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Physical Review A

Effects of spectral cross relaxation and collisional dephasing on the absorption of light by organic-dye solutions

David W. Vahey, Amnon Yariv

Published Nov 1, 1974• 25 citations
Publisher

Abstract

The nonlinear absorption of intense laser light is used to obtain information about the times required for spectral cross relaxation (${T}_{3}$) and collisional dephasing (${T}_{2}$) in organic-dye solutions. The steady-state transmission experiment of Huff and DeShazer indicates that ${T}_{2}$ is in the range 0.1-1 psec for cryptocyanine-methanol, while ${T}_{3}$ is in the range 1-10 psec. For DDI-glycerin (1, 1\ensuremath{'}-diethyl-2, 2\ensuremath{'}-dicarbocyanine-iodide-glycerin), measurements of ultrashort-pulse transmission as a function of incident energy and optical polarization suggest a value ${T}_{2}=0.4\ifmmode\pm\else\textpm\fi{}0.2$ psec for that system. These results are based on a new set of rate equations for absorption by inhomogeneously broadened organic-dye solutions. The equations display the effects of spectral cross relaxation and are not limited to light intensities less than those at which coherent interaction effects become important. For the case of ultrashort incident pulses, the equations are extended to account for excited-state absorption, the dependence of absorption on optical polarization, and the existence of a multiplicity of interacting excited states.

DOI
10.1103/physreva.10.1578

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Study of ultra-fast relaxation processes by resonant Rayleigh-type optical mixing. I. Theory

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A comprehensive theory of a new type of three-wave optical mixing (${\ensuremath{\omega}}_{3}=2{\ensuremath{\omega}}_{1}\ensuremath{-}{\ensuremath{\omega}}_{2}$) is given for the spectroscopic purpose of determining ultra-short relaxation times (picoseconds or less) associated with excited states of condensed matter. Two incident light frequencies ${\ensuremath{\omega}}_{1}$ and ${\ensuremath{\omega}}_{2}$ are chosen so that they are both resonant with an inhomogeneously broadened optical transition and $|{\ensuremath{\omega}}_{2}\ensuremath{-}{\ensuremath{\omega}}_{1}|$ is in the vicinity of inverse relaxation times. The theory is developed for its major application to broad-band electronic transitions. The nonlinear susceptibility ${\ensuremath{\chi}}_{R}^{(3)}$ for this process (named resonant Rayleigh-type mixing) is calculated by a density-matrix formalism with the model of a two-level atomic system incorporating the distribution of resonance frequencies and longitudinal (${T}_{1}$) and transverse (${T}_{2}$) relaxation times. The result shows a frequency characteristic depending only on ${T}_{1}$, ${T}_{2}$, and ${\ensuremath{\omega}}_{2}\ensuremath{-}{\ensuremath{\omega}}_{1}$ in a broad-band limit, which serves for the determination of ${T}_{1}$ and ${T}_{2}$ in the frequency domain. The analysis is further extended to include the various effects as follows. The calculation of the saturation effect in the lowest order reveals that it modifies the shape of the frequency characteristic of the nonlinear susceptibility but does not affect seriously the determination of relaxation times. The effect of spectral cross relaxation within the inhomogeneous broadening is incorporated by a generalized density-matrix formalism. The resultant ${\ensuremath{\chi}}_{R}^{(3)}$ consists of two terms. The dominant term is the same as before except that ${T}_{1}$ is replaced by a combined relaxation time ${T}_{1}^{\ensuremath{'}}={({T}_{1}^{\ensuremath{-}1}+{T}_{3}^{\ensuremath{-}1})}^{\ensuremath{-}1}$, where ${T}_{3}$ is the cross-relaxation time. The other term originates from the inverse spectral-diffusion process. The effect of other energy levels located between the two levels under study is analyzed with a simple three-level model. The resultant ${\ensuremath{\chi}}_{R}^{(3)}$ also consists of two terms, the dominant one being the same as that for the two-level model except that ${T}_{1}^{\ensuremath{-}1}$ should be interpreted as total population decay rate of the upper level. For accurate derivation of ${\ensuremath{\chi}}_{R}^{(3)}$ from experiment, we discuss the interference between the resonant and nonresonant susceptibility terms and the light-wave propagation effect. Finally, the close analogy between this type of optical mixing and the photon echo is discussed.

