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Physical Review A

Spectral line shape of resonant four-wave mixing induced by broad-bandwidth lasers

Peter G. R. Smith, P. Ewart

Published Sep 1, 1996• 12 citations

Abstract

We present a theoretical and experimental study of the line shape of resonant four-wave mixing induced by broad-bandwidth laser radiation that revises the theory of Meacher, Smith, Ewart, and Cooper (MSEC) [Phys. Rev. A 46, 2718 (1992)]. We adopt the same method as MSEC but correct for an invalid integral used to average over the distribution of atomic velocities. The revised theory predicts a Voigt line shape composed of a homogeneous, Lorentzian component, defined by the collisional rate \ensuremath{\Gamma}, and an inhomogeneous, Doppler component, which is a squared Gaussian. The width of the inhomogeneous component is reduced by a factor of \ensuremath{\surd}2 compared to the simple Doppler width predicted by MSEC. In the limit of dominant Doppler broadening, the width of the homogeneous component is predicted to be 4\ensuremath{\Gamma}, whereas in the limit of dominant homogeneous broadening, the predicted width is 2\ensuremath{\Gamma}. An experimental measurement is reported of the line shape of the four-wave-mixing signal using a broad-bandwidth, ``modeless,'' laser resonant with the ${\mathit{Q}}_{1}$(6) line of the A${\mathrm{}}^{2}$\ensuremath{\Sigma}-X${\mathrm{}}^{2}$\ensuremath{\Pi}(0,0) system of the hydroxyl radical. The measured widths of the Voigt components were found to be consistent with the predictions of the revised theory. \textcopyright{} 1996 The American Physical Society.

DOI: 10.1103/physreva.54.2347

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Proceedings of the Royal Society of London A Mathematical and Physical Sciences•Jun 30, 1978•12 citations

We present a theoretical and experimental study of the line shape of resonant four-wave mixing induced by broad-bandwidth laser radiation that revises the theory of Meacher, Smith, Ewart, and Cooper (MSEC) [Phys. Rev. A 46, 2718 (1992)]. We adopt the same method as MSEC but correct for an invalid integral used to average over the distribution of atomic velocities. The revised theory predicts a Voigt line shape composed of a homogeneous, Lorentzian component, defined by the collisional rate \ensuremath{\Gamma}, and an inhomogeneous, Doppler component, which is a squared Gaussian. The width of the inhomogeneous component is reduced by a factor of \ensuremath{\surd}2 compared to the simple Doppler width predicted by MSEC. In the limit of dominant Doppler broadening, the width of the homogeneous component is predicted to be 4\ensuremath{\Gamma}, whereas in the limit of dominant homogeneous broadening, the predicted width is 2\ensuremath{\Gamma}. An experimental measurement is reported of the line shape of the four-wave-mixing signal using a broad-bandwidth, ``modeless,'' laser resonant with the

{\mathit{Q}}_{1}

(6) line of the A

{\mathrm{}}^{2}

\ensuremath{\Sigma}-X

{\mathrm{}}^{2}

\ensuremath{\Pi}(0,0) system of the hydroxyl radical. The measured widths of the Voigt components were found to be consistent with the predictions of the revised theory. \textcopyright{} 1996 The American Physical Society.

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/

{b}^{2}

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 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.

The spectral line shape of the four-wave mixing (FWM) signal, induced by broad bandwidth laser fields, is derived in the case of a nonresonant interaction. The laser fields are assumed to have a Lorentzian spectrum arising from Markovian stochastic fluctuations described by the phase-diffusion, chaotic field, and Gaussian-amplitude models. The usual ``phase conjugating'' geometry for FWM is considered where the nonresonant interaction is determined only by the statistics of, and the correlations between, the pump and probe fields. For each one of the four possible correlation states between the three input fields and for each of the three fluctuating field models, the line shape of the FWM signal is calculated and shown to be particularly sensitive to the field statistics.

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

{I}_{\mathrm{sat}}

, 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 I

{I}_{\mathrm{sat}}

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.

It is well known that the spectral line shape of the narrow change signal induced at one Doppler-broadened transition by an intense laser field resonating with a second one can be extremely complex and nonclassical in form, the exact details depending on such factors as the direction of observation relative to the propagation direction of the laser field and its spatial form (traveling wave or standing wave). Nevertheless, the area under the change signal line shape, i.e. its frequency-integrated intensity, is always the same for a given laser field, and is independent of the coupling between laser and change-signal fields. A proof of this equal area property, based on quantum-mechanical transition-rate theory, is given for a three-level system composed of degenerate or near-degenerate states. The resulting expression is also applicable to line-shape calculations for experiments which study resonance fluorescence induced by intense laser or resonance radiation or other forms of excitation.

Abstract The development with time of the excitation of a transition between two atomic or molecular energy levels under the influence of monochromatic laser radiation is examined under conditions of strictly inhomogeneous line broadening due to such causes as doppler shift arising from translational velocity. The ratio of the number of molecules, N2, in an excited state to the total number, N, is calculated for various ratios of the intensity parameter β ═ E0μ/ħ to the half line-width ∆, where E0 is the amplitude of the electric field in the incident radiation and is the dipole moment matrix element for the transition. Excitation functions obtained in a previous paper (I) for various values of the ratio γ/β, where γ is the half line-width in the absence of inhomogeneous broadening, are used to obtain the variation of N2/N with time under conditions of mixed broadening for various values of the ratio γ/∆ and γ/β, when the exciting radiation is in exact resonance with the central frequency of the transition.