We present a theoretical analysis of the spatial shape of two symmetric signals of degenerate four-wave mixing induced by Gaussian beams in a thin sample of two-level atoms. Our calculations take into account the full spatial and spectral dependences of the relevant nonlinear susceptibilities that govern the two processes. This reveals two interesting effects. The first one is that the total power of incident beams affects the transverse profile of the four-wave-mixing signals at the medium exit and their free propagation. The second one is the influence of the spectral characteristics of the medium on the longitudinal profile of both generated signals upon free propagation. We argue that the first effect can be seen as the saturation of the medium in regions of higher intensity, while the second can be understood as the result of a nonlinear contribution to the refractive index inside the atomic medium. These effects can be symmetric between the two signals, with asymmetries induced by different detunings from the resonance of the incident fields.
We present a theoretical analysis of the spatial shape of two symmetric signals of degenerate four-wave mixing induced by Gaussian beams in a thin sample of two-level atoms. Our calculations take into account the full spatial and spectral dependences of the relevant nonlinear susceptibilities that govern the two processes. This reveals two interesting effects. The first one is that the total power of incident beams affects the transverse profile of the four-wave-mixing signals at the medium exit and their free propagation. The second one is the influence of the spectral characteristics of the medium on the longitudinal profile of both generated signals upon free propagation. We argue that the first effect can be seen as the saturation of the medium in regions of higher intensity, while the second can be understood as the result of a nonlinear contribution to the refractive index inside the atomic medium. These effects can be symmetric between the two signals, with asymmetries induced by different detunings from the resonance of the incident fields.
We present a theoretical analysis of the spatial shape (transverse and longitudinal) of two degenerate four-wave mixing (FWM) signals induced by fields Ea and Eb, with nonuniform transverse profiles, and wave-vectors ka and kb. The two signals are generated in a cold sample of two-level atoms, in the forward directions 2ka − kb and 2kb − ka. In all cases, the transverse shapes of the generated signals (both near- and far-field) are dictated by the projection of the nonlinear coherence onto the Laguerre-Gaussian (LG) function space. Our calculations show that the longitudinal profile of each generated beam is affected by the spectral characteristics of the medium, described by the third order nonlinear susceptibility. For Gaussian incident fields with equal detunings from resonance, the FWM beam shapes are completely symmetric, with an asymmetry induced by different detunings of incident fields. By considering incident beams with LG distributions, the topological charge conservation determines the orbital angular momentum of each signal and the longitudinal profiles show similar behaviors to the case with Gaussian beams. The phase discontinuities of LG modes reveal a rotation and distortion of the phase distribution of the FWM beams caused by the detunings. We discuss for selected configurations, the relations between both generated beams and their propagation properties. We also present preliminary experimental results of the power spectra of the two FWM signals generated in our magneto-optical trap of rubidium atoms.
In this work, we present an experimental study on the spatial profile of coherent light generated in a forward four-wave mixing (FWM) process. We are interested in investigating how the transverse structure of the incident beams, more specifically of the Hermite-Gauss (HG) modes, is transferred to the generated signal in this nonlinear process. For the experiment, we used a continuous wave diode laser to excite a sample of Rubidium atoms, on the ⁸⁵Rb D2 line, tuned to the frequency of the 5²S_(1/2) ( = 3) → 5²P_(3/2) transition. With a Spatial Light Modulator, we were able to split the laser beam in two and structure them in the desired modes. These two beams, A and B, with wave vectors kA and kB, and linear orthogonal polarizations, with quasi-copropagated configuration, were used to induce two FWM signals in the directions 2kA-kB and 2kB-kA. The spatial profile of incident beams and the two nonlinear signals were captured by a CMOS camera, and the regimes of a thin or extensive medium were investigated. We were able to obtain results for a combination of gaussian-HG beams, as well as HG-HG beams. The profiles obtained by computer simulation were compared with the experimental measurements. Our theoretical model relies on solving the paraxial equation of the generated field. It leads to the overlap integrals of the incident beams, which unfold into some interesting selection rules in the transversal plane, for the thin and extensive medium regimes, and in the longitudinal axis, only for the long medium regime. Although the extensive medium regime presents some additional selection rules, the experimental requirements are difficult to achieve in our system, among them, we highlight the Gouy phase matching. On the other hand, a good agreement between theory and experiment was observed in the thin medium regime.
