Modulation Instability Induced Frequency Comb Generation in a Continuously Pumped Optical Parametric Oscillator

Continuously pumped passive nonlinear cavities can be harnessed for the creation of novel optical frequency combs. While most research has focused on third-order “Kerr” nonlinear interactions, recent studies have shown that frequency comb formation can also occur via second-order nonlinear effects. Here, we report on the formation of quadratic combs in optical parametric oscillator (OPO) configurations. Specifically, we demonstrate that optical frequency combs can be generated in the parametric region around half of the pump frequency in a continuously driven OPO. We also model the OPO dynamics through a single time-domain mean-field equation, identifying previously unknown dynamical regimes, induced by modulation instabilities, which lead to comb formation. Numerical simulation results are in good agreement with experimentally observed spectra. Moreover, the analysis of the coherence properties of the simulated spectra shows the existence of correlated and phase-locked combs. Our results reveal previously unnoticed dynamics of an apparently well assessed optical system, and can lead to a new class of frequency comb sources that may stimulate novel applications by enabling straightforward access to elusive spectral regions, such as the midinfrared.

Figure 1

MI gain of (a) the nonzero constant solution and (b) trivial zero solution, as a function of the offset frequency $\mathrm{\Omega }/2\pi$ and the detuning $\mathrm{\Delta }$, calculated for our experimental parameters: ${\alpha }_1=0.017$, $L=15\mathrm{mm}$, $k_1^{\prime \prime }=0.234{\mathrm{ps}}^2/\mathrm{m}$, $k_2^{\prime \prime }=0.714{\mathrm{ps}}^2/\mathrm{m}$, $\mathrm{\Delta }{k}^{\prime }=792\mathrm{ps}/\mathrm{m}$, $\kappa =6.58{\mathrm{W}}^{-1/2}{\mathrm{m}}^{-1}$, $\xi =0$, $P_{\mathrm{in}}=300\mathrm{mW}$. White curves highlight the maxima of the instability gain.

Figure 2

(a) Schematic of the experimental setup. Beam splitter (BS), electro-optic phase modulator (EOM); piezoelectric actuator (PZT), photodiode (PD). (b) and (c) Experimental infrared comb spectra with zero cavity detuning for different values of the pump power (resolution bandwidth, $\mathrm{RBW}=6\mathrm{GHz}$). (d) and (e) rf spectra showing beat notes at 505 MHz, in the IR and visible region, respectively, corresponding to the comb spectrum shown in (b) ($\mathrm{RBW}=1\mathrm{kHz}$).

Figure 3

Normalized infrared OPO spectra for different cavity detunings. (a)–(c) experimental spectra for relative detunings $\mathrm{\Delta }=-0.3$, 0, $+0.3$, respectively. (d)–(f) corresponding numerically calculated spectra. (g) Map of calculated OPO spectra as a function of the offset from the degeneracy frequency and the relative detuning. Measurements and simulations refer to 300 mW of pump power.

Figure 4

Coherence analysis. (a) Temporal evolution of the parametric field envelope ($t_R=111\mathrm{ps}$, $P_{\mathrm{in}}=300\mathrm{mW}$, and $\mathrm{\Delta }=-2$). (b) Traces (1) are the first order correlation function $g^{(1)}(m,\lambda )$ as a function of wavelength, calculated for different time delays up to $100\mu \mathrm{s}$; trace (2) is the comb spectrum profile. (c) Offset phase evolution for two symmetric modes $\pm 18$ and their average phase. (d) Steady state offset phase (blue open circles) and symmetric average phase (red dots), as a function of mode number $n$.