Characterizing second-harmonic generation of complex structured pulses resembling quantum Green’s function

Structural variations during second harmonic (SH) generation for complex pulsed fields with transverse morphologies that resemble the quantum Green’s functions (GFs) of harmonic oscillators are explored. By collimating and reimaging the fundamental-wave GF pulses emitted from a near-hemispherical Nd:YVO 4 /Cr 4+ :YAG resonator into an extra-cavity KTP crystal via a custom 4 f system, SH fields with complete and stable beam structures are realized, even when the central order of input mode reaches up to N = 49. All experimental transverse patterns for the SH modes—ranging from the near field to the far field—reveal intricate morphologies with order-doubling features, which are confirmed to be accurately characterized by the derived analytical wave function. Additionally, by reconstructing the experimental propagation evolution of the frequency-doubled complex pulses, the corresponding phase structures are numerically resolved. This reveals the emergence of more abundant vortex singularities in the SH mode compared to the fundamental-wave GF pulses. Furthermore, owing to the temporal filtering effect inherent to the nonlinear process, the pulse width of the SH field is effectively narrowed to approximately two-thirds of that of the fundamental GF pulse. Consequently, the overall peak power of the SH pulse is maintained at several watts, demonstrating significant potential for applications in nanophotonics.