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Physical Review A

Multiple resonance-enhanced four-wave-mixing processes in I2

Ilse Aben, Wim Ubachs, Pieternel Felicitas Levelt, Gert van der Zwan, W. Hogervorst

Published Nov 1, 1991• 11 citations
Publisher

Abstract

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 ${\mathit{X}}^{1}$${\mathrm{\ensuremath{\Sigma}}}_{\mathit{g}}^{+}$ and the electronic excited state ${\mathit{B}}^{3}$${\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 ${\mathrm{I}}_{2}$, 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.

DOI
10.1103/physreva.44.5881

Related papers

Saturation effects in coherent anti-Stokes Raman scattering

A. D. Wilson‐Gordon, R. Klimovsky-Barid, Harry Friedmann

Saturation effects in coherent anti-Stokes Raman scattering (CARS) spectroscopy are discussed. The discussion is limited to Raman-resonant CARS (${\ensuremath{\omega}}_{l}\ensuremath{-}{\ensuremath{\omega}}_{s}\ensuremath{\simeq}{\ensuremath{\omega}}_{21}$, where ${\ensuremath{\omega}}_{l,s}$ are the frequencies of the pump fields with powers ${P}_{l,s}$, and ${\ensuremath{\omega}}_{21}$ is the frequency of the Raman transition $|1〉\ensuremath{\rightarrow}|2〉$) with the possible addition of a one-photon resonance (${\ensuremath{\omega}}_{l}\ensuremath{\simeq}{\ensuremath{\omega}}_{31}$ is the frequency of the electronic transition $|1〉\ensuremath{\rightarrow}|3〉$). For these cases we show that the CARS polarization ${\mathcal{p}}_{\mathrm{CARS}}$ is proportional to the off-diagonal density-matrix element ${\ensuremath{\rho}}_{21}$. In order to determine ${\ensuremath{\rho}}_{21}$, we use Laplace transforms to solve the Bloch equations for the effective two-level system $|1〉$ and $|2〉$ when ${\ensuremath{\omega}}_{l}$ is far from resonance, or for the three-level system $|1〉,|2〉$, and $|3〉$, when ${\ensuremath{\omega}}_{l}\ensuremath{\simeq}{\ensuremath{\omega}}_{31}$. The steady-state expression for ${\mathcal{p}}_{\mathrm{CARS}}$ in the former case gives ${\mathcal{p}}_{\mathrm{CARS}}\ensuremath{\propto}{P}_{l}{P}_{s}^{\frac{1}{2}}$ at low powers and ${\mathcal{p}}_{\mathrm{CARS}}\ensuremath{\propto}{P}_{l}^{0}{P}_{s}^{\frac{\ensuremath{-}1}{2}}$ at high powers. In the three-level system, we show that when the pressure is low and at least one field is weak, the slow time dependence of ${\ensuremath{\rho}}_{21}$ must be considered. When one field is strong and the other weak, the CARS spectrum is Stark split. When ${P}_{l}$ is high, for example, ${\mathcal{p}}_{\mathrm{CARS}}\ensuremath{\propto}{P}_{l}^{0}{P}_{s}^{\frac{1}{2}}$ for ${\ensuremath{\omega}}_{s}\ensuremath{\simeq}{\ensuremath{\omega}}_{32}$ and ${\mathcal{p}}_{\mathrm{CARS}}\ensuremath{\propto}{({P}_{l}{P}_{s})}^{\frac{1}{2}}$ when ${\ensuremath{\omega}}_{s}\ensuremath{\simeq}{\ensuremath{\omega}}_{32}\ifmmode\pm\else\textpm\fi{}{V}_{13}$ where ${V}_{13}$ is the one-photon Rabi frequency for the $|1〉\ensuremath{\rightarrow}|3〉$ transition. The Wilcox-Lamb approximation is used to reduce the three-level Bloch equations to rate equations containing one- and two-photon terms. When the fields are so weak that both one- and two-photon terms are small compared to the decay terms, the usual expression for ${\mathcal{p}}_{\mathrm{CARS}}$ is reproduced. If only the direct two-photon processes are important, the effective-two-system results are reproduced. When both fields are intense and nearly resonant, the steady state is rapidly achieved. The results for the case where one field is much stronger than the other are essentially the same as those for one strong and one weak field. When the fields are of comparable strength, ${\mathcal{p}}_{\mathrm{CARS}}\ensuremath{\propto}{P}_{l}^{\frac{1}{2}}{P}_{s}^{0}$ for ${\ensuremath{\omega}}_{l}\ensuremath{\simeq}{\ensuremath{\omega}}_{31}$ and ${\ensuremath{\omega}}_{s}\ensuremath{\simeq}{\ensuremath{\omega}}_{32}$, and the CARS spectrum is split into five components.

