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IEEE Journal of Quantum Electronics

Quantum theory of third-harmonic generation in a gaseous system - Part I

N. Nayak

Published Aug 1, 1980• 6 citations
Publisher

Abstract

We present a theoretical treatment for third-harmonic generation in gaseous systems by describing the atomic system as well as the radiation field quantum mechanically. The atomic system is represented by two energy levels. The effects of detuning in the two-photon interactions and the atomic motion on the third-harmonic generation are taken into account. The photon statistics of the third-harmonic field are obtained. The average photon number, second-order coherence, and output variance of the third-harmonic field are discussed. The above results show a nonlinear dependence on the detuning in the two-photon interactions and the Doppler parameter.

DOI
10.1109/jqe.1980.1070581

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We present a theoretical treatment for third-harmonic generation in gaseous systems by describing the atomic system as well as the radiation field quantum mechanically. The atomic system is represented by two energy levels. The effects of detuning in the two-photon interactions and the atomic motion on the third-harmonic generation are taken into account. The photon statistics of the third-harmonic field are obtained. The average photon number, second-order coherence, and output variance of the third-harmonic field are discussed. The above results show a nonlinear dependence on the detuning in the two-photon interactions and the Doppler parameter.
In Part I [1], we presented a quantum theory of third-harmonic generation (THG) under two-photon absorption in a two-level atomic medium. We studied the time evolution of the third-harmonic field and its coherence properties by neglecting the depletion of the pump power. In this paper we extend the theory by taking the depletion of pump power into account. The density matrix for the third-harmonic field is derived. The study of the growth rate of the density matrix in time shows the quantum mechanical effects more clearly. The effects of detuning in the two-photon interactions and atomic motion are pointed out. The results are compared with the earlier results and differences are brought out.
Second-harmonic generation in a degenerate two-level gaseous system is considered with atomic motion taken into account. The diagonal elements of the dipole matrix are assumed to be present. We find the photon statistics of the second-harmonic field to be considerably influenced by the detuning in the two-photon absorption. The threshold condition is discussed. We obtain the average second-harmonic photon number and thus calculate the dimensionless intensity parameter. The output variance of the second-harmonic field is discussed.
A fully quantum theory for the general case of any higher order optical harmonic generation associated with resonance absorption of pump photons is presented. The treatment utilizes density-matrix formalism based on the Scully-Lamb model, both atomic system and radiation fields being quantized. Detuning, homogeneous and inhomogeneous broadening and relaxation parameters are included. Coherence characteristics and build up of the harmonic fields are discussed for different initial distributions. The Doppler limit restriction has been lifted and the possibility of a relatively efficient conversion of harmonic photons of higher orders is discussed.
A quantum mechanical treatment of stimulated Raman effect in a gaseous system is presented. The atomic system is described by a two-level system. The radiation fields also obey the laws of quantum mechanics. The effects of atomic motion and the detuning in one-photon absorption and two-photon emission are incorporated in the analysis. The equation of motion describing the photon-statistics of all the fields are obtained. As a special case, the time evolution of the average photon number, second order coherence and output variance of the Stokes field is studied. The results show a non-linear dependence on the parameters representing atomic motion, detuning and atomic relaxation constants.
Several extensions are made in theoretical studies of multiphoton ionization associated with third-harmonic fields that are generated near three-photon resonances in gaseous media. The treatment includes the strong coherent cancellation effect which occurs between three-photon pumping of a resonant level and pumping by the third-harmonic field that is generated in the same medium. The incomplete spoiling of the cancellation effect with counterpropagating beams is quantitatively described. Also, treatment of the multiphoton-ionization problem involving phase-matched, third-harmonic fields is generalized to include quantitative effects of index changes due to a buffer gas, absorption due to dimers, and collisional dephasing of coherent excitations. Finally, we show how measurements of linewidths, phase-matching positions, and pressure dependences of ionization yields associated with third-harmonic generation and subsequent absorption can be used to determine oscillator strengths of a resonant level, absorption coefficients for a resonant species and a buffer-gas species, and the refractive index of the buffer gas at the third-harmonic frequency.
The interaction of electrons with strong laser fields is usually treated with semiclassical theory, where the laser is represented by an external field. There are analytic solutions for the free electron wave functions, which incorporate the interaction with the laser field exactly, but the joint effect of the atomic binding potential presents an obstacle for the analysis. Moreover, the radiation is a dynamical system, the number of photons changes during the interactions. Thus, it is legitimate to ask how can one treat the high order processes nonperturbatively, in such a way that the electron-atom interaction and the quantized nature of radiation be simultaneously taken into account? An analytic method is proposed to answer this question in the framework of nonrelativistic quantum electrodynamics. As an application, a quantum optical generalization of the strong-field Kramers-Heisenberg formula is derived for describing high-harmonic generation. Our formalism is suitable to analyse, among various quantal effects, the possible role of arbitrary photon statistics of the incoming field. The present paper is dedicated to the memory of Prof. Dr. Fritz Ehlotzky, who had significantly contributed to the theory of strong-field phenomena over many decades.