Conversion Efficiency in Kerr-Microresonator Optical Parametric Oscillators: From Three Modes to Many Modes

Optical parametric oscillation in a Kerr nonlinear microresonator can generate coherent laser light with frequencies that are widely separated from the pump frequency, allowing, for example, visible light to be generated using a near-infrared pump. To be practically useful, the pump-to-signal conversion efficiency must be far higher than what has been demonstrated in microresonator-based oscillators with widely separated output frequencies. To address this challenge, here we theoretically and numerically study parametric oscillations in Kerr nonlinear microresonators, revealing an intricate solution space that arises from an interplay of nonlinear processes. As a start, we use a three-mode approximation to derive an efficiency-maximizing relation between pump power and frequency mismatch. However, realistic devices, such as integrated microring resonators, support far more than three modes. Hence, a more accurate model that includes the entire modal landscape is necessary to determine potential inefficiencies arising from unwanted competing nonlinear processes. To this end, we numerically simulate the Lugiato-Lefever equation that accounts for the full spectrum of nonlinearly coupled resonator modes. We observe and characterize two nonlinear phenomena linked to parametric oscillations in multimode resonators: mode competition and cross-phase modulation-induced modulation instability. Both processes may impact conversion efficiency. Finally, we show how to increase the conversion efficiency to approximately 25% by tuning the microresonator loss rates. Our analysis will guide microresonator designs that aim for high conversion efficiency and output power.

Figure 1

An introduction to the microresonator-based degenerately pumped optical parametric oscillator ($\mu \mathrm{OPO}$). (a) A schematic of a microring resonator coupled to an access waveguide that carries the input and output fields. (b) The energy diagram for the degenerate FWM process that drives parametric oscillation. (c) A depiction of an ideal output spectrum. Energy from the pump laser is exclusively distributed to a single pair of signal and idler modes with mode numbers ${\mu }_s\gg 0$ and ${\mu }_i=-{\mu }_s$, respectively. (d) Depictions of mode spectra, nonlinear couplings, and frequency shifts in both a three-mode approximation (TMA) and a multimode model. The dashed lines correspond to zero frequency mismatch. The red arrows indicate mode-frequency shifts induced by Kerr nonlinearity. The TMA considers FWM between only the pump, signal, and idler modes, while multimode models account for the nonlinear couplings (orange, open arrows) between all mode sets conducive to FWM.

Figure 2

A study of the $\mu \mathrm{OPO}$ conversion efficiency using a three-mode approximation. (a) The conversion efficiency, $CE$ (gray), and the effective mismatch, ${\mathrm{\Delta }}_s$ (green), versus the pump-resonator frequency detuning, $\alpha$, for a pump-laser power $P_{\mathrm{in}}=20\mathrm{mW}$. From top to bottom, the mismatch values are $\delta =1.25$, 2, 4, and 7, respectively. The dashed sections indicate unstable or oscillatory solutions. (b) A generalized $CE$ map showing the relationship between maximum $CE$, normalized pump power, $X$, and $\delta$. The white dashed line is expressed as $X=8\delta$ and closely follows the contour of highest maximum $CE$. (c) $CE$ (gray) and the effective pump-resonator detuning, ${\mathrm{\Delta }}_p$ (green), versus $\alpha$, for $P_{\mathrm{in}}=20$ mW and $\delta =2$. (d) ${\mathrm{\Delta }}_p$ versus $\delta$ at values of $\alpha$ that maximize $CE$ for $P_{\mathrm{in}}=15$ mW (red, $X\approx 5$) and $P_{\mathrm{in}}=25$ mW (orange, $X\approx 8$).

Figure 3

Depictions of microresonator dispersion. (a) ${\delta }_{\mu }$ (green circles) and the integrated dispersion, $D_{\mathrm{int}}$ (purple stripe), for the TE1 mode family, calculated using finite-element-method eigenfrequency simulations. The microresonator has ring radius $RR=15\mu \mathrm{m}$, ring width $RW=800$ nm, height $H=600$ nm, and pump frequency ${\omega }_p=2\pi \times 388$ THz. The inset illustrates the test-device cross section. (b) The group-velocity refractive index, $n_g$, versus $\omega /2\pi$ for the same device as in (a). The dashed lines illustrate how a turning point in ${\delta }_{\mu }$ occurs when $n_g(-\mu )=n_g(\mu )$.

Figure 4

The transition from a TMA to multimode LLE simulations. (a) A depiction of the FWM processes studied in this work. Mode competition (orange arrows) suppresses FWM to the target signal and idler mode pair (with mode numbers $\pm \mu$) in favor of spectrally adjacent mode pairs [with mode numbers $\pm (\mu \pm 1)$] and modulation instability induced by cross-phase modulation (XPM MI, purple arrows) reroutes energy in the pump mode to its spectrally nearest neighbors. (b) $CE$ map for mode $\mu =46$; the resonator parameters are $RR=15\mu \mathrm{m}$, $RW=800$ nm, $H=600$ nm, and ${\omega }_p=2\pi \times 382$ THz. (c) The difference in $CE$ between simulations based on a TMA and $CE$ map in (b).

