Abstract

We propose optimal designs for triply-resonant optical parametric oscillators (OPOs) based on degenerate four-wave mixing (FWM) in microcavities. We show that optimal designs in general call for different external coupling to pump and signal/idler resonances. We provide a number of normalized performance metrics including threshold pump power and maximum achievable conversion efficiency for OPOs with and without two-photon (TPA) and free-carrier absorption (FCA). We find that the maximum achievable conversion efficiency is bound to an upper limit by nonlinear and free-carrier losses independent of pump power, while linear losses only increase the pump power required to achieve a certain conversion efficiency. The results of this work suggest unique advantages in on-chip implementations that allow explicit engineering of resonances, mode field overlaps, dispersion, and wavelength-and mode-selective coupling. We provide universal design curves that yield optimum designs, and give example designs of microring-resonator-based OPOs in silicon at the wavelengths 1.55 μm (with TPA) and 2.3 μm (no TPA) as well as in silicon nitride (Si3N4) at 1.55 μm. For typical microcavity quality factor of 106, we show that the oscillation threshold in excitation bus can be well into the sub-mW regime for silicon microrings and a few mW for silicon nitride microrings. The conversion efficiency can be a few percent when pumped at 10 times of the threshold. Next, based on our results, we suggest a family of synthetic “photonic molecule”-like, coupled-cavity systems to implement optimum FWM, where structure design for control of resonant wavelengths can be separated from that of optimizing nonlinear conversion efficiency, and where furthermore pump, signal, and idler coupling to bus waveguides can be controlled independently, using interferometric cavity supermode coupling as an example. Finally, consideration of these complex geometries calls for a generalization of the nonlinear figure of merit (NFOM) as a metric for performance in nonlinear photonic systems, and shows different efficiencies for single and multi-cavity geometries, as well as for standing and traveling wave excitations.

© 2014 Optical Society of America

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2013

2012

2011

B. Kuyken, H. Ji, S. Clemmen, S. Selvaraja, H. Hu, M. Pu, M. Galili, P. Jeppesen, G. Morthier, S. Massar, L. K. Oxenløwe, G. Roelkens, and R. Baets, “Nonlinear properties of and nonlinear processing in hydrogenated amorphous silicon waveguides,” Opt. Express19, B146–B153 (2011).
[CrossRef]

M. A. Foster, J. S. Levy, O. Kuzucu, K. Saha, M. Lipson, and A. L. Gaeta, “Silicon-based monolithic optical frequency comb source,” Opt. Express19, 14233–14239 (2011).
[CrossRef] [PubMed]

D. M. Ramirez, A. W. Rodriguez, H. Hashemi, J. D. Joannopoulos, M. Soljačić, and S. G. Johnson, “Degenerate four-wave mixing in triply resonant kerr cavities,” Phys. Rev. A83, 033834 (2011).
[CrossRef]

D. A. Fishman, C. M. Cirloganu, S. Webster, L. A. Padilha, M. Monroe, D. J. Hagan, and E. W. Van Stryland, “Sensitive mid-infrared detection in wide-bandgap semiconductors using extreme non-degenerate two-photon absorption,” Nat. Photonics5, 561–565 (2011).
[CrossRef]

2010

2009

H. Hashemi, A. W. Rodriguez, J. D. Joannopoulos, M. Soljačić, and S. G. Johnson, “Nonlinear harmonic generation and devices in doubly resonant kerr cavities,” Phys. Rev. A79, 013812 (2009).
[CrossRef]

L. Razzari, D. Duchesne, M. Ferrera, R. Morandotti, S. Chu, B. Little, and D. Moss, “Cmos-compatible integrated optical hyper-parametric oscillator,” Nat. Photonics4, 41–45 (2009).
[CrossRef]

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. Photonics4, 37–40 (2009).
[CrossRef]

2008

2007

Q. Lin, O. J. Painter, and G. P. Agrawal, “Nonlinear optical phenomena in silicon waveguides: modeling and applications,” Opt. Express15, 16604–16644 (2007).
[CrossRef] [PubMed]

Q. Lin, J. Zhang, G. Piredda, R. Boyd, P. Fauchet, and G. Agrawal, “Dispersion of silicon nonlinearities in the near infrared region,” Appl. Phys. Lett.91, 021111 (2007).
[CrossRef]

A. Rodriguez, M. Soljačić, J. D. Joannopoulos, and S. G. Johnson, “χ(2) and χ(3) harmonic generation at a critical power in inhomogeneous doublyresonant cavities,” Opt. Express15, 7303–7318 (2007).
[CrossRef] [PubMed]

P. Dell’Haye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, and T. Kippenberg, “Optical frequency comb generation from a monolithic microresonator,” Nature450, 1214–1217 (2007).
[CrossRef]

2006

2005

D. Dimitropoulos, R. Jhaveri, R. Claps, J. Woo, and B. Jalali, “Lifetime of photogenerated carriers in silicon-on-insulator rib waveguides,” Appl. Phys. Lett.86, 071115 (2005).
[CrossRef]

H. Rong, R. Jones, A. Liu, O. Cohen, D. Hak, A. Fang, and M. Paniccia, “A continuous-wave raman silicon laser,” Nature433, 725–728 (2005).
[CrossRef] [PubMed]

2004

M. F. Yanik, W. Suh, Z. Wang, and S. Fan, “Stopping light in a waveguide with an all-optical analog of electromagnetically induced transparency,” Phys. Rev. Lett.93, 233903 (2004).
[CrossRef] [PubMed]

T. Kippenberg, S. Spillane, and K. Vahala, “Kerr-nonlinearity optical parametric oscillation in an ultrahigh-q toroid microcavity,” Phys. Rev. Lett.93, 83904 (2004).
[CrossRef]

1997

B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J.-P. Laine, “Microring resonator channel dropping filters,” J. Lightwave Technol.15, 998–1005 (1997).
[CrossRef]

Adibi, A.

A. H. Atabaki and A. Adibi, “Ultra-compact coupled-resonator device for four-wave-mixing applications,” in “CLEO: Science and Innovations,” (Optical Society of America, 2011), p. CTuS6.

Agha, I.

M. Davanço, J. R. Ong, A. B. Shehata, A. Tosi, I. Agha, S. Assefa, F. Xia, W. M. Green, S. Mookherjea, and K. Srinivasan, “Telecommunications-band heralded single photons from a silicon nanophotonic chip,” Appl. Phys. Lett.100, 261104 (2012).
[CrossRef]

Agrawal, G.

Q. Lin, J. Zhang, G. Piredda, R. Boyd, P. Fauchet, and G. Agrawal, “Dispersion of silicon nonlinearities in the near infrared region,” Appl. Phys. Lett.91, 021111 (2007).
[CrossRef]

Agrawal, G. P.

Arcizet, O.

P. Dell’Haye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, and T. Kippenberg, “Optical frequency comb generation from a monolithic microresonator,” Nature450, 1214–1217 (2007).
[CrossRef]

Assefa, S.

M. Davanço, J. R. Ong, A. B. Shehata, A. Tosi, I. Agha, S. Assefa, F. Xia, W. M. Green, S. Mookherjea, and K. Srinivasan, “Telecommunications-band heralded single photons from a silicon nanophotonic chip,” Appl. Phys. Lett.100, 261104 (2012).
[CrossRef]

Atabaki, A. H.

A. H. Atabaki and A. Adibi, “Ultra-compact coupled-resonator device for four-wave-mixing applications,” in “CLEO: Science and Innovations,” (Optical Society of America, 2011), p. CTuS6.

Azzini, S.

Baets, R.

Bajoni, D.

Barwicz, T.

