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Physical review. B, Condensed matter

Non-Markovian theory of resonant nonlinear-optical phenomena in the frequency domain

Masaki Aihara

Published May 15, 1983• 6 citations
Publisher

Abstract

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 ${T}_{2}$. These provide us the general guide for the study of several relaxation phenomena in condensed materials by means of resonant coherent nonlinear-optical methods.

DOI
10.1103/physrevb.27.5904

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Soviet Journal of Quantum Electronics•Oct 31, 1982•2 citations
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.
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.
Selected topics related to the fundamentals of nonlinear optics are discussed. First, a brief description on the theory of nonlinear optical susceptibilities is given. Emphasis is on the basic properties, on the recently developed bond theory, and on the dispersion of nonlinear susceptibilities. Then the coupled wave approach is discussed. It is shown that the transverse resonant excitation in material can be treated as a material excitational wave, and most nonlinear optics problems can be understood as results of coupling between em waves or em and material excitational waves. A number of examples of current interest are presented. These include sum- and difference-frequency generation, Raman scattering by coherent excitation, stimulated Raman and stimulated polariton scattering, four-wave mixing, and vibrational relaxation measurements. Finally, the physical distinction between laser emission and resonant stimulated Raman scattering in an optically pumped three-level system are considered. A similar problem on the distinction of hot luminescence and resonant Raman scattering has recently attracted much attention. It is shown that the hot laser action is connected directly with the population inversion, while the resonant stimulated Raman scattering is connected with transverse excitation. The problem is, however, more complicated for a multi-level system. (auth)
A unified theory of time-domain and frequency-domain four-wave mixing processes, which is based on the nonlinear response function R(t3, t2, t1), is developed. The response function is expressed in terms of the four-point correlation function of the dipole operator F(τ1, τ2, τ3, τ4) and is evaluated explicitly for a stochastic model of line broadening that holds for any correlation time of the bath. Our results interpolate between the fast-modulation limit, in which the optical Bloch equations are valid, and the static limit of inhomogeneous line broadening. As an example of the relationship between time-domain and frequency-domain four-wave mixing, we compare the capabilities of steady-state and transient coherent anti-Stokes Raman spectroscopy experiments to probe the vibrational dynamics in ground and excited electronic states.
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.
The equivalence between the Hamiltonians representing nonlinear optical phenomena and N two-level atoms interacting via the electromagnetic field is exploited to find a solution to a parametric amplifier, a frequency converter and to the problems of Brillouin and Raman scattering. The n-photon probability and average number of photons as a function of time are obtained for all these cases. The results of the parametric amplifier can be readily applied to the Brillouin and Raman effects. In particular the results obtained for spontaneous fluorescence are applicable to the Stokes line of the stimulated Raman effect. Other nonlinear effects are discussed in the context of the present theory.
It is shown that in order to obtain a correct interpretation of experiments on four-photon nonlinear spectroscopy within the absorption line, it is necessary to allow not only for a coherent four-photon process due to a resonant electronic nonlinearity but also for a similar process involving generation of optically induced diffraction gratings followed by the diffraction of one of the probe beams from these gratings. It is also necessary to allow for the contribution of highly excited molecular states to the resonant electronic nonlinearity. General expressions are derived and investigated for the cubic susceptibility in an isotropic medium. It is found that all nonzero components of this tensor may be expressed in terms of a single scalar function of the interacting wave frequencies. The explicit form of this function is determined in the lowest order of perturbation theory.