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Journal of the Physical Society of Japan

Two-Photon Resonant Rayleigh-Type Optical Mixing and Relaxation

Tatsuo Yajima

Published Mar 15, 1978• 5 citations
Publisher

Abstract

The dispersion characteristics of the nonlinear susceptibility χ ( n ) ( n =3, 5, 7) describing a new type of optical mixing have been calculated for their application to the study of ultra-fast relaxation dynamics in resonant media. The process is that two incident light waves at frequencies ω 1 and ω 2 produce new light waves at ω 1 - m ε and ω 1 + n ε ( ε =ω 2 -ω 1 ; m , n =1, 2, 3, …) under the conditions of two-photon resonance (2ω 1 , 2ω 2 ≃optical transition frequency) and Rayleigh-type resonance (|ω 1 -ω 2 |≃relaxation rate). Results of analysis based on an inhomogeneously broadened three-level model revealed that Rayleigh-type dispersions involving the longitudinal and transverse relaxation rates appear in χ ( n ) ( n ≧5) and χ ( n ) ( n ≧7), respectively. Their analogy with two-photon resonant photon echo and various precautions in applying them to actual situations are also discussed.

DOI
10.1143/jpsj.44.948

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The dispersion characteristics of the nonlinear susceptibility χ ( n ) ( n =3, 5, 7) describing a new type of optical mixing have been calculated for their application to the study of ultra-fast relaxation dynamics in resonant media. The process is that two incident light waves at frequencies ω 1 and ω 2 produce new light waves at ω 1 - m ε and ω 1 + n ε ( ε =ω 2 -ω 1 ; m , n =1, 2, 3, …) under the conditions of two-photon resonance (2ω 1 , 2ω 2 ≃optical transition frequency) and Rayleigh-type resonance (|ω 1 -ω 2 |≃relaxation rate). Results of analysis based on an inhomogeneously broadened three-level model revealed that Rayleigh-type dispersions involving the longitudinal and transverse relaxation rates appear in χ ( n ) ( n ≧5) and χ ( n ) ( n ≧7), respectively. Their analogy with two-photon resonant photon echo and various precautions in applying them to actual situations are also discussed.
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.
Two-photon transitions have provided the basis for numerous useful nonlinear interactions with radiation. In this paper we calculate the absorption and coupling coefficients for one-, two-, and three-wave mixing in two-photon homogeneously broadened media. The resulting field equations form the basis for understanding saturation spectroscopy, phase conjugation,1 and single-mode instabilities2 in lasers and optical bistability in two-photon media. We assume that the wave frequencies are not resonant with any intermediate state j and that the probe and conjugate waves cannot saturate the two-photon transition. This allows us to use the two-photon two-level model and to transfer much of our knowledge of the one-photon two-level systems to the two-photon case. Two major differences occur between the two models. First, dynamic Stark shifts of the active levels play an important role in generating asymmetric spectra and nonzero dispersion (≃ω s T1) at line center. Second, high pump intensities do not bleach the absorption coefficient at the Rabi sidebands.
