Isolation of coherent and incoherent nonlinear spectroscopic signals by phase modulation

We investigate the effect of phase modulation of laser beams on the coherent and incoherent non-linear signals arising from the interaction of femtosecond pulses with matter. We observe that two collinear beams, whose phases are modulated by frequencies $\phi_1$ and $\phi_2$, produce two second harmonic signals from non-linear crystals whose intensities at the detector are modulated at the frequencies $\phi_2-\phi_1$ and $2(\phi_2-\phi_1)$. We also observe that an incoherent action signal, such as fluorescence and photocurrent, which arises from the absorption of two photons, is modulated at the same frequencies as in the case of second harmonic generation. We present a theoretical analysis to explain our observations. These results are important to understand how phase modulation techniques can be used to isolate different field-matter interaction pathways in a non-linear process. Because the method uses modulation of the signal intensity rather than wave-vector matching to isolate different signals, it could be useful to perform multi-photon absorption studies on single molecules or nanoparticles.