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Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE

Quantum theory of higher-order optical harmonic generation

Partha Sarathi Gupta

Published Apr 1, 1992• 0 citations
Publisher

Abstract

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.

DOI
10.1117/12.58088

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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.
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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.
Optical harmonic generation and mixing in the gas phase has been proposed as a technique for the generation of coherent radiation in the vacuum ultraviolet and soft x-ray spectral region. At the high field strengths required by these processes the interaction between atoms and the electromagnetic field shows intensity-dependent resonances. In this paper we modify harmonic generation theory to include the effect of these frequency shifts. Closed-form expressions for generated dipole moment, absorption probability, and coherence length are presented. The most important consequences of frequency shifts on resonantly enhanced processes are that the pump laser must be tuned away from the small-field resonance frequency, that the conversion efficiency may saturate, and that the dispersion of the medium may change sign. As an example, the generation of 198-\AA{} radiation by a five-photon mixing process in Li+{\mathrm{Li}}^{+}Li+ is considered.
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 general quantum theory of nonlinear mixing in multimode fields is formulated. The theory is valid for arbitrary media and thus includes various diverse cases such as those corresponding to nonlinear mixing in multilevel systems, optical fibers, etc. Thus Zeeman coherence effects are automatically included. It is applicable to both degenerate and nondegenerate mixing experiments. The theory is in terms of two distinct sets of correlation functions of the polarization operators. One set of correlations is related to the nonlinear susceptibilities. The other set has no counterpart in semiclassical theory and is related to the quantum mechanical flunctuations of the polarization operator. The general structure of the evolution of the density matrix of the generated fields is discussed. The quantum statistics of the generated fields can be studied in terms of the Wigner distribution function which is explicitly evaluated. Higher-order squeezing characteristics of the field are discussed. Conditions in which the quantization of the semiclassical equations is adequate are examined. The application of the quantum theory to nonlinear mixing in two-photon media is also presented. The theory is also capable of accounting for the interparticle correlations.