Visible Kerr comb generation in a high-Q silica microdisk resonator with a large wedge angle
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摘要
This paper describes the specially designed geometry of a dry-etched large-wedge-angle silica microdisk resonator that enables anomalous dispersion in the 780 nm wavelength regime. This anomalous dispersion occurs naturally without the use of a mode-hybridization technique to control the geometrical dispersion. By fabricating a1-μm-thick silica microdisk with a wedge angle as large as 56° and an optical Q-factor larger than 107, we achieve a visible Kerr comb that covers the wavelength interval of 700–897 nm. The wide optical frequency range and the closeness to the clock transition at 698 nm of 87 Sr atoms make our visible comb a potentially useful tool in optical atomic clock applications.
This paper describes the specially designed geometry of a dry-etched large-wedge-angle silica microdisk resonator that enables anomalous dispersion in the 780 nm wavelength regime. This anomalous dispersion occurs naturally without the use of a mode-hybridization technique to control the geometrical dispersion. By fabricating a1-μm-thick silica microdisk with a wedge angle as large as 56° and an optical Q-factor larger than 107, we achieve a visible Kerr comb that covers the wavelength interval of 700–897 nm. The wide optical frequency range and the closeness to the clock transition at 698 nm of 87 Sr atoms make our visible comb a potentially useful tool in optical atomic clock applications.
This paper describes the specially designed geometry of a dry-etched large-wedge-angle silica microdisk resonator that enables anomalous dispersion in the 780 nm wavelength regime. This anomalous dispersion occurs naturally without the use of a mode-hybridization technique to control the geometrical dispersion. By fabricating a1-μm-thick silica microdisk with a wedge angle as large as 56° and an optical Q-factor larger than 107, we achieve a visible Kerr comb that covers the wavelength interval of 700–897 nm. The wide optical frequency range and the closeness to the clock transition at 698 nm of 87 Sr atoms make our visible comb a potentially useful tool in optical atomic clock applications.
引文
1.T.Udem,R.Holzwarth,and T.W.H?nsch,“Optical frequency metrology,”Nature 416,233-237(2002).
2.A.D.Ludlow,M.M.Boyd,and J.Ye,“Optical atomic clocks,”Rev.Mod.Phys.87,637-701(2015).
3.R.Holzwarth,T.Udem,T.W.H?nsch,J.C.Knight,W.J.Wadsworth,and P.St.J.Russell,“Optical frequency synthesizer for precision spectroscopy,”Phys.Rev.Lett.85,2264-2267(2000).
4.T.Steinmetz,T.Wilken,C.Araujo-Hauck,R.Holzwarth,T.W.H?nsch,L.Pasquini,A.Manescau,S.D’Odorico,M.T.Murphy,T.Kentischer,W.Schmidt,and T.Udem,“Laser frequency combs for astronomical observations,”Science 321,1335-1337(2008).
5.T.J.Kippenberg,R.Holzwarth,and S.A.Diddams,“Microresonatorbased optical frequency combs,”Science 332,555-559(2011).
6.P.Del’Haye,A.Schliesser,O.Arcizet,T.Wilken,R.Holzwarth,and T.J.Kippenberg,“Optical frequency comb generation from a monolithic microresonator,”Nature 450,1214-1217(2007).
7.Y.Li,X.Jiang,G.Zhao,and L.Yang,“Whispering gallery mode microresonator for nonlinear photonics,”arXiv:1809.04878(2017).
8.G.Lin,A.Coillet,and Y.K.Chembo,“Nonlinear photonics with high-Qwhispering-gallery-mode resonators,”Adv.Opt.Photon.9,828-890(2017).
9.A.A.Savchenkov,A.B.Matsko,V.S.Ilchenko,I.Solomatine,D.Seidel,and L.Maleki,“Tunable optical frequency comb with a crystalline whispering gallery mode resonator,”Phys.Rev.Lett.101,093902(2008).
10.I.S.Grudinin,N.Yu,and L.Maleki,“Generation of optical frequency combs with a CaF2resonator,”Opt.Lett.34,878-880(2009).
11.S.B.Papp and S.A.Diddams,“Spectral and temporal characterization of a fused-quartz-microresonator optical frequency comb,”Phys.Rev.A 84,053833(2011).
