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International Symposium on Molecular Spectroscopy (ISMS)

THEORETICAL ANALYSIS OF THE RESONANCE FOUR-WAVE MIXING AMPLITUDES: A FULLY NON-DEGENERATE CASE.

А. П. Коузов

Published Jun 26, 2015• 0 citations
Publisher

Abstract

Degenerate (one-color) and two-color variants of the resonant four-wave mixing (RFWM) have developed into a sensitive and nonintrusive spectroscopic tool to study molecules in different gaseous environments. Yet, the fully non-degenerate (four-color, 4C) RFWM was scrutinized and implemented only for the Coherent AntiStokes Raman Scattering (CARS) excitation schemefootnote{B. Attal-Tr'{e}tout, P. Berlemont, and J. P. Taran, Mol. Phys. textbf{70}, 1 (1990).},footnote{J.P. Kuehner, S.V. Naik, W.D. Kulatilaka, N. Chai, N.M. Laurendeau, R.P. Lucht, M.O. Scully, S. Roy, A.K. Patnaik, and J.R. Gord, J. Chem. Phys. textbf{128}, 174308 (2008).} . Here, by using the line-space approachfootnote{A. Kouzov and P. Radi, J. Chem. Phys. textbf{140}, 194302 (2014).}, we analyze other 4C-RFWM schemes potentially interesting for the efficient up- and down-frequency conversion as well as for studies of molecular states. Decoupled expressions of the 4C-RFWM amplitudes are derived which allows to predict their polarization dependence.

DOI
10.15278/isms.2015.fe08

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Degenerate (one-color) and two-color variants of the resonant four-wave mixing (RFWM) have developed into a sensitive and nonintrusive spectroscopic tool to study molecules in different gaseous environments. Yet, the fully non-degenerate (four-color, 4C) RFWM was scrutinized and implemented only for the Coherent AntiStokes Raman Scattering (CARS) excitation schemefootnote{B. Attal-Tr'{e}tout, P. Berlemont, and J. P. Taran, Mol. Phys. textbf{70}, 1 (1990).},footnote{J.P. Kuehner, S.V. Naik, W.D. Kulatilaka, N. Chai, N.M. Laurendeau, R.P. Lucht, M.O. Scully, S. Roy, A.K. Patnaik, and J.R. Gord, J. Chem. Phys. textbf{128}, 174308 (2008).} . Here, by using the line-space approachfootnote{A. Kouzov and P. Radi, J. Chem. Phys. textbf{140}, 194302 (2014).}, we analyze other 4C-RFWM schemes potentially interesting for the efficient up- and down-frequency conversion as well as for studies of molecular states. Decoupled expressions of the 4C-RFWM amplitudes are derived which allows to predict their polarization dependence.
Resonance-enhanced four-wave-mixing processes were studied in molecular iodine by detection of a wave generated at frequency ${\mathrm{\ensuremath{\omega}}}_{3}$=2${\mathrm{\ensuremath{\omega}}}_{1}$-${\mathrm{\ensuremath{\omega}}}_{2}$ for a wide range of Raman frequencies (${\mathrm{\ensuremath{\omega}}}_{1}$-${\mathrm{\ensuremath{\omega}}}_{2}$). On the basis of known spectroscopic constants of the two electronic states involved, the electronic ground state X1{\mathit{X}}^{1}X1${\mathrm{\ensuremath{\Sigma}}}_{\mathit{g}}^{+}$ and the electronic excited state B3{\mathit{B}}^{3}B3${\mathrm{\ensuremath{\Pi}}}_{\mathit{u}0}^{+}$, an interpretation of the spectral features in terms of three essentially different four-wave-mixing processes is given. Resonance enhancement through bound states with discrete energy levels and the enhancement effect by dissociative continuum states are considered. In addition to resonance-enhanced coherent anti-Stokes Raman-scattering (CARS) processes, in which the Raman resonances are in the electronic ground state, two resonant schemes are identified in I2{\mathrm{I}}_{2}I2​, in which Raman resonances in an excited state are probed: excited-state parametric and nonparametric CARS. The latter process is an example of electronic population transfer in a four-wave-mixing cycle.
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.
The recent evolution toward resonant conditions of the two-color resonant four-wave-mixing (TC-RFWM) spectroscopy has been dictated by the high sensitivity required experimentally. While some models have been used in different contexts such as light pressure forces in strong polychromatic fields, magnetically assisted Sisyphus effect, or multiphoton resonances in \ensuremath{\Lambda} atoms, the existence in molecules of additional processes such as nonradiative transitions and rotational or vibrational dephasings requires the extension of previous models. For this reason, we give here a general description of the internal dynamics for a molecule undergoing two strong grating beams, acting either on two different transitions sharing a common level or on the same transition, and one weak probe beam to reproduce the ground-state grating and stimulated emission pumping configurations of TC-RFWM spectroscopy. By combining high spectral resolution and strong grating beams, we show that the TC-RFWM spectrum is very sensitive to the transition constants, dephasing constants, as well as to the transverse velocity of the molecules in the jet. The last case corresponding to a bichromatic field acting on a single transition is used to explain the origin of the line-shape asymmetry observed experimentally on jet-cooled molecules.
We have investigated the dependence of degenerate four-wave mixing (DFWM) spectra of gas-phase I2I_{2}I2​ on laser intensity, molecular transition moments and molecular concentration. The experimental results are interpreted by using a weak-field theory of DFWM derived with a density-matrix method. For comparison, the direct absorption spectra of I2I_{2}I2​ are also presented. We have demonstrated, in our observed DFWM spectra of I2I_{2}I2​, that the spectral resolution in DFWM can be better than that of linear optical spectroscopy. The outcome has also been elucidated by the theoretical result.
An analytical expression for the degenerate four-wave mixing signal produced in a medium composed of two-level atoms by pump and probe lasers of arbitrary intensities and parallel linear polarization, suffering negligible absorption and depletion, has been derived. The Abrams–Lind approach [Opt. Lett.2, 94 and 3, 205 (1978)] for deriving the phase-matched polarization has been extended to find all the other higher-order polarization terms contributing to the signal. The signal field amplitude is expressed as an infinite series whose convergence rate depends on the intensity of the input fields and their detuning from resonance. The solution is successfully verified by comparison with a full numerical nonperturbative calculation and is found to be equivalent to the Abrams–Lind result in the limit of weak probe intensity. The result is used to calculate the saturation effects on spectral line shape that are relevant to simulation of molecular spectra obtained by degenerate four-wave mixing with saturating pump and probe fields. The result is extended to treat atomic motion in the case of forward geometry of the input beams.