Abstract

We report time domain observations of optical instability in high Q silicon nitride whispering gallery disk resonators. At low laser power the transmitted optical power through the disk looks chaotic. At higher power, the optical output settles into a stable self-pulsing regime with periodicity ranging from hundreds of milliseconds to hundreds of seconds. This phenomenon is explained by the interplay between a fast thermo-optic nonlinearity within the disk and a slow thermo-mechanic nonlinearity of the structure. A model for this interplay is developed which provides good agreement with experimental data and points out routes to control this instability.

© 2012 OSA

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2012

2011

2010

K. H. Lee, T. G. McRae, G. I. Harris, J. Knittel, and W. P. Bowen, “Cooling and control of a cavity optoelectromechanical system,” Phys. Rev. Lett.104(12), 123604 (2010).
[CrossRef] [PubMed]

L. Ding, C. Baker, P. Senellart, A. Lemaitre, S. Ducci, G. Leo, and I. Favero, “High frequency GaAs nano-optomechanical disk resonator,” Phys. Rev. Lett.105(26), 263903 (2010).
[CrossRef] [PubMed]

W. H. P. Pernice, M. Li, and H. X. Tang, “Time-domain measurement of optical transport in silicon micro-ring resonators,” Opt. Express18(17), 18438–18452 (2010), doi:.
[CrossRef] [PubMed]

J. Zhu, S. K. Ozdemir, Y. F. Xiao, L. Li, L. He, D. R. Chen, and L. Yang, “On-chip single nanoparticle detection and sizing by mode splitting in an ultrahigh-Q microresonator,” Nat. Photonics4(1), 46–49 (2010).
[CrossRef]

Q. P. Unterreithmeier, T. Faust, and J. P. Kotthaus, “Damping of nanomechanical resonators,” Phys. Rev. Lett.105(2), 027205 (2010).
[CrossRef] [PubMed]

D. T. H. Tan, K. Ikeda, P. C. Sun, and Y. Fainman, “Group velocity dispersion and self phase modulation in silicon nitride waveguides,” Appl. Phys. Lett.96(6), 061101 (2010), doi:.
[CrossRef]

L. Ding, C. Belacel, S. Ducci, G. Leo, and I. Favero, “Ultralow loss single-mode silica tapers manufactured by a microheater,” Appl. Opt.49(13), 2441–2445 (2010).
[CrossRef]

2009

2008

2007

M. Oxborrow, “How to simulate the whispering gallery modes of dielectric microresonator in FEMLAB/COMSOL,” Proc. SPIE6452(64520J), 64520J (2007).
[CrossRef]

2006

S. S. Verbridge, J. M. Parpia, R. B. Reichenbach, L. M. Bellan, and H. G. Craighead, “High quality factor resonance at room temperature with nanostrings under high tensile stress,” J. Appl. Phys.99(12), 124304 (2006).
[CrossRef]

T. J. Johnson, M. Borselli, and O. Painter, “Self-induced optical modulation of the transmission through a high-Q silicon microdisk resonator,” Opt. Express14(2), 817–831 (2006).
[CrossRef] [PubMed]

T. Aoki, B. Dayan, E. Wilcut, W. P. Bowen, A. S. Parkins, T. J. Kippenberg, K. J. Vahala, and H. J. Kimble, “Observation of strong coupling between one atom and a monolithic microresonator,” Nature443(7112), 671–674 (2006).
[CrossRef] [PubMed]

2005

T. J. Kippenberg, H. Rokhsari, T. Carmon, A. Scherer, and K. J. Vahala, “Analysis of radiation-pressure induced mechanical oscillation of an optical microcavity,” Phys. Rev. Lett.95(3), 033901 (2005).
[CrossRef] [PubMed]

E. Peter, P. Senellart, D. Martrou, A. Lemaître, J. Hours, J. M. Gérard, and J. Bloch, “Exciton-photon strong-coupling regime for a single quantum dot embedded in a microcavity,” Phys. Rev. Lett.95(6), 067401 (2005).
[CrossRef] [PubMed]

A. E. Fomin, M. L. Gorodetsky, I. S. Grudinin, and V. S. Ilchenko, “Nonstationary nonlinear effects in optical microspheres,” J. Opt. Soc. Am. B22(2), 459–465 (2005).
[CrossRef]

A. De Rossi, V. Ortiz, M. Calligaro, L. Lanco, S. Ducci, V. Berger, and I. Sagnes, “Measuring propagation loss in a multimode semiconductor waveguide,” J. Appl. Phys.97(7), 073105 (2005).
[CrossRef]

2004

2002

F. Vollmer, D. Braun, A. Libchaber, M. Khoshsima, I. Teraoka, and S. Arnold, “Protein detection by optical shift of a resonant microcavity,” Appl. Phys. Lett.80(21), 4057 (2002), doi:.
[CrossRef]

2000

1999

C. Manolatou, M. J. Khan, S. Fan, P. R. Villeneuve, H. A. Haus, and J. D. Joannopoulos, “Coupling of modes analysis of resonant channel add-drop filters,” IEEE J. Quantum Electron.35(9), 1322–1331 (1999).
[CrossRef]

S. T. Chu, B. E. Little, W. Pan, T. A. Kaneko, S. A. Sato, and Y. A. Kokubun, “An eight-channel add-drop filter using vertically coupled microring resonators over a cross grid,” IEEE Photon. Technol. Lett.11(6), 691–693 (1999).
[CrossRef]

1996

J. G. E. Gardeniers, H. A. C. Tilmans, and C. C. G. Visser, “LPCVD silicon-rich silicon nitride films for applications in micromechanics, studied with statistical experimental design,” J. Vac. Sci. Technol. A14(5), 2879–2892 (1996).
[CrossRef]

1987

1985

R. Regener and W. Sohler, “Loss in low-finesse Ti: LiNbO3 optical waveguide resonators,” Appl. Phys. B36(3), 143–147 (1985), doi:.
[CrossRef]

1983

Almeida, V. R.

Andronico, A.

Aoki, T.

T. Aoki, B. Dayan, E. Wilcut, W. P. Bowen, A. S. Parkins, T. J. Kippenberg, K. J. Vahala, and H. J. Kimble, “Observation of strong coupling between one atom and a monolithic microresonator,” Nature443(7112), 671–674 (2006).
[CrossRef] [PubMed]

Arnold, S.