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Evolution of four-wave mixing by controlling Raman coherence in a multi-dressed atomic system

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The evolution of enhanced four-wave mixing based on Raman coherence as the probe intensity increases is observed experimentally in an 85 R b atomic vapor system. We propose that for a Doppler-broadened Λ -type level system, the generated anti-Stokes field from one-photon resonance will be suppressed by the dressed-states resonance and can be controlled by the probe intensity, which influences the effective transverse relaxation rate of Raman coherence. When the probe field is not so weak, the double resonance for dressed states will greatly suppress the anti-Stokes emission at the central frequency, and the signal peak will be split. We show that this phenomenon is a result of the destructive polarization interference between atoms with different velocities from the two types of resonance. Based on our model, we present a new, to the best of our knowledge, interpretation of four-wave mixing enhancement in the conventional Λ -type electromagnetically induced transparency system.

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The delayed or time-domain four-wave-mixing experiment is treated in the regime of intense near-resonant pulses. The interaction with the radiation during both pump and probe pulses is considered to all powers of the electric field amplitude. Analytical results are obtained for an effective four-level system. These include the dependence of the coherence amplitudes on the ratio of the pump-field intensities when there is a large vibrational discrepancy between ground and excited electronic states and a general solution for the unitary time development during the probe pulse. For the first time, delayed coherent anti-Stokes Raman scattering is detected from highly dilute (10-ppm) guest molecules. Illustrative examples are presented for the system of pentacene in benzoic acid at low temperature. Vibronic-free induction decay and the effect of field inhomogeneity across the beam profile are found to be essential for understanding the observed intensity and spectral distribution of the signal beam in the region of optimum pulse intensity.

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Time resolved coherent anti-Stokes Raman scattering of I2 isolated in matrix argon: Vibrational dynamics on the ground electronic state

Michael Karavitis, R. Zadoyan, Vartkess Ara Apkarian

Time-resolved, electronically resonant, coherent anti-Stokes Raman scattering is used to prepare and interrogate vibronic coherences of molecular iodine in matrix Ar. Coherences that involve evolution on the excited B(3Π0u) state, first- and third-order coherences, decay in less than one vibrational period (τ<300 fs). In contrast, as many as 200 vibrational periods of motion can be observed for Raman-prepared wave packets consisting of zero-phonon vibrational superpositions on the ground electronic state (second-order coherence). Packets consisting of v=4, 5 and v=3, 4, 5 on the X(1Σg) state decay with a half-life of 10±1 ps at 31 K, allowing a more accurate measure of vibrational level spacings and decoherence time than has been possible in frequency domain. The harmonic frequency of the molecule is reduced by 1.5 cm−1 (0.7%) in the matrix. The lack of recurrence in the excited electronic state ensures that the resonant anti-Stokes scattering arises only from the negative momentum component of the Raman packet. This momentum filter, which should be ubiquitous in condensed media, leads to a signal with deeper modulation than in the gas phase.

