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Annual Meeting Optical Society of America

High-order harmonic spectra computed for a model atom

L. Roso, Joseph H. Eberly

Published Jan 1, 1989• 0 citations
Publisher

Abstract

We compute numerically the scattered light spectra accompanying MPI of a very simple atomic model. The model atom is similar to the hydrogen negative ion and has a single bound state and N + 1 continua (corresponding to angular momenta L = 0,…,N). The model is designed to focus attention on two parameters, a single representative dipole matrix element between the ground state and continuum states and a single dipole matrix element between continuum states. Each continua is modeled as a flat quasicontinuum. The bound state is coupled to any level in the L = 1 continuum with the same rate. Any level in the Lth continuum is coupled to any other level in the (L–1) st and (L+1)st continua with equal rates. We have calculated numerically the response of this atom to a laser pulse without introducing the rotating wave approximation.

DOI
10.1364/oam.1989.fu4

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We compute numerically the scattered light spectra accompanying MPI of a very simple atomic model. The model atom is similar to the hydrogen negative ion and has a single bound state and N + 1 continua (corresponding to angular momenta L = 0,…,N). The model is designed to focus attention on two parameters, a single representative dipole matrix element between the ground state and continuum states and a single dipole matrix element between continuum states. Each continua is modeled as a flat quasicontinuum. The bound state is coupled to any level in the L = 1 continuum with the same rate. Any level in the Lth continuum is coupled to any other level in the (L–1) st and (L+1)st continua with equal rates. We have calculated numerically the response of this atom to a laser pulse without introducing the rotating wave approximation.
The coherent scattering spectrum of a three-level atom driven by an intense laser field with multiphoton resonant excitation is considered. The spectrum corresponding to harmonic generation is investigated both analytically and numerically for different initial atomic states. The analytical calculations are based on the generalized rotating wave approximation, which allows one to explain obtained spectra containing in general both even and odd harmonics. The results applied to the hydrogen atom show that one can achieve efficient generation of moderately high harmonics using multiphoton resonant excitation by appropriate laser pulses.
We solve a model with one bound state and two continua.1 Two lasers are used to study the above threshold ionization process. The first laser couples the electron in a bound state to a continuum state. The photons are assumed to be energetic enough to promote the electron directly into the continuum far from the threshold; this avoids complications due to the Rydberg levels and the ponderomotive shifts of the threshold. The second laser mixes the electron wavefunction between two continuum states (C—C transitions); since we avoid the rotating wave approximation, multiple transitions between these continua are allowed. We find that the ionization rate is unaffected by the C—C transitions. Our model of hydrogenic atoms allows redistribution among the continua only during the ionization process and the photoelectron spectra exhibits peak suppression; i.e., the population is not the strongest at the energy of the ionization laser.
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 investigate the third order harmonic generation for model atoms interacting with intense short laser pulses. We focus on the cases where the ponderomotive energy is not big enough to generate high-order harmonics, and where resonant transitions between bound states are also as important as multi-photon ionization and low-order harmonic generation. We find that the third order harmonic intensity from a model hydrogen atom can decrease as the laser frequency draws near a resonance between bound states. We explain this result by looking into the ionization and Rabi oscillation from the numerical calculation of the time-dependent 1-D Schrödinger equation. In order to support our explanation we also investigate the populations of bound states that play an important role in this atomic process.