F. Vollmer, D. Braun, A. Libchaber, M. Khoshsima, I. Teraoka, and S. Arnold, “Protein detection by optical shift of a resonant microcavity,” Appl. Phys. Lett.80(21), 4057 (2002), doi:.
[CrossRef]

Baker, C.

L. Ding, C. Baker, P. Senellart, A. Lemaitre, S. Ducci, G. Leo, and I. Favero, “Wavelength-sized GaAs optomechanical resonators with gigahertz frequency,” Appl. Phys. Lett.98, 113801 (2011).

L. Ding, C. Baker, P. Senellart, A. Lemaitre, S. Ducci, G. Leo, and I. Favero, “High frequency GaAs nano-optomechanical disk resonator,” Phys. Rev. Lett.105(26), 263903 (2010).
[CrossRef] [PubMed]

Bauters, J. F.

Belacel, C.

Bellan, L. M.

S. S. Verbridge, J. M. Parpia, R. B. Reichenbach, L. M. Bellan, and H. G. Craighead, “High quality factor resonance at room temperature with nanostrings under high tensile stress,” J. Appl. Phys.99(12), 124304 (2006).
[CrossRef]

Berger, V.

A. De Rossi, V. Ortiz, M. Calligaro, L. Lanco, S. Ducci, V. Berger, and I. Sagnes, “Measuring propagation loss in a multimode semiconductor waveguide,” J. Appl. Phys.97(7), 073105 (2005).
[CrossRef]

Bloch, J.

E. Peter, P. Senellart, D. Martrou, A. Lemaître, J. Hours, J. M. Gérard, and J. Bloch, “Exciton-photon strong-coupling regime for a single quantum dot embedded in a microcavity,” Phys. Rev. Lett.95(6), 067401 (2005).
[CrossRef] [PubMed]

Blumenthal, D. J.

Borselli, M.

Bowen, W. P.

K. H. Lee, T. G. McRae, G. I. Harris, J. Knittel, and W. P. Bowen, “Cooling and control of a cavity optoelectromechanical system,” Phys. Rev. Lett.104(12), 123604 (2010).
[CrossRef] [PubMed]

T. Aoki, B. Dayan, E. Wilcut, W. P. Bowen, A. S. Parkins, T. J. Kippenberg, K. J. Vahala, and H. J. Kimble, “Observation of strong coupling between one atom and a monolithic microresonator,” Nature443(7112), 671–674 (2006).
[CrossRef] [PubMed]

Bowers, J. E.

Braun, D.

F. Vollmer, D. Braun, A. Libchaber, M. Khoshsima, I. Teraoka, and S. Arnold, “Protein detection by optical shift of a resonant microcavity,” Appl. Phys. Lett.80(21), 4057 (2002), doi:.
[CrossRef]

Calligaro, M.

A. De Rossi, V. Ortiz, M. Calligaro, L. Lanco, S. Ducci, V. Berger, and I. Sagnes, “Measuring propagation loss in a multimode semiconductor waveguide,” J. Appl. Phys.97(7), 073105 (2005).
[CrossRef]

Camacho, R. M.

Cao, T.

Carmon, T.

T. J. Kippenberg, H. Rokhsari, T. Carmon, A. Scherer, and K. J. Vahala, “Analysis of radiation-pressure induced mechanical oscillation of an optical microcavity,” Phys. Rev. Lett.95(3), 033901 (2005).
[CrossRef] [PubMed]

T. Carmon, L. Yang, and K. Vahala, “Dynamical thermal behavior and thermal self-stability of microcavities,” Opt. Express12(20), 4742–4750 (2004).
[CrossRef] [PubMed]

Chan, J.

Chen, D. R.

J. Zhu, S. K. Ozdemir, Y. F. Xiao, L. Li, L. He, D. R. Chen, and L. Yang, “On-chip single nanoparticle detection and sizing by mode splitting in an ultrahigh-Q microresonator,” Nat. Photonics4(1), 46–49 (2010).
[CrossRef]

Chen, L.

G. S. Wiederhecker, L. Chen, A. Gondarenko, and M. Lipson, “Controlling photonic structures using optical forces,” Nature462(7273), 633–636 (2009).
[CrossRef] [PubMed]

Chen, S.

Chipouline, A.

Chu, S. T.

S. T. Chu, B. E. Little, W. Pan, T. A. Kaneko, S. A. Sato, and Y. A. Kokubun, “An eight-channel add-drop filter using vertically coupled microring resonators over a cross grid,” IEEE Photon. Technol. Lett.11(6), 691–693 (1999).
[CrossRef]

Craighead, H. G.

S. S. Verbridge, J. M. Parpia, R. B. Reichenbach, L. M. Bellan, and H. G. Craighead, “High quality factor resonance at room temperature with nanostrings under high tensile stress,” J. Appl. Phys.99(12), 124304 (2006).
[CrossRef]

Dayan, B.

T. Aoki, B. Dayan, E. Wilcut, W. P. Bowen, A. S. Parkins, T. J. Kippenberg, K. J. Vahala, and H. J. Kimble, “Observation of strong coupling between one atom and a monolithic microresonator,” Nature443(7112), 671–674 (2006).
[CrossRef] [PubMed]

De Rossi, A.

A. De Rossi, V. Ortiz, M. Calligaro, L. Lanco, S. Ducci, V. Berger, and I. Sagnes, “Measuring propagation loss in a multimode semiconductor waveguide,” J. Appl. Phys.97(7), 073105 (2005).
[CrossRef]

Deych, L.

Ding, L.

L. Ding, C. Baker, P. Senellart, A. Lemaitre, S. Ducci, G. Leo, and I. Favero, “Wavelength-sized GaAs optomechanical resonators with gigahertz frequency,” Appl. Phys. Lett.98, 113801 (2011).

L. Ding, C. Belacel, S. Ducci, G. Leo, and I. Favero, “Ultralow loss single-mode silica tapers manufactured by a microheater,” Appl. Opt.49(13), 2441–2445 (2010).
[CrossRef]

L. Ding, C. Baker, P. Senellart, A. Lemaitre, S. Ducci, G. Leo, and I. Favero, “High frequency GaAs nano-optomechanical disk resonator,” Phys. Rev. Lett.105(26), 263903 (2010).
[CrossRef] [PubMed]

Dong, C.

L. He, Y.-F. Xiao, C. Dong, J. Zhu, V. Gaddam, and L. Yang, “Compensation of thermal refraction effect in high-Q toroidal microresonator by polydimethylsiloxane coating,” Appl. Phys. Lett.93(20), 201102 (2008).
[CrossRef]

Ducci, S.

