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Journal of the Optical Society of America B

Coherent dips revisited

Murray Sargent

Published May 1, 1988• 2 citations
Publisher

Abstract

The semiclassical two-level multiwave coefficients are presented in the limit that the population-difference lifetime T1, is much greater than the dipole lifetime T2. These coefficients are easy to understand analytically and show how pump-probe detuning shifts the phase and the amplitude of pump scattering off induced population pulsations. In particular, a simple formula is found that shows how energy can be exchanged by two waves in an apparently pure-index regime. An FM wave is shown to propagate unchanged through such a medium, and the beat-note component of an AM wave is shown to be unmodified by a thin medium. Simple formulas for T1 of the order of T2 are given that reveal intense pump-dispersive and Lorentzian features at a glance. The results are of use in saturation spectroscopy, modulation spectroscopy, optical instabilities, and phase conjugation.

DOI
10.1364/josab.5.000987

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The semiclassical two-level multiwave coefficients are presented in the limit that the population-difference lifetime T1, is much greater than the dipole lifetime T2. These coefficients are easy to understand analytically and show how pump-probe detuning shifts the phase and the amplitude of pump scattering off induced population pulsations. In particular, a simple formula is found that shows how energy can be exchanged by two waves in an apparently pure-index regime. An FM wave is shown to propagate unchanged through such a medium, and the beat-note component of an AM wave is shown to be unmodified by a thin medium. Simple formulas for T1 of the order of T2 are given that reveal intense pump-dispersive and Lorentzian features at a glance. The results are of use in saturation spectroscopy, modulation spectroscopy, optical instabilities, and phase conjugation.
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 consider an intrinsic semiconductor medium subject to an injection current, an arbitrarily intense electromagnetic wave, and one or two non-saturating waves. We assume that the saturating wave intensity is constant throughout the interaction region and ignore transverse variations. We further assume that the carrier–carrier scattering is sufficiently fast that the carriers are described approximately by Fermi-Dirac distributions. We find that the probe-wave propagation is described by Beer’s law or coupled-mode equations. These equations are determined by a Fourier expansion technique highly developed for two-level atoms. The absorption and coupling coefficients experience asymmetric dips generated by pump scattering off carrier-density pulsations induced by the pump–probe interference. These effects are reminiscent of coherent-dip phenomena well-known in two-level systems but differ significantly in form.
We study unidirectional three-wave spectroscopy for which one pump and two probe waves beat in a nonlinear medium at the frequencies ±Δ with respect to the pump. Field transmission spectra change as the fields propagate through the medium due to probe variations in both amplitude and phase. In a homogeneously broadened two-level system, the rates of change in sidemode amplitudes and phases are determined by the effectiveness of the population pulsations in coupling the fields. For cases in which the population difference lifetime T1 is comparable with the dipole lifetime T2, the strong coupling between the sidemodes and the pump leads to fast amplitude change, while strong coupling between the two sidemodes leads to fast phase change. When the pump Rabi flopping frequency Ω ≅ Δ and Δ ≅ 1/T1, the population pulsations couple the two sidemodes most efficiently, causing the sidemode phases relative to the pump phase to vary rapidly. When Ω ≅ 2Δ, the pump field scatters strongly off the population pulsations, thereby amplifying the sidemode fields. When Ω ≫ Δ, the population pulsations become too weak to couple the fields significantly, and phase, amplitude, and spectral features in general change very slowly.
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.
This dissertation presents theories of multiwave mixing in two- and three-level media. The first part of the dissertation treats the semiclassical theories in two-level media. Chapter 2 gives the simple semiclassical theory of four-wave mixing when the two pump frequences differ by more than the reciprocal of the population-difference lifetime. This difference washes out the pump spatial holes as well as one of the two reflection gratings. We compare the results to the degenerate treatment of Abrams and Lind and find significant differences in the reflection coefficient spectra. Chapter 3 presents the semiclassical theory of multiwave in a squeezed vacuum characterized by unequal in-phase and in-quadrature dipole decay times. For a highly squeezed vacuum, we find sharp resonances in both probe absorption and reflection coefficients, which provide sensitive ways to measure the amount of squeezing in the vacuum. The second part of the dissertation treats the quantum theories in two- and three-level media. Chapter 4 develops the fourth-order quantum theory of multiwave mixing to describe the effects of sidemode saturation in two-level media. We derive explicit formulas for the fourth-order quantum coefficients and show that the fourth-order quantum theory reproduces the third-order semiclassical coefficient obtained by truncating a continued fraction. We apply the results to cavity problems and find significant differences in the sideband spectra given by the second- and fourth-order treatments, particularly as the sidemode approaches the laser threshold. The final chapter presents a quantum theory of multiwave mixing in three-level cascades with a two-photon pump. The explicit formulas for the resonance fluorescence spectrum and the quantum combination-tone source term are derived. The theory is applied to the generation of squeezed states of light. We find almost perfect squeezing for some strong pump intensities and good broad-band squeezing for low pump intensities.
This dissertation formulates and applies a theory describing how one or two strong classical waves and one or two weak quantum mechanical waves interact in a two-level medium. The theory unifies many topics in quantum optics, such as resonance fluorescence, saturation spectroscopy, modulation spectroscopy, the build up of laser and optical bistability instabilities, and phase conjugation. The theory is based on a quantum population pulsation approach that resembles the semiclassical theories, but is substantially more detailed. Calculations are performed to include the effects of inhomogeneous broadening, spatial hole burning, and Gaussian transverse variations. The resonance fluorescence spectrum in a high finesse optical cavity is analyzed in detail, demonstrating how stimulated emission and multiwave processes alter the spectrum from the usual three peaks. The effects of quantum noise during the propagation of weak signal and conjugate fields in phase conjugation and modulation spectroscopy are studied. Our analysis demonstrates that quantum noise affects not only the intensities of the signal and conjugate, but also their relative phase, and in particular we determine a quantum limit to the semiclassical theory of FM modulation spectroscopy. Finally, we derive the corresponding theory for the two-photon, two-level medium. This yields the first calculation of the two-photon resonance fluorescence spectrum. Because of the greater number of possible interactions in the two-photon two-level model, the theoretical formalism is considerably more complex, and many effects arise that are absent in the one-photon problem. We discuss the role of the Stark shifts on the emission spectrum and show how the Rayleigh scattering is markedly different.