The effects of Gaussian-beam averaging with various apertures are derived for three-wave mixing in nonlinear two-level media. The model studies the phase transition from plane-wave to pure Gaussian saturation. Applications to phase conjugation, beat-frequency spectroscopy, and laser–optical-bistability instability phenomena are considered. The Gaussian beams are seen to reduce or wash out dynamic Stark splittings observed for plane-wave saturation. For strong saturation, the side-mode gain/absorption coefficient versus detuning yields slightly larger gain for the homogeneously broadened laser instability but substantially curtailed gain for the corresponding absorptive optical-bistability instability.
The effects of Gaussian-beam averaging with various apertures are derived for three-wave mixing in nonlinear two-level media. The model studies the phase transition from plane-wave to pure Gaussian saturation. Applications to phase conjugation, beat-frequency spectroscopy, and laser–optical-bistability instability phenomena are considered. The Gaussian beams are seen to reduce or wash out dynamic Stark splittings observed for plane-wave saturation. For strong saturation, the side-mode gain/absorption coefficient versus detuning yields slightly larger gain for the homogeneously broadened laser instability but substantially curtailed gain for the corresponding absorptive optical-bistability instability.
The effects of inhomogeneous broadening, spatial hole burning, and Gaussian beams are calculated for the four coefficients derived in previous papers in this series [Phys. Rev. A 31, 3112 (1985); 31, 3124 (1985)] on the quantum theory of multiwave mixing. These coefficients uniformly describe how one or two strong classical-field modes and one or two weak quantum-field modes interact in a two-level medium. The phenomena for which this theory is applicable includes resonance fluorescence, Rayleigh scattering, laser and/or optical bistability instabilities, and phase conjugation. The averaged formulas are illustrated by resonance-fluorescence spectra. We find the sidepeaks can be ``washed out'' and that asymmetries can occur.
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.
We use the semiclassical strong-signal theory of the laser to predict and explain the onset of side-mode buildup in lasers with one oscillating mode. Two general categories are considered: one for which the side modes and the oscillating mode all have the same wavelength and the other for which they have different wavelengths. The treatments include an arbitrary amount of inhomogeneous broadening. Our approach unifies the treatments of the side-mode instabilities presented earlier and extends them to handle standing waves in addition to the previously treated running waves. We write the field and the population matrix elements as Fourier series in the adjacent-mode beat frequency. This approach has been used extensively in both multimode laser theory and saturation spectroscopy. This technique coincides with linear stability analyses used by others, provided that our beat frequency includes a contribution that is proportional to the complex side-mode gain. We give a solution that allows for detuned operation along with its simpler, centrally tuned special case. The connection with saturation spectroscopy clearly reveals that the side-mode instabilities require side-mode gain. For the single-wavelength case, nonlinear anomalous dispersion is also required. The side-mode gain and dispersion result from both inhomogeneous broadening and population pulsations. The lowest instability thresholds occur when both of these mechanisms play a role. The approach can also be used to treat instabilities in optical bistability by substituting the appropriate equation of state for the strong-mode intensity and by changing the sign of the absorption coefficient. In homogeneously broadened, standing-wave lasers, we show that multiwavelength instabilities depend strongly on the position of the medium in the cavity. We illustrate the theory by giving numerical results for the output pulsation frequency and for the instability threshold by using parameters that are appropriate for the He–Xe laser. These results correlate well with experimental observations.
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.
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.
Nonlinear optics normally involves the change in the susceptibility of a material due to the presence of optical radiation. In many cases, the light-matter interaction produces a change in the refractive index of the medium, as occurs in photorefractive media, as well as in Kerr-like media, Brillouin-active media and absorbing media with thermallyinduced refractive index changes. Such nonlinear refractive index variation corresponds to a change in the real part of the optical susceptibility %. In resonant materials, the optical radiation induces stimulated emission (or absorption) causing transfer of population between different energy states of the medium. This can lead to saturation of the amplification (or absorption) corresponding to changes in the imaginary part of the susceptibility x= i2odk where the amplitude gain coefficient a(I) is a saturable function of the intensity /, and k is the magnitude of the wavevector of the optical radiation. The interference of two (or more) coherent beams I(x) in the laser amplifier can cause a modulation of the population inversion N(x). This modulation of the inversion is commonly known as spatial hole burning in lasers and leads to formation of a gain-grating a(x) = 1/2GN(X) where G is the stimulated emission cross-section. The gain grating can act as a very efficient diffractive optical element exhibiting high diffraction efficiency that may be even greater than unity [16] and also very high phase conjugate reflectivity using four-wave mixing (FWM) geometries [5-7].