M. A. Popović, T. Barwicz, M. R. Watts, P. T. Rakich, L. Socci, E. P. Ippen, F. X. Kärtner, and H. I. Smith, “Multistage high-order microring-resonator add-drop filters,” Opt. Lett.31, 2571–2573 (2006).
[CrossRef]

M. A. Popović, T. Barwicz, M. R. Watts, P. T. Rakich, L. Socci, E. P. Ippen, F. X. Kärtner, and H. I. Smith, “Multistage high-order microring-resonator add-drop filters,” Opt. Lett.31, 2571–2573 (2006).
[CrossRef]

M. A. Popović, T. Barwicz, F. Gan, M. S. Dahlem, C. W. Holzwarth, P. T. Rakich, H. I. Smith, E. P. Ippen, and F. X. Kärtner, “Transparent wavelength switching of resonant filters,” in “Conference on Lasers and Electro-Optics,” (Optical Society of America, 2007).

M. A. Popović, M. R. Watts, T. Barwicz, P. T. Rakich, L. Socci, E. P. Ippen, F. X. Kärtner, and H. I. Smith, “High-index-contrast, wide-fsr microring-resonator filter design and realization with frequency-shift compensation,” in “Optical Fiber Communication Conference,” (Optical Society of America, 2005).

M. A. Popović, M. R. Watts, T. Barwicz, P. T. Rakich, L. Socci, E. P. Ippen, F. X. Kärtner, and H. I. Smith, “High-index-contrast and wide-FSR microring-resonator filter design and realization with frequency-shift compensation,” in “Technical Digest of the Optical Fiber Communication Conference,” (Anaheim, CA, 2005). Paper OFK1.

M. A. Popović, T. Barwicz, P. T. Rakich, M. S. Dahlem, C. W. Holzwarth, F. Gan, L. Socci, M. R. Watts, H. I. Smith, F. X. Kärtner, and E. P. Ippen, “Experimental demonstration of loop-coupled microring resonators for optimally sharp optical filters,” in “Conference on Lasers and Electro-Optics,” (OSA, 2008).

Bloch, J.

C. Diederichs, J. Tignon, G. Dasbach, C. Ciuti, A. Lemaitre, J. Bloch, P. Roussignol, and C. Delalande, “Parametric oscillation in vertical triple microcavities,” Nature440, 904–907 (2006).
[CrossRef] [PubMed]

Bonneau, D.

Boyd, R.

Q. Lin, J. Zhang, G. Piredda, R. Boyd, P. Fauchet, and G. Agrawal, “Dispersion of silicon nonlinearities in the near infrared region,” Appl. Phys. Lett.91, 021111 (2007).
[CrossRef]

Boyd, R. W.

R. W. Boyd, Nonlinear optics, 3rd ed. (Academic Press, 2008).

Chu, S.

L. Razzari, D. Duchesne, M. Ferrera, R. Morandotti, S. Chu, B. Little, and D. Moss, “Cmos-compatible integrated optical hyper-parametric oscillator,” Nat. Photonics4, 41–45 (2009).
[CrossRef]

Chu, S. T.

B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J.-P. Laine, “Microring resonator channel dropping filters,” J. Lightwave Technol.15, 998–1005 (1997).
[CrossRef]

Cirloganu, C. M.

D. A. Fishman, C. M. Cirloganu, S. Webster, L. A. Padilha, M. Monroe, D. J. Hagan, and E. W. Van Stryland, “Sensitive mid-infrared detection in wide-bandgap semiconductors using extreme non-degenerate two-photon absorption,” Nat. Photonics5, 561–565 (2011).
[CrossRef]

Ciuti, C.

C. Diederichs, J. Tignon, G. Dasbach, C. Ciuti, A. Lemaitre, J. Bloch, P. Roussignol, and C. Delalande, “Parametric oscillation in vertical triple microcavities,” Nature440, 904–907 (2006).
[CrossRef] [PubMed]

Claps, R.

D. Dimitropoulos, R. Jhaveri, R. Claps, J. Woo, and B. Jalali, “Lifetime of photogenerated carriers in silicon-on-insulator rib waveguides,” Appl. Phys. Lett.86, 071115 (2005).
[CrossRef]

Clark, A. S.

Clemmen, S.

Cohen, O.

H. Rong, R. Jones, A. Liu, O. Cohen, D. Hak, A. Fang, and M. Paniccia, “A continuous-wave raman silicon laser,” Nature433, 725–728 (2005).
[CrossRef] [PubMed]

Dahlem, M. S.

M. S. Dahlem, C. W. Holzwarth, H. I. Smith, E. P. Ippen, and M. A. Popović, “Dynamical slow light cell based on controlled far-field interference of microring resonators,” in “Integrated Photonics Research, Silicon and Nanophotonics,” (OSA, 2010).
[CrossRef]

M. A. Popović, T. Barwicz, P. T. Rakich, M. S. Dahlem, C. W. Holzwarth, F. Gan, L. Socci, M. R. Watts, H. I. Smith, F. X. Kärtner, and E. P. Ippen, “Experimental demonstration of loop-coupled microring resonators for optimally sharp optical filters,” in “Conference on Lasers and Electro-Optics,” (OSA, 2008).

M. A. Popović, T. Barwicz, F. Gan, M. S. Dahlem, C. W. Holzwarth, P. T. Rakich, H. I. Smith, E. P. Ippen, and F. X. Kärtner, “Transparent wavelength switching of resonant filters,” in “Conference on Lasers and Electro-Optics,” (Optical Society of America, 2007).

Dasbach, G.

C. Diederichs, J. Tignon, G. Dasbach, C. Ciuti, A. Lemaitre, J. Bloch, P. Roussignol, and C. Delalande, “Parametric oscillation in vertical triple microcavities,” Nature440, 904–907 (2006).
[CrossRef] [PubMed]

Dašic, M.

M. Dašić and M. A. Popović, “Minimum drop-loss design of microphotonic microring-resonator channel add-drop filters,” in “Telecommunications Forum (TELFOR), 2012 20th,” (IEEE, 2012), pp. 927–930.
[CrossRef]

Davanço, M.

M. Davanço, J. R. Ong, A. B. Shehata, A. Tosi, I. Agha, S. Assefa, F. Xia, W. M. Green, S. Mookherjea, and K. Srinivasan, “Telecommunications-band heralded single photons from a silicon nanophotonic chip,” Appl. Phys. Lett.100, 261104 (2012).
[CrossRef]

Delalande, C.

C. Diederichs, J. Tignon, G. Dasbach, C. Ciuti, A. Lemaitre, J. Bloch, P. Roussignol, and C. Delalande, “Parametric oscillation in vertical triple microcavities,” Nature440, 904–907 (2006).
[CrossRef] [PubMed]

Dell’Haye, P.

P. Dell’Haye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, and T. Kippenberg, “Optical frequency comb generation from a monolithic microresonator,” Nature450, 1214–1217 (2007).
[CrossRef]

Diederichs, C.

C. Diederichs, J. Tignon, G. Dasbach, C. Ciuti, A. Lemaitre, J. Bloch, P. Roussignol, and C. Delalande, “Parametric oscillation in vertical triple microcavities,” Nature440, 904–907 (2006).
[CrossRef] [PubMed]

Dimitropoulos, D.

D. Dimitropoulos, R. Jhaveri, R. Claps, J. Woo, and B. Jalali, “Lifetime of photogenerated carriers in silicon-on-insulator rib waveguides,” Appl. Phys. Lett.86, 071115 (2005).
[CrossRef]

Dorenbos, S. N.

Duchesne, D.

L. Razzari, D. Duchesne, M. Ferrera, R. Morandotti, S. Chu, B. Little, and D. Moss, “Cmos-compatible integrated optical hyper-parametric oscillator,” Nat. Photonics4, 41–45 (2009).
[CrossRef]

Engin, E.

Ezaki, M.

Fan, S.

