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Physical Review A

Coherent Raman scattering with incoherent light for a multiply resonant mixture: A factorized time correlator diagram analysis

Jason C. Kirkwood, A. C. Albrecht, Darin J. Ulness, Michael J. Stimson

Published Dec 1, 1998• 19 citations
Publisher

Abstract

Coherent Raman scattering using broadband, incoherent (noisy) light (termed ${\mathrm{I}}^{(2)}\mathrm{CRS}$---the I for interferometric and the superscript 2 for twice acting noisy fields) is treated for a multiply resonant, multicomponent mixture of molecules. Both the resonant-resonant and resonant-nonresonant hyperpolarizability contributions to the signal intensity are included in this formalism. The emphasis is on the physical understanding of this spectroscopy. To this end, factorized time correlator (FTC) diagram analysis is used to characterize the ${\mathrm{I}}^{(2)}\mathrm{CRS}$ signal. The FTC diagram technique also serves to organize the terms that arise through the stochastic averaging of the noise at the intensity level. In addition to the FTC diagram analysis, a spectral filter analogy is employed to provide further insight into the physics behind ${\mathrm{I}}^{(2)}\mathrm{CRS}.$

DOI
10.1103/physreva.58.4910

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Coherent Raman scattering using broadband, incoherent (noisy) light (termed I(2)CRS{\mathrm{I}}^{(2)}\mathrm{CRS}I(2)CRS---the I for interferometric and the superscript 2 for twice acting noisy fields) is treated for a multiply resonant, multicomponent mixture of molecules. Both the resonant-resonant and resonant-nonresonant hyperpolarizability contributions to the signal intensity are included in this formalism. The emphasis is on the physical understanding of this spectroscopy. To this end, factorized time correlator (FTC) diagram analysis is used to characterize the I(2)CRS{\mathrm{I}}^{(2)}\mathrm{CRS}I(2)CRS signal. The FTC diagram technique also serves to organize the terms that arise through the stochastic averaging of the noise at the intensity level. In addition to the FTC diagram analysis, a spectral filter analogy is employed to provide further insight into the physics behind I(2)CRS.{\mathrm{I}}^{(2)}\mathrm{CRS}.I(2)CRS.
The theory for coherent Raman scattering (CRS) with broadband incoherent light is presented for a multiply resonant, multicomponent mixture of molecules that exhibits simultaneous multiple resonances with the frequencies of the driving fields. All possible pairwise hyperpolarizability contributions to the signal intensity are included in the theoretical treatment---(resonant-resonant, resonant-nonresonant, and nonresonant-nonresonant correlations between chromophores) and it is shown how the different types of correlations manifest themselves as differently behaved components of the signal intensity. The Raman resonances are modeled as Lorentzians in the frequency domain, as is the spectral density of the incoherent light. The analytic results for this multiply resonant mixture are presented and applied to a specific binary mixture. These analytic results will be used to recover frequencies and dephasing times in a series of experiments on multiply resonant mixtures.
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Degenerate four-wave mixing using broadband, non-transform-limited (incoherent) laser light having a Lorentzian spectral density is treated analytically for a Bloch two-level system. We consider the configuration in which an incoherent beam from a Lorentzian source is split into two beams having distinct k vectors, k and ${\mathit{k}}^{\ensuremath{'}}$. The twin beam along ${\mathit{k}}^{\ensuremath{'}}$ has been subjected to a controllable delay, \ensuremath{\tau}, relative to k. These twin beams, focused into the sample of interest, generate new fourth fields, the signal at ks{\mathit{k}}_{\mathit{s}}ks​=2k-${\mathit{k}}^{\ensuremath{'}}$ being examined here as a function of \ensuremath{\tau}. A set of factorized time correlation (FTC) diagrams is introduced to organize the calculation and to yield both analytic expressions as well as physical insight. Not only is the signal not symmetric in \ensuremath{\tau} but the signal can peak at \ensuremath{\tau}\ensuremath{\ne}0 even in the absence of inhomogeneous broadening. Finally, the physical insights and the use of the FTC diagrams are extended to rationalize several other incoherent light spectroscopies. \textcopyright{} 1996 The American Physical Society.
The theory for electronically nonresonant coherent Raman scattering using incoherent light (I(2)CRS) is presented for a Brownian oscillator material response. The exploration of Raman line shapes is based on two different approaches, both of which incorporate the Brownian oscillator model. The first takes the Raman transition as a two-level system that is indirectly coupled to the bath through “primary” Brownian oscillators. The I(2)CRS signal in the overdamped, high temperature regime is calculated analytically using this approach. The second approach is the standard one in which the Raman transition itself is represented by primary Brownian oscillators that are directly coupled to the bath. This method permits analytic calculation of the I(2)CRS signal for the entire parameter space and also for a distribution of oscillators to account for inhomogeneous linewidth broadening. These two methods allow for different interpretations of Raman line shapes using noisy light spectroscopy.
This work examines the properties and mathematical structure of factorized time correlation (FTC) diagram analysis in a general context. The goal is to extract general principles and analytic behavior that are not tied to any particular phenomenon in physics. It is hoped that this will provide a basis for expanded use of FTC diagram analysis beyond its current employment in the study of noisy light-based nonlinear optical spectroscopy. Furthermore, the concept of indirect correlation in a two-channel system driven by twin stationary circular Gaussian stochastic inputs is defined and discussed both analytically and through FTC diagram analysis.