Abstract

Microring resonators made from silicon are becoming a popular microscale device format for generating photon pairs at telecommunications wavelengths at room temperature. In compact devices with a footprint less than 5 × 104 mm2, we demonstrate pair generation using only a few microwatts of average pump power. We discuss the role played by important parameters such as the loss, group-velocity dispersion and the ring-waveguide coupling coefficient in finding the optimum operating point for silicon microring pair generation. Silicon photonics can be fabricated using deep ultraviolet lithography wafer-scale fabrication processes, which is scalable and cost-effective. Such small devices and low pump power requirements, and the side-coupled waveguide geometry which uses an integrated waveguide, could be beneficial for future scaled-up architectures where many pair-generation devices are required on the same chip.

© 2016 Optical Society of America

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2015 (6)

2014 (3)

2013 (4)

J. R. Ong, R. Kumar, and S. Mookherjea, “Ultra-high-contrast and tunable-bandwidth filter using cascaded high-order silicon microring filters,” Photon. Technol. Lett., IEEE 25, 1543–1546 (2013).
[Crossref]

J. R. Ong, R. Kumar, R. Aguinaldo, and S. Mookherjea, “Efficient cw four-wave mixing in silicon-on-insulator micro-rings with active carrier removal,” Photon. Technol. Lett., IEEE 25, 1699–1702 (2013).
[Crossref]

R. Kumar, J. R. Ong, J. Recchio, K. Srinivasan, and S. Mookherjea, “Spectrally multiplexed and tunable-wavelength photon pairs at 1.55 µ m from a silicon coupled-resonator optical waveguide,” Opt. Lett. 38, 2969–2971 (2013).
[Crossref] [PubMed]

E. Engin, D. Bonneau, C. M. Natarajan, A. S. Clark, M. G. Tanner, R. H. 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. Express 21, 27826–27834 (2013).
[Crossref]

2012 (6)

L. G. Helt, M. Liscidini, and J. E. Sipe, “How does it scale? comparing quantum and classical nonlinear optical processes in integrated devices,” J. Opt. Soc. Am. B 29, 2199–2212 (2012).
[Crossref]

S. Azzini, D. Grassani, M. Galli, L. C. Andreani, M. Sorel, M. J. Strain, L. G. Helt, J. E. Sipe, M. Liscidini, and D. Bajoni, “From classical four-wave mixing to parametric fluorescence in silicon microring resonators,” Opt. Lett. 37, 3807–3809 (2012).
[Crossref] [PubMed]

A. Biberman, M. J. Shaw, E. Timurdogan, J. B. Wright, and M. R. Watts, “Ultralow-loss silicon ring resonators,” Opt. Lett. 37, 4236–4238 (2012).
[Crossref] [PubMed]

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

A. Tosi, A. DellaFrera, A. Bahgat Shehata, and C. Scarcella, “Fully programmable single-photon detection module for InGaAs/InP single-photon avalanche diodes with clean and sub-nanosecond gating transitions,” Rev. Sci. Instrum. 83, 013104 (2012).
[Crossref] [PubMed]

R. Aguinaldo, Y. Shen, and S. Mookherjea, “Large dispersion of silicon directional couplers obtained via wide-band microring parametric characterization,” IEEE Photon. Technol. Lett. 24, 1242–1244 (2012).
[Crossref]

2011 (3)

F. Morichetti, A. Canciamilla, C. Ferrari, A. Samarelli, M. Sorel, and A. Melloni, “Travelling-wave resonant four-wave mixing breaks the limits of cavity-enhanced all-optical wavelength conversion,” Nat. Commun. 2, 296 (2011).
[Crossref] [PubMed]

C. Xiong, G. D. Marshall, A. Peruzzo, M. Lobino, A. S. Clark, D. Y. Choi, S. J. Madden, C. M. Natarajan, M. G. Tanner, R. H. Hadfield, S. N. Dorenbos, T. Zijlstra, V. Zwiller, M. G. Thompson, J. G. Rarity, M. J. Steel, B. Luther-Davies, B. J. Eggleton, and J. L. O’Brien, “Generation of correlated photon pairs in a chalcogenide As2S3 waveguide,” Appl. Phys. Lett. 98, 051101 (2011).
[Crossref]

J. Chen, Z. H. Levine, J. Fan, and A. L. Migdall, “Frequency-bin entangled comb of photon pairs from a silicon-on-insulator micro-resonator,” Opt. Express 19, 1470–1483 (2011).
[Crossref] [PubMed]

2010 (3)

M. L. Cooper, G. Gupta, M. A. Schneider, W. M. J. Green, S. Assefa, F. Xia, Y. A. Vlasov, and S. Mookherjea, “Statistics of light transport in 235-ring silicon coupled-resonator optical waveguides,” Opt. Express 18, 26505–26516 (2010).
[Crossref] [PubMed]

M. Soltani, S. Yegnanarayanan, L. Qing, and A. Adibi, “Systematic engineering of waveguide-resonator coupling for silicon microring/microdisk/racetrack resonators: theory and experiment,” IEEE J. Quant. Electron. 46, 1158–1169 (2010).
[Crossref]

D. Höckel, E. Martin, and O. Benson, “Note: An ultranarrow bandpass filter system for single-photon experiments in quantum optics,” Rev. Sci. Instrum. 81, 026108 (2010).
[Crossref] [PubMed]

2009 (1)

2008 (2)

K.-I. Harada, H. Takesue, H. Fukuda, T. Tsuchizawa, T. Watanabe, K. Yamada, Y. Tokura, and S.-I. Itabashi, “Generation of high-purity entangled photon pairs using silicon wirewaveguide,” Opt. Express 16, 20368–20373 (2008).
[Crossref] [PubMed]

P. J. Mosley, J. S. Lundeen, B. J. Smith, P. Wasylczyk, A. B. U’Ren, C. Silberhorn, and I. A. Walmsley, “Heralded generation of ultrafast single photons in pure quantum states,” Phys. Rev. Lett. 100, 133601 (2008).
[Crossref] [PubMed]

2006 (3)

2005 (2)

2004 (3)

2003 (1)

K. J. Vahala, “Optical microcavities,” Nature 424, 829 (2003).
[Crossref]

2000 (2)

A. Yariv, “Universal relations for coupling of optical power between microresonators and dielectric waveguides,” Electron. Lett. 36, 321–322 (2000).
[Crossref]

P. P. Absil, J. V. Hryniewicz, B. E. Little, P. S. Cho, R. A. Wilson, L. G. Joneckis, and P.-T. Ho, “Wavelength conversion in GaAs micro-ring resonators,” Opt. Lett. 25, 554–556 (2000).
[Crossref]

Absil, P. P.

