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Physical Review A

Effect of redistributed radiation on degenerate four-wave mixing

N. J. Mulgan, R. J. Ballagh

Published Oct 1, 1993• 6 citations
Publisher

Abstract

A model is formulated to investigate quantitatively how near-resonant degenerate four-wave mixing (DFWM) in a vapor of homogeneously broadened two-state atoms is affected by radiation scattered from the pump beams. The spectral intensity distribution of the scattered radiation in the interaction volume is obtained by solving the equation of radiative transfer and a detailed treatment of the modification this fluctuating field causes on the optical dipole induced by the coherent laser fields is presented. For a broad parameter regime, this modification can be expressed via a single quantity that incorporates the energy of the fluctuating field together with bandwidths and frequency offsets from the atomic resonance. Numerical solutions are given which show that the scattered field causes the DFWM reflectivity to be depressed by a signficant amount, depending most strongly on a characteristic transverse length of the interaction volume. Approximate analytic expressions for the reflectivity are also given which assist in physically understanding the underlying mechanisms.

DOI
10.1103/physreva.48.3321

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Physical Review A•Sep 1, 1990•47 citations
A model is formulated to investigate quantitatively how near-resonant degenerate four-wave mixing (DFWM) in a vapor of homogeneously broadened two-state atoms is affected by radiation scattered from the pump beams. The spectral intensity distribution of the scattered radiation in the interaction volume is obtained by solving the equation of radiative transfer and a detailed treatment of the modification this fluctuating field causes on the optical dipole induced by the coherent laser fields is presented. For a broad parameter regime, this modification can be expressed via a single quantity that incorporates the energy of the fluctuating field together with bandwidths and frequency offsets from the atomic resonance. Numerical solutions are given which show that the scattered field causes the DFWM reflectivity to be depressed by a signficant amount, depending most strongly on a characteristic transverse length of the interaction volume. Approximate analytic expressions for the reflectivity are also given which assist in physically understanding the underlying mechanisms.
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 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.
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.
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].
In recent years, the collisional redistribution of radiation and collision-induced broadening of Rydberg atomic spectral lines by buffer gas perturbation have aroused the renewed interest. Rydberg atoms having a large dipole moment and long lifetime can interact with each other coherently for relatively long time, which makes them a potential candidate for quantum information processing. Besides, collisional redistribution has an important potential application in laser cooling and trapping. Based on previous experimental data, in this paper, two-nondegenerate four-wave mixing (NFWM) for studying atom collision, composed of two-photon resonant NFWM and collisional redistribution NFWM, is reported. The spectrum variation of the two-NFWM affected by the pressure, temperature, detuning and collision-broadening rate coefficient is analyzed. The principle of two-NFWM involving three incident beams is explained as follows. Consider two-NFWM in a |0-|1-|2 cascade three-level system, where states between |0 and |1 and between |1 and |2 ightangle are coupled by resonant frequencies 1 and 2 , respectively. Beam 1 with frequency 1 propagates along the direction opposite to the direction of beam 2, beams 2 and 2' have the same frequency 2, and between their directions there exists a small angle. Assuming that 1 1 and 2 2 so that 1 drives the transition from |0 to |1 while 2 drives the transition from |1 to |2, the simultaneous interactions of atoms with beams 1 and 2 will induce atomic coherence between |0 and |2 through two-photon excitation. This coherence is probed by beam 2', and as a result a two-photon resonant NFWM signal of frequency 1 is generated in the direction almost opposite to the direction of beam 2'. To avoid strong absorption at the resonant frequency of transition from |0 to |1, here the wavelength of beam1 is detuned from the exact resonance. An atom population of level |1 caused by collisional redistribution can be induced when a certain buffer gas pressure is imposed. The collisional redistribution NFWM process also exists in this case. Beam 2 drives the transition from |1 to |2 to induce an atomic coherence which is probed by beam 2' for giving rise to an atomic population grating. A collisional redistribution NFWM signal propagating along the same direction as the two-photon resonant NFWM signal is generated when beam 1 is scattered by the grating. Much information about atomic collisions can be obtained by analyzing the two NFWM signals. In a cascade three-level system composed of ground state, intermediate state and Rydberg state, and the two-NFWM can be used to investigate not only the broadening and shifting of the Rydberg level but also the collisional redistribution of the intermediate state. Unlike other experiments studying the pressure dependence of the longitudinal relaxation rate of atom states, this technique is a purely optical coherent means, and can measure the transverse relaxation rate 20 between Rydberg state and ground state as well as the pressure dependence of the transverse relaxation rate 21 between Rydberg state and intermediate state.
The steady-state theory of degenerate four-wave mixing (DFWM) with broad-bandwidth lasers is extended to treat the time dependence of the interaction allowing calculation of the intensity dependence and saturation behavior of DFWM with pulsed lasers. An experimental study of this intensity dependence as a function of laser bandwidth is reported, showing qualitative agreement with the theory. The line shape of DFWM induced by broad-bandwidth pump fields, probed by a monochromatic probe, is shown to have a Doppler-broadened profile in agreement with the theoretical predictions.