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.
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.
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.
In this article, we analyse a new sensitive spectroscopic technique in which a sample cell is irradiated by a two-frequency pump beam (ω + Δ ± δ/2) and a counter-propagating probe beam (ω). The phase-conjugate fields re-emitted at frequencies ω ± δ, via four-wave mixing processes, are detected through their heterodyne beating with the probe field. The properties of the heterodyne beat signal are studied by means of a perturbation calculation of the nonlinear response of the medium, carried out to the third order in the incident fields. In resonant gas media, the emission lineshape exhibits a Doppler-free doublet structure [frequency splitting 3 δ/2 for single-photon resonances, and δ/2 for two-photon transitions]. By using a phase-sensitive detection of the beat signal, one can separately analyse real (dispersion) or imaginary parts (absorption) of the nonlinear susceptibility. For single-photon transitions, the signal phase shift is related to the atomic levels lifetime and allows one to study the relaxation dynamics in gas phase (like quenching collisions, velocity diffusion, etc...). This is particularly interesting for cross-over resonances, in which only the common level contributes. In three-level systems, one also predicts a selective resonant enhancement of the phase-conjugate emission through Raman-type two-photon processes. Most of these predictions have been experimentally verified.
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.
We investigate the properties of a homogeneously broadened three-level system interacting with two optical fields, Eω1(x,t) and Eω2(x,t), with central frequencies ω1 and ω2 whose amplitudes are assumed to be slowly varying in time compared with dephasing times of the optical transitions. We calculate Raman susceptibilities allowing each transition to be induced by both fields and without employing the rotating wave approximation (RWA); this enables correct calculation of the optical response of the medium when the fields are on-resonance or arbitrarily far off-resonance with the Raman transition. In some frequency regions the RWA causes substantial errors. A distinction is drawn between two different types of RWA: (a) a neglect of off-resonant components of the polarization for direct optical transitions, and (b) neglect of off-resonant components of off-diagonal density matrix elements of order Eω1(x,t)Eω2(x,t)* in the field amplitudes. The first RWA is known to become inaccurate when the pump frequency is significantly off-resonance with the pump transition. We show that the second RWA becomes inaccurate when ‖ω1−ω2‖ is small or comparable to the active Raman mode frequency (even when the field are close to resonance). Calculations based upon our expressions show that when the pump frequency is close to resonance with the optical transition from the ground state, a nonlinear absorption at the Stokes shifted frequency occurs. This absorption can significantly decrease the Raman gain. Also, we find that Rabi splittings in a three-level system can occur even when the pulse durations of the incident fields are short compared with the T1 times of the system. We solve analytically for the optical response of the system to two fields of arbitrary strength and frequency. We analyze this solution and its perturbative expansion in the field amplitudes. We use the Maxwell–Bloch equations developed here to propagate time-dependent pulses.
The analytic solutions of equations, which describe the time behavior of density-matrix elements of a two-level medium driven by weak-probe tunable and strong fields, namely, scrE(t)=E(t)cos[\ensuremath{\omega}t+\ensuremath{\varphi}(t)]+\ensuremath{\varepsilon} cos[(\ensuremath{\omega}+\ensuremath{\delta})t], the amplitude E(t) and phase \ensuremath{\varphi}(t) of which are any arbitrary functions of time, are obtained. We obtain explicit expressions of the medium polarizability in two cases of the strong-field parameters modulation: first, E(t)=E0cos(\ensuremath{\xi} cos\ensuremath{\Omega}t), where E0, \ensuremath{\Omega}, and \ensuremath{\xi} are constants, and \ensuremath{\varphi}=0; and second, both the amplitude E(t) and the phase \ensuremath{\varphi}(t) are periodic functions of time, and E(t)=C\ensuremath{\varphi}\ifmmode \dot{}\else \.{}\fi{}, where C is an arbitrary constant. These cases of modulation are generalizations of cases that were considered earlier. The first case of modulation causes a shift of the well-known Rabi resonances in the weak-probe absorption and dispersion spectra by the factor J0(\ensuremath{\xi}) (J0 is the zero-order Bessel function of the first kind) in comparison with a nonmodulated exciting field. In the second case of the modulation, the Rabi resonances are present but the effect of such a modulation results in an attenuation of the amplitude-modulation effect. The values of the parameters E0, \ensuremath{\Omega}, C, and \ensuremath{\xi} and the frequency ranges in which the weak-probe gain, caused by resonant Rayleigh scattering and by three-photon scattering, takes place are determined for these two cases of the modulation.
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.