The resonant nonlinear-optical phenomenon in condensed matter due to two-frequency excitation is investigated by a first-principles theory without using the phenomenological relaxation-time description. A new generalized nonlinear susceptibility is introduced in order to describe the non-Markovian process in the condensed phase associated with the resonant four-photon parametric effect. It is shown that the present non-Markovian version of the theory of the resonant Rayleigh-type optical mixing (RROM) naturally includes the resonant coherent anti-Stokes Raman scattering (RCARS) as the memory effect of the reservoir. That is, the two phenomena PROM and RCARS are unified from the microscopic viewpoint. The general expressions for nonlinear susceptibility for RROM and RCARS are obtained by taking into consideration the frequency dependence of the transverse relaxation time T2{T}_{2}T2​. These provide us the general guide for the study of several relaxation phenomena in condensed materials by means of resonant coherent nonlinear-optical methods.
The dynamical properties of a third-order nonlinear optical process, in which two incident light pulses with wave vectors k 1 and k 2 generate a light wave with 2 k 2 - k 1 in resonant media, are discussed with particular attention to the effect of transverse relaxation time T 2 . It is shown that the correlation trace (output light energy vs incident pulse separation) for this process provides a general means of obtaining information on ultrashort T 2 for both the homogeneously and inhomogeneously broadened transitions, and for both t p (pulse width)< T 2 and t p > T 2 . An experimental demonstration for t p > T 2 is given by using a dye molecule and a picosecond light source.
We calculate the third-order nonlinear absorption and dispersion (self-phase-modulation and cross-phase modulation) of a homogeneously broadened two-level molecular system with permanent dipole moments interacting with (1) an optical field of frequency \ensuremath{\omega}, and (2) two optical fields of frequency ${\mathrm{\ensuremath{\omega}}}_{1}$ and ${\mathrm{\ensuremath{\omega}}}_{2}$, i.e., the susceptibilities ${\mathrm{\ensuremath{\chi}}}^{(3)}$(-\ensuremath{\omega};\ensuremath{\omega},-\ensuremath{\omega},\ensuremath{\omega}) and ${\mathrm{\ensuremath{\chi}}}^{(3)}$(-${\mathrm{\ensuremath{\omega}}}_{1}$;${\mathrm{\ensuremath{\omega}}}_{1}$,-${\mathrm{\ensuremath{\omega}}}_{2}$,${\mathrm{\ensuremath{\omega}}}_{2}$). Sharp features in the nonlinear absorption and dispersion appear whenever one- or two-photon resonance conditions are met. Only features corresponding to one-photon resonance arise if the difference between the permanent dipole moments of the levels, \ensuremath{\Delta}\ensuremath{\mu}, vanishes. The ${\mathrm{\ensuremath{\chi}}}^{(3)}$(-\ensuremath{\omega};\ensuremath{\omega},-\ensuremath{\omega},\ensuremath{\omega}) spectrum has two features, one at frequency ${\mathrm{\ensuremath{\omega}}}_{\mathit{b}\mathit{a}}$ dominated by contributions proportional to ${\mathrm{\ensuremath{\mu}}}_{\mathit{a}\mathit{b}}^{4}$, where ${\mathrm{\ensuremath{\mu}}}_{\mathit{a}\mathit{b}}$ is the transition dipole moment of the two levels with transition frequency ${\mathrm{\ensuremath{\omega}}}_{\mathit{b}\mathit{a}}$, and the other at frequency ${\mathrm{\ensuremath{\omega}}}_{\mathit{b}\mathit{a}}$/2 dominated by contributions proportional to (\ensuremath{\Delta}\ensuremath{\mu})2{)}^{2})2${\mathrm{\ensuremath{\mu}}}_{\mathit{a}\mathit{b}}^{2}$. Sharp features in ${\mathrm{\ensuremath{\chi}}}^{(3)}$(-${\mathrm{\ensuremath{\omega}}}_{1}$;${\mathrm{\ensuremath{\omega}}}_{2}$,-${\mathrm{\ensuremath{\omega}}}_{2}$,${\mathrm{\ensuremath{\omega}}}_{1}$) appear when ${\mathrm{\ensuremath{\omega}}}_{2}$ is near ${\mathrm{\ensuremath{\omega}}}_{\mathit{b}\mathit{a}}$,${\mathrm{\ensuremath{\omega}}}_{1}$,${\mathrm{\ensuremath{\omega}}}_{1}$+${\mathrm{\ensuremath{\omega}}}_{\mathit{b}\mathit{a}}$, and \ensuremath{\Vert}${\mathrm{\ensuremath{\omega}}}_{1}$-${\mathrm{\ensuremath{\omega}}}_{\mathit{b}\mathit{a}}$\ensuremath{\Vert}.
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.