12.B.J.M.Hausmann,I.Bulu,V.Venkataraman,P.Deotare,and M.Lonar,“Diamond nonlinear photonics,”Nat.Photonics 8,369-374(2014).
13.H.Jung,C.Xiong,K.Y.Fong,X.Zhang,and H.X.Tang,“Optical frequency comb generation from aluminum nitride microring resonator,”Opt.Lett.38,2810-2813(2013).
14.T.Herr,K.Hartinger,J.Riemensberger,C.Y.Wang,E.Gavartin,R.Holzwarth,M.L.Gorodetsky,and T.J.Kippenberg,“Universal formation dynamics and noise of Kerr-frequency combs in microresonators,”Nat.Photonics 6,480-487(2012).
15.M.A.Foster,J.S.Levy,O.Kuzucu,K.Saha,M.Lipson,and A.L.Gaeta,“Silicon-based monolithic optical frequency comb source,”Opt.Express 19,14233-14239(2011).
16.Q.Lu,S.Liu,X.Wu,L.Liu,and L.Xu,“Stimulated Brillouin laser and frequency comb generation in high-Q microbubble resonators,”Opt.Lett.41,1736-1739(2016).
17.J.S.Levy,A.Gondarenko,M.A.Foster,A.C.Turner-Foster,A.L.Gaeta,and M.Lipson,“CMOS-compatible multiple-wavelength oscillator for on-chip optical interconnects,”Nat.Photonics 4,37-40(2010).
18.L.Razzari,D.Duchesne,M.Ferrera,R.Morandotti,S.Chu,B.E.Little,and D.J.Moss,“CMOS-compatible integrated optical hyperparametric oscillator,”Nat.Photonics 4,41-45(2010).
19.F.Ferdous,H.Miao,D.E.Leaird,K.Srinivasan,J.Wang,L.Chen,L.T.Vargheses,and A.M.Weiner,“Spectral line-by-line pulse shaping of on-chip microresonator frequency combs,”Nat.Photonics 5,770-776(2011).
20.J.Li,H.Lee,T.Chen,and K.J.Vahala,“Low-pump-power,low phase-noise,and microwave to millimeter-wave repetition rate operation in microcombs,”Phys.Rev.Lett.109,233901(2012).
21.J.Ma,X.Jiang,and M.Xiao,“Kerr frequency combs in large-size,ultra-high-Q toroid microcavities with low repetition rates,”Photon.Res.5,B54-B58(2017).
22.T.Herr,V.Brasch,J.D.Jost,C.Y.Wang,N.M.Kondratiev,M.L.Gorodetsky,and T.J.Kippenberg,“Temporal solitons in optical microresonators,”Nat.Photonics 8,145-152(2014).
23.P.M.Palomo,J.N.Kemal,M.Karpov,A.Kordts,J.Pfeifle,M.H.P.Pfeiffer,P.Trocha,S.Wolf,V.Brasch,M.H.Anderson,R.Rosenberger,K.Vijayan,W.Freude,T.J.Kippenberg,and C.Koos,“Microresonator-based solitons for massively parallel coherent optical communications,”Nature 546,274-279(2017).
24.D.T.Spencer,T.Drake,T.C.Briles,J.Stone,L.C.Sinclair,C.Fredrick,Q.Li,D.Westly,B.R.Ilic,A.Bluestone,N.Volet,T.Komljenovic,L.Chang,S.H.Lee,D.Y.Oh,M.-G.Suh,K.Y.Yang,M.H.P.Pfeiffer,T.J.Kippenberg,E.Norberg,L.Theogarajan,K.Vahala,N.R.Newbury,K.Srinivasan,J.E.Bowers,S.A.Diddams,and S.B.Papp,“An optical-frequency synthesizer using integrated photonics,”Nature 557,81-85(2018).
25.M.-G.Suh and K.J.Vahala,“Soliton microcomb range measurement,”Science 359,884-887(2018).
26.P.Trocha,M.Karpov,D.Ganin,M.H.P.Pfeiffer,A.Kordts,S.Wolf,J.Krockenberger,P.Marin-Palomo,C.Weimann,S.Randel,W.Freude,T.J.Kippenberg,and C.Koos,“Ultrafast optical ranging using microresonator soliton frequency combs,”Science 359,887-891(2018).
