Femtosecond covariance spectroscopy.

The success of nonlinear optics relies largely on pulse-to-pulse consistency. In contrast, covariance-based techniques used in photoionization electron spectroscopy and mass spectrometry have shown that a wealth of information can be extracted from noise that is lost when averaging multiple measurements. Here, we apply covariance-based detection to nonlinear optical spectroscopy, and show that noise in a femtosecond laser is not necessarily a liability to be mitigated, but can act as a unique and powerful asset. As a proof of principle we apply this approach to the process of stimulated Raman scattering in α-quartz. Our results demonstrate how nonlinear processes in the sample can encode correlations between the spectral components of ultrashort pulses with uncorrelated stochastic fluctuations. This in turn provides richer information compared with the standard nonlinear optics techniques that are based on averages over many repetitions with well-behaved laser pulses. These proof-of-principle results suggest that covariance-based nonlinear spectroscopy will improve the applicability of fs nonlinear spectroscopy in wavelength ranges where stable, transform-limited pulses are not available, such as X-ray free-electron lasers which naturally have spectrally noisy pulses ideally suited for this approach.