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Raman processes

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The traditional Raman process alluded to in Chap. 13 is a three-state sequence of transitions in which radiative excitation (induced by a pump field) is followed by spontaneous emission that produces a final state differing from the initial state [Her50b] [Sho90, § 17.5]. When the final state of the sequence is more energetic than the initial state the resulting emission line (to the red of the pump wavelength) is known as a Stokes spectral line. The difference between the pump frequency and the Stokes frequency, the Raman frequency , defines the excitation energy of the final state relative to the initial state. When, instead, the final state has lower energy than the initial state, as can occur when the initial quantum state is already excited, then the emission is an anti-Stokes line, at a bluer wavelength than the pump field. The overall Raman scattering is a two-photon process. Typically Raman spectroscopy deals with molecules; the two-photon transitions are then between vibrational-rotational states, through electronically excited intermediate states, that are characterized in part by vibrational quantum number v and rotational angular momentum quantum numbers J, M . From any given excited electronic state there are many fluorescing transitions, corresponding to various vibrational and rotational quantum numbers of the final state. The wavelengths of the various Stokes and anti-Stokes lines (i.e. the Raman frequencies) characterize the particular molecular species, and so they have provided a valuable diagnostic tool for spectroscopists.

Cambridge University Press eBooks•Sep 29, 2011•0 citations

Raman-resonant four-wave mixing and energy transfer

Bradley L. Bobbs, Charles Warner

When nonlinear four-wave mixing (FWM) is resonant with a Raman transition, it causes coupling between lightwaves separated by two Raman frequency shifts. This coupling is manifested in effects such as Raman gain suppression, higher-order Stokes generation, and coherent Raman spectroscopies (e.g., CARS). An understanding of these effects, however, has been impeded by a widespread misconception that Raman-resonant FWM does not transfer energy between the lightwaves and the medium. The equations for the photon densities at each light frequency show that, in actuality, two medium excitations are created or destroyed for each photon produced by FWM. This point may be further clarified by consideration of the driving nonlinear polarization and by comparisons with broadband Raman conversion and nonresonant FWM. The Raman gain suppression effect may be understood qualitatively in terms of destructive interference between FWM and the usual stimulated Raman process.

