Fig. 2: Generic cases of SHG in a PTC.

a, b Case I: intensity of the SH as a function of time, with both \(k\) and \(2k\) in the band, with (a) and without (b) phase-matching, respectively. When the process is phase-matched (a) the SH intensity grows parabolically with time, whereas for a phase-mismatched process (b) the SH intensity oscillates in time. c Case II: phase-mismatched SHG with the SH in the momentum gap: The SH grows exponentially despite the phase mismatch. The vertical axis is logarithmic. d Case III: phase mismatched SHG with the fundamental in the gap: both the fundamental and second harmonics grow exponentially, despite the phase mismatch. The second harmonic grows with an exponential rate twice as large as the first harmonic, even though its mode belongs to the band. e, f Case IV: phase mismatched SHG with both fundamental and SH in the gap, where the dominant exponent is the exponent of the fundamental mode (e) and where the dominant exponent is the exponent of the SH mode (f)