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Journal of the Optical Society of America B

Symmetry properties of the Rayleigh-type optical mixing signal in frequency space

Héctor J. Franco, José Luis Paz, A. Marcano O., M. C. Salazar, Isaac Reif

Published Jan 1, 1990• 19 citations
Publisher

Abstract

The dependence of the Rayleigh-type optical mixing signal intensity on the incident pump and probe frequencies for different values of T1/T2 is shown in a three-dimensional plot for a homogeneously broadened line. The symmetry properties of the signal in this space and of the nonlinear process that generates the line are related. These properties are used to show the dependence of the measured spectra on the way in which the frequencies of the incident fields are varied. Furthermore, this plot in frequency space yields information that may be used for the determination of the relaxation times T1 and T2.

DOI
10.1364/josab.7.000057

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In a propagation treatment at all orders in the pump field and to first order in the probe and signal fields, for a homogeneously broadened line, we obtain three approximation levels for the Rayleigh-type optical mixing signal intensity: the first level is an analytical solution valid only for constant pump intensity; the analytical solution corresponding to the second approximation level considers pump propagation and yields good results for a large number of cases; the third level is an exact but numerical solution. We present and compare the results obtained with the different approximations.

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Polarization and Rayleigh type optical mixing spectra of a homogeneously and inhomogeneously broadened two-level system are calculated. We show that the nonlinear signal has contributions from the population pulsation and saturation effect. In the homogeneous system these effects can induce an asymmetric spectrum when the probe frequency is different from the resonant value. In the inhomogeneous system the spectral response is always symmetric independent of the probe frequency value. Line saturation and dynamic stark splitting are substantially reduced in the inhomogeneous case. The mentioned differences can be used to identify the type of broadening of a given spectral line.

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Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE•Nov 18, 2013•0 citations
The dependence of the Rayleigh-type optical mixing signal intensity on the incident pump and probe frequencies for different values of T1/T2 is shown in a three-dimensional plot for a homogeneously broadened line. The symmetry properties of the signal in this space and of the nonlinear process that generates the line are related. These properties are used to show the dependence of the measured spectra on the way in which the frequencies of the incident fields are varied. Furthermore, this plot in frequency space yields information that may be used for the determination of the relaxation times T1 and T2.
In a propagation treatment at all orders in the pump field and to first order in the probe and signal fields, for a homogeneously broadened line, we obtain three approximation levels for the Rayleigh-type optical mixing signal intensity: the first level is an analytical solution valid only for constant pump intensity; the analytical solution corresponding to the second approximation level considers pump propagation and yields good results for a large number of cases; the third level is an exact but numerical solution. We present and compare the results obtained with the different approximations.
Polarization and Rayleigh type optical mixing spectra of a homogeneously and inhomogeneously broadened two-level system are calculated. We show that the nonlinear signal has contributions from the population pulsation and saturation effect. In the homogeneous system these effects can induce an asymmetric spectrum when the probe frequency is different from the resonant value. In the inhomogeneous system the spectral response is always symmetric independent of the probe frequency value. Line saturation and dynamic stark splitting are substantially reduced in the inhomogeneous case. The mentioned differences can be used to identify the type of broadening of a given spectral line.
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.
Abstract: Previous work by one of the authors has shown the dependence of the Rayleigh-type optical mixing (RTOM) signal intensity on the incident pump and probe frequencies for different values of the ratio of relaxation times k = T1 / T2. In this work an inverse problem methodology is defined to determine the ratio of the relaxation times k from sparse and/or noisy incident pump and probe frequency data in order to reconstruct the full field RTOM signal intensity. The simulated results show a robust procedure which points to potential efficient experimental application, to be pursued in the near future. Resumen: Trabajo previo de uno de los autores ha demostrado la dependencia de la intensidad de la señal de mezcla óptica tipo Rayleigh (RTOM) en la frecuencia de bombeo y de prueba para valores diferentes de relación de tiempos de relajación k = T1 / T2. En este trabajo se define una metodología de problema inverso para determinar la relación de tiempos de relajación k de datos incidentes, que son escasos y/o con ruido, de frecuencia de bombeo y de prueba que son usados para reconstruir la intensidad de la señal de RTOM de campo entero. Los resultados simulados demuestran un procedimiento robusto que apunta a una potencial aplicación experimental, que se la perseguirá en un futuro cercano.
In this work are studied the symmetry properties of the Rayleigh-type optical mixing signal of a two-level molecular system immersed in a thermal bath and irradiated by a classical electromagnetic field. The solvent induces a random shift of the Bohr frequency in the molecular system. A methodology based in cumulant expansions is employed to obtain the average of the coherences, populations, and susceptibilities of Fourier components associated, calculated by the optical stochastic Bloch equations. These symmetry properties show the dependence of the measured spectra with the variations in the frequencies of the incident fields. Our results show that the inclusion of the thermal bath diminishes the intensity response as well it promotes the loss of the symmetry properties, compared with the same results in the absence of the bath.
In this work we studied the symmetry properties of the Rayleigh-type optical mixing signal, of a two-level molecular system immersed in a thermal bath and irradiated by a classical electromagnetic field. Taking into account that the solvent produces an inhomogeneous displacement of the Bohr frequency and, employing a cumulant expansion to solve the stochastical equations derived, we obtained the mean values of the coherence, populations and susceptibilities Fourier components of the molecular system, from the optical stochastical Bloch equations (OSBE). These symmetry properties are used to show the dependence of the spectra with the frequency shifts of the incident fields, and we obtained that the inclusion of the thermal bath diminishes the intensity response as well it promotes the loss of the symmetry properties, compared with the same results in absence of the bath.