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Journal of the Optical Society of America

Spectral-diffusion effects in polarization spectroscopy

F. García–Golding

Published Jan 1, 1983• 10 citations
Publisher

Abstract

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.

DOI
10.1364/josa.73.000059

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Physical Review A•Nov 1, 1974•25 citations
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.
We have incorporated spectral diffusion in a system of three quantum states immersed in a thermal bath for the study of optical responses in four-wave mixing spectroscopy, using the density matrix formalism and the development of quantum relaxation theory. The primary objective of this work is to establish a generalized theoretical framework that evaluates how internal relaxation within all three quantum states—including a non-resonant third state—modifies the macroscopic nonlinear polarization of the medium. By introducing specific scaling parameters, we aim to isolate and quantify the influence of spectral diffusion in the third state, a feature typically omitted in existing literature. The results found allow giving explanations about the parametric amplification, as well as the possibility of generation of sensors in regions close to the degeneracy of optical frequencies. In our study we have been able to recognize previously published results for the optical susceptibility responses of absorption and scattering. Our model, unlike those already presented in this topic, assumes all states of the quantum system with internal relaxations. We found that the inclusion of spectral diffusion in the third state significantly impacts the system’s kinetic behavior, specifically eliminating “hole burning” effects in dispersion and parametric amplification in nonlinear absorption. These findings suggest that interband and intraband kinetic constants must be considered simultaneously to accurately model the ultra-fast relaxation times in complex molecular systems like organic dyes. The development of the problem is subject to rotating wave considerations in the selection of the Fourier component terms associated with both the coherences induced in the system and those coming from the canonical populations of the states. In addition, it involves the development of perturbation theory in the amplitudes of the incident electromagnetic fields associated with this nonlinear FWM type technique. Our treatments have been performed for frequency differences in the absorption range of organic dyes, without any restriction in the cases of near-zero tunings. Our proposal for the third state with spectral diffusion effects completely eliminates both the “hole burning” created in the dispersion and the parametric amplification generated in nonlinear absorption, when compared with results with the state outside resonance treated without spectral diffusion effects within the band. This highlights the importance of interband kinetic constants in comparison with those within the band.
Effects of spectral diffusion on the third-order nonlinear susceptibility for two- and three-level quantum systems, immersed in a thermal reservoir, are evaluated with a four-wave mixing (FWM) signal through the density matrix formalism. For this, inhomogeneously broadened two- and three-level quantum systems are used. In these models, the distribution of natural frequencies follows a Lorentzian function. The absorptive and dispersive nonlinear optical responses are determined for the considered quantum models. The results presented in this work show potential applications in the development of optical switches and provide a potential method to measure spectral diffusion relaxation times in three-level quantum systems.
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.
A nonperturbative analysis (in one of the fields) of the Rayleigh-type optical-mixing signal in an absorbing medium shows that it is more sensible to saturation than the effect that could be expected from ``absorption saturation arguments.'' We also report the observation of such high-order effects in the organic dye malachite green. We have found that an effective two-level system model with a longitudinal relaxation time of 1.1\ifmmode\pm\else\textpm\fi{}0.2 ps, a transverse relaxation time of 0.13\ifmmode\pm\else\textpm\fi{}0.05 ps, and a transition moment of 1.2 D account for the deviation from the third-order regime. However, the frequency region of the data that follows the predicted spectral deformation is power dependent and it shrinks as power increases.
A detailed theoretical analysis is given for the ultrafast relaxation study using polarization spectroscopy in the frequency domain to clarify the effect of cross relaxation on the measured results. It is shown that the calculated nonlinear spectral response function reflects the inhomogeneous width rather than the homogeneous one if the rate of cross relaxation of the transition becomes so singnificant. This prevents us from precise determination of phase relaxation Time T 2 of the transition by the stationary polarization spectroscopy. It is further shown that the cross relaxation effect onto the measured spectral response function can effectively be diminished by introducing a partially transient condition. An experiment using picosecond light pulses is presented to demonstrate this fact.
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 ${T}_{2}=0.4\ifmmode\pm\else\textpm\fi{}0.2$ 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.