L. Ding, C. Baker, P. Senellart, A. Lemaitre, S. Ducci, G. Leo, and I. Favero, “Wavelength-sized GaAs optomechanical resonators with gigahertz frequency,” Appl. Phys. Lett.98, 113801 (2011).

L. Ding, C. Belacel, S. Ducci, G. Leo, and I. Favero, “Ultralow loss single-mode silica tapers manufactured by a microheater,” Appl. Opt.49(13), 2441–2445 (2010).
[CrossRef]

L. Ding, C. Baker, P. Senellart, A. Lemaitre, S. Ducci, G. Leo, and I. Favero, “High frequency GaAs nano-optomechanical disk resonator,” Phys. Rev. Lett.105(26), 263903 (2010).
[CrossRef] [PubMed]

A. De Rossi, V. Ortiz, M. Calligaro, L. Lanco, S. Ducci, V. Berger, and I. Sagnes, “Measuring propagation loss in a multimode semiconductor waveguide,” J. Appl. Phys.97(7), 073105 (2005).
[CrossRef]

Egorov, O.

Eichenfield, M.

Fainman, Y.

D. T. H. Tan, K. Ikeda, P. C. Sun, and Y. Fainman, “Group velocity dispersion and self phase modulation in silicon nitride waveguides,” Appl. Phys. Lett.96(6), 061101 (2010), doi:.
[CrossRef]

Fan, S.

C. Manolatou, M. J. Khan, S. Fan, P. R. Villeneuve, H. A. Haus, and J. D. Joannopoulos, “Coupling of modes analysis of resonant channel add-drop filters,” IEEE J. Quantum Electron.35(9), 1322–1331 (1999).
[CrossRef]

Fang, W.

Faust, T.

Q. P. Unterreithmeier, T. Faust, and J. P. Kotthaus, “Damping of nanomechanical resonators,” Phys. Rev. Lett.105(2), 027205 (2010).
[CrossRef] [PubMed]

Favero, I.

L. Ding, C. Baker, P. Senellart, A. Lemaitre, S. Ducci, G. Leo, and I. Favero, “Wavelength-sized GaAs optomechanical resonators with gigahertz frequency,” Appl. Phys. Lett.98, 113801 (2011).

L. Ding, C. Belacel, S. Ducci, G. Leo, and I. Favero, “Ultralow loss single-mode silica tapers manufactured by a microheater,” Appl. Opt.49(13), 2441–2445 (2010).
[CrossRef]

L. Ding, C. Baker, P. Senellart, A. Lemaitre, S. Ducci, G. Leo, and I. Favero, “High frequency GaAs nano-optomechanical disk resonator,” Phys. Rev. Lett.105(26), 263903 (2010).
[CrossRef] [PubMed]

A. Andronico, I. Favero, and G. Leo, “Difference frequency generation in GaAs microdisks,” Opt. Lett.33(18), 2026–2028 (2008).
[CrossRef] [PubMed]

Fei, Y.

Fomin, A. E.

Gaddam, V.

L. He, Y.-F. Xiao, C. Dong, J. Zhu, V. Gaddam, and L. Yang, “Compensation of thermal refraction effect in high-Q toroidal microresonator by polydimethylsiloxane coating,” Appl. Phys. Lett.93(20), 201102 (2008).
[CrossRef]

Gaeta, A. L.

Gardeniers, J. G. E.

J. G. E. Gardeniers, H. A. C. Tilmans, and C. C. G. Visser, “LPCVD silicon-rich silicon nitride films for applications in micromechanics, studied with statistical experimental design,” J. Vac. Sci. Technol. A14(5), 2879–2892 (1996).
[CrossRef]

Gérard, J. M.

E. Peter, P. Senellart, D. Martrou, A. Lemaître, J. Hours, J. M. Gérard, and J. Bloch, “Exciton-photon strong-coupling regime for a single quantum dot embedded in a microcavity,” Phys. Rev. Lett.95(6), 067401 (2005).
[CrossRef] [PubMed]

Gondarenko, A.

G. S. Wiederhecker, L. Chen, A. Gondarenko, and M. Lipson, “Controlling photonic structures using optical forces,” Nature462(7273), 633–636 (2009).
[CrossRef] [PubMed]

A. Gondarenko, J. S. Levy, and M. Lipson, “High confinement micron-scale silicon nitride high Q ring resonator,” Opt. Express17(14), 11366–11370 (2009).
[CrossRef] [PubMed]

Gorodetsky, M. L.

Grudinin, I. S.

Harris, G. I.

K. H. Lee, T. G. McRae, G. I. Harris, J. Knittel, and W. P. Bowen, “Cooling and control of a cavity optoelectromechanical system,” Phys. Rev. Lett.104(12), 123604 (2010).
[CrossRef] [PubMed]

Haus, H. A.

C. Manolatou, M. J. Khan, S. Fan, P. R. Villeneuve, H. A. Haus, and J. D. Joannopoulos, “Coupling of modes analysis of resonant channel add-drop filters,” IEEE J. Quantum Electron.35(9), 1322–1331 (1999).
[CrossRef]

He, L.

J. Zhu, S. K. Ozdemir, Y. F. Xiao, L. Li, L. He, D. R. Chen, and L. Yang, “On-chip single nanoparticle detection and sizing by mode splitting in an ultrahigh-Q microresonator,” Nat. Photonics4(1), 46–49 (2010).
[CrossRef]

L. He, Y.-F. Xiao, C. Dong, J. Zhu, V. Gaddam, and L. Yang, “Compensation of thermal refraction effect in high-Q toroidal microresonator by polydimethylsiloxane coating,” Appl. Phys. Lett.93(20), 201102 (2008).
[CrossRef]

Heck, M. J. R.

Henry, C. H.

Hours, J.

E. Peter, P. Senellart, D. Martrou, A. Lemaître, J. Hours, J. M. Gérard, and J. Bloch, “Exciton-photon strong-coupling regime for a single quantum dot embedded in a microcavity,” Phys. Rev. Lett.95(6), 067401 (2005).
[CrossRef] [PubMed]

Ikeda, K.

D. T. H. Tan, K. Ikeda, P. C. Sun, and Y. Fainman, “Group velocity dispersion and self phase modulation in silicon nitride waveguides,” Appl. Phys. Lett.96(6), 061101 (2010), doi:.
[CrossRef]

Ilchenko, V. S.

