I. INTRODUCTION

Section:

ChooseTop of pageABSTRACTI. INTRODUCTION <<II. DESIGN AND TEST OF WA...III. μOPO GENERATION AND ...IV. LIMITATIONS ON OUTPUT...V. DISCUSSIONSUPPLEMENTARY MATERIALREFERENCESLasers operating at visible and near-infrared (NIR) wavelengths are essential to modern science and technology,1–41. T. Bothwell, C. J. Kennedy, A. Aeppli, D. Kedar, J. M. Robinson, E. Oelker, A. Staron, and J. Ye, “Resolving the gravitational redshift across a millimetre-scale atomic sample,” Nature 602, 420–424 (2022). https://doi.org/10.1038/s41586-021-04349-72. D. Awschalom, K. K. Berggren, H. Bernien, S. Bhave, L. D. Carr, P. Davids, S. E. Economou, D. Englund, A. Faraon, M. Fejer et al., “Development of quantum interconnects (QuICs) for next-generation information technologies,” PRX Quantum 2, 017002 (2021). https://doi.org/10.1103/prxquantum.2.0170023. L. E. M. Matheus, A. B. Vieira, L. F. M. Vieira, M. A. M. Vieira, and O. Gnawali, “Visible light communication: Concepts, applications and challenges,” IEEE Commun. Surv. Tutorials 21, 3204–3237 (2019). https://doi.org/10.1109/comst.2019.29133484. R. Weissleder, “A clearer vision for in vivo imaging,” Nat. Biotechnol. 19, 316–317 (2001). https://doi.org/10.1038/86684 but affordable systems typically suffer from poor spectral purity and gaps in spectral coverage, while higher-performance options are large and expensive. The latter often rely on bulk nonlinear optics to spectrally translate longer-wavelength lasers to the targeted frequency, employing either sum-frequency or second-harmonic generation in χ(2)-nonlinear media.5–75. J. N. Tinsley, S. Bandarupally, J.-P. Penttinen, S. Manzoor, S. Ranta, L. Salvi, M. Guina, and N. Poli, “Watt-level blue light for precision spectroscopy, laser cooling and trapping of strontium and cadmium atoms,” Opt. Express 29, 25462–25476 (2021). https://doi.org/10.1364/oe.4298986. D. C. Parrotta, A. J. Kemp, M. D. Dawson, and J. E. Hastie, “Multiwatt, continuous-wave, tunable diamond Raman laser with intracavity frequency-doubling to the visible region,” IEEE J. Sel. Top. Quantum Electron. 19, 1400108 (2013). https://doi.org/10.1109/jstqe.2013.22490467. R. P. Mildren, H. M. Pask, H. Ogilvy, and J. A. Piper, “Discretely tunable, all-solid-state laser in the green, yellow, and red,” Opt. Lett. 30, 1500–1502 (2005). https://doi.org/10.1364/ol.30.001500 Their operational complexity and substantial power consumption (they often require liquid cooling systems) render them impractical in many situations. Hence, it is desirable to transition the nonlinear wavelength conversion to a more scalable nonlinear optics platform, e.g., integrated photonics.One approach is to leverage the wavelength flexibility inherent to optical parametric oscillators using nonlinear microresonators, which possess large optical quality factors (Q) and small mode volumes to intensify circulating light and promote efficient nonlinear interactions.8,98. K. J. Vahala, “Optical microcavities,” Nature 424, 839–846 (2003). https://doi.org/10.1038/nature019399. T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Kerr-nonlinearity optical parametric oscillation in an ultrahigh-Q toroid microcavity,” Phys. Rev. Lett. 93, 083904 (2004). https://doi.org/10.1103/PhysRevLett.93.083904 Recent studies of microresonator-based optical parametric oscillators (μOPOs) have demonstrated broad spectral separation between pump and generated light,10–1210. N. L. B. Sayson, T. Bi, V. Ng, H. Pham, L. S. Trainor, H. G. L. Schwefel, S. Coen, M. Erkintalo, and S. G. Murdoch, “Octave-spanning tunable parametric oscillation in crystalline kerr microresonators,” Nat. Photonics 13, 701–706 (2019). https://doi.org/10.1038/s41566-019-0485-411. C. S. Werner, T. Beckmann, K. Buse, and I. Breunig, “Blue-pumped whispering gallery optical parametric oscillator,” Opt. Lett. 37, 4224–4226 (2012). https://doi.org/10.1364/ol.37.00422412. Y. Jia, K. Hanka, K. T. Zawilski, P. G. Schunemann, K. Buse, and I. Breunig, “Continuous-wave whispering-gallery optical parametric oscillator based on CdSiP2,” Opt. Express 26, 10833–10841 (2018). https://doi.org/10.1364/oe.26.010833 low-power operation,13,1413. X. Lu, G. Moille, A. Singh, Q. Li, D. A. Westly, A. Rao, S.