Physical Review A•Jan 1, 1978•146 citations
The nonlinear absorption of intense laser light is used to obtain information about the times required for spectral cross relaxation (T3{T}_{3}T3​) and collisional dephasing (T2{T}_{2}T2​) in organic-dye solutions. The steady-state transmission experiment of Huff and DeShazer indicates that T2{T}_{2}T2​ is in the range 0.1-1 psec for cryptocyanine-methanol, while T3{T}_{3}T3​ is in the range 1-10 psec. For DDI-glycerin (1, 1\ensuremath{'}-diethyl-2, 2\ensuremath{'}-dicarbocyanine-iodide-glycerin), measurements of ultrashort-pulse transmission as a function of incident energy and optical polarization suggest a value T2=0.4\ifmmode±\else\textpm\fi0.2{T}_{2}=0.4\ifmmode\pm\else\textpm\fi{}0.2T2​=0.4\ifmmode±\else\textpm\fi0.2 psec for that system. These results are based on a new set of rate equations for absorption by inhomogeneously broadened organic-dye solutions. The equations display the effects of spectral cross relaxation and are not limited to light intensities less than those at which coherent interaction effects become important. For the case of ultrashort incident pulses, the equations are extended to account for excited-state absorption, the dependence of absorption on optical polarization, and the existence of a multiplicity of interacting excited states.
Intracavity measurements of degenerate four-wave mixing have been performed with femtosecond laser pulses to determine the phase relaxation time of the absorbing dye, diethyloxadicarbocyanine iodide (DODCI), in ethylene glycol. The influence of various broadening mechanisms (homogeneous, inhomogeneous, cross relaxation) and pulse parameters (frequency, pulse shape, phase modulation, and energy) is investigated. The dependence of the signal energy on probe delay is found to be strongly affected by the phase modulation of the pulse; hence determination of the phase relaxation time requires accurate diagnostics of the laser source. The rise of the signal provides an upper limit of 10 fsec for the phase relaxation time of the main absorption band of the dye DODCI. The decay of the signal is found to be related to the phase relaxation time of a two-photon absorption in the same dye. Accurate fitting of the data with the theory yields 50 ± 10 fsec for the dephasing time of the two-photon excitation in DODCI.
We model the response of a molecular medium to an intense pulse of light. We begin by reducing the Maxwell-Bloch equations for a multilevel molecular model system to a set of simplified propagation equations. We thereby obtain expressions for the frequency, temperature, and field intensity dependence of the ground and excited electronic state absorption cross sections, and the emission cross section, as well as the temperature dependence of the decay rate of the excited state. The phenomena of saturation, excited-state absorption, optically induced dichroism and birefringence, vibronic energy transfer, induced molecular alignment, decay of the matter-field interaction energy, and rotational diffusion are considered within the general formalism for describing the propagation of an intense pulse of light in the molecular medium. We find that the cross section for absorption (emission) of a photon is not identical to the cross section for molecular excitation (deexcitation) via the matter-field interaction. This is explained in terms of an energy balance between the matter, radiation, the matter-radiation interaction, and the bath responsible for the broadening of the molecular transitions. The inequality of these cross sections determines whether the system supplies or draws energy from the thermal bath. The rise in the vibrational temperature of molecules subjected to an intense short pulse is shown to significantly affect propagation of the pulse. The alignment of the molecular system resulting from saturation of the ground-state transition by a polarized laser pulse and the effects of the orientational dependence of the matter-field interaction including effects of rotational diffusion are determined. As an example of the formalism we consider pulse propagation in a Rhodamine 6G solution. We fit the parameters of the model to determine the frequency dependence of the ground- and excited-state absorption cross sections and the emission cross section and compare with experimentally determined cross sections. We describe the propagation of a pulse whose temporal width is comparable to the rotational diffusion time of the molecules in the dye solution and calculate polarization pulse probe signals.
