Resolutions of degenerate four-wave mixing with forward and phase-conjugate configurations (FDFWM and PCDFWM) are investigated and compared theoretically and experimentally in hot rubidium (Rb) atomic vapor. The theoretical simulations indicate that PCDFWM is of much higher resolution than FDFWM. The resolution of PCDFWM is less dependent on Doppler broadening. The experimental results are in good agreement with the theoretical expectation. PCDFWM can resolve the hyperfine transitions and crossover resonances of 87Rb which cannot be achieved by FDFWM. Additionally, with sample temperature increasing, the linewidth of FDFWM spectrum obviously broadens. In comparison, no obvious broadening can be observed in the PCDFWM spectrum.
Resolutions of degenerate four-wave mixing with forward and phase-conjugate configurations (FDFWM and PCDFWM) are investigated and compared theoretically and experimentally in hot rubidium (Rb) atomic vapor. The theoretical simulations indicate that PCDFWM is of much higher resolution than FDFWM. The resolution of PCDFWM is less dependent on Doppler broadening. The experimental results are in good agreement with the theoretical expectation. PCDFWM can resolve the hyperfine transitions and crossover resonances of 87Rb which cannot be achieved by FDFWM. Additionally, with sample temperature increasing, the linewidth of FDFWM spectrum obviously broadens. In comparison, no obvious broadening can be observed in the PCDFWM spectrum.
In this work, we present a set of experiments involving four-wave mixing processes in rubidium atoms. We employ several experimental configurations exploring two- and three-level systems in cascade or Λ configurations, all of which are in a copropagating geometry, to investigate different physical phenomena present in each scenario. In the first experiment, we combine a cw diode laser and a 1 GHz femtosecond pulse train to generate coherent blue light through the four-wave mixing process in hot rubidium vapor. The generated coherent signal is analyzed by scanning either the diode frequency or the repetition rate of the frequency comb. In both cases, the cw laser induces an Autler- Townes splitting with a wide separation in the doublet structure due to the configuration of copropagating fields. To model these results, we employ the density matrix formalism to write the Bloch equations and solve them numerically. The solutions allow us to explore other cases beyond those involving our experimental setup, comparing the response of the upper-level population and the generated signal coherence for homogeneously and non-homogeneously broadened systems. Moreover, in this same experiment, we also observed an unexpected interference effect between excitation routes of the four-wave mixing process when the repetition rate of the pulse train is scanned. The theoretical model demonstrates that the interfering effect appears only when the repetition rate is carefully chosen to produce a pair of modes of the frequency comb resonant with both one- and two-photon transitions for the same atomic velocity group. In the second experiment, still with a hot rubidium vapor, we explore two symmetrical degenerate four- wave mixing signals induced by two almost copropagating laser beams, with~ka and~kb wave-vectors, and detected simultaneously in the 2~ka −~kb and 2~kb −~ka directions. Each spectrum has a single peak with a short frequency separation between them, which our model indicates is due to the nonlinear behavior of the refractive indexes, a consequence of the high intensity of both input fields. Finally, we use the same experimental setup for the two symmetrical signals to study the field intensity fluctuations of the degenerate four-wave mixing signals and the transmission signals. In this case, the sample is an ensemble of cold rubidium atoms in a magneto-optical trap. There is an intensity- intensity cross-correlation between the signals due to the resonant phase-noise to amplitude-noise conversion. In this type of cold system, we can neglect the Doppler- broadening, allowing a straightforward study of the dependency of the correlations on the laser detuning. In these measurements, we observe a type of oscillation in the correlation curves that the theoretical model indicates to be connected with Rabi oscillations, revealed by the fluctuations long after the transient period.
The observation of a new type of saturation effects in degenerate four-wave mixing in Na vapor is reported. The experimental configuration is neither counterpropagating nor velocity selective, and yet sub-Doppler high-resolution lines are observed at the atomic-transition frequencies, and at the weighted center of a Doppler-broadened distribution. These new features are the result of an interplay between absorption and saturation, and the inhomogeneous broadening is essential for their existence.