M. F. Yanik, W. Suh, Z. Wang, and S. Fan, “Stopping light in a waveguide with an all-optical analog of electromagnetically induced transparency,” Phys. Rev. Lett.93, 233903 (2004).
[CrossRef] [PubMed]

Fang, A.

H. Rong, R. Jones, A. Liu, O. Cohen, D. Hak, A. Fang, and M. Paniccia, “A continuous-wave raman silicon laser,” Nature433, 725–728 (2005).
[CrossRef] [PubMed]

Fauchet, P.

Q. Lin, J. Zhang, G. Piredda, R. Boyd, P. Fauchet, and G. Agrawal, “Dispersion of silicon nonlinearities in the near infrared region,” Appl. Phys. Lett.91, 021111 (2007).
[CrossRef]

Ferrera, M.

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M. A. Popović, T. Barwicz, F. Gan, M. S. Dahlem, C. W. Holzwarth, P. T. Rakich, H. I. Smith, E. P. Ippen, and F. X. Kärtner, “Transparent wavelength switching of resonant filters,” in “Conference on Lasers and Electro-Optics,” (Optical Society of America, 2007).

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D. A. Fishman, C. M. Cirloganu, S. Webster, L. A. Padilha, M. Monroe, D. J. Hagan, and E. W. Van Stryland, “Sensitive mid-infrared detection in wide-bandgap semiconductors using extreme non-degenerate two-photon absorption,” Nat. Photonics5, 561–565 (2011).
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M. S. Dahlem, C. W. Holzwarth, H. I. Smith, E. P. Ippen, and M. A. Popović, “Dynamical slow light cell based on controlled far-field interference of microring resonators,” in “Integrated Photonics Research, Silicon and Nanophotonics,” (OSA, 2010).
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M. A. Popović, T. Barwicz, F. Gan, M. S. Dahlem, C. W. Holzwarth, P. T. Rakich, H. I. Smith, E. P. Ippen, and F. X. Kärtner, “Transparent wavelength switching of resonant filters,” in “Conference on Lasers and Electro-Optics,” (Optical Society of America, 2007).

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P. Dell’Haye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, and T. Kippenberg, “Optical frequency comb generation from a monolithic microresonator,” Nature450, 1214–1217 (2007).
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M. A. Popović, T. Barwicz, M. R. Watts, P. T. Rakich, L. Socci, E. P. Ippen, F. X. Kärtner, and H. I. Smith, “Multistage high-order microring-resonator add-drop filters,” Opt. Lett.31, 2571–2573 (2006).
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H. Hashemi, A. W. Rodriguez, J. D. Joannopoulos, M. Soljačić, and S. G. Johnson, “Nonlinear harmonic generation and devices in doubly resonant kerr cavities,” Phys. Rev. A79, 013812 (2009).
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A. Rodriguez, M. Soljačić, J. D. Joannopoulos, and S. G. Johnson, “χ(2) and χ(3) harmonic generation at a critical power in inhomogeneous doublyresonant cavities,” Opt. Express15, 7303–7318 (2007).
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M. A. Popović, M. R. Watts, T. Barwicz, P. T. Rakich, L. Socci, E. P. Ippen, F. X. Kärtner, and H. I. Smith, “High-index-contrast, wide-fsr microring-resonator filter design and realization with frequency-shift compensation,” in “Optical Fiber Communication Conference,” (Optical Society of America, 2005).

M. A. Popović, T. Barwicz, F. Gan, M. S. Dahlem, C. W. Holzwarth, P. T. Rakich, H. I. Smith, E. P. Ippen, and F. X. Kärtner, “Transparent wavelength switching of resonant filters,” in “Conference on Lasers and Electro-Optics,” (Optical Society of America, 2007).

M. A. Popović, M. R. Watts, T. Barwicz, P. T. Rakich, L. Socci, E. P. Ippen, F. X. Kärtner, and H. I. Smith, “High-index-contrast and wide-FSR microring-resonator filter design and realization with frequency-shift compensation,” in “Technical Digest of the Optical Fiber Communication Conference,” (Anaheim, CA, 2005). Paper OFK1.

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L. Razzari, D. Duchesne, M. Ferrera, R. Morandotti, S. Chu, B. Little, and D. Moss, “Cmos-compatible integrated optical hyper-parametric oscillator,” Nat. Photonics4, 41–45 (2009).
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L. Razzari, D. Duchesne, M. Ferrera, R. Morandotti, S. Chu, B. Little, and D. Moss, “Cmos-compatible integrated optical hyper-parametric oscillator,” Nat. Photonics4, 41–45 (2009).
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M. A. Popović, T. Barwicz, M. R. Watts, P. T. Rakich, L. Socci, E. P. Ippen, F. X. Kärtner, and H. I. Smith, “Multistage high-order microring-resonator add-drop filters,” Opt. Lett.31, 2571–2573 (2006).
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M. A. Popović, M. R. Watts, T. Barwicz, P. T. Rakich, L. Socci, E. P. Ippen, F. X. Kärtner, and H. I. Smith, “High-index-contrast, wide-fsr microring-resonator filter design and realization with frequency-shift compensation,” in “Optical Fiber Communication Conference,” (Optical Society of America, 2005).

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M. A. Popović, M. R. Watts, T. Barwicz, P. T. Rakich, L. Socci, E. P. Ippen, F. X. Kärtner, and H. I. Smith, “High-index-contrast and wide-FSR microring-resonator filter design and realization with frequency-shift compensation,” in “Technical Digest of the Optical Fiber Communication Conference,” (Anaheim, CA, 2005). Paper OFK1.

C. Gentry and M. A. Popović, “Dark state lasers,” in “CLEO: Science and Innovations,” (Optical Society of America, 2013), pp. CM3F–1.

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M. A. Popović, T. Barwicz, M. R. Watts, P. T. Rakich, L. Socci, E. P. Ippen, F. X. Kärtner, and H. I. Smith, “Multistage high-order microring-resonator add-drop filters,” Opt. Lett.31, 2571–2573 (2006).
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M. A. Popović, T. Barwicz, M. R. Watts, P. T. Rakich, L. Socci, E. P. Ippen, F. X. Kärtner, and H. I. Smith, “Multistage high-order microring-resonator add-drop filters,” Opt. Lett.31, 2571–2573 (2006).
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M. A. Popović, M. R. Watts, T. Barwicz, P. T. Rakich, L. Socci, E. P. Ippen, F. X. Kärtner, and H. I. Smith, “High-index-contrast, wide-fsr microring-resonator filter design and realization with frequency-shift compensation,” in “Optical Fiber Communication Conference,” (Optical Society of America, 2005).

M. A. Popović, T. Barwicz, F. Gan, M. S. Dahlem, C. W. Holzwarth, P. T. Rakich, H. I. Smith, E. P. Ippen, and F. X. Kärtner, “Transparent wavelength switching of resonant filters,” in “Conference on Lasers and Electro-Optics,” (Optical Society of America, 2007).

M. A. Popović, M. R. Watts, T. Barwicz, P. T. Rakich, L. Socci, E. P. Ippen, F. X. Kärtner, and H. I. Smith, “High-index-contrast and wide-FSR microring-resonator filter design and realization with frequency-shift compensation,” in “Technical Digest of the Optical Fiber Communication Conference,” (Anaheim, CA, 2005). Paper OFK1.

M. A. Popović, T. Barwicz, P. T. Rakich, M. S. Dahlem, C. W. Holzwarth, F. Gan, L. Socci, M. R. Watts, H. I. Smith, F. X. Kärtner, and E. P. Ippen, “Experimental demonstration of loop-coupled microring resonators for optimally sharp optical filters,” in “Conference on Lasers and Electro-Optics,” (OSA, 2008).