Joannopoulos, J. D.

C. Manolatou, M. J. Khan, S. Fan, P. R. Villeneuve, H. A. Haus, and J. D. Joannopoulos, “Coupling of modes analysis of resonant channel add-drop filters,” IEEE J. Quantum Electron.35(9), 1322–1331 (1999).
[CrossRef]

Johnson, T. J.

Kaneko, T. A.

S. T. Chu, B. E. Little, W. Pan, T. A. Kaneko, S. A. Sato, and Y. A. Kokubun, “An eight-channel add-drop filter using vertically coupled microring resonators over a cross grid,” IEEE Photon. Technol. Lett.11(6), 691–693 (1999).
[CrossRef]

Katz, L. E.

Kazarinov, R. F.

Khan, M. J.

C. Manolatou, M. J. Khan, S. Fan, P. R. Villeneuve, H. A. Haus, and J. D. Joannopoulos, “Coupling of modes analysis of resonant channel add-drop filters,” IEEE J. Quantum Electron.35(9), 1322–1331 (1999).
[CrossRef]

Khoshsima, M.

F. Vollmer, D. Braun, A. Libchaber, M. Khoshsima, I. Teraoka, and S. Arnold, “Protein detection by optical shift of a resonant microcavity,” Appl. Phys. Lett.80(21), 4057 (2002), doi:.
[CrossRef]

Kimble, H. J.

T. Aoki, B. Dayan, E. Wilcut, W. P. Bowen, A. S. Parkins, T. J. Kippenberg, K. J. Vahala, and H. J. Kimble, “Observation of strong coupling between one atom and a monolithic microresonator,” Nature443(7112), 671–674 (2006).
[CrossRef] [PubMed]

Kippenberg, T. J.

T. Aoki, B. Dayan, E. Wilcut, W. P. Bowen, A. S. Parkins, T. J. Kippenberg, K. J. Vahala, and H. J. Kimble, “Observation of strong coupling between one atom and a monolithic microresonator,” Nature443(7112), 671–674 (2006).
[CrossRef] [PubMed]

T. J. Kippenberg, H. Rokhsari, T. Carmon, A. Scherer, and K. J. Vahala, “Analysis of radiation-pressure induced mechanical oscillation of an optical microcavity,” Phys. Rev. Lett.95(3), 033901 (2005).
[CrossRef] [PubMed]

Knittel, J.

K. H. Lee, T. G. McRae, G. I. Harris, J. Knittel, and W. P. Bowen, “Cooling and control of a cavity optoelectromechanical system,” Phys. Rev. Lett.104(12), 123604 (2010).
[CrossRef] [PubMed]

Kokubun, Y. A.

S. T. Chu, B. E. Little, W. Pan, T. A. Kaneko, S. A. Sato, and Y. A. Kokubun, “An eight-channel add-drop filter using vertically coupled microring resonators over a cross grid,” IEEE Photon. Technol. Lett.11(6), 691–693 (1999).
[CrossRef]

Kotthaus, J. P.

Q. P. Unterreithmeier, T. Faust, and J. P. Kotthaus, “Damping of nanomechanical resonators,” Phys. Rev. Lett.105(2), 027205 (2010).
[CrossRef] [PubMed]

Kuo, P. S.

Lanco, L.

A. De Rossi, V. Ortiz, M. Calligaro, L. Lanco, S. Ducci, V. Berger, and I. Sagnes, “Measuring propagation loss in a multimode semiconductor waveguide,” J. Appl. Phys.97(7), 073105 (2005).
[CrossRef]

Lederer, F.

Lee, H. J.

Lee, K. H.

K. H. Lee, T. G. McRae, G. I. Harris, J. Knittel, and W. P. Bowen, “Cooling and control of a cavity optoelectromechanical system,” Phys. Rev. Lett.104(12), 123604 (2010).
[CrossRef] [PubMed]

Lemaitre, A.

L. Ding, C. Baker, P. Senellart, A. Lemaitre, S. Ducci, G. Leo, and I. Favero, “Wavelength-sized GaAs optomechanical resonators with gigahertz frequency,” Appl. Phys. Lett.98, 113801 (2011).

L. Ding, C. Baker, P. Senellart, A. Lemaitre, S. Ducci, G. Leo, and I. Favero, “High frequency GaAs nano-optomechanical disk resonator,” Phys. Rev. Lett.105(26), 263903 (2010).
[CrossRef] [PubMed]

Lemaître, A.

E. Peter, P. Senellart, D. Martrou, A. Lemaître, J. Hours, J. M. Gérard, and J. Bloch, “Exciton-photon strong-coupling regime for a single quantum dot embedded in a microcavity,” Phys. Rev. Lett.95(6), 067401 (2005).
[CrossRef] [PubMed]

Leo, G.

L. Ding, C. Baker, P. Senellart, A. Lemaitre, S. Ducci, G. Leo, and I. Favero, “Wavelength-sized GaAs optomechanical resonators with gigahertz frequency,” Appl. Phys. Lett.98, 113801 (2011).

L. Ding, C. Belacel, S. Ducci, G. Leo, and I. Favero, “Ultralow loss single-mode silica tapers manufactured by a microheater,” Appl. Opt.49(13), 2441–2445 (2010).
[CrossRef]

L. Ding, C. Baker, P. Senellart, A. Lemaitre, S. Ducci, G. Leo, and I. Favero, “High frequency GaAs nano-optomechanical disk resonator,” Phys. Rev. Lett.105(26), 263903 (2010).
[CrossRef] [PubMed]

A. Andronico, I. Favero, and G. Leo, “Difference frequency generation in GaAs microdisks,” Opt. Lett.33(18), 2026–2028 (2008).
[CrossRef] [PubMed]

Levy, J. S.

Li, L.

J. Zhu, S. K. Ozdemir, Y. F. Xiao, L. Li, L. He, D. R. Chen, and L. Yang, “On-chip single nanoparticle detection and sizing by mode splitting in an ultrahigh-Q microresonator,” Nat. Photonics4(1), 46–49 (2010).
[CrossRef]

Li, M.

Libchaber, A.

F. Vollmer, D. Braun, A. Libchaber, M. Khoshsima, I. Teraoka, and S. Arnold, “Protein detection by optical shift of a resonant microcavity,” Appl. Phys. Lett.80(21), 4057 (2002), doi:.
[CrossRef]

Lipson, M.

Little, B. E.

