Broadband CARS spectral phase retrieval using a time-domain Kramers–Kronig transform

We describe a closed-form approach for performing a Kramers-Kronig (KK) transform that can be used to rapidly and reliably retrieve the phase, and thus the resonant imaginary component, from a broadband coherent anti-Stokes Raman scattering (CARS) spectrum with a nonflat background. In this approach we transform the frequency-domain data to the time domain, perform an operation that ensures a causality criterion is met, then transform back to the frequency domain. The fact that this method handles causality in the time domain allows us to conveniently account for spectrally varying nonresonant background from CARS as a response function with a finite rise time. A phase error accompanies KK transform of data with finite frequency range. In examples shown here, that phase error leads to small (<1%) errors in the retrieved resonant spectra.