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L. Razzari, D. Duchesne, M. Ferrera, R. Morandotti, S. Chu, B. Little, and D. Moss, “Cmos-compatible integrated optical hyper-parametric oscillator,” Nat. Photonics4, 41–45 (2009).
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Rodriguez, A. W.

D. M. Ramirez, A. W. Rodriguez, H. Hashemi, J. D. Joannopoulos, M. Soljačić, and S. G. Johnson, “Degenerate four-wave mixing in triply resonant kerr cavities,” Phys. Rev. A83, 033834 (2011).
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H. Rong, R. Jones, A. Liu, O. Cohen, D. Hak, A. Fang, and M. Paniccia, “A continuous-wave raman silicon laser,” Nature433, 725–728 (2005).
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C. Diederichs, J. Tignon, G. Dasbach, C. Ciuti, A. Lemaitre, J. Bloch, P. Roussignol, and C. Delalande, “Parametric oscillation in vertical triple microcavities,” Nature440, 904–907 (2006).
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P. Dell’Haye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, and T. Kippenberg, “Optical frequency comb generation from a monolithic microresonator,” Nature450, 1214–1217 (2007).
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M. A. Popović, T. Barwicz, M. R. Watts, P. T. Rakich, L. Socci, E. P. Ippen, F. X. Kärtner, and H. I. Smith, “Multistage high-order microring-resonator add-drop filters,” Opt. Lett.31, 2571–2573 (2006).
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M. A. Popović, T. Barwicz, M. R. Watts, P. T. Rakich, L. Socci, E. P. Ippen, F. X. Kärtner, and H. I. Smith, “Multistage high-order microring-resonator add-drop filters,” Opt. Lett.31, 2571–2573 (2006).
[CrossRef]

M. A. Popović, T. Barwicz, F. Gan, M. S. Dahlem, C. W. Holzwarth, P. T. Rakich, H. I. Smith, E. P. Ippen, and F. X. Kärtner, “Transparent wavelength switching of resonant filters,” in “Conference on Lasers and Electro-Optics,” (Optical Society of America, 2007).

M. A. Popović, M. R. Watts, T. Barwicz, P. T. Rakich, L. Socci, E. P. Ippen, F. X. Kärtner, and H. I. Smith, “High-index-contrast, wide-fsr microring-resonator filter design and realization with frequency-shift compensation,” in “Optical Fiber Communication Conference,” (Optical Society of America, 2005).

M. A. Popović, M. R. Watts, T. Barwicz, P. T. Rakich, L. Socci, E. P. Ippen, F. X. Kärtner, and H. I. Smith, “High-index-contrast and wide-FSR microring-resonator filter design and realization with frequency-shift compensation,” in “Technical Digest of the Optical Fiber Communication Conference,” (Anaheim, CA, 2005). Paper OFK1.

M. A. Popović, T. Barwicz, P. T. Rakich, M. S. Dahlem, C. W. Holzwarth, F. Gan, L. Socci, M. R. Watts, H. I. Smith, F. X. Kärtner, and E. P. Ippen, “Experimental demonstration of loop-coupled microring resonators for optimally sharp optical filters,” in “Conference on Lasers and Electro-Optics,” (OSA, 2008).

M. S. Dahlem, C. W. Holzwarth, H. I. Smith, E. P. Ippen, and M. A. Popović, “Dynamical slow light cell based on controlled far-field interference of microring resonators,” in “Integrated Photonics Research, Silicon and Nanophotonics,” (OSA, 2010).
[CrossRef]

Socci, L.

M. A. Popović, T. Barwicz, M. R. Watts, P. T. Rakich, L. Socci, E. P. Ippen, F. X. Kärtner, and H. I. Smith, “Multistage high-order microring-resonator add-drop filters,” Opt. Lett.31, 2571–2573 (2006).
[CrossRef]

M. A. Popović, T. Barwicz, M. R. Watts, P. T. Rakich, L. Socci, E. P. Ippen, F. X. Kärtner, and H. I. Smith, “Multistage high-order microring-resonator add-drop filters,” Opt. Lett.31, 2571–2573 (2006).
[CrossRef]

M. A. Popović, M. R. Watts, T. Barwicz, P. T. Rakich, L. Socci, E. P. Ippen, F. X. Kärtner, and H. I. Smith, “High-index-contrast, wide-fsr microring-resonator filter design and realization with frequency-shift compensation,” in “Optical Fiber Communication Conference,” (Optical Society of America, 2005).

M. A. Popović, M. R. Watts, T. Barwicz, P. T. Rakich, L. Socci, E. P. Ippen, F. X. Kärtner, and H. I. Smith, “High-index-contrast and wide-FSR microring-resonator filter design and realization with frequency-shift compensation,” in “Technical Digest of the Optical Fiber Communication Conference,” (Anaheim, CA, 2005). Paper OFK1.

M. A. Popović, T. Barwicz, P. T. Rakich, M. S. Dahlem, C. W. Holzwarth, F. Gan, L. Socci, M. R. Watts, H. I. Smith, F. X. Kärtner, and E. P. Ippen, “Experimental demonstration of loop-coupled microring resonators for optimally sharp optical filters,” in “Conference on Lasers and Electro-Optics,” (OSA, 2008).

Soljacic, M.

D. M. Ramirez, A. W. Rodriguez, H. Hashemi, J. D. Joannopoulos, M. Soljačić, and S. G. Johnson, “Degenerate four-wave mixing in triply resonant kerr cavities,” Phys. Rev. A83, 033834 (2011).
[CrossRef]

H. Hashemi, A. W. Rodriguez, J. D. Joannopoulos, M. Soljačić, and S. G. Johnson, “Nonlinear harmonic generation and devices in doubly resonant kerr cavities,” Phys. Rev. A79, 013812 (2009).
[CrossRef]

A. Rodriguez, M. Soljačić, J. D. Joannopoulos, and S. G. Johnson, “χ(2) and χ(3) harmonic generation at a critical power in inhomogeneous doublyresonant cavities,” Opt. Express15, 7303–7318 (2007).
[CrossRef] [PubMed]

Sorel, M.

Spillane, S.

T. Kippenberg, S. Spillane, and K. Vahala, “Kerr-nonlinearity optical parametric oscillation in an ultrahigh-q toroid microcavity,” Phys. Rev. Lett.93, 83904 (2004).
[CrossRef]

Srinivasan, K.

M. Davanço, J. R. Ong, A. B. Shehata, A. Tosi, I. Agha, S. Assefa, F. Xia, W. M. Green, S. Mookherjea, and K. Srinivasan, “Telecommunications-band heralded single photons from a silicon nanophotonic chip,” Appl. Phys. Lett.100, 261104 (2012).
[CrossRef]

Strain, M. J.

Suh, W.

M. F. Yanik, W. Suh, Z. Wang, and S. Fan, “Stopping light in a waveguide with an all-optical analog of electromagnetically induced transparency,” Phys. Rev. Lett.93, 233903 (2004).
[CrossRef] [PubMed]

Suzuki, N.

Sze, S. M.

S. M. Sze and K. K. Ng, Physics of Semiconductor Devices (Wiley, 2006).

Tanner, M.

Thompson, M. G.

Tignon, J.

C. Diederichs, J. Tignon, G. Dasbach, C. Ciuti, A. Lemaitre, J. Bloch, P. Roussignol, and C. Delalande, “Parametric oscillation in vertical triple microcavities,” Nature440, 904–907 (2006).
[CrossRef] [PubMed]

Tosi, A.

M. Davanço, J. R. Ong, A. B. Shehata, A. Tosi, I. Agha, S. Assefa, F. Xia, W. M. Green, S. Mookherjea, and K. Srinivasan, “Telecommunications-band heralded single photons from a silicon nanophotonic chip,” Appl. Phys. Lett.100, 261104 (2012).
[CrossRef]

Turner, A. C.

