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Journal of Luminescence

Phase conjugate degenerate four-wave mixing in molecular aggregates

Francis C. Spano, Shaul Mukamel

Published Jan 1, 1990• 4 citations
Publisher

Abstract

We investigate phase conjugate degenerate four-wave mixing (DFWM) in molecular aggregates consisting of N interacting, homogeneously broadened two-level systems in a cyclic configuration with a dimension much smaller than an optical wavelength. The interaction includes both the static dipole-dipole coupling and superradiant coupling. We show that, in general the size dependence of χ(3) is determined by the relative magnitudes of the homogeneous dephasing rate and the superradiant decay rate. We predict the existence of a narrow dephasing-induced resonance superposed on the broader superradiant exciton lineshape.

DOI
10.1016/0022-2313(90)90208-s

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Journal of the Optical Society of America B•Dec 1, 1996•17 citations
We investigate phase conjugate degenerate four-wave mixing (DFWM) in molecular aggregates consisting of N interacting, homogeneously broadened two-level systems in a cyclic configuration with a dimension much smaller than an optical wavelength. The interaction includes both the static dipole-dipole coupling and superradiant coupling. We show that, in general the size dependence of χ(3) is determined by the relative magnitudes of the homogeneous dephasing rate and the superradiant decay rate. We predict the existence of a narrow dephasing-induced resonance superposed on the broader superradiant exciton lineshape.
We investigate the third-order nonlinear-optical susceptibility ${\ensuremath{\chi}}^{(3)}$ in molecular aggregates and its scaling with aggregate size. The aggregate is modeled as a collection of N-interacting, homogeneously broadened two-level systems, where the interaction includes both the static dipole-dipole coupling and superradiant coupling. Our expression for ${\ensuremath{\chi}}^{(3)}$, derived using Liouville-space Green-function techniques, contains contributions from excitons as well as biexcitons. The scaling of ${\ensuremath{\chi}}^{(3)}$ with aggregate size and its dependence on the laser detuning and homogeneous and inhomogeneous dephasing rates are analyzed. ${\ensuremath{\chi}}^{(3)}$ contains terms that scale as \ensuremath{\sim}N2{N}^{2}N2 and \ensuremath{\sim}N(N-1). However, under off-resonant conditions these terms interfere destructively, resulting in a linear \ensuremath{\sim}N scaling, which is precisely what is expected for N monomers. Detailed calculations of phase-conjugate degenerate four-wave mixing and pump-probe spectroscopy are performed. In the first case we predict the existence of a narrow dephasing-induced resonance superposed on the broader superradiant exciton line shape. Pump-probe spectroscopy shows a series of biexciton absorption lines, some of which are superradiantly broadened.
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.
Degenerate four-wave mixing (DFWM) spectroscopy is a potentially powerful technique for quantitative measurement of minor species in gas-phase media [1-3]. In this paper we compare the DFWM signal levels and lineshapes obtained using two different phase-matching geometries: the phase-conjugate geometry and the forward phase-matched geometry. DFWM signal levels are calculated by solving the time-dependent density matrix equations using direct numerical integration (DNI) [4, 5] for a two-level system interacting with three laser beams. This numerical analysis includes the effects of molecular motion, intermolecular collisions, and high laser intensities, all of which have been shown to affect the DFWM signal level.
An analytic nonperturbative treatment of degenerate four-wave mixing is presented to study Zeeman-coherence effects in phase conjugation. The counterpropagating pump waves are taken to be orthogonally circularly polarized and interact with different Zeeman sublevels. The nonlinear medium is modeled in terms of a homogeneously broadened, folded three-level system. Particular attention is paid to the phase-conjugate spectrum obtained by varying the Larmor frequency associated with the Zeeman splitting. A level-crossing resonance is found to occur in the saturation regime and manifests itself as a narrow peak or dip depending on various level decay rates. Even for a purely absorptive system, the Zeeman coupling of the ground-state degenerate sublevels is shown to produce significant phase-conjugate reflectivities at pump intensities much lower than those required for a two-level system.
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.
The use of picosecond lasers for degenerate four-wave mixing (DFWM) diagnostics of atmospheric pressure flames has been suggested as a means of overcoming the complicated collision-rate dependence of the DFWM signal in the nanosecond-laser pulse-length regime. In this paper DFWM spectroscopy in the regime in which the laser pulse length (τL) is less than the characteristic collisional time (τC) is investigated theoretically. DFWM signal levels for the phase-conjugate geometry (counterpropagating pump beams) are calculated by integration of the time-dependent density-matrix equations at numerous grid points along the phase-matching axis and by summing of the polarization contribution from each of these grid points. Both purely homogeneously broadened resonances and resonances that are both collision broadened and Doppler broadened are considered. DFWM signal generation for these two types of resonances in the fully transient regime (τL≪τC) is strikingly different. For a homogeneously broadened resonance, the DFWM signal persists long after the laser beam has passed through the medium. The decay rate of the DFWM signal is determined by the collision rate in the medium, and the integrated DFWM signal is still dependent on the collision rate even in this fully transient regime. However, when the Doppler broadening of the resonance is much greater than the collisional broadening, the DFWM signal decays with a fall time close to that of the laser pulse. For τL≪τC, we find that it is the laser spectral width, rather than the collisional width, that dictates the range of velocity groups with which the pulses effectively interact. The signal decays with a fall time close to that of the laser pulse because the excited velocity groups radiate out of phase with one another. In a medium in which the Doppler width is much greater than the collisional width, the DFWM reflectivity is independent of collision rate for τL≪τC, but the signal level is dependent on the Doppler width. Finally, we investigate the effect of different laser pulse lengths on reflectivity saturation curves for resonances with different levels of Doppler broadening.