-P. Yu, T. C. Briles, S. B. Papp, and K. Srinivasan, “Milliwatt-threshold visible–telecom optical parametric oscillation using silicon nanophotonics,” Optica 6, 1535–1541 (2019). https://doi.org/10.1364/optica.6.00153514. J. Lu, A. Al Sayem, Z. Gong, J. B. Surya, C.-L. Zou, and H. X. Tang, “Ultralow-threshold thin-film lithium niobate optical parametric oscillator,” Optica 8, 539–544 (2021). https://doi.org/10.1364/optica.418984 and visible-wavelength access.1515. X. Lu, G. Moille, A. Rao, D. A. Westly, and K. Srinivasan, “On-chip optical parametric oscillation into the visible: Generating red, orange, yellow, and green from a near-infrared pump,” Optica 7, 1417–1425 (2020). https://doi.org/10.1364/optica.393810 In particular, operating in the visible spectrum presents specific challenges: stronger material dispersion, larger scattering losses that reduce quality factors, and shorter evanescent field decay lengths that impede waveguide coupling. While both χ(2) and χ(3) μOPOs offer some wavelength flexibility, χ(3) systems are useful to generate visible light from a NIR pump; moreover, their natural availability in the popular silicon photonics platform1616. D. J. Moss, R. Morandotti, A. L. Gaeta, and M. Lipson, “New CMOS-compatible platforms based on silicon nitride and Hydex for nonlinear optics,” Nat. Photonics 7, 597–607 (2013). https://doi.org/10.1038/nphoton.2013.183 can enable their scalable fabrication and integration with other components, including pump lasers.1717. H. Park, C. Zhang, M. A. Tran, and T. Komljenovic, “Heterogeneous silicon nitride photonics,” Optica 7, 336 (2020). https://doi.org/10.1364/optica.391809 On the other hand, the reported or inferred (e.g., from optical spectra) conversion efficiencies are ≲0.1%,10,13,15,18–2010. N. L. B. Sayson, T. Bi, V. Ng, H. Pham, L. S. Trainor, H. G. L. Schwefel, S. Coen, M. Erkintalo, and S. G. Murdoch, “Octave-spanning tunable parametric oscillation in crystalline kerr microresonators,” Nat. Photonics 13, 701–706 (2019). https://doi.org/10.1038/s41566-019-0485-413. X. Lu, G. Moille, A. Singh, Q. Li, D. A. Westly, A. Rao, S.-P. Yu, T. C. Briles, S. B. Papp, and K. Srinivasan, “Milliwatt-threshold visible–telecom optical parametric oscillation using silicon nanophotonics,” Optica 6, 1535–1541 (2019). https://doi.org/10.1364/optica.6.00153515. X. Lu, G. Moille, A. Rao, D. A. Westly, and K. Srinivasan, “On-chip optical parametric oscillation into the visible: Generating red, orange, yellow, and green from a near-infrared pump,” Optica 7, 1417–1425 (2020). https://doi.org/10.1364/optica.39381018. S. Fujii, S. Tanaka, M. Fuchida, H. Amano, Y. Hayama, R. Suzuki, Y. Kakinuma, and T. Tanabe, “Octave-wide phase-matched four-wave mixing in dispersion-engineered crystalline microresonators,” Opt. Lett. 44, 3146–3149 (2019). https://doi.org/10.1364/ol.44.00314619. Y. Tang, Z. Gong, X. Liu, and H. X. Tang, “Widely separated optical Kerr parametric oscillation in AlN microrings,” Opt. Lett. 45, 1124–1127 (2020). https://doi.org/10.1364/ol.38431720. R. R. Domeneguetti, Y. Zhao, X. Ji, M. Martinelli, M. Lipson, A. L. Gaeta, and P. Nussenzveig, “Parametric sideband generation in CMOS-compatible oscillators from visible to telecom wavelengths,” Optica 8, 316–322 (2021). https://doi.org/10.1364/optica.404755 and the available output power is far too low for many applications (e.g., <10 μW for previous visible μOPOs1515. X. Lu, G. Moille, A. Rao, D. A. Westly, and K. Srinivasan, “On-chip optical parametric oscillation into the visible: Generating red, orange, yellow, and green from a near-infrared pump,” Optica 7, 1417–1425 (2020). https://doi.org/10.1364/optica.393810). Realizing higher conversion efficiencies and output powers would enable a wide range of on-chip applications and broaden the reach of silicon photonics in the visible spectrum.Here, we demonstrate efficient μOPOs that generate coherent light within the spectral window between 260 and 510 THz (590 and 1150 nm). We measure conversion efficiencies between 3.5% and 14.5% with corresponding on-chip output powers greater than 1 mW (and as high as 5 mW). Our results spring from efficient broadband waveguide-resonator coupling, which we realize with pulley-waveguide geometries designed using coupled-mode simulations. Notably, previous μOPO works have implemented such geometries to minimize threshold power;1313. X. Lu, G. Moille, A. Singh, Q. Li, D. A. Westly, A. Rao, S.