Turner-Foster, A. C.

A. C. Turner-Foster, M. A. Foster, J. S. Levy, C. B. Poitras, R. Salem, A. L. Gaeta, and M. Lipson, “Ultrashort free-carrier lifetime in low-loss silicon nanowaveguides,” Opt. Express18, 3582–3591 (2010).
[CrossRef] [PubMed]

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. Photonics4, 37–40 (2009).
[CrossRef]

Vahala, K.

T. Kippenberg, S. Spillane, and K. Vahala, “Kerr-nonlinearity optical parametric oscillation in an ultrahigh-q toroid microcavity,” Phys. Rev. Lett.93, 83904 (2004).
[CrossRef]

Van Stryland, E. W.

D. A. Fishman, C. M. Cirloganu, S. Webster, L. A. Padilha, M. Monroe, D. J. Hagan, and E. W. Van Stryland, “Sensitive mid-infrared detection in wide-bandgap semiconductors using extreme non-degenerate two-photon absorption,” Nat. Photonics5, 561–565 (2011).
[CrossRef]

Wang, Z.

M. F. Yanik, W. Suh, Z. Wang, and S. Fan, “Stopping light in a waveguide with an all-optical analog of electromagnetically induced transparency,” Phys. Rev. Lett.93, 233903 (2004).
[CrossRef] [PubMed]

Watts, M. R.

M. A. Popović, T. Barwicz, M. R. Watts, P. T. Rakich, L. Socci, E. P. Ippen, F. X. Kärtner, and H. I. Smith, “Multistage high-order microring-resonator add-drop filters,” Opt. Lett.31, 2571–2573 (2006).
[CrossRef]

M. A. Popović, T. Barwicz, M. R. Watts, P. T. Rakich, L. Socci, E. P. Ippen, F. X. Kärtner, and H. I. Smith, “Multistage high-order microring-resonator add-drop filters,” Opt. Lett.31, 2571–2573 (2006).
[CrossRef]

M. A. Popović, M. R. Watts, T. Barwicz, P. T. Rakich, L. Socci, E. P. Ippen, F. X. Kärtner, and H. I. Smith, “High-index-contrast, wide-fsr microring-resonator filter design and realization with frequency-shift compensation,” in “Optical Fiber Communication Conference,” (Optical Society of America, 2005).

M. A. Popović, M. R. Watts, T. Barwicz, P. T. Rakich, L. Socci, E. P. Ippen, F. X. Kärtner, and H. I. Smith, “High-index-contrast and wide-FSR microring-resonator filter design and realization with frequency-shift compensation,” in “Technical Digest of the Optical Fiber Communication Conference,” (Anaheim, CA, 2005). Paper OFK1.

M. A. Popović, T. Barwicz, P. T. Rakich, M. S. Dahlem, C. W. Holzwarth, F. Gan, L. Socci, M. R. Watts, H. I. Smith, F. X. Kärtner, and E. P. Ippen, “Experimental demonstration of loop-coupled microring resonators for optimally sharp optical filters,” in “Conference on Lasers and Electro-Optics,” (OSA, 2008).

Webster, S.

D. A. Fishman, C. M. Cirloganu, S. Webster, L. A. Padilha, M. Monroe, D. J. Hagan, and E. W. Van Stryland, “Sensitive mid-infrared detection in wide-bandgap semiconductors using extreme non-degenerate two-photon absorption,” Nat. Photonics5, 561–565 (2011).
[CrossRef]

Wilken, T.

P. Dell’Haye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, and T. Kippenberg, “Optical frequency comb generation from a monolithic microresonator,” Nature450, 1214–1217 (2007).
[CrossRef]

Woo, J.

D. Dimitropoulos, R. Jhaveri, R. Claps, J. Woo, and B. Jalali, “Lifetime of photogenerated carriers in silicon-on-insulator rib waveguides,” Appl. Phys. Lett.86, 071115 (2005).
[CrossRef]

Xia, F.

M. Davanço, J. R. Ong, A. B. Shehata, A. Tosi, I. Agha, S. Assefa, F. Xia, W. M. Green, S. Mookherjea, and K. Srinivasan, “Telecommunications-band heralded single photons from a silicon nanophotonic chip,” Appl. Phys. Lett.100, 261104 (2012).
[CrossRef]

Yanik, M. F.

M. F. Yanik, W. Suh, Z. Wang, and S. Fan, “Stopping light in a waveguide with an all-optical analog of electromagnetically induced transparency,” Phys. Rev. Lett.93, 233903 (2004).
[CrossRef] [PubMed]

Yoshida, H.

Zeng, X.

X. Zeng and M. A. Popović, “Optimum micro-optical parametric oscillators based on third-order nonlinearity,” in “CLEO: Science and Innovations,” (Optical Society of America, 2013), p. CTh1F7.

Zhang, J.

Q. Lin, J. Zhang, G. Piredda, R. Boyd, P. Fauchet, and G. Agrawal, “Dispersion of silicon nonlinearities in the near infrared region,” Appl. Phys. Lett.91, 021111 (2007).
[CrossRef]

Zwiller, V.

Appl. Phys. Lett.

M. Davanço, J. R. Ong, A. B. Shehata, A. Tosi, I. Agha, S. Assefa, F. Xia, W. M. Green, S. Mookherjea, and K. Srinivasan, “Telecommunications-band heralded single photons from a silicon nanophotonic chip,” Appl. Phys. Lett.100, 261104 (2012).
[CrossRef]

D. Dimitropoulos, R. Jhaveri, R. Claps, J. Woo, and B. Jalali, “Lifetime of photogenerated carriers in silicon-on-insulator rib waveguides,” Appl. Phys. Lett.86, 071115 (2005).
[CrossRef]

Q. Lin, J. Zhang, G. Piredda, R. Boyd, P. Fauchet, and G. Agrawal, “Dispersion of silicon nonlinearities in the near infrared region,” Appl. Phys. Lett.91, 021111 (2007).
[CrossRef]

J. Lightwave Technol.

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[CrossRef]

Nat. Photonics

D. A. Fishman, C. M. Cirloganu, S. Webster, L. A. Padilha, M. Monroe, D. J. Hagan, and E. W. Van Stryland, “Sensitive mid-infrared detection in wide-bandgap semiconductors using extreme non-degenerate two-photon absorption,” Nat. Photonics5, 561–565 (2011).
[CrossRef]

L. Razzari, D. Duchesne, M. Ferrera, R. Morandotti, S. Chu, B. Little, and D. Moss, “Cmos-compatible integrated optical hyper-parametric oscillator,” Nat. Photonics4, 41–45 (2009).
[CrossRef]

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. Photonics4, 37–40 (2009).
[CrossRef]

Nature

C. Diederichs, J. Tignon, G. Dasbach, C. Ciuti, A. Lemaitre, J. Bloch, P. Roussignol, and C. Delalande, “Parametric oscillation in vertical triple microcavities,” Nature440, 904–907 (2006).
[CrossRef] [PubMed]

H. Rong, R. Jones, A. Liu, O. Cohen, D. Hak, A. Fang, and M. Paniccia, “A continuous-wave raman silicon laser,” Nature433, 725–728 (2005).
[CrossRef] [PubMed]

P. Dell’Haye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, and T. Kippenberg, “Optical frequency comb generation from a monolithic microresonator,” Nature450, 1214–1217 (2007).
[CrossRef]

Opt. Express

M. A. Foster, J. S. Levy, O. Kuzucu, K. Saha, M. Lipson, and A. L. Gaeta, “Silicon-based monolithic optical frequency comb source,” Opt. Express19, 14233–14239 (2011).
[CrossRef] [PubMed]

A. C. Turner, M. A. Foster, A. L. Gaeta, and M. Lipson, “Ultra-low power parametric frequency conversion in a silicon microring resonator,” Opt. Express16, 4881–4887 (2008).
[CrossRef] [PubMed]