Adibi, A.

M. Soltani, S. Yegnanarayanan, L. Qing, and A. Adibi, “Systematic engineering of waveguide-resonator coupling for silicon microring/microdisk/racetrack resonators: theory and experiment,” IEEE J. Quant. Electron. 46, 1158–1169 (2010).
[Crossref]

Agha, I.

M. Davanco, J. R. Ong, A. B. Shehata, A. Tosi, I. Agha, S. Assefa, F. Xia, W. M. J. 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. P.

Aguinaldo, R.

J. R. Ong, R. Kumar, R. Aguinaldo, and S. Mookherjea, “Efficient cw four-wave mixing in silicon-on-insulator micro-rings with active carrier removal,” Photon. Technol. Lett., IEEE 25, 1699–1702 (2013).
[Crossref]

R. Aguinaldo, Y. Shen, and S. Mookherjea, “Large dispersion of silicon directional couplers obtained via wide-band microring parametric characterization,” IEEE Photon. Technol. Lett. 24, 1242–1244 (2012).
[Crossref]

Andreani, L. C.

Aoki, T.

Assefa, S.

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

M. L. Cooper, G. Gupta, M. A. Schneider, W. M. J. Green, S. Assefa, F. Xia, Y. A. Vlasov, and S. Mookherjea, “Statistics of light transport in 235-ring silicon coupled-resonator optical waveguides,” Opt. Express 18, 26505–26516 (2010).
[Crossref] [PubMed]

Atikian, H. A.

Azzini, S.

Baehr-Jones, T.

N. C. Harris, D. Grassani, A. Simbula, M. Pant, M. Galli, T. Baehr-Jones, M. Hochberg, D. Englund, D. Bajoni, and C. Galland, “Integrated source of spectrally filtered correlated photons for large-scale quantum photonic systems,” Phys. Rev. X 4, 041047 (2014).

Y. Zhang, T. Baehr-Jones, R. Ding, T. Pinguet, Z. Xuan, and M. Hochberg, “Silicon multi-project wafer platforms for optoelectronic system integration,” in “Group IV Photonics (GFP), 2012 IEEE 9th International Conference on,” (2012), pp. 63–65.

Baets, R.

Bajoni, D.

Benson, O.

D. Höckel, E. Martin, and O. Benson, “Note: An ultranarrow bandpass filter system for single-photon experiments in quantum optics,” Rev. Sci. Instrum. 81, 026108 (2010).
[Crossref] [PubMed]

Biberman, A.

Bogaerts, W.

Bolivar, P. H.

Bonneau, D.

D. Bonneau, G. J. Mendoza, J. L. O’Brien, and M. G. Thompson, “Effect of loss on multiplexed single-photon sources,” New J. Phys. 17, 043057 (2015).
[Crossref]

E. Engin, D. Bonneau, C. M. Natarajan, A. S. Clark, M. G. Tanner, R. H. 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. Express 21, 27826–27834 (2013).
[Crossref]

J. W. Silverstone, R. Santagati, D. Bonneau, M. J. Strain, M. Sorel, J. L. O’Brien, and M. G. Thompson, “Qubit entanglement between ring-resonator photon-pair sources on a silicon chip,” Nat. Commun.6 (2015).
[Crossref] [PubMed]

Borselli, M.

Burek, M. J.

Canciamilla, A.

F. Morichetti, A. Canciamilla, C. Ferrari, A. Samarelli, M. Sorel, and A. Melloni, “Travelling-wave resonant four-wave mixing breaks the limits of cavity-enhanced all-optical wavelength conversion,” Nat. Commun. 2, 296 (2011).
[Crossref] [PubMed]

Carmon, T.

Chen, J.

Cho, P. S.

Choi, D. Y.

C. Xiong, G. D. Marshall, A. Peruzzo, M. Lobino, A. S. Clark, D. Y. Choi, S. J. Madden, C. M. Natarajan, M. G. Tanner, R. H. Hadfield, S. N. Dorenbos, T. Zijlstra, V. Zwiller, M. G. Thompson, J. G. Rarity, M. J. Steel, B. Luther-Davies, B. J. Eggleton, and J. L. O’Brien, “Generation of correlated photon pairs in a chalcogenide As2S3 waveguide,” Appl. Phys. Lett. 98, 051101 (2011).
[Crossref]

Clark, A. S.

E. Engin, D. Bonneau, C. M. Natarajan, A. S. Clark, M. G. Tanner, R. H. 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. Express 21, 27826–27834 (2013).
[Crossref]

C. Xiong, G. D. Marshall, A. Peruzzo, M. Lobino, A. S. Clark, D. Y. Choi, S. J. Madden, C. M. Natarajan, M. G. Tanner, R. H. Hadfield, S. N. Dorenbos, T. Zijlstra, V. Zwiller, M. G. Thompson, J. G. Rarity, M. J. Steel, B. Luther-Davies, B. J. Eggleton, and J. L. O’Brien, “Generation of correlated photon pairs in a chalcogenide As2S3 waveguide,” Appl. Phys. Lett. 98, 051101 (2011).
[Crossref]

Clemmen, S.

Cooper, M. L.

Davanco, M.

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

DellaFrera, A.

