Skip to main content
U.S. flag

An official website of the United States government

Official websites use .gov
A .gov website belongs to an official government organization in the United States.

Secure .gov websites use HTTPS
A lock ( ) or https:// means you’ve safely connected to the .gov website. Share sensitive information only on official, secure websites.

Broadband CARS Spectral Phase Retrieval Using a Time-Domain Kramers-Kronig Transform



Yuexin Liu, Young Jong Lee, Marcus T. Cicerone


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.
Optics Letters


Kramers-Kronig relations, spectroscopy, coherent anti-Stokes Raman scattering, phase retrieval


Liu, Y. , Lee, Y. and Cicerone, M. (2009), Broadband CARS Spectral Phase Retrieval Using a Time-Domain Kramers-Kronig Transform, Optics Letters, [online],, (Accessed May 30, 2024)


If you have any questions about this publication or are having problems accessing it, please contact

Created April 21, 2009, Updated October 12, 2021