Annual Meeting Optical Society of America•Jan 1, 1987•0 citations
Resonance-enhanced four-wave-mixing processes were studied in molecular iodine by detection of a wave generated at frequency \ensuremathω3{\mathrm{\ensuremath{\omega}}}_{3}\ensuremathω3​=2\ensuremathω1{\mathrm{\ensuremath{\omega}}}_{1}\ensuremathω1​-\ensuremathω2{\mathrm{\ensuremath{\omega}}}_{2}\ensuremathω2​ for a wide range of Raman frequencies (\ensuremathω1{\mathrm{\ensuremath{\omega}}}_{1}\ensuremathω1​-\ensuremathω2{\mathrm{\ensuremath{\omega}}}_{2}\ensuremathω2​). On the basis of known spectroscopic constants of the two electronic states involved, the electronic ground state {\mathit{X}}^{1}$${\mathrm{\ensuremath{\Sigma}}}_{\mathit{g}}^{+} and the electronic excited state {\mathit{B}}^{3}$${\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.
Saturation effects in coherent anti-Stokes Raman scattering (CARS) spectroscopy are discussed. The discussion is limited to Raman-resonant CARS (${\ensuremath{\omega}}_{l}\ensuremath{-}{\ensuremath{\omega}}_{s}\ensuremath{\simeq}{\ensuremath{\omega}}_{21}$, where ${\ensuremath{\omega}}_{l,s}$ are the frequencies of the pump fields with powers Pl,s{P}_{l,s}Pl,s​, and ${\ensuremath{\omega}}_{21}$ is the frequency of the Raman transition $|1〉\ensuremath{\rightarrow}|2〉$) with the possible addition of a one-photon resonance (${\ensuremath{\omega}}_{l}\ensuremath{\simeq}{\ensuremath{\omega}}_{31}$ is the frequency of the electronic transition $|1〉\ensuremath{\rightarrow}|3〉$). For these cases we show that the CARS polarization pCARS{\mathcal{p}}_{\mathrm{CARS}}pCARS​ is proportional to the off-diagonal density-matrix element ${\ensuremath{\rho}}_{21}$. In order to determine ${\ensuremath{\rho}}_{21}$, we use Laplace transforms to solve the Bloch equations for the effective two-level system ∣1〉|1〉∣1〉 and ∣2〉|2〉∣2〉 when ${\ensuremath{\omega}}_{l}$ is far from resonance, or for the three-level system ∣1〉,∣2〉|1〉,|2〉∣1〉,∣2〉, and ∣3〉|3〉∣3〉, when ${\ensuremath{\omega}}_{l}\ensuremath{\simeq}{\ensuremath{\omega}}_{31}$. The steady-state expression for pCARS{\mathcal{p}}_{\mathrm{CARS}}pCARS​ in the former case gives ${\mathcal{p}}_{\mathrm{CARS}}\ensuremath{\propto}{P}_{l}{P}_{s}^{\frac{1}{2}}$ at low powers and ${\mathcal{p}}_{\mathrm{CARS}}\ensuremath{\propto}{P}_{l}^{0}{P}_{s}^{\frac{\ensuremath{-}1}{2}}$ at high powers. In the three-level system, we show that when the pressure is low and at least one field is weak, the slow time dependence of ${\ensuremath{\rho}}_{21}$ must be considered. When one field is strong and the other weak, the CARS spectrum is Stark split. When Pl{P}_{l}Pl​ is high, for example, ${\mathcal{p}}_{\mathrm{CARS}}\ensuremath{\propto}{P}_{l}^{0}{P}_{s}^{\frac{1}{2}}$ for ${\ensuremath{\omega}}_{s}\ensuremath{\simeq}{\ensuremath{\omega}}_{32}$ and ${\mathcal{p}}_{\mathrm{CARS}}\ensuremath{\propto}{({P}_{l}{P}_{s})}^{\frac{1}{2}}$ when ${\ensuremath{\omega}}_{s}\ensuremath{\simeq}{\ensuremath{\omega}}_{32}\ifmmode\pm\else\textpm\fi{}{V}_{13}$ where V13{V}_{13}V13​ is the one-photon Rabi frequency for the $|1〉\ensuremath{\rightarrow}|3〉$ transition. The Wilcox-Lamb approximation is used to reduce the three-level Bloch equations to rate equations containing one- and two-photon terms. When the fields are so weak that both one- and two-photon terms are small compared to the decay terms, the usual expression for pCARS{\mathcal{p}}_{\mathrm{CARS}}pCARS​ is reproduced. If only the direct two-photon processes are important, the effective-two-system results are reproduced. When both fields are intense and nearly resonant, the steady state is rapidly achieved. The results for the case where one field is much stronger than the other are essentially the same as those for one strong and one weak field. When the fields are of comparable strength, ${\mathcal{p}}_{\mathrm{CARS}}\ensuremath{\propto}{P}_{l}^{\frac{1}{2}}{P}_{s}^{0}$ for ${\ensuremath{\omega}}_{l}\ensuremath{\simeq}{\ensuremath{\omega}}_{31}$ and ${\ensuremath{\omega}}_{s}\ensuremath{\simeq}{\ensuremath{\omega}}_{32}$, and the CARS spectrum is split into five components.
The evolution of enhanced four-wave mixing based on Raman coherence as the probe intensity increases is observed experimentally in an 85 R b atomic vapor system. We propose that for a Doppler-broadened Λ -type level system, the generated anti-Stokes field from one-photon resonance will be suppressed by the dressed-states resonance and can be controlled by the probe intensity, which influences the effective transverse relaxation rate of Raman coherence. When the probe field is not so weak, the double resonance for dressed states will greatly suppress the anti-Stokes emission at the central frequency, and the signal peak will be split. We show that this phenomenon is a result of the destructive polarization interference between atoms with different velocities from the two types of resonance. Based on our model, we present a new, to the best of our knowledge, interpretation of four-wave mixing enhancement in the conventional Λ -type electromagnetically induced transparency system.