A. Tosi, A. DellaFrera, A. Bahgat Shehata, and C. Scarcella, “Fully programmable single-photon detection module for InGaAs/InP single-photon avalanche diodes with clean and sub-nanosecond gating transitions,” Rev. Sci. Instrum. 83, 013104 (2012).
[Crossref] [PubMed]

Ding, R.

Y. Zhang, T. Baehr-Jones, R. Ding, T. Pinguet, Z. Xuan, and M. Hochberg, “Silicon multi-project wafer platforms for optoelectronic system integration,” in “Group IV Photonics (GFP), 2012 IEEE 9th International Conference on,” (2012), pp. 63–65.

Dorenbos, S. N.

E. Engin, D. Bonneau, C. M. Natarajan, A. S. Clark, M. G. Tanner, R. H. 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. Express 21, 27826–27834 (2013).
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C. Xiong, G. D. Marshall, A. Peruzzo, M. Lobino, A. S. Clark, D. Y. Choi, S. J. Madden, C. M. Natarajan, M. G. Tanner, R. H. Hadfield, S. N. Dorenbos, T. Zijlstra, V. Zwiller, M. G. Thompson, J. G. Rarity, M. J. Steel, B. Luther-Davies, B. J. Eggleton, and J. L. O’Brien, “Generation of correlated photon pairs in a chalcogenide As2S3 waveguide,” Appl. Phys. Lett. 98, 051101 (2011).
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Dyer, S. D.

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C. Xiong, G. D. Marshall, A. Peruzzo, M. Lobino, A. S. Clark, D. Y. Choi, S. J. Madden, C. M. Natarajan, M. G. Tanner, R. H. Hadfield, S. N. Dorenbos, T. Zijlstra, V. Zwiller, M. G. Thompson, J. G. Rarity, M. J. Steel, B. Luther-Davies, B. J. Eggleton, and J. L. O’Brien, “Generation of correlated photon pairs in a chalcogenide As2S3 waveguide,” Appl. Phys. Lett. 98, 051101 (2011).
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N. C. Harris, D. Grassani, A. Simbula, M. Pant, M. Galli, T. Baehr-Jones, M. Hochberg, D. Englund, D. Bajoni, and C. Galland, “Integrated source of spectrally filtered correlated photons for large-scale quantum photonic systems,” Phys. Rev. X 4, 041047 (2014).

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Fan, J.

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F. Morichetti, A. Canciamilla, C. Ferrari, A. Samarelli, M. Sorel, and A. Melloni, “Travelling-wave resonant four-wave mixing breaks the limits of cavity-enhanced all-optical wavelength conversion,” Nat. Commun. 2, 296 (2011).
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J. Heebner, R. Grover, and T. Ibrahim, Optical Microresonators–Theory, Fabrication, and Applications (Springer, 2008).

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E. Engin, D. Bonneau, C. M. Natarajan, A. S. Clark, M. G. Tanner, R. H. 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. Express 21, 27826–27834 (2013).
[Crossref]

C. Xiong, G. D. Marshall, A. Peruzzo, M. Lobino, A. S. Clark, D. Y. Choi, S. J. Madden, C. M. Natarajan, M. G. Tanner, R. H. Hadfield, S. N. Dorenbos, T. Zijlstra, V. Zwiller, M. G. Thompson, J. G. Rarity, M. J. Steel, B. Luther-Davies, B. J. Eggleton, and J. L. O’Brien, “Generation of correlated photon pairs in a chalcogenide As2S3 waveguide,” Appl. Phys. Lett. 98, 051101 (2011).
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Harris, N. C.

N. C. Harris, D. Grassani, A. Simbula, M. Pant, M. Galli, T. Baehr-Jones, M. Hochberg, D. Englund, D. Bajoni, and C. Galland, “Integrated source of spectrally filtered correlated photons for large-scale quantum photonic systems,” Phys. Rev. X 4, 041047 (2014).

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J. Heebner, R. Grover, and T. Ibrahim, Optical Microresonators–Theory, Fabrication, and Applications (Springer, 2008).

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Ho, P.-T.

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N. C. Harris, D. Grassani, A. Simbula, M. Pant, M. Galli, T. Baehr-Jones, M. Hochberg, D. Englund, D. Bajoni, and C. Galland, “Integrated source of spectrally filtered correlated photons for large-scale quantum photonic systems,” Phys. Rev. X 4, 041047 (2014).

Y. Zhang, T. Baehr-Jones, R. Ding, T. Pinguet, Z. Xuan, and M. Hochberg, “Silicon multi-project wafer platforms for optoelectronic system integration,” in “Group IV Photonics (GFP), 2012 IEEE 9th International Conference on,” (2012), pp. 63–65.

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Inoue, K.

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Kumar, P.

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M. Savanier, R. Kumar, and S. Mookherjea, “Optimizing photon-pair generation electronically using a p-i-n diode incorporated in a silicon microring resonator,” Appl. Phys. Lett. 107, 131101 (2015).
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J. R. Ong, R. Kumar, and S. Mookherjea, “Ultra-high-contrast and tunable-bandwidth filter using cascaded high-order silicon microring filters,” Photon. Technol. Lett., IEEE 25, 1543–1546 (2013).
[Crossref]

J. R. Ong, R. Kumar, R. Aguinaldo, and S. Mookherjea, “Efficient cw four-wave mixing in silicon-on-insulator micro-rings with active carrier removal,” Photon. Technol. Lett., IEEE 25, 1699–1702 (2013).
[Crossref]