Abstract We report experimental and theoretical results of two symmetrical signals of degenerate four-wave mixing generated in rubidium vapor. Both nonlinear signals are induced by two almost copropagating laser beams, with k ⃗ a and k ⃗ b wave-vectors, and detected simultaneously in the 2 k ⃗ a − k ⃗ b and 2 k ⃗ b − k ⃗ a directions. In each direction, we observe a single peak when the two beams are tuned on the closed transition 85 Rb 5 S 1/2 ( F = 3) → 5 P 3/2 ( F = 4). The excitation spectra reveal a small frequency separation between the two peaks, which is explained when propagation effects are taken into account. Furthermore, our theoretical analysis shows that a correct description of the frequency position of each peak is achieved with a variable refractive index for both lasers, in which the scanning laser experiences an anomalous window in the refractive index near resonance.
We present in this paper a theory of the saturation effects in degenerate four-wave mixing which makes use of the dressed-atom method. We mainly study the case of a Doppler-broadened line, but several results can be applied to the case of nearly degenerate four-wave mixing for a Doppler-free medium. We show that a close relationship does exist between the two problems : the generated beam is mainly created when the detuning between two incident beams is equal to the Rabi frequency. In the degenerate case, this condition is fulfilled by two velocity groups because of the Doppler shift. We analytically calculate the lineshapes for two-level atoms in the case of forward saturation (the intense pump beam propagates in the same direction than the probe beam) and in the case of backward saturation (the intense pump beam and the probe beam propagate in opposite directions). We also study collisional effects and the influence of the intensity of the weak pump beam when perturbation theory can no longer be applied.
In this work we describe an experimental investigation on the spatial intensity distribution of the two transmitted beams and the two beams generated in a four-wave mixing (FWM) process, using rubidium vapor as a nonlinear medium. The two FWM signals are detected in the 2⃗kF − ⃗kP and 2⃗kP − ⃗kF directions, where ⃗kF and ⃗kP are the incident beam directions. A diode laser, responsible by the two Ąelds that driven the FWM process, is tuned to the 5S1⇑2(F ≙ 3) Ð→ 5P3⇑2 of the 85Rb. The images of the four beams are obtained using a CMOS camera for a Ąxed atomic density and a Ąxed frequency of the input beams. From each image we can determine the spatial distribution of the intensity of the respective beam. We analyzed the spatial intensity distribution of the signals, as well as the spatial intensity Ćuctuation. The spatial intensity distribution of the transmitted beams shows a decrease of the beam waist for intensities close to the saturation intensity, while for high intensities the spatial proĄle is almost constant. This behavior indicates an autofocusing effect, related to the variation of the refractive index of the medium. The spatial distribution of the intensity Ćuctuation is obtained considering an average surface for each image, and is used to calculate the autocorrelation of the transmitted beams and the two generated FWM beams. Working with the intensity Ćuctuation, we can deĄne a transverse spatial correlation length, which is used to compare the behavior of the four beams with that of a free beam. In this sense, the autocorrelations of intensity Ćuctuations of the transmitted beams show a decrease in the correlation length in the same region of intensities where we observe the self-focusing effect, and as we increase the intensity, the correlation length approaches the value obtained for the free beam. On the other hand, the correlation length for the autocorrelations of the signals generated by FWM increases as the intensity of the incident beams increases.
In this paper we analyze theoretically the influence of the local-field effects on reflection spectra of a degenerate four-wave mixing in a dense medium of two-level atoms. We discuss the spectral dependence of the nonlinear absorption and dispersion of the signal and probe fields as well as their coupling for various strengths of the external field and atomic densities. For weak external fields the local-field effects reduce to the well known Lorentz redshift of the reflection spectra. When the external field saturates the transition in a dense sample significant modifications of the probe reflectivity and of the wave mixing spectra occur, resulting in asymmetric, multipeak line shapes. \textcopyright{} 1996 The American Physical Society.