W. D. Sacher and J. K. S. Poon, “Dynamics of microring resonator modulators,” Opt. Express16, 15741–15753 (2008).
[CrossRef] [PubMed]

E. Engin, D. Bonneau, C. M. Natarajan, A. S. Clark, M. Tanner, R. Hadfield, S. N. Dorenbos, V. Zwiller, K. Ohira, N. Suzuki, H. Yoshida, N. Iizuka, M. Ezaki, J. L. O’Brien, and M. G. Thompson, “Photon pair generation in a silicon micro-ring resonator with reverse bias enhancement,” Opt. Express21, 27826–27834 (2013).
[CrossRef]

S. Azzini, D. Grassani, M. J. Strain, M. Sorel, L. Helt, J. Sipe, M. Liscidini, M. Galli, and D. Bajoni, “Ultra-low power generation of twin photons in a compact silicon ring resonator,” Opt. Express20, 23100–23107 (2012).
[CrossRef] [PubMed]

A. C. Turner-Foster, M. A. Foster, J. S. Levy, C. B. Poitras, R. Salem, A. L. Gaeta, and M. Lipson, “Ultrashort free-carrier lifetime in low-loss silicon nanowaveguides,” Opt. Express18, 3582–3591 (2010).
[CrossRef] [PubMed]

A. Rodriguez, M. Soljačić, J. D. Joannopoulos, and S. G. Johnson, “χ(2) and χ(3) harmonic generation at a critical power in inhomogeneous doublyresonant cavities,” Opt. Express15, 7303–7318 (2007).
[CrossRef] [PubMed]

B. Kuyken, H. Ji, S. Clemmen, S. Selvaraja, H. Hu, M. Pu, M. Galili, P. Jeppesen, G. Morthier, S. Massar, L. K. Oxenløwe, G. Roelkens, and R. Baets, “Nonlinear properties of and nonlinear processing in hydrogenated amorphous silicon waveguides,” Opt. Express19, B146–B153 (2011).
[CrossRef]

Q. Lin, O. J. Painter, and G. P. Agrawal, “Nonlinear optical phenomena in silicon waveguides: modeling and applications,” Opt. Express15, 16604–16644 (2007).
[CrossRef] [PubMed]

A. C. Turner, M. A. Foster, A. L. Gaeta, and M. Lipson, “Ultra-low power parametric frequency conversion in a silicon microring resonator,” Opt. Express16, 4881–4887 (2008).
[CrossRef] [PubMed]

A. R. Motamedi, A. H. Nejadmalayeri, A. Khilo, F. X. Kärtner, and E. P. Ippen, “Ultrafast nonlinear optical studies of silicon nanowaveguides,” Opt. Express20, 4085–4101 (2012).
[CrossRef] [PubMed]

Opt. Lett.

Phys. Rev. A

H. Hashemi, A. W. Rodriguez, J. D. Joannopoulos, M. Soljačić, and S. G. Johnson, “Nonlinear harmonic generation and devices in doubly resonant kerr cavities,” Phys. Rev. A79, 013812 (2009).
[CrossRef]

D. M. Ramirez, A. W. Rodriguez, H. Hashemi, J. D. Joannopoulos, M. Soljačić, and S. G. Johnson, “Degenerate four-wave mixing in triply resonant kerr cavities,” Phys. Rev. A83, 033834 (2011).
[CrossRef]

Phys. Rev. Lett.

T. Kippenberg, S. Spillane, and K. Vahala, “Kerr-nonlinearity optical parametric oscillation in an ultrahigh-q toroid microcavity,” Phys. Rev. Lett.93, 83904 (2004).
[CrossRef]

M. F. Yanik, W. Suh, Z. Wang, and S. Fan, “Stopping light in a waveguide with an all-optical analog of electromagnetically induced transparency,” Phys. Rev. Lett.93, 233903 (2004).
[CrossRef] [PubMed]

Other

M. S. Dahlem, C. W. Holzwarth, H. I. Smith, E. P. Ippen, and M. A. Popović, “Dynamical slow light cell based on controlled far-field interference of microring resonators,” in “Integrated Photonics Research, Silicon and Nanophotonics,” (OSA, 2010).
[CrossRef]

H. A. Haus, Waves and Fields in Optoelectronics (Prentice-HallNew Jersey, 1984).

M. A. Popović, “Resonant optical modulators beyond conventional energy-efficiency and modulation frequency limitations,” in “Integrated Photonics Research, Silicon and Nanophotonics,” (OSA, 2010).
[CrossRef]

M. A. Popović, “Sharply-defined optical filters and dispersionless delay lines based on loop-coupled resonators and,” in “Lasers and Electro-Optics, 2007. CLEO 2007. Conference on,” (IEEE, 2007), pp. 1–2.

M. A. Popović, T. Barwicz, P. T. Rakich, M. S. Dahlem, C. W. Holzwarth, F. Gan, L. Socci, M. R. Watts, H. I. Smith, F. X. Kärtner, and E. P. Ippen, “Experimental demonstration of loop-coupled microring resonators for optimally sharp optical filters,” in “Conference on Lasers and Electro-Optics,” (OSA, 2008).

X. Zeng and M. A. Popović, “Optimum micro-optical parametric oscillators based on third-order nonlinearity,” in “CLEO: Science and Innovations,” (Optical Society of America, 2013), p. CTh1F7.

A. H. Atabaki and A. Adibi, “Ultra-compact coupled-resonator device for four-wave-mixing applications,” in “CLEO: Science and Innovations,” (Optical Society of America, 2011), p. CTuS6.

R. W. Boyd, Nonlinear optics, 3rd ed. (Academic Press, 2008).

M. A. Popović, M. R. Watts, T. Barwicz, P. T. Rakich, L. Socci, E. P. Ippen, F. X. Kärtner, and H. I. Smith, “High-index-contrast and wide-FSR microring-resonator filter design and realization with frequency-shift compensation,” in “Technical Digest of the Optical Fiber Communication Conference,” (Anaheim, CA, 2005). Paper OFK1.

M. Dašić and M. A. Popović, “Minimum drop-loss design of microphotonic microring-resonator channel add-drop filters,” in “Telecommunications Forum (TELFOR), 2012 20th,” (IEEE, 2012), pp. 927–930.
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M. A. Popović, M. R. Watts, T. Barwicz, P. T. Rakich, L. Socci, E. P. Ippen, F. X. Kärtner, and H. I. Smith, “High-index-contrast, wide-fsr microring-resonator filter design and realization with frequency-shift compensation,” in “Optical Fiber Communication Conference,” (Optical Society of America, 2005).

C. Gentry and M. A. Popović, “Dark state lasers,” in “CLEO: Science and Innovations,” (Optical Society of America, 2013), pp. CM3F–1.

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Figures (11)

Fig. 1
Fig. 1

(a) Illustration of the micro-OPO model including a multimode resonator; (b) a traveling-wave resonant structure enables separated input and output ports; (c) example proposed multimode resonator based on 3 coupled microring cavities, showing an approach to unequal pump and signal/idler external coupling [22].

Fig. 2
Fig. 2

Normalized design curves for optimum OPO (using a “partial-TPA” model with pump-assisted TPA terms only and no FCA included): (a) maximum pump-to-signal/idler conversion efficiency versus pump power (normalized by oscillation threshold when loss due to TPA is ignored) and nonlinear loss sine [defined in Eq. (5)]; (b) corresponding optimum pump resonance coupling normalized by cavity intrinsic loss; (c) corresponding optimum ratio of signal/idler relative to pump resonance coupling.

Fig. 3
Fig. 3

Performance comparison of OPO designs with optimum unequal pump and signal/idler couplings and with optimized equal couplings (assuming no FCA): (a) power conversion efficiency; (b) optimum coupling values.