-P. Yu, T. C. Briles, S. B. Papp, and K. Srinivasan, “Milliwatt-threshold visible–telecom optical parametric oscillation using silicon nanophotonics,” Optica 6, 1535–1541 (2019). https://doi.org/10.1364/optica.6.001535 in contrast, our goal is to maximize output power using pulleys that operate in a fundamentally different regime. In the rest of this paper, we first introduce the key μOPO physics and specify our experimental procedures. Then, we explain our coupled-mode simulations and present measurements to confirm their accuracy. Next, we present the optical spectra of 16 different μOPOs from which we determine output powers and conversion efficiencies. Finally, we show how parasitic nonlinear processes currently constrain the maximum realizable output power. Our work is an important step forward in the quest for practical, chip-based sources of visible laser light using nonlinear optics.The μOPOs we consider generate monochromatic signal and idler waves from a continuous-wave (CW) pump laser through resonantly enhanced degenerate four wave mixing (FWM), as depicted in Fig. 1(a).2121. R. W. Boyd, Nonlinear Optics (Academic Press, 2020). In experiments, we pump a fundamental transverse-electric (TE0) eigenmode of a silicon nitride microring near 385 THz, and FWM transfers energy to TE0 signal and idler modes. In principle, the range of accessible output frequencies, as constrained only by energy conservation, is DC to 2ωp, where ωp is the pump frequency. However, in practice, this range is dictated by the group velocity dispersion (GVD), which must be engineered such that FWM to the targeted signal and idler modes is favored (simultaneously phase- and frequency-matched), but FWM to other modes is suppressed. In the supplementary material, we recount our approach to dispersion engineering that is also described in Ref. 1515. X. Lu, G. Moille, A. Rao, D. A. Westly, and K. Srinivasan, “On-chip optical parametric oscillation into the visible: Generating red, orange, yellow, and green from a near-infrared pump,” Optica 7, 1417–1425 (2020). https://doi.org/10.1364/optica.393810.A separate challenge is to ensure that pump power is efficiently converted into output signal or idler power. Hence, we define the on-chip conversion efficiency aswhere Ps(i) is the signal (idler) power in the waveguide output and Pin is the pump power in the waveguide input. Recent theoretical work has derived the maximum obtainable CE asCEs(i)max=12κs(i)κpΓs(i)Γpωs(i)ωp,(2)where κs(i) and κp are the waveguide-resonator coupling rates of the signal (idler) and pump modes, Γs(i) and Γp are the total loss rates (i.e., loaded linewidths) of the signal (idler) and pump modes, and ωs(i) and ωp are the frequencies of the signal (idler) and pump light, respectively.1010. N. L. B. Sayson, T. Bi, V. Ng, H. Pham, L. S. Trainor, H. G. L. Schwefel, S. Coen, M. Erkintalo, and S. G. Murdoch, “Octave-spanning tunable parametric oscillation in crystalline kerr microresonators,” Nat. Photonics 13, 701–706 (2019). https://doi.org/10.1038/s41566-019-0485-4 Clearly, obtaining large CE involves engineering κ for both the pump mode and targeted signal/idler modes. Such coupling engineering is a common problem in the nonlinear optics of Kerr microresonators; it arises, for example, in the efficient extraction of octave-spanning Kerr microcombs.2222. G. Moille, Q. Li, T. C. Briles, S.-P. Yu, T. Drake, X. Lu, A. Rao, D. Westly, S. B. Papp, and K. Srinivasan, “Broadband resonator-waveguide coupling for efficient extraction of octave-spanning microcombs,” Opt. Lett. 44, 4737–4740 (2019). https://doi.org/10.1364/ol.44.004737 The problem is that, given a straight waveguide evanescently coupled to a microring resonator, κ decreases exponentially with frequency due to decreasing overlap of the microring mode with the waveguide mode. Hence, when the pump mode is critically coupled, the signal mode is undercoupled, resulting in low CE. Moreover, when ωs is a visible frequency, the smaller evanescent decay length compared to NIR frequencies exacerbates the challenge. One solution is to utilize so-called pulley waveguides, which increase the physical distance over which the waveguide and microring can exchange energy.22–2422. G. Moille, Q. Li, T. C. Briles, S.