The effect of cross relaxation on the third-order nonlinear susceptibility, χ(3) (−ω2, ω1, −ω1, ω2), describing polarization spectroscopy experiments is calculated with a density matrix formalism. By using a model of inhomogeneously broadened two-level systems with a reservoir state and allowing for cross relaxation within the inhomogeneous line, we find that both the ratio of the maximum signal at ω1 = ω2 to the background and the broad component resembling the absorption band depend on the cross-relaxation rate. Also, one additional component with a characteristic width of the order of the spectral-diffusion rate Γ3 is expected to appear, depending on the value of this parameter. Furthermore, we estimate from published data [ Phys. Rev. A17, 1439 ( 1978)] the product of the transverse-relaxation time times the cross-relaxation time T2T3 to be 0.014 psec2 for cyanine dye and 0.0009 psec2 for malachite green oxalate.
A novel transient three-pulse scattering technique for measuring ultrafast dephasing times in condensed matter is analyzed using a perturbative solution of the density matrix equation. The advantages of this technique include subpulsewidth resolution, a clear distinction between homogeneous and inhomogeneous broadening, and sensitivity to spectral cross-relaxation. Its application to the case of a multilevel resonance is also considered. We report results of femtosecond dephasing experiments with dye molecules in liquids and in a polymer host. The dephasing time is determined to be less than 20 fsec for dyes in solution at room temperature. At low temperatures in polymers, a transition from homogeneous to inhomogeneous broadening has been observed and studied as a function of temperature.
A comprehensive theory of a new type of three-wave optical mixing (${\ensuremath{\omega}}_{3}=2{\ensuremath{\omega}}_{1}\ensuremath{-}{\ensuremath{\omega}}_{2}$) is given for the spectroscopic purpose of determining ultra-short relaxation times (picoseconds or less) associated with excited states of condensed matter. Two incident light frequencies ${\ensuremath{\omega}}_{1}$ and ${\ensuremath{\omega}}_{2}$ are chosen so that they are both resonant with an inhomogeneously broadened optical transition and $|{\ensuremath{\omega}}_{2}\ensuremath{-}{\ensuremath{\omega}}_{1}|$ is in the vicinity of inverse relaxation times. The theory is developed for its major application to broad-band electronic transitions. The nonlinear susceptibility ${\ensuremath{\chi}}_{R}^{(3)}$ for this process (named resonant Rayleigh-type mixing) is calculated by a density-matrix formalism with the model of a two-level atomic system incorporating the distribution of resonance frequencies and longitudinal (T1{T}_{1}T1​) and transverse (T2{T}_{2}T2​) relaxation times. The result shows a frequency characteristic depending only on T1{T}_{1}T1​, T2{T}_{2}T2​, and ${\ensuremath{\omega}}_{2}\ensuremath{-}{\ensuremath{\omega}}_{1}$ in a broad-band limit, which serves for the determination of T1{T}_{1}T1​ and T2{T}_{2}T2​ in the frequency domain. The analysis is further extended to include the various effects as follows. The calculation of the saturation effect in the lowest order reveals that it modifies the shape of the frequency characteristic of the nonlinear susceptibility but does not affect seriously the determination of relaxation times. The effect of spectral cross relaxation within the inhomogeneous broadening is incorporated by a generalized density-matrix formalism. The resultant ${\ensuremath{\chi}}_{R}^{(3)}$ consists of two terms. The dominant term is the same as before except that T1{T}_{1}T1​ is replaced by a combined relaxation time ${T}_{1}^{\ensuremath{'}}={({T}_{1}^{\ensuremath{-}1}+{T}_{3}^{\ensuremath{-}1})}^{\ensuremath{-}1}$, where T3{T}_{3}T3​ is the cross-relaxation time. The other term originates from the inverse spectral-diffusion process. The effect of other energy levels located between the two levels under study is analyzed with a simple three-level model. The resultant ${\ensuremath{\chi}}_{R}^{(3)}$ also consists of two terms, the dominant one being the same as that for the two-level model except that ${T}_{1}^{\ensuremath{-}1}$ should be interpreted as total population decay rate of the upper level. For accurate derivation of ${\ensuremath{\chi}}_{R}^{(3)}$ from experiment, we discuss the interference between the resonant and nonresonant susceptibility terms and the light-wave propagation effect. Finally, the close analogy between this type of optical mixing and the photon echo is discussed.