Fig. 4
Fig. 4

Normalized design curves for optimum OPO (I) using a “full-TPA” model (with all TPA terms but no FCA included) and (II) comparison of “partial-TPA” and “full-TPA” models (assuming no FCA): (a) maximum efficiency versus pump power and nonlinear loss sine, and corresponding (b) pump resonance coupling and (c) ratio of signal/idler relative to pump resonance coupling in (I) and signal/idler resonance coupling in (II). See Fig. 2 for parameter definitions.

Fig. 5
Fig. 5

Optimum OPO design curves for nonlinear media with and without TPA loss (assuming no FCA), representative of, e.g., of silicon nitride at 1550 nm and Si at 2.3 μm (linear), and Si at 1550 nm (σ3 = 0.23).

Fig. 6
Fig. 6

Performance of silicon microcavity at 1550 nm resonance with various free-carrier lifetime and intrinsic cavity quality factors.

Fig. 7
Fig. 7

The OPO threshold vs (a) normalized free carrier lifetime and σ3; (b) free carrier lifetime for silicon cavity resonant near 1550 nm with linear unloaded Q of 106 and effective volume of 8.4 μm3.

Fig. 8
Fig. 8

Example microring cavity topology for illustration of effective figure of merit: (a) single-ring cavity with traveling-wave mode; (b) single-ring cavity with standing-wave mode; (c) triple-ring cavity with traveling-wave mode; (d) triple-ring cavity with standing-wave mode.

Fig. 9
Fig. 9

Mode fields of the pump, signal and idler resonances for the configurations (a)–(d) in Fig. 8 (color–coded intensity scales are different in single and triple-cavity cases in order to show the mode features clearly).

Fig. 10
Fig. 10

Mode overlap integrand for the FWM and various TPA coefficients for configurations (a)–(d) in Fig. 8. It shows that a vector of FOM is needed to account for the ratio of FWM relative to various TPA terms. Besides, different cavity topologies have different FOM (see Table 3).

Fig. 11
Fig. 11

Small signal gain and loss in an optical parametric oscillator based on degenerate four wave mixing.

Tables (4)

Tables Icon

Table 1 Third-order nonlinear properties of some common on-chip nonlinear material

Tables Icon

Table 2 Predicted performance of optical parametric oscillators based on some common on-chip nonlinear material in a single-ring cavity with traveling-wave mode

Tables Icon

Table 3 Comparison of FWM and TPA coefficients in various cavity topologies

Tables Icon

Table 4 Predicted performance of optical parametric oscillators based on 3-ring photonic molecule with traveling-wave mode

Equations (77)

Equations on this page are rendered with MathJax. Learn more.