-P. Yu, T. Drake, X. Lu, A. Rao, D. Westly, S. B. Papp, and K. Srinivasan, “Broadband resonator-waveguide coupling for efficient extraction of octave-spanning microcombs,” Opt. Lett. 44, 4737–4740 (2019). https://doi.org/10.1364/ol.44.00473723. E. Shah Hosseini, S. Yegnanarayanan, A. H. Atabaki, M. Soltani, and A. Adibi, “Systematic design and fabrication of high-Q single-mode pulley-coupled planar silicon nitride microdisk resonators at visible wavelengths,” Opt. Express 18, 2127 (2010). https://doi.org/10.1364/oe.18.00212724. Q. Li, M. Davanço, and K. Srinivasan, “Efficient and low-noise single-photon-level frequency conversion interfaces using silicon nanophotonics,” Nat. Photonics 10, 406–414 (2016). https://doi.org/10.1038/nphoton.2016.64 Figure 1(b) illustrates the physical differences between straight-waveguide couplers (top panel) and pulley-waveguide couplers (bottom panel), and it depicts the three geometric parameters that define such couplers in our study: The waveguide width, WW; the waveguide-resonator gap, G; and the pulley length, Lp (which approaches zero in the straight-waveguide limit). In Fig. 1(c), we present measurements of μOPO spectra extracted from nominally identical microrings using either a straight-waveguide coupler (top panel) or a pulley-waveguide coupler (bottom panel). These measurements are representative of other comparisons between the two coupling schemes. While Pi is roughly the same in each case, Ps is ∼20× greater in the pulley waveguide. To explain the result, we show in the same panels the simulated coupling ratio, defined as K = κ/γ (equivalently, K = Qi/Qc, where Qi and Qc are the intrinsic and coupling quality factors, respectively), where γ = Γ − κ is the intrinsic loss rate. Near ωs, K is ≈8× higher for the pulley-waveguide coupler.

II. DESIGN AND TEST OF WAVEGUIDE COUPLERS

Section:

ChooseTop of pageABSTRACTI. INTRODUCTIONII. DESIGN AND TEST OF WA... <<III. μOPO GENERATION AND ...IV. LIMITATIONS ON OUTPUT...V. DISCUSSIONSUPPLEMENTARY MATERIALREFERENCESTo design couplers for testing, we simulate κ spectra for a variety of coupling geometries with the goal of achieving K ≈ 1 (i.e., critical coupling) at frequencies between 260 and 510 THz. Notably, achieving high CE only requires that κ be optimized at ωp and either ωs or ωi, depending on which output wave (signal or idler) is to be used. (Moreover, ideally, the unused wave is undercoupled to reduce threshold power.) At the same time, broadband near-critical coupling is preferable, since then a single coupling geometry is robust against design imperfections, and it may be used for many different μOPOs. Our simulations are based on a coupled mode theory for optical waveguides,2525. W.-P. Huang, “Coupled-mode theory for optical waveguides: An overview,” J. Opt. Soc. Am. A 11, 963–983 (1994). https://doi.org/10.1364/josaa.11.000963 which calculates κ according to the following equation:where R is the microring outer radius, c is the speed of light, ng is the group refractive index, and kt is a coupling coefficient defined askt=iω4∫L∫A(εWG−εR)ER*⋅EWGdrdzeiϕdl,(4)where ɛWG(R) is the dielectric permittivity of the access waveguide (microring), EWG(R) is the electric field of the waveguide (microring) eigenmode, and ϕ is an accumulated phase accounting for the difference in the waveguide and microring propagation constants. The coordinates r and z are horizontal and vertical coordinates in-plane with the microring/waveguide cross section, as labeled in the inset of Fig. 2(a), and l follows the direction of light propagation. The central integral in Eq. (4) evaluates the evanescent overlap between the microring and waveguide modes at the frequency ω. For more details, see Ref. 2222. G. Moille, Q. Li, T. C. Briles, S.