R. Kumar, J. R. Ong, J. Recchio, K. Srinivasan, and S. Mookherjea, “Spectrally multiplexed and tunable-wavelength photon pairs at 1.55 µ m from a silicon coupled-resonator optical waveguide,” Opt. Lett. 38, 2969–2971 (2013).
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C. Xiong, G. D. Marshall, A. Peruzzo, M. Lobino, A. S. Clark, D. Y. Choi, S. J. Madden, C. M. Natarajan, M. G. Tanner, R. H. Hadfield, S. N. Dorenbos, T. Zijlstra, V. Zwiller, M. G. Thompson, J. G. Rarity, M. J. Steel, B. Luther-Davies, B. J. Eggleton, and J. L. O’Brien, “Generation of correlated photon pairs in a chalcogenide As2S3 waveguide,” Appl. Phys. Lett. 98, 051101 (2011).
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P. J. Mosley, J. S. Lundeen, B. J. Smith, P. Wasylczyk, A. B. U’Ren, C. Silberhorn, and I. A. Walmsley, “Heralded generation of ultrafast single photons in pure quantum states,” Phys. Rev. Lett. 100, 133601 (2008).
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C. Xiong, G. D. Marshall, A. Peruzzo, M. Lobino, A. S. Clark, D. Y. Choi, S. J. Madden, C. M. Natarajan, M. G. Tanner, R. H. Hadfield, S. N. Dorenbos, T. Zijlstra, V. Zwiller, M. G. Thompson, J. G. Rarity, M. J. Steel, B. Luther-Davies, B. J. Eggleton, and J. L. O’Brien, “Generation of correlated photon pairs in a chalcogenide As2S3 waveguide,” Appl. Phys. Lett. 98, 051101 (2011).
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Madden, S. J.

C. Xiong, G. D. Marshall, A. Peruzzo, M. Lobino, A. S. Clark, D. Y. Choi, S. J. Madden, C. M. Natarajan, M. G. Tanner, R. H. Hadfield, S. N. Dorenbos, T. Zijlstra, V. Zwiller, M. G. Thompson, J. G. Rarity, M. J. Steel, B. Luther-Davies, B. J. Eggleton, and J. L. O’Brien, “Generation of correlated photon pairs in a chalcogenide As2S3 waveguide,” Appl. Phys. Lett. 98, 051101 (2011).
[Crossref]

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C. Xiong, G. D. Marshall, A. Peruzzo, M. Lobino, A. S. Clark, D. Y. Choi, S. J. Madden, C. M. Natarajan, M. G. Tanner, R. H. Hadfield, S. N. Dorenbos, T. Zijlstra, V. Zwiller, M. G. Thompson, J. G. Rarity, M. J. Steel, B. Luther-Davies, B. J. Eggleton, and J. L. O’Brien, “Generation of correlated photon pairs in a chalcogenide As2S3 waveguide,” Appl. Phys. Lett. 98, 051101 (2011).
[Crossref]

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D. Höckel, E. Martin, and O. Benson, “Note: An ultranarrow bandpass filter system for single-photon experiments in quantum optics,” Rev. Sci. Instrum. 81, 026108 (2010).
[Crossref] [PubMed]

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M. Savanier, R. Kumar, and S. Mookherjea, “Optimizing photon-pair generation electronically using a p-i-n diode incorporated in a silicon microring resonator,” Appl. Phys. Lett. 107, 131101 (2015).
[Crossref]

J. R. Ong, R. Kumar, and S. Mookherjea, “Ultra-high-contrast and tunable-bandwidth filter using cascaded high-order silicon microring filters,” Photon. Technol. Lett., IEEE 25, 1543–1546 (2013).
[Crossref]

J. R. Ong, R. Kumar, R. Aguinaldo, and S. Mookherjea, “Efficient cw four-wave mixing in silicon-on-insulator micro-rings with active carrier removal,” Photon. Technol. Lett., IEEE 25, 1699–1702 (2013).
[Crossref]

R. Kumar, J. R. Ong, J. Recchio, K. Srinivasan, and S. Mookherjea, “Spectrally multiplexed and tunable-wavelength photon pairs at 1.55 µ m from a silicon coupled-resonator optical waveguide,” Opt. Lett. 38, 2969–2971 (2013).
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R. Aguinaldo, Y. Shen, and S. Mookherjea, “Large dispersion of silicon directional couplers obtained via wide-band microring parametric characterization,” IEEE Photon. Technol. Lett. 24, 1242–1244 (2012).
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M. L. Cooper, G. Gupta, M. A. Schneider, W. M. J. Green, S. Assefa, F. Xia, Y. A. Vlasov, and S. Mookherjea, “Statistics of light transport in 235-ring silicon coupled-resonator optical waveguides,” Opt. Express 18, 26505–26516 (2010).
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P. J. Mosley, J. S. Lundeen, B. J. Smith, P. Wasylczyk, A. B. U’Ren, C. Silberhorn, and I. A. Walmsley, “Heralded generation of ultrafast single photons in pure quantum states,” Phys. Rev. Lett. 100, 133601 (2008).
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Nambu, Y.

Natarajan, C. M.

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[Crossref]

C. Xiong, G. D. Marshall, A. Peruzzo, M. Lobino, A. S. Clark, D. Y. Choi, S. J. Madden, C. M. Natarajan, M. G. Tanner, R. H. Hadfield, S. N. Dorenbos, T. Zijlstra, V. Zwiller, M. G. Thompson, J. G. Rarity, M. J. Steel, B. Luther-Davies, B. J. Eggleton, and J. L. O’Brien, “Generation of correlated photon pairs in a chalcogenide As2S3 waveguide,” Appl. Phys. Lett. 98, 051101 (2011).
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O’Brien, J. L.

D. Bonneau, G. J. Mendoza, J. L. O’Brien, and M. G. Thompson, “Effect of loss on multiplexed single-photon sources,” New J. Phys. 17, 043057 (2015).
[Crossref]

E. Engin, D. Bonneau, C. M. Natarajan, A. S. Clark, M. G. Tanner, R. H. 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. Express 21, 27826–27834 (2013).
[Crossref]

C. Xiong, G. D. Marshall, A. Peruzzo, M. Lobino, A. S. Clark, D. Y. Choi, S. J. Madden, C. M. Natarajan, M. G. Tanner, R. H. Hadfield, S. N. Dorenbos, T. Zijlstra, V. Zwiller, M. G. Thompson, J. G. Rarity, M. J. Steel, B. Luther-Davies, B. J. Eggleton, and J. L. O’Brien, “Generation of correlated photon pairs in a chalcogenide As2S3 waveguide,” Appl. Phys. Lett. 98, 051101 (2011).
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J. W. Silverstone, R. Santagati, D. Bonneau, M. J. Strain, M. Sorel, J. L. O’Brien, and M. G. Thompson, “Qubit entanglement between ring-resonator photon-pair sources on a silicon chip,” Nat. Commun.6 (2015).
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Ong, J. R.

