Considering Photoinduced Second-Harmonic Generation as a dc Kerr Optical Parametric Oscillation or Amplification Process

Photoinduced second-harmonic generation (SHG) in centrosymmetric materials like silica and silicon nitride has been commonly explained as an effective second-order (${\chi }^{(2)}$) process mediated by a dc electric field and the medium’s third-order (${\chi }^{(3)}$) nonlinearity. In this explanation, the coherent photogalvanic effect is the source of a dc electric field whose spatial periodicity naturally enables quasi-phase-matching. While successful in explaining many observations from experiment, the behavior at low input powers, and in particular, the apparent existence of a threshold for efficient photoinduced SHG observed in some experiments has largely been overlooked theoretically. In this Paper, we reconsider photoinduced SHG within the framework of four-wave mixing (FWM), involving degenerate pump, second-harmonic signal, and dc electric field. We propose a hypothesis that photoinduced SHG is a FWM-mediated dc Kerr optical parametric oscillation or amplification process. This hypothesis can explain the threshold behavior, and moreover, predicts unconventional light amplification, both of which we verify by experiments in silicon-nitride microresonators. Finally, we discuss the physical implications of our work in various platforms and future directions.

Figure 1

History and physical mechanisms behind photoinduced SHG. (a) Timeline of photoinduced SHG, where dots are aligned to the year when events were reported. Photoinduced SHG was discovered in 1986 in doped fiber and subsequently extensively studied in the next decade, through the incorporation of three related theoretical models: (b) quasi-phase-matching (QPM), (c), field-induced SHG (FISH), and (d) the coherent photogalvanic effect (CPG). Various schemes have been explored, mostly in doped optical fibers, in writing, initiating, and erasing the photoinduced grating for QPM and the dc electric field that enables phase and frequency matching of the SHG process based on an effective ${\chi }^{(2)}$ framework. Photoinduced SHG has seen a revival of interest in recent years, in particular, in ${\text{Si}}_3{\text{N}}_4$ waveguides and microrings. (e) In this work, we study photoinduced SHG in a ${\text{Si}}_3{\text{N}}_4$ microring, where efficiency is high, QPM is not required, and a continuous-wave pump is used. We show that photoinduced SHG can be considered as a dc Kerr optical parametric oscillation and amplification process, from which a threshold behavior is naturally expected, and which matches well with experimental results.

Figure 2

dc Kerr optical parametric oscillation (OPO). (a) Schematic shows an integrated microring with coupling waveguides and related physical parameters for the pump ($p$), signal ($s$), and dc fields, in particular, intracavity energies $U_{p,s,\text{dc}}$, waveguide-resonator coupling rates ${\mathrm{\Gamma }}_{cp,cs}$ and intrinsic loss rates ${\mathrm{\Gamma }}_{0p,0s,\text{dc}}$. (b) Forward tuning of the pump laser shows thermal bistability. The kink (at $\lambda \approx 1558.0$ nm) is where the pump and SHG modes are frequency matched. The inset shows the cold-cavity transmission (i.e., without thermal bistability) of the pump mode with mode number of $m=154$. (c) Optical spectrum of the second-harmonic signal at $\lambda \approx 779$ nm. The inset shows the cold-cavity transmission trace. The SHG mode has a mode number of $m=308$. (d), Adiabatic forward tuning is used to record the threshold behavior of the SHG process. The threshold power is ($2.3\pm 0.1$) mW, as shown in the inset. The decrease of the power after crossing the threshold is due to the decrease of intracavity pump energy after conversion, which corresponds to an approximately 8 % intracavity conversion efficiency. The dashed lines (labeled I–V) have colors that correspond to traces in (e) with I–II below threshold, and III–V above threshold. (e) The temporal response shows distinctive properties of the dc Kerr OPO. The time constant observed in these traces agrees with a first-principles estimate of the $RC$ time constant in silicon nitride (e.g., $\tau \approx$ 16 s for trace II). Both growth and decay are observed for traces I1-I3 (with different initial SHG signals), with the final power only dependent on the final laser-cavity detuning and the dropped pump power.

Figure 3

Unconventional light amplification. (a) The system is prepared near (but below) the dc Kerr OPO threshold by fine control of pump detuning. (b) A cw input broadband seed signal is gradually increased from approximately equal to $-53$ dBm (red) to approximately equal $\mathrm{to}-43$ dBm (blue) to activate light amplification of a narrowband signal around 779 nm. The spectral oscillation (8 to 10 dB) in the cw seed is from the amplified spontaneous emission source. The dashed and solid green traces (plotted within the light blue shaded region centered at 779 nm) represent the initial and final signal with the same cw seed. The peak signal, once the amplification is activated, is identical to the above-threshold case (approximately equal to $-19$ dBm, see the dashed red curve). This amplification regime is unique, and agrees with the theoretical prediction. The inset shows the optical spectrum without the cw seed, which is due to weak second-harmonic generation resulting from the small intrinsic SHG.