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Journal of the Physical Society of Japan

Spatial Optical Parametric Coupling of Picosecond Light Pulses and Transverse Relaxation Effect in Resonant Media

Tatsuo Yajima, Yoichi Taira

Published Nov 1, 1979• 487 citations
Publisher

Abstract

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.

DOI
10.1143/jpsj.47.1620

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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.
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.
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.
Analytical and numerical methods are used to study the transverse reshaping that occurs when ultrashort optical pulses propagate through media resonant to the pulse-carrier frequency. The field is described by the scalar quasi-optic equation, and the material system by the full Bloch equations describing a distribution of homogeneously broadened two-level molecules. Transverse derivatives in the field equation provide the coupling leading to the development of radially dependent frequency variations and phase curvature. We have studied the origin and evolution of transverse energy flow and investigated the conditions under which this leads to significant coherent self-focusing. The physical mechanisms responsible for the onset and development of the focusing phenomenon are discussed and related to other types of self-confinement of optical pulses.
The optical density of a resonant medium significantly affects the output signal in time resolved nonlinear optical interactions. The resonant interaction with a medium reshapes any propagating short pulse, and the mutual interaction of several such pulses, as in four-wave mixing, must be treated self-consistently. We present a theoretical framework for the proper handling of the propagation of all pulses, input as well as generated, in the small area limit. For optically thick resonant absorbers, negative time delay signals are observed, and the apparent decay rate of the induced polarization is faster than the rate observed for thin samples. We experimentally measure degenerate four-wave mixing in an atomic medium as an example, and demonstrate the quality of the theoretical model by the excellent fit to measured signals over several orders of magnitude. The improved understanding enables us to provide a simple, but surprisingly accurate, estimate for the apparent decay rate in homogeneously broadened optically thick media: If the absorption is given by \ensuremath{\alpha}, the propagation length is L, and the transverse relaxation time is T2{\mathit{T}}_{2}T2​, the apparent decay rate 2/Ta{\mathit{T}}_{\mathit{a}}Ta​ of a time resolved four-wave mixing signal is given by 2/Ta{\mathit{T}}_{\mathit{a}}Ta​=(2/T2{\mathit{T}}_{2}T2​)(1+\ensuremath{\alpha}L/2).
We show that four-wave mixing in optically dense media is very different than in an optically thin medium. In time-resolved degenerate four-wave mixing we experimentally observe ``negative'' time delay response, fast decay rates at short delay times, and broad shoulders at long delays. The observations are explained very well in terms of pulse propagation effects, and a theoretical framework is presented for the analysis of the nonlinear interaction. We treat both the incident and generated fields self-consistently, and solve for the interaction of short pulses with a resonantly absorbing medium. The implications for the extraction of relaxation rates from four-wave-mixing experiments involving ultrashort pulses in optically dense matter are discussed.
Our recent studies of methods for obtaining ultrashort light pulses and analysis of the impact of ultrashort optical pulses on classical and quantum micro-objects with the identification of the determining role of the degree of unipolarity of pulses (maximum for strictly unipolar pulses) in the efficiency of direct laser acceleration of charged particles and excitation of atoms and molecules have been reviewed. Special attention has been paid to the coherent mode locking regime in lasers, which is implemented when the laser pulse duration is less than the relaxation time of the working transition in media with resonance amplification and absorption (laser analog of the phenomenon of self-induced transparency). Experimental data on coherent mode locking in a titanium-sapphire laser with a cell with rubidium vapor at self-induced transparency in rubidium have been presented. The interaction of ultrashort pulses in resonant media has also been analyzed.