-P. Yu, T. Drake, X. Lu, A. Rao, D. Westly, S. B. Papp, and K. Srinivasan, “Broadband resonator-waveguide coupling for efficient extraction of octave-spanning microcombs,” Opt. Lett. 44, 4737–4740 (2019). https://doi.org/10.1364/ol.44.004737. Figure 2(a) shows simulated κ spectra for a SiN microring with R = 25 μm, ring width RW = 825 nm, and height H = 500 nm, which are chosen to suitably engineer the GVD (see the supplementary material). In addition, we choose WW = 375 nm, G = 150 nm, and Lp from 0 μm (i.e., a straight-waveguide coupler) to 5 μm. Increasing Lp results in larger κ at higher frequencies compared to the straight-waveguide coupler (dark gray line), but it introduces resonances in the κ spectra (i.e., regions where κ → 0) that arise from the coherent energy exchange between the microring and waveguide. These resonances blueshift when Lp is increased. Based on these data, we select Lp = 3 μm for further study because it minimizes variations in κ over the targeted spectral region.Next, we optimize WW and G. The predominant effect of changing G is to vertically shift the entire κ spectrum; i.e., G has little impact on the spectral location of the coupling resonances. However, the relationship between κ and WW is more complex. Within the range of values considered, larger WW increases κ because the waveguide propagation constant shifts closer to the microring propagation constant, and the evanescent overlap between the microring and waveguide modes is not appreciably changed. At the same time, coupling resonances are redshifted. Figure 2(b) shows κ spectra for G = 150 nm, Lp = 3 μm, and three values of WW. Apparently, the flattest κ spectra are realized for WW between 375 and 400 nm. After choosing WW, G may be chosen to realize critical coupling near ωp.To assess the accuracy of our simulations, we fabricate an array of SiN microrings with systematic coupling parameter variations (see the supplementary material for details on our fabrication process), and we measure κ for each device. Specifically, we use either Lp = 0 or Lp = 3 μm, G between 110 and 160 nm, and WW between 350 and 400 nm. We carry out mode spectroscopy using a CW titanium sapphire (TiS) laser, which is tunable from 305 to 415 THz (720–980 nm), and from the microresonator transmission spectra, we calculate K and κ values following Pfieffer et al.2626. M. H. Pfeiffer, J. Liu, M. Geiselmann, and T. J. Kippenberg, “Coupling ideality of integrated planar high-Q microresonators,” Phys. Rev. Appl. 7, 024026 (2017). https://doi.org/10.1103/physrevapplied.7.024026 In addition, we can perform sum frequency generation with the TiS laser and a 154 THz (1950 nm) laser to generate coherent light from 460 to 510 THz (590–650 nm). We find that simulations predict slightly larger κ values than we measure; to compensate, we reduce G by ∼20 nm in experiments. In Fig. 2(a), we present κ measurements (gray circles) of a straight-waveguide-coupled device with WW = 375 nm and G = 125 nm. The measured κ values are slightly lower than the corresponding simulation with G = 150 nm, but both data decrease exponentially with frequency, which is a known characteristic of straight-waveguide couplers.2222. G. Moille, Q. Li, T. C. Briles, S.-P. Yu, T. Drake, X. Lu, A. Rao, D. Westly, S. B. Papp, and K. Srinivasan, “Broadband resonator-waveguide coupling for efficient extraction of octave-spanning microcombs,” Opt. Lett. 44, 4737–4740 (2019). https://doi.org/10.1364/ol.44.004737 Specifically, we measure κ ≈ 800 MHz near 350 THz, but it drops sharply to κ ≈ 40 MHz near 500 THz. In contrast, we observe a more achromatic κ spectrum in a pulley-waveguide-coupled device with WW = 375 nm, G = 135 nm, and Lp = 3 μm. Our measurements (green circles) are shown in Fig. 2(b), and they agree with the corresponding simulation with G = 150 nm. Our measurements indicate that between 300 and 500 THz, κ takes on values over the relatively narrow range (compared to the straight-waveguide coupler) of 180–400 MHz. We also measure γ ≈ 300 MHz that is approximately independent of optical frequency for the wavelengths of interest (we have observed that γ increases at higher frequencies, but this behavior varies between different fabrication runs and requires further examination). Therefore, according to Eq. (2), our best pulley-waveguide-coupled devices should support many different μOPOs with CE > 1%.