The delayed or time-domain four-wave-mixing experiment is treated in the regime of intense near-resonant pulses. The interaction with the radiation during both pump and probe pulses is considered to all powers of the electric field amplitude. Analytical results are obtained for an effective four-level system. These include the dependence of the coherence amplitudes on the ratio of the pump-field intensities when there is a large vibrational discrepancy between ground and excited electronic states and a general solution for the unitary time development during the probe pulse. For the first time, delayed coherent anti-Stokes Raman scattering is detected from highly dilute (10-ppm) guest molecules. Illustrative examples are presented for the system of pentacene in benzoic acid at low temperature. Vibronic-free induction decay and the effect of field inhomogeneity across the beam profile are found to be essential for understanding the observed intensity and spectral distribution of the signal beam in the region of optimum pulse intensity.
Time-resolved, electronically resonant, coherent anti-Stokes Raman scattering is used to prepare and interrogate vibronic coherences of molecular iodine in matrix Ar. Coherences that involve evolution on the excited B(3Π0u) state, first- and third-order coherences, decay in less than one vibrational period (τ<300 fs). In contrast, as many as 200 vibrational periods of motion can be observed for Raman-prepared wave packets consisting of zero-phonon vibrational superpositions on the ground electronic state (second-order coherence). Packets consisting of v=4, 5 and v=3, 4, 5 on the X(1Σg) state decay with a half-life of 10±1 ps at 31 K, allowing a more accurate measure of vibrational level spacings and decoherence time than has been possible in frequency domain. The harmonic frequency of the molecule is reduced by 1.5 cm−1 (0.7%) in the matrix. The lack of recurrence in the excited electronic state ensures that the resonant anti-Stokes scattering arises only from the negative momentum component of the Raman packet. This momentum filter, which should be ubiquitous in condensed media, leads to a signal with deeper modulation than in the gas phase.
The traditional Raman process alluded to in Chap. 13 is a three-state sequence of transitions in which radiative excitation (induced by a pump field) is followed by spontaneous emission that produces a final state differing from the initial state [Her50b] [Sho90, § 17.5]. When the final state of the sequence is more energetic than the initial state the resulting emission line (to the red of the pump wavelength) is known as a Stokes spectral line. The difference between the pump frequency and the Stokes frequency, the Raman frequency , defines the excitation energy of the final state relative to the initial state. When, instead, the final state has lower energy than the initial state, as can occur when the initial quantum state is already excited, then the emission is an anti-Stokes line, at a bluer wavelength than the pump field. The overall Raman scattering is a two-photon process. Typically Raman spectroscopy deals with molecules; the two-photon transitions are then between vibrational-rotational states, through electronically excited intermediate states, that are characterized in part by vibrational quantum number v and rotational angular momentum quantum numbers J, M . From any given excited electronic state there are many fluorescing transitions, corresponding to various vibrational and rotational quantum numbers of the final state. The wavelengths of the various Stokes and anti-Stokes lines (i.e. the Raman frequencies) characterize the particular molecular species, and so they have provided a valuable diagnostic tool for spectroscopists.
When nonlinear four-wave mixing (FWM) is resonant with a Raman transition, it causes coupling between lightwaves separated by two Raman frequency shifts. This coupling is manifested in effects such as Raman gain suppression, higher-order Stokes generation, and coherent Raman spectroscopies (e.g., CARS). An understanding of these effects, however, has been impeded by a widespread misconception that Raman-resonant FWM does not transfer energy between the lightwaves and the medium. The equations for the photon densities at each light frequency show that, in actuality, two medium excitations are created or destroyed for each photon produced by FWM. This point may be further clarified by consideration of the driving nonlinear polarization and by comparisons with broadband Raman conversion and nonresonant FWM. The Raman gain suppression effect may be understood qualitatively in terms of destructive interference between FWM and the usual stimulated Raman process.