Figure 5

Mode competition and switching. (a) An illustration of mode competition in a $\mu \mathrm{OPO}$. Oscillations on the target signal and idler mode pair (with mode numbers $\pm \mu$) are suppressed in favor of spectrally adjacent mode pairs [with mode numbers $\pm (\mu \pm 1)$]. (b) ${\delta }_{\mu }$ for the test device with ${\omega }_p=2\pi \times 382$ THz. The pale orange stripe overlaps modes that are predicted to oscillate when $P_{\mathrm{in}}=20$ mW. (c) $CE$ maps for modes $\mu =46$, $\mu =45$, and $\mu =44$. The maps indicate a mode competition in which the mode pair with smallest $\delta$ is favored for oscillation. (d) $CE$ versus $\alpha$ for modes $\mu =46$ (blue) and $\mu =45$ (green), where $P_{\mathrm{in}}=20$ mW and $\beta =5\times {10}^{-4}$. The pale stripes come from a TMA. (e) Optical spectra that correspond to different values of $\alpha$ in (d). The blue, orange, green, and red spectra correspond to $\alpha =1$, $1.5$, $2.5$, and $3.5$, respectively.

Figure 6

The characterization of XPM MI in microresonators with normal dispersion. (a) An illustration of the XPM-MI FWM process (purple arrows), in which XPM between the pump, signal, and idler fields induces MI that distributes pump energy into spectrally nearby sidebands. (b) ${\delta }_{\mu }$ for $\beta =0.0625$ (green circles), $\beta =1.25$ (cyan upward-pointing triangles), and $\beta =5$ (blue downward-pointing triangles). One mode pair ($\mu =\pm 50$) is frequency shifted to be frequency matched to the pump mode. (c) $CE$ versus $\alpha$ for the ${\delta }_{\mu }$ spectra in (b), with $P_{\mathrm{in}}=20$ mW and $\delta =3$. (d) The optical spectrum associated with (c), with $\alpha =4$ and $\beta =0.0625$. (e) $CE$ maps for microresonators with the ${\delta }_{\mu }$ spectra in (b), ordered from least to greatest $\beta$.

Figure 7

The theory of XPM MI. (a) The MI gain parameter, $\lambda$, versus $\alpha$ for $\mu =1$. The orange and gold curves correspond to $P_{\mathrm{in}}=15$ mW, $\delta =1.5$, $\beta =0.25$, and $P_{\mathrm{in}}=25$ mW, $\delta =2.5$, $\beta =0.25$, respectively. The red curves both use $P_{\mathrm{in}}=20$ mW, with the bold and dashed curves corresponding to $\delta =2.5$, $\beta =1$ and $\delta =5$, $\beta =0.25$, respectively. (b) The value of $\alpha$ at which $CE$ deviates from a TMA, ${\alpha }_{\mathrm{on}}$, versus $\beta$ for $P_{\mathrm{in}}=20$ mW and $\delta =2.5$. The dark blue diamonds mark values predicted by LLE simulations, while the pale blue stripe corresponds to $\lambda ({\alpha }_{\mathrm{on}})=0$ calculated from Eq. (5). (c) ${\alpha }_{\mathrm{on}}$ map for $\mu =1$ and $\beta =0.25$. ${\alpha }_{\mathrm{on}}$ is normalized to the detuning, ${\alpha }_{\max }$, at which $CE$ is maximized in a TMA. The purple region indicates that XPM MI is absent and the border between the purple and gold regions indicates threshold $P_{\mathrm{in}}$ values for XPM MI.

Figure 8

Strategies to achieve high $CE$ and increase power output from a $\mu \mathrm{OPO}$. (a) A depiction of a microring resonator and its loss rates. (b) Simulated mode transmission versus ${\omega }_p$ for ${\kappa }_i=2\pi \times 125$ MHz (red), ${\kappa }_i=2\pi \times 200$ MHz (orange), and ${\kappa }_i=2\pi \times 275\mathrm{MHz}$ (gold). In each case, ${\kappa }_c={\kappa }_i$. (c) $CE$ maps corresponding to (b), for a resonator with $RR=15\mu \mathrm{m}$, $RW=800$ nm, $H=600$ nm, and ${\omega }_p=2\pi \times 385$ THz. (d) Simulated mode transmissions for ${\kappa }_c={\kappa }_i$ (green) and ${\kappa }_c=10\times {\kappa }_i$ (blue), where ${\kappa }_i=2\pi \times 200$ MHz. (e) The maximum $CE$ versus ${\kappa }_c/{\kappa }_i$ and $\delta$, where ${\kappa }_c$ is the signal-mode coupling rate. The pump and idler modes are critically coupled.