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The Journal of Chemical Physics

Line-space description of resonant four-wave mixing: Theory for isotropic molecular states

А. П. Коузов, Peter Radi

Published May 15, 2014• 5 citations
Publisher

Abstract

Based on the quantum Liouville formalism, a theory of the two-color, triply resonant four-wave mixing is developed for molecules with isotropically oriented angular momenta. The approach allows to strictly incorporate the relaxation matrices Γ((r)) (r = 0, 1, 2) into the third-order susceptibility χ((3)) whose expression acquires therewith the form of a scalar product in the line space. Thanks to this representation, isolation of all resonance terms from χ((3)) becomes a routine task. Some of these terms correspond to the case when a molecule initially interacts with two pump photons of the same frequency. Such interactions give rise to the grating line-space vectors which have the same (zero) eigenfrequency. Due to this degeneracy, the latter are easily mixed by rotationally inelastic collisions which shows up in a state-resolved coherence transfer. The satellite signals induced thereby provide a great scope to study the state-to-state inelastic rates in situ by purely optical means. If the diagonal form of Γ is assumed, the satellites become forbidden and our results reduce to conventional expressions for the main resonances. Polarization configurations are designed for direct measurements of the population (r = 0), orientation (r = 1), and alignment (r = 2) contributions to χ((3)). Finally, depending on the photon-molecule interaction sequence, the resonance terms of χ((3)) are shown to be differently affected by velocity averaging, the effect which conspicuously manifests itself when Doppler broadening becomes paramount.