Nowadays degenerate four wave mixing (DFWM) with forward geometry (FDFWM) has found increasing usage, for example, in the gas phase detection. In the fields of theoretical development of DFWM, most of the studies have been concentrated on the understanding of phase-conjugated DFWM. However, few theories have dealt with FDFWM. These models calculated the signal directly from the off-diagonal element of the density matrix by assuming optically thin medium. In the paper we obtain a theoretical model for FDFWM in Doppler broadened two-level absorbers including propagation effect and discuss the Doppler effect on the signal intensity and lineshape. Assuming that probe and signal field amplitudes are small compared with those of the pump fields and pump field amplitudes are constant in the interaction volume, we obtain the following coupled equations from the Maxwell equation using the slowly varying envelope approximation and the paraxial approximation.
We present in this paper a theoretical study of the angular dependence of phase-conjugate emission via resonant degenerate four-wave mixing (FWM) in Doppler-broadened gas media. The nonlinear optical polarization is calculated through an expansion of the atomic density matrix up to the third order in the incident fields. One finds that, when the incident frequency is resonant for a single-photon transition, both strength and lineshape of the phase-conjugate emission strongly depend on the angle 0 between standing pump wave and probe wave. The emission intensity, maximum at grazing incidences, decreases rapidly by several orders of magnitude when 0 increases. Simultaneously the emission linewidth increases from homogeneous to Doppler width. Similar effects are predicted for nearly-degenerate FWM (different pump and probe frequencies). In particular, one shows that the bandwidth of the equivalent optical reflection filter undergoes a Doppler-broadening linearly increasing with sin θ. On the opposite, none of the above effects is predicted in two-photon-resonant FWM which does not exhibit any angular dependence.
We present experimental observations of phase-conjugate degenerate four-wave mixing in a two-level atomic system (the ${3s}^{2}{S}_{1/2}, F=2, {m}_{F}=\stackrel{\ensuremath{\rightarrow}}{2}{3p}^{2}{P}_{3/2}, F=3, {m}_{F}=3$ transition in a diffuse, collision-free, thermal beam of atomic sodium) driven by cw, narrow-band $(\ensuremath{\sim}250 \mathrm{kHz}),$ stabilized pump and probe fields. The primary measurements that we report are of the peak signal strength of the four-wave-mixing interaction and the bandwidth of the four-wave-mixing spectrum (as the laser frequency is tuned through the resonant frequency) as a function of the intensity of the pump field. We compare these measurements with the predictions of a theoretical analysis in which extension to include the effect of the small transverse velocity $(\ensuremath{\sim}3 \mathrm{m}/\mathrm{s}\mathrm{e}\mathrm{c})$ of the atomic beam is critical. These measurements provide a direct verification of the theory of this important nonlinear interaction.
We present a theoretical study of phase-conjugate (PC) emission by backward degenerate four-wave mixing (DFWM) processes in a resonant gas medium. In this study, high-intensity (pump) saturation and Doppler broadening are simultaneously taken into account. We develop a general formalism to deal with the saturation effects in the case of a three-level atom (with degenerate frequencies) and obtain analytical expressions of the PC field amplitude (after integration over the velocity distribution) under the following conditions: (1) the probe beam and one of the pump beams have intensities weak enough to be treated at the lowest perturbation order, whereas the other pump intensity is arbitrarily high; (2) the probe crosses the standing pump wave at grazing incidence. It is shown that the saturation induced by the forward pump (copropagating with the probe beam) leads to entirely different behavior of the nonlinear susceptibility from that induced by the backward pump (counterpropagating with the probe beam). This saturation anisotropy is specific to the inhomogeneous broadening of the gas medium, which is also responsible for the frequency splitting experimentally observed in the intensity line shape of the PC emission for large pump intensities. At saturation, this splitting is theoretically shown to be proportional to the Rabi frequency, with a proportionality coefficient critically dependent on the relaxation processes; in the case of a single-relaxation model, the splitting is equal to 1.5 times the Rabi frequency. The PC intensity line shape is demonstrated to be of dispersive origin in the case of backward saturation, whereas for forward saturation, narrow structures are predicted in both real and imaginary parts of the nonlinear susceptibility, also leading to the intensity line-shape splitting. One also finds that the reflectivity of PC mirrors based on resonant DFWM strongly depends on the origin of the saturation. For a saturating forward pump, the PC emission intensity tends toward zero with increasing pump intensities, and it reaches a finite (nonzero) limit in the case of backward saturation. Most of these predictions have been observed experimentally [ BlochD., Phys. Rev. Lett.49, 719 ( 1982)].