d A s d t = r s , tot A s j ω s β fwm , s A p 2 A i *
d A p d t = r p , tot A p 2 j ω p β fwm , p A p * A s A i j 2 r p , ext S p , +
d A i d t = r i , tot A i j ω i β fwm , i A p 2 A s *
S s , = j 2 r s , ext A s
S p , = S p , + j 2 r p , ext A p
S i , = j 2 r i , ext A i
β fwm , s = β fwm , p * = β fwm , i β fwm .
r s , tot = r s , o + r s , ext + r FC + ω s ( β tpa , ss | A s | 2 + 2 β tpa , sp | A p | 2 + 2 β tpa , si | A i | 2 ) r p , tot = r p , o + r p , ext + r FC + ω p ( 2 β tpa , sp | A s | 2 + β tpa , pp | A p | 2 + 2 β tpa , ip | A i | 2 ) r i , tot = r i , o + r i , ext + r FC + ω i ( 2 β tpa , si | A s | 2 + 2 β tpa , ip | A p | 2 + β tpa , ii | A i | 2 ) .
r FC = τ FC σ a v g 2 V eff ( β tpa , ss | A s | 4 + β tpa , pp | A p | 4 + β tpa , ii | A i | 4 + 4 β tpa , sp | A s | 2 | A p | 2 + 4 β tpa , ip | A i | 2 | A p | 2 + 4 β tpa , si | A s | 2 | A i | 2 )
σ 3 [ χ ( 3 ) ] | χ ( 3 ) | = β tpa β fwm
d B s d τ = ρ s , tot B s j 2 B p 2 B i *
d B p d τ = ρ p , tot B p j 4 B p * B s B i j 2 ρ p , ext T p , +
d B i d τ = ρ i , tot B i j 2 B p 2 B s *
T s , = j 2 ρ s , ext B s
T p , = T p , + j 2 ρ p , ext B p
T i , = j 2 ρ i , ext B i .
τ r o t
B k A k A o , with A o 2 r o ω β fwm
T k , ± S k , ± S o , with S o 2 r o 2 ω β fwm
ρ s , tot 1 + ρ s , ext + 2 σ 3 ( d ss | B s | 2 + 2 d sp | B p | 2 + 2 d si | B i | 2 ) + ρ FC
ρ p , tot 1 + ρ p , ext + 2 σ 3 ( 2 d sp | B s | 2 + d pp | B p | 2 + 2 d ip | B i | 2 ) + ρ FC
ρ i , tot 1 + ρ i , ext + 2 σ 3 ( 2 d si | B s | 2 + 2 d ip | B p | 2 + d ii | B i | 2 ) + ρ FC .
d mn β tpa , mn σ 3 β fwm
ρ FC r FC r o = σ 3 ρ FC ( d ss | B s | 4 + d pp | B p | 4 + d ii | B i | 4 + 4 d sp | B s | 2 | B p | 2 + 4 d ip | B i | 2 | B p | 2 + 4 d si | B s | 2 | B i | 2 )
ρ FC τ FC σ a v g V eff β fwm 2 4 r o ( ω β fwm ) 2 = ( σ a n nl 2 ω n g n 2 ) τ FC Q o .
η | S s , | 2 | S p , + | 2 .
B s = 2 j ρ s , tot 1 B p 2 B i *
B i = 2 j ρ i , tot 1 B p 2 B s *
T p , + = j ρ p , tot B p + 4 j B p * B s B i 2 ρ p , ext
d ss = d pp = d ii = d sp = d si = d ip = 1 .
ρ s , tot = 1 + ρ s , ext + 4 σ 3 | B p | 2 ρ p , tot = 1 + ρ p , ext + 2 σ 3 | B p | 2 ρ i , tot = 1 + ρ s , ext + 4 σ 3 | B p | 2
| B s | 2 = | B i | 2 = 0 ( below threshold )
| B p | 2 = ( 1 + ρ s , ext ) 2 ( 1 2 σ 3 ) ( above threshold ) .
P th , min = 1 σ 3 ( 1 2 σ 3 ) 2 2 r o 2 ω β fwm = 1 σ 3 ( 1 2 σ 3 ) 2 P th , lin , min
η | S s , | 2 | S p , + | 2 = 2 r s , ext | A s | 2 | S p , + | 2 = 2 ρ s , ext | B s | 2 | T p , + | 2 .
ρ p , ext , opt = 1 2 σ 3 1 + ρ s , ext | T p , + | 2 .
( 1 + ρ s , ext ) 2 ( 2 σ 3 ρ s , ext + 1 σ 3 ) ( 1 2 σ 3 ) 2 | T p , + | 2 = 0 .
ρ s , ext , opt = 1 6 σ 3 ( 1 3 σ 3 + ( D E ) 1 / 3 + ( D + E ) 1 / 3 ) D ( 3 σ 3 1 ) 3 + 54 σ 3 2 ( 1 2 σ 3 ) 2 | T p , + | 2 E = 3 σ 3 ( 1 2 σ 3 ) 6 | T p , + | 2 ( 3 σ 3 1 ) 3 + D ]
ρ s , ext , opt = | T p , + | 2 1
ρ p , ext , opt = | T p , + | 2 .
η max ( | T p , + | 2 , σ 3 = 0 ) = ( | T p , + | 2 1 ) 2 2 | T p , + | 2
| T p , + ( ec ) | 2 = ( 1 + ρ ext , opt ) 3 ( 1 + 2 ρ ext , opt ) 2 ρ ext , opt ( 3 + 2 ρ ext , opt ) 2 .
η < 1 2 σ 3 .
P th , min ( ec ) = 27 ( 1 σ 3 ) 2 16 ( 1 2 σ 3 ) 3 P th , lin , min .
ρ s , tot = 1 + ρ s , ext + 2 σ 3 ( | B s | 2 + 2 | B p | 2 + 2 | B i | 2 ) ρ p , tot = 1 + ρ p , ext + 2 σ 3 ( 2 | B s | 2 + | B p | 2 + 2 | B i | 2 ) ρ i , tot = 1 + ρ s , ext + 2 σ 3 ( 2 | B s | 2 + 2 | B p | 2 + | B i | 2 )
| B p | 2 = ( 1 + ρ s , ext + 6 σ 3 | B s | 2 ) 2 ( 1 2 σ 3 )
4 ( 1 2 σ 3 ) 3 ρ p , ext | T p , + | 2 = ( 6 σ 3 | B s | 2 + 1 + ρ s , ext ) × [ ( 1 2 σ 3 ) ( 1 + ρ p , ext ) + σ 3 ( 1 + ρ s , ext ) + 2 ( 2 5 σ 3 2 ) | B s | 2 ] 2
σ 3 ρ FC B p 4 + ( 8 σ 3 ρ FC B s 2 + 4 σ 3 2 ) B p 2 = ( 6 σ 3 ρ FC B s 4 + 6 σ 3 B s 2 + 1 + ρ s , ext )
[ ( 2 2 σ 3 ) B p 2 + ( 4 + 2 σ 3 ) B s 2 + ρ p , ext ρ s , ext ] 2 B p 2 = 2 ρ p , ext , opt T p , + 2
P th , min = 4 ( 1 σ 3 ) [ ( 1 2 σ 3 ) + ( 1 2 σ 3 ) 2 σ 3 ρ FC ] 2 P th , lin , min
β fwm , s = 3 16 ε 0 d 3 x ( E s * χ ¯ ¯ ( 3 ) E p 2 E i * ) d 3 x ( 1 2 ε | E s | 2 ) d 3 x ( 1 2 ε | E i | 2 ) d 3 x ( 1 2 ε | E p | 2 ) 3 χ 1111 ( 3 ) 4 n nl 4 ε 0 V eff
V eff χ 1111 ( 3 ) d 3 x ( ε | E s | 2 ) d 3 x ( ε | E i | 2 ) d 3 x ( ε | E p | 2 ) ε 0 2 n n l 4 d 3 x ( E s * χ ¯ ¯ ( 3 ) : E p 2 E i * ) .
β tpa , sp = 3 16 ε 0 d 3 x ( E s * [ χ ¯ ¯ ( 3 ) ] : E s E p E p * ) d 3 x ( 1 2 ε | E s | 2 ) d 3 x ( 1 2 ε | E p | 2 ) .
N ν t = G N ν τ ν + D ν 2 N ν s ν μ ν ( N ν E dc ) G N ν τ ν , eff
G = 1 2 ω Δ E Δ t Δ V = 1 4 ω [ E tot * J ] = 1 4 ω [ j ω ε 0 E tot * χ ¯ ¯ ( 3 ) : E tot 3 ]
N ν = G τ ν , eff .
α ν = σ ν N ν
r k , FC ν = j ω 4 d 3 x ( E k * δ P k ( FCA , ν ) ) d 3 x ( 1 2 ε | E k | 2 ) = ω 4 d 3 x ( ε 0 n nl α ν k 0 | E k | 2 ) d 3 x ( 1 2 ε | E k | 2 ) = ε 0 n nl ω σ ν 4 k 0 d 3 x ( G τ ν , eff | E k | 2 ) d 3 x ( 1 2 ε | E k | 2 ) = c ε 0 2 n nl σ ν 16 d 3 x ( τ ν , eff ( E tot * [ χ ¯ ¯ ( 3 ) ] : E tot 3 ) | E k | 2 ) d 3 x ( 1 2 ε | E k | 2 ) .
N ν = τ eff V eff ( β tpa , ss | A s | 4 + β tpa , pp | A p | 4 + β tpa , ii | A i | 4 + 4 β tpa , sp | A s | 2 | A p | 2 + 4 β tpa , ip | A i | 2 | A p | 2 + 4 β tpa , si | A s | 2 | A i | 2 ) .
r FC = α FC v g 2 = σ a N ν v g 2 = τ eff σ a v g 2 V eff ( β tpa , ss | A s | 4 + β tpa , pp | A p | 4 + β tpa , ii | A i | 4 + 4 β tpa , sp | A s | 2 | A p | 2 + 4 β tpa , ip | A i | 2 | A p | 2 + 4 β tpa , si | A s | 2 | A i | 2 )
T p , + = j ( 2 ρ p , ext ) 1 ( ρ p , tot + 8 ρ i , tot 1 | B s | 2 | B p | 2 ) B p .
| T p , + | 2 = ρ p , tot 2 2 ρ p , ext | B p | 2
ρ p , ext = 1 + 2 σ 3 | B p | 2 + σ 3 ρ F C | B p | 4
P th = ( 2 ( 1 + 2 σ 3 | B p | 2 + σ 3 ρ F C | B p | 4 ) | B p | 2 ) min
2 | B p | 2 = ρ s , tot = 1 + ρ s , ext + 4 σ 3 | B p | 2 + σ 3 ρ F C | B p | 4
| B p | 2 = ( 1 2 σ 3 ) ( 1 2 σ 3 ) 2 σ 3 ρ F C ( 1 + ρ s , ext ) σ 3 ρ F C
ρ F C ( 1 2 σ 3 ) 3 σ 3
P th = 4 ( 1 σ 3 ) ( ( 1 2 σ 3 ) + ( 1 2 σ 3 ) 2 σ 3 ρ FC ) 2
| T p , + | 2 = ρ p , tot 3 4 ( 1 σ 3 ) ρ ext
P th = ( 1 σ 3 ) 2 4 ( 1 2 σ 3 ) 3 ( 1 + ρ ext ) 3 ρ ext | min = 27 ( 1 σ 3 ) 2 16 ( 1 2 σ 3 ) 3
T p , + = j ( 2 ρ p , ext ) 1 ( ρ p , tot + 8 ρ i , tot 1 | B s | 2 | B p | 2 ) B p = A + B | B s | 2
η = 2 ρ s , ext | B s | 2 | T p , + | 2 = ( 2 ρ s , ext B ) T p , + A | T p , + | 2
η = ( 2 ρ s , ext | B | ) | T p , + | | A | | T p , + | 2
| T p , + | 2 = 4 A 2 = 2 ρ p , tot 2 ρ p , ext | B p | 2
η max = ρ s , ext 2 A B = ( 1 2 σ 3 ) 2 2 ρ s , ext ρ p , ext ρ i , tot ρ p , tot ( 1 + ρ s , ext ) 2 = ( 1 2 σ 3 ) ρ p , ext ρ p , tot ρ s , ext ( 1 + ρ i , ext ) ( 1 + ρ s , ext ) 2 < 1 2 σ 3 .
d B s d τ = ρ s , tot B s + 4 ρ i , tot 1 | B p | 4 B s ρ loss B s + ρ gain B s
d B p d τ = ρ p , tot B p 8 ρ i , tot 1 | B p | 2 B p j 2 ρ p , ext T p , +

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