S. T. Chu, B. E. Little, W. Pan, T. A. Kaneko, S. A. Sato, and Y. A. Kokubun, “An eight-channel add-drop filter using vertically coupled microring resonators over a cross grid,” IEEE Photon. Technol. Lett.11(6), 691–693 (1999).
[CrossRef]

Luo, L. W.

Manolatou, C.

C. Manolatou, M. J. Khan, S. Fan, P. R. Villeneuve, H. A. Haus, and J. D. Joannopoulos, “Coupling of modes analysis of resonant channel add-drop filters,” IEEE J. Quantum Electron.35(9), 1322–1331 (1999).
[CrossRef]

Martrou, D.

E. Peter, P. Senellart, D. Martrou, A. Lemaître, J. Hours, J. M. Gérard, and J. Bloch, “Exciton-photon strong-coupling regime for a single quantum dot embedded in a microcavity,” Phys. Rev. Lett.95(6), 067401 (2005).
[CrossRef] [PubMed]

McRae, T. G.

K. H. Lee, T. G. McRae, G. I. Harris, J. Knittel, and W. P. Bowen, “Cooling and control of a cavity optoelectromechanical system,” Phys. Rev. Lett.104(12), 123604 (2010).
[CrossRef] [PubMed]

Okamura, Y.

Okawachi, Y.

Orlowsky, K. J.

Ortiz, V.

A. De Rossi, V. Ortiz, M. Calligaro, L. Lanco, S. Ducci, V. Berger, and I. Sagnes, “Measuring propagation loss in a multimode semiconductor waveguide,” J. Appl. Phys.97(7), 073105 (2005).
[CrossRef]

Oxborrow, M.

M. Oxborrow, “How to simulate the whispering gallery modes of dielectric microresonator in FEMLAB/COMSOL,” Proc. SPIE6452(64520J), 64520J (2007).
[CrossRef]

Ozdemir, S. K.

J. Zhu, S. K. Ozdemir, Y. F. Xiao, L. Li, L. He, D. R. Chen, and L. Yang, “On-chip single nanoparticle detection and sizing by mode splitting in an ultrahigh-Q microresonator,” Nat. Photonics4(1), 46–49 (2010).
[CrossRef]

Painter, O.

Pan, W.

S. T. Chu, B. E. Little, W. Pan, T. A. Kaneko, S. A. Sato, and Y. A. Kokubun, “An eight-channel add-drop filter using vertically coupled microring resonators over a cross grid,” IEEE Photon. Technol. Lett.11(6), 691–693 (1999).
[CrossRef]

Parkins, A. S.

T. Aoki, B. Dayan, E. Wilcut, W. P. Bowen, A. S. Parkins, T. J. Kippenberg, K. J. Vahala, and H. J. Kimble, “Observation of strong coupling between one atom and a monolithic microresonator,” Nature443(7112), 671–674 (2006).
[CrossRef] [PubMed]

Parpia, J. M.

S. S. Verbridge, J. M. Parpia, R. B. Reichenbach, L. M. Bellan, and H. G. Craighead, “High quality factor resonance at room temperature with nanostrings under high tensile stress,” J. Appl. Phys.99(12), 124304 (2006).
[CrossRef]

Pernice, W. H. P.

Pertsch, T.

Peter, E.

E. Peter, P. Senellart, D. Martrou, A. Lemaître, J. Hours, J. M. Gérard, and J. Bloch, “Exciton-photon strong-coupling regime for a single quantum dot embedded in a microcavity,” Phys. Rev. Lett.95(6), 067401 (2005).
[CrossRef] [PubMed]

Preston, K.

Pryamikov, A. D.

Regener, R.

R. Regener and W. Sohler, “Loss in low-finesse Ti: LiNbO3 optical waveguide resonators,” Appl. Phys. B36(3), 143–147 (1985), doi:.
[CrossRef]

Reichenbach, R. B.

S. S. Verbridge, J. M. Parpia, R. B. Reichenbach, L. M. Bellan, and H. G. Craighead, “High quality factor resonance at room temperature with nanostrings under high tensile stress,” J. Appl. Phys.99(12), 124304 (2006).
[CrossRef]

Rokhsari, H.

T. J. Kippenberg, H. Rokhsari, T. Carmon, A. Scherer, and K. J. Vahala, “Analysis of radiation-pressure induced mechanical oscillation of an optical microcavity,” Phys. Rev. Lett.95(3), 033901 (2005).
[CrossRef] [PubMed]

Sagnes, I.

A. De Rossi, V. Ortiz, M. Calligaro, L. Lanco, S. Ducci, V. Berger, and I. Sagnes, “Measuring propagation loss in a multimode semiconductor waveguide,” J. Appl. Phys.97(7), 073105 (2005).
[CrossRef]

Saha, K.

Sato, S. A.

S. T. Chu, B. E. Little, W. Pan, T. A. Kaneko, S. A. Sato, and Y. A. Kokubun, “An eight-channel add-drop filter using vertically coupled microring resonators over a cross grid,” IEEE Photon. Technol. Lett.11(6), 691–693 (1999).
[CrossRef]

Scherer, A.

T. J. Kippenberg, H. Rokhsari, T. Carmon, A. Scherer, and K. J. Vahala, “Analysis of radiation-pressure induced mechanical oscillation of an optical microcavity,” Phys. Rev. Lett.95(3), 033901 (2005).
[CrossRef] [PubMed]

Schmidt, C.

Senellart, P.

L. Ding, C. Baker, P. Senellart, A. Lemaitre, S. Ducci, G. Leo, and I. Favero, “Wavelength-sized GaAs optomechanical resonators with gigahertz frequency,” Appl. Phys. Lett.98, 113801 (2011).

L. Ding, C. Baker, P. Senellart, A. Lemaitre, S. Ducci, G. Leo, and I. Favero, “High frequency GaAs nano-optomechanical disk resonator,” Phys. Rev. Lett.105(26), 263903 (2010).
[CrossRef] [PubMed]

E. Peter, P. Senellart, D. Martrou, A. Lemaître, J. Hours, J. M. Gérard, and J. Bloch, “Exciton-photon strong-coupling regime for a single quantum dot embedded in a microcavity,” Phys. Rev. Lett.95(6), 067401 (2005).
[CrossRef] [PubMed]

Sohler, W.

R. Regener and W. Sohler, “Loss in low-finesse Ti: LiNbO3 optical waveguide resonators,” Appl. Phys. B36(3), 143–147 (1985), doi:.
[CrossRef]

Solomon, G. S.