III. μOPO GENERATION AND CONVERSION EFFICIENCY MEASUREMENTS

Section:

ChooseTop of pageABSTRACTI. INTRODUCTIONII. DESIGN AND TEST OF WA...III. μOPO GENERATION AND ... <<IV. LIMITATIONS ON OUTPUT...V. DISCUSSIONSUPPLEMENTARY MATERIALREFERENCESTo test our prediction, we record μOPO spectra with a calibrated optical spectrum analyzer (OSA) and calculate Ps, Pi, and CE values after accounting for optical losses between the waveguide and OSA (for details, see the supplementary material). To generate μOPOs, we tune ωp into resonance, starting blue detuned and decreasing ωp until Ps and Pi are maximized. We repeat this procedure for different Pin values with the goal of maximizing CE. To ensure a variety of ωs(i) values, we engineer the GVD by systematically varying RW in different devices (see Ref. 1515. X. Lu, G. Moille, A. Rao, D. A. Westly, and K. Srinivasan, “On-chip optical parametric oscillation into the visible: Generating red, orange, yellow, and green from a near-infrared pump,” Optica 7, 1417–1425 (2020). https://doi.org/10.1364/optica.393810 and the supplementary material). We utilize pulley-waveguide couplers such as those characterized in Fig. 2, and we find that CE is maximized for Lp = 3 μm, WW between 375 and 400 nm, and G between 125 and 135 nm. In Fig. 3(a), we present a compiled set of μOPO spectra that we extract from pulley-waveguide couplers with parameters in the above optimum range. In most cases, both Ps and Pi are greater than 1 mW, and Pin is typically between 30 and 45 mW. However, there are atypical μOPO spectra for which either Ps or Pi is ≪1 mW. Most likely, these result from mode interactions that locally alter the microring GVD and Q.27,2827. T. Herr, V. Brasch, J. D. Jost, I. Mirgorodskiy, G. Lihachev, M. L. Gorodetsky, and T. J. Kippenberg, “Mode spectrum and temporal soliton formation in optical microresonators,” Phys. Rev. Lett. 113, 123901 (2014). https://doi.org/10.1103/physrevlett.113.12390128. S. Ramelow, A. Farsi, S. Clemmen, J. S. Levy, A. R. Johnson, Y. Okawachi, M. R. E. Lamont, M. Lipson, and A. L. Gaeta, “Strong polarization mode coupling in microresonators,” Opt. Lett. 39, 5134–5137 (2014). https://doi.org/10.1364/ol.39.005134 Table II of the supplementary material lists the individual pump, signal, and idler frequencies and powers for each μOPO spectrum shown in Fig. 3(a).To characterize the μOPO performance, we calculate from Fig. 3(a) the largest values of Ps and Pi in spectral bins spanning ∼20 THz each, and we plot the results in Fig. 3(b) along with the corresponding CE values. We find Pi > 1 mW from 264 to 346 THz and Pi > 2 mW from 275 to 346 THz. In the best case, we generate 4 mW of idler power at 315 THz using Pin = 34 mW, which equates to CE ≈ 12%. Meanwhile, Ps > 1.9 mW from 416 to 506 THz, and in the best case, we generate 5 mW of signal power at 454 THz using Pin = 34 mW, which equates to CE ≈ 14.5%. Moreover, as expected from our simulations and measurements of κ and Eq. (2), CE decreases in the spectral wings as a result of smaller κ. Indeed, we can make a useful comparison between our CE measurements and a theoretical prediction (gray dashed line) based on our coupling measurements and Eq. (2). The central portion of data indicates CE values smaller than our theoretical prediction due to competing nonlinear processes that are exacerbated for narrow-band spectra.2929. J. R. Stone, G. Moille, X. Lu, and K. Srinivasan, “Conversion efficiency in kerr-microresonator optical parametric oscillators: From three modes to many modes,” Phys. Rev. Appl. 17, 024038 (2022). https://doi.org/10.1103/physrevapplied.17.024038 Discrepancies at higher frequencies are primarily due to imperfect frequency mismatch arising from the stronger impact of higher-order dispersion. In between these extremes, we observe good agreement between our measurements and theory.