R. Kumar, J. R. Ong, J. Recchio, K. Srinivasan, and S. Mookherjea, “Spectrally multiplexed and tunable-wavelength photon pairs at 1.55 µ m from a silicon coupled-resonator optical waveguide,” Opt. Lett. 38, 2969–2971 (2013).
[Crossref] [PubMed]

J. R. Ong, R. Kumar, and S. Mookherjea, “Ultra-high-contrast and tunable-bandwidth filter using cascaded high-order silicon microring filters,” Photon. Technol. Lett., IEEE 25, 1543–1546 (2013).
[Crossref]

J. R. Ong, R. Kumar, R. Aguinaldo, and S. Mookherjea, “Efficient cw four-wave mixing in silicon-on-insulator micro-rings with active carrier removal,” Photon. Technol. Lett., IEEE 25, 1699–1702 (2013).
[Crossref]

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

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Peng, J.

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C. Xiong, G. D. Marshall, A. Peruzzo, M. Lobino, A. S. Clark, D. Y. Choi, S. J. Madden, C. M. Natarajan, M. G. Tanner, R. H. Hadfield, S. N. Dorenbos, T. Zijlstra, V. Zwiller, M. G. Thompson, J. G. Rarity, M. J. Steel, B. Luther-Davies, B. J. Eggleton, and J. L. O’Brien, “Generation of correlated photon pairs in a chalcogenide As2S3 waveguide,” Appl. Phys. Lett. 98, 051101 (2011).
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Y. Zhang, T. Baehr-Jones, R. Ding, T. Pinguet, Z. Xuan, and M. Hochberg, “Silicon multi-project wafer platforms for optoelectronic system integration,” in “Group IV Photonics (GFP), 2012 IEEE 9th International Conference on,” (2012), pp. 63–65.

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Qing, L.

M. Soltani, S. Yegnanarayanan, L. Qing, and A. Adibi, “Systematic engineering of waveguide-resonator coupling for silicon microring/microdisk/racetrack resonators: theory and experiment,” IEEE J. Quant. Electron. 46, 1158–1169 (2010).
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C. Xiong, G. D. Marshall, A. Peruzzo, M. Lobino, A. S. Clark, D. Y. Choi, S. J. Madden, C. M. Natarajan, M. G. Tanner, R. H. Hadfield, S. N. Dorenbos, T. Zijlstra, V. Zwiller, M. G. Thompson, J. G. Rarity, M. J. Steel, B. Luther-Davies, B. J. Eggleton, and J. L. O’Brien, “Generation of correlated photon pairs in a chalcogenide As2S3 waveguide,” Appl. Phys. Lett. 98, 051101 (2011).
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F. Morichetti, A. Canciamilla, C. Ferrari, A. Samarelli, M. Sorel, and A. Melloni, “Travelling-wave resonant four-wave mixing breaks the limits of cavity-enhanced all-optical wavelength conversion,” Nat. Commun. 2, 296 (2011).
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Santagati, R.

J. W. Silverstone, R. Santagati, D. Bonneau, M. J. Strain, M. Sorel, J. L. O’Brien, and M. G. Thompson, “Qubit entanglement between ring-resonator photon-pair sources on a silicon chip,” Nat. Commun.6 (2015).
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Sasaki, M.

Savanier, M.

M. Savanier, R. Kumar, and S. Mookherjea, “Optimizing photon-pair generation electronically using a p-i-n diode incorporated in a silicon microring resonator,” Appl. Phys. Lett. 107, 131101 (2015).
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M. Davanco, J. R. Ong, A. B. Shehata, A. Tosi, I. Agha, S. Assefa, F. Xia, W. M. J. Green, S. Mookherjea, and K. Srinivasan, “Telecommunications-band heralded single photons from a silicon nanophotonic chip,” Appl. Phys. Lett. 100, 261104 (2012).
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Shehata, A. Bahgat

A. Tosi, A. DellaFrera, A. Bahgat Shehata, and C. Scarcella, “Fully programmable single-photon detection module for InGaAs/InP single-photon avalanche diodes with clean and sub-nanosecond gating transitions,” Rev. Sci. Instrum. 83, 013104 (2012).
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R. Aguinaldo, Y. Shen, and S. Mookherjea, “Large dispersion of silicon directional couplers obtained via wide-band microring parametric characterization,” IEEE Photon. Technol. Lett. 24, 1242–1244 (2012).
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P. J. Mosley, J. S. Lundeen, B. J. Smith, P. Wasylczyk, A. B. U’Ren, C. Silberhorn, and I. A. Walmsley, “Heralded generation of ultrafast single photons in pure quantum states,” Phys. Rev. Lett. 100, 133601 (2008).
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J. W. Silverstone, R. Santagati, D. Bonneau, M. J. Strain, M. Sorel, J. L. O’Brien, and M. G. Thompson, “Qubit entanglement between ring-resonator photon-pair sources on a silicon chip,” Nat. Commun.6 (2015).
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Smith, B. J.

