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Physical Review A

Frequency spectrum of the signal wave in resonant four-wave mixing induced by broad-bandwidth lasers

D. R. Meacher, Peter G. R. Smith, P. Ewart, J. Cooper

Published Sep 1, 1992• 17 citations
Publisher

Abstract

We develop a theory of near-resonant four-wave mixing induced by broad-bandwidth chaotic fields in a medium composed of two-level atoms. By solving the equations of motion for the elements of the atomic density matrix using an appropriate decorrelation approximation, we derive an analytic expression for the frequency spectrum of the signal wave for the case of the bandwidth of the fluctuations of the pump field exceeding the other relaxation rates in the problem. The probe wave is considered weak and of arbitrary bandwidth. The theory is valid for pump intensities up to and exceeding the bandwidth-dependent saturation value. Finally, we show how the frequency spectrum is modified if account is taken of atomic motion. The theoretical results are of direct relevance to practical experiments involving broad-bandwidth pulsed lasers and employing an atomic vapor as the nonlinear medium.

DOI
10.1103/physreva.46.2718

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We develop a theory of near-resonant four-wave mixing induced by broad-bandwidth chaotic fields in a medium composed of two-level atoms. By solving the equations of motion for the elements of the atomic density matrix using an appropriate decorrelation approximation, we derive an analytic expression for the frequency spectrum of the signal wave for the case of the bandwidth of the fluctuations of the pump field exceeding the other relaxation rates in the problem. The probe wave is considered weak and of arbitrary bandwidth. The theory is valid for pump intensities up to and exceeding the bandwidth-dependent saturation value. Finally, we show how the frequency spectrum is modified if account is taken of atomic motion. The theoretical results are of direct relevance to practical experiments involving broad-bandwidth pulsed lasers and employing an atomic vapor as the nonlinear medium.
We present a revision of the theory of finite laser bandwidth effects on degenerate four-wave mixing (DFWM) of Alber, Cooper, and Ewart (ACE) [Phys. Rev. A 31, 2344 (1985)]. The same model is used for intense, broad-bandwidth pump lasers interacting with a weak monochromatic probe in a medium composed of two-level atoms. The density-matrix equations describing the time evolution of the atomic polarization coupled to fluctuating fields are solved using an appropriate decorrelation approximation. A steady-state analytical solution is found for resonant and near-resonant DFWM, for intensities that do not saturate the medium. For more intense fields it is found to be necessary to form a more complicated set of differential equations for all products of the field and atomic density-matrix elements. The results show that as in the ACE theory, increasing the bandwidth b leads to an increased effective saturation intensity. However, we find the DFWM reflectivity scales as 1/b2{b}^{2}b2 when the probe is monochromatic and not 1/b as in the ACE theory. Furthermore, the saturation behavior of nonresonant DFWM is found to differ from the predictions of ACE. Atomic motion effects are shown to yield a Doppler-broadened line shape and also to affect the saturation behavior of the signal.
The effects of finite laser bandwidth on resonant degenerate four-wave mixing (DFWM) are calculated with use of a model in which the intense, counterpropagating pump beams are characterized by a chaotic field, the probe beam is weak and monochromatic, and the medium consists of a gas of two-level atoms. We present a steady-state solution in the limit where the pump-laser bandwidth exceeds all other atomic relaxation rates. Although the mean intensity due to the fluctuating fields is spatially independent (no steady-state standing-wave pattern is established), the analytic results indicate that, for intensities above the saturation intensity Isat{I}_{\mathrm{sat}}Isat​, spatially periodic saturation effects are important. Increasing bandwidth is shown to lead to an increase in the effective saturation intensity resulting in lower phase-conjugate reflectivity for IIsat{I}_{\mathrm{sat}}Isat​ than for coherent pump fields, in contrast to the results for narrow-bandwidth chaotic fields. The resonant DFWM line shape is also calculated and compared to the coherent result. We comment on the application of the model to other four-wave-mixing processes employing broad-bandwidth lasers.
We consider the dependence of the four-wave-mixing signal from an ensemble of homogeneously broadened, two-level atoms on the statistics of the pump field. Two models for the pump field are considered: the chaotic field and the phase-diffusing field. A Monte Carlo simulation procedure is used to integrate the equations describing the four-wave-mixing process numerically. This technique allows consideration of arbitrary bandwidths for the pump and incorporates pump-induced saturation effects. Results are presented for the strength of the four-wave-mixing signal as a function of the pump intensity for both resonant and off-resonant cases. It is shown that the four-wave-mixing signal is sensitive to the statistics of the pump, particularly for strong fields.
We explore theoretically the signal generated by a fluctuating radiation field via a four-wave mixing process in an atomic vapor undergoing Markovian dephasing collisions. We consider specifically a broadband laser field well detuned from the atomic resonance and calculate the signal generated using counterpropagating pump fields as a function of the time delay of the incident probe field. We assume that the pulse duration is long compared to the correlation time of the incident fields. By considering the various time-ordered pathways and treating the incident-field amplitudes as random Gaussian variables, we find that the signal generated is dominated by two specific pathways that depend only on the incident-field bandwidth and the decay rate of the ground to excited state coherence. An interesting aspect of the far-impact-detuning case is that the signal is dominated by resonances involving spatially separated atoms. Our formalism provides a basis for describing the effects of mode correlations on four-wave-mixing signals generated by stochastic fields.
Resonant Degenerate Four Wave Mixing, DFWM, is a nonlinear optical process arising from the third order susceptibility X(3). Resonance enhancement of X(3)in absorbing media makes the process useful for spectroscopic applications[1]. In many situations pulsed, high power lasers are used which have broad spectral bandwiths. The stochastic fluctuations of amplitude and phase associated with this bandwidth complicates the theoretical description in such cases. An averaging of the atom-field interaction is required simultaneously over the temporal and spatial variations of the fields. A solution has been found in the broad bandwidth limit which uses an appropriate decorrelation approximation. Theoretical and experimental results will be presented for the effects of laser bandwidth on the power dependence and temporal behaviour of resonant DFWM[2].
We develop a field-nonlinear theory of sub-Doppler spectroscopy in a gas of two-level atoms, based on a self-consistent solution of the Maxwell-Bloch equations in the mean field and single-atom density matrix approximations. This makes it possible to correctly take into account the effects caused by the free motion of atoms in a gas, which lead to a nonlinear dependence of the spectroscopic signal on the atomic density even in the absent of a direct interatomic interaction (e.g., dipole-dipole interaction). Within the framework of this approach, analytical expressions for the light field were obtained for an arbitrary number of resonant waves and arbitrary optical thickness of a gas medium. Sub-Doppler spectroscopy in the transmission signal for two counterpropagating and co-propagating waves has been studied in detail. A previously unknown red shift of a narrow sub-Doppler resonance is predicted in a counterpropagating waves scheme, when the frequency of one wave is fixed and the frequency of the other wave is varied. The magnitude of this shift depends on the atomic density and can be more than an order of magnitude greater than the known shift from the interatomic dipole-dipole interaction (Lorentz-Lorenz shift). The found effects, caused by the free motion of atoms, require a significant revision of the existing picture of spectroscopic effects depending on the density of atoms in a gas. Apart of fundamental aspect, obtained results are important for precision laser spectroscopy and optical atomic clocks.