27.E.Obrzud,M.Rainer,A.Harutyunyan,M.H.Anderson,J.Liu,M.Geiselmann,B.Chazelas,S.Kundermann,S.Lecomte,M.Cecconi,A.Ghedina,E.Molinari,F.Pepe,F.Wildi,F.Bouchy,T.J.Kippenberg,and T.Herr,“A microphotonic astrocomb,”Nat.Photonics 13,31-35(2019).
28.M.-G.Suh,X.Yi,Y.-H.Lai,S.Leifer,I.S.Grudinin,G.Vasisht,E.C.Martin,M.P.Fitzgerald,G.Doppmann,J.Wang,D.Mawet,S.B.Papp,S.A.Diddams,C.Beichman,and K.Vahala,“Searching for exoplanets using a microresonator astrocomb,”Nat.Photonics 13,25-30(2019).
29.X.Xue,F.Leo,Y.Xuan,J.A.Jaramillo-Villegas,P.-H.Wang,D.E.Leaird,M.Erkintalo,M.Qi,and A.M.Weiner,“Second-harmonicassisted four-wave mixing in chip-based microresonator frequency comb generation,”Light Sci.Appl.6,e16253(2017).
30.H.Jung,R.Stoll,X.Guo,D.Fischer,and H.X.Tang,“Green,red,and IR frequency comb line generation from single IR pump in AlN microring resonator,”Optica 1,396-399(2014).
31.L.Wang,L.Chang,N.Volet,M.H.P.Pfeiffer,M.Zervas,H.Guo,T.J.Kippenberg,and J.E.Bowers,“Frequency comb generation in the green using silicon nitride microresonators,”Laser Photon.Rev.10,631-638(2016).
32.X.Liu,C.Sun,B.Xiong,L.Wang,Y.Han,Z.Hao,H.Li,Y.Luo,J.Yan,T.Wei,Y.Zhang,and J.Wang,“Generation of multiple near-visible comb lines in an AlN microring viaχ(2)andχ(3)optical nonlinearities,”Appl.Phys.Lett.113,171106(2018).
33.X.Guo,C.-L.Zou,H.Jung,Z.Gong,A.Bruch,L.Jiang,and H.X.Tang,“Efficient generation of a near-visible frequency comb via Cherenkov-like radiation from a Kerr microcomb,”Phys.Rev.Appl.10,014012(2018).
34.A.A.Savchenkov,A.B.Matsko,W.Liang,V.S.Ilchenko,D.Seidel,and L.Maleki,“Kerr combs with selectable central frequency,”Nat.Photonics 5,293-296(2011).
35.Y.Yang,X.Jiang,S.Kasumie,G.Zhao,L.Xu,J.M.Ward,L.Yang,and S.N.Chormaic,“Four-wave mixing parametric oscillation and frequency comb generation at visible wavelengths in a silica microbubble resonator,”Opt.Lett.41,5266-5269(2016).
36.S.H.Lee,D.Y.Oh,Q.-F.Yang,B.Shen,H.Wang,K.Y.Yang,Y.H.Lai,X.Yi,and K.Vahala,“Towards visible soliton microcomb generation,”Nat.Commun.8,1295(2017).
37.M.Karpov,M.H.P.Pfeiffer,J.Liu,A.Lukashchuk,and T.J.Kippenberg,“Photonic chip-based soliton frequency combs covering the biological imaging window,”Nat.Commun.9,1146(2018).
38.P.S.Donvalkar,F.A.S.Barbosa,X.Ji,Y.Okawachi,R.Mcnally,A.Farsi,A.Klenner,M.Lipson,and A.L.Gaeta,“Broadband frequency comb generation in the near-visible using higher-order modes in silicon nitride microresonators,”in Conference on Lasers and ElectroOptics(2017),paper STu4J.5.
39.G.Li,P.Liu,X.Jiang,C.Yang,J.Ma,H.Wu,and M.Xiao,“High-Qsilica microdisk optical resonators with large wedge angles on a silicon chip,”Photon.Res.3,279-282(2015).
40.X.Jiang,Q.Lin,J.Rosenberg,K.Vahala,and O.Painter,“High-Qdouble-disk microcavities for cavity optomechanics,”Opt.Express17,20911-20919(2009).
41.H.Lee,T.Chen,J.Li,K.Y.Yang,S.Jeon,O.Painter,and K.J.Vahala,“Chemically etched ultrahigh-Q wedge-resonator on a silicon chip,”Nat.Photonics 6,369-373(2012).