DOI
10.1063/1.4874159

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Based on the quantum Liouville formalism, a theory of the two-color, triply resonant four-wave mixing is developed for molecules with isotropically oriented angular momenta. The approach allows to strictly incorporate the relaxation matrices Γ((r)) (r = 0, 1, 2) into the third-order susceptibility χ((3)) whose expression acquires therewith the form of a scalar product in the line space. Thanks to this representation, isolation of all resonance terms from χ((3)) becomes a routine task. Some of these terms correspond to the case when a molecule initially interacts with two pump photons of the same frequency. Such interactions give rise to the grating line-space vectors which have the same (zero) eigenfrequency. Due to this degeneracy, the latter are easily mixed by rotationally inelastic collisions which shows up in a state-resolved coherence transfer. The satellite signals induced thereby provide a great scope to study the state-to-state inelastic rates in situ by purely optical means. If the diagonal form of Γ is assumed, the satellites become forbidden and our results reduce to conventional expressions for the main resonances. Polarization configurations are designed for direct measurements of the population (r = 0), orientation (r = 1), and alignment (r = 2) contributions to χ((3)). Finally, depending on the photon-molecule interaction sequence, the resonance terms of χ((3)) are shown to be differently affected by velocity averaging, the effect which conspicuously manifests itself when Doppler broadening becomes paramount.
It is shown that in order to obtain a correct interpretation of experiments on four-photon nonlinear spectroscopy within the absorption line, it is necessary to allow not only for a coherent four-photon process due to a resonant electronic nonlinearity but also for a similar process involving generation of optically induced diffraction gratings followed by the diffraction of one of the probe beams from these gratings. It is also necessary to allow for the contribution of highly excited molecular states to the resonant electronic nonlinearity. General expressions are derived and investigated for the cubic susceptibility in an isotropic medium. It is found that all nonzero components of this tensor may be expressed in terms of a single scalar function of the interacting wave frequencies. The explicit form of this function is determined in the lowest order of perturbation theory.
Applying a recently developed ns Two-Colour Resonant Four-Wave Mixing (TC-RFWM) technique, a number of resonance lines forbidden by conventional three-level schemes are observed. A theoretical frequency-domain picture is developed with the help of exact (off-diagonal) relaxation matrices and general four-level schemes. The newly observed extra resonances are induced by collisional transfer of rotational energy. The role of the population, orientation and alignment gratings that are formed in both electronic states coupled by laser fields is elucidated. Relaxation models using the fast-collision approximation are elaborated for Hund's cases a and b giving quantitative arguments for the proposed interpretation. The potential of the TC-RFWM technique to study state-to-state transfer rates is emphasized.
We present the theory of resonant backward degenerate four-wave mixing in gaseous media for Doppler-broadened degenerate two-level systems. The degeneracy of the atomic levels as well as the influence of the polarization of the incident fields are taken into account within the framework of a tensorial formalism. The amplitude, line shape, and polarization of the reemitted phase-conjugate field are calculated up to the third order in the incident fields. The velocity average is performed in the approximation of large Doppler linewidths. One considers in detail the influence of depolarizing relaxation processes and the effects of pump-probe angular separation. In particular, the importance of the residual Doppler effect (shortening of the lifetime of the optically induced gratings by the atomic motion) associated with the pump-probe angle is emphasized. We show that the polarization of the reemitted fields is governed by very simple laws as soon as the residual Doppler effect overcomes the lifetimes of the atomic levels. Phase-conjugate mirrors can be simply characterized by a linear dichroism (parallel pump polarizations) or a birefringence (cross-polarized pumps). Vectorial phase conjugation is also analyzed, in the case of counter-rotating circular polarizations of the pumps. Finally, the case of two-photon transition is studied and compared with the case of a resonant transition.
The interaction of two closely spaced resonances in degenerate four-wave-mixing spectra is investigated theoretically. Degenerate four-wave-mixing spectra for the phase-conjugate geometry (counterpropagating pump beams) are calculated by integrating the time-dependent density-matrix equations at numerous grid points along the phase-matching axis and summing the polarization contribution from each of these grid points. Both purely homogeneously broadened resonances and resonances that are both collision and Doppler broadened are considered. The homogeneously broadened results are found to agree with the results of the widely used Abrams–Lind expression for the case of stationary absorbers exposed to a steady-state excitation. We examine the effects of power broadening, pressure broadening, transition dipole moment, and population on the spectra of two neighboring homogeneously broadened resonances. As the laser power is increased, the real parts of the resonance susceptibilities destructively interfere with each other in the region between the lines, resulting in a significant and persistent dip in reflectivity. When Doppler broadening is included in the calculations, similar interference effects are observed, although at first the interference dip begins to disappear until the homogeneous part of the line shapes is broadened significantly by saturation. Using this numerical approach, we obtain good agreement with experimental results of primarily Doppler-broadened neighboring resonances.
The physics of the degenerate four-wave mixing process for resonant transitions between two degenerate energy levels is investigated by direct numerical integration of the time-dependent density matrix equations. The Zeeman structure of the upper and lower energy levels is included in a multistate formulation of the density matrix equations. The inclusion of the Zeeman structure enables the investigation of the degenerate four-wave mixing process for different polarization configurations of the forward pump, backward pump, and probe beams. Saturation curves and lineshapes are calculated for different polarization configurations and for numerous low-J transitions. At low laser intensity, the results of our calculations are in excellent agreement with perturbation theory in terms of the relative intensities of the degenerate four-wave mixing signal for linear polarization configurations. As the laser intensity increases and the resonance starts to saturate, we find in general that the relative degenerate four-wave mixing reflectivity increases for the crossed polarization configurations compared to the parallel polarization configuration because the saturation intensity is higher. However, for some resonance transitions, some of the crossed polarization configurations saturate at lower laser intensities than the parallel polarization configuration, even though the reflectivity for these crossed polarization configurations is much lower than for the parallel polarization configuration in the perturbative intensity limit. This result is explained in terms of the coupling of the various Zeeman states during the degenerate four-wave mixing interaction for specific polarization configurations. The effect of saturation on the resonance line shapes for the different polarization configurations is also investigated. Finally, a limited number of calculations are performed for resonances that are Doppler broadened as well as collision broadened. The effect of saturation on the reflectivity of the crossed polarization configurations compared to the parallel polarization configuration is even more significant for resonances with comparable Doppler and collisional broadening.
We present a combined theoretical and experimental study of the effects of laser polarization on optical coherences produced in two-color, resonant four-wave mixing (TC-RFWM). A time-dependent model incorporating diagrammatic perturbation theory and spherical tensor formalism is used to interpret observations of quantum beats due to molecular hyperfine structure in time-resolved TC-RFWM in nitric oxide. Good agreement is found between the model and the observed time-resolved signals for two distinct excitation schemes and a variety of polarization configurations including both polarization and population gratings. Measured hyperfine energy intervals are reported for the $X{}^{2}{\ensuremath{\Pi}}_{1/2},$ v=0v=0v=0 ground state and the $A{}^{2}{\ensuremath{\Sigma}}^{+},$ v=0v=0v=0 excited state of NO. The experimental results demonstrate that TC-RFWM can be used to perform state-selective, quantum beat spectroscopy in three-level systems by suitably designing three experimental features: the excitation scheme for the matter-field interaction, the time ordering of the laser pulses, and the polarization of the incident laser beams.