Spencer, D. T.

Sun, P. C.

D. T. H. Tan, K. Ikeda, P. C. Sun, and Y. Fainman, “Group velocity dispersion and self phase modulation in silicon nitride waveguides,” Appl. Phys. Lett.96(6), 061101 (2010), doi:.
[CrossRef]

Tan, D. T. H.

D. T. H. Tan, K. Ikeda, P. C. Sun, and Y. Fainman, “Group velocity dispersion and self phase modulation in silicon nitride waveguides,” Appl. Phys. Lett.96(6), 061101 (2010), doi:.
[CrossRef]

Tang, H. X.

Teraoka, I.

F. Vollmer, D. Braun, A. Libchaber, M. Khoshsima, I. Teraoka, and S. Arnold, “Protein detection by optical shift of a resonant microcavity,” Appl. Phys. Lett.80(21), 4057 (2002), doi:.
[CrossRef]

Tien, M. C.

Tilmans, H. A. C.

J. G. E. Gardeniers, H. A. C. Tilmans, and C. C. G. Visser, “LPCVD silicon-rich silicon nitride films for applications in micromechanics, studied with statistical experimental design,” J. Vac. Sci. Technol. A14(5), 2879–2892 (1996).
[CrossRef]

Tünnermann, A.

Unterreithmeier, Q. P.

Q. P. Unterreithmeier, T. Faust, and J. P. Kotthaus, “Damping of nanomechanical resonators,” Phys. Rev. Lett.105(2), 027205 (2010).
[CrossRef] [PubMed]

Vahala, K.

Vahala, K. J.

T. Aoki, B. Dayan, E. Wilcut, W. P. Bowen, A. S. Parkins, T. J. Kippenberg, K. J. Vahala, and H. J. Kimble, “Observation of strong coupling between one atom and a monolithic microresonator,” Nature443(7112), 671–674 (2006).
[CrossRef] [PubMed]

T. J. Kippenberg, H. Rokhsari, T. Carmon, A. Scherer, and K. J. Vahala, “Analysis of radiation-pressure induced mechanical oscillation of an optical microcavity,” Phys. Rev. Lett.95(3), 033901 (2005).
[CrossRef] [PubMed]

Verbridge, S. S.

S. S. Verbridge, J. M. Parpia, R. B. Reichenbach, L. M. Bellan, and H. G. Craighead, “High quality factor resonance at room temperature with nanostrings under high tensile stress,” J. Appl. Phys.99(12), 124304 (2006).
[CrossRef]

Villeneuve, P. R.

C. Manolatou, M. J. Khan, S. Fan, P. R. Villeneuve, H. A. Haus, and J. D. Joannopoulos, “Coupling of modes analysis of resonant channel add-drop filters,” IEEE J. Quantum Electron.35(9), 1322–1331 (1999).
[CrossRef]

Visser, C. C. G.

J. G. E. Gardeniers, H. A. C. Tilmans, and C. C. G. Visser, “LPCVD silicon-rich silicon nitride films for applications in micromechanics, studied with statistical experimental design,” J. Vac. Sci. Technol. A14(5), 2879–2892 (1996).
[CrossRef]

Vollmer, F.

F. Vollmer, D. Braun, A. Libchaber, M. Khoshsima, I. Teraoka, and S. Arnold, “Protein detection by optical shift of a resonant microcavity,” Appl. Phys. Lett.80(21), 4057 (2002), doi:.
[CrossRef]

Wen, Y. H.

Wiederhecker, G. S.

L. W. Luo, G. S. Wiederhecker, K. Preston, and M. Lipson, “Power insensitive silicon microring resonators,” Opt. Lett.37(4), 590–592 (2012).
[CrossRef] [PubMed]

G. S. Wiederhecker, L. Chen, A. Gondarenko, and M. Lipson, “Controlling photonic structures using optical forces,” Nature462(7273), 633–636 (2009).
[CrossRef] [PubMed]

Wilcut, E.

T. Aoki, B. Dayan, E. Wilcut, W. P. Bowen, A. S. Parkins, T. J. Kippenberg, K. J. Vahala, and H. J. Kimble, “Observation of strong coupling between one atom and a monolithic microresonator,” Nature443(7112), 671–674 (2006).
[CrossRef] [PubMed]

Xiao, Y. F.

J. Zhu, S. K. Ozdemir, Y. F. Xiao, L. Li, L. He, D. R. Chen, and L. Yang, “On-chip single nanoparticle detection and sizing by mode splitting in an ultrahigh-Q microresonator,” Nat. Photonics4(1), 46–49 (2010).
[CrossRef]

Xiao, Y.-F.

L. He, Y.-F. Xiao, C. Dong, J. Zhu, V. Gaddam, and L. Yang, “Compensation of thermal refraction effect in high-Q toroidal microresonator by polydimethylsiloxane coating,” Appl. Phys. Lett.93(20), 201102 (2008).
[CrossRef]

Yamamoto, S.

Yang, L.

J. Zhu, S. K. Ozdemir, Y. F. Xiao, L. Li, L. He, D. R. Chen, and L. Yang, “On-chip single nanoparticle detection and sizing by mode splitting in an ultrahigh-Q microresonator,” Nat. Photonics4(1), 46–49 (2010).
[CrossRef]

L. He, Y.-F. Xiao, C. Dong, J. Zhu, V. Gaddam, and L. Yang, “Compensation of thermal refraction effect in high-Q toroidal microresonator by polydimethylsiloxane coating,” Appl. Phys. Lett.93(20), 201102 (2008).
[CrossRef]

T. Carmon, L. Yang, and K. Vahala, “Dynamical thermal behavior and thermal self-stability of microcavities,” Opt. Express12(20), 4742–4750 (2004).
[CrossRef] [PubMed]

Yoshinaka, S.

Zhang, L.

Zhu, J.

J. Zhu, S. K. Ozdemir, Y. F. Xiao, L. Li, L. He, D. R. Chen, and L. Yang, “On-chip single nanoparticle detection and sizing by mode splitting in an ultrahigh-Q microresonator,” Nat. Photonics4(1), 46–49 (2010).
[CrossRef]

L. He, Y.-F. Xiao, C. Dong, J. Zhu, V. Gaddam, and L. Yang, “Compensation of thermal refraction effect in high-Q toroidal microresonator by polydimethylsiloxane coating,” Appl. Phys. Lett.93(20), 201102 (2008).
[CrossRef]

Appl. Opt.