P. J. Mosley, J. S. Lundeen, B. J. Smith, P. Wasylczyk, A. B. U’Ren, C. Silberhorn, and I. A. Walmsley, “Heralded generation of ultrafast single photons in pure quantum states,” Phys. Rev. Lett. 100, 133601 (2008).
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M. Soltani, S. Yegnanarayanan, L. Qing, and A. Adibi, “Systematic engineering of waveguide-resonator coupling for silicon microring/microdisk/racetrack resonators: theory and experiment,” IEEE J. Quant. Electron. 46, 1158–1169 (2010).
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D. Grassani, S. Azzini, M. Liscidini, M. Galli, M. J. Strain, M. Sorel, J. E. Sipe, and D. Bajoni, “Micrometer-scale integrated silicon source of time-energy entangled photons,” Optica 2, 88–94 (2015).
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[Crossref] [PubMed]

J. W. Silverstone, R. Santagati, D. Bonneau, M. J. Strain, M. Sorel, J. L. O’Brien, and M. G. Thompson, “Qubit entanglement between ring-resonator photon-pair sources on a silicon chip,” Nat. Commun.6 (2015).
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R. Kumar, J. R. Ong, J. Recchio, K. Srinivasan, and S. Mookherjea, “Spectrally multiplexed and tunable-wavelength photon pairs at 1.55 µ m from a silicon coupled-resonator optical waveguide,” Opt. Lett. 38, 2969–2971 (2013).
[Crossref] [PubMed]

M. Davanco, J. R. Ong, A. B. Shehata, A. Tosi, I. Agha, S. Assefa, F. Xia, W. M. J. Green, S. Mookherjea, and K. Srinivasan, “Telecommunications-band heralded single photons from a silicon nanophotonic chip,” Appl. Phys. Lett. 100, 261104 (2012).
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C. Xiong, G. D. Marshall, A. Peruzzo, M. Lobino, A. S. Clark, D. Y. Choi, S. J. Madden, C. M. Natarajan, M. G. Tanner, R. H. Hadfield, S. N. Dorenbos, T. Zijlstra, V. Zwiller, M. G. Thompson, J. G. Rarity, M. J. Steel, B. Luther-Davies, B. J. Eggleton, and J. L. O’Brien, “Generation of correlated photon pairs in a chalcogenide As2S3 waveguide,” Appl. Phys. Lett. 98, 051101 (2011).
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[Crossref]

Thompson, M. G.

D. Bonneau, G. J. Mendoza, J. L. O’Brien, and M. G. Thompson, “Effect of loss on multiplexed single-photon sources,” New J. Phys. 17, 043057 (2015).
[Crossref]

E. Engin, D. Bonneau, C. M. Natarajan, A. S. Clark, M. G. Tanner, R. H. 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. Express 21, 27826–27834 (2013).
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[Crossref]

J. W. Silverstone, R. Santagati, D. Bonneau, M. J. Strain, M. Sorel, J. L. O’Brien, and M. G. Thompson, “Qubit entanglement between ring-resonator photon-pair sources on a silicon chip,” Nat. Commun.6 (2015).
[Crossref] [PubMed]

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Tokura, Y.

Tosi, A.

A. Tosi, A. DellaFrera, A. Bahgat Shehata, and C. Scarcella, “Fully programmable single-photon detection module for InGaAs/InP single-photon avalanche diodes with clean and sub-nanosecond gating transitions,” Rev. Sci. Instrum. 83, 013104 (2012).
[Crossref] [PubMed]

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

Wang, C.

Wasylczyk, P.

P. J. Mosley, J. S. Lundeen, B. J. Smith, P. Wasylczyk, A. B. U’Ren, C. Silberhorn, and I. A. Walmsley, “Heralded generation of ultrafast single photons in pure quantum states,” Phys. Rev. Lett. 100, 133601 (2008).
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Y. Zhang, T. Baehr-Jones, R. Ding, T. Pinguet, Z. Xuan, and M. Hochberg, “Silicon multi-project wafer platforms for optoelectronic system integration,” in “Group IV Photonics (GFP), 2012 IEEE 9th International Conference on,” (2012), pp. 63–65.

Zhou, Q.

Zijlstra, T.

C. Xiong, G. D. Marshall, A. Peruzzo, M. Lobino, A. S. Clark, D. Y. Choi, S. J. Madden, C. M. Natarajan, M. G. Tanner, R. H. Hadfield, S. N. Dorenbos, T. Zijlstra, V. Zwiller, M. G. Thompson, J. G. Rarity, M. J. Steel, B. Luther-Davies, B. J. Eggleton, and J. L. O’Brien, “Generation of correlated photon pairs in a chalcogenide As2S3 waveguide,” Appl. Phys. Lett. 98, 051101 (2011).
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Zwiller, V.

E. Engin, D. Bonneau, C. M. Natarajan, A. S. Clark, M. G. Tanner, R. H. 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. Express 21, 27826–27834 (2013).
[Crossref]

C. Xiong, G. D. Marshall, A. Peruzzo, M. Lobino, A. S. Clark, D. Y. Choi, S. J. Madden, C. M. Natarajan, M. G. Tanner, R. H. Hadfield, S. N. Dorenbos, T. Zijlstra, V. Zwiller, M. G. Thompson, J. G. Rarity, M. J. Steel, B. Luther-Davies, B. J. Eggleton, and J. L. O’Brien, “Generation of correlated photon pairs in a chalcogenide As2S3 waveguide,” Appl. Phys. Lett. 98, 051101 (2011).
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Appl. Phys. Lett. (3)

M. Savanier, R. Kumar, and S. Mookherjea, “Optimizing photon-pair generation electronically using a p-i-n diode incorporated in a silicon microring resonator,” Appl. Phys. Lett. 107, 131101 (2015).
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C. Xiong, G. D. Marshall, A. Peruzzo, M. Lobino, A. S. Clark, D. Y. Choi, S. J. Madden, C. M. Natarajan, M. G. Tanner, R. H. Hadfield, S. N. Dorenbos, T. Zijlstra, V. Zwiller, M. G. Thompson, J. G. Rarity, M. J. Steel, B. Luther-Davies, B. J. Eggleton, and J. L. O’Brien, “Generation of correlated photon pairs in a chalcogenide As2S3 waveguide,” Appl. Phys. Lett. 98, 051101 (2011).
[Crossref]

