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Physical Review A

Resonant four-wave parametric interactions: Adiabatic formulation

Thomas A. DeTemple, M. Gurnick, François Henri Julien

Published May 1, 1988• 10 citations
Publisher

Abstract

The density-matrix approach, in the quasistatic or adiabatic limit, is applied to the case of resonant four-wave interactions in four-level systems resulting in explicit analytical expressions for the nonlinear optical parametric and nonparametric polarizations, including ac Stark effects. The results are applicable to any closed-linkage diagram of this type but are applied only to the specific case of resonant frequency tripling. The results revealed the presence of up to six Stark-shifted nonparametric resonances and the presence of saturation and interference effects on the parametric susceptibility. In particular, the third-order susceptibility is shown to saturate to a very small value under high-field conditions which restricts the range of useful pump intensities. Under near-resonant to resonant conditions, these imply that because of either nonparametric loss present at the pump or harmonic frequencies or power-dependent dispersive and saturation effects, conversion efficiencies higher than 25% may be difficult to realize in practice.

DOI
10.1103/physreva.37.3358

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The density-matrix approach, in the quasistatic or adiabatic limit, is applied to the case of resonant four-wave interactions in four-level systems resulting in explicit analytical expressions for the nonlinear optical parametric and nonparametric polarizations, including ac Stark effects. The results are applicable to any closed-linkage diagram of this type but are applied only to the specific case of resonant frequency tripling. The results revealed the presence of up to six Stark-shifted nonparametric resonances and the presence of saturation and interference effects on the parametric susceptibility. In particular, the third-order susceptibility is shown to saturate to a very small value under high-field conditions which restricts the range of useful pump intensities. Under near-resonant to resonant conditions, these imply that because of either nonparametric loss present at the pump or harmonic frequencies or power-dependent dispersive and saturation effects, conversion efficiencies higher than 25% may be difficult to realize in practice.
Motivated by recent experiments on second-harmonic generation in three-level systems [K. S. Yngvesson and E. L. Kollberg, Appl. Phys. Lett. 36, 104 (1980)], a semiclassical description of resonant three-wave parametric interactions in three nondegenerate levels is considered. Within the limitations of the rotating-wave approximation, analytical nonperturbative solutions for the nonlinear polarization are derived, which are then coupled to Maxwell's equations and integrated numerically. The results of sample calculations for second-harmonic generation show (a) a maximum harmonic conversion occurs on full resonance, (b) the conversion efficiency can be > 50% but requires a strongly saturating pump, (c) the conversion distance increases with pumping intensity, and (d) the inclusion of ac Stark splittings is crucial for the achievement of high conversion efficiency. The role of level splittings is further explored for the case of a resonant parametric oscillator.
A nonperturbative theoretical study of four-wave-mixing processes in four-level systems is presented. Using the density-matrix formalism, numerical results are obtained for two arbitrarily intense pump fields with the third field a weak probe. The fourth wave, radiated by the system, is obtained analytically for some special cases. We find, for different frequency detunings, Rabi frequencies, and relaxation rates that the line shapes exhibit: (i) dynamic Stark splitting and shifting, and merging of resonances; (ii) power broadening with two-photon antiresonance behavior resulting from a destructive interference between different contributions to the optical polarization; and (iii) extra resonances induced by field intensities and saturation processes.
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 semiclassical treatment is given for the serial, near-resonant interaction of three waves in seven equivalent four-level atomic or molecular systems. Explicit equations are presented for the complex susceptibilties for the three waves, valid for $t\ensuremath{\gg}{T}_{2}$, which contain implicit ac Stark-shift and -splitting information, and can be separated into one-, two-, and three-photon contributions. For the case of a weak third or terminal wave, equations for the degree of ac Stark shifting and splitting are found and indicate that the three-photon or triple-resonance contribution is immune to the optical Stark effect if the remaining two waves are tuned to at least two-photon resonance conditions. The conditions for Doppler-free triple resonance are outlined, and it is shown that an equivalent ac Stark broadening may exist but may also be minimized by an appropriate choice of field intensities.
In earlier chapters we have considered in detail the susceptibility formalism of nonlinear optics, which is perhaps the most familiar approach and has a wide range of application. Starting from the constitutive relations of Chapter 2, the susceptibility formalism is quite general. In many practical applications, a particular phenomenon can be described accurately by a single order of nonlinearity, and the susceptibility then provides a useful and convenient description. However, this is not always the case. Some of the most interesting phenomena in nonlinear optics involve close resonance with the transition frequencies of the medium, and perhaps also the use of very intense optical fields. As remarked in §4.5, in these circumstances the resonant susceptibilities display mathematical divergences which are clearly unphysical. Strictly, these divergences occur only because higher-order nonlinearities have been neglected. Successive orders of nonlinearity take into account such effects as saturation, power broadening and level shifts (optical Stark effect). For intense fields or very close resonance, the contributions from several orders of nonlinearity may be comparable in magnitude. Therefore, despite its generality, the susceptibility formalism does not necessarily provide the most practical approach for the description of resonant processes. This chapter is concerned with the problem of deriving alternative and more manageable descriptions of resonant nonlinear processes, illustrated by examples. For much of the chapter, the nonlinear medium is treated as a two-level system (described in §6.2).
An expression is obtained for the nonlinear susceptibility governing the addition of frequencies under resonance conditions. Allowance is made for the Stark shift, changes in populations, and transient response of the medium. Analysis is made of the stimulated emission of the third harmonic as a result of two-photon resonance in parallel and focused beams. It is shown that the saturation effect limits seriously the conversion coefficient of a nonlinear medium of fixed length. Optimal intensities of the fundamental-frequency radiation are found. It is shown that when a beam is focused sharply in a nonlinear medium, the saturation effect and Stark shift make it possible to add frequencies in media with normal dispersion. The relationship between four-photon parametric effects and coherent resonant processes such as self-induced transparency is established for ultrashort pulses.