42.M.Oxborrow,“Traceable 2D finite-element simulation of the whispering gallery modes of axisymmetric electromagnetic resonators,”IEEETrans.Microwave Theory Tech.55,1209-1218(2007).
43.M.L.Gorodetsky and V.S.Ilchenko,“High-Q optical whisperinggallery microresonators:precession approach for spherical mode analysis and emission patterns with prism couplers,”Opt.Commun.113,133-143(1994).
44.T.J.Kippenberg,S.M.Spillane,and K.J.Vahala,“Modal coupling in traveling-wave resonators,”Opt.Lett.27,1669-1671(2002).
45.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).
46.Y.K.Chembo and N.Yu,“Modal expansion approach to opticalfrequency-comb generation with monolithic whispering-gallery-mode resonators,”Phys.Rev.A 82,033801(2010).
47.T.J.Kippenberg,A.L.Gaeta,M.Lipson,and M.L.Gorodetsky,“Dissipative Kerr solitons in optical microresonators,”Science 361,eaan8083(2018).
2.A.D.Ludlow,M.M.Boyd,and J.Ye,“Optical atomic clocks,”Rev.Mod.Phys.87,637-701(2015).
3.R.Holzwarth,T.Udem,T.W.H?nsch,J.C.Knight,W.J.Wadsworth,and P.St.J.Russell,“Optical frequency synthesizer for precision spectroscopy,”Phys.Rev.Lett.85,2264-2267(2000).
4.T.Steinmetz,T.Wilken,C.Araujo-Hauck,R.Holzwarth,T.W.H?nsch,L.Pasquini,A.Manescau,S.D’Odorico,M.T.Murphy,T.Kentischer,W.Schmidt,and T.Udem,“Laser frequency combs for astronomical observations,”Science 321,1335-1337(2008).
5.T.J.Kippenberg,R.Holzwarth,and S.A.Diddams,“Microresonatorbased optical frequency combs,”Science 332,555-559(2011).
6.P.Del’Haye,A.Schliesser,O.Arcizet,T.Wilken,R.Holzwarth,and T.J.Kippenberg,“Optical frequency comb generation from a monolithic microresonator,”Nature 450,1214-1217(2007).
7.Y.Li,X.Jiang,G.Zhao,and L.Yang,“Whispering gallery mode microresonator for nonlinear photonics,”arXiv:1809.04878(2017).
8.G.Lin,A.Coillet,and Y.K.Chembo,“Nonlinear photonics with high-Qwhispering-gallery-mode resonators,”Adv.Opt.Photon.9,828-890(2017).
9.A.A.Savchenkov,A.B.Matsko,V.S.Ilchenko,I.Solomatine,D.Seidel,and L.Maleki,“Tunable optical frequency comb with a crystalline whispering gallery mode resonator,”Phys.Rev.Lett.101,093902(2008).
10.I.S.Grudinin,N.Yu,and L.Maleki,“Generation of optical frequency combs with a CaF2resonator,”Opt.Lett.34,878-880(2009).
11.S.B.Papp and S.A.Diddams,“Spectral and temporal characterization of a fused-quartz-microresonator optical frequency comb,”Phys.Rev.A 84,053833(2011).
12.B.J.M.Hausmann,I.Bulu,V.Venkataraman,P.Deotare,and M.Lonar,“Diamond nonlinear photonics,”Nat.Photonics 8,369-374(2014).
13.H.Jung,C.Xiong,K.Y.Fong,X.Zhang,and H.X.Tang,“Optical frequency comb generation from aluminum nitride microring resonator,”Opt.Lett.38,2810-2813(2013).
14.T.Herr,K.Hartinger,J.Riemensberger,C.Y.Wang,E.Gavartin,R.Holzwarth,M.L.Gorodetsky,and T.J.Kippenberg,“Universal formation dynamics and noise of Kerr-frequency combs in microresonators,”Nat.Photonics 6,480-487(2012).
15.M.A.Foster,J.S.Levy,O.Kuzucu,K.Saha,M.Lipson,and A.L.Gaeta,“Silicon-based monolithic optical frequency comb source,”Opt.Express 19,14233-14239(2011).
16.Q.Lu,S.Liu,X.Wu,L.Liu,and L.Xu,“Stimulated Brillouin laser and frequency comb generation in high-Q microbubble resonators,”Opt.Lett.41,1736-1739(2016).
