Coherent Raman scattering spectroscopy is studied purposely, with the Gaussian ultrashort pulses as a hands-on elucidatory extraction tool of the clean coherent Raman resonant spectra from the overall measured data contaminated with the non-resonant four wave mixing background. The integral formulae for both the coherent anti-Stokes and Stokes Raman scattering are given in the semiclassical picture, and the closed-form solutions in terms of a complex error function are obtained. An analytic form of maximum enhancement of pure coherent Raman spectra at threshold time delay depending on bandwidth of probe pulse is also obtained. The observed experimental data for pyridine in liquid-phase are quantitatively elucidated and the inferred time-resolved coherent Raman resonant results are reconstructed with a new insight.
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