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IEEE Journal of Quantum Electronics

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

N. Nayak

Published Feb 1, 1982• 2 citations
Publisher

Abstract

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.

DOI
10.1109/jqe.1982.1071500

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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.
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.
A numerical investigation is made of third harmonic generation (THG) under conditions of two-photon resonance between the fundamental frequency and a selected transition in the atoms of a medium. It is shown that under conditions of coherent interaction between the pump pulse and the medium, a THG efficiency of ∼20% or higher may be achieved. In this case, THG in the form of high-power ultrashort pulses may be obtained. It is shown that experimental observation of similar effects in specific media should not present particular difficulties.
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.
Ttie build - up of the third harmonic and the depletion of the pump mode are investigated d escribing both the fields quantum mechani- cally. The Heisenberg equationsof motion describing the dynamics of the pump and oF the third harmonic mode are solved in the short-time appro- ximation. It is found that the pump mode shows antibunching effect due to the nonlinear interaction between the modes. l'hird harmonic generation has been studied both theoretically as well as experimental 1 y by severa1 workersl-l1 in the f ield of nonl i- near optics. The third harmonic is being used to produce ultraviolet ra - diation not available from existing lasers by nonlinear optical mixing . The initial build-up of the third harmonic has been studied theoretically, in general, using the parametric approximation i.e. considering the pump depletion to be negligible. It is a reasonable approximation so long as the conversion efficiency is quite low, but recent experiments have shom increased irnprovement in the conversion ef f iciency l l. Under these c i r - cumstances, it is necessary to include pump depletion while studying the harmonic generation. The present communication aims at investigating the dynamics of the third harmonic generation including pump depletion and treating both the pump and the harmonic fields quantum mechanically. The model Hamiltonian of the system is written along the line of shen12 and ~alls'~. The Heisenberg equations of motion describing the time evolution of the pump and the harmonic mode are solved in the short - time approxima- tion14. Our analysis shows thatasaresultof nonlinear interaction, the pump photons manifest antibunching, if the pump mode is in the coherent state and the harmonic mode in the vacum state initially. The photons generated in the harmonic mode are uncorrelated for the quoted i n i tia l conditions.
Density-matrix formalism is applied to construct a model for third- and fifth-harmonic generation in a four-photon resonant medium. The transition rates and coherent superposition of states associated with the resonant levels are treated rigorously, while the adiabatic following approximation is used to calculate the contribution of all nonresonant levels to the interaction. Expressions for the quadratic and quartic Stark shift and multiphoton ionization emerge naturally from the theory. At the power level required for appreciable four-photon absorption, the intensity-dependent Stark shift and ionization become overwhelming and limit the efficiency of harmonic generation. Because of the Stark shift, an intense pulse suffers a ``self-detuning'' effect and propagates through the resonant medium with minimal absorption. The interference of the phase-dependent Raman scattering with the multiphoton transition is found to be responsible for the vanishing of ionization in the presence of the generated third-harmonic field, an effect which was observed in earlier experiments.
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.