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Physical Review A

Higher-harmonic production in a model atom with short-range potential

Wilhelm Becker, S. Long, John K. McIver

Published Apr 1, 1990• 148 citations
Publisher

Abstract

Production of higher harmonics in a model atom described by a three-dimensional \ensuremath{\delta}-function potential in the presence of a monochromatic linearly polarized field is investigated. The rates for production of the individual harmonics can be expressed as one-dimensional integrals. The only property of the atom that enters the model is its ionization energy. The results are in qualitative agreement with many of the data. In particular, the model exhibits a ``rising plateau.''

DOI
10.1103/physreva.41.4112

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Production of higher harmonics in a model atom described by a three-dimensional \ensuremath{\delta}-function potential in the presence of a monochromatic linearly polarized field is investigated. The rates for production of the individual harmonics can be expressed as one-dimensional integrals. The only property of the atom that enters the model is its ionization energy. The results are in qualitative agreement with many of the data. In particular, the model exhibits a ``rising plateau.''
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].
We study the harmonics generated by a two-level system in which we include the effects of ionization and recombination. The ionized wavepackets are treated as free classical particles moving inside the electric field. We recover the basic features of the single-atom harmonic generation in the tunnelling limit, finding a good agreement between our model and the one-dimensional Schrödinger equation results. The model allows us to investigate the role of bound states in the harmonic generation in the tunnelling regime. In particular, we show that the harmonic efficiency in the plateau region is strongly dependent on the strength of the dipole moment of bound-bound transitions.
We present a study of the effect of time-dependent ionization on harmonic generation by bound–bound transitions. The introduction of tunneling and above-barrier ionization rates in a simple bound system, a two-level atom, induces an unexpected richness in the harmonic spectra, leading to the production of radiation at high frequencies. Such frequencies are not reached when no ionization or steady ionization is considered. Time-dependent ionization is, therefore, a key process in the understanding of generation of middle plateau frequency harmonics from atoms in the tunneling ionization regime and shows that bound–free transitions are not uniquely responsible for the higher-harmonic frequencies in realistic atoms. Following previous authors’ studies, we use an adiabatic approach to understand the presence of these higher frequencies.
The photoionization of atoms in a strong electromagnetic field leads to the effect of "above-threshold" ionization, displaying a number of peaks in the energy spectrum of the photoelectrons. In the case of a one-electron model atom in a circularly polarized field, we derive expressions that describe the position as well as the intensity of those peaks as a function of the field parameters by means of a nonperturbative method. It is immediately clear from our formulas that the lowest-energy peaks in general disappear with increasing field intensity, for Coulomb-like as well as shortrange potentials. For a zero-range potential, a numerical evaluation is easily performed, leading to results that are in every respect qualitatively similar to the experimental results obtained thus far on these kinds of processes.
A simple, "exactly solvable" atomic model will be presented that helps to discuss some of the interesting features of multiphoton ionization of atoms. The electromagnetic field is treated non-perturbatively, it can be strong. The atomic electrons are assumed to be non-interacting (Hartree model). They are bound in a harmonic oscillator potential (one- or three-dimensional). At the ionization energy, the potential is cut off and the free electron has no Coulomb attraction. The Schrodinger equation has exact analytic solutions valid in the inner or outer regions. If the field is turned on adiabatically, the exact solution can be found by matching these solutions at the boundary; both of their value and the normal derivative at all times. Thus the machine calculation reduces to repeated application of fast Fourier transforms and to the solution of linear equations (unfortunately a lot of them).