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Journal of the Optical Society of America B

Population pulsations and the dynamic Stark effect

Robert W. Boyd, Murray Sargent

Published Jan 1, 1988• 53 citations
Publisher

Abstract

We present a theoretical description of the interaction of optical waves due to the resonant nonlinear response of an atomic system. We emphasize how the resonant nature of the nonlinear coupling is modified by the shifting of the atomic energy levels as a consequence of the dynamic Stark effect and show the equivalent role played by population pulsations in determining the nature of the nonlinear coupling. A general formalism is developed to treat these effects and is explicitly applied to several examples of current interest, including single-beam saturation spectroscopy, pump–probe saturation spectroscopy, modulation spectroscopy, degenerate four-wave mixing for phase conjugation, and instabilities in the beam propagation through resonant media.

DOI
10.1364/josab.5.000099

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Journal of the Optical Society of America B•Sep 1, 1985•22 citations
We present a theoretical description of the interaction of optical waves due to the resonant nonlinear response of an atomic system. We emphasize how the resonant nature of the nonlinear coupling is modified by the shifting of the atomic energy levels as a consequence of the dynamic Stark effect and show the equivalent role played by population pulsations in determining the nature of the nonlinear coupling. A general formalism is developed to treat these effects and is explicitly applied to several examples of current interest, including single-beam saturation spectroscopy, pump–probe saturation spectroscopy, modulation spectroscopy, degenerate four-wave mixing for phase conjugation, and instabilities in the beam propagation through resonant media.
Steady-state interactions of one, two, and three waves with two-photon two-level media are treated semiclassically. Dynamic Stark shifts as well as dynamic Stark splittings are included in the theory. The intermediate levels are assumed to be nonresonant, and one wave is allowed to have arbitrarily large intensity. The theory parallels the one-photon theory of multiwave mixing couched in terms of population pulsations. We find the one-, two-, and three-wave two-photon absorption coefficients and the three-wave coupling coefficients. We find similarities and significant differences between the one- and two-photon polarizations of the medium. The work treats two-photon saturation spectroscopy, laser- and optical-bistability instabilities, and phase conjugation with a unified formalism.
We discuss the effects of population pulsations on the spectral response of backward nearly degenerate four-wave mixing (NDFWM). We include the effects of state-specific decay and spontaneous emission on both two-level and inverted-v-type three-level systems. We compare the results with experiments in both the gas phase and the condensed phase that show ultranarrow resonances. In addition, we examine the response in the presence of strong pump fields. We show that in a simple two-level system, the NDFWM response results in an ac-Stark splitting of the spectral response, whereas in a simple three-level system with decay to a metastable state, a single narrow resonance is observed, which broadens in the presence of strong pumps.
Two-photon transitions have provided the basis for numerous useful nonlinear interactions with radiation. In this paper we calculate the absorption and coupling coefficients for one-, two-, and three-wave mixing in two-photon homogeneously broadened media. The resulting field equations form the basis for understanding saturation spectroscopy, phase conjugation,1 and single-mode instabilities2 in lasers and optical bistability in two-photon media. We assume that the wave frequencies are not resonant with any intermediate state j and that the probe and conjugate waves cannot saturate the two-photon transition. This allows us to use the two-photon two-level model and to transfer much of our knowledge of the one-photon two-level systems to the two-photon case. Two major differences occur between the two models. First, dynamic Stark shifts of the active levels play an important role in generating asymmetric spectra and nonzero dispersion (≃ω s T1) at line center. Second, high pump intensities do not bleach the absorption coefficient at the Rabi sidebands.
The theory of nondegenerate four-wave mixing (NDFWM) in semiconductor lasers and amplifiers is presented with particular emphasis on the physical processes that lead to population pulsations. In the case of nearly degenerate four-wave mixing, modulation of the carrier density at the beat frequency Ω of the pump and probe waves creates a dynamic population grating whose effectiveness is governed by the spontaneous carrier lifetime τs. Such a grating affects both the gain and the refractive index of the probe wave. In particular, the probe gain exhibits features analogous to those observed in a detuned atomic system arising from the optical Stark effect. Both the gain grating and the index grating contribute to NDFWM, with the dominant contribution coming from the index grating. For detunings such that Ωτs ≫ 1, population pulsations correspond to modulation of the intraband population arising from spectral hole burning. Our results show that NDFWM is then limited by the phase-mismatch effects governed by the transit time τ rather than by the intraband population-relaxation time T1. Significant NDFWM is expected to occur for detunings up to about 300 GHz for typical transit-time values of 3 psec in semiconductor lasers.
Population pulsations are fluctuations in the population difference (of a two level system) due to the presence of two or more coherent waves interfering in the medium. In this work, we show that population pulsations generated by three waves, a central wave and two mode-locked sidebands, are responsible for both the multiwavelength and the single-wavelength instabilities of single-mode lasers containing homogeneously-broadened media. The role of the population pulsations in establishing these instabilities, however, diminish as the central mode is detuned away from the atomic resonance frequency. For homogeneously-broadened lasers, we find two regions of single-wavelength instability. The first is at line center, for which population pulsations are solely responsible, and the second is off line center where the unsaturated medium provides the required gain and anomalous dispersion. For the case of inhomogeneously-broadened lasers, we show that population pulsations significantly increase the instability range over that predicted by Casperson for single-mode bad-cavity lasers. Both the unidirectional ring and the standing-wave cavities are treated. The Fourier expansion technique, used in this work, for treating three-frequency operation in saturation spectroscopy is shown to be equivalent (in appropriate limits) to the linear stability analysis in laser theory and optical bistability. We also show, in single-sideband saturation spectroscopy, that for long interaction lengths propagation effects can significantly influence the absorption and dispersion coefficients of the medium. Finally, we show that under certain conditions the pronounced splittind effects of the population pulsations develop into regions of intense absorption.
The propagation of a weakly modulated light beam through a nonlinear material is treated. The optical field representing such a beam consists of a strong carrier-frequency component and two weak, symmetrically displaced sidebands that combine to form a field, which may be purely amplitude modulated (AM), frequency modulated (FM), or some combination of these two modulation forms. It is shown that, for any optical nonlinearity, two modulation forms exist that have the property that the form of modulation is invariant under propagation of the beam. If a modulation form other than one of these natural modes is injected into the nonlinear medium, the modulation form will change as the beam propagates, asymptotically approaching the natural mode that experiences the lower attenuation. Explicit expressions for these natural modes are presented for the case in which the nonlinear medium can be modeled as a collection of two-level atoms. For the special case of an on-resonance pump beam, the natural modes correspond to pure amplitude modulation and pure frequency modulation. This formalism provides a general description of saturation spectroscopy for both AM and FM fields. Formulas are derived for the rate at which a pure FM beam is converted to an AM beam owing to its interaction with a two-level atomic medium. We also consider the variation that is due to propagation of the depth of modulation of an AM wave interacting resonantly with two-level atoms. The formalism predicts the existence of spectral features whose shape depends sensitively on the internal relaxation processes of the material. Experimental spectra are presented for ruby, alexandrite, and fluorescein in glass and are interpreted.