17.J.S.Levy,A.Gondarenko,M.A.Foster,A.C.Turner-Foster,A.L.Gaeta,and M.Lipson,“CMOS-compatible multiple-wavelength oscillator for on-chip optical interconnects,”Nat.Photonics 4,37-40(2010).
18.L.Razzari,D.Duchesne,M.Ferrera,R.Morandotti,S.Chu,B.E.Little,and D.J.Moss,“CMOS-compatible integrated optical hyperparametric oscillator,”Nat.Photonics 4,41-45(2010).
19.F.Ferdous,H.Miao,D.E.Leaird,K.Srinivasan,J.Wang,L.Chen,L.T.Vargheses,and A.M.Weiner,“Spectral line-by-line pulse shaping of on-chip microresonator frequency combs,”Nat.Photonics 5,770-776(2011).
20.J.Li,H.Lee,T.Chen,and K.J.Vahala,“Low-pump-power,low phase-noise,and microwave to millimeter-wave repetition rate operation in microcombs,”Phys.Rev.Lett.109,233901(2012).
21.J.Ma,X.Jiang,and M.Xiao,“Kerr frequency combs in large-size,ultra-high-Q toroid microcavities with low repetition rates,”Photon.Res.5,B54-B58(2017).
22.T.Herr,V.Brasch,J.D.Jost,C.Y.Wang,N.M.Kondratiev,M.L.Gorodetsky,and T.J.Kippenberg,“Temporal solitons in optical microresonators,”Nat.Photonics 8,145-152(2014).
23.P.M.Palomo,J.N.Kemal,M.Karpov,A.Kordts,J.Pfeifle,M.H.P.Pfeiffer,P.Trocha,S.Wolf,V.Brasch,M.H.Anderson,R.Rosenberger,K.Vijayan,W.Freude,T.J.Kippenberg,and C.Koos,“Microresonator-based solitons for massively parallel coherent optical communications,”Nature 546,274-279(2017).
24.D.T.Spencer,T.Drake,T.C.Briles,J.Stone,L.C.Sinclair,C.Fredrick,Q.Li,D.Westly,B.R.Ilic,A.Bluestone,N.Volet,T.Komljenovic,L.Chang,S.H.Lee,D.Y.Oh,M.-G.Suh,K.Y.Yang,M.H.P.Pfeiffer,T.J.Kippenberg,E.Norberg,L.Theogarajan,K.Vahala,N.R.Newbury,K.Srinivasan,J.E.Bowers,S.A.Diddams,and S.B.Papp,“An optical-frequency synthesizer using integrated photonics,”Nature 557,81-85(2018).
25.M.-G.Suh and K.J.Vahala,“Soliton microcomb range measurement,”Science 359,884-887(2018).
26.P.Trocha,M.Karpov,D.Ganin,M.H.P.Pfeiffer,A.Kordts,S.Wolf,J.Krockenberger,P.Marin-Palomo,C.Weimann,S.Randel,W.Freude,T.J.Kippenberg,and C.Koos,“Ultrafast optical ranging using microresonator soliton frequency combs,”Science 359,887-891(2018).
27.E.Obrzud,M.Rainer,A.Harutyunyan,M.H.Anderson,J.Liu,M.Geiselmann,B.Chazelas,S.Kundermann,S.Lecomte,M.Cecconi,A.Ghedina,E.Molinari,F.Pepe,F.Wildi,F.Bouchy,T.J.Kippenberg,and T.Herr,“A microphotonic astrocomb,”Nat.Photonics 13,31-35(2019).
28.M.-G.Suh,X.Yi,Y.-H.Lai,S.Leifer,I.S.Grudinin,G.Vasisht,E.C.Martin,M.P.Fitzgerald,G.Doppmann,J.Wang,D.Mawet,S.B.Papp,S.A.Diddams,C.Beichman,and K.Vahala,“Searching for exoplanets using a microresonator astrocomb,”Nat.Photonics 13,25-30(2019).
29.X.Xue,F.Leo,Y.Xuan,J.A.Jaramillo-Villegas,P.-H.Wang,D.E.Leaird,M.Erkintalo,M.Qi,and A.M.Weiner,“Second-harmonicassisted four-wave mixing in chip-based microresonator frequency comb generation,”Light Sci.Appl.6,e16253(2017).
