Derivation of new equations for phase-shift cavity ring-down spectroscopy

Current phase-shift cavity ring-down spectroscopy (PS-CRDS) experiments make use of equations originally developed for fluorescence studies. As these equations fail to take the length of the optical cavity and the superposition of reflecting beams into account, they lose validity as the length of the cavity increases. A new set of equations, based solely on the principles of PS-CRDS, is developed for determining the ring-down time from either the phase shift or the intensity of the waveform exiting the cavity. It is shown that the PS-CRDS equations reduce to those developed for fluorescence study for short cavities. The new equations provide a more accurate method in determining the characteristic ring-down time and phase shift for long cavities, especially fiber optic cavities, which is promising in on-site chemical sensing.