Appl. Phys. B

R. Regener and W. Sohler, “Loss in low-finesse Ti: LiNbO3 optical waveguide resonators,” Appl. Phys. B36(3), 143–147 (1985), doi:.
[CrossRef]

Appl. Phys. Lett.

L. He, Y.-F. Xiao, C. Dong, J. Zhu, V. Gaddam, and L. Yang, “Compensation of thermal refraction effect in high-Q toroidal microresonator by polydimethylsiloxane coating,” Appl. Phys. Lett.93(20), 201102 (2008).
[CrossRef]

D. T. H. Tan, K. Ikeda, P. C. Sun, and Y. Fainman, “Group velocity dispersion and self phase modulation in silicon nitride waveguides,” Appl. Phys. Lett.96(6), 061101 (2010), doi:.
[CrossRef]

F. Vollmer, D. Braun, A. Libchaber, M. Khoshsima, I. Teraoka, and S. Arnold, “Protein detection by optical shift of a resonant microcavity,” Appl. Phys. Lett.80(21), 4057 (2002), doi:.
[CrossRef]

L. Ding, C. Baker, P. Senellart, A. Lemaitre, S. Ducci, G. Leo, and I. Favero, “Wavelength-sized GaAs optomechanical resonators with gigahertz frequency,” Appl. Phys. Lett.98, 113801 (2011).

IEEE J. Quantum Electron.

C. Manolatou, M. J. Khan, S. Fan, P. R. Villeneuve, H. A. Haus, and J. D. Joannopoulos, “Coupling of modes analysis of resonant channel add-drop filters,” IEEE J. Quantum Electron.35(9), 1322–1331 (1999).
[CrossRef]

IEEE Photon. Technol. Lett.

S. T. Chu, B. E. Little, W. Pan, T. A. Kaneko, S. A. Sato, and Y. A. Kokubun, “An eight-channel add-drop filter using vertically coupled microring resonators over a cross grid,” IEEE Photon. Technol. Lett.11(6), 691–693 (1999).
[CrossRef]

J. Appl. Phys.

S. S. Verbridge, J. M. Parpia, R. B. Reichenbach, L. M. Bellan, and H. G. Craighead, “High quality factor resonance at room temperature with nanostrings under high tensile stress,” J. Appl. Phys.99(12), 124304 (2006).
[CrossRef]

A. De Rossi, V. Ortiz, M. Calligaro, L. Lanco, S. Ducci, V. Berger, and I. Sagnes, “Measuring propagation loss in a multimode semiconductor waveguide,” J. Appl. Phys.97(7), 073105 (2005).
[CrossRef]

J. Opt. Soc. Am. B

J. Vac. Sci. Technol. A

J. G. E. Gardeniers, H. A. C. Tilmans, and C. C. G. Visser, “LPCVD silicon-rich silicon nitride films for applications in micromechanics, studied with statistical experimental design,” J. Vac. Sci. Technol. A14(5), 2879–2892 (1996).
[CrossRef]

Nat. Photonics

J. Zhu, S. K. Ozdemir, Y. F. Xiao, L. Li, L. He, D. R. Chen, and L. Yang, “On-chip single nanoparticle detection and sizing by mode splitting in an ultrahigh-Q microresonator,” Nat. Photonics4(1), 46–49 (2010).
[CrossRef]

Nature

T. Aoki, B. Dayan, E. Wilcut, W. P. Bowen, A. S. Parkins, T. J. Kippenberg, K. J. Vahala, and H. J. Kimble, “Observation of strong coupling between one atom and a monolithic microresonator,” Nature443(7112), 671–674 (2006).
[CrossRef] [PubMed]

G. S. Wiederhecker, L. Chen, A. Gondarenko, and M. Lipson, “Controlling photonic structures using optical forces,” Nature462(7273), 633–636 (2009).
[CrossRef] [PubMed]

Opt. Express

R. M. Camacho, J. Chan, M. Eichenfield, and O. Painter, “Characterization of radiation pressure and thermal effects in a nanoscale optomechanical cavity,” Opt. Express17(18), 15726–15735 (2009).
[CrossRef] [PubMed]

T. Carmon, L. Yang, and K. Vahala, “Dynamical thermal behavior and thermal self-stability of microcavities,” Opt. Express12(20), 4742–4750 (2004).
[CrossRef] [PubMed]

C. Schmidt, A. Chipouline, T. Pertsch, A. Tünnermann, O. Egorov, F. Lederer, and L. Deych, “Nonlinear thermal effects in optical microspheres at different wavelength sweeping speeds,” Opt. Express16(9), 6285–6301 (2008), doi:.
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T. J. Johnson, M. Borselli, and O. Painter, “Self-induced optical modulation of the transmission through a high-Q silicon microdisk resonator,” Opt. Express14(2), 817–831 (2006).
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W. H. P. Pernice, M. Li, and H. X. Tang, “Time-domain measurement of optical transport in silicon micro-ring resonators,” Opt. Express18(17), 18438–18452 (2010), doi:.
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S. Chen, L. Zhang, Y. Fei, and T. Cao, “Bistability and self-pulsation phenomena in silicon microring resonators based on nonlinear optical effects,” Opt. Express20(7), 7454–7468 (2012).
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A. Gondarenko, J. S. Levy, and M. Lipson, “High confinement micron-scale silicon nitride high Q ring resonator,” Opt. Express17(14), 11366–11370 (2009).
[CrossRef] [PubMed]

M. C. Tien, J. F. Bauters, M. J. R. Heck, D. T. Spencer, D. J. Blumenthal, and J. E. Bowers, “Ultra-high quality factor planar Si3N4 ring resonators on Si substrates,” Opt. Express19(14), 13551–13556 (2011).
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Opt. Lett.

Phys. Rev. Lett.

