A quantitative approach to evaluate the problem of coherence of spectral components of the third-order susceptibility generating coherent anti-Stokes Raman signals

Precise interpretation of spectral measurements is central to the development of the full extent of the applicative potential of coherent anti-Stokes Raman spectroscopy (CARS). One recognized problem that jeopardizes the achievement of high precision is the determination of the best spectral convolution over the relevant bandwidths when degeneracy of laser frequencies is involved. Although the analytical solutions of CARS signals generated by pump and Stokes lasers with standard (i.e. Gaussian or Lorentzian) lineshapes are well known, research in this field has overlooked the criterion on how to discern coherence between spectral components of the third-order nonlinear susceptibility. Understandably, the ordinary approach is based on an intuitive comparison between the spectral width sigma(1) of the pump laser with respect to the width Gamma of the relevant Raman transitions. More precisely, if sigma(1) << Gamma, then the spectral synthesis can be obtained in the limit of narrowband pump; otherwise spectral coherence has to be included in the calculation leading to problematic spectral analysis. In an attempt to clarify this qualitative criterion better, the present work demonstrates that the limit between the two opposite regimes can have a clearer and neater definition than that accepted so far. In this case, this paper shows that for nonoverlapping Raman transitions determined by a Lorentzian susceptibility, the issue is governed by the analytic function, root pi Gamma exp[(Gamma/sigma(1))(2)] erfc(Gamma/sigma(1))/sigma(1), which depends uniquely on the ratio Gamma/sigma(1). The unitary limit of this function for sigma(1) << Gamma justifies the incoherent or the narrowband-pump approach. Copyright (c) 2006 John Wiley & Sons, Ltd.