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We present a novel spectroscopic technique for second harmonic generation(SHG) using femtosecond laser pulses at 30~kHz repetition rate, whichnevertheless provides high spectral resolution limited only by thespectrometer. The potential of this method is demonstrated by applying it tothe yellow exciton series of Cu$_2$O. Besides even parity states with $S-$ and$D-$ envelope, we also observe odd parity, $P-$ excitons with linewidths downto 100 $\mu$eV, despite of the broad excitation laser spectrum with a fullwidth at half maximum of 14~meV. The underlying light-matter interactionmechanisms of SHG are elaborated by a group theoretical analysis which allowsus to determine the linear and circular polarization dependences, in goodagreement with experiment.

High-Resolution Second Harmonic Generation Spectroscopy with Femtosecond Laser Pulses on Excitons in Cu$_2$O