Q. P. Unterreithmeier, T. Faust, and J. P. Kotthaus, “Damping of nanomechanical resonators,” Phys. Rev. Lett.105(2), 027205 (2010).
[CrossRef] [PubMed]

E. Peter, P. Senellart, D. Martrou, A. Lemaître, J. Hours, J. M. Gérard, and J. Bloch, “Exciton-photon strong-coupling regime for a single quantum dot embedded in a microcavity,” Phys. Rev. Lett.95(6), 067401 (2005).
[CrossRef] [PubMed]

T. J. Kippenberg, H. Rokhsari, T. Carmon, A. Scherer, and K. J. Vahala, “Analysis of radiation-pressure induced mechanical oscillation of an optical microcavity,” Phys. Rev. Lett.95(3), 033901 (2005).
[CrossRef] [PubMed]

L. Ding, C. Baker, P. Senellart, A. Lemaitre, S. Ducci, G. Leo, and I. Favero, “High frequency GaAs nano-optomechanical disk resonator,” Phys. Rev. Lett.105(26), 263903 (2010).
[CrossRef] [PubMed]

K. H. Lee, T. G. McRae, G. I. Harris, J. Knittel, and W. P. Bowen, “Cooling and control of a cavity optoelectromechanical system,” Phys. Rev. Lett.104(12), 123604 (2010).
[CrossRef] [PubMed]

Proc. SPIE

M. Oxborrow, “How to simulate the whispering gallery modes of dielectric microresonator in FEMLAB/COMSOL,” Proc. SPIE6452(64520J), 64520J (2007).
[CrossRef]

Other

COMSOL material library.

Supplementary Material (1)

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Figures (7)

Fig. 1
Fig. 1

(a) Optical microscope top view of a sample containing 10 waveguide/disk resonator sets. Inset: tapered part of the waveguide in the disk vicinity. (b) SEM micrograph side view of 3 SiN disk resonators and coupling waveguides. (c) SEM top view of the evanescent coupling region between tapered waveguide and disk. (d) SEM side view of the cleaved waveguide input facet. In (b), (c) and (d) SiN is colored in purple, SiO2 in gray and Si in blue.

Fig. 2
Fig. 2

(a) Optical transmission spectrum of a SiN disk resonator. (b) A fine optical resonance measured in a weakly coupled configuration far from critical coupling, showing an optical Q of 400 000, typical of the highest optical Qs in this device. (c) A broader optical resonance, measured at low power (solid black line) and high power (red dash-dot line).

Fig. 3
Fig. 3

(a) Normalized optical power measured as a function of time at the output of a waveguide coupled to a SiN disk resonator. The employed WGM resonance at λ = 1539.31 nm has an on-resonance normalized transmission of 7% and a FWHM of 46 pm. It is identified as a p = 1, m = 57 WGM (radial and azimuthal orders). Inset: initial position of laser wavelength λ (red arrow) with respect to the WGM resonance in the transmission T spectrum. (b) Oscillatory behavior shown with greater detail. The inset pictures are single frames from an infrared camera video of the output waveguide transmission taken during measurement (See Media 1). (c) Characteristic time of the oscillatory transmission as a function of the optical power dissipated by the disk. The line through the high power data points is an exponential fit.

Fig. 4
Fig. 4

Experimental (left) and model (right) results for the normalized optical transmission of a WGM resonator for three distinct pump laser configurations. (a) Experiments and (b) model for an initial blue detuned pump configuration with λ = 1539.35 nm and A⋅Pguide = 77 µW. (c) Experiments and (d) model for the same optical power but with initial laser blue detuning 2 pm greater. (e) Experiments and (f) model for a lower power (A⋅Pguide = 65 µW) and for a detuning similar to (a) and (b). In (b), (d) and (f) the temperature oscillations are on the order of a few Kelvin for T1 and a few tens of mK for T2.

Fig. 5
Fig. 5

(a) Schematic description of the regime of self-pulsating optical transmission. In step 1 the laser line starts on the blue-detuned flank of the WGM thermo-optically distorted resonance. The sample substrate slowly heats up, gradually blue-shifting the resonance. Once the resonance wavelength moves past the fixed laser wavelength, the laser line reaches the unstable flank of the thermo-optically distorted resonance: at this stage the WGM snaps out of laser resonance and the transmission jumps back to unity (step 2). The WGM is out of resonance with the laser and the substrate slowly cools down, gradually red-shifting the optical resonance (step 3). Finally when the red-shift is sufficient, the WGM snaps back into resonance with the laser in the thermo-optically distorted regime (step 4). In the self-pulsating regime, the output optical power cycles through the hysteresis curve. (b) 2D axi-symmetric FEM simulation of the thermal strain in a SiN disk resonator structure. The surface color code shows the magnitude of the radial displacement. The deformed shape displays strongly exaggerated displacement amplitude.

Fig. 6
Fig. 6

(a) False color SEM micrograph of a straight SiN waveguide. (b) CCD image top view of an entire SiN waveguide fabricated using protocol 1, with 633 nm wavelength laser light injected at the left sample facet (bright spot at center left) and travelling along the waveguide (bright horizontal line with decaying intensity). (c) Corresponding upwards scattered light intensity as a function of the position along the waveguide, with an exponential decay fit (solid red line). The insets show SEM pictures featuring typical waveguide defects leading to locally increased scattering. The left inset shows an inhomogeneous waveguide width resulting from lithography errors. The right inset shows a crack in the SiN resulting from the high tensile stress. This waveguide was processed using protocol 1. (d) Same waveguide measured after 4h of annealing at 1050°C under N2 atmosphere. The scattering is essentially caused by isolated defects like the ones shown in the inset of (c), but the output power is now higher than in (b).

Fig. 7
Fig. 7

Transmission spectrum of a 8 µm wide and 2.5 cm long straight waveguide fabricated using protocol 2. Due to the reflectivity of the sample facets, the transmission shows Fabry-Perot interference fringes with 26.6 pm periodicity. Linear losses can be extracted knowing the contrast of the fringes, the facet reflectivity and the sample length. The precision of this method is set by the accuracy in the evaluation of the facet reflectivity and sample length. If the waveguide is multimode, some beating in the fringes’ amplitude leads to an overestimation of losses [38].

Tables (2)

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Table 1 Physical parameters used in the models

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Table 2 Optical losses measured on straight SiN waveguides at different wavelengths

Equations (5)

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λ res ( T 1 , T 2 )= λ res ( T 0 , T 0 )+( λ res T 1 )( T 1 T 0 )+( λ res T 2 )( T 2 T 0 )
Λ( λ, T 1 , T 2 )=1 C 1+ ( λ λ res ( T 1 , T 2 ) δλ/2 ) 2
d T 1 dt = P guide A 1Λ( λ, T 1 , T 2 ) m 1 c 1 G 1,2 m 1 c 1 ( T 1 T 2 )
d T 2 dt = G 1,2 ( T 1 T 2 ) m 2 c 2 G 2 m 2 c 2 ( T 2 T 0 )
Q=2π 1 1 e α λ 0 / n eff

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