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

Electron. Lett. (1)

A. Yariv, “Universal relations for coupling of optical power between microresonators and dielectric waveguides,” Electron. Lett. 36, 321–322 (2000).
[Crossref]

IEEE J. Quant. Electron. (1)

M. Soltani, S. Yegnanarayanan, L. Qing, and A. Adibi, “Systematic engineering of waveguide-resonator coupling for silicon microring/microdisk/racetrack resonators: theory and experiment,” IEEE J. Quant. Electron. 46, 1158–1169 (2010).
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IEEE Photon. Technol. Lett. (1)

R. Aguinaldo, Y. Shen, and S. Mookherjea, “Large dispersion of silicon directional couplers obtained via wide-band microring parametric characterization,” IEEE Photon. Technol. Lett. 24, 1242–1244 (2012).
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J. Opt. Soc. Am. B (1)

Nat. Commun. (1)

F. Morichetti, A. Canciamilla, C. Ferrari, A. Samarelli, M. Sorel, and A. Melloni, “Travelling-wave resonant four-wave mixing breaks the limits of cavity-enhanced all-optical wavelength conversion,” Nat. Commun. 2, 296 (2011).
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Nature (1)

K. J. Vahala, “Optical microcavities,” Nature 424, 829 (2003).
[Crossref]

New J. Phys. (1)

D. Bonneau, G. J. Mendoza, J. L. O’Brien, and M. G. Thompson, “Effect of loss on multiplexed single-photon sources,” New J. Phys. 17, 043057 (2015).
[Crossref]

Opt. Express (14)

E. Engin, D. Bonneau, C. M. Natarajan, A. S. Clark, M. G. Tanner, R. H. 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. Express 21, 27826–27834 (2013).
[Crossref]

Y. Guo, W. Zhang, N. Lv, Q. Zhou, Y. Huang, and J. Peng, “The impact of nonlinear losses in the silicon microring cavities on cw pumping correlated photon pair generation,” Opt. Express 22, 2620–2631 (2014).
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C. Wang, M. J. Burek, Z. Lin, H. A. Atikian, V. Venkataraman, I. Huang, P. Stark, and M. Lončar, “Integrated high quality factor lithium niobate microdisk resonators,” Opt. express 22, 30924–30933 (2014).
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R. Wakabayashi, M. Fujiwara, K.-I. Yoshino, Y. Nambu, M. Sasaki, and T. Aoki, “Time-bin entangled photon pair generation from Si micro-ring resonator,” Opt. Express 23, 1103–1113 (2015).
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M. Borselli, T. Johnson, and O. Painter, “Beyond the Rayleigh scattering limit in high-Q silicon microdisks: theory and experiment,” Opt. Express 13, 1515–1530 (2005).
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H. Takesue and K. Inoue, “1.5-µ m band quantum-correlated photon pair generation in dispersion-shifted fiber: suppression of noise photons by cooling fiber,” Opt. Express 13, 7832–7839 (2005).
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E. Dulkeith, F. Xia, L. Schares, W. M. J. Green, and Y. A. Vlasov, “Group index and group velocity dispersion in silicon-on-insulator photonic wires,” Opt. Express 14, 3853–3863 (2006).
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J. E. Sharping, K. F. Lee, M. A. Foster, A. C. Turner, B. S. Schmidt, M. Lipson, A. L. Gaeta, and P. Kumar, “Generation of correlated photons in nanoscale silicon waveguides,” Opt. Express 14, 12388–12393 (2006).
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K.-I. Harada, H. Takesue, H. Fukuda, T. Tsuchizawa, T. Watanabe, K. Yamada, Y. Tokura, and S.-I. Itabashi, “Generation of high-purity entangled photon pairs using silicon wirewaveguide,” Opt. Express 16, 20368–20373 (2008).
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S. Clemmen, K. P. Huy, W. Bogaerts, R. Baets, P. Emplit, and S. Massar, “Continuous wave photon pair generation in silicon-on-insulator waveguides and ring resonators,” Opt. Express 17, 16558–16570 (2009).
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M. L. Cooper, G. Gupta, M. A. Schneider, W. M. J. Green, S. Assefa, F. Xia, Y. A. Vlasov, and S. Mookherjea, “Statistics of light transport in 235-ring silicon coupled-resonator optical waveguides,” Opt. Express 18, 26505–26516 (2010).
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J. Chen, Z. H. Levine, J. Fan, and A. L. Migdall, “Frequency-bin entangled comb of photon pairs from a silicon-on-insulator micro-resonator,” Opt. Express 19, 1470–1483 (2011).
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T. Carmon, L. Yang, and K. Vahala, “Dynamical thermal behavior and thermal self-stability of microcavities,” Opt. Express 12, 4742–4750 (2004).
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W. C. Jiang, X. Lu, J. Zhang, O. Painter, and Q. Lin, “Silicon-chip source of bright photon pairs,” Opt. Express 23, 20884–20904 (2015).
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Optica (2)

Photon. Technol. Lett., IEEE (2)

J. R. Ong, R. Kumar, and S. Mookherjea, “Ultra-high-contrast and tunable-bandwidth filter using cascaded high-order silicon microring filters,” Photon. Technol. Lett., IEEE 25, 1543–1546 (2013).
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J. R. Ong, R. Kumar, R. Aguinaldo, and S. Mookherjea, “Efficient cw four-wave mixing in silicon-on-insulator micro-rings with active carrier removal,” Photon. Technol. Lett., IEEE 25, 1699–1702 (2013).
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Phys. Rev. Lett. (1)

P. J. Mosley, J. S. Lundeen, B. J. Smith, P. Wasylczyk, A. B. U’Ren, C. Silberhorn, and I. A. Walmsley, “Heralded generation of ultrafast single photons in pure quantum states,” Phys. Rev. Lett. 100, 133601 (2008).
[Crossref] [PubMed]

Phys. Rev. X (1)

N. C. Harris, D. Grassani, A. Simbula, M. Pant, M. Galli, T. Baehr-Jones, M. Hochberg, D. Englund, D. Bajoni, and C. Galland, “Integrated source of spectrally filtered correlated photons for large-scale quantum photonic systems,” Phys. Rev. X 4, 041047 (2014).

