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Soviet Journal of Quantum Electronics

Generation of optical harmonics in atoms

Nikolai L. Manakov, V. D. Ovsyannikov, Л. П. Рапопорт

Published Jan 31, 1975• 4 citations
Publisher

Abstract

A method is suggested for the calculation of the nonlinear susceptibility corresponding to the generation of second and third optical harmonics in atomic gases. The method is based on the use of the Green's function of an atomic optical electron in the computation of the composite matrix elements. The results are given of a numerical calculation of χ(–3ω;ω,ω,ω) and χ(–2ω;0,ω,ω) for hydrogen, alkali atoms, and inert gases at the frequencies of ruby and neodymium laser radiations and their second harmonics. A two-photon resonance in the generation of the third harmonics is considered.

DOI
10.1070/qe1975v005n01abeh010667

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A method is suggested for the calculation of the nonlinear susceptibility corresponding to the generation of second and third optical harmonics in atomic gases. The method is based on the use of the Green's function of an atomic optical electron in the computation of the composite matrix elements. The results are given of a numerical calculation of χ(–3ω;ω,ω,ω) and χ(–2ω;0,ω,ω) for hydrogen, alkali atoms, and inert gases at the frequencies of ruby and neodymium laser radiations and their second harmonics. A two-photon resonance in the generation of the third harmonics is considered.
This paper presents calculations of the third-order susceptibility in the alkali metal vapors involving spin-orbit coupling in the energy spectrum of the alkali atoms and some power transfer between the atoms in the ensemble and their surroundings, including the radiation field. The results are plotted as a function of wavelength in the vicinity of the neodymium and iodine laser lines, and their second, third, and fourth harmonics for each of the alkalies; i.e., 10 520 Å\lambda, 2\lambda, 3\lambda, 4\lambda)\lambda, 2\lambda, 3\lambda, 4\lambda) < 13 230 Å, respectively.
In the previous chapter we examined nonlinear propagation in a two-level medium, which leads to amplification or absorption of the radiation field. As we will see further on in this volume, when the medium is placed in an optical cavity one realizes a laser in the case of amplification or an optically bistable system in the absorptive configuration. On the other hand, the nonlinear response of the atomic polarization to the electric field leads to further fascinating scenarios [29, 51–60]. For example, a field with central ω 0 can generate harmonic components with central frequencies 2 ω 0 , 3 ω 0 etc., processes that are called second-harmonic generation, third-harmonic generation etc. Furthermore, the interaction of two fields with central frequencies ω 1 and ω 2 , with ω 1 > ω 2 , can generate components with central frequency ω 1 + ω 2 (sum-frequency generation) and ω 1 − ω 2 (difference-frequency generation). Such phenomena have not yet appeared in this volume because we have assumed that the electric field corresponds to a narrow frequency band around a central frequency ω 0 only, whereas to describe them it is necessary to assume that the electric field is the superposition of a number of frequency bands. In this chapter we will provide a description of the basic elements which are necessary to describe these phenomena and we will discuss a small number of them for the case of materials with a quadratic nonlinearity, referring to textbooks in nonlinear optics for the description of other phenomena. The following chapter is devoted, instead, to the case of materials with cubic nonlinearities. The starting point of nonlinear optics is the expansion of the atomic polarization in powers of the electric field, when the electric field is not too intense. In our treatment in Chapter 5 we saw that the expansion of the nonlinear susceptibility contains only even powers of the electric field (see Eq. (5.27)), and therefore the expansion of the polarization includes only odd powers.
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.
A semi parabolic quantum well model to study the third order optical nonlinearities was proposed. A concise formulation of the third harmonic generation was presented. Using a compact density matrix approach, the extremely large third order nonlinearities were obtained. The reason is the possibility of tuning independently the frequency of the semi parabolic potential and the triply resonance effect. The value of | χ (3) max (3 ω )| is as high as 2.7×10 -12 (m/V) 2 at 10.7 μm.
Optical second harmonic generation has been observed on or near the 4s 2 S 1/2 -6s 2 S 1/2 , 4s 2 S 1/2 -np 2 P 1/2, 3/2 (n=5, 6) and 4s 2 S 1/2 -nd 2 D 5/2, 3/2 (n=3,4) two-photon resonances in potassium vapour, as laser scanning in the 660-940 nm wavelength range. The second harmonic intensities and ionisation currents have been measured as a function of pumping laser detuning from the two-photon resonances. On or near the two-photon resonances, the intensity and vapour-density dependence of second harmonic generation have also been investigated. The saturation of second harmonic generation has been observed. Some calculations and analyses have been made with the chi (3) model. The results have been compared with the measurements. Based on these analyses, the mechanisms for second harmonic generation in isotropic vapours are discussed.
We present a theoretical study of the influence of laser-induced continuum structure on the third-harmonic generation in atomic gases. We employ the density-matrix formalism including the spatiotemporal structure of laser pulses. The theoretical model is then applied to third-harmonic generation in sodium experiments. Extensive numerical calculations using realistic atomic parameters in combination with the parameters in actual experimental situations demonstrate the subtle interplay of various effects and pinpoint causes of possible complications. We obtain good agreement with the few existing experimental data.