30.H.Jung,R.Stoll,X.Guo,D.Fischer,and H.X.Tang,“Green,red,and IR frequency comb line generation from single IR pump in AlN microring resonator,”Optica 1,396-399(2014).
31.L.Wang,L.Chang,N.Volet,M.H.P.Pfeiffer,M.Zervas,H.Guo,T.J.Kippenberg,and J.E.Bowers,“Frequency comb generation in the green using silicon nitride microresonators,”Laser Photon.Rev.10,631-638(2016).
32.X.Liu,C.Sun,B.Xiong,L.Wang,Y.Han,Z.Hao,H.Li,Y.Luo,J.Yan,T.Wei,Y.Zhang,and J.Wang,“Generation of multiple near-visible comb lines in an AlN microring viaχ(2)andχ(3)optical nonlinearities,”Appl.Phys.Lett.113,171106(2018).
33.X.Guo,C.-L.Zou,H.Jung,Z.Gong,A.Bruch,L.Jiang,and H.X.Tang,“Efficient generation of a near-visible frequency comb via Cherenkov-like radiation from a Kerr microcomb,”Phys.Rev.Appl.10,014012(2018).
34.A.A.Savchenkov,A.B.Matsko,W.Liang,V.S.Ilchenko,D.Seidel,and L.Maleki,“Kerr combs with selectable central frequency,”Nat.Photonics 5,293-296(2011).
35.Y.Yang,X.Jiang,S.Kasumie,G.Zhao,L.Xu,J.M.Ward,L.Yang,and S.N.Chormaic,“Four-wave mixing parametric oscillation and frequency comb generation at visible wavelengths in a silica microbubble resonator,”Opt.Lett.41,5266-5269(2016).
36.S.H.Lee,D.Y.Oh,Q.-F.Yang,B.Shen,H.Wang,K.Y.Yang,Y.H.Lai,X.Yi,and K.Vahala,“Towards visible soliton microcomb generation,”Nat.Commun.8,1295(2017).
37.M.Karpov,M.H.P.Pfeiffer,J.Liu,A.Lukashchuk,and T.J.Kippenberg,“Photonic chip-based soliton frequency combs covering the biological imaging window,”Nat.Commun.9,1146(2018).
38.P.S.Donvalkar,F.A.S.Barbosa,X.Ji,Y.Okawachi,R.Mcnally,A.Farsi,A.Klenner,M.Lipson,and A.L.Gaeta,“Broadband frequency comb generation in the near-visible using higher-order modes in silicon nitride microresonators,”in Conference on Lasers and ElectroOptics(2017),paper STu4J.5.
39.G.Li,P.Liu,X.Jiang,C.Yang,J.Ma,H.Wu,and M.Xiao,“High-Qsilica microdisk optical resonators with large wedge angles on a silicon chip,”Photon.Res.3,279-282(2015).
40.X.Jiang,Q.Lin,J.Rosenberg,K.Vahala,and O.Painter,“High-Qdouble-disk microcavities for cavity optomechanics,”Opt.Express17,20911-20919(2009).
41.H.Lee,T.Chen,J.Li,K.Y.Yang,S.Jeon,O.Painter,and K.J.Vahala,“Chemically etched ultrahigh-Q wedge-resonator on a silicon chip,”Nat.Photonics 6,369-373(2012).
42.M.Oxborrow,“Traceable 2D finite-element simulation of the whispering gallery modes of axisymmetric electromagnetic resonators,”IEEETrans.Microwave Theory Tech.55,1209-1218(2007).
43.M.L.Gorodetsky and V.S.Ilchenko,“High-Q optical whisperinggallery microresonators:precession approach for spherical mode analysis and emission patterns with prism couplers,”Opt.Commun.113,133-143(1994).
44.T.J.Kippenberg,S.M.Spillane,and K.J.Vahala,“Modal coupling in traveling-wave resonators,”Opt.Lett.27,1669-1671(2002).
45.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).
46.Y.K.Chembo and N.Yu,“Modal expansion approach to opticalfrequency-comb generation with monolithic whispering-gallery-mode resonators,”Phys.Rev.A 82,033801(2010).
47.T.J.Kippenberg,A.L.Gaeta,M.Lipson,and M.L.Gorodetsky,“Dissipative Kerr solitons in optical microresonators,”Science 361,eaan8083(2018).