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D. Höckel, E. Martin, and O. Benson, “Note: An ultranarrow bandpass filter system for single-photon experiments in quantum optics,” Rev. Sci. Instrum. 81, 026108 (2010).
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A. Tosi, A. DellaFrera, A. Bahgat Shehata, and C. Scarcella, “Fully programmable single-photon detection module for InGaAs/InP single-photon avalanche diodes with clean and sub-nanosecond gating transitions,” Rev. Sci. Instrum. 83, 013104 (2012).
[Crossref] [PubMed]

Other (6)

Z. Vernon, M. Liscidini, and J. E. Sipe, “No free lunch: the tradeoff between heralding rate and efficiency in microresonator-based heralded single photon sources,” arXiv:1512.04879 (2015).

J. W. Silverstone, R. Santagati, D. Bonneau, M. J. Strain, M. Sorel, J. L. O’Brien, and M. G. Thompson, “Qubit entanglement between ring-resonator photon-pair sources on a silicon chip,” Nat. Commun.6 (2015).
[Crossref] [PubMed]

A. Yariv and P. Yeh, Photonics (Oxford, 2007).

K. Sakoda, Optical Properties of Photonic Crystals (Springer, 2001).
[Crossref]

J. Heebner, R. Grover, and T. Ibrahim, Optical Microresonators–Theory, Fabrication, and Applications (Springer, 2008).

Y. Zhang, T. Baehr-Jones, R. Ding, T. Pinguet, Z. Xuan, and M. Hochberg, “Silicon multi-project wafer platforms for optoelectronic system integration,” in “Group IV Photonics (GFP), 2012 IEEE 9th International Conference on,” (2012), pp. 63–65.

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Figures (7)

Fig. 1
Fig. 1

(a) Microscope plan view of a microring resonator with embedded p-i-n junction across the waveguide. (b) Cross-section schematic drawing; measured devices have A = 650 nm, B = 220 nm or B = 340 nm, C = 70 nm, D = 900 nm. (c) Transmission spectrum for B = 220 nm and B = 340 nm microring resonators (the solid circles indicate the chosen pump wavelengths, and the dotted circles the measured signal-idler resonances). (d) Group velocity dispersion (calculation) for the lowest-order TE modes of waveguides with the cross-section shown in panel (b).

Fig. 2
Fig. 2

Nonlinear relationship between the quality factor Q and the coupling coefficient |κ|2, calculated using Eq. (3a) and the parameters specific to the two structures studied here (B = 220 nm: blue, B = 340 nm: red). The |κ|2 values for the fabricated devices are estimated from the measured loaded Q’s (dots and dashed lines)

Fig. 3
Fig. 3

The appropriate choice of the coupling coefficient |κ|2 is guided by numerical calculations of the coincidences counted in 900 s (with assumed values of the fiber-to-chip coupling efficiency, filter insertion loss and detector efficiency that simulate the experiments performed in this report). A study of two cross-sections (B = 220 nm and 340 nm – blue and red curves respectively) for increasing pump powers: (a) 10 µW, (b) 100 µW and (c) 1 mW, shows the critical role of GVD and self-modulation on the coincidences count, through the SFWM phase-mismatch.

Fig. 4
Fig. 4

Measurements of PGR as the pump wavelength was finely tuned across the resonance (the error bars are smaller than the markers); input pump power was approximately −10 dBm. The dashed line is a squared-Lorentzian fit.

Fig. 5
Fig. 5

(a) Measured pair generation rate (PGR) using pump pulses of 3 ns duration at a repetition rate of 5 MHz (the error bars are too small to be visible) and (b) coincidences-to-accidentals ratio (CAR) versus average optical pump power in the feeder waveguide.

Fig. 6
Fig. 6

Measured on-chip pair generation rate (PGR) versus average optical pump power (P) for the devices presented here (dots), along with the recent experimental reports of Table 1 in microrings (squares), and microdisk (diamond). The quadratic dependence of PGR with P is highlighted by the fits PGR = KQ3/R2 × P2 of each experimental dataset at low pump power (dashed lines), while saturation occurs at higher pump powers.

Fig. 7
Fig. 7

(a) Schematic of microring resonator coupled to a waveguide, with electric field amplitude coefficients which are used in the matrix model of propagation. (b) Calculation of the phase mismatch versus finesse, using two approaches as discussed in the text.

Tables (1)

Tables Icon

Table 1 Recent results of photon pair generation using silicon microring resonators (comparison: recent microdisk result)

Equations (12)

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

r = Δ ν [ γ P r e s ( λ ) L eff r e s ] 2 sin c 2 ( β 2 Δ ω 2 L r e s 2 + γ P r e s L eff res )
P r e s ( λ ) = P × F π × ( λ P / 2 Q ) 2 ( λ λ P ) 2 + ( λ P / 2 Q ) 2 L r e s = L × F π L eff r e s = 1 e α L α × F π
Q L = π 2 arcsin ( 1 a τ 2 a τ ) n g L λ
a τ 1 π a τ 1 a τ n g L λ
Q U = 2 π n g λ α .
Q cpl = 2 π n g L λ | κ | 2 .
H A 2 A 1 = κ a e i ϕ 1 τ a e i ϕ ,
( B 1 B 2 ) = ( τ κ * κ τ * ) ( A 1 A 2 ) .
Φ arg ( H ) = π 2 + ϕ + tan 1 ( τ a sin ϕ 1 τ a cos ϕ ) .
Φ π 2 + ϕ + ( τ a 1 τ a ) ϕ = π 2 + ( 1 1 τ a ) ϕ .
Δ k L = 2 k p L k s L k i L β 2 Δ ω 2 L
Δ Φ = 2 Φ p Φ s Φ i .

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