsic a property of nonlinear systems as are the plateau and the cutoff.
Optical harmonic generation and mixing in the gas phase has been proposed as a technique for the generation of coherent radiation in the vacuum ultraviolet and soft x-ray spectral region. At the high field strengths required by these processes the interaction between atoms and the electromagnetic field shows intensity-dependent resonances. In this paper we modify harmonic generation theory to include the effect of these frequency shifts. Closed-form expressions for generated dipole moment, absorption probability, and coherence length are presented. The most important consequences of frequency shifts on resonantly enhanced processes are that the pump laser must be tuned away from the small-field resonance frequency, that the conversion efficiency may saturate, and that the dispersion of the medium may change sign. As an example, the generation of 198-\AA{} radiation by a five-photon mixing process in Li+{\mathrm{Li}}^{+}Li+ is considered.
We show that a simple nonperturbative two-level model of an atom driven by a very strong periodic field, results in a rich picture of very high harmonic generation and related phenomena. It reproduces experimentally observed plateau, yields for the first time simple analytic formulas for the plateau cutoff frequency, critical driving intensity, and saturation, and predicts intensity-induced multi-resonances. One of the most fascinating phenomenon discovered recently in nonlinear interaction of light with atoms and ions, is very high-order (up to 135) odd harmonic generation (HHG) by intense (∼ 1013 W/cm2 and higher) optical laser radiation in rare gasses and some ions [1], The spectra of generated harmonics drasticly deviate from the perturbation theory predictions [2], In particular, intensity of harmonics, falling monotonically with their orders only up to a certain point, levels off forming a so-called “plateau", and falls monotonically again beyond it. Generally, harmonte generation depends on phase-matching conditions and nonlinear response of individual atoms. It has recently become clear however [3,4] that the major features of HHG, in particular the plateau, result mainly from general properties of atomic nonlinear response. The most direct and apparently successful way so far to approach the problem theoretically has been numerical simulation of the Schrödinger [2,5-7] (including an empiric rule [6]) for many-electron atoms using Hartry-Slater approximation. This approach requires, however, tremendous amount of calculations and involves many processes, making it difficult to gain simple insights. An interesting simplified model [8] based on 3-D delta-potential with a single (ground) level [9] produces results in integration form. The idea of retaining a single energy scale (ionization energy) brings one close to an even simpler system: a two-level model atom. A two-level model of HHG [10], however, due to various complications introduced into it, failed to generate simple results, whereas an analytic solution [11] holds for a virtually degenerate two-level model only (see below) which is unapplicable to the experimental conditions [1-4,7,9]; besides, no relaxation was considered in [10,11].
This paper presents an analytical study of high-order harmonic generation (HHG) by quantum systems possessing permanent dipole moments (PDMs), such as atoms/molecules placed in a static electric field or polar molecules. HHG by conventional, non-PDM systems typically demonstrated a plateau in the envelope of harmonics followed by a steep fall in harmonic intensities when the harmonic number exceeds a certain threshold. This paper shows that the HHG by PDM systems results in a significant extension of the plateau to higher frequencies and a slower decline of intensities at frequencies higher than the end of the plateau . Moreover, there occurs a substantial growth of the total, summed intensity of all components of the scattering spectrum . Other physically interesting distinctive features of the HHG by PDM systems are: the presence of both odd and even harmonics of the laser frequency in the scattering spectrum; a triplet structure of both odd and even harmonics; and the appearance of a component at the doubled Rabi frequency (i.e. at a frequency much lower than the laser frequency). The results are obtained by using the adiabatic basis for the analysis of a multi-quantum resonance in a two-level system possessing PDMs.
In the limits of weak and strong excitation, we solve analytically the Riccati-type differential equation for a two-level system in the field of two different lasers and then use these solutions to find easy to handle formulae for the induced electric-dipole moment of the system.We apply these formulae to describe the effects of strong excitation in both the higher harmonics generation and multiphoton frequency mixing, namely the diminishing of the generation of harmonics of a given laser when an additional laser is turned on and the diminishing of the radiation at mixed frequencies with increasing strength of lasers under the condition of multiphoton excitation.