IV. LIMITATIONS ON OUTPUT POWER

Section:

ChooseTop of pageABSTRACTI. INTRODUCTIONII. DESIGN AND TEST OF WA...III. μOPO GENERATION AND ...IV. LIMITATIONS ON OUTPUT... <<V. DISCUSSIONSUPPLEMENTARY MATERIALREFERENCESFinally, we discuss the limits to μOPO output power and analyze an example. The relationship between Ps(i), Pin, and other experimental parameters has been analyzed theoretically.2929. J. R. Stone, G. Moille, X. Lu, and K. Srinivasan, “Conversion efficiency in kerr-microresonator optical parametric oscillators: From three modes to many modes,” Phys. Rev. Appl. 17, 024038 (2022). https://doi.org/10.1103/physrevapplied.17.024038 Therein, it was predicted that Ps(i) increases with Pin for Pin ≳ Pth, but further increases in Pin lead to saturation or even reduction of Ps(i). The reason is that parasitic FWM processes compete with the targeted μOPO process. The predominant parasitic FWM process that we observe in experiments is mode competition between the targeted signal/idler modes and their spectral neighbors.2929. J. R. Stone, G. Moille, X. Lu, and K. Srinivasan, “Conversion efficiency in kerr-microresonator optical parametric oscillators: From three modes to many modes,” Phys. Rev. Appl. 17, 024038 (2022). https://doi.org/10.1103/physrevapplied.17.024038 In Fig. 4, we show the μOPO spectrum for a single device as Pin is increased. For Pin = 32 mW, only the pump, signal, and idler modes oscillate, and Ps(i) > 1 mW (top panel). The idler power in this case is marked by the red line in each panel. When Pin = 50 mW (middle panel), ωs and ωi shift to higher and lower frequencies, respectively. This behavior was predicted in Ref. 2929. J. R. Stone, G. Moille, X. Lu, and K. Srinivasan, “Conversion efficiency in kerr-microresonator optical parametric oscillators: From three modes to many modes,” Phys. Rev. Appl. 17, 024038 (2022). https://doi.org/10.1103/physrevapplied.17.024038 and termed “mode switching.” In addition, other modes with frequencies close to ωs(i) begin to oscillate and steal energy from the μOPO. Hence, Pi decreases to a level below the red line, despite the increase in Pin. When Pin is further increased to 70 mW (bottom panel), Ps(i) remains approximately the same, but the power in the competing modes increases. The behavior demonstrated in this example is ubiquitous within our μOPO devices and explains why Ps(i) cannot be increased arbitrarily by increasing Pin. Still, increasing CE and Ps(i) beyond the levels we demonstrate may be possible using alternate phase- and frequency-matching strategies, such as that reported in Ref. 3030. F. Zhou, X. Lu, A. Rao, J. Stone, G. Moille, E. Perez, D. Westly, and K. Srinivasan, “Hybrid-mode-family kerr optical parametric oscillation for robust coherent light generation on chip,” Laser Photonics Rev. 16, 2100582 (2022). https://doi.org/10.1002/lpor.202100582. As it stands, the achieved power levels are relevant for some applications, such as spectroscopy of various coherent near-infrared and visible systems.31,3231. X. Xu, B. Sun, P. R. Berman, D. G. Steel, A. S. Bracker, D. Gammon, and L. J. Sham, “Coherent optical spectroscopy of a strongly driven quantum dot,” Science 317, 929–932 (2007). https://doi.org/10.1126/science.114297932. D. V. Brazhnikov, S. M. Ignatovich, I. S. Mesenzova, A. M. Mikhailov, R. Boudot, and M. N. Skvortsov, “Two-frequency sub-Doppler spectroscopy of the caesium D1 line in various configurations of counterpropagating laser beams,” Quantum Electron. 50, 1015 (2020). https://doi.org/10.1070/qel17433