Fig. 1: Harmonic generation in a nonlinear PTC.

a Sketch of the system. A wave with wavenumber k enters a nonlinear spatially—homogeneous PTC. The nonlinearity generates a second harmonic wave, with wavenumber 2k. b Band structure of a linear PTC. In blue is the real component of the Floquet frequency \({\omega }_{F}(k)\), and in red is the imaginary component. The regions with a gray background in b are the momentum gaps. The stars with different colors represent different choices of momentum for the fundamental and second harmonic signal: Orange—phase-matched process between band modes. Green—phase-mismatched process between band modes. Purple—phase-mismatched process with the fundamental in a band and the second harmonic in the gap. Yellow—phase-mismatched process with the fundamental in the gap and the second harmonic in the band