We describe a closed-form approach for performing a Kramers-Kronig (KK) transform on data of finite frequency range; the method is analytically exact and computationally simple and robust. 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. We demonstrate that this approach is equivalent to a KK transform, and can be used to rapidly and reliably retrieve the phase, and thus the resonant component, from a broadband coherent anti-Stokes Raman scattering (CARS) spectrum. The fact that this method handles causality in the time domain allows us to account for spectrally varying nonresonant background (NRB) from CARS in a natural way, as a response function with a finite rise-time. This method also eliminates many of the difficulties typically encountered in applying KK transforms, including the need to use numerical integration and the consideration of Cauchy principal parts for dealing with infinities.
Citation: Optics Letters
Pub Type: Journals
Kramers-Kronig relations, spectroscopy, coherent anti-Stokes Raman scattering, phase retrieval