Characterization method of unusual second-order-harmonic generation based on vortex transformation

When a few-cycle laser beam nonresonantly propagates through an inversion-symmetric medium, beyond the usual odd-order harmonics in the transmission spectra, a well-defined spectral peak at twice the incident laser central frequency is disclosed [T. Tritschler et al., Phys. Rev. Lett. 90, 217404 (2003)]. Beyond the characterization method via the rf measurement of its carrier-envelope phase dependence, here a more direct mode of characterization is proposed by means of vortex transformation where a few-cycle vortex laser is adopted instead: One can easily clarify its origin as an usual third-order harmonic that appears to be a second-order harmonic, based on the